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

Percentage Accurate: 78.3% → 99.7%

Time: 7.6s

Alternatives: 9

Speedup: 0.5×

Specification

Math

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

FPCore

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

C

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 78.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(3 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right) - z \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (- (* x (* 3.0 (log (/ (cbrt x) (cbrt y))))) z))

C

double code(double x, double y, double z) {
	return (x * (3.0 * log((cbrt(x) / cbrt(y))))) - z;
}

Java

public static double code(double x, double y, double z) {
	return (x * (3.0 * Math.log((Math.cbrt(x) / Math.cbrt(y))))) - z;
}

Julia

function code(x, y, z)
	return Float64(Float64(x * Float64(3.0 * log(Float64(cbrt(x) / cbrt(y))))) - z)
end

Wolfram

code[x_, y_, z_] := N[(N[(x * N[(3.0 * N[Log[N[(N[Power[x, 1/3], $MachinePrecision] / N[Power[y, 1/3], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]

Derivation

1. Initial program 73.9%

   \[x \cdot \log \left(\frac{x}{y}\right) - z \]
2. Step-by-step derivation

   1. add-cube-cbrt 73.9%

      \[\leadsto x \cdot \log \color{blue}{\left(\left(\sqrt[3]{\frac{x}{y}} \cdot \sqrt[3]{\frac{x}{y}}\right) \cdot \sqrt[3]{\frac{x}{y}}\right)} - z \]

   2. log-prod 73.9%

      \[\leadsto x \cdot \color{blue}{\left(\log \left(\sqrt[3]{\frac{x}{y}} \cdot \sqrt[3]{\frac{x}{y}}\right) + \log \left(\sqrt[3]{\frac{x}{y}}\right)\right)} - z \]

   3. pow2 73.9%

      \[\leadsto x \cdot \left(\log \color{blue}{\left({\left(\sqrt[3]{\frac{x}{y}}\right)}^{2}\right)} + \log \left(\sqrt[3]{\frac{x}{y}}\right)\right) - z \]

3. Applied egg-rr 73.9%

   \[\leadsto x \cdot \color{blue}{\left(\log \left({\left(\sqrt[3]{\frac{x}{y}}\right)}^{2}\right) + \log \left(\sqrt[3]{\frac{x}{y}}\right)\right)} - z \]
4. Step-by-step derivation

   1. log-pow 73.9%

      \[\leadsto x \cdot \left(\color{blue}{2 \cdot \log \left(\sqrt[3]{\frac{x}{y}}\right)} + \log \left(\sqrt[3]{\frac{x}{y}}\right)\right) - z \]

   2. distribute-lft1-in 73.9%

      \[\leadsto x \cdot \color{blue}{\left(\left(2 + 1\right) \cdot \log \left(\sqrt[3]{\frac{x}{y}}\right)\right)} - z \]

   3. metadata-eval 73.9%

      \[\leadsto x \cdot \left(\color{blue}{3} \cdot \log \left(\sqrt[3]{\frac{x}{y}}\right)\right) - z \]

5. Simplified 73.9%

   \[\leadsto x \cdot \color{blue}{\left(3 \cdot \log \left(\sqrt[3]{\frac{x}{y}}\right)\right)} - z \]
6. Step-by-step derivation

   1. cbrt-div 99.7%

      \[\leadsto x \cdot \left(3 \cdot \log \color{blue}{\left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)}\right) - z \]

   2. div-inv 99.7%

      \[\leadsto x \cdot \left(3 \cdot \log \color{blue}{\left(\sqrt[3]{x} \cdot \frac{1}{\sqrt[3]{y}}\right)}\right) - z \]

7. Applied egg-rr 99.7%

   \[\leadsto x \cdot \left(3 \cdot \log \color{blue}{\left(\sqrt[3]{x} \cdot \frac{1}{\sqrt[3]{y}}\right)}\right) - z \]
8. Step-by-step derivation

   1. associate-*r/ 99.7%

      \[\leadsto x \cdot \left(3 \cdot \log \color{blue}{\left(\frac{\sqrt[3]{x} \cdot 1}{\sqrt[3]{y}}\right)}\right) - z \]

   2. *-rgt-identity 99.7%

      \[\leadsto x \cdot \left(3 \cdot \log \left(\frac{\color{blue}{\sqrt[3]{x}}}{\sqrt[3]{y}}\right)\right) - z \]

9. Simplified 99.7%

   \[\leadsto x \cdot \left(3 \cdot \log \color{blue}{\left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)}\right) - z \]

10. Final simplification 99.7%

    \[\leadsto x \cdot \left(3 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right) - z \]

Alternative 2: 86.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;t_0 \leq -\infty \lor \neg \left(t_0 \leq 5 \cdot 10^{+294}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(x, \log \left(\frac{y}{x}\right), z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 5e+294)))
     (- z)
     (- (fma x (log (/ y x)) z)))))

