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

Percentage Accurate: 77.2% → 99.7%

Time: 8.0s

Alternatives: 5

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: 77.2% 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 76.9%

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

   1. add-cube-cbrt 76.9%

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

   2. log-prod 76.8%

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

   3. pow2 76.8%

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

3. Applied egg-rr 76.8%

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

   1. log-pow 76.8%

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

   2. distribute-lft1-in 76.8%

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

   3. metadata-eval 76.8%

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

   4. *-commutative 76.8%

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

5. Simplified 76.8%

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

   1. cbrt-div 99.7%

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

   2. div-inv 99.7%

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

7. Applied egg-rr 99.7%

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

   1. associate-*r/ 99.7%

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

   2. *-rgt-identity 99.7%

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

9. Simplified 99.7%

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

10. Final simplification 99.7%

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

Alternative 2: 86.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;t_0 \leq -\infty \lor \neg \left(t_0 \leq 10^{+292}\right):\\ \;\;\;\;x \cdot 0 - z\\ \mathbf{else}:\\ \;\;\;\;t_0 - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 1e+292)))
     (- (* x 0.0) 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+292)) {
		tmp = (x * 0.0) - 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+292)) {
		tmp = (x * 0.0) - 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+292):
		tmp = (x * 0.0) - 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+292))
		tmp = Float64(Float64(x * 0.0) - 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+292)))
		tmp = (x * 0.0) - 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+292]], $MachinePrecision]], N[(N[(x * 0.0), $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 1e292 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 6.2%

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

      1. add-cube-cbrt 6.2%

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

      2. log-prod 6.2%

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

      3. pow2 6.2%

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

   3. Applied egg-rr 6.2%

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

      1. log-pow 6.2%

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

      2. distribute-lft1-in 6.2%

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

      3. metadata-eval 6.2%

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

      4. *-commutative 6.2%

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

   5. Simplified 6.2%

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

      1. add-log-exp 6.2%

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

      2. exp-to-pow 6.2%

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

      3. pow3 6.2%

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

      4. add-cube-cbrt 6.2%

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

      5. add-sqr-sqrt 5.0%

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

      6. sqrt-unprod 5.7%

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

      7. clear-num 5.7%

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

      8. neg-log 5.7%

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

      9. clear-num 5.7%

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

      10. neg-log 10.4%

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

      11. sqr-neg 10.4%

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

      12. sqrt-unprod 0.8%

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

      13. add-sqr-sqrt 2.9%

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

      14. clear-num 1.4%

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

      15. add-sqr-sqrt 1.4%

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

      16. associate-/r* 1.4%

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

      17. metadata-eval 1.4%

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

      18. sqrt-div 1.4%

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

      19. clear-num 1.4%

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

      20. add-exp-log 1.4%

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

      21. add-sqr-sqrt 0.6%

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

      22. sqrt-unprod 0.6%

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

   7. Applied egg-rr 0.0%

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

      1. +-inverses 46.6%

         \[\leadsto x \cdot \color{blue}{0} - z \]

   9. Simplified 46.6%

      \[\leadsto x \cdot \color{blue}{0} - z \]

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 1e292

   1. Initial program 99.5%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log \left(\frac{x}{y}\right) \leq -\infty \lor \neg \left(x \cdot \log \left(\frac{x}{y}\right) \leq 10^{+292}\right):\\ \;\;\;\;x \cdot 0 - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \end{array} \]

Alternative 3: 90.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.25 \cdot 10^{-212}:\\ \;\;\;\;x \cdot \left(-\log \left(\frac{y}{x}\right)\right) - z\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-307}:\\ \;\;\;\;x \cdot 0 - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -2.25e-212)
   (- (* x (- (log (/ y x)))) z)
   (if (<= x -2e-307) (- (* x 0.0) z) (- (* x (- (log x) (log y))) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.25e-212) {
		tmp = (x * -log((y / x))) - z;
	} else if (x <= -2e-307) {
		tmp = (x * 0.0) - 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 <= (-2.25d-212)) then
        tmp = (x * -log((y / x))) - z
    else if (x <= (-2d-307)) then
        tmp = (x * 0.0d0) - 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 <= -2.25e-212) {
		tmp = (x * -Math.log((y / x))) - z;
	} else if (x <= -2e-307) {
		tmp = (x * 0.0) - z;
	} else {
		tmp = (x * (Math.log(x) - Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -2.25e-212:
		tmp = (x * -math.log((y / x))) - z
	elif x <= -2e-307:
		tmp = (x * 0.0) - z
	else:
		tmp = (x * (math.log(x) - math.log(y))) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -2.25e-212)
		tmp = Float64(Float64(x * Float64(-log(Float64(y / x)))) - z);
	elseif (x <= -2e-307)
		tmp = Float64(Float64(x * 0.0) - 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 <= -2.25e-212)
		tmp = (x * -log((y / x))) - z;
	elseif (x <= -2e-307)
		tmp = (x * 0.0) - z;
	else
		tmp = (x * (log(x) - log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -2.25e-212], N[(N[(x * (-N[Log[N[(y / x), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[x, -2e-307], N[(N[(x * 0.0), $MachinePrecision] - z), $MachinePrecision], 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 < -2.2499999999999999e-212

