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

Percentage Accurate: 77.7% → 99.7%

Time: 24.4s

Alternatives: 8

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: 77.7% 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 74.9%

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

   1. add-cube-cbrt 74.8%

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

   2. log-prod 74.8%

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

   3. pow2 74.8%

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

   4. metadata-eval 74.8%

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

   5. log-pow 74.8%

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

   6. metadata-eval 74.8%

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

4. Applied egg-rr 74.8%

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

   1. distribute-lft1-in 74.8%

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

   2. metadata-eval 74.8%

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

   3. *-commutative 74.8%

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

6. Simplified 74.8%

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

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

8. Applied egg-rr 99.7%

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

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

10. Simplified 99.7%

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

11. Final simplification 99.7%

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

Alternative 2: 83.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ t_1 := x \cdot \log \left(x \cdot y\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;t\_1 - z\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;t\_0 - z\\ \mathbf{else}:\\ \;\;\;\;\left|t\_1\right| - z\\ \end{array} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 4.5%

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

      1. add-cube-cbrt 4.5%

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

      2. log-prod 4.5%

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

      3. pow2 4.5%

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

      4. metadata-eval 4.5%

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

      5. log-pow 4.5%

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

      6. metadata-eval 4.5%

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

   4. Applied egg-rr 4.5%

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

      1. distribute-lft1-in 4.5%

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

      2. metadata-eval 4.5%

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

      3. *-commutative 4.5%

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

   6. Simplified 4.5%

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

      1. add-log-exp 4.5%

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

      2. exp-to-pow 4.5%

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

      3. pow3 4.5%

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

      4. add-cube-cbrt 4.5%

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

      5. log-div 51.7%

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

      6. add-cube-cbrt 51.7%

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

      7. unpow2 51.7%

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

      8. fma-neg 51.7%

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

      9. add-log-exp 47.0%

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

      10. add0 47.0%

          \[\leadsto \color{blue}{\left(\log \left(e^{x \cdot \mathsf{fma}\left({\left(\sqrt[3]{\log x}\right)}^{2}, \sqrt[3]{\log x}, -\log y\right)}\right) + 0\right)} - z \]

   8. Applied egg-rr 49.7%

      \[\leadsto \color{blue}{\left(\log \left({\left(x \cdot y\right)}^{x}\right) + 0\right)} - z \]
   9. Step-by-step derivation

      1. log-pow 55.5%

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

      2. add0 55.5%

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

   10. Simplified 55.5%

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

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

   1. Initial program 83.8%

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

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

   1. Initial program 74.9%

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

      1. add-cube-cbrt 74.8%

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

      2. log-prod 74.8%

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

      3. pow2 74.8%

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

      4. metadata-eval 74.8%

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

      5. log-pow 74.8%

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

      6. metadata-eval 74.8%

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

   4. Applied egg-rr 74.8%

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

      1. distribute-lft1-in 74.8%

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

      2. metadata-eval 74.8%

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

      3. *-commutative 74.8%

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

   6. Simplified 74.8%

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

      1. add-log-exp 74.8%

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

      2. exp-to-pow 74.8%

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

      3. pow3 74.8%

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

      4. add-cube-cbrt 74.9%

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

      5. log-div 48.2%

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

      6. add-cube-cbrt 47.9%

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

      7. unpow2 47.9%

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

      8. fma-neg 47.9%

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

      9. add-sqr-sqrt 25.8%

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

      10. sqrt-unprod 26.7%

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

      11. pow2 26.7%

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

   8. Applied egg-rr 28.5%

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

      1. unpow2 28.5%

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

      2. rem-sqrt-square 28.5%

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

      3. log-pow 40.7%

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

   10. Simplified 40.7%

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

4. Final simplification 80.6%

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

Alternative 3: 83.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ t_1 := \log \left(x \cdot y\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;x \cdot \left|t\_1\right| - z\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;t\_0 - z\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot t\_1\right| - z\\ \end{array} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 4.5%

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

      1. add-cube-cbrt 4.5%

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

      2. log-prod 4.5%

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

      3. pow2 4.5%

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

      4. metadata-eval 4.5%

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

      5. log-pow 4.5%

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

      6. metadata-eval 4.5%

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

   4. Applied egg-rr 4.5%

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

      1. distribute-lft1-in 4.5%

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

      2. metadata-eval 4.5%

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

      3. *-commutative 4.5%

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

   6. Simplified 4.5%

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

      1. add-log-exp 4.5%

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

      2. exp-to-pow 4.5%

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

      3. pow3 4.5%

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

      4. add-cube-cbrt 4.5%

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

      5. add-sqr-sqrt 3.2%

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

      6. pow1/2 3.2%

         \[\leadsto x \cdot \left(\color{blue}{{\log \left(\frac{x}{y}\right)}^{0.5}} \cdot \sqrt{\log \left(\frac{x}{y}\right)}\right) - z \]