C

double code(double x, double y, double z) {
	double t_0 = x * log((x / y));
	double tmp;
	if ((t_0 <= -((double) INFINITY)) || !(t_0 <= 5e+294)) {
		tmp = -z;
	} else {
		tmp = -fma(x, log((y / x)), z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x * log(Float64(x / y)))
	tmp = 0.0
	if ((t_0 <= Float64(-Inf)) || !(t_0 <= 5e+294))
		tmp = Float64(-z);
	else
		tmp = Float64(-fma(x, log(Float64(y / x)), z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, (-Infinity)], N[Not[LessEqual[t$95$0, 5e+294]], $MachinePrecision]], (-z), (-N[(x * N[Log[N[(y / x), $MachinePrecision]], $MachinePrecision] + z), $MachinePrecision])]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x (log.f64 (/.f64 x y))) < -inf.0 or 4.9999999999999999e294 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 6.6%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   2. Step-by-step derivation

      1. remove-double-neg 6.6%

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

      2. sub0-neg 6.6%

         \[\leadsto -\color{blue}{\left(0 - \left(x \cdot \log \left(\frac{x}{y}\right) - z\right)\right)} \]

      3. associate--r- 6.6%

         \[\leadsto -\color{blue}{\left(\left(0 - x \cdot \log \left(\frac{x}{y}\right)\right) + z\right)} \]

      4. neg-sub0 6.6%

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

      5. distribute-rgt-neg-in 6.6%

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

      6. neg-sub0 6.6%

         \[\leadsto -\left(x \cdot \color{blue}{\left(0 - \log \left(\frac{x}{y}\right)\right)} + z\right) \]

      7. log-div 43.6%

         \[\leadsto -\left(x \cdot \left(0 - \color{blue}{\left(\log x - \log y\right)}\right) + z\right) \]

      8. associate-+l- 43.6%

         \[\leadsto -\left(x \cdot \color{blue}{\left(\left(0 - \log x\right) + \log y\right)} + z\right) \]

      9. neg-sub0 43.6%

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

      10. +-commutative 43.6%

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

      11. sub-neg 43.6%

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

      12. log-div 12.0%

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

      13. fma-udef 12.0%

          \[\leadsto -\color{blue}{\mathsf{fma}\left(x, \log \left(\frac{y}{x}\right), z\right)} \]

   3. Simplified 12.0%

      \[\leadsto \color{blue}{-\mathsf{fma}\left(x, \log \left(\frac{y}{x}\right), z\right)} \]

   4. Taylor expanded in x around 0 45.4%

      \[\leadsto -\color{blue}{z} \]

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 4.9999999999999999e294

   1. Initial program 99.8%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   2. Step-by-step derivation

      1. remove-double-neg 99.8%

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

      2. sub0-neg 99.8%

         \[\leadsto -\color{blue}{\left(0 - \left(x \cdot \log \left(\frac{x}{y}\right) - z\right)\right)} \]

      3. associate--r- 99.8%

         \[\leadsto -\color{blue}{\left(\left(0 - x \cdot \log \left(\frac{x}{y}\right)\right) + z\right)} \]

      4. neg-sub0 99.8%

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

      5. distribute-rgt-neg-in 99.8%

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

      6. neg-sub0 99.8%

         \[\leadsto -\left(x \cdot \color{blue}{\left(0 - \log \left(\frac{x}{y}\right)\right)} + z\right) \]

      7. log-div 45.7%

         \[\leadsto -\left(x \cdot \left(0 - \color{blue}{\left(\log x - \log y\right)}\right) + z\right) \]

      8. associate-+l- 45.7%

         \[\leadsto -\left(x \cdot \color{blue}{\left(\left(0 - \log x\right) + \log y\right)} + z\right) \]

      9. neg-sub0 45.7%

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

      10. +-commutative 45.7%

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

      11. sub-neg 45.7%

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

      12. log-div 99.8%

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

      13. fma-udef 99.8%

          \[\leadsto -\color{blue}{\mathsf{fma}\left(x, \log \left(\frac{y}{x}\right), z\right)} \]

   3. Simplified 99.8%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log \left(\frac{x}{y}\right) \leq -\infty \lor \neg \left(x \cdot \log \left(\frac{x}{y}\right) \leq 5 \cdot 10^{+294}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(x, \log \left(\frac{y}{x}\right), z\right)\\ \end{array} \]

Alternative 3: 87.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;t_0 \leq -\infty \lor \neg \left(t_0 \leq 5 \cdot 10^{+294}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;t_0 - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 5e+294))) (- z) (- t_0 z))))

