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

Percentage Accurate: 77.6% → 99.7%

Time: 7.2s

Alternatives: 7

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

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

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 80.1%

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

   1. add-cube-cbrt 80.1%

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

   2. log-prod 80.1%

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

   3. pow2 80.1%

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

3. Applied egg-rr 80.1%

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

   1. log-pow 80.1%

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

   2. distribute-lft1-in 80.1%

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

   3. metadata-eval 80.1%

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

   4. *-commutative 80.1%

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

5. Simplified 80.1%

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

   1. cbrt-div 99.8%

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

   2. div-inv 99.7%

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

7. Applied egg-rr 99.7%

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

   1. associate-*r/ 99.8%

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

   2. *-rgt-identity 99.8%

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

9. Simplified 99.8%

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

10. Final simplification 99.8%

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

Alternative 2: 88.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\frac{x}{y}\right)\\ t_1 := x \cdot t_0\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;-z\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+307}:\\ \;\;\;\;x \cdot \left(\left(t_0 + 1\right) + -1\right) - z\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (log (/ x y))) (t_1 (* x t_0)))
   (if (<= t_1 (- INFINITY))
     (- z)
     (if (<= t_1 5e+307) (- (* x (+ (+ t_0 1.0) -1.0)) z) (- z)))))

C

double code(double x, double y, double z) {
	double t_0 = log((x / y));
	double t_1 = x * t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -z;
	} else if (t_1 <= 5e+307) {
		tmp = (x * ((t_0 + 1.0) + -1.0)) - z;
	} else {
		tmp = -z;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.log((x / y));
	double t_1 = x * t_0;
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = -z;
	} else if (t_1 <= 5e+307) {
		tmp = (x * ((t_0 + 1.0) + -1.0)) - z;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.log((x / y))
	t_1 = x * t_0
	tmp = 0
	if t_1 <= -math.inf:
		tmp = -z
	elif t_1 <= 5e+307:
		tmp = (x * ((t_0 + 1.0) + -1.0)) - z
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	t_0 = log(Float64(x / y))
	t_1 = Float64(x * t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-z);
	elseif (t_1 <= 5e+307)
		tmp = Float64(Float64(x * Float64(Float64(t_0 + 1.0) + -1.0)) - z);
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = log((x / y));
	t_1 = x * t_0;
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = -z;
	elseif (t_1 <= 5e+307)
		tmp = (x * ((t_0 + 1.0) + -1.0)) - z;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(x * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], (-z), If[LessEqual[t$95$1, 5e+307], N[(N[(x * N[(N[(t$95$0 + 1.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], (-z)]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x (log.f64 (/.f64 x y))) < -inf.0 or 5e307 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 8.4%

