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

Percentage Accurate: 76.9% → 99.7%

Time: 12.4s

Alternatives: 9

Speedup: 0.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.9% 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

\[\begin{array}{l} \\ 0 \cdot z + \left(3 \cdot \left(x \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right) - z\right) \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+ (* 0.0 z) (- (* 3.0 (* x (log (/ (cbrt x) (cbrt y))))) z)))

C

double code(double x, double y, double z) {
	return (0.0 * z) + ((3.0 * (x * log((cbrt(x) / cbrt(y))))) - z);
}

Java

public static double code(double x, double y, double z) {
	return (0.0 * z) + ((3.0 * (x * Math.log((Math.cbrt(x) / Math.cbrt(y))))) - z);
}

Julia

function code(x, y, z)
	return Float64(Float64(0.0 * z) + Float64(Float64(3.0 * Float64(x * log(Float64(cbrt(x) / cbrt(y))))) - z))
end

Wolfram

code[x_, y_, z_] := N[(N[(0.0 * z), $MachinePrecision] + N[(N[(3.0 * N[(x * N[Log[N[(N[Power[x, 1/3], $MachinePrecision] / N[Power[y, 1/3], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.7%

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

   1. add-sqr-sqrt 37.0%

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

   2. pow2 37.0%

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

4. Applied egg-rr 37.0%

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

   1. unpow2 37.0%

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

   2. add-sqr-sqrt 78.7%

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

   3. *-commutative 78.7%

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

   4. *-un-lft-identity 78.7%

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

   5. *-un-lft-identity 78.7%

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

   6. metadata-eval 78.7%

      \[\leadsto \left(\color{blue}{\left(3 \cdot 0.3333333333333333\right)} \cdot \log \left(\frac{x}{y}\right)\right) \cdot x - 1 \cdot z \]

   7. associate-*r* 78.5%

      \[\leadsto \color{blue}{\left(3 \cdot \left(0.3333333333333333 \cdot \log \left(\frac{x}{y}\right)\right)\right)} \cdot x - 1 \cdot z \]

   8. log-pow 78.5%

      \[\leadsto \left(3 \cdot \color{blue}{\log \left({\left(\frac{x}{y}\right)}^{0.3333333333333333}\right)}\right) \cdot x - 1 \cdot z \]

   9. unpow1/3 78.6%

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

   10. cbrt-undiv 99.6%

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

   11. associate-*l* 99.7%

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

   12. prod-diff 99.7%

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

   13. cbrt-undiv 78.6%

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

   14. *-commutative 78.6%

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

   15. *-un-lft-identity 78.6%

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

   16. *-commutative 78.6%

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

6. Applied egg-rr 78.6%

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

   1. fma-udef 78.6%

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

   2. associate-+l+ 78.6%

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

   3. *-commutative 78.6%

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

   4. associate-*r* 78.6%

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

   5. *-commutative 78.6%

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

   6. associate-+l+ 78.6%

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

   7. fma-udef 78.6%

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

   8. +-commutative 78.6%

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

   9. fma-udef 78.6%

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

   10. *-rgt-identity 78.6%

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

   11. neg-mul-1 78.6%

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

   12. distribute-lft1-in 78.6%

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

   13. metadata-eval 78.6%

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

   14. fma-neg 78.6%

       \[\leadsto 0 \cdot z + \color{blue}{\left(\left(x \cdot 3\right) \cdot \log \left(\sqrt[3]{\frac{x}{y}}\right) - z\right)} \]

   15. *-commutative 78.6%

       \[\leadsto 0 \cdot z + \left(\color{blue}{\left(3 \cdot x\right)} \cdot \log \left(\sqrt[3]{\frac{x}{y}}\right) - z\right) \]

   16. associate-*r* 78.6%

       \[\leadsto 0 \cdot z + \left(\color{blue}{3 \cdot \left(x \cdot \log \left(\sqrt[3]{\frac{x}{y}}\right)\right)} - z\right) \]

8. Simplified 78.6%

   \[\leadsto \color{blue}{0 \cdot z + \left(3 \cdot \left(x \cdot \log \left(\sqrt[3]{\frac{x}{y}}\right)\right) - z\right)} \]
9. Step-by-step derivation