C

double code(double x, double y, double z) {
	double t_0 = x * log((x / y));
	double tmp;
	if ((t_0 <= -((double) INFINITY)) || !(t_0 <= 5e+294)) {
		tmp = -z;
	} else {
		tmp = t_0 - z;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = x * Math.log((x / y));
	double tmp;
	if ((t_0 <= -Double.POSITIVE_INFINITY) || !(t_0 <= 5e+294)) {
		tmp = -z;
	} else {
		tmp = t_0 - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.log((x / y))
	tmp = 0
	if (t_0 <= -math.inf) or not (t_0 <= 5e+294):
		tmp = -z
	else:
		tmp = t_0 - z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * log(Float64(x / y)))
	tmp = 0.0
	if ((t_0 <= Float64(-Inf)) || !(t_0 <= 5e+294))
		tmp = Float64(-z);
	else
		tmp = Float64(t_0 - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * log((x / y));
	tmp = 0.0;
	if ((t_0 <= -Inf) || ~((t_0 <= 5e+294)))
		tmp = -z;
	else
		tmp = t_0 - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, (-Infinity)], N[Not[LessEqual[t$95$0, 5e+294]], $MachinePrecision]], (-z), N[(t$95$0 - z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x (log.f64 (/.f64 x y))) < -inf.0 or 4.9999999999999999e294 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 6.6%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   2. Step-by-step derivation

      1. remove-double-neg 6.6%

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

      2. sub0-neg 6.6%

         \[\leadsto -\color{blue}{\left(0 - \left(x \cdot \log \left(\frac{x}{y}\right) - z\right)\right)} \]

      3. associate--r- 6.6%

         \[\leadsto -\color{blue}{\left(\left(0 - x \cdot \log \left(\frac{x}{y}\right)\right) + z\right)} \]

      4. neg-sub0 6.6%

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

      5. distribute-rgt-neg-in 6.6%

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

      6. neg-sub0 6.6%

         \[\leadsto -\left(x \cdot \color{blue}{\left(0 - \log \left(\frac{x}{y}\right)\right)} + z\right) \]

      7. log-div 43.6%

         \[\leadsto -\left(x \cdot \left(0 - \color{blue}{\left(\log x - \log y\right)}\right) + z\right) \]

      8. associate-+l- 43.6%

         \[\leadsto -\left(x \cdot \color{blue}{\left(\left(0 - \log x\right) + \log y\right)} + z\right) \]

      9. neg-sub0 43.6%

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

      10. +-commutative 43.6%

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

      11. sub-neg 43.6%

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

      12. log-div 12.0%

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

      13. fma-udef 12.0%

          \[\leadsto -\color{blue}{\mathsf{fma}\left(x, \log \left(\frac{y}{x}\right), z\right)} \]

   3. Simplified 12.0%

      \[\leadsto \color{blue}{-\mathsf{fma}\left(x, \log \left(\frac{y}{x}\right), z\right)} \]

   4. Taylor expanded in x around 0 45.4%

      \[\leadsto -\color{blue}{z} \]

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 4.9999999999999999e294

   1. Initial program 99.8%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log \left(\frac{x}{y}\right) \leq -\infty \lor \neg \left(x \cdot \log \left(\frac{x}{y}\right) \leq 5 \cdot 10^{+294}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \end{array} \]

Alternative 4: 87.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{+184}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right)\\ \mathbf{elif}\;x \leq -2.1 \cdot 10^{-209}:\\ \;\;\;\;-\mathsf{fma}\left(x, \log \left(\frac{y}{x}\right), z\right)\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-210}:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{+176}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -9e+184)
   (* x (- (log (- x)) (log (- y))))
   (if (<= x -2.1e-209)
     (- (fma x (log (/ y x)) z))
     (if (<= x 5.2e-210)
       (- z)
       (if (<= x 8.2e+176)
         (- (* x (log (/ x y))) z)
         (* x (- (log x) (log y))))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -9e+184) {
		tmp = x * (log(-x) - log(-y));
	} else if (x <= -2.1e-209) {
		tmp = -fma(x, log((y / x)), z);
	} else if (x <= 5.2e-210) {
		tmp = -z;
	} else if (x <= 8.2e+176) {
		tmp = (x * log((x / y))) - z;
	} else {
		tmp = x * (log(x) - log(y));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -9e+184)
		tmp = Float64(x * Float64(log(Float64(-x)) - log(Float64(-y))));
	elseif (x <= -2.1e-209)
		tmp = Float64(-fma(x, log(Float64(y / x)), z));
	elseif (x <= 5.2e-210)
		tmp = Float64(-z);
	elseif (x <= 8.2e+176)
		tmp = Float64(Float64(x * log(Float64(x / y))) - z);
	else
		tmp = Float64(x * Float64(log(x) - log(y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -9e+184], N[(x * N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2.1e-209], (-N[(x * N[Log[N[(y / x), $MachinePrecision]], $MachinePrecision] + z), $MachinePrecision]), If[LessEqual[x, 5.2e-210], (-z), If[LessEqual[x, 8.2e+176], N[(N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(x * N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -9.00000000000000072e184