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

      1. clear-num 8.4%

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

      2. log-rec 13.8%

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

   3. Applied egg-rr 13.8%

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

      1. neg-log 8.4%

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

      2. clear-num 8.4%

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

      3. expm1-log1p-u 7.1%

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

      4. expm1-udef 7.1%

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

      5. log1p-udef 7.1%

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

      6. add-exp-log 8.4%

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

      7. associate--l+ 8.4%

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

      8. distribute-lft-in 8.4%

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

      9. *-commutative 8.4%

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

      10. *-un-lft-identity 8.4%

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

      11. sub-neg 8.4%

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

      12. add-sqr-sqrt 7.1%

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

      13. sqrt-unprod 8.0%

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

      14. clear-num 8.0%

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

      15. neg-log 8.0%

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

      16. clear-num 8.0%

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

      17. neg-log 13.3%

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

      18. sqr-neg 13.3%

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

      19. sqrt-unprod 0.9%

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

      20. add-sqr-sqrt 1.2%

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

      21. metadata-eval 1.2%

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

   5. Applied egg-rr 1.2%

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

      1. +-commutative 1.2%

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

      2. *-rgt-identity 1.2%

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

      3. distribute-lft-in 1.2%

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

      4. associate-+l+ 1.2%

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

      5. metadata-eval 1.2%

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

      6. +-rgt-identity 1.2%

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

   7. Simplified 1.2%

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

   8. Taylor expanded in x around 0 44.8%

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

      1. neg-mul-1 44.8%

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

   10. Simplified 44.8%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 5e307

   1. Initial program 99.8%

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

      1. expm1-log1p-u 50.1%

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

      2. expm1-udef 50.1%

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

      3. log1p-udef 50.1%

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

      4. add-exp-log 99.8%

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

   3. Applied egg-rr 99.8%

      \[\leadsto x \cdot \color{blue}{\left(\left(1 + \log \left(\frac{x}{y}\right)\right) - 1\right)} - z \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log \left(\frac{x}{y}\right) \leq -\infty:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \cdot \log \left(\frac{x}{y}\right) \leq 5 \cdot 10^{+307}:\\ \;\;\;\;x \cdot \left(\left(\log \left(\frac{x}{y}\right) + 1\right) + -1\right) - z\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 3: 88.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;-z\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+307}:\\ \;\;\;\;t_0 - z\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))))
   (if (<= t_0 (- INFINITY)) (- z) (if (<= t_0 5e+307) (- t_0 z) (- z)))))

C

double code(double x, double y, double z) {
	double t_0 = x * log((x / y));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = -z;
	} else if (t_0 <= 5e+307) {
		tmp = t_0 - z;
	} else {
		tmp = -z;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = x * Math.log((x / y));
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = -z;
	} else if (t_0 <= 5e+307) {
		tmp = t_0 - z;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.log((x / y))
	tmp = 0
	if t_0 <= -math.inf:
		tmp = -z
	elif t_0 <= 5e+307:
		tmp = t_0 - z
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * log(Float64(x / y)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(-z);
	elseif (t_0 <= 5e+307)
		tmp = Float64(t_0 - z);
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * log((x / y));
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = -z;
	elseif (t_0 <= 5e+307)
		tmp = t_0 - z;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], (-z), If[LessEqual[t$95$0, 5e+307], N[(t$95$0 - z), $MachinePrecision], (-z)]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x (log.f64 (/.f64 x y))) < -inf.0 or 5e307 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 8.4%

