Numeric.SpecFunctions.Extra:bd0 from math-functions-0.1.5.2

Average Error: 15.4 → 0.4

Time: 4.5s

Precision: binary64

Math

\[x \cdot \log \left(\frac{x}{y}\right) - z \]

↓

\[\begin{array}{l} \mathbf{if}\;y \leq 0:\\ \;\;\;\;\left(\log \left(\frac{-1}{y}\right) \cdot x + x \cdot \log \left(-x\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- (* x (log (/ x y))) z))

↓

(FPCore (x y z)
 :precision binary64
 (if (<= y 0.0)
   (- (+ (* (log (/ -1.0 y)) x) (* x (log (- x)))) z)
   (- (* x (- (log x) (log y))) z)))

C

double code(double x, double y, double z) {
	return (x * log((x / y))) - z;
}

↓

double code(double x, double y, double z) {
	double tmp;
	if (y <= 0.0) {
		tmp = ((log((-1.0 / y)) * x) + (x * log(-x))) - z;
	} else {
		tmp = (x * (log(x) - log(y))) - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x * log((x / y))) - z
end function

↓

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 0.0d0) then
        tmp = ((log(((-1.0d0) / y)) * x) + (x * log(-x))) - z
    else
        tmp = (x * (log(x) - log(y))) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	return (x * Math.log((x / y))) - z;
}

↓

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 0.0) {
		tmp = ((Math.log((-1.0 / y)) * x) + (x * Math.log(-x))) - z;
	} else {
		tmp = (x * (Math.log(x) - Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	return (x * math.log((x / y))) - z

↓

def code(x, y, z):
	tmp = 0
	if y <= 0.0:
		tmp = ((math.log((-1.0 / y)) * x) + (x * math.log(-x))) - z
	else:
		tmp = (x * (math.log(x) - math.log(y))) - z
	return tmp

Julia

function code(x, y, z)
	return Float64(Float64(x * log(Float64(x / y))) - z)
end

↓

function code(x, y, z)
	tmp = 0.0
	if (y <= 0.0)
		tmp = Float64(Float64(Float64(log(Float64(-1.0 / y)) * x) + Float64(x * log(Float64(-x)))) - z);
	else
		tmp = Float64(Float64(x * Float64(log(x) - log(y))) - z);
	end
	return tmp
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * log((x / y))) - z;
end

↓

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 0.0)
		tmp = ((log((-1.0 / y)) * x) + (x * log(-x))) - z;
	else
		tmp = (x * (log(x) - log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := N[(N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]

↓

code[x_, y_, z_] := If[LessEqual[y, 0.0], N[(N[(N[(N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision] + N[(x * N[Log[(-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(N[(x * N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	15.4
Target	8.0
Herbie	0.4

\[\begin{array}{l} \mathbf{if}\;y < 7.595077799083773 \cdot 10^{-308}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \]

Derivation

1. Split input into 2 regimes

2. if y < 0.0

   1. Initial program 15.5

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]

   2. Taylor expanded in y around -inf 0.4

      \[\leadsto \color{blue}{\left(\log \left(\frac{-1}{y}\right) \cdot x + \log \left(-x\right) \cdot x\right) - z} \]

   if 0.0 < y

   1. Initial program 15.3

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]

   2. Taylor expanded in y around 0 0.3

      \[\leadsto x \cdot \color{blue}{\left(\log x - \log y\right)} - z \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0:\\ \;\;\;\;\left(\log \left(\frac{-1}{y}\right) \cdot x + x \cdot \log \left(-x\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \]

Reproduce

herbie shell --seed 2022162 
(FPCore (x y z)
  :name "Numeric.SpecFunctions.Extra:bd0 from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (if (< y 7.595077799083773e-308) (- (* x (log (/ x y))) z) (- (* x (- (log x) (log y))) z))

  (- (* x (log (/ x y))) z))