      7. pow1/2 4.4%

         \[\leadsto x \cdot \left({\log \left(\frac{x}{y}\right)}^{0.5} \cdot \color{blue}{{\log \left(\frac{x}{y}\right)}^{0.5}}\right) - z \]

      8. pow-prod-down 4.4%

         \[\leadsto x \cdot \color{blue}{{\left(\log \left(\frac{x}{y}\right) \cdot \log \left(\frac{x}{y}\right)\right)}^{0.5}} - z \]

   8. Applied egg-rr 57.6%

      \[\leadsto x \cdot \color{blue}{{\left({\log \left(x \cdot y\right)}^{2}\right)}^{0.5}} - z \]
   9. Step-by-step derivation

      1. unpow1/2 57.6%

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

      2. unpow2 57.6%

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

      3. rem-sqrt-square 57.6%

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

   10. Simplified 57.6%

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

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

   1. Initial program 83.8%

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

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

   1. Initial program 74.9%

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

      1. add-cube-cbrt 74.8%

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

      2. log-prod 74.8%

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

      3. pow2 74.8%

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

      4. metadata-eval 74.8%

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

      5. log-pow 74.8%

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

      6. metadata-eval 74.8%

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

   4. Applied egg-rr 74.8%

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

      1. distribute-lft1-in 74.8%

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

      2. metadata-eval 74.8%

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

      3. *-commutative 74.8%

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

   6. Simplified 74.8%

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

      1. add-log-exp 74.8%

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

      2. exp-to-pow 74.8%

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

      3. pow3 74.8%

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

      4. add-cube-cbrt 74.9%

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

      5. log-div 48.2%

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

      6. add-cube-cbrt 47.9%

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

      7. unpow2 47.9%

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

      8. fma-neg 47.9%

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

      9. add-sqr-sqrt 25.8%

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

      10. sqrt-unprod 26.7%

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

      11. pow2 26.7%

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

   8. Applied egg-rr 28.5%

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

      1. unpow2 28.5%

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

      2. rem-sqrt-square 28.5%

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

      3. log-pow 40.7%

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

   10. Simplified 40.7%

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

4. Final simplification 80.9%

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

Alternative 4: 83.0% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 INFINITY)))
     (- (* 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 <= ((double) INFINITY))) {
		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 <= Double.POSITIVE_INFINITY)) {
		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 <= math.inf):
		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 <= Inf))
		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 <= Inf)))
		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, Infinity]], $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 +inf.0 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 4.5%

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

      1. add-cube-cbrt 4.5%

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

      2. log-prod 4.5%

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

      3. pow2 4.5%

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

      4. metadata-eval 4.5%

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

      5. log-pow 4.5%

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

      6. metadata-eval 4.5%

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

   4. Applied egg-rr 4.5%

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

      1. distribute-lft1-in 4.5%

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

      2. metadata-eval 4.5%

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

      3. *-commutative 4.5%

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

   6. Simplified 4.5%

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

      1. add-log-exp 4.5%

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

      2. exp-to-pow 4.5%

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

      3. pow3 4.5%

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

      4. add-cube-cbrt 4.5%

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

      5. log-div 51.7%

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

      6. add-cube-cbrt 51.7%

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

      7. unpow2 51.7%

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

      8. fma-neg 51.7%

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

      9. add-log-exp 47.0%

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

      10. add0 47.0%

          \[\leadsto \color{blue}{\left(\log \left(e^{x \cdot \mathsf{fma}\left({\left(\sqrt[3]{\log x}\right)}^{2}, \sqrt[3]{\log x}, -\log y\right)}\right) + 0\right)} - z \]

   8. Applied egg-rr 49.7%

      \[\leadsto \color{blue}{\left(\log \left({\left(x \cdot y\right)}^{x}\right) + 0\right)} - z \]
   9. Step-by-step derivation