   1. cbrt-div 99.7%

      \[\leadsto 0 \cdot z + \left(3 \cdot \left(x \cdot \log \color{blue}{\left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)}\right) - z\right) \]

   2. div-inv 99.7%

      \[\leadsto 0 \cdot z + \left(3 \cdot \left(x \cdot \log \color{blue}{\left(\sqrt[3]{x} \cdot \frac{1}{\sqrt[3]{y}}\right)}\right) - z\right) \]

10. Applied egg-rr 99.7%

    \[\leadsto 0 \cdot z + \left(3 \cdot \left(x \cdot \log \color{blue}{\left(\sqrt[3]{x} \cdot \frac{1}{\sqrt[3]{y}}\right)}\right) - z\right) \]
11. Step-by-step derivation

    1. associate-*r/ 99.7%

       \[\leadsto 0 \cdot z + \left(3 \cdot \left(x \cdot \log \color{blue}{\left(\frac{\sqrt[3]{x} \cdot 1}{\sqrt[3]{y}}\right)}\right) - z\right) \]

    2. *-rgt-identity 99.7%

       \[\leadsto 0 \cdot z + \left(3 \cdot \left(x \cdot \log \left(\frac{\color{blue}{\sqrt[3]{x}}}{\sqrt[3]{y}}\right)\right) - z\right) \]

12. Simplified 99.7%

    \[\leadsto 0 \cdot z + \left(3 \cdot \left(x \cdot \log \color{blue}{\left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)}\right) - z\right) \]

13. Final simplification 99.7%

    \[\leadsto 0 \cdot z + \left(3 \cdot \left(x \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right) - z\right) \]
14. Add Preprocessing

Alternative 2: 87.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;t_0 \leq -\infty \lor \neg \left(t_0 \leq 5 \cdot 10^{+305}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;t_0 - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 5e+305))) (- 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 <= 5e+305)) {
		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 <= 5e+305)) {
		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 <= 5e+305):
		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 <= 5e+305))
		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 <= 5e+305)))
		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, 5e+305]], $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 5.00000000000000009e305 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 5.1%

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

      1. remove-double-neg 5.1%

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

      2. sub-neg 5.1%

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

      3. distribute-neg-in 5.1%

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

      4. distribute-rgt-neg-in 5.1%

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

      5. remove-double-neg 5.1%

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

      6. fma-udef 5.1%

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

      7. log-div 43.8%

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

      8. sub-neg 43.8%

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

      9. distribute-neg-in 43.8%

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

      10. remove-double-neg 43.8%

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

      11. +-commutative 43.8%

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

      12. sub-neg 43.8%

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

      13. log-div 6.9%

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

   3. Simplified 6.9%

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

   5. Taylor expanded in x around 0 56.7%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 5.00000000000000009e305

   1. Initial program 99.8%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log \left(\frac{x}{y}\right) \leq -\infty \lor \neg \left(x \cdot \log \left(\frac{x}{y}\right) \leq 5 \cdot 10^{+305}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(x, \log y - \log x, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -4e-310)
   (- (* x (- (log (- x)) (log (- y)))) z)
   (- (fma x (- (log y) (log x)) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -4e-310) {
		tmp = (x * (log(-x) - log(-y))) - z;
	} else {
		tmp = -fma(x, (log(y) - log(x)), z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -4e-310)
		tmp = Float64(Float64(x * Float64(log(Float64(-x)) - log(Float64(-y)))) - z);
	else
		tmp = Float64(-fma(x, Float64(log(y) - log(x)), z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -4e-310], N[(N[(x * N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], (-N[(x * N[(N[Log[y], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if y < -3.999999999999988e-310

   1. Initial program 75.9%

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

      1. frac-2neg 42.8%

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

      2. log-div 51.9%

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

   4. Applied egg-rr 99.4%

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

   if -3.999999999999988e-310 < y

   1. Initial program 81.5%

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

      1. remove-double-neg 81.5%

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

      2. sub-neg 81.5%

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

      3. distribute-neg-in 81.5%

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

      4. distribute-rgt-neg-in 81.5%

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

      5. remove-double-neg 81.5%

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

      6. fma-udef 81.5%

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

      7. log-div 99.4%

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

      8. sub-neg 99.4%

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

      9. distribute-neg-in 99.4%

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

      10. remove-double-neg 99.4%

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

      11. +-commutative 99.4%

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

      12. sub-neg 99.4%

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

      13. log-div 80.4%

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

   3. Simplified 80.4%

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

      1. log-div 99.4%

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

   6. Applied egg-rr 99.4%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(x, \log y - \log x, z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 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 78.7%