   1. Initial program 53.6%

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

   2. Taylor expanded in z around 0 53.6%

      \[\leadsto \color{blue}{x \cdot \log \left(\frac{x}{y}\right)} \]
   3. Step-by-step derivation

      1. frac-2neg 53.6%

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

      2. log-div 90.3%

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

   4. Applied egg-rr 90.3%

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

   if -9.00000000000000072e184 < x < -2.09999999999999996e-209

   1. Initial program 82.1%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   2. Step-by-step derivation

      1. remove-double-neg 82.1%

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

      2. sub0-neg 82.1%

         \[\leadsto -\color{blue}{\left(0 - \left(x \cdot \log \left(\frac{x}{y}\right) - z\right)\right)} \]

      3. associate--r- 82.1%

         \[\leadsto -\color{blue}{\left(\left(0 - x \cdot \log \left(\frac{x}{y}\right)\right) + z\right)} \]

      4. neg-sub0 82.1%

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

      5. distribute-rgt-neg-in 82.1%

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

      6. neg-sub0 82.1%

         \[\leadsto -\left(x \cdot \color{blue}{\left(0 - \log \left(\frac{x}{y}\right)\right)} + z\right) \]

      7. log-div 0.0%

         \[\leadsto -\left(x \cdot \left(0 - \color{blue}{\left(\log x - \log y\right)}\right) + z\right) \]

      8. associate-+l- 0.0%

         \[\leadsto -\left(x \cdot \color{blue}{\left(\left(0 - \log x\right) + \log y\right)} + z\right) \]

      9. neg-sub0 0.0%

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

      10. +-commutative 0.0%

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

      11. sub-neg 0.0%

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

      12. log-div 85.2%

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

      13. fma-udef 85.2%

          \[\leadsto -\color{blue}{\mathsf{fma}\left(x, \log \left(\frac{y}{x}\right), z\right)} \]

   3. Simplified 85.2%

      \[\leadsto \color{blue}{-\mathsf{fma}\left(x, \log \left(\frac{y}{x}\right), z\right)} \]

   if -2.09999999999999996e-209 < x < 5.1999999999999997e-210

   1. Initial program 44.4%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   2. Step-by-step derivation

      1. remove-double-neg 44.4%

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

      2. sub0-neg 44.4%

         \[\leadsto -\color{blue}{\left(0 - \left(x \cdot \log \left(\frac{x}{y}\right) - z\right)\right)} \]

      3. associate--r- 44.4%

         \[\leadsto -\color{blue}{\left(\left(0 - x \cdot \log \left(\frac{x}{y}\right)\right) + z\right)} \]

      4. neg-sub0 44.4%

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

      5. distribute-rgt-neg-in 44.4%

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

      6. neg-sub0 44.4%

         \[\leadsto -\left(x \cdot \color{blue}{\left(0 - \log \left(\frac{x}{y}\right)\right)} + z\right) \]

      7. log-div 40.5%

         \[\leadsto -\left(x \cdot \left(0 - \color{blue}{\left(\log x - \log y\right)}\right) + z\right) \]

      8. associate-+l- 40.5%

         \[\leadsto -\left(x \cdot \color{blue}{\left(\left(0 - \log x\right) + \log y\right)} + z\right) \]

      9. neg-sub0 40.5%

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

      10. +-commutative 40.5%

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

      11. sub-neg 40.5%

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

      12. log-div 44.4%

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

      13. fma-udef 44.4%

          \[\leadsto -\color{blue}{\mathsf{fma}\left(x, \log \left(\frac{y}{x}\right), z\right)} \]

   3. Simplified 44.4%

      \[\leadsto \color{blue}{-\mathsf{fma}\left(x, \log \left(\frac{y}{x}\right), z\right)} \]

   4. Taylor expanded in x around 0 91.3%

      \[\leadsto -\color{blue}{z} \]

   if 5.1999999999999997e-210 < x < 8.1999999999999998e176

   1. Initial program 89.6%

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

   if 8.1999999999999998e176 < x

   1. Initial program 60.0%

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

   2. Taylor expanded in z around 0 52.6%

      \[\leadsto \color{blue}{x \cdot \log \left(\frac{x}{y}\right)} \]
   3. Step-by-step derivation

      1. log-div 91.7%

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

   4. Applied egg-rr 91.7%

      \[\leadsto x \cdot \color{blue}{\left(\log x - \log y\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{+184}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right)\\ \mathbf{elif}\;x \leq -2.1 \cdot 10^{-209}:\\ \;\;\;\;-\mathsf{fma}\left(x, \log \left(\frac{y}{x}\right), z\right)\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-210}:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{+176}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right)\\ \end{array} \]

