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

Percentage Accurate: 77.0% → 99.7%

Time: 14.1s

Alternatives: 10

Speedup: 0.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 74.5%

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

   1. add-cube-cbrt 74.5%

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

   2. log-prod 74.5%

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

   3. pow2 74.5%

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

4. Applied egg-rr 74.5%

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

   1. log-pow 74.5%

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

   2. distribute-lft1-in 74.5%

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

   3. metadata-eval 74.5%

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

6. Simplified 74.5%

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

   1. cbrt-div 99.7%

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

   2. div-inv 99.7%

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

8. Applied egg-rr 99.7%

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

   1. associate-*r/ 99.7%

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

   2. *-rgt-identity 99.7%

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

10. Simplified 99.7%

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

11. Final simplification 99.7%

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

Alternative 2: 86.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;x \cdot \log \left(x \cdot y\right) - z\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+273}:\\ \;\;\;\;t\_0 - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))))
   (if (<= t_0 (- INFINITY))
     (- (* x (log (* x y))) z)
     (if (<= t_0 4e+273) (- t_0 z) (* x (- (log x) (log y)))))))

C

double code(double x, double y, double z) {
	double t_0 = x * log((x / y));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (x * log((x * y))) - z;
	} else if (t_0 <= 4e+273) {
		tmp = t_0 - z;
	} else {
		tmp = x * (log(x) - log(y));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = x * Math.log((x / y));
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = (x * Math.log((x * y))) - z;
	} else if (t_0 <= 4e+273) {
		tmp = t_0 - z;
	} else {
		tmp = x * (Math.log(x) - Math.log(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.log((x / y))
	tmp = 0
	if t_0 <= -math.inf:
		tmp = (x * math.log((x * y))) - z
	elif t_0 <= 4e+273:
		tmp = t_0 - z
	else:
		tmp = x * (math.log(x) - math.log(y))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * log(Float64(x / y)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(x * log(Float64(x * y))) - z);
	elseif (t_0 <= 4e+273)
		tmp = Float64(t_0 - z);
	else
		tmp = Float64(x * Float64(log(x) - log(y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * log((x / y));
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = (x * log((x * y))) - z;
	elseif (t_0 <= 4e+273)
		tmp = t_0 - z;
	else
		tmp = x * (log(x) - log(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(x * N[Log[N[(x * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[t$95$0, 4e+273], N[(t$95$0 - z), $MachinePrecision], N[(x * N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 4.8%

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

      1. add-cube-cbrt 4.8%

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

      2. log-prod 4.8%

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

      3. pow2 4.8%

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

   4. Applied egg-rr 4.8%

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

      1. log-pow 4.8%

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

      2. distribute-lft1-in 4.8%

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

      3. metadata-eval 4.8%

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

   6. Simplified 4.8%

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

      1. *-commutative 4.8%

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

      2. add-log-exp 4.8%

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

      3. exp-to-pow 4.8%

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

      4. pow3 4.8%

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

      5. add-cube-cbrt 4.8%

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

      6. frac-2neg 4.8%

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

      7. diff-log 56.9%

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

      8. sub-neg 56.9%

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

      9. distribute-rgt-in 56.9%

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

      10. add-sqr-sqrt 56.9%

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

      11. sqrt-unprod 19.8%

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

      12. sqr-neg 19.8%

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

      13. sqrt-unprod 0.0%

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

      14. add-sqr-sqrt 0.0%

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

   8. Applied egg-rr 42.9%

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

      1. distribute-rgt-out 42.9%

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

      2. sub-neg 42.9%

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

      3. log-div 4.8%

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

      4. *-commutative 4.8%

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

      5. log-div 42.9%

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

      6. sub-neg 42.9%

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

      7. add-log-exp 42.9%

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

      8. sum-log 1.3%

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

      9. add-sqr-sqrt 0.0%

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

      10. sqrt-unprod 39.6%

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

      11. sqr-neg 39.6%

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

      12. sqrt-unprod 39.6%

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

      13. add-sqr-sqrt 39.6%

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

      14. add-exp-log 58.9%

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

   10. Applied egg-rr 58.9%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 3.99999999999999978e273

   1. Initial program 99.6%

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

   if 3.99999999999999978e273 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 15.6%

