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

Percentage Accurate: 78.0% → 99.7%

Time: 8.2s

Alternatives: 12

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: 78.0% 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

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 74.6%

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

   1. add-cube-cbrt 74.6%

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

   2. associate-*l* 74.6%

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

   3. log-prod 74.6%

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

   4. pow2 74.6%

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

   5. metadata-eval 74.6%

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

   6. log-pow 74.6%

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

   7. metadata-eval 74.6%

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

3. Applied egg-rr 74.6%

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

   1. distribute-rgt1-in 74.6%

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

   2. metadata-eval 74.6%

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

   3. *-commutative 74.6%

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

5. Simplified 74.6%

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

   1. cbrt-div 99.7%

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

   2. div-inv 99.7%

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

7. Applied egg-rr 99.7%

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

   1. associate-*r/ 99.7%

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

   2. *-rgt-identity 99.7%

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

9. Simplified 99.7%

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

10. Final simplification 99.7%

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

Alternative 2: 87.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\frac{x}{y}\right)\\ t_1 := x \cdot t_0\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;-z\\ \mathbf{elif}\;t_1 \leq 10^{+265}:\\ \;\;\;\;\mathsf{fma}\left(t_0, x, -z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (log (/ x y))) (t_1 (* x t_0)))
   (if (<= t_1 (- INFINITY))
     (- z)
     (if (<= t_1 1e+265) (fma t_0 x (- z)) (* x (- (log x) (log y)))))))

C

double code(double x, double y, double z) {
	double t_0 = log((x / y));
	double t_1 = x * t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -z;
	} else if (t_1 <= 1e+265) {
		tmp = fma(t_0, x, -z);
	} else {
		tmp = x * (log(x) - log(y));
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = log(Float64(x / y))
	t_1 = Float64(x * t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-z);
	elseif (t_1 <= 1e+265)
		tmp = fma(t_0, x, Float64(-z));
	else
		tmp = Float64(x * Float64(log(x) - log(y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(x * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], (-z), If[LessEqual[t$95$1, 1e+265], N[(t$95$0 * x + (-z)), $MachinePrecision], N[(x * N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 4.0%

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

   2. Taylor expanded in x around 0 57.9%

      \[\leadsto \color{blue}{-1 \cdot z} \]
   3. Step-by-step derivation

      1. neg-mul-1 57.9%

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

   4. Simplified 57.9%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 1.00000000000000007e265

   1. Initial program 99.5%

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

      1. *-commutative 99.5%

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

      2. fma-neg 99.5%

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

   3. Applied egg-rr 99.5%

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

   if 1.00000000000000007e265 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 8.7%

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

      1. add-cube-cbrt 8.7%

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

      2. associate-*l* 8.7%

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

      3. log-prod 8.8%

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

      4. pow2 8.8%

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

      5. metadata-eval 8.8%

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

      6. log-pow 8.8%

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

      7. metadata-eval 8.8%

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

   3. Applied egg-rr 8.8%

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

      1. distribute-rgt1-in 8.8%

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

      2. metadata-eval 8.8%

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

      3. *-commutative 8.8%

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

   5. Simplified 8.8%

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

   6. Taylor expanded in z around 0 8.8%

      \[\leadsto \color{blue}{3 \cdot \left(x \cdot \log \left({\left(\frac{x}{y}\right)}^{0.3333333333333333}\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative 8.8%

         \[\leadsto 3 \cdot \color{blue}{\left(\log \left({\left(\frac{x}{y}\right)}^{0.3333333333333333}\right) \cdot x\right)} \]

      2. unpow1/3 8.8%

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

      3. associate-*r* 8.8%

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

      4. unpow1/3 8.7%

         \[\leadsto \left(3 \cdot \log \color{blue}{\left({\left(\frac{x}{y}\right)}^{0.3333333333333333}\right)}\right) \cdot x \]

      5. log-pow 8.7%

         \[\leadsto \left(3 \cdot \color{blue}{\left(0.3333333333333333 \cdot \log \left(\frac{x}{y}\right)\right)}\right) \cdot x \]

      6. associate-*r* 8.7%

         \[\leadsto \color{blue}{\left(\left(3 \cdot 0.3333333333333333\right) \cdot \log \left(\frac{x}{y}\right)\right)} \cdot x \]

      7. metadata-eval 8.7%

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

      8. *-lft-identity 8.7%

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

      9. *-commutative 8.7%

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

   8. Simplified 8.7%

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

      1. log-div 52.2%

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

   10. Applied egg-rr 52.2%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log \left(\frac{x}{y}\right) \leq -\infty:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \cdot \log \left(\frac{x}{y}\right) \leq 10^{+265}:\\ \;\;\;\;\mathsf{fma}\left(\log \left(\frac{x}{y}\right), x, -z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right)\\ \end{array} \]