   1. Initial program 83.7%

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

      1. clear-num 82.8%

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

      2. log-rec 83.9%

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

   3. Applied egg-rr 83.9%

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

   if -2.2499999999999999e-212 < x < -1.99999999999999982e-307

   1. Initial program 53.7%

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

      1. add-cube-cbrt 53.7%

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

      2. log-prod 53.7%

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

      3. pow2 53.7%

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

   3. Applied egg-rr 53.7%

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

      1. log-pow 53.7%

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

      2. distribute-lft1-in 53.7%

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

      3. metadata-eval 53.7%

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

      4. *-commutative 53.7%

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

   5. Simplified 53.7%

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

      1. add-log-exp 53.7%

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

      2. exp-to-pow 53.7%

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

      3. pow3 53.7%

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

      4. add-cube-cbrt 53.7%

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

      5. add-sqr-sqrt 11.1%

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

      6. sqrt-unprod 47.3%

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

      7. clear-num 47.1%

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

      8. neg-log 47.1%

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

      9. clear-num 47.1%

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

      10. neg-log 47.1%

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

      11. sqr-neg 47.1%

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

      12. sqrt-unprod 36.0%

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

      13. add-sqr-sqrt 47.1%

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

      14. clear-num 47.1%

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

      15. add-sqr-sqrt 47.1%

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

      16. associate-/r* 47.1%

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

      17. metadata-eval 47.1%

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

      18. sqrt-div 47.1%

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

      19. clear-num 47.1%

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

      20. add-exp-log 47.1%

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

      21. add-sqr-sqrt 11.1%

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

      22. sqrt-unprod 46.2%

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

   7. Applied egg-rr 46.5%

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

      1. +-inverses 85.8%

         \[\leadsto x \cdot \color{blue}{0} - z \]

   9. Simplified 85.8%

      \[\leadsto x \cdot \color{blue}{0} - z \]

   if -1.99999999999999982e-307 < x

   1. Initial program 74.7%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.25 \cdot 10^{-212}:\\ \;\;\;\;x \cdot \left(-\log \left(\frac{y}{x}\right)\right) - z\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-307}:\\ \;\;\;\;x \cdot 0 - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \]

Alternative 4: 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 79.3%

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

      1. frac-2neg 79.3%

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

      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 74.7%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 99.5%

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

Alternative 5: 50.8% accurate, 21.4× speedup

Math

\[\begin{array}{l} \\ x \cdot 0 - z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- (* x 0.0) z))

C

double code(double x, double y, double z) {
	return (x * 0.0) - 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 * 0.0d0) - z
end function

Java

public static double code(double x, double y, double z) {
	return (x * 0.0) - z;
}

Python

def code(x, y, z):
	return (x * 0.0) - z

Julia

function code(x, y, z)
	return Float64(Float64(x * 0.0) - z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * 0.0) - z;
end

Wolfram

code[x_, y_, z_] := N[(N[(x * 0.0), $MachinePrecision] - z), $MachinePrecision]

Derivation

1. Initial program 76.9%

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

   1. add-cube-cbrt 76.9%

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

   2. log-prod 76.8%

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

   3. pow2 76.8%

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

3. Applied egg-rr 76.8%

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

   1. log-pow 76.8%

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

   2. distribute-lft1-in 76.8%

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

   3. metadata-eval 76.8%

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

   4. *-commutative 76.8%

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

5. Simplified 76.8%

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

   1. add-log-exp 76.8%

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

   2. exp-to-pow 76.9%

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

   3. pow3 76.9%

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

   4. add-cube-cbrt 76.9%

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

   5. add-sqr-sqrt 36.9%

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

   6. sqrt-unprod 60.5%

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

   7. clear-num 59.7%

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

   8. neg-log 59.7%

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

   9. clear-num 59.7%

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

   10. neg-log 60.9%

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

   11. sqr-neg 60.9%

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

   12. sqrt-unprod 22.7%

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

   13. add-sqr-sqrt 38.3%

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

   14. clear-num 37.9%

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

   15. add-sqr-sqrt 37.9%

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

   16. associate-/r* 37.9%

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

   17. metadata-eval 37.9%

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

   18. sqrt-div 37.9%

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

   19. clear-num 37.9%

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

   20. add-exp-log 37.9%

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

   21. add-sqr-sqrt 15.2%

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

   22. sqrt-unprod 38.4%

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

7. Applied egg-rr 39.4%

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

   1. +-inverses 50.7%

      \[\leadsto x \cdot \color{blue}{0} - z \]

9. Simplified 50.7%

   \[\leadsto x \cdot \color{blue}{0} - z \]

10. Final simplification 50.7%

    \[\leadsto x \cdot 0 - z \]

Developer target: 88.4% 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 2023192 
(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))