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

Percentage Accurate: 77.6% → 99.7%

Time: 10.9s

Alternatives: 10

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: 77.6% 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

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

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(Float64(x * log(Float64(cbrt(x) / cbrt(y)))) * 3.0) - z)
end

Wolfram

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

Derivation

1. Initial program 77.3%

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

   1. log1p-expm1-u 45.8%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(x \cdot \log \left(\frac{x}{y}\right)\right)\right)} - z \]

   2. expm1-udef 33.7%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{e^{x \cdot \log \left(\frac{x}{y}\right)} - 1}\right) - z \]

   3. *-commutative 33.7%

      \[\leadsto \mathsf{log1p}\left(e^{\color{blue}{\log \left(\frac{x}{y}\right) \cdot x}} - 1\right) - z \]

   4. exp-to-pow 33.7%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{{\left(\frac{x}{y}\right)}^{x}} - 1\right) - z \]

4. Applied egg-rr 33.7%

   \[\leadsto \color{blue}{\mathsf{log1p}\left({\left(\frac{x}{y}\right)}^{x} - 1\right)} - z \]
5. Step-by-step derivation

   1. pow-to-exp 33.7%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{e^{\log \left(\frac{x}{y}\right) \cdot x}} - 1\right) - z \]

   2. *-commutative 33.7%

      \[\leadsto \mathsf{log1p}\left(e^{\color{blue}{x \cdot \log \left(\frac{x}{y}\right)}} - 1\right) - z \]

   3. expm1-udef 45.8%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(x \cdot \log \left(\frac{x}{y}\right)\right)}\right) - z \]

   4. log1p-expm1-u 77.3%

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

   5. add-cube-cbrt 77.3%

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

   6. pow3 77.3%

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

   7. exp-to-pow 77.2%

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

   8. add-log-exp 77.2%

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

   9. associate-*r* 77.2%

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

6. Applied egg-rr 77.2%

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

   1. cbrt-div 99.6%

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

   2. div-inv 99.6%

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

8. Applied egg-rr 99.6%

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

   1. associate-*r/ 99.6%

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

   2. *-rgt-identity 99.6%

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

10. Simplified 99.6%

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

11. Final simplification 99.6%

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

Alternative 2: 86.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;-z\\ \mathbf{elif}\;t_0 \leq 10^{+288}:\\ \;\;\;\;t_0 - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))))
   (if (<= t_0 (- INFINITY))
     (- z)
     (if (<= t_0 1e+288) (- 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+288) {
		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+288) {
		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+288:
		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+288)
		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+288)
		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+288], 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 9.1%

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

      1. remove-double-neg 9.1%

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

      2. sub-neg 9.1%

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

      3. distribute-neg-in 9.1%

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

      4. distribute-rgt-neg-in 9.1%

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

      5. remove-double-neg 9.1%

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

      6. fma-udef 9.1%

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

      7. log-div 50.0%

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

      8. sub-neg 50.0%

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

      9. distribute-neg-in 50.0%

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

      10. remove-double-neg 50.0%

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

      11. +-commutative 50.0%

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

      12. sub-neg 50.0%

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

      13. log-div 11.6%

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

   3. Simplified 11.6%

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

   5. Taylor expanded in x around 0 42.9%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 1e288

   1. Initial program 99.6%

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

   if 1e288 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 5.3%

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

   3. Taylor expanded in z around 0 5.3%

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

      1. log-div 49.5%

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

   5. Applied egg-rr 49.5%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log \left(\frac{x}{y}\right) \leq -\infty:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \cdot \log \left(\frac{x}{y}\right) \leq 10^{+288}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;t_0 \leq -\infty \lor \neg \left(t_0 \leq 10^{+288}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;t_0 - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 1e+288))) (- 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 <= 1e+288)) {
		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 <= 1e+288)) {
		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 <= 1e+288):
		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 <= 1e+288))
		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 <= 1e+288)))
		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, 1e+288]], $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 1e288 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 7.5%