Alternative 3: 87.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;-z\\ \mathbf{elif}\;t_0 \leq 10^{+265}:\\ \;\;\;\;t_0 - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))))
   (if (<= t_0 (- INFINITY))
     (- z)
     (if (<= t_0 1e+265) (- t_0 z) (* x (- (log x) (log y)))))))

C

double code(double x, double y, double z) {
	double t_0 = x * log((x / y));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = -z;
	} else if (t_0 <= 1e+265) {
		tmp = t_0 - z;
	} else {
		tmp = x * (log(x) - log(y));
	}
	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) {
		tmp = -z;
	} else if (t_0 <= 1e+265) {
		tmp = t_0 - z;
	} else {
		tmp = x * (Math.log(x) - Math.log(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.log((x / y))
	tmp = 0
	if t_0 <= -math.inf:
		tmp = -z
	elif t_0 <= 1e+265:
		tmp = t_0 - z
	else:
		tmp = x * (math.log(x) - math.log(y))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * log(Float64(x / y)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(-z);
	elseif (t_0 <= 1e+265)
		tmp = Float64(t_0 - z);
	else
		tmp = Float64(x * Float64(log(x) - log(y)));
	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)
		tmp = -z;
	elseif (t_0 <= 1e+265)
		tmp = t_0 - z;
	else
		tmp = x * (log(x) - log(y));
	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[LessEqual[t$95$0, (-Infinity)], (-z), If[LessEqual[t$95$0, 1e+265], N[(t$95$0 - z), $MachinePrecision], N[(x * N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 4.0%

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

   2. Taylor expanded in x around 0 57.9%

      \[\leadsto \color{blue}{-1 \cdot z} \]
   3. Step-by-step derivation

      1. neg-mul-1 57.9%

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

   4. Simplified 57.9%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 1.00000000000000007e265

   1. Initial program 99.5%

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

   if 1.00000000000000007e265 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 8.7%

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

      1. add-cube-cbrt 8.7%

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

      2. associate-*l* 8.7%

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

      3. log-prod 8.8%

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

      4. pow2 8.8%

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

      5. metadata-eval 8.8%

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

      6. log-pow 8.8%

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

      7. metadata-eval 8.8%

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

   3. Applied egg-rr 8.8%

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

      1. distribute-rgt1-in 8.8%

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

      2. metadata-eval 8.8%

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

      3. *-commutative 8.8%

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

   5. Simplified 8.8%

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

   6. Taylor expanded in z around 0 8.8%

      \[\leadsto \color{blue}{3 \cdot \left(x \cdot \log \left({\left(\frac{x}{y}\right)}^{0.3333333333333333}\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative 8.8%

         \[\leadsto 3 \cdot \color{blue}{\left(\log \left({\left(\frac{x}{y}\right)}^{0.3333333333333333}\right) \cdot x\right)} \]

      2. unpow1/3 8.8%

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

      3. associate-*r* 8.8%

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

      4. unpow1/3 8.7%

         \[\leadsto \left(3 \cdot \log \color{blue}{\left({\left(\frac{x}{y}\right)}^{0.3333333333333333}\right)}\right) \cdot x \]

      5. log-pow 8.7%

         \[\leadsto \left(3 \cdot \color{blue}{\left(0.3333333333333333 \cdot \log \left(\frac{x}{y}\right)\right)}\right) \cdot x \]

      6. associate-*r* 8.7%

         \[\leadsto \color{blue}{\left(\left(3 \cdot 0.3333333333333333\right) \cdot \log \left(\frac{x}{y}\right)\right)} \cdot x \]

      7. metadata-eval 8.7%

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

      8. *-lft-identity 8.7%

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

      9. *-commutative 8.7%

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

   8. Simplified 8.7%

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

      1. log-div 52.2%

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

   10. Applied egg-rr 52.2%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log \left(\frac{x}{y}\right) \leq -\infty:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \cdot \log \left(\frac{x}{y}\right) \leq 10^{+265}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right)\\ \end{array} \]

Alternative 4: 86.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;-z\\ \mathbf{elif}\;t_0 \leq 4 \cdot 10^{+269}:\\ \;\;\;\;t_0 - z\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))))
   (if (<= t_0 (- INFINITY)) (- z) (if (<= t_0 4e+269) (- t_0 z) (- z)))))

C

double code(double x, double y, double z) {
	double t_0 = x * log((x / y));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = -z;
	} else if (t_0 <= 4e+269) {
		tmp = t_0 - z;
	} else {
		tmp = -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) {
		tmp = -z;
	} else if (t_0 <= 4e+269) {
		tmp = t_0 - z;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.log((x / y))
	tmp = 0
	if t_0 <= -math.inf:
		tmp = -z
	elif t_0 <= 4e+269:
		tmp = t_0 - z
	else:
		tmp = -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))
		tmp = Float64(-z);
	elseif (t_0 <= 4e+269)
		tmp = Float64(t_0 - z);
	else
		tmp = Float64(-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)
		tmp = -z;
	elseif (t_0 <= 4e+269)
		tmp = t_0 - z;
	else
		tmp = -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[LessEqual[t$95$0, (-Infinity)], (-z), If[LessEqual[t$95$0, 4e+269], N[(t$95$0 - z), $MachinePrecision], (-z)]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x (log.f64 (/.f64 x y))) < -inf.0 or 4.0000000000000002e269 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 4.6%

