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

Percentage Accurate: 76.8% → 99.7%

Time: 11.0s

Alternatives: 9

Speedup: 0.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 82.3%

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

   1. add-log-exp 30.3%

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

   2. *-commutative 30.3%

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

   3. exp-to-pow 30.4%

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

3. Applied egg-rr 30.4%

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

   1. add-cube-cbrt 30.4%

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

   2. pow3 30.4%

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

   3. exp-to-pow 30.4%

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

   4. exp-prod 30.3%

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

   5. *-commutative 30.3%

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

   6. add-log-exp 82.2%

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

   7. associate-*r* 82.2%

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

5. Applied egg-rr 82.2%

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

   1. cbrt-div 99.7%

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

   2. div-inv 99.7%

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

7. Applied egg-rr 99.7%

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

   1. associate-*r/ 99.7%

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

   2. *-rgt-identity 99.7%

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

9. Simplified 99.7%

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

10. Final simplification 99.7%

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

Alternative 2: 88.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ t_1 := x \cdot \log \left(x \cdot y\right)\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;t_1 - z\\ \mathbf{elif}\;t_0 \leq 10^{+306}:\\ \;\;\;\;t_0 - z\\ \mathbf{else}:\\ \;\;\;\;\left|t_1\right| - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))) (t_1 (* x (log (* x y)))))
   (if (<= t_0 (- INFINITY))
     (- t_1 z)
     (if (<= t_0 1e+306) (- t_0 z) (- (fabs t_1) z)))))

C

double code(double x, double y, double z) {
	double t_0 = x * log((x / y));
	double t_1 = x * log((x * y));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = t_1 - z;
	} else if (t_0 <= 1e+306) {
		tmp = t_0 - z;
	} else {
		tmp = fabs(t_1) - z;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = x * Math.log((x / y));
	double t_1 = x * Math.log((x * y));
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = t_1 - z;
	} else if (t_0 <= 1e+306) {
		tmp = t_0 - z;
	} else {
		tmp = Math.abs(t_1) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.log((x / y))
	t_1 = x * math.log((x * y))
	tmp = 0
	if t_0 <= -math.inf:
		tmp = t_1 - z
	elif t_0 <= 1e+306:
		tmp = t_0 - z
	else:
		tmp = math.fabs(t_1) - z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * log(Float64(x / y)))
	t_1 = Float64(x * log(Float64(x * y)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(t_1 - z);
	elseif (t_0 <= 1e+306)
		tmp = Float64(t_0 - z);
	else
		tmp = Float64(abs(t_1) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * log((x / y));
	t_1 = x * log((x * y));
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = t_1 - z;
	elseif (t_0 <= 1e+306)
		tmp = t_0 - z;
	else
		tmp = abs(t_1) - 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]}, Block[{t$95$1 = N[(x * N[Log[N[(x * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(t$95$1 - z), $MachinePrecision], If[LessEqual[t$95$0, 1e+306], N[(t$95$0 - z), $MachinePrecision], N[(N[Abs[t$95$1], $MachinePrecision] - z), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 5.4%

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

      1. add-cube-cbrt 5.4%

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

      2. log-prod 5.4%

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

      3. pow2 5.4%

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

   3. Applied egg-rr 5.4%

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

      1. log-pow 5.4%

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

      2. distribute-lft1-in 5.4%

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

      3. metadata-eval 5.4%

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

      4. *-commutative 5.4%

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

   5. Simplified 5.4%

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

      1. add-log-exp 5.4%

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

      2. *-commutative 5.4%

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

      3. exp-prod 5.4%

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

      4. exp-to-pow 5.4%

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

      5. pow3 5.4%

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

      6. add-cube-cbrt 5.4%

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

      7. log-pow 5.4%

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

      8. frac-2neg 5.4%

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

      9. diff-log 74.7%

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

      10. sub-neg 74.7%

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

      11. distribute-rgt-in 74.6%

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

      12. add-sqr-sqrt 74.7%

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

      13. sqrt-unprod 28.6%

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

      14. sqr-neg 28.6%

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

      15. sqrt-unprod 0.0%

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

      16. add-sqr-sqrt 0.0%

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

   7. Applied egg-rr 25.0%

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

      1. distribute-rgt-out 25.0%

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

      2. sub-neg 25.0%

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

      3. *-commutative 25.0%

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

      4. sub-neg 25.0%

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

      5. add-sqr-sqrt 0.0%

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

      6. sqrt-unprod 25.0%

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

      7. sqr-neg 25.0%

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

      8. sqrt-unprod 25.0%

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

      9. add-sqr-sqrt 25.0%

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

      10. sum-log 43.0%

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

   9. Applied egg-rr 43.0%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 1.00000000000000002e306

   1. Initial program 99.8%

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

   if 1.00000000000000002e306 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 3.6%

