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

Percentage Accurate: 77.3% → 99.7%

Time: 9.5s

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: 77.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.3%

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

   1. add-cube-cbrt 79.3%

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

   2. log-prod 79.3%

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

   3. pow2 79.3%

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

3. Applied egg-rr 79.3%

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

   1. log-pow 79.3%

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

   2. distribute-lft1-in 79.3%

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

   3. metadata-eval 79.3%

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

5. Simplified 79.3%

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

   1. cbrt-div 99.7%

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

   2. div-inv 99.7%

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

7. Applied egg-rr 99.7%

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

   1. associate-*r/ 99.7%

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

   2. *-rgt-identity 99.7%

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

9. Simplified 99.7%

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

10. Final simplification 99.7%

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

Alternative 2: 87.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;t_0 \leq -\infty \lor \neg \left(t_0 \leq 5 \cdot 10^{+302}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;t_0 - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 5e+302))) (- z) (- t_0 z))))

C

double code(double x, double y, double z) {
	double t_0 = x * log((x / y));
	double tmp;
	if ((t_0 <= -((double) INFINITY)) || !(t_0 <= 5e+302)) {
		tmp = -z;
	} else {
		tmp = t_0 - z;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = x * Math.log((x / y));
	double tmp;
	if ((t_0 <= -Double.POSITIVE_INFINITY) || !(t_0 <= 5e+302)) {
		tmp = -z;
	} else {
		tmp = t_0 - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.log((x / y))
	tmp = 0
	if (t_0 <= -math.inf) or not (t_0 <= 5e+302):
		tmp = -z
	else:
		tmp = t_0 - z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * log(Float64(x / y)))
	tmp = 0.0
	if ((t_0 <= Float64(-Inf)) || !(t_0 <= 5e+302))
		tmp = Float64(-z);
	else
		tmp = Float64(t_0 - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * log((x / y));
	tmp = 0.0;
	if ((t_0 <= -Inf) || ~((t_0 <= 5e+302)))
		tmp = -z;
	else
		tmp = t_0 - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, (-Infinity)], N[Not[LessEqual[t$95$0, 5e+302]], $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 5e302 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 4.7%

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

      1. remove-double-neg 4.7%

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

      2. sub0-neg 4.7%

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

      3. associate--r- 4.7%

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

      4. neg-sub0 4.7%

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

      5. distribute-rgt-neg-in 4.7%

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

      6. neg-sub0 4.7%

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

      7. log-div 42.4%

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

      8. associate-+l- 42.4%

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

      9. neg-sub0 42.4%

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

      10. +-commutative 42.4%

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

      11. sub-neg 42.4%

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

      12. log-div 7.6%

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

      13. fma-udef 7.6%

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

   3. Simplified 7.6%

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

   4. Taylor expanded in x around 0 44.2%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 5e302

   1. Initial program 99.3%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log \left(\frac{x}{y}\right) \leq -\infty \lor \neg \left(x \cdot \log \left(\frac{x}{y}\right) \leq 5 \cdot 10^{+302}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \end{array} \]

Alternative 3: 86.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right)\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+302}:\\ \;\;\;\;t_0 - z\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))))
   (if (<= t_0 (- INFINITY))
     (* x (- (log (- x)) (log (- y))))
     (if (<= t_0 5e+302) (- 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 = x * (log(-x) - log(-y));
	} else if (t_0 <= 5e+302) {
		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 = x * (Math.log(-x) - Math.log(-y));
	} else if (t_0 <= 5e+302) {
		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 = x * (math.log(-x) - math.log(-y))
	elif t_0 <= 5e+302:
		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(x * Float64(log(Float64(-x)) - log(Float64(-y))));
	elseif (t_0 <= 5e+302)
		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 = x * (log(-x) - log(-y));
	elseif (t_0 <= 5e+302)
		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)], N[(x * N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+302], N[(t$95$0 - z), $MachinePrecision], (-z)]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 5.3%