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

   2. Taylor expanded in x around 0 48.5%

      \[\leadsto \color{blue}{-1 \cdot z} \]
   3. Step-by-step derivation

      1. neg-mul-1 48.5%

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

   4. Simplified 48.5%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 4.0000000000000002e269

   1. Initial program 99.5%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log \left(\frac{x}{y}\right) \leq -\infty:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \cdot \log \left(\frac{x}{y}\right) \leq 4 \cdot 10^{+269}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 5: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\mathsf{fma}\left(\log \left(-x\right) - \log \left(-y\right), x, -z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1e-310)
   (fma (- (log (- x)) (log (- y))) x (- z))
   (- (* x (- (log x) (log y))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1e-310) {
		tmp = fma((log(-x) - log(-y)), x, -z);
	} else {
		tmp = (x * (log(x) - log(y))) - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1e-310)
		tmp = fma(Float64(log(Float64(-x)) - log(Float64(-y))), x, Float64(-z));
	else
		tmp = Float64(Float64(x * Float64(log(x) - log(y))) - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1e-310], N[(N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision] * x + (-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 < -9.999999999999969e-311

   1. Initial program 76.8%

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

      1. *-commutative 76.8%

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

      2. fma-neg 76.8%

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

   3. Applied egg-rr 76.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\log \left(\frac{x}{y}\right), x, -z\right)} \]
   4. Step-by-step derivation

      1. frac-2neg 41.6%

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

      2. log-div 55.2%

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

   5. Applied egg-rr 99.6%

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

   if -9.999999999999969e-311 < y

   1. Initial program 72.6%

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

      1. log-div 50.7%

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

   3. Applied egg-rr 99.5%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\mathsf{fma}\left(\log \left(-x\right) - \log \left(-y\right), x, -z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \]

Alternative 6: 86.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;x \leq -3.4 \cdot 10^{+99}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right)\\ \mathbf{elif}\;x \leq -1.4 \cdot 10^{-186}:\\ \;\;\;\;x \cdot t_0 - z\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-139}:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{+166}:\\ \;\;\;\;\mathsf{fma}\left(t_0, x, -z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (log (/ x y))))
   (if (<= x -3.4e+99)
     (* x (- (log (- x)) (log (- y))))
     (if (<= x -1.4e-186)
       (- (* x t_0) z)
       (if (<= x 1.6e-139)
         (- z)
         (if (<= x 1.85e+166) (fma t_0 x (- z)) (* x (- (log x) (log y)))))))))

C

double code(double x, double y, double z) {
	double t_0 = log((x / y));
	double tmp;
	if (x <= -3.4e+99) {
		tmp = x * (log(-x) - log(-y));
	} else if (x <= -1.4e-186) {
		tmp = (x * t_0) - z;
	} else if (x <= 1.6e-139) {
		tmp = -z;
	} else if (x <= 1.85e+166) {
		tmp = fma(t_0, x, -z);
	} else {
		tmp = x * (log(x) - log(y));
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = log(Float64(x / y))
	tmp = 0.0
	if (x <= -3.4e+99)
		tmp = Float64(x * Float64(log(Float64(-x)) - log(Float64(-y))));
	elseif (x <= -1.4e-186)
		tmp = Float64(Float64(x * t_0) - z);
	elseif (x <= 1.6e-139)
		tmp = Float64(-z);
	elseif (x <= 1.85e+166)
		tmp = fma(t_0, x, Float64(-z));
	else
		tmp = Float64(x * Float64(log(x) - log(y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -3.4e+99], N[(x * N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.4e-186], N[(N[(x * t$95$0), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[x, 1.6e-139], (-z), If[LessEqual[x, 1.85e+166], N[(t$95$0 * x + (-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 < -3.39999999999999984e99

   1. Initial program 67.0%

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

      1. add-cube-cbrt 67.0%

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

      2. associate-*l* 67.0%

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

      3. log-prod 67.0%

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

      4. pow2 67.0%

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

      5. metadata-eval 67.0%

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

      6. log-pow 67.0%

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

      7. metadata-eval 67.0%

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

   3. Applied egg-rr 67.0%

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

      1. distribute-rgt1-in 67.0%

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

      2. metadata-eval 67.0%

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

      3. *-commutative 67.0%

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

   5. Simplified 67.0%

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

   6. Taylor expanded in z around 0 58.8%

      \[\leadsto \color{blue}{3 \cdot \left(x \cdot \log \left({\left(\frac{x}{y}\right)}^{0.3333333333333333}\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative 58.8%

         \[\leadsto 3 \cdot \color{blue}{\left(\log \left({\left(\frac{x}{y}\right)}^{0.3333333333333333}\right) \cdot x\right)} \]