Alternative 5: 93.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{+180}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right)\\ \mathbf{elif}\;x \leq -4.5 \cdot 10^{-212}:\\ \;\;\;\;-\mathsf{fma}\left(x, \log \left(\frac{y}{x}\right), z\right)\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-308}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -9e+180)
   (* x (- (log (- x)) (log (- y))))
   (if (<= x -4.5e-212)
     (- (fma x (log (/ y x)) z))
     (if (<= x -2e-308) (- z) (- (* x (- (log x) (log y))) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -9e+180) {
		tmp = x * (log(-x) - log(-y));
	} else if (x <= -4.5e-212) {
		tmp = -fma(x, log((y / x)), z);
	} else if (x <= -2e-308) {
		tmp = -z;
	} else {
		tmp = (x * (log(x) - log(y))) - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -9e+180)
		tmp = Float64(x * Float64(log(Float64(-x)) - log(Float64(-y))));
	elseif (x <= -4.5e-212)
		tmp = Float64(-fma(x, log(Float64(y / x)), z));
	elseif (x <= -2e-308)
		tmp = Float64(-z);
	else
		tmp = Float64(Float64(x * Float64(log(x) - log(y))) - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -9e+180], N[(x * N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -4.5e-212], (-N[(x * N[Log[N[(y / x), $MachinePrecision]], $MachinePrecision] + z), $MachinePrecision]), If[LessEqual[x, -2e-308], (-z), N[(N[(x * N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -8.99999999999999962e180

   1. Initial program 53.6%

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

   2. Taylor expanded in z around 0 53.6%

      \[\leadsto \color{blue}{x \cdot \log \left(\frac{x}{y}\right)} \]
   3. Step-by-step derivation

      1. frac-2neg 53.6%

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

      2. log-div 90.3%

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

   4. Applied egg-rr 90.3%

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

   if -8.99999999999999962e180 < x < -4.4999999999999999e-212

   1. Initial program 82.1%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   2. Step-by-step derivation

      1. remove-double-neg 82.1%

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

      2. sub0-neg 82.1%

         \[\leadsto -\color{blue}{\left(0 - \left(x \cdot \log \left(\frac{x}{y}\right) - z\right)\right)} \]

      3. associate--r- 82.1%

         \[\leadsto -\color{blue}{\left(\left(0 - x \cdot \log \left(\frac{x}{y}\right)\right) + z\right)} \]

      4. neg-sub0 82.1%

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

      5. distribute-rgt-neg-in 82.1%

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

      6. neg-sub0 82.1%

         \[\leadsto -\left(x \cdot \color{blue}{\left(0 - \log \left(\frac{x}{y}\right)\right)} + z\right) \]

      7. log-div 0.0%

         \[\leadsto -\left(x \cdot \left(0 - \color{blue}{\left(\log x - \log y\right)}\right) + z\right) \]

      8. associate-+l- 0.0%

         \[\leadsto -\left(x \cdot \color{blue}{\left(\left(0 - \log x\right) + \log y\right)} + z\right) \]

      9. neg-sub0 0.0%

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

      10. +-commutative 0.0%

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

      11. sub-neg 0.0%

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

      12. log-div 85.2%

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

      13. fma-udef 85.2%

          \[\leadsto -\color{blue}{\mathsf{fma}\left(x, \log \left(\frac{y}{x}\right), z\right)} \]

   3. Simplified 85.2%

      \[\leadsto \color{blue}{-\mathsf{fma}\left(x, \log \left(\frac{y}{x}\right), z\right)} \]

   if -4.4999999999999999e-212 < x < -1.9999999999999998e-308

   1. Initial program 55.1%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   2. Step-by-step derivation

      1. remove-double-neg 55.1%

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

      2. sub0-neg 55.1%

         \[\leadsto -\color{blue}{\left(0 - \left(x \cdot \log \left(\frac{x}{y}\right) - z\right)\right)} \]

      3. associate--r- 55.1%

         \[\leadsto -\color{blue}{\left(\left(0 - x \cdot \log \left(\frac{x}{y}\right)\right) + z\right)} \]

      4. neg-sub0 55.1%

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

      5. distribute-rgt-neg-in 55.1%

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

      6. neg-sub0 55.1%

         \[\leadsto -\left(x \cdot \color{blue}{\left(0 - \log \left(\frac{x}{y}\right)\right)} + z\right) \]

      7. log-div 0.0%

         \[\leadsto -\left(x \cdot \left(0 - \color{blue}{\left(\log x - \log y\right)}\right) + z\right) \]

      8. associate-+l- 0.0%

         \[\leadsto -\left(x \cdot \color{blue}{\left(\left(0 - \log x\right) + \log y\right)} + z\right) \]