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

      1. log-div 36.9%

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

   3. Applied egg-rr 36.9%

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

   5. Applied egg-rr 46.4%

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

      1. unpow2 46.4%

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

      2. rem-sqrt-square 50.5%

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

   7. Simplified 50.5%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log \left(\frac{x}{y}\right) \leq -\infty:\\ \;\;\;\;x \cdot \log \left(x \cdot y\right) - z\\ \mathbf{elif}\;x \cdot \log \left(\frac{x}{y}\right) \leq 10^{+306}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \log \left(x \cdot y\right)\right| - z\\ \end{array} \]

Alternative 3: 89.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ t_1 := \log \left(x \cdot y\right)\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;x \cdot \left|t_1\right| - z\\ \mathbf{elif}\;t_0 \leq 10^{+306}:\\ \;\;\;\;t_0 - z\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot t_1\right| - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))) (t_1 (log (* x y))))
   (if (<= t_0 (- INFINITY))
     (- (* x (fabs t_1)) z)
     (if (<= t_0 1e+306) (- t_0 z) (- (fabs (* x t_1)) z)))))

C

double code(double x, double y, double z) {
	double t_0 = x * log((x / y));
	double t_1 = log((x * y));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (x * fabs(t_1)) - z;
	} else if (t_0 <= 1e+306) {
		tmp = t_0 - z;
	} else {
		tmp = fabs((x * t_1)) - z;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = x * Math.log((x / y));
	double t_1 = Math.log((x * y));
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = (x * Math.abs(t_1)) - z;
	} else if (t_0 <= 1e+306) {
		tmp = t_0 - z;
	} else {
		tmp = Math.abs((x * t_1)) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.log((x / y))
	t_1 = math.log((x * y))
	tmp = 0
	if t_0 <= -math.inf:
		tmp = (x * math.fabs(t_1)) - z
	elif t_0 <= 1e+306:
		tmp = t_0 - z
	else:
		tmp = math.fabs((x * t_1)) - z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * log(Float64(x / y)))
	t_1 = log(Float64(x * y))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(x * abs(t_1)) - z);
	elseif (t_0 <= 1e+306)
		tmp = Float64(t_0 - z);
	else
		tmp = Float64(abs(Float64(x * t_1)) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * log((x / y));
	t_1 = log((x * y));
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = (x * abs(t_1)) - z;
	elseif (t_0 <= 1e+306)
		tmp = t_0 - z;
	else
		tmp = abs((x * t_1)) - 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]}, Block[{t$95$1 = N[Log[N[(x * y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(x * N[Abs[t$95$1], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[t$95$0, 1e+306], N[(t$95$0 - z), $MachinePrecision], N[(N[Abs[N[(x * t$95$1), $MachinePrecision]], $MachinePrecision] - z), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 5.4%

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

      1. add-cube-cbrt 5.4%

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

      2. log-prod 5.4%

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

      3. pow2 5.4%

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

   3. Applied egg-rr 5.4%

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

      1. log-pow 5.4%

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

      2. distribute-lft1-in 5.4%

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

      3. metadata-eval 5.4%

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

      4. *-commutative 5.4%

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

   5. Simplified 5.4%

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

      1. cbrt-div 99.6%

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

      2. div-inv 99.6%

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

   7. Applied egg-rr 99.5%

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

      1. associate-*r/ 99.6%

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

      2. *-rgt-identity 99.6%

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

   9. Simplified 99.5%

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

       1. add-sqr-sqrt 74.3%

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

       2. sqrt-unprod 99.5%

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

       3. pow2 99.5%

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

       4. add-log-exp 5.1%

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

       5. cbrt-div 5.1%

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

       6. exp-to-pow 5.1%

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

       7. pow3 5.1%

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

       8. add-cube-cbrt 5.1%

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

       9. diff-log 25.0%

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

   11. Applied egg-rr 46.4%

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

       1. unpow2 46.4%

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

       2. rem-sqrt-square 46.4%

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

   13. Simplified 46.4%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 1.00000000000000002e306

   1. Initial program 99.8%

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

   if 1.00000000000000002e306 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 3.6%