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

   3. Taylor expanded in z around 0 15.6%

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

      1. log-div 55.4%

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

   5. Applied egg-rr 55.4%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log \left(\frac{x}{y}\right) \leq -\infty:\\ \;\;\;\;x \cdot \log \left(x \cdot y\right) - z\\ \mathbf{elif}\;x \cdot \log \left(\frac{x}{y}\right) \leq 4 \cdot 10^{+273}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;t\_0 \leq -\infty \lor \neg \left(t\_0 \leq 10^{+306}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;t\_0 - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 1e+306))) (- 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 = -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 = -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 = -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(-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 = -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]], (-z), N[(t$95$0 - z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 6.5%

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

      1. remove-double-neg 6.5%

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

      2. sub-neg 6.5%

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

      3. distribute-neg-in 6.5%

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

      4. sub-neg 6.5%

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

      5. distribute-rgt-neg-in 6.5%

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

      6. fma-neg 6.5%

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

      7. log-div 56.5%

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

      8. sub-neg 56.5%

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

      9. distribute-neg-in 56.5%

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

      10. remove-double-neg 56.5%

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

      11. +-commutative 56.5%

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

      12. sub-neg 56.5%

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

      13. log-div 13.5%

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

      14. remove-double-neg 13.5%

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

   3. Simplified 13.5%

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

   5. Taylor expanded in x around 0 46.1%

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

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

   1. Initial program 99.6%

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

4. Final simplification 85.2%

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

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 6.5%

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

      1. add-cube-cbrt 6.5%

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

      2. log-prod 6.5%

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

      3. pow2 6.5%

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

   4. Applied egg-rr 6.5%

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

      1. log-pow 6.5%

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

      2. distribute-lft1-in 6.5%

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

      3. metadata-eval 6.5%

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

   6. Simplified 6.5%

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

      1. *-commutative 6.5%

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

      2. add-log-exp 6.5%

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

      3. exp-to-pow 6.5%

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

      4. pow3 6.5%

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

      5. add-cube-cbrt 6.5%

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

      6. frac-2neg 6.5%

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

      7. diff-log 43.3%

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

      8. sub-neg 43.3%

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

      9. distribute-rgt-in 43.3%

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

      10. add-sqr-sqrt 43.3%

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

      11. sqrt-unprod 17.5%

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

      12. sqr-neg 17.5%

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

      13. sqrt-unprod 0.0%

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

      14. add-sqr-sqrt 0.0%

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

   8. Applied egg-rr 56.5%

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

      1. distribute-rgt-out 56.5%

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

      2. sub-neg 56.5%

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

      3. log-div 6.5%

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

      4. *-commutative 6.5%

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

      5. log-div 56.5%

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

      6. sub-neg 56.5%

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

      7. add-log-exp 56.5%

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

      8. sum-log 4.2%

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

      9. add-sqr-sqrt 3.6%

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

      10. sqrt-unprod 23.6%

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

      11. sqr-neg 23.6%

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

      12. sqrt-unprod 20.1%

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

      13. add-sqr-sqrt 32.3%

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

      14. add-exp-log 50.9%

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

   10. Applied egg-rr 50.9%

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

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

   1. Initial program 99.6%

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

4. Final simplification 86.5%

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

Alternative 5: 93.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{+215}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right)\\ \mathbf{elif}\;x \leq -9 \cdot 10^{-175}:\\ \;\;\;\;-\mathsf{fma}\left(x, \log \left(\frac{y}{x}\right), z\right)\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.5e+215)
   (* x (- (log (- x)) (log (- y))))
   (if (<= x -9e-175)
     (- (fma x (log (/ y x)) z))
     (if (<= x -5e-310) (- z) (- (* x (- (log x) (log y))) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.5e+215) {
		tmp = x * (log(-x) - log(-y));
	} else if (x <= -9e-175) {
		tmp = -fma(x, log((y / x)), z);
	} else if (x <= -5e-310) {
		tmp = -z;
	} else {
		tmp = (x * (log(x) - log(y))) - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.5e+215)
		tmp = Float64(x * Float64(log(Float64(-x)) - log(Float64(-y))));
	elseif (x <= -9e-175)
		tmp = Float64(-fma(x, log(Float64(y / x)), z));
	elseif (x <= -5e-310)
		tmp = Float64(-z);
	else
		tmp = Float64(Float64(x * Float64(log(x) - log(y))) - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.5e+215], N[(x * N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -9e-175], (-N[(x * N[Log[N[(y / x), $MachinePrecision]], $MachinePrecision] + z), $MachinePrecision]), If[LessEqual[x, -5e-310], (-z), N[(N[(x * N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.5e215