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

   1. add-cube-cbrt 78.7%

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

   2. log-prod 78.6%

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

   3. pow2 78.6%

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

4. Applied egg-rr 78.6%

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

   1. log-pow 78.6%

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

   2. distribute-lft1-in 78.6%

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

   3. metadata-eval 78.6%

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

6. Simplified 78.6%

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

   1. cbrt-div 99.7%

      \[\leadsto 0 \cdot z + \left(3 \cdot \left(x \cdot \log \color{blue}{\left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)}\right) - z\right) \]

   2. div-inv 99.7%

      \[\leadsto 0 \cdot z + \left(3 \cdot \left(x \cdot \log \color{blue}{\left(\sqrt[3]{x} \cdot \frac{1}{\sqrt[3]{y}}\right)}\right) - z\right) \]

8. Applied egg-rr 99.6%

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

   1. associate-*r/ 99.7%

      \[\leadsto 0 \cdot z + \left(3 \cdot \left(x \cdot \log \color{blue}{\left(\frac{\sqrt[3]{x} \cdot 1}{\sqrt[3]{y}}\right)}\right) - z\right) \]

   2. *-rgt-identity 99.7%

      \[\leadsto 0 \cdot z + \left(3 \cdot \left(x \cdot \log \left(\frac{\color{blue}{\sqrt[3]{x}}}{\sqrt[3]{y}}\right)\right) - z\right) \]

10. Simplified 99.6%

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

11. Final simplification 99.6%

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

Alternative 5: 93.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{+138}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right)\\ \mathbf{elif}\;x \leq -1.86 \cdot 10^{-218}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -2.2e+138)
   (* x (- (log (- x)) (log (- y))))
   (if (<= x -1.86e-218)
     (- (* x (log (/ x y))) 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 <= -2.2e+138) {
		tmp = x * (log(-x) - log(-y));
	} else if (x <= -1.86e-218) {
		tmp = (x * log((x / y))) - z;
	} else if (x <= -5e-310) {
		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 <= (-2.2d+138)) then
        tmp = x * (log(-x) - log(-y))
    else if (x <= (-1.86d-218)) then
        tmp = (x * log((x / y))) - z
    else if (x <= (-5d-310)) 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 <= -2.2e+138) {
		tmp = x * (Math.log(-x) - Math.log(-y));
	} else if (x <= -1.86e-218) {
		tmp = (x * Math.log((x / y))) - z;
	} else if (x <= -5e-310) {
		tmp = -z;
	} else {
		tmp = (x * (Math.log(x) - Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -2.2e+138:
		tmp = x * (math.log(-x) - math.log(-y))
	elif x <= -1.86e-218:
		tmp = (x * math.log((x / y))) - z
	elif x <= -5e-310:
		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 <= -2.2e+138)
		tmp = Float64(x * Float64(log(Float64(-x)) - log(Float64(-y))));
	elseif (x <= -1.86e-218)
		tmp = Float64(Float64(x * log(Float64(x / y))) - z);
	elseif (x <= -5e-310)
		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 <= -2.2e+138)
		tmp = x * (log(-x) - log(-y));
	elseif (x <= -1.86e-218)
		tmp = (x * log((x / y))) - z;
	elseif (x <= -5e-310)
		tmp = -z;
	else
		tmp = (x * (log(x) - log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -2.2e+138], N[(x * N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.86e-218], N[(N[(x * N[Log[N[(x / y), $MachinePrecision]], $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 < -2.2000000000000001e138