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

      1. log-div 36.9%

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

   3. Applied egg-rr 36.9%

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

   5. Applied egg-rr 46.4%

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

      1. unpow2 46.4%

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

      2. rem-sqrt-square 50.5%

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

   7. Simplified 50.5%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log \left(\frac{x}{y}\right) \leq -\infty:\\ \;\;\;\;x \cdot \left|\log \left(x \cdot y\right)\right| - z\\ \mathbf{elif}\;x \cdot \log \left(\frac{x}{y}\right) \leq 10^{+306}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \log \left(x \cdot y\right)\right| - z\\ \end{array} \]

Alternative 4: 88.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;t_0 \leq -\infty \lor \neg \left(t_0 \leq 10^{+306}\right):\\ \;\;\;\;x \cdot \log \left(x \cdot y\right) - z\\ \mathbf{else}:\\ \;\;\;\;t_0 - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 1e+306)))
     (- (* x (log (* x y))) z)
     (- t_0 z))))

C

double code(double x, double y, double z) {
	double t_0 = x * log((x / y));
	double tmp;
	if ((t_0 <= -((double) INFINITY)) || !(t_0 <= 1e+306)) {
		tmp = (x * log((x * y))) - z;
	} else {
		tmp = t_0 - z;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = x * Math.log((x / y));
	double tmp;
	if ((t_0 <= -Double.POSITIVE_INFINITY) || !(t_0 <= 1e+306)) {
		tmp = (x * Math.log((x * y))) - z;
	} else {
		tmp = t_0 - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.log((x / y))
	tmp = 0
	if (t_0 <= -math.inf) or not (t_0 <= 1e+306):
		tmp = (x * math.log((x * y))) - z
	else:
		tmp = t_0 - z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * log(Float64(x / y)))
	tmp = 0.0
	if ((t_0 <= Float64(-Inf)) || !(t_0 <= 1e+306))
		tmp = Float64(Float64(x * log(Float64(x * y))) - z);
	else
		tmp = Float64(t_0 - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * log((x / y));
	tmp = 0.0;
	if ((t_0 <= -Inf) || ~((t_0 <= 1e+306)))
		tmp = (x * log((x * y))) - z;
	else
		tmp = t_0 - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, (-Infinity)], N[Not[LessEqual[t$95$0, 1e+306]], $MachinePrecision]], N[(N[(x * N[Log[N[(x * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], 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 1.00000000000000002e306 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 4.4%

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

      1. add-cube-cbrt 4.4%

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

      2. log-prod 4.4%

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

      3. pow2 4.4%

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

   3. Applied egg-rr 4.4%

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

      1. log-pow 4.4%

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

      2. distribute-lft1-in 4.4%

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

      3. metadata-eval 4.4%

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

      4. *-commutative 4.4%

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

   5. Simplified 4.4%

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

      1. add-log-exp 4.4%

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

      2. *-commutative 4.4%

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

      3. exp-prod 4.4%

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

      4. exp-to-pow 4.4%

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

      5. pow3 4.4%

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

      6. add-cube-cbrt 4.4%

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

      7. log-pow 4.4%

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

      8. frac-2neg 4.4%

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

      9. diff-log 67.9%

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

      10. sub-neg 67.9%

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

      11. distribute-rgt-in 67.9%

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

      12. add-sqr-sqrt 67.9%

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

      13. sqrt-unprod 21.3%

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

      14. sqr-neg 21.3%

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

      15. sqrt-unprod 0.0%

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

      16. add-sqr-sqrt 0.0%

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

   7. Applied egg-rr 31.9%

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

      1. distribute-rgt-out 31.8%

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

      2. sub-neg 31.8%

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

      3. *-commutative 31.8%

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

      4. sub-neg 31.8%

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

      5. add-sqr-sqrt 21.1%

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

      6. sqrt-unprod 31.8%

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

      7. sqr-neg 31.8%

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

      8. sqrt-unprod 10.6%

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

      9. add-sqr-sqrt 12.1%

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

      10. sum-log 44.1%

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

   9. Applied egg-rr 44.1%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 1.00000000000000002e306

   1. Initial program 99.8%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log \left(\frac{x}{y}\right) \leq -\infty \lor \neg \left(x \cdot \log \left(\frac{x}{y}\right) \leq 10^{+306}\right):\\ \;\;\;\;x \cdot \log \left(x \cdot y\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \end{array} \]