   1. Initial program 58.9%

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

   3. Taylor expanded in z around 0 52.1%

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

      1. frac-2neg 52.1%

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

      2. log-div 88.4%

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

   5. Applied egg-rr 88.4%

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

   if -1.5e215 < x < -8.99999999999999996e-175

   1. Initial program 84.6%

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

      1. remove-double-neg 84.6%

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

      2. sub-neg 84.6%

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

      3. distribute-neg-in 84.6%

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

      4. sub-neg 84.6%

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

      5. distribute-rgt-neg-in 84.6%

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

      6. fma-neg 84.6%

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

      7. log-div 0.0%

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

      8. sub-neg 0.0%

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

      9. distribute-neg-in 0.0%

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

      10. remove-double-neg 0.0%

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

      11. +-commutative 0.0%

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

      12. sub-neg 0.0%

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

      13. log-div 87.3%

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

      14. remove-double-neg 87.3%

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

   3. Simplified 87.3%

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

   if -8.99999999999999996e-175 < x < -4.999999999999985e-310

   1. Initial program 78.2%

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

      1. remove-double-neg 78.2%

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

      2. sub-neg 78.2%

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

      3. distribute-neg-in 78.2%

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

      4. sub-neg 78.2%

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

      5. distribute-rgt-neg-in 78.2%

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

      6. fma-neg 78.2%

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

      7. log-div 0.0%

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

      8. sub-neg 0.0%

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

      9. distribute-neg-in 0.0%

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

      10. remove-double-neg 0.0%

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

      11. +-commutative 0.0%

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

      12. sub-neg 0.0%

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

      13. log-div 78.2%

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

      14. remove-double-neg 78.2%

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

   3. Simplified 78.2%

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

   5. Taylor expanded in x around 0 98.6%

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

   if -4.999999999999985e-310 < x

   1. Initial program 70.4%

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

      1. log-div 45.5%

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

   4. Applied egg-rr 99.6%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{+215}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right)\\ \mathbf{elif}\;x \leq -9 \cdot 10^{-175}:\\ \;\;\;\;-\mathsf{fma}\left(x, \log \left(\frac{y}{x}\right), z\right)\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-2d-310)) then
        tmp = (x * (log(-x) - log(-y))) - z
    else
        tmp = (x * (log(x) - log(y))) - z
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1.999999999999994e-310

   1. Initial program 78.3%

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

      1. frac-2neg 42.0%

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

      2. log-div 55.1%

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

   4. Applied egg-rr 99.4%

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

   if -1.999999999999994e-310 < y

   1. Initial program 70.4%

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

      1. log-div 45.5%

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

   4. Applied egg-rr 99.6%

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

4. Final simplification 99.5%

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

Alternative 7: 66.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.6 \cdot 10^{-25} \lor \neg \left(z \leq -3.9 \cdot 10^{-97} \lor \neg \left(z \leq -2.2 \cdot 10^{-126}\right) \land z \leq 2.6 \cdot 10^{-34}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -3.6e-25)
         (not
          (or (<= z -3.9e-97) (and (not (<= z -2.2e-126)) (<= z 2.6e-34)))))
   (- z)
   (* x (log (/ x y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -3.6e-25) || !((z <= -3.9e-97) || (!(z <= -2.2e-126) && (z <= 2.6e-34)))) {
		tmp = -z;
	} else {
		tmp = x * log((x / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-3.6d-25)) .or. (.not. (z <= (-3.9d-97)) .or. (.not. (z <= (-2.2d-126))) .and. (z <= 2.6d-34))) then
        tmp = -z
    else
        tmp = x * log((x / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -3.6e-25) || !((z <= -3.9e-97) || (!(z <= -2.2e-126) && (z <= 2.6e-34)))) {
		tmp = -z;
	} else {
		tmp = x * Math.log((x / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -3.6e-25) or not ((z <= -3.9e-97) or (not (z <= -2.2e-126) and (z <= 2.6e-34))):
		tmp = -z
	else:
		tmp = x * math.log((x / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -3.6e-25) || !((z <= -3.9e-97) || (!(z <= -2.2e-126) && (z <= 2.6e-34))))
		tmp = Float64(-z);
	else
		tmp = Float64(x * log(Float64(x / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -3.6e-25) || ~(((z <= -3.9e-97) || (~((z <= -2.2e-126)) && (z <= 2.6e-34)))))
		tmp = -z;
	else
		tmp = x * log((x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -3.6e-25], N[Not[Or[LessEqual[z, -3.9e-97], And[N[Not[LessEqual[z, -2.2e-126]], $MachinePrecision], LessEqual[z, 2.6e-34]]]], $MachinePrecision]], (-z), N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.5999999999999999e-25 or -3.8999999999999998e-97 < z < -2.20000000000000014e-126 or 2.5999999999999999e-34 < z