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

      1. clear-num 8.4%

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

      2. log-rec 13.8%

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

   3. Applied egg-rr 13.8%

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

      1. neg-log 8.4%

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

      2. clear-num 8.4%

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

      3. expm1-log1p-u 7.1%

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

      4. expm1-udef 7.1%

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

      5. log1p-udef 7.1%

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

      6. add-exp-log 8.4%

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

      7. associate--l+ 8.4%

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

      8. distribute-lft-in 8.4%

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

      9. *-commutative 8.4%

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

      10. *-un-lft-identity 8.4%

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

      11. sub-neg 8.4%

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

      12. add-sqr-sqrt 7.1%

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

      13. sqrt-unprod 8.0%

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

      14. clear-num 8.0%

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

      15. neg-log 8.0%

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

      16. clear-num 8.0%

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

      17. neg-log 13.3%

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

      18. sqr-neg 13.3%

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

      19. sqrt-unprod 0.9%

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

      20. add-sqr-sqrt 1.2%

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

      21. metadata-eval 1.2%

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

   5. Applied egg-rr 1.2%

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

      1. +-commutative 1.2%

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

      2. *-rgt-identity 1.2%

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

      3. distribute-lft-in 1.2%

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

      4. associate-+l+ 1.2%

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

      5. metadata-eval 1.2%

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

      6. +-rgt-identity 1.2%

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

   7. Simplified 1.2%

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

   8. Taylor expanded in x around 0 44.8%

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

      1. neg-mul-1 44.8%

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

   10. Simplified 44.8%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 5e307

   1. Initial program 99.8%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log \left(\frac{x}{y}\right) \leq -\infty:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \cdot \log \left(\frac{x}{y}\right) \leq 5 \cdot 10^{+307}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 4: 91.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{-170}:\\ \;\;\;\;\left(-x\right) \cdot \log \left(\frac{y}{x}\right) - z\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-305}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -4.5e-170)
   (- (* (- x) (log (/ y x))) z)
   (if (<= x -5e-305) (- z) (- (* x (- (log x) (log y))) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -4.5e-170) {
		tmp = (-x * log((y / x))) - z;
	} else if (x <= -5e-305) {
		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 <= (-4.5d-170)) then
        tmp = (-x * log((y / x))) - z
    else if (x <= (-5d-305)) 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 <= -4.5e-170) {
		tmp = (-x * Math.log((y / x))) - z;
	} else if (x <= -5e-305) {
		tmp = -z;
	} else {
		tmp = (x * (Math.log(x) - Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -4.5e-170:
		tmp = (-x * math.log((y / x))) - z
	elif x <= -5e-305:
		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 <= -4.5e-170)
		tmp = Float64(Float64(Float64(-x) * log(Float64(y / x))) - z);
	elseif (x <= -5e-305)
		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 <= -4.5e-170)
		tmp = (-x * log((y / x))) - z;
	elseif (x <= -5e-305)
		tmp = -z;
	else
		tmp = (x * (log(x) - log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -4.5e-170], N[(N[((-x) * N[Log[N[(y / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[x, -5e-305], (-z), N[(N[(x * N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.50000000000000002e-170

   1. Initial program 81.9%

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

      1. clear-num 81.9%

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

      2. log-rec 84.1%

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

   3. Applied egg-rr 84.1%

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

   if -4.50000000000000002e-170 < x < -4.99999999999999985e-305

   1. Initial program 59.5%

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

      1. clear-num 59.5%

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

      2. log-rec 59.5%

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

   3. Applied egg-rr 59.5%

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

      1. neg-log 59.5%

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

      2. clear-num 59.5%

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

      3. expm1-log1p-u 8.3%

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

      4. expm1-udef 8.3%

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

      5. log1p-udef 8.3%

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

      6. add-exp-log 59.5%

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

      7. associate--l+ 59.5%

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

      8. distribute-lft-in 59.5%

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

      9. *-commutative 59.5%

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

      10. *-un-lft-identity 59.5%

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

      11. sub-neg 59.5%

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

      12. add-sqr-sqrt 8.3%

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

      13. sqrt-unprod 59.4%

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

      14. clear-num 59.4%

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

      15. neg-log 59.4%

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

      16. clear-num 59.4%

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

      17. neg-log 59.4%

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

      18. sqr-neg 59.4%

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

      19. sqrt-unprod 51.1%

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

      20. add-sqr-sqrt 59.4%

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

      21. metadata-eval 59.4%

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

   5. Applied egg-rr 59.4%

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

      1. +-commutative 59.4%

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

      2. *-rgt-identity 59.4%

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

      3. distribute-lft-in 59.4%

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

      4. associate-+l+ 59.4%

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

      5. metadata-eval 59.4%

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

      6. +-rgt-identity 59.4%

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

   7. Simplified 59.4%

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

   8. Taylor expanded in x around 0 89.5%

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

      1. neg-mul-1 89.5%

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

   10. Simplified 89.5%

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

   if -4.99999999999999985e-305 < x

   1. Initial program 82.6%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{-170}:\\ \;\;\;\;\left(-x\right) \cdot \log \left(\frac{y}{x}\right) - z\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-305}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \]

Alternative 5: 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 77.5%

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

      1. frac-2neg 77.5%

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

      2. log-div 99.5%

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

   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 82.6%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 6: 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 77.5%