      9. neg-sub0 0.0%

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

      10. +-commutative 0.0%

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

      11. sub-neg 0.0%

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

      12. log-div 55.1%

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

      13. fma-udef 55.1%

          \[\leadsto -\color{blue}{\mathsf{fma}\left(x, \log \left(\frac{y}{x}\right), z\right)} \]

   3. Simplified 55.1%

      \[\leadsto \color{blue}{-\mathsf{fma}\left(x, \log \left(\frac{y}{x}\right), z\right)} \]

   4. Taylor expanded in x around 0 89.5%

      \[\leadsto -\color{blue}{z} \]

   if -1.9999999999999998e-308 < x

   1. Initial program 75.1%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   2. Step-by-step derivation

      1. log-div 46.9%

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

   3. Applied egg-rr 99.6%

      \[\leadsto x \cdot \color{blue}{\left(\log x - \log y\right)} - z \]
3. Recombined 4 regimes into one program.

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{+180}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right)\\ \mathbf{elif}\;x \leq -4.5 \cdot 10^{-212}:\\ \;\;\;\;-\mathsf{fma}\left(x, \log \left(\frac{y}{x}\right), z\right)\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-308}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \]

Alternative 6: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -5e-310)
   (- (* x (- (log (- x)) (log (- y)))) z)
   (- (* x (- (log x) (log y))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -5e-310) {
		tmp = (x * (log(-x) - log(-y))) - 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
    real(8) :: tmp
    if (y <= (-5d-310)) then
        tmp = (x * (log(-x) - log(-y))) - 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) {
	double tmp;
	if (y <= -5e-310) {
		tmp = (x * (Math.log(-x) - Math.log(-y))) - z;
	} else {
		tmp = (x * (Math.log(x) - Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -5e-310:
		tmp = (x * (math.log(-x) - math.log(-y))) - z
	else:
		tmp = (x * (math.log(x) - math.log(y))) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -5e-310)
		tmp = Float64(Float64(x * Float64(log(Float64(-x)) - log(Float64(-y)))) - z);
	else
		tmp = Float64(Float64(x * Float64(log(x) - log(y))) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -5e-310)
		tmp = (x * (log(-x) - log(-y))) - z;
	else
		tmp = (x * (log(x) - log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -5e-310], N[(N[(x * N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(N[(x * N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.999999999999985e-310

   1. Initial program 73.0%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   2. Step-by-step derivation

      1. frac-2neg 39.1%

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

      2. log-div 52.2%

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

   3. Applied egg-rr 99.6%

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

   if -4.999999999999985e-310 < y

   1. Initial program 75.1%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   2. Step-by-step derivation

      1. log-div 46.9%

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

   3. Applied egg-rr 99.6%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \]

Alternative 7: 67.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -650000000000 \lor \neg \left(z \leq 125000\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\left(-x\right) \cdot \log \left(\frac{y}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -650000000000.0) (not (<= z 125000.0)))
   (- z)
   (* (- x) (log (/ y x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -650000000000.0) || !(z <= 125000.0)) {
		tmp = -z;
	} else {
		tmp = -x * log((y / x));
	}
	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
    real(8) :: tmp
    if ((z <= (-650000000000.0d0)) .or. (.not. (z <= 125000.0d0))) then
        tmp = -z
    else
        tmp = -x * log((y / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -650000000000.0) || !(z <= 125000.0)) {
		tmp = -z;
	} else {
		tmp = -x * Math.log((y / x));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -650000000000.0) or not (z <= 125000.0):
		tmp = -z
	else:
		tmp = -x * math.log((y / x))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -650000000000.0) || !(z <= 125000.0))
		tmp = Float64(-z);
	else
		tmp = Float64(Float64(-x) * log(Float64(y / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -650000000000.0) || ~((z <= 125000.0)))
		tmp = -z;
	else
		tmp = -x * log((y / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -650000000000.0], N[Not[LessEqual[z, 125000.0]], $MachinePrecision]], (-z), N[((-x) * N[Log[N[(y / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -6.5e11 or 125000 < z

   1. Initial program 74.9%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   2. Step-by-step derivation

      1. remove-double-neg 74.9%

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

      2. sub0-neg 74.9%

         \[\leadsto -\color{blue}{\left(0 - \left(x \cdot \log \left(\frac{x}{y}\right) - z\right)\right)} \]

      3. associate--r- 74.9%

         \[\leadsto -\color{blue}{\left(\left(0 - x \cdot \log \left(\frac{x}{y}\right)\right) + z\right)} \]

      4. neg-sub0 74.9%

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

      5. distribute-rgt-neg-in 74.9%

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

      6. neg-sub0 74.9%

         \[\leadsto -\left(x \cdot \color{blue}{\left(0 - \log \left(\frac{x}{y}\right)\right)} + z\right) \]