      1. log-pow 55.5%

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

      2. add0 55.5%

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

   10. Simplified 55.5%

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

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

   1. Initial program 83.8%

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

4. Final simplification 80.6%

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

Alternative 5: 91.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.4 \cdot 10^{-195}:\\ \;\;\;\;\left(-z\right) - x \cdot \log \left(\frac{y}{x}\right)\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-309}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -8.4e-195)
   (- (- z) (* x (log (/ y x))))
   (if (<= x -5e-309) (- z) (- (* x (- (log x) (log y))) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -8.4e-195) {
		tmp = -z - (x * log((y / x)));
	} else if (x <= -5e-309) {
		tmp = -z;
	} else {
		tmp = (x * (log(x) - log(y))) - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-8.4d-195)) then
        tmp = -z - (x * log((y / x)))
    else if (x <= (-5d-309)) then
        tmp = -z
    else
        tmp = (x * (log(x) - log(y))) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -8.4e-195) {
		tmp = -z - (x * Math.log((y / x)));
	} else if (x <= -5e-309) {
		tmp = -z;
	} else {
		tmp = (x * (Math.log(x) - Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -8.4e-195:
		tmp = -z - (x * math.log((y / x)))
	elif x <= -5e-309:
		tmp = -z
	else:
		tmp = (x * (math.log(x) - math.log(y))) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -8.4e-195)
		tmp = Float64(Float64(-z) - Float64(x * log(Float64(y / x))));
	elseif (x <= -5e-309)
		tmp = Float64(-z);
	else
		tmp = Float64(Float64(x * Float64(log(x) - log(y))) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -8.4e-195)
		tmp = -z - (x * log((y / x)));
	elseif (x <= -5e-309)
		tmp = -z;
	else
		tmp = (x * (log(x) - log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -8.4e-195], N[((-z) - N[(x * N[Log[N[(y / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -5e-309], (-z), N[(N[(x * N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.4e-195

   1. Initial program 83.1%

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

      1. clear-num 83.1%

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

      2. neg-log 84.4%

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

   4. Applied egg-rr 84.4%

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

   if -8.4e-195 < x < -4.9999999999999995e-309

   1. Initial program 41.3%

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

      1. add-cube-cbrt 41.3%

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

      2. log-prod 41.3%

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

      3. pow2 41.3%

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

      4. metadata-eval 41.3%

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

      5. log-pow 41.3%

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

      6. metadata-eval 41.3%

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

   4. Applied egg-rr 41.3%

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

      1. distribute-lft1-in 41.3%

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

      2. metadata-eval 41.3%

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

      3. *-commutative 41.3%

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

   6. Simplified 41.3%

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

      1. add-log-exp 41.3%

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

      2. exp-to-pow 41.3%

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

      3. pow3 41.3%

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

      4. add-cube-cbrt 41.3%

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

      5. log-div 0.0%

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

      6. add-cube-cbrt 0.0%

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

      7. unpow2 0.0%

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

      8. fma-neg 0.0%

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

      9. add-log-exp 0.0%

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

      10. add0 0.0%

          \[\leadsto \color{blue}{\left(\log \left(e^{x \cdot \mathsf{fma}\left({\left(\sqrt[3]{\log x}\right)}^{2}, \sqrt[3]{\log x}, -\log y\right)}\right) + 0\right)} - z \]

   8. Applied egg-rr 76.2%

      \[\leadsto \color{blue}{\left(\log \left({\left(x \cdot y\right)}^{x}\right) + 0\right)} - z \]
   9. Step-by-step derivation

      1. log-pow 76.6%

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

      2. add0 76.6%

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

   10. Simplified 76.6%

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

   11. Taylor expanded in x around 0 90.6%

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

       1. neg-mul-1 90.6%

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

   13. Simplified 90.6%

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

   if -4.9999999999999995e-309 < x

   1. Initial program 72.8%

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

      1. log-div 99.4%

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

   4. Applied egg-rr 99.4%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.4 \cdot 10^{-195}:\\ \;\;\;\;\left(-z\right) - x \cdot \log \left(\frac{y}{x}\right)\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-309}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -4.999999999999985e-310

   1. Initial program 76.7%

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

      1. frac-2neg 76.7%

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

      2. log-div 99.4%

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

   4. Applied egg-rr 99.4%

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

   if -4.999999999999985e-310 < x

   1. Initial program 72.8%

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

      1. log-div 99.4%

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

   4. Applied egg-rr 99.4%

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

4. Final simplification 99.4%

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

Alternative 7: 50.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{-29}:\\ \;\;\;\;x \cdot \log \left(x \cdot y\right) - z\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -4.8e-29) (- (* x (log (* x y))) z) (- z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -4.8e-29) {
		tmp = (x * log((x * y))) - z;
	} else {
		tmp = -z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-4.8d-29)) then
        tmp = (x * log((x * y))) - z
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -4.8e-29) {
		tmp = (x * Math.log((x * y))) - z;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -4.8e-29:
		tmp = (x * math.log((x * y))) - z
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -4.8e-29)
		tmp = Float64(Float64(x * log(Float64(x * y))) - z);
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -4.8e-29)
		tmp = (x * log((x * y))) - z;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -4.8e-29], N[(N[(x * N[Log[N[(x * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], (-z)]