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

      1. remove-double-neg 7.5%

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

      2. sub-neg 7.5%

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

      3. distribute-neg-in 7.5%

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

      4. distribute-rgt-neg-in 7.5%

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

      5. remove-double-neg 7.5%

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

      6. fma-udef 7.5%

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

      7. log-div 54.7%

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

      8. sub-neg 54.7%

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

      9. distribute-neg-in 54.7%

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

      10. remove-double-neg 54.7%

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

      11. +-commutative 54.7%

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

      12. sub-neg 54.7%

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

      13. log-div 9.6%

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

   3. Simplified 9.6%

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

   5. Taylor expanded in x around 0 43.2%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 1e288

   1. Initial program 99.6%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log \left(\frac{x}{y}\right) \leq -\infty \lor \neg \left(x \cdot \log \left(\frac{x}{y}\right) \leq 10^{+288}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 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 77.3%

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

   1. add-cube-cbrt 77.3%

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

   2. log-prod 77.2%

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

   3. pow2 77.2%

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

4. Applied egg-rr 77.2%

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

   1. log-pow 77.2%

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

   2. distribute-lft1-in 77.2%

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

   3. metadata-eval 77.2%

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

6. Simplified 77.2%

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

   1. cbrt-div 99.6%

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

   2. div-inv 99.6%

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

8. Applied egg-rr 99.6%

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

   1. associate-*r/ 99.6%

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

   2. *-rgt-identity 99.6%

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

10. Simplified 99.6%

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

11. Final simplification 99.6%

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

Alternative 5: 99.7% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 77.3%

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

   1. log1p-expm1-u 45.8%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(x \cdot \log \left(\frac{x}{y}\right)\right)\right)} - z \]

   2. expm1-udef 33.7%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{e^{x \cdot \log \left(\frac{x}{y}\right)} - 1}\right) - z \]

   3. *-commutative 33.7%

      \[\leadsto \mathsf{log1p}\left(e^{\color{blue}{\log \left(\frac{x}{y}\right) \cdot x}} - 1\right) - z \]

   4. exp-to-pow 33.7%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{{\left(\frac{x}{y}\right)}^{x}} - 1\right) - z \]

4. Applied egg-rr 33.7%

   \[\leadsto \color{blue}{\mathsf{log1p}\left({\left(\frac{x}{y}\right)}^{x} - 1\right)} - z \]
5. Step-by-step derivation

   1. pow-to-exp 33.7%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{e^{\log \left(\frac{x}{y}\right) \cdot x}} - 1\right) - z \]

   2. *-commutative 33.7%

      \[\leadsto \mathsf{log1p}\left(e^{\color{blue}{x \cdot \log \left(\frac{x}{y}\right)}} - 1\right) - z \]

   3. expm1-udef 45.8%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(x \cdot \log \left(\frac{x}{y}\right)\right)}\right) - z \]

   4. log1p-expm1-u 77.3%

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

   5. add-cube-cbrt 77.3%

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

   6. pow3 77.3%

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

   7. exp-to-pow 77.2%

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

   8. add-log-exp 77.2%

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

   9. *-commutative 77.2%

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

   10. associate-*r* 77.2%

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

6. Applied egg-rr 77.2%

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

   1. cbrt-div 99.6%

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

   2. div-inv 99.6%

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

8. Applied egg-rr 99.6%

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

   1. associate-*r/ 99.6%

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

   2. *-rgt-identity 99.6%

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

10. Simplified 99.6%

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

11. Final simplification 99.6%

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

Alternative 6: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(x, \log y - \log x, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -4e-310)
   (- (* x (- (log (- x)) (log (- y)))) z)
   (- (fma x (- (log y) (log x)) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -4e-310) {
		tmp = (x * (log(-x) - log(-y))) - z;
	} else {
		tmp = -fma(x, (log(y) - log(x)), z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -4e-310)
		tmp = Float64(Float64(x * Float64(log(Float64(-x)) - log(Float64(-y)))) - z);
	else
		tmp = Float64(-fma(x, Float64(log(y) - log(x)), z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -4e-310], N[(N[(x * N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], (-N[(x * N[(N[Log[y], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if y < -3.999999999999988e-310