      2. unpow1/3 58.7%

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

      3. associate-*r* 58.8%

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

      4. unpow1/3 58.8%

         \[\leadsto \left(3 \cdot \log \color{blue}{\left({\left(\frac{x}{y}\right)}^{0.3333333333333333}\right)}\right) \cdot x \]

      5. log-pow 58.7%

         \[\leadsto \left(3 \cdot \color{blue}{\left(0.3333333333333333 \cdot \log \left(\frac{x}{y}\right)\right)}\right) \cdot x \]

      6. associate-*r* 58.8%

         \[\leadsto \color{blue}{\left(\left(3 \cdot 0.3333333333333333\right) \cdot \log \left(\frac{x}{y}\right)\right)} \cdot x \]

      7. metadata-eval 58.8%

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

      8. *-lft-identity 58.8%

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

      9. *-commutative 58.8%

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

   8. Simplified 58.8%

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

      1. frac-2neg 58.8%

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

      2. log-div 88.4%

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

   10. Applied egg-rr 88.4%

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

   if -3.39999999999999984e99 < x < -1.39999999999999992e-186

   1. Initial program 91.0%

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

   if -1.39999999999999992e-186 < x < 1.6e-139

   1. Initial program 58.4%

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

   2. Taylor expanded in x around 0 95.5%

      \[\leadsto \color{blue}{-1 \cdot z} \]
   3. Step-by-step derivation

      1. neg-mul-1 95.5%

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

   4. Simplified 95.5%

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

   if 1.6e-139 < x < 1.85000000000000011e166

   1. Initial program 89.2%

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

      1. *-commutative 89.2%

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

      2. fma-neg 89.3%

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

   3. Applied egg-rr 89.3%

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

   if 1.85000000000000011e166 < x

   1. Initial program 61.9%

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

      1. add-cube-cbrt 61.8%

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

      2. associate-*l* 61.8%

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

      3. log-prod 61.9%

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

      4. pow2 61.9%

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

      5. metadata-eval 61.9%

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

      6. log-pow 61.9%

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

      7. metadata-eval 61.9%

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

   3. Applied egg-rr 61.9%

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

      1. distribute-rgt1-in 61.9%

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

      2. metadata-eval 61.9%

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

      3. *-commutative 61.9%

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

   5. Simplified 61.9%

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

   6. Taylor expanded in z around 0 55.6%

      \[\leadsto \color{blue}{3 \cdot \left(x \cdot \log \left({\left(\frac{x}{y}\right)}^{0.3333333333333333}\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative 55.6%

         \[\leadsto 3 \cdot \color{blue}{\left(\log \left({\left(\frac{x}{y}\right)}^{0.3333333333333333}\right) \cdot x\right)} \]

      2. unpow1/3 55.7%

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

      3. associate-*r* 55.7%

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

      4. unpow1/3 55.6%

         \[\leadsto \left(3 \cdot \log \color{blue}{\left({\left(\frac{x}{y}\right)}^{0.3333333333333333}\right)}\right) \cdot x \]

      5. log-pow 55.7%

         \[\leadsto \left(3 \cdot \color{blue}{\left(0.3333333333333333 \cdot \log \left(\frac{x}{y}\right)\right)}\right) \cdot x \]

      6. associate-*r* 55.7%

         \[\leadsto \color{blue}{\left(\left(3 \cdot 0.3333333333333333\right) \cdot \log \left(\frac{x}{y}\right)\right)} \cdot x \]

      7. metadata-eval 55.7%

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

      8. *-lft-identity 55.7%

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

      9. *-commutative 55.7%

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

   8. Simplified 55.7%

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

      1. log-div 90.7%

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

   10. Applied egg-rr 90.7%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{+99}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right)\\ \mathbf{elif}\;x \leq -1.4 \cdot 10^{-186}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-139}:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{+166}:\\ \;\;\;\;\mathsf{fma}\left(\log \left(\frac{x}{y}\right), x, -z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right)\\ \end{array} \]