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

   2. Taylor expanded in z around 0 5.3%

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

      1. frac-2neg 5.3%

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

      2. log-div 57.8%

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

   4. Applied egg-rr 57.8%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 5e302

   1. Initial program 99.3%

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

   if 5e302 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 4.2%

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

      1. remove-double-neg 4.2%

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

      2. sub0-neg 4.2%

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

      3. associate--r- 4.2%

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

      4. neg-sub0 4.2%

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

      5. distribute-rgt-neg-in 4.2%

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

      6. neg-sub0 4.2%

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

      7. log-div 47.7%

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

      8. associate-+l- 47.7%

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

      9. neg-sub0 47.7%

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

      10. +-commutative 47.7%

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

      11. sub-neg 47.7%

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

      12. log-div 5.7%

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

      13. fma-udef 5.7%

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

   3. Simplified 5.7%

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

   4. Taylor expanded in x around 0 53.1%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log \left(\frac{x}{y}\right) \leq -\infty:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right)\\ \mathbf{elif}\;x \cdot \log \left(\frac{x}{y}\right) \leq 5 \cdot 10^{+302}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 4: 99.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-310}:\\ \;\;\;\;-\mathsf{fma}\left(x, \log \left(-y\right) - \log \left(-x\right), z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(2 \cdot \log \left(\frac{\sqrt{x}}{\sqrt{y}}\right)\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2e-310)
   (- (fma x (- (log (- y)) (log (- x))) z))
   (- (* x (* 2.0 (log (/ (sqrt x) (sqrt y))))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2e-310) {
		tmp = -fma(x, (log(-y) - log(-x)), z);
	} else {
		tmp = (x * (2.0 * log((sqrt(x) / sqrt(y))))) - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2e-310)
		tmp = Float64(-fma(x, Float64(log(Float64(-y)) - log(Float64(-x))), z));
	else
		tmp = Float64(Float64(x * Float64(2.0 * log(Float64(sqrt(x) / sqrt(y))))) - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2e-310], (-N[(x * N[(N[Log[(-y)], $MachinePrecision] - N[Log[(-x)], $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision]), N[(N[(x * N[(2.0 * N[Log[N[(N[Sqrt[x], $MachinePrecision] / N[Sqrt[y], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.999999999999994e-310

   1. Initial program 78.4%

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

      1. remove-double-neg 78.4%

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

      2. sub0-neg 78.4%

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

      3. associate--r- 78.4%

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

      4. neg-sub0 78.4%

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

      5. distribute-rgt-neg-in 78.4%

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

      6. neg-sub0 78.4%

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

      7. log-div 0.0%

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

      8. associate-+l- 0.0%

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

      9. neg-sub0 0.0%

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

      10. +-commutative 0.0%

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

      11. sub-neg 0.0%

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

      12. log-div 77.5%

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

      13. fma-udef 77.5%

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

   3. Simplified 77.5%

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

      1. frac-2neg 77.5%

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

      2. log-div 99.5%

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

   5. Applied egg-rr 99.5%

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

   if -1.999999999999994e-310 < y

   1. Initial program 80.5%

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

      1. add-sqr-sqrt 80.5%

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

      2. log-prod 80.5%

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

   3. Applied egg-rr 80.5%

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

      1. count-2 80.5%

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

   5. Simplified 80.5%

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

      1. sqrt-div 99.8%

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

      2. div-inv 99.8%

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

   7. Applied egg-rr 99.8%

      \[\leadsto x \cdot \left(2 \cdot \log \color{blue}{\left(\sqrt{x} \cdot \frac{1}{\sqrt{y}}\right)}\right) - z \]
   8. Step-by-step derivation

      1. associate-*r/ 99.8%

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

      2. *-rgt-identity 99.8%

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

   9. Simplified 99.8%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-310}:\\ \;\;\;\;-\mathsf{fma}\left(x, \log \left(-y\right) - \log \left(-x\right), z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(2 \cdot \log \left(\frac{\sqrt{x}}{\sqrt{y}}\right)\right) - z\\ \end{array} \]