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

      1. frac-2neg 77.5%

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

      2. log-div 99.5%

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

   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 82.6%

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

      1. clear-num 82.6%

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

      2. log-rec 83.1%

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

   3. Applied egg-rr 83.1%

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

      1. neg-log 82.6%

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

      2. clear-num 82.6%

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

      3. expm1-log1p-u 41.4%

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

      4. expm1-udef 41.4%

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

      5. log1p-udef 41.4%

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

      6. add-exp-log 82.6%

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

      7. associate--l+ 82.6%

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

      8. distribute-lft-in 82.6%

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

      9. *-commutative 82.6%

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

      10. *-un-lft-identity 82.6%

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

      11. sub-neg 82.6%

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

      12. add-sqr-sqrt 41.7%

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

      13. sqrt-unprod 65.0%

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

      14. clear-num 65.0%

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

      15. neg-log 65.0%

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

      16. clear-num 65.0%

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

      17. neg-log 65.6%

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

      18. sqr-neg 65.6%

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

      19. sqrt-unprod 23.2%

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

      20. add-sqr-sqrt 36.3%

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

      21. metadata-eval 36.3%

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

   5. Applied egg-rr 36.3%

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

      1. +-commutative 36.3%

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

      2. *-rgt-identity 36.3%

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

      3. distribute-lft-in 36.3%

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

      4. associate-+l+ 36.3%

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

      5. metadata-eval 36.3%

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

      6. +-rgt-identity 36.3%

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

   7. Simplified 36.3%

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

      1. add-cube-cbrt 36.3%

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

      2. pow3 36.3%

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

      3. exp-to-pow 36.3%

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

      4. add-log-exp 36.3%

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

      5. add-sqr-sqrt 23.2%

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

      6. sqrt-unprod 65.6%

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

      7. sqr-neg 65.6%

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

      8. sqrt-unprod 42.2%

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

      9. add-sqr-sqrt 83.1%

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

      10. add-log-exp 82.5%

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

      11. exp-neg 82.5%

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

      12. exp-to-pow 82.6%

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

      13. pow3 82.6%

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

      14. add-cube-cbrt 82.6%

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

      15. clear-num 82.6%

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

      16. diff-log 99.3%

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

      17. sub-neg 99.3%

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

   9. Applied egg-rr 99.3%

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

4. Final simplification 99.4%

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

Alternative 7: 51.5% 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 80.1%

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

   1. clear-num 80.1%

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

   2. log-rec 81.3%

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

3. Applied egg-rr 81.3%

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

   1. neg-log 80.1%

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

   2. clear-num 80.1%

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

   3. expm1-log1p-u 40.9%

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

   4. expm1-udef 40.9%

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

   5. log1p-udef 40.9%

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

   6. add-exp-log 80.1%

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

   7. associate--l+ 80.1%

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

   8. distribute-lft-in 80.1%

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

   9. *-commutative 80.1%

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

   10. *-un-lft-identity 80.1%

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

   11. sub-neg 80.1%

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

   12. add-sqr-sqrt 41.1%

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

   13. sqrt-unprod 62.1%

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

   14. clear-num 62.1%

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

   15. neg-log 62.1%

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

   16. clear-num 62.1%

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

   17. neg-log 63.2%

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

   18. sqr-neg 63.2%

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

   19. sqrt-unprod 20.9%

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

   20. add-sqr-sqrt 35.2%

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

   21. metadata-eval 35.2%

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

5. Applied egg-rr 35.2%

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

   1. +-commutative 35.2%

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

   2. *-rgt-identity 35.2%

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

   3. distribute-lft-in 35.2%

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

   4. associate-+l+ 35.2%

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

   5. metadata-eval 35.2%

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

   6. +-rgt-identity 35.2%

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

7. Simplified 35.2%

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

8. Taylor expanded in x around 0 45.7%

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

   1. neg-mul-1 45.7%

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

10. Simplified 45.7%

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

11. Final simplification 45.7%

    \[\leadsto -z \]

Developer target: 88.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	double tmp;
	if (y < 7.595077799083773e-308) {
		tmp = (x * log((x / y))) - z;
	} else {
		tmp = (x * (log(x) - log(y))) - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y < 7.595077799083773d-308) then
        tmp = (x * log((x / y))) - z
    else
        tmp = (x * (log(x) - log(y))) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y < 7.595077799083773e-308) {
		tmp = (x * Math.log((x / y))) - z;
	} else {
		tmp = (x * (Math.log(x) - Math.log(y))) - z;
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023196 
(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))