Alternative 5: 82.0% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))))
   (if (<= t_0 (- INFINITY)) (- z) (if (<= t_0 INFINITY) (- 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 <= ((double) INFINITY)) {
		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 <= Double.POSITIVE_INFINITY) {
		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 <= math.inf:
		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 <= Inf)
		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 <= Inf)
		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, Infinity], 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 +inf.0 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 5.4%

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

      1. add-log-exp 5.4%

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

      2. *-commutative 5.4%

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

      3. exp-to-pow 5.4%

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

   3. Applied egg-rr 5.4%

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

   4. Taylor expanded in x around 0 36.4%

      \[\leadsto \log \color{blue}{1} - z \]

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

   1. Initial program 88.8%

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

4. Final simplification 84.7%

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

Alternative 6: 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 82.3%

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

   1. add-cube-cbrt 82.3%

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

   2. log-prod 82.2%

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

   3. pow2 82.2%

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

3. Applied egg-rr 82.2%

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

   1. log-pow 82.2%

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

   2. distribute-lft1-in 82.2%

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

   3. metadata-eval 82.2%

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

   4. *-commutative 82.2%

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

5. Simplified 82.2%

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

   1. cbrt-div 99.7%

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

   2. div-inv 99.7%

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

7. Applied egg-rr 99.7%

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

   1. associate-*r/ 99.7%

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

   2. *-rgt-identity 99.7%

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

9. Simplified 99.7%

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

10. Final simplification 99.7%

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

Alternative 7: 99.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -5e-310) {
		tmp = (x * (log(-x) - log(-y))) - z;
	} else {
		tmp = (x * (log(x) - log(y))) - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-5d-310)) then
        tmp = (x * (log(-x) - log(-y))) - z
    else
        tmp = (x * (log(x) - log(y))) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -5e-310) {
		tmp = (x * (Math.log(-x) - Math.log(-y))) - z;
	} else {
		tmp = (x * (Math.log(x) - Math.log(y))) - z;
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -4.999999999999985e-310

   1. Initial program 79.0%

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

      1. frac-2neg 79.0%

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

      2. log-div 99.6%

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

   3. Applied egg-rr 99.6%

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

   if -4.999999999999985e-310 < y

   1. Initial program 86.7%

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

      1. log-div 99.5%

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

   3. Applied egg-rr 99.5%

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

4. Final simplification 99.6%

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

Alternative 8: 99.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -5e-310) {
		tmp = (x * (log(-x) - log(-y))) - z;
	} else {
		tmp = ((x * log(x)) - (x * log(y))) - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-5d-310)) then
        tmp = (x * (log(-x) - log(-y))) - z
    else
        tmp = ((x * log(x)) - (x * log(y))) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -5e-310) {
		tmp = (x * (Math.log(-x) - Math.log(-y))) - z;
	} else {
		tmp = ((x * Math.log(x)) - (x * Math.log(y))) - z;
	}
	return tmp;
}

Python

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

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -5e-310)
		tmp = Float64(Float64(x * Float64(log(Float64(-x)) - log(Float64(-y)))) - z);
	else
		tmp = Float64(Float64(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 <= -5e-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, -5e-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 < -4.999999999999985e-310

   1. Initial program 79.0%

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

      1. frac-2neg 79.0%

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

      2. log-div 99.6%

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

   3. Applied egg-rr 99.6%

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

   if -4.999999999999985e-310 < y

   1. Initial program 86.7%

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

      1. log-div 99.5%

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

   3. Applied egg-rr 99.5%

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

      1. sub-neg 99.5%

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

      2. distribute-rgt-out 99.6%

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

      3. distribute-lft-neg-out 99.6%

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

      4. unsub-neg 99.6%

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

      5. *-commutative 99.6%

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

      6. *-commutative 99.6%

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

   5. Applied egg-rr 99.6%

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

4. Final simplification 99.6%

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

Alternative 9: 49.8% 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 82.3%

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

   1. add-log-exp 30.3%

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

   2. *-commutative 30.3%

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

   3. exp-to-pow 30.4%

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

3. Applied egg-rr 30.4%

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

4. Taylor expanded in x around 0 46.9%

   \[\leadsto \log \color{blue}{1} - z \]

5. Final simplification 46.9%

   \[\leadsto -z \]

Developer target: 88.2% 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 2023199 
(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))