Alternative 5: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-310}:\\ \;\;\;\;-\mathsf{fma}\left(x, \log \left(-y\right) - \log \left(-x\right), z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \log x - x \cdot \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2e-310)
   (- (fma x (- (log (- y)) (log (- x))) z))
   (- (- (* x (log x)) (* x (log y))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2e-310) {
		tmp = -fma(x, (log(-y) - log(-x)), z);
	} else {
		tmp = ((x * log(x)) - (x * log(y))) - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2e-310)
		tmp = Float64(-fma(x, Float64(log(Float64(-y)) - log(Float64(-x))), z));
	else
		tmp = Float64(Float64(Float64(x * log(x)) - Float64(x * log(y))) - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2e-310], (-N[(x * N[(N[Log[(-y)], $MachinePrecision] - N[Log[(-x)], $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision]), N[(N[(N[(x * N[Log[x], $MachinePrecision]), $MachinePrecision] - N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.999999999999994e-310

   1. Initial program 78.4%

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

      1. remove-double-neg 78.4%

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

      2. sub0-neg 78.4%

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

      3. associate--r- 78.4%

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

      4. neg-sub0 78.4%

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

      5. distribute-rgt-neg-in 78.4%

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

      6. neg-sub0 78.4%

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

      7. log-div 0.0%

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

      8. associate-+l- 0.0%

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

      9. neg-sub0 0.0%

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

      10. +-commutative 0.0%

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

      11. sub-neg 0.0%

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

      12. log-div 77.5%

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

      13. fma-udef 77.5%

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

   3. Simplified 77.5%

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

      1. frac-2neg 77.5%

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

      2. log-div 99.5%

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

   5. Applied egg-rr 99.5%

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

   if -1.999999999999994e-310 < y

   1. Initial program 80.5%

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

      1. add-cbrt-cube 54.0%

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

      2. pow3 54.0%

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

   3. Applied egg-rr 54.0%

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

      1. rem-cbrt-cube 80.5%

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

      2. add-cube-cbrt 80.5%

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

      3. pow3 80.5%

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

      4. exp-to-pow 80.4%

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

      5. add-log-exp 80.4%

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

      6. *-commutative 80.4%

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

      7. associate-*r* 80.4%

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

   5. Applied egg-rr 80.4%

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

      1. cbrt-div 99.7%

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

      2. div-inv 99.7%

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

   7. Applied egg-rr 99.7%

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

      1. associate-*r/ 99.7%

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

      2. *-rgt-identity 99.7%

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

   9. Simplified 99.7%

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

       1. cbrt-div 80.4%

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

       2. associate-*r* 80.4%

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

       3. add-log-exp 80.4%

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

       4. *-commutative 80.4%

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

       5. exp-to-pow 80.5%

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

       6. pow3 80.5%

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

       7. add-cube-cbrt 80.5%

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

       8. diff-log 99.3%

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

       9. sub-neg 99.3%

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

       10. distribute-rgt-in 99.4%

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

   11. Applied egg-rr 99.4%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-310}:\\ \;\;\;\;-\mathsf{fma}\left(x, \log \left(-y\right) - \log \left(-x\right), z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \log x - x \cdot \log y\right) - z\\ \end{array} \]

