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

Average Error: 15.0 → 0.3

Time: 15.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -4.3772205150223 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log x - \left(z + x \cdot \log y\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= y -4.3772205150223e-310)
   (- (* x (- (log (- x)) (log (- y)))) z)
   (- (* x (log x)) (+ z (* x (log y))))))

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 <= -4.3772205150223e-310) {
		tmp = (x * (log(-x) - log(-y))) - z;
	} else {
		tmp = (x * log(x)) - (z + (x * log(y)));
	}
	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 <= (-4.3772205150223d-310)) then
        tmp = (x * (log(-x) - log(-y))) - z
    else
        tmp = (x * log(x)) - (z + (x * log(y)))
    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 <= -4.3772205150223e-310) {
		tmp = (x * (Math.log(-x) - Math.log(-y))) - z;
	} else {
		tmp = (x * Math.log(x)) - (z + (x * Math.log(y)));
	}
	return tmp;
}

Python

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

↓

def code(x, y, z):
	tmp = 0
	if y <= -4.3772205150223e-310:
		tmp = (x * (math.log(-x) - math.log(-y))) - z
	else:
		tmp = (x * math.log(x)) - (z + (x * math.log(y)))
	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 <= -4.3772205150223e-310)
		tmp = Float64(Float64(x * Float64(log(Float64(-x)) - log(Float64(-y)))) - z);
	else
		tmp = Float64(Float64(x * log(x)) - Float64(z + Float64(x * log(y))));
	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 <= -4.3772205150223e-310)
		tmp = (x * (log(-x) - log(-y))) - z;
	else
		tmp = (x * log(x)) - (z + (x * log(y)));
	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, -4.3772205150223e-310], N[(N[(x * N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(N[(x * N[Log[x], $MachinePrecision]), $MachinePrecision] - N[(z + N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	15.0
Target	7.5
Herbie	0.3

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

Derivation

1. Split input into 2 regimes

2. if y < -4.377220515022299e-310

   1. Initial program 14.7

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

   2. Applied egg-rr 0.3

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

   if -4.377220515022299e-310 < y

   1. Initial program 15.3

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

   2. Taylor expanded in y around 0 0.4

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.3772205150223 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log x - \left(z + x \cdot \log y\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022133 
(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))