   1. Initial program 81.8%

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

      1. frac-2neg 42.1%

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

      2. log-div 51.4%

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

   4. Applied egg-rr 99.5%

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

   if -3.999999999999988e-310 < y

   1. Initial program 72.0%

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

      1. remove-double-neg 72.0%

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

      2. sub-neg 72.0%

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

      3. distribute-neg-in 72.0%

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

      4. distribute-rgt-neg-in 72.0%

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

      5. remove-double-neg 72.0%

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

      6. fma-udef 72.0%

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

      7. log-div 99.6%

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

      8. sub-neg 99.6%

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

      9. distribute-neg-in 99.6%

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

      10. remove-double-neg 99.6%

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

      11. +-commutative 99.6%

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

      12. sub-neg 99.6%

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

      13. log-div 71.0%

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

   3. Simplified 71.0%

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

      1. log-div 99.6%

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

   6. Applied egg-rr 99.6%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(x, \log y - \log x, z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 93.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{+188}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right)\\ \mathbf{elif}\;x \leq -7.5 \cdot 10^{-187}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-308}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -5.8e+188)
   (* x (- (log (- x)) (log (- y))))
   (if (<= x -7.5e-187)
     (- (* x (log (/ x y))) 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 <= -5.8e+188) {
		tmp = x * (log(-x) - log(-y));
	} else if (x <= -7.5e-187) {
		tmp = (x * log((x / y))) - z;
	} else if (x <= -2e-308) {
		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 <= (-5.8d+188)) then
        tmp = x * (log(-x) - log(-y))
    else if (x <= (-7.5d-187)) then
        tmp = (x * log((x / y))) - z
    else if (x <= (-2d-308)) 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 <= -5.8e+188) {
		tmp = x * (Math.log(-x) - Math.log(-y));
	} else if (x <= -7.5e-187) {
		tmp = (x * Math.log((x / y))) - z;
	} else if (x <= -2e-308) {
		tmp = -z;
	} else {
		tmp = (x * (Math.log(x) - Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -5.8e+188:
		tmp = x * (math.log(-x) - math.log(-y))
	elif x <= -7.5e-187:
		tmp = (x * math.log((x / y))) - z
	elif x <= -2e-308:
		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 <= -5.8e+188)
		tmp = Float64(x * Float64(log(Float64(-x)) - log(Float64(-y))));
	elseif (x <= -7.5e-187)
		tmp = Float64(Float64(x * log(Float64(x / y))) - z);
	elseif (x <= -2e-308)
		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 <= -5.8e+188)
		tmp = x * (log(-x) - log(-y));
	elseif (x <= -7.5e-187)
		tmp = (x * log((x / y))) - z;
	elseif (x <= -2e-308)
		tmp = -z;
	else
		tmp = (x * (log(x) - log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -5.8e+188], N[(x * N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -7.5e-187], N[(N[(x * N[Log[N[(x / y), $MachinePrecision]], $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 < -5.7999999999999999e188