   1. Initial program 69.5%

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

      1. remove-double-neg 69.5%

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

      2. sub-neg 69.5%

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

      3. distribute-neg-in 69.5%

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

      4. sub-neg 69.5%

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

      5. distribute-rgt-neg-in 69.5%

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

      6. fma-neg 69.5%

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

      7. log-div 52.3%

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

      8. sub-neg 52.3%

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

      9. distribute-neg-in 52.3%

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

      10. remove-double-neg 52.3%

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

      11. +-commutative 52.3%

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

      12. sub-neg 52.3%

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

      13. log-div 72.7%

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

      14. remove-double-neg 72.7%

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

   3. Simplified 72.7%

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

   5. Taylor expanded in x around 0 74.2%

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

   if -3.5999999999999999e-25 < z < -3.8999999999999998e-97 or -2.20000000000000014e-126 < z < 2.5999999999999999e-34

   1. Initial program 80.6%

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

   3. Taylor expanded in z around 0 65.1%

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

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.6 \cdot 10^{-25} \lor \neg \left(z \leq -3.9 \cdot 10^{-97} \lor \neg \left(z \leq -2.2 \cdot 10^{-126}\right) \land z \leq 2.6 \cdot 10^{-34}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 66.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{-24}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq -4 \cdot 10^{-97}:\\ \;\;\;\;\left(-x\right) \cdot \log \left(\frac{y}{x}\right)\\ \mathbf{elif}\;z \leq -1.12 \cdot 10^{-125} \lor \neg \left(z \leq 9.5 \cdot 10^{-28}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2e-24)
   (- z)
   (if (<= z -4e-97)
     (* (- x) (log (/ y x)))
     (if (or (<= z -1.12e-125) (not (<= z 9.5e-28)))
       (- z)
       (* x (log (/ x y)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2e-24) {
		tmp = -z;
	} else if (z <= -4e-97) {
		tmp = -x * log((y / x));
	} else if ((z <= -1.12e-125) || !(z <= 9.5e-28)) {
		tmp = -z;
	} else {
		tmp = x * log((x / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-2d-24)) then
        tmp = -z
    else if (z <= (-4d-97)) then
        tmp = -x * log((y / x))
    else if ((z <= (-1.12d-125)) .or. (.not. (z <= 9.5d-28))) then
        tmp = -z
    else
        tmp = x * log((x / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -2e-24) {
		tmp = -z;
	} else if (z <= -4e-97) {
		tmp = -x * Math.log((y / x));
	} else if ((z <= -1.12e-125) || !(z <= 9.5e-28)) {
		tmp = -z;
	} else {
		tmp = x * Math.log((x / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -2e-24:
		tmp = -z
	elif z <= -4e-97:
		tmp = -x * math.log((y / x))
	elif (z <= -1.12e-125) or not (z <= 9.5e-28):
		tmp = -z
	else:
		tmp = x * math.log((x / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2e-24)
		tmp = Float64(-z);
	elseif (z <= -4e-97)
		tmp = Float64(Float64(-x) * log(Float64(y / x)));
	elseif ((z <= -1.12e-125) || !(z <= 9.5e-28))
		tmp = Float64(-z);
	else
		tmp = Float64(x * log(Float64(x / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -2e-24)
		tmp = -z;
	elseif (z <= -4e-97)
		tmp = -x * log((y / x));
	elseif ((z <= -1.12e-125) || ~((z <= 9.5e-28)))
		tmp = -z;
	else
		tmp = x * log((x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2e-24], (-z), If[LessEqual[z, -4e-97], N[((-x) * N[Log[N[(y / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[z, -1.12e-125], N[Not[LessEqual[z, 9.5e-28]], $MachinePrecision]], (-z), N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.99999999999999985e-24 or -4.00000000000000014e-97 < z < -1.11999999999999997e-125 or 9.50000000000000001e-28 < z