Alternative 7: 92.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{+98}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right)\\ \mathbf{elif}\;x \leq -4.4 \cdot 10^{-184}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-309}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -3e+98)
   (* x (- (log (- x)) (log (- y))))
   (if (<= x -4.4e-184)
     (- (* x (log (/ x y))) z)
     (if (<= x -1e-309) (- z) (- (* x (- (log x) (log y))) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -3e+98) {
		tmp = x * (log(-x) - log(-y));
	} else if (x <= -4.4e-184) {
		tmp = (x * log((x / y))) - z;
	} else if (x <= -1e-309) {
		tmp = -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 (x <= (-3d+98)) then
        tmp = x * (log(-x) - log(-y))
    else if (x <= (-4.4d-184)) then
        tmp = (x * log((x / y))) - z
    else if (x <= (-1d-309)) then
        tmp = -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 (x <= -3e+98) {
		tmp = x * (Math.log(-x) - Math.log(-y));
	} else if (x <= -4.4e-184) {
		tmp = (x * Math.log((x / y))) - z;
	} else if (x <= -1e-309) {
		tmp = -z;
	} else {
		tmp = (x * (Math.log(x) - Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -3e+98:
		tmp = x * (math.log(-x) - math.log(-y))
	elif x <= -4.4e-184:
		tmp = (x * math.log((x / y))) - z
	elif x <= -1e-309:
		tmp = -z
	else:
		tmp = (x * (math.log(x) - math.log(y))) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -3e+98)
		tmp = Float64(x * Float64(log(Float64(-x)) - log(Float64(-y))));
	elseif (x <= -4.4e-184)
		tmp = Float64(Float64(x * log(Float64(x / y))) - z);
	elseif (x <= -1e-309)
		tmp = Float64(-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 (x <= -3e+98)
		tmp = x * (log(-x) - log(-y));
	elseif (x <= -4.4e-184)
		tmp = (x * log((x / y))) - z;
	elseif (x <= -1e-309)
		tmp = -z;
	else
		tmp = (x * (log(x) - log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -3e+98], N[(x * N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -4.4e-184], N[(N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[x, -1e-309], (-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 < -3.0000000000000001e98

   1. Initial program 67.0%

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

      1. add-cube-cbrt 67.0%

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

      2. associate-*l* 67.0%

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

      3. log-prod 67.0%

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

      4. pow2 67.0%

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

      5. metadata-eval 67.0%

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

      6. log-pow 67.0%

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

      7. metadata-eval 67.0%

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

   3. Applied egg-rr 67.0%

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

      1. distribute-rgt1-in 67.0%

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

      2. metadata-eval 67.0%

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

      3. *-commutative 67.0%

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

   5. Simplified 67.0%

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

   6. Taylor expanded in z around 0 58.8%

      \[\leadsto \color{blue}{3 \cdot \left(x \cdot \log \left({\left(\frac{x}{y}\right)}^{0.3333333333333333}\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative 58.8%

         \[\leadsto 3 \cdot \color{blue}{\left(\log \left({\left(\frac{x}{y}\right)}^{0.3333333333333333}\right) \cdot x\right)} \]

      2. unpow1/3 58.7%

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

      3. associate-*r* 58.8%

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

      4. unpow1/3 58.8%

         \[\leadsto \left(3 \cdot \log \color{blue}{\left({\left(\frac{x}{y}\right)}^{0.3333333333333333}\right)}\right) \cdot x \]

      5. log-pow 58.7%

         \[\leadsto \left(3 \cdot \color{blue}{\left(0.3333333333333333 \cdot \log \left(\frac{x}{y}\right)\right)}\right) \cdot x \]

      6. associate-*r* 58.8%

         \[\leadsto \color{blue}{\left(\left(3 \cdot 0.3333333333333333\right) \cdot \log \left(\frac{x}{y}\right)\right)} \cdot x \]

      7. metadata-eval 58.8%

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

      8. *-lft-identity 58.8%

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

      9. *-commutative 58.8%

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

   8. Simplified 58.8%

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

      1. frac-2neg 58.8%

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

      2. log-div 88.4%

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

   10. Applied egg-rr 88.4%

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

   if -3.0000000000000001e98 < x < -4.39999999999999984e-184

   1. Initial program 91.0%

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

   if -4.39999999999999984e-184 < x < -1.000000000000002e-309

   1. Initial program 60.4%

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

   2. Taylor expanded in x around 0 96.0%

      \[\leadsto \color{blue}{-1 \cdot z} \]
   3. Step-by-step derivation

      1. neg-mul-1 96.0%

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

   4. Simplified 96.0%

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

   if -1.000000000000002e-309 < x

   1. Initial program 72.6%

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

      1. log-div 50.7%

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

   3. Applied egg-rr 99.5%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{+98}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right)\\ \mathbf{elif}\;x \leq -4.4 \cdot 10^{-184}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-309}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \]

