Version 9 (modified by cowan, 5 years ago) (diff) |
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Flonums package
Flonums are a subset of the inexact real numbers provided by a Scheme implementation. In most Schemes, the flonums and the inexact reals are the same.
The procedures in this package don't have hard-coded prefixes. The intent is that they be placed in a module. Users can then import this module with their own prefix such as fl, or ƒ if their Scheme implementation supports that character, or with no prefix at all if the intent is to override the normal Scheme arithmetic routines.
Basic procedures
(flonum? obj)
Returns #t if obj is a flonum, #f otherwise.
(real->flonum x)
Returns the best flonum representation of x.
R7RS-small-compatible procedures
The following R7RS-small procedures have flonum equivalents:
= < <= > >= integer? zero? positive? negative? odd? even? finite? infinite? nan? + - * / abs numerator denominator floor ceiling truncate round log sin cos tan asin acos atan sqrt expt
TODO: need to see about integer division procedures
C99 <math.h> constants
The following constants are defined in terms of the constants of the <math.h> header of ISO/IEC 9899:1999 (C language).
Scheme name | C name | Comments |
e | M_E | Value of e |
log2-e | M_LOG2E | Value of log_{2} e |
log10-e | M_LOG10E | Value of log_{10} e |
ln-2 | M_LN2 | Value of log_{e} 2 |
ln-10 | M_LN10 | Value of log_{e} 10 |
pi | M_PI | Value of pi |
pi/2 | M_PI_2 | Value of pi/2 |
pi/4 | M_PI_4 | Value of pi/4 |
one-over-pi | M_1_PI | Value of 1/pi |
two-over-pi | M_2_PI | Value of 2/pi |
two-over-sqrt-pi | M_2_SQRTPI | Value of 2/sqrt(pi) |
sqrt-2 | M_SQRT2 | Value of sqrt(2) |
one-over-sqrt-2 | M_SQRT1_2 | Value of 1/sqrt(2) |
maximum-flonum | HUGE_VAL | +inf.0 or else the largest finite flonum |
fast-multiply-add | #t if FP_FAST_FMA is 1, or #f otherwise | multiply-add is fast |
integer-exponent-zero | FP_ILOGB0 | what (integer-binary-log 0) returns |
integer-exponent-nan | FP_ILOGBNAN | what (integer-binary-log +0.nan) returns |
C99 <math.h> procedures
The following procedures are defined in terms of the functions of the <math.h> header of ISO/IEC 9899:1999 (C language). In the C signatures, the types "double" and "int" are mapped to Scheme flonums and (suitably bounded) Scheme exact integers respectively.
Scheme name | C signature | Comments |
acosh | double acosh(double) | hyperbolic arc cosine |
asinh | double asinh(double) | hyperbolic arc sine |
atanh | double atanh(double) | hyperbolic arc tangent |
cbrt | double cbrt(double); | cube root |
complementary-error-function | double erfc(double) | - |
copy-sign | double copysign(double x, double y) | result has magnitude of x and sign of y |
cosh | double cosh(double) | hyperbolic cosine |
make-flonum | double ldexp(double x, int n) | x*2^{n} |
error-function | double erf(double) | - |
exp | double exp(double) | e^{x} |
exp-binary | double exp,,2,,(double) | base-2 exponential |
exp-minus-1 | double expm1(double) | e^{x}-1 |
exponent | double logb(double x) | the exponent of x, which is the integral part of log_r(|x|), as a signed floating-point value, for non-zero x, where r is the radix of the machine's floating-point arithmetic |
first-bessel-order-0 | double j0(double) | bessel function of the first kind, order 0 |
first-bessel-order-1 | double j1(double) | bessel function of the first kind, order 1 |
first-bessel | double jn(int n, double) | bessel function of the first kind, order n |
fraction-exponent | double modf(double, double *) | returns two values, fraction and int exponent |
gamma | double tgamma(double) | - |
hypotenuse | double hypot(double, double) | sqrt(x^{2}+y^{2}) |
integer-exponent | int ilogb(double) | binary log as int |
log-binary | double log2(double) | log base 2 |
log-decimal | double log10(double) | log base 10 |
log-gamma | double lgamma(double) | returns two values, log(|gamma(x)|) and sgn(gamma(x)) |
log-one-plus | double log1p(double x) | log (1+x) |
multiply-add | double fma(double a, double b, double c) | a*b+c |
next-after | double nextafter(double, double) | next flonum following x in the direction of y |
normalized-fraction-exponent | double frexp(double, int *) | returns two values, fraction in range [1/2,1) and int exponent |
positive-difference | double fdim(double, double) | - |
remquo | double remquo(double, double, int *) | returns two values, rounded remainder and low-order n bits of the quotient (n is implementation-defined) |
scalbn | double scalbn(double x, int y) | x*r^{y}, where r is the machine float radix |
second-bessel-order-0 | double y0(double) | bessel function of the second kind, order 0 |
second-bessel-order-1 | double y1(double) | bessel function of the second kind, order 1 |
second-bessel | double yn(int n, double) | bessel function of the second kind, order n |
sinh | double cosh(double) | hyperbolic sine |
tanh | double cosh(double) | hyperbolic tangent |
General remarks
In the event that these operations do not yield a real result for the given arguments, the result may be +nan.0, or may be some unspecified flonum.
Implementations that use IEEE binary floating-point arithmetic should follow the relevant standards for these procedures.
Compnum procedures from <complex.h>
A compnum is a general complex number whose real-part and imag-part are both flonums. The following procedures should be in a different library from the flonum procedures, since they will only be relevant to Schemes that support general complex numbers, and since there are conflicting names.
Scheme name | C signature | Comments |
abs | double cabs(double complex) | same as magnitude |
acos | double complex cacos(double complex) | - |
acosh | double complex cacosh(double complex) | - |
angle | double carg(double complex) | - |
asin | double complex casin(double complex) | - |
asinh | double complex casinh(double complex) | - |
atan | double complex catan(double complex) | - |
atanh | double complex catanh(double complex) | - |
conjugate | double complex conj(double complex) | complex conjugate |
cos | double complex ccos(double complex) | - |
cosh | double complex ccosh(double complex) | - |
exp | double complex cexp(double complex) | - |
expt | double complex cpow(double complex, double complex) | - |
imag-part | double cimag(double complex) | - |
log | double complex clog(double complex) | - |
magnitude | double cabs(double complex) | same as abs |
projection | double complex cproj(double complex) | projects to Riemann sphere |
real-part | double creal(double complex) | - |
sin | double complex csin(double complex) | - |
sinh | double complex csinh(double complex) | - |
sqrt | double complex csqrt(double complex) | - |
tan | double complex ctan(double complex) | - |
tanh | double complex ctanh(double complex) | - |
Derived procedures
TODO: what library do these go in?
(cis z)
Returns e^{iz}, a complex number whose real part is cos z and whose imaginary part is sin z.
(signum z)
Returns a complex number whose phase is the same as z but whose magnitude is 1, unless z is zero, in which case it returns z. As a consequence of this definition, negative real numbers return -1, positive real numbers return 1, and zero returns zero.
(decode-float z) and friends