   1. Initial program 69.5%

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

      1. remove-double-neg 69.5%

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

      2. sub-neg 69.5%

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

      3. distribute-neg-in 69.5%

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

      4. sub-neg 69.5%

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

      5. distribute-rgt-neg-in 69.5%

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

      6. fma-neg 69.5%

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

      7. log-div 52.3%

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

      8. sub-neg 52.3%

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

      9. distribute-neg-in 52.3%

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

      10. remove-double-neg 52.3%

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

      11. +-commutative 52.3%

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

      12. sub-neg 52.3%

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

      13. log-div 72.7%

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

      14. remove-double-neg 72.7%

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

   3. Simplified 72.7%

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

   5. Taylor expanded in x around 0 74.2%

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

   if -1.99999999999999985e-24 < z < -4.00000000000000014e-97

   1. Initial program 90.9%

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

      1. remove-double-neg 90.9%

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

      2. sub-neg 90.9%

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

      3. distribute-neg-in 90.9%

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

      4. sub-neg 90.9%

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

      5. distribute-rgt-neg-in 90.9%

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

      6. fma-neg 90.9%

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

      7. log-div 27.1%

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

      8. sub-neg 27.1%

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

      9. distribute-neg-in 27.1%

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

      10. remove-double-neg 27.1%

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

      11. +-commutative 27.1%

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

      12. sub-neg 27.1%

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

      13. log-div 94.2%

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

      14. remove-double-neg 94.2%

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

   3. Simplified 94.2%

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

   5. Taylor expanded in x around inf 18.3%

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

      1. log-rec 18.3%

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

      2. sub-neg 18.3%

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

      3. log-div 85.2%

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

   7. Simplified 85.2%

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

   if -1.11999999999999997e-125 < z < 9.50000000000000001e-28

   1. Initial program 79.5%

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

   3. Taylor expanded in z around 0 63.3%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{-24}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq -4 \cdot 10^{-97}:\\ \;\;\;\;\left(-x\right) \cdot \log \left(\frac{y}{x}\right)\\ \mathbf{elif}\;z \leq -1.12 \cdot 10^{-125} \lor \neg \left(z \leq 9.5 \cdot 10^{-28}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 50.4% 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 74.5%

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

   1. remove-double-neg 74.5%

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

   2. sub-neg 74.5%

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

   3. distribute-neg-in 74.5%

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

   4. sub-neg 74.5%

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

   5. distribute-rgt-neg-in 74.5%

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

   6. fma-neg 74.5%

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

   7. log-div 48.3%

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

   8. sub-neg 48.3%

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

   9. distribute-neg-in 48.3%

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

   10. remove-double-neg 48.3%

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

   11. +-commutative 48.3%

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

   12. sub-neg 48.3%

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

   13. log-div 75.7%

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

   14. remove-double-neg 75.7%

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

3. Simplified 75.7%

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

5. Taylor expanded in x around 0 49.6%

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

6. Final simplification 49.6%

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

Alternative 10: 2.2% accurate, 107.0× 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 z
end

MATLAB

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

Wolfram

code[x_, y_, z_] := z

Derivation

1. Initial program 74.5%

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

   1. add-cube-cbrt 74.5%

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

   2. log-prod 74.5%

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

   3. pow2 74.5%

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

4. Applied egg-rr 74.5%

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

   1. log-pow 74.5%

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

   2. distribute-lft1-in 74.5%

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

   3. metadata-eval 74.5%

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

6. Simplified 74.5%

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

   1. fma-neg 74.4%

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

   2. *-commutative 74.4%

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

   3. add-log-exp 74.4%

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

   4. exp-to-pow 74.5%

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

   5. pow3 74.5%

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

   6. add-cube-cbrt 74.5%

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

   7. fma-neg 74.5%

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

   8. rem-cube-cbrt 73.0%

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

   9. sqr-pow 33.0%

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

   10. pow-prod-down 33.5%

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

8. Applied egg-rr 9.5%

   \[\leadsto \color{blue}{{\left(\sqrt[3]{{\left(\log \left({\left(\frac{x}{y}\right)}^{x}\right) - z\right)}^{2}}\right)}^{1.5}} \]
9. Step-by-step derivation

   1. log-pow 21.2%

      \[\leadsto {\left(\sqrt[3]{{\left(\color{blue}{x \cdot \log \left(\frac{x}{y}\right)} - z\right)}^{2}}\right)}^{1.5} \]

10. Simplified 21.2%

    \[\leadsto \color{blue}{{\left(\sqrt[3]{{\left(x \cdot \log \left(\frac{x}{y}\right) - z\right)}^{2}}\right)}^{1.5}} \]

11. Taylor expanded in x around 0 2.1%

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

12. Final simplification 2.1%

    \[\leadsto z \]
13. Add Preprocessing

Developer target: 87.9% 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 2024039 
(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))