Alternative 6: 93.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.6 \cdot 10^{+160}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right)\\ \mathbf{elif}\;x \leq -1.15 \cdot 10^{-129}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-308}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -9.6e+160)
   (* x (- (log (- x)) (log (- y))))
   (if (<= x -1.15e-129)
     (- (* x (log (/ x y))) z)
     (if (<= x -5e-308) (- z) (- (* x (- (log x) (log y))) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -9.6e+160) {
		tmp = x * (log(-x) - log(-y));
	} else if (x <= -1.15e-129) {
		tmp = (x * log((x / y))) - z;
	} else if (x <= -5e-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 <= (-9.6d+160)) then
        tmp = x * (log(-x) - log(-y))
    else if (x <= (-1.15d-129)) then
        tmp = (x * log((x / y))) - z
    else if (x <= (-5d-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 <= -9.6e+160) {
		tmp = x * (Math.log(-x) - Math.log(-y));
	} else if (x <= -1.15e-129) {
		tmp = (x * Math.log((x / y))) - z;
	} else if (x <= -5e-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 <= -9.6e+160:
		tmp = x * (math.log(-x) - math.log(-y))
	elif x <= -1.15e-129:
		tmp = (x * math.log((x / y))) - z
	elif x <= -5e-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 <= -9.6e+160)
		tmp = Float64(x * Float64(log(Float64(-x)) - log(Float64(-y))));
	elseif (x <= -1.15e-129)
		tmp = Float64(Float64(x * log(Float64(x / y))) - z);
	elseif (x <= -5e-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 <= -9.6e+160)
		tmp = x * (log(-x) - log(-y));
	elseif (x <= -1.15e-129)
		tmp = (x * log((x / y))) - z;
	elseif (x <= -5e-308)
		tmp = -z;
	else
		tmp = (x * (log(x) - log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -9.6e+160], N[(x * N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.15e-129], N[(N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[x, -5e-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 < -9.6000000000000006e160

   1. Initial program 59.2%

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

   2. Taylor expanded in z around 0 52.2%

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

      1. frac-2neg 52.2%

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

      2. log-div 91.4%

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

   4. Applied egg-rr 91.4%

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

   if -9.6000000000000006e160 < x < -1.15e-129

   1. Initial program 96.9%

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

   if -1.15e-129 < x < -4.99999999999999955e-308

   1. Initial program 62.4%

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

      1. remove-double-neg 62.4%

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

      2. sub0-neg 62.4%

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

      3. associate--r- 62.4%

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

      4. neg-sub0 62.4%

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

      5. distribute-rgt-neg-in 62.4%

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

      6. neg-sub0 62.4%

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

      7. log-div 0.0%

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

      8. associate-+l- 0.0%

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

      9. neg-sub0 0.0%

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

      10. +-commutative 0.0%

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

      11. sub-neg 0.0%

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

      12. log-div 61.2%

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

      13. fma-udef 61.2%

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

   3. Simplified 61.2%

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

   4. Taylor expanded in x around 0 85.7%

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

   if -4.99999999999999955e-308 < x

   1. Initial program 80.5%

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

      1. log-div 99.3%

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

   3. Applied egg-rr 99.3%

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

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.6 \cdot 10^{+160}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right)\\ \mathbf{elif}\;x \leq -1.15 \cdot 10^{-129}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-308}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \]

Alternative 7: 93.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.6 \cdot 10^{+160}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right)\\ \mathbf{elif}\;x \leq -1.42 \cdot 10^{-130}:\\ \;\;\;\;x \cdot \left(2 \cdot \log \left(\sqrt{\frac{x}{y}}\right)\right) - z\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -6.6e+160)
   (* x (- (log (- x)) (log (- y))))
   (if (<= x -1.42e-130)
     (- (* x (* 2.0 (log (sqrt (/ x y))))) z)
     (if (<= x -5e-310) (- z) (- (* x (- (log x) (log y))) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -6.6e+160) {
		tmp = x * (log(-x) - log(-y));
	} else if (x <= -1.42e-130) {
		tmp = (x * (2.0 * log(sqrt((x / y))))) - z;
	} else if (x <= -5e-310) {
		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 <= (-6.6d+160)) then
        tmp = x * (log(-x) - log(-y))
    else if (x <= (-1.42d-130)) then
        tmp = (x * (2.0d0 * log(sqrt((x / y))))) - z
    else if (x <= (-5d-310)) 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 <= -6.6e+160) {
		tmp = x * (Math.log(-x) - Math.log(-y));
	} else if (x <= -1.42e-130) {
		tmp = (x * (2.0 * Math.log(Math.sqrt((x / y))))) - z;
	} else if (x <= -5e-310) {
		tmp = -z;
	} else {
		tmp = (x * (Math.log(x) - Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -6.6e+160:
		tmp = x * (math.log(-x) - math.log(-y))
	elif x <= -1.42e-130:
		tmp = (x * (2.0 * math.log(math.sqrt((x / y))))) - z
	elif x <= -5e-310:
		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 <= -6.6e+160)
		tmp = Float64(x * Float64(log(Float64(-x)) - log(Float64(-y))));
	elseif (x <= -1.42e-130)
		tmp = Float64(Float64(x * Float64(2.0 * log(sqrt(Float64(x / y))))) - z);
	elseif (x <= -5e-310)
		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 <= -6.6e+160)
		tmp = x * (log(-x) - log(-y));
	elseif (x <= -1.42e-130)
		tmp = (x * (2.0 * log(sqrt((x / y))))) - z;
	elseif (x <= -5e-310)
		tmp = -z;
	else
		tmp = (x * (log(x) - log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -6.6e+160], N[(x * N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.42e-130], N[(N[(x * N[(2.0 * N[Log[N[Sqrt[N[(x / y), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[x, -5e-310], (-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 < -6.5999999999999994e160