   1. Initial program 64.3%

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

   3. Taylor expanded in z around 0 61.4%

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

      1. frac-2neg 61.4%

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

      2. log-div 90.1%

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

   5. Applied egg-rr 90.1%

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

   if -2.2000000000000001e138 < x < -1.8600000000000001e-218

   1. Initial program 90.1%

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

   if -1.8600000000000001e-218 < x < -4.999999999999985e-310

   1. Initial program 51.5%

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

      1. remove-double-neg 51.5%

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

      2. sub-neg 51.5%

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

      3. distribute-neg-in 51.5%

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

      4. distribute-rgt-neg-in 51.5%

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

      5. remove-double-neg 51.5%

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

      6. fma-udef 51.5%

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

      7. log-div 0.0%

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

      8. sub-neg 0.0%

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

      9. distribute-neg-in 0.0%

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

      10. remove-double-neg 0.0%

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

      11. +-commutative 0.0%

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

      12. sub-neg 0.0%

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

      13. log-div 47.4%

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

   3. Simplified 47.4%

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

   5. Taylor expanded in x around 0 100.0%

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

   if -4.999999999999985e-310 < x

   1. Initial program 81.5%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{+138}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right)\\ \mathbf{elif}\;x \leq -1.86 \cdot 10^{-218}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -4e-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 <= -4e-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 <= (-4d-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 <= -4e-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 <= -4e-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 <= -4e-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 <= -4e-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, -4e-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 < -3.999999999999988e-310

   1. Initial program 75.9%

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

      1. frac-2neg 42.8%

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

      2. log-div 51.9%

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

   4. Applied egg-rr 99.4%

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

   if -3.999999999999988e-310 < y

   1. Initial program 81.5%

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

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -980000 \lor \neg \left(z \leq 2.3 \cdot 10^{-99}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\left(-x\right) \cdot \log \left(\frac{y}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -980000.0) (not (<= z 2.3e-99))) (- z) (* (- x) (log (/ y x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -980000.0) || !(z <= 2.3e-99)) {
		tmp = -z;
	} else {
		tmp = -x * log((y / x));
	}
	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 <= (-980000.0d0)) .or. (.not. (z <= 2.3d-99))) then
        tmp = -z
    else
        tmp = -x * log((y / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -980000.0) || !(z <= 2.3e-99)) {
		tmp = -z;
	} else {
		tmp = -x * Math.log((y / x));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -980000.0) or not (z <= 2.3e-99):
		tmp = -z
	else:
		tmp = -x * math.log((y / x))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -980000.0) || !(z <= 2.3e-99))
		tmp = Float64(-z);
	else
		tmp = Float64(Float64(-x) * log(Float64(y / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -980000.0) || ~((z <= 2.3e-99)))
		tmp = -z;
	else
		tmp = -x * log((y / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -980000.0], N[Not[LessEqual[z, 2.3e-99]], $MachinePrecision]], (-z), N[((-x) * N[Log[N[(y / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -9.8e5 or 2.2999999999999998e-99 < z

   1. Initial program 79.2%

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

      1. remove-double-neg 79.2%

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

      2. sub-neg 79.2%

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

      3. distribute-neg-in 79.2%

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

      4. distribute-rgt-neg-in 79.2%

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

      5. remove-double-neg 79.2%

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

      6. fma-udef 79.2%

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

      7. log-div 50.5%

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

      8. sub-neg 50.5%

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

      9. distribute-neg-in 50.5%

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

      10. remove-double-neg 50.5%

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

      11. +-commutative 50.5%

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

      12. sub-neg 50.5%

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

      13. log-div 76.2%

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

   3. Simplified 76.2%

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

   5. Taylor expanded in x around 0 70.7%

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

   if -9.8e5 < z < 2.2999999999999998e-99

   1. Initial program 78.1%

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

      1. remove-double-neg 78.1%

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

      2. sub-neg 78.1%

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

      3. distribute-neg-in 78.1%

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

      4. distribute-rgt-neg-in 78.1%

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

      5. remove-double-neg 78.1%

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

      6. fma-udef 78.1%

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

      7. log-div 49.5%

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

      8. sub-neg 49.5%

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

      9. distribute-neg-in 49.5%

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

      10. remove-double-neg 49.5%

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

      11. +-commutative 49.5%

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

      12. sub-neg 49.5%

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

      13. log-div 78.4%

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

   3. Simplified 78.4%

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

   5. Taylor expanded in x around inf 41.1%

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

      1. log-rec 41.1%

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

      2. neg-mul-1 41.1%

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

      3. neg-mul-1 41.1%

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

      4. sub-neg 41.1%

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

      5. log-div 68.3%

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

   7. Simplified 68.3%

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

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -980000 \lor \neg \left(z \leq 2.3 \cdot 10^{-99}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\left(-x\right) \cdot \log \left(\frac{y}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1400000 \lor \neg \left(z \leq 1.