Alternative 8: 92.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.08 \cdot 10^{+99}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right)\\ \mathbf{elif}\;x \leq -1.55 \cdot 10^{-183}:\\ \;\;\;\;x \cdot \left(3 \cdot \log \left(\sqrt[3]{\frac{x}{y}}\right)\right) - z\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-309}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.08e+99)
   (* x (- (log (- x)) (log (- y))))
   (if (<= x -1.55e-183)
     (- (* x (* 3.0 (log (cbrt (/ x y))))) z)
     (if (<= x -1e-309) (- z) (- (* x (- (log x) (log y))) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.08e+99) {
		tmp = x * (log(-x) - log(-y));
	} else if (x <= -1.55e-183) {
		tmp = (x * (3.0 * log(cbrt((x / y))))) - z;
	} else if (x <= -1e-309) {
		tmp = -z;
	} else {
		tmp = (x * (log(x) - log(y))) - z;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.08e+99) {
		tmp = x * (Math.log(-x) - Math.log(-y));
	} else if (x <= -1.55e-183) {
		tmp = (x * (3.0 * Math.log(Math.cbrt((x / y))))) - z;
	} else if (x <= -1e-309) {
		tmp = -z;
	} else {
		tmp = (x * (Math.log(x) - Math.log(y))) - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.08e+99)
		tmp = Float64(x * Float64(log(Float64(-x)) - log(Float64(-y))));
	elseif (x <= -1.55e-183)
		tmp = Float64(Float64(x * Float64(3.0 * log(cbrt(Float64(x / y))))) - z);
	elseif (x <= -1e-309)
		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, -1.08e+99], N[(x * N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.55e-183], N[(N[(x * N[(3.0 * N[Log[N[Power[N[(x / y), $MachinePrecision], 1/3], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[x, -1e-309], (-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 < -1.08e99

   1. Initial program 67.0%

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

      1. add-cube-cbrt 67.0%

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

      2. associate-*l* 67.0%

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

      3. log-prod 67.0%

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

      4. pow2 67.0%

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

      5. metadata-eval 67.0%

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

      6. log-pow 67.0%

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

      7. metadata-eval 67.0%

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

   3. Applied egg-rr 67.0%

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

      1. distribute-rgt1-in 67.0%

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

      2. metadata-eval 67.0%

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

      3. *-commutative 67.0%

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

   5. Simplified 67.0%

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

   6. Taylor expanded in z around 0 58.8%

      \[\leadsto \color{blue}{3 \cdot \left(x \cdot \log \left({\left(\frac{x}{y}\right)}^{0.3333333333333333}\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative 58.8%

         \[\leadsto 3 \cdot \color{blue}{\left(\log \left({\left(\frac{x}{y}\right)}^{0.3333333333333333}\right) \cdot x\right)} \]

      2. unpow1/3 58.7%

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

      3. associate-*r* 58.8%

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

      4. unpow1/3 58.8%

         \[\leadsto \left(3 \cdot \log \color{blue}{\left({\left(\frac{x}{y}\right)}^{0.3333333333333333}\right)}\right) \cdot x \]

      5. log-pow 58.7%

         \[\leadsto \left(3 \cdot \color{blue}{\left(0.3333333333333333 \cdot \log \left(\frac{x}{y}\right)\right)}\right) \cdot x \]

      6. associate-*r* 58.8%

         \[\leadsto \color{blue}{\left(\left(3 \cdot 0.3333333333333333\right) \cdot \log \left(\frac{x}{y}\right)\right)} \cdot x \]

      7. metadata-eval 58.8%

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

      8. *-lft-identity 58.8%

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

      9. *-commutative 58.8%

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

   8. Simplified 58.8%

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

      1. frac-2neg 58.8%

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

      2. log-div 88.4%

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

   10. Applied egg-rr 88.4%

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

   if -1.08e99 < x < -1.55e-183

   1. Initial program 91.0%

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

      1. add-cube-cbrt 91.0%

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

      2. associate-*l* 91.0%

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

      3. log-prod 91.0%

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

      4. pow2 91.0%

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

      5. metadata-eval 91.0%

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

      6. log-pow 91.0%

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

      7. metadata-eval 91.0%

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

   3. Applied egg-rr 91.0%

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

      1. distribute-rgt1-in 91.0%

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

      2. metadata-eval 91.0%

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

      3. *-commutative 91.0%

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

   5. Simplified 91.0%

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

   if -1.55e-183 < x < -1.000000000000002e-309

   1. Initial program 60.4%

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

   2. Taylor expanded in x around 0 96.0%

      \[\leadsto \color{blue}{-1 \cdot z} \]
   3. Step-by-step derivation

      1. neg-mul-1 96.0%

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

   4. Simplified 96.0%

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

   if -1.000000000000002e-309 < x

   1. Initial program 72.6%

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

      1. log-div 50.7%

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

   3. Applied egg-rr 99.5%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.08 \cdot 10^{+99}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right)\\ \mathbf{elif}\;x \leq -1.55 \cdot 10^{-183}:\\ \;\;\;\;x \cdot \left(3 \cdot \log \left(\sqrt[3]{\frac{x}{y}}\right)\right) - z\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-309}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \]

Alternative 9: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1e-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 <= -1e-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 <= (-1d-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 <= -1e-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 <= -1e-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 <= -1e-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 <= -1e-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, -1e-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 < -9.999999999999969e-311