   1. Initial program 59.2%

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

   2. Taylor expanded in z around 0 52.2%

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

      1. frac-2neg 52.2%

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

      2. log-div 91.4%

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

   4. Applied egg-rr 91.4%

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

   if -6.5999999999999994e160 < x < -1.4199999999999999e-130

   1. Initial program 96.9%

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

      1. add-sqr-sqrt 96.9%

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

      2. log-prod 96.9%

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

   3. Applied egg-rr 96.9%

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

      1. count-2 96.9%

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

   5. Simplified 96.9%

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

   if -1.4199999999999999e-130 < x < -4.999999999999985e-310

   1. Initial program 62.4%

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

      1. remove-double-neg 62.4%

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

      2. sub0-neg 62.4%

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

      3. associate--r- 62.4%

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

      4. neg-sub0 62.4%

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

      5. distribute-rgt-neg-in 62.4%

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

      6. neg-sub0 62.4%

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

      7. log-div 0.0%

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

      8. associate-+l- 0.0%

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

      9. neg-sub0 0.0%

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

      10. +-commutative 0.0%

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

      11. sub-neg 0.0%

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

      12. log-div 61.2%

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

      13. fma-udef 61.2%

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

   3. Simplified 61.2%

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

   4. Taylor expanded in x around 0 85.7%

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

   if -4.999999999999985e-310 < x

   1. Initial program 80.5%

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

      1. log-div 99.3%

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

   3. Applied egg-rr 99.3%

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

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.6 \cdot 10^{+160}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right)\\ \mathbf{elif}\;x \leq -1.42 \cdot 10^{-130}:\\ \;\;\;\;x \cdot \left(2 \cdot \log \left(\sqrt{\frac{x}{y}}\right)\right) - z\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \]

Alternative 8: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2e-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 <= -2e-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 <= (-2d-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 <= -2e-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 <= -2e-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 <= -2e-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 <= -2e-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, -2e-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 < -1.999999999999994e-310

   1. Initial program 78.4%

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

      1. frac-2neg 40.8%

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

      2. log-div 52.7%

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

   3. Applied egg-rr 99.5%

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

   if -1.999999999999994e-310 < y

   1. Initial program 80.5%

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

      1. log-div 99.3%

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

   3. Applied egg-rr 99.3%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \]

Alternative 9: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \log x - x \cdot \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2e-310)
   (- (* x (- (log (- x)) (log (- y)))) z)
   (- (- (* x (log x)) (* x (log y))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2e-310) {
		tmp = (x * (log(-x) - log(-y))) - z;
	} else {
		tmp = ((x * log(x)) - (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 <= (-2d-310)) then
        tmp = (x * (log(-x) - log(-y))) - z
    else
        tmp = ((x * log(x)) - (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 <= -2e-310) {
		tmp = (x * (Math.log(-x) - Math.log(-y))) - z;
	} else {
		tmp = ((x * Math.log(x)) - (x * Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -2e-310:
		tmp = (x * (math.log(-x) - math.log(-y))) - z
	else:
		tmp = ((x * math.log(x)) - (x * math.log(y))) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2e-310)
		tmp = Float64(Float64(x * Float64(log(Float64(-x)) - log(Float64(-y)))) - z);
	else
		tmp = Float64(Float64(Float64(x * log(x)) - Float64(x * log(y))) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -2e-310)
		tmp = (x * (log(-x) - log(-y))) - z;
	else
		tmp = ((x * log(x)) - (x * log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2e-310], N[(N[(x * N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(N[(N[(x * N[Log[x], $MachinePrecision]), $MachinePrecision] - N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.999999999999994e-310