Derivation

1. Split input into 2 regimes

2. if x < -4.79999999999999984e-29

   1. Initial program 82.8%

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

      1. add-cube-cbrt 82.7%

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

      2. log-prod 82.7%

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

      3. pow2 82.7%

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

      4. metadata-eval 82.7%

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

      5. log-pow 82.7%

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

      6. metadata-eval 82.7%

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

   4. Applied egg-rr 82.7%

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

      1. distribute-lft1-in 82.7%

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

      2. metadata-eval 82.7%

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

      3. *-commutative 82.7%

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

   6. Simplified 82.7%

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

      1. add-log-exp 82.7%

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

      2. exp-to-pow 82.7%

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

      3. pow3 82.7%

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

      4. add-cube-cbrt 82.8%

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

      5. log-div 0.0%

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

      6. add-cube-cbrt 0.0%

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

      7. unpow2 0.0%

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

      8. fma-neg 0.0%

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

      9. add-log-exp 0.0%

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

      10. add0 0.0%

          \[\leadsto \color{blue}{\left(\log \left(e^{x \cdot \mathsf{fma}\left({\left(\sqrt[3]{\log x}\right)}^{2}, \sqrt[3]{\log x}, -\log y\right)}\right) + 0\right)} - z \]

   8. Applied egg-rr 5.5%

      \[\leadsto \color{blue}{\left(\log \left({\left(x \cdot y\right)}^{x}\right) + 0\right)} - z \]
   9. Step-by-step derivation

      1. log-pow 30.5%

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

      2. add0 30.5%

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

   10. Simplified 30.5%

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

   if -4.79999999999999984e-29 < x

   1. Initial program 71.4%

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

      1. add-cube-cbrt 71.4%

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

      2. log-prod 71.4%

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

      3. pow2 71.4%

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

      4. metadata-eval 71.4%

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

      5. log-pow 71.4%

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

      6. metadata-eval 71.4%

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

   4. Applied egg-rr 71.4%

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

      1. distribute-lft1-in 71.4%

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

      2. metadata-eval 71.4%

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

      3. *-commutative 71.4%

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

   6. Simplified 71.4%

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

      1. add-log-exp 71.4%

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

      2. exp-to-pow 71.4%

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

      3. pow3 71.4%

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

      4. add-cube-cbrt 71.4%

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

      5. log-div 68.9%

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

      6. add-cube-cbrt 68.5%

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

      7. unpow2 68.5%

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

      8. fma-neg 68.5%

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

      9. add-log-exp 26.8%

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

      10. add0 26.8%

          \[\leadsto \color{blue}{\left(\log \left(e^{x \cdot \mathsf{fma}\left({\left(\sqrt[3]{\log x}\right)}^{2}, \sqrt[3]{\log x}, -\log y\right)}\right) + 0\right)} - z \]

   8. Applied egg-rr 38.5%

      \[\leadsto \color{blue}{\left(\log \left({\left(x \cdot y\right)}^{x}\right) + 0\right)} - z \]
   9. Step-by-step derivation

      1. log-pow 45.5%

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

      2. add0 45.5%

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

   10. Simplified 45.5%

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

   11. Taylor expanded in x around 0 53.6%

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

       1. neg-mul-1 53.6%

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

   13. Simplified 53.6%

       \[\leadsto \color{blue}{-z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{-29}:\\ \;\;\;\;x \cdot \log \left(x \cdot y\right) - z\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 50.0% 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 74.9%

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

   1. add-cube-cbrt 74.8%

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

   2. log-prod 74.8%

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

   3. pow2 74.8%

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

   4. metadata-eval 74.8%

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

   5. log-pow 74.8%

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

   6. metadata-eval 74.8%

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

4. Applied egg-rr 74.8%

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

   1. distribute-lft1-in 74.8%

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

   2. metadata-eval 74.8%

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

   3. *-commutative 74.8%

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

6. Simplified 74.8%

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

   1. add-log-exp 74.8%

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

   2. exp-to-pow 74.8%

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

   3. pow3 74.8%

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

   4. add-cube-cbrt 74.9%

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

   5. log-div 48.2%

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

   6. add-cube-cbrt 47.9%

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

   7. unpow2 47.9%

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

   8. fma-neg 47.9%

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

   9. add-log-exp 18.7%

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

   10. add0 18.7%

       \[\leadsto \color{blue}{\left(\log \left(e^{x \cdot \mathsf{fma}\left({\left(\sqrt[3]{\log x}\right)}^{2}, \sqrt[3]{\log x}, -\log y\right)}\right) + 0\right)} - z \]

8. Applied egg-rr 28.6%

   \[\leadsto \color{blue}{\left(\log \left({\left(x \cdot y\right)}^{x}\right) + 0\right)} - z \]
9. Step-by-step derivation

   1. log-pow 41.0%

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

   2. add0 41.0%

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

10. Simplified 41.0%

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

11. Taylor expanded in x around 0 44.9%

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

    1. neg-mul-1 44.9%

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

13. Simplified 44.9%

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

14. Final simplification 44.9%

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

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 2024034 
(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))