      7. log-div 47.6%

         \[\leadsto -\left(x \cdot \left(0 - \color{blue}{\left(\log x - \log y\right)}\right) + z\right) \]

      8. associate-+l- 47.6%

         \[\leadsto -\left(x \cdot \color{blue}{\left(\left(0 - \log x\right) + \log y\right)} + z\right) \]

      9. neg-sub0 47.6%

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

      10. +-commutative 47.6%

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

      11. sub-neg 47.6%

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

      12. log-div 75.7%

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

      13. fma-udef 75.7%

          \[\leadsto -\color{blue}{\mathsf{fma}\left(x, \log \left(\frac{y}{x}\right), z\right)} \]

   3. Simplified 75.7%

      \[\leadsto \color{blue}{-\mathsf{fma}\left(x, \log \left(\frac{y}{x}\right), z\right)} \]

   4. Taylor expanded in x around 0 78.0%

      \[\leadsto -\color{blue}{z} \]

   if -6.5e11 < z < 125000

   1. Initial program 72.9%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   2. Step-by-step derivation

      1. remove-double-neg 72.9%

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

      2. sub0-neg 72.9%

         \[\leadsto -\color{blue}{\left(0 - \left(x \cdot \log \left(\frac{x}{y}\right) - z\right)\right)} \]

      3. associate--r- 72.9%

         \[\leadsto -\color{blue}{\left(\left(0 - x \cdot \log \left(\frac{x}{y}\right)\right) + z\right)} \]

      4. neg-sub0 72.9%

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

      5. distribute-rgt-neg-in 72.9%

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

      6. neg-sub0 72.9%

         \[\leadsto -\left(x \cdot \color{blue}{\left(0 - \log \left(\frac{x}{y}\right)\right)} + z\right) \]

      7. log-div 42.5%

         \[\leadsto -\left(x \cdot \left(0 - \color{blue}{\left(\log x - \log y\right)}\right) + z\right) \]

      8. associate-+l- 42.5%

         \[\leadsto -\left(x \cdot \color{blue}{\left(\left(0 - \log x\right) + \log y\right)} + z\right) \]

      9. neg-sub0 42.5%

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

      10. +-commutative 42.5%

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

      11. sub-neg 42.5%

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

      12. log-div 75.2%

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

      13. fma-udef 75.2%

          \[\leadsto -\color{blue}{\mathsf{fma}\left(x, \log \left(\frac{y}{x}\right), z\right)} \]

   3. Simplified 75.2%

      \[\leadsto \color{blue}{-\mathsf{fma}\left(x, \log \left(\frac{y}{x}\right), z\right)} \]

   4. Taylor expanded in x around inf 33.7%

      \[\leadsto -\color{blue}{x \cdot \left(\log y + \log \left(\frac{1}{x}\right)\right)} \]
   5. Step-by-step derivation

      1. log-rec 33.7%

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

      2. neg-mul-1 33.7%

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

      3. neg-mul-1 33.7%

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

      4. sub-neg 33.7%

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

      5. log-div 63.9%

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

   6. Simplified 63.9%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -650000000000 \lor \neg \left(z \leq 125000\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\left(-x\right) \cdot \log \left(\frac{y}{x}\right)\\ \end{array} \]

Alternative 8: 67.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1650000000 \lor \neg \left(z \leq 47000000\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1650000000.0) (not (<= z 47000000.0)))
   (- z)
   (* x (log (/ x y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1650000000.0) || !(z <= 47000000.0)) {
		tmp = -z;
	} else {
		tmp = x * log((x / 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
    real(8) :: tmp
    if ((z <= (-1650000000.0d0)) .or. (.not. (z <= 47000000.0d0))) then
        tmp = -z
    else
        tmp = x * log((x / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1650000000.0) || !(z <= 47000000.0)) {
		tmp = -z;
	} else {
		tmp = x * Math.log((x / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1650000000.0) or not (z <= 47000000.0):
		tmp = -z
	else:
		tmp = x * math.log((x / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1650000000.0) || !(z <= 47000000.0))
		tmp = Float64(-z);
	else
		tmp = Float64(x * log(Float64(x / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1650000000.0) || ~((z <= 47000000.0)))
		tmp = -z;
	else
		tmp = x * log((x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1650000000.0], N[Not[LessEqual[z, 47000000.0]], $MachinePrecision]], (-z), N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.65e9 or 4.7e7 < z

   1. Initial program 74.9%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   2. Step-by-step derivation

      1. remove-double-neg 74.9%

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

      2. sub0-neg 74.9%

         \[\leadsto -\color{blue}{\left(0 - \left(x \cdot \log \left(\frac{x}{y}\right) - z\right)\right)} \]

      3. associate--r- 74.9%

         \[\leadsto -\color{blue}{\left(\left(0 - x \cdot \log \left(\frac{x}{y}\right)\right) + z\right)} \]