   1. Initial program 76.8%

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

      1. frac-2neg 41.6%

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

      2. log-div 55.2%

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

   3. Applied egg-rr 99.5%

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

   if -9.999999999999969e-311 < y

   1. Initial program 72.6%

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

      1. log-div 50.7%

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

   3. Applied egg-rr 99.5%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \]

Alternative 10: 64.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.29 \lor \neg \left(x \leq 1.15 \cdot 10^{+22}\right):\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -0.29) (not (<= x 1.15e+22))) (* x (log (/ x y))) (- z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -0.29) || !(x <= 1.15e+22)) {
		tmp = x * log((x / y));
	} else {
		tmp = -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 ((x <= (-0.29d0)) .or. (.not. (x <= 1.15d+22))) then
        tmp = x * log((x / y))
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -0.29) || !(x <= 1.15e+22)) {
		tmp = x * Math.log((x / y));
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -0.29) or not (x <= 1.15e+22):
		tmp = x * math.log((x / y))
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -0.29) || !(x <= 1.15e+22))
		tmp = Float64(x * log(Float64(x / y)));
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -0.29) || ~((x <= 1.15e+22)))
		tmp = x * log((x / y));
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -0.29], N[Not[LessEqual[x, 1.15e+22]], $MachinePrecision]], N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], (-z)]

Derivation

1. Split input into 2 regimes

2. if x < -0.28999999999999998 or 1.1500000000000001e22 < x

   1. Initial program 76.4%

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

      1. add-cube-cbrt 76.4%

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

      2. associate-*l* 76.4%

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

      3. log-prod 76.4%

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

      4. pow2 76.4%

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

      5. metadata-eval 76.4%

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

      6. log-pow 76.4%

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

      7. metadata-eval 76.4%

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

   3. Applied egg-rr 76.4%

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

      1. distribute-rgt1-in 76.4%

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

      2. metadata-eval 76.4%

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

      3. *-commutative 76.4%

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

   5. Simplified 76.4%

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

   6. Taylor expanded in z around 0 61.1%

      \[\leadsto \color{blue}{3 \cdot \left(x \cdot \log \left({\left(\frac{x}{y}\right)}^{0.3333333333333333}\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative 61.1%

         \[\leadsto 3 \cdot \color{blue}{\left(\log \left({\left(\frac{x}{y}\right)}^{0.3333333333333333}\right) \cdot x\right)} \]

      2. unpow1/3 61.1%

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

      3. associate-*r* 61.3%

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

      4. unpow1/3 61.2%

         \[\leadsto \left(3 \cdot \log \color{blue}{\left({\left(\frac{x}{y}\right)}^{0.3333333333333333}\right)}\right) \cdot x \]

      5. log-pow 61.1%

         \[\leadsto \left(3 \cdot \color{blue}{\left(0.3333333333333333 \cdot \log \left(\frac{x}{y}\right)\right)}\right) \cdot x \]

      6. associate-*r* 61.3%

         \[\leadsto \color{blue}{\left(\left(3 \cdot 0.3333333333333333\right) \cdot \log \left(\frac{x}{y}\right)\right)} \cdot x \]

      7. metadata-eval 61.3%

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

      8. *-lft-identity 61.3%

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

      9. *-commutative 61.3%

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

   8. Simplified 61.3%

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

   if -0.28999999999999998 < x < 1.1500000000000001e22

   1. Initial program 72.9%

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

   2. Taylor expanded in x around 0 78.1%

      \[\leadsto \color{blue}{-1 \cdot z} \]
   3. Step-by-step derivation

      1. neg-mul-1 78.1%

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

   4. Simplified 78.1%

      \[\leadsto \color{blue}{-z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.29 \lor \neg \left(x \leq 1.15 \cdot 10^{+22}\right):\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 11: 65.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0045:\\ \;\;\;\;\left(-x\right) \cdot \log \left(\frac{y}{x}\right)\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{+23}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -0.0045)
   (* (- x) (log (/ y x)))
   (if (<= x 4.5e+23) (- z) (* x (log (/ x y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -0.0045) {
		tmp = -x * log((y / x));
	} else if (x <= 4.5e+23) {
		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 (x <= (-0.0045d0)) then
        tmp = -x * log((y / x))
    else if (x <= 4.5d+23) 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 (x <= -0.0045) {
		tmp = -x * Math.log((y / x));
	} else if (x <= 4.5e+23) {
		tmp = -z;
	} else {
		tmp = x * Math.log((x / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -0.0045:
		tmp = -x * math.log((y / x))
	elif x <= 4.5e+23:
		tmp = -z
	else:
		tmp = x * math.log((x / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -0.0045)
		tmp = Float64(Float64(-x) * log(Float64(y / x)));
	elseif (x <= 4.5e+23)
		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 (x <= -0.0045)
		tmp = -x * log((y / x));
	elseif (x <= 4.5e+23)
		tmp = -z;
	else
		tmp = x * log((x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -0.0045], N[((-x) * N[Log[N[(y / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.5e+23], (-z), N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.00449999999999999966