   1. Initial program 78.4%

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

      1. frac-2neg 40.8%

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

      2. log-div 52.7%

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

   3. Applied egg-rr 99.5%

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

   if -1.999999999999994e-310 < y

   1. Initial program 80.5%

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

      1. add-cbrt-cube 54.0%

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

      2. pow3 54.0%

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

   3. Applied egg-rr 54.0%

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

      1. rem-cbrt-cube 80.5%

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

      2. add-cube-cbrt 80.5%

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

      3. pow3 80.5%

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

      4. exp-to-pow 80.4%

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

      5. add-log-exp 80.4%

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

      6. *-commutative 80.4%

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

      7. associate-*r* 80.4%

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

   5. Applied egg-rr 80.4%

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

      1. cbrt-div 99.7%

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

      2. div-inv 99.7%

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

   7. Applied egg-rr 99.7%

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

      1. associate-*r/ 99.7%

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

      2. *-rgt-identity 99.7%

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

   9. Simplified 99.7%

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

       1. cbrt-div 80.4%

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

       2. associate-*r* 80.4%

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

       3. add-log-exp 80.4%

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

       4. *-commutative 80.4%

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

       5. exp-to-pow 80.5%

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

       6. pow3 80.5%

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

       7. add-cube-cbrt 80.5%

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

       8. diff-log 99.3%

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

       9. sub-neg 99.3%

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

       10. distribute-rgt-in 99.4%

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

   11. Applied egg-rr 99.4%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \log x - x \cdot \log y\right) - z\\ \end{array} \]

Alternative 10: 66.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{+15} \lor \neg \left(z \leq 7.2 \cdot 10^{+25}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\left(-x\right) \cdot \log \left(\frac{y}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -5.5e+15) (not (<= z 7.2e+25))) (- z) (* (- x) (log (/ y x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -5.5e+15) || !(z <= 7.2e+25)) {
		tmp = -z;
	} else {
		tmp = -x * log((y / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-5.5d+15)) .or. (.not. (z <= 7.2d+25))) then
        tmp = -z
    else
        tmp = -x * log((y / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -5.5e+15) || !(z <= 7.2e+25)) {
		tmp = -z;
	} else {
		tmp = -x * Math.log((y / x));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -5.5e+15) or not (z <= 7.2e+25):
		tmp = -z
	else:
		tmp = -x * math.log((y / x))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -5.5e+15) || !(z <= 7.2e+25))
		tmp = Float64(-z);
	else
		tmp = Float64(Float64(-x) * log(Float64(y / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -5.5e+15) || ~((z <= 7.2e+25)))
		tmp = -z;
	else
		tmp = -x * log((y / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -5.5e+15], N[Not[LessEqual[z, 7.2e+25]], $MachinePrecision]], (-z), N[((-x) * N[Log[N[(y / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5.5e15 or 7.20000000000000031e25 < z