      4. neg-sub0 74.9%

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

      5. distribute-rgt-neg-in 74.9%

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

      6. neg-sub0 74.9%

         \[\leadsto -\left(x \cdot \color{blue}{\left(0 - \log \left(\frac{x}{y}\right)\right)} + z\right) \]

      7. log-div 47.6%

         \[\leadsto -\left(x \cdot \left(0 - \color{blue}{\left(\log x - \log y\right)}\right) + z\right) \]

      8. associate-+l- 47.6%

         \[\leadsto -\left(x \cdot \color{blue}{\left(\left(0 - \log x\right) + \log y\right)} + z\right) \]

      9. neg-sub0 47.6%

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

      10. +-commutative 47.6%

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

      11. sub-neg 47.6%

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

      12. log-div 75.7%

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

      13. fma-udef 75.7%

          \[\leadsto -\color{blue}{\mathsf{fma}\left(x, \log \left(\frac{y}{x}\right), z\right)} \]

   3. Simplified 75.7%

      \[\leadsto \color{blue}{-\mathsf{fma}\left(x, \log \left(\frac{y}{x}\right), z\right)} \]

   4. Taylor expanded in x around 0 78.0%

      \[\leadsto -\color{blue}{z} \]

   if -1.65e9 < z < 4.7e7

   1. Initial program 72.9%

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

   2. Taylor expanded in z around 0 61.6%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1650000000 \lor \neg \left(z \leq 47000000\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \end{array} \]

Alternative 9: 49.7% accurate, 53.5× speedup

Math

\[\begin{array}{l} \\ -z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- z))

C

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

Fortran

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

Java

public static double code(double x, double y, double z) {
	return -z;
}

Python

def code(x, y, z):
	return -z

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := (-z)

Derivation

1. Initial program 73.9%

   \[x \cdot \log \left(\frac{x}{y}\right) - z \]
2. Step-by-step derivation

   1. remove-double-neg 73.9%

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

   2. sub0-neg 73.9%

      \[\leadsto -\color{blue}{\left(0 - \left(x \cdot \log \left(\frac{x}{y}\right) - z\right)\right)} \]

   3. associate--r- 73.9%

      \[\leadsto -\color{blue}{\left(\left(0 - x \cdot \log \left(\frac{x}{y}\right)\right) + z\right)} \]

   4. neg-sub0 73.9%

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

   5. distribute-rgt-neg-in 73.9%

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

   6. neg-sub0 73.9%

      \[\leadsto -\left(x \cdot \color{blue}{\left(0 - \log \left(\frac{x}{y}\right)\right)} + z\right) \]

   7. log-div 45.1%

      \[\leadsto -\left(x \cdot \left(0 - \color{blue}{\left(\log x - \log y\right)}\right) + z\right) \]

   8. associate-+l- 45.1%

      \[\leadsto -\left(x \cdot \color{blue}{\left(\left(0 - \log x\right) + \log y\right)} + z\right) \]

   9. neg-sub0 45.1%

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

   10. +-commutative 45.1%

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

   11. sub-neg 45.1%

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

   12. log-div 75.4%

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

   13. fma-udef 75.5%

       \[\leadsto -\color{blue}{\mathsf{fma}\left(x, \log \left(\frac{y}{x}\right), z\right)} \]

3. Simplified 75.5%

   \[\leadsto \color{blue}{-\mathsf{fma}\left(x, \log \left(\frac{y}{x}\right), z\right)} \]

4. Taylor expanded in x around 0 50.4%

   \[\leadsto -\color{blue}{z} \]

5. Final simplification 50.4%

   \[\leadsto -z \]

Developer target: 88.9% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (if (< y 7.595077799083773e-308)
   (- (* x (log (/ x y))) z)
   (- (* x (- (log x) (log y))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y < 7.595077799083773e-308) {
		tmp = (x * log((x / y))) - 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
    real(8) :: tmp
    if (y < 7.595077799083773d-308) then
        tmp = (x * log((x / y))) - 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) {
	double tmp;
	if (y < 7.595077799083773e-308) {
		tmp = (x * Math.log((x / y))) - z;
	} else {
		tmp = (x * (Math.log(x) - Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y < 7.595077799083773e-308:
		tmp = (x * math.log((x / y))) - z
	else:
		tmp = (x * (math.log(x) - math.log(y))) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y < 7.595077799083773e-308)
		tmp = Float64(Float64(x * log(Float64(x / y))) - z);
	else
		tmp = Float64(Float64(x * Float64(log(x) - log(y))) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y < 7.595077799083773e-308)
		tmp = (x * log((x / y))) - z;
	else
		tmp = (x * (log(x) - log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Less[y, 7.595077799083773e-308], N[(N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(N[(x * N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]

Reproduce

herbie shell --seed 2023336 
(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))