   1. Initial program 76.5%

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

      1. add-cube-cbrt 76.5%

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

      2. associate-*l* 76.5%

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

      3. log-prod 76.5%

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

      4. pow2 76.5%

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

      5. metadata-eval 76.5%

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

      6. log-pow 76.5%

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

      7. metadata-eval 76.5%

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

   3. Applied egg-rr 76.5%

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

      1. distribute-rgt1-in 76.5%

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

      2. metadata-eval 76.5%

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

      3. *-commutative 76.5%

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

   5. Simplified 76.5%

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

   6. Taylor expanded in z around 0 62.7%

      \[\leadsto \color{blue}{3 \cdot \left(x \cdot \log \left({\left(\frac{x}{y}\right)}^{0.3333333333333333}\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative 62.7%

         \[\leadsto 3 \cdot \color{blue}{\left(\log \left({\left(\frac{x}{y}\right)}^{0.3333333333333333}\right) \cdot x\right)} \]

      2. unpow1/3 62.7%

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

      3. associate-*r* 62.9%

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

      4. unpow1/3 62.9%

         \[\leadsto \left(3 \cdot \log \color{blue}{\left({\left(\frac{x}{y}\right)}^{0.3333333333333333}\right)}\right) \cdot x \]

      5. log-pow 62.8%

         \[\leadsto \left(3 \cdot \color{blue}{\left(0.3333333333333333 \cdot \log \left(\frac{x}{y}\right)\right)}\right) \cdot x \]

      6. associate-*r* 62.9%

         \[\leadsto \color{blue}{\left(\left(3 \cdot 0.3333333333333333\right) \cdot \log \left(\frac{x}{y}\right)\right)} \cdot x \]

      7. metadata-eval 62.9%

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

      8. *-lft-identity 62.9%

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

      9. *-commutative 62.9%

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

   8. Simplified 62.9%

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

      1. clear-num 62.9%

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

      2. neg-log 64.7%

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

   10. Applied egg-rr 64.7%

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

   if -0.00449999999999999966 < x < 4.49999999999999979e23

   1. Initial program 72.9%

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

   2. Taylor expanded in x around 0 78.1%

      \[\leadsto \color{blue}{-1 \cdot z} \]
   3. Step-by-step derivation

      1. neg-mul-1 78.1%

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

   4. Simplified 78.1%

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

   if 4.49999999999999979e23 < x

   1. Initial program 76.3%

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

      1. add-cube-cbrt 76.3%

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

      2. associate-*l* 76.3%

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

      3. log-prod 76.3%

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

      4. pow2 76.3%

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

      5. metadata-eval 76.3%

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

      6. log-pow 76.3%

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

      7. metadata-eval 76.3%

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

   3. Applied egg-rr 76.3%

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

      1. distribute-rgt1-in 76.3%

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

      2. metadata-eval 76.3%

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

      3. *-commutative 76.3%

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

   5. Simplified 76.3%

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

   6. Taylor expanded in z around 0 59.1%

      \[\leadsto \color{blue}{3 \cdot \left(x \cdot \log \left({\left(\frac{x}{y}\right)}^{0.3333333333333333}\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative 59.1%

         \[\leadsto 3 \cdot \color{blue}{\left(\log \left({\left(\frac{x}{y}\right)}^{0.3333333333333333}\right) \cdot x\right)} \]

      2. unpow1/3 59.3%

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

      3. associate-*r* 59.4%

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

      4. unpow1/3 59.2%

         \[\leadsto \left(3 \cdot \log \color{blue}{\left({\left(\frac{x}{y}\right)}^{0.3333333333333333}\right)}\right) \cdot x \]

      5. log-pow 59.2%

         \[\leadsto \left(3 \cdot \color{blue}{\left(0.3333333333333333 \cdot \log \left(\frac{x}{y}\right)\right)}\right) \cdot x \]

      6. associate-*r* 59.4%

         \[\leadsto \color{blue}{\left(\left(3 \cdot 0.3333333333333333\right) \cdot \log \left(\frac{x}{y}\right)\right)} \cdot x \]

      7. metadata-eval 59.4%

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

      8. *-lft-identity 59.4%

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

      9. *-commutative 59.4%

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

   8. Simplified 59.4%

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

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0045:\\ \;\;\;\;\left(-x\right) \cdot \log \left(\frac{y}{x}\right)\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{+23}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \end{array} \]

Alternative 12: 50.2% 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 74.6%

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

2. Taylor expanded in x around 0 47.9%

   \[\leadsto \color{blue}{-1 \cdot z} \]
3. Step-by-step derivation

   1. neg-mul-1 47.9%

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

4. Simplified 47.9%

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

5. Final simplification 47.9%

   \[\leadsto -z \]

Developer target: 88.5% 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 2023290 
(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))