   1. Initial program 83.1%

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

      1. remove-double-neg 83.1%

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

      2. sub0-neg 83.1%

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

      3. associate--r- 83.1%

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

      4. neg-sub0 83.1%

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

      5. distribute-rgt-neg-in 83.1%

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

      6. neg-sub0 83.1%

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

      7. log-div 39.8%

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

      8. associate-+l- 39.8%

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

      9. neg-sub0 39.8%

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

      10. +-commutative 39.8%

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

      11. sub-neg 39.8%

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

      12. log-div 81.0%

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

      13. fma-udef 81.0%

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

   3. Simplified 81.0%

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

   4. Taylor expanded in x around 0 80.8%

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

   if -5.5e15 < z < 7.20000000000000031e25

   1. Initial program 76.6%

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

      1. remove-double-neg 76.6%

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

      2. sub0-neg 76.6%

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

      3. associate--r- 76.6%

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

      4. neg-sub0 76.6%

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

      5. distribute-rgt-neg-in 76.6%

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

      6. neg-sub0 76.6%

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

      7. log-div 48.8%

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

      8. associate-+l- 48.8%

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

      9. neg-sub0 48.8%

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

      10. +-commutative 48.8%

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

      11. sub-neg 48.8%

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

      12. log-div 76.6%

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

      13. fma-udef 76.6%

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

   3. Simplified 76.6%

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

   4. Taylor expanded in x around inf 38.3%

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

      1. log-rec 38.3%

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

      2. neg-mul-1 38.3%

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

      3. neg-mul-1 38.3%

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

      4. sub-neg 38.3%

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

      5. log-div 61.1%

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

   6. Simplified 61.1%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{+15} \lor \neg \left(z \leq 7.2 \cdot 10^{+25}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\left(-x\right) \cdot \log \left(\frac{y}{x}\right)\\ \end{array} \]

Alternative 11: 66.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.5 \cdot 10^{+15} \lor \neg \left(z \leq 4.5 \cdot 10^{+27}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -6.5e+15) (not (<= z 4.5e+27))) (- z) (* x (log (/ x y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -6.5e+15) || !(z <= 4.5e+27)) {
		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 <= (-6.5d+15)) .or. (.not. (z <= 4.5d+27))) 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 <= -6.5e+15) || !(z <= 4.5e+27)) {
		tmp = -z;
	} else {
		tmp = x * Math.log((x / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -6.5e+15) or not (z <= 4.5e+27):
		tmp = -z
	else:
		tmp = x * math.log((x / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -6.5e+15) || !(z <= 4.5e+27))
		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 <= -6.5e+15) || ~((z <= 4.5e+27)))
		tmp = -z;
	else
		tmp = x * log((x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -6.5e+15], N[Not[LessEqual[z, 4.5e+27]], $MachinePrecision]], (-z), N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -6.5e15 or 4.4999999999999999e27 < z

   1. Initial program 83.1%

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

      1. remove-double-neg 83.1%

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

      2. sub0-neg 83.1%

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

      3. associate--r- 83.1%

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

      4. neg-sub0 83.1%

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

      5. distribute-rgt-neg-in 83.1%

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

      6. neg-sub0 83.1%

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

      7. log-div 39.8%

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

      8. associate-+l- 39.8%

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

      9. neg-sub0 39.8%

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

      10. +-commutative 39.8%

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

      11. sub-neg 39.8%

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

      12. log-div 81.0%

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

      13. fma-udef 81.0%

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

   3. Simplified 81.0%

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

   4. Taylor expanded in x around 0 80.8%

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

   if -6.5e15 < z < 4.4999999999999999e27

   1. Initial program 76.6%

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

   2. Taylor expanded in z around 0 61.1%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.5 \cdot 10^{+15} \lor \neg \left(z \leq 4.5 \cdot 10^{+27}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \end{array} \]

Alternative 12: 50.6% 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 79.3%

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

   1. remove-double-neg 79.3%

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

   2. sub0-neg 79.3%

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

   3. associate--r- 79.3%

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

   4. neg-sub0 79.3%

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

   5. distribute-rgt-neg-in 79.3%

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

   6. neg-sub0 79.3%

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

   7. log-div 45.0%

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

   8. associate-+l- 45.0%

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

   9. neg-sub0 45.0%

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

   10. +-commutative 45.0%

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

   11. sub-neg 45.0%

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

   12. log-div 78.4%

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

   13. fma-udef 78.4%

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

3. Simplified 78.4%

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

4. Taylor expanded in x around 0 48.7%

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

5. Final simplification 48.7%

   \[\leadsto -z \]

Developer target: 88.3% 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 2023318 
(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))