   1. Initial program 62.6%

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

   3. Taylor expanded in z around 0 54.9%

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

      1. frac-2neg 54.9%

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

      2. log-div 89.3%

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

   5. Applied egg-rr 89.3%

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

   if -5.7999999999999999e188 < x < -7.5000000000000004e-187

   1. Initial program 89.9%

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

   if -7.5000000000000004e-187 < x < -1.9999999999999998e-308

   1. Initial program 72.7%

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

      1. remove-double-neg 72.7%

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

      2. sub-neg 72.7%

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

      3. distribute-neg-in 72.7%

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

      4. distribute-rgt-neg-in 72.7%

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

      5. remove-double-neg 72.7%

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

      6. fma-udef 72.7%

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

      7. log-div 0.0%

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

      8. sub-neg 0.0%

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

      9. distribute-neg-in 0.0%

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

      10. remove-double-neg 0.0%

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

      11. +-commutative 0.0%

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

      12. sub-neg 0.0%

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

      13. log-div 72.7%

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

   3. Simplified 72.7%

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

   5. Taylor expanded in x around 0 95.5%

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

   if -1.9999999999999998e-308 < x

   1. Initial program 72.0%

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

      1. log-div 53.8%

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

   4. 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 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{+188}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right)\\ \mathbf{elif}\;x \leq -7.5 \cdot 10^{-187}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-308}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -4e-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 <= -4e-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 <= (-4d-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 <= -4e-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 <= -4e-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 <= -4e-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 <= -4e-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, -4e-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 < -3.999999999999988e-310

   1. Initial program 81.8%

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

      1. frac-2neg 42.1%

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

      2. log-div 51.4%

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

   4. Applied egg-rr 99.5%

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

   if -3.999999999999988e-310 < y

   1. Initial program 72.0%

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

      1. log-div 53.8%

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

   4. 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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 67.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.8 \cdot 10^{-20} \lor \neg \left(z \leq 5.2 \cdot 10^{+41}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -3.8e-20) (not (<= z 5.2e+41))) (- z) (* x (log (/ x y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -3.8e-20) || !(z <= 5.2e+41)) {
		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 <= (-3.8d-20)) .or. (.not. (z <= 5.2d+41))) 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 <= -3.8e-20) || !(z <= 5.2e+41)) {
		tmp = -z;
	} else {
		tmp = x * Math.log((x / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -3.8e-20) or not (z <= 5.2e+41):
		tmp = -z
	else:
		tmp = x * math.log((x / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -3.8e-20) || !(z <= 5.2e+41))
		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 <= -3.8e-20) || ~((z <= 5.2e+41)))
		tmp = -z;
	else
		tmp = x * log((x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -3.8e-20], N[Not[LessEqual[z, 5.2e+41]], $MachinePrecision]], (-z), N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.7999999999999998e-20 or 5.2000000000000001e41 < z

   1. Initial program 77.3%

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

      1. remove-double-neg 77.3%

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

      2. sub-neg 77.3%

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

      3. distribute-neg-in 77.3%

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

      4. distribute-rgt-neg-in 77.3%

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

      5. remove-double-neg 77.3%

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

      6. fma-udef 77.3%

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

      7. log-div 47.5%

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

      8. sub-neg 47.5%

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

      9. distribute-neg-in 47.5%

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

      10. remove-double-neg 47.5%

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

      11. +-commutative 47.5%

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

      12. sub-neg 47.5%

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

      13. log-div 76.5%

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

   3. Simplified 76.5%

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

   5. Taylor expanded in x around 0 75.4%

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

   if -3.7999999999999998e-20 < z < 5.2000000000000001e41

   1. Initial program 77.3%

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

   3. Taylor expanded in z around 0 65.3%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.8 \cdot 10^{-20} \lor \neg \left(z \leq 5.2 \cdot 10^{+41}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 50.7% 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 77.3%

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

   1. remove-double-neg 77.3%

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

   2. sub-neg 77.3%

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

   3. distribute-neg-in 77.3%

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

   4. distribute-rgt-neg-in 77.3%

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

   5. remove-double-neg 77.3%

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

   6. fma-udef 77.3%

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

   7. log-div 45.9%

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

   8. sub-neg 45.9%

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

   9. distribute-neg-in 45.9%

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

   10. remove-double-neg 45.9%

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

   11. +-commutative 45.9%

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

   12. sub-neg 45.9%

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

   13. log-div 76.8%

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

3. Simplified 76.8%

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

5. Taylor expanded in x around 0 48.4%

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

6. Final simplification 48.4%

   \[\leadsto -z \]
7. Add Preprocessing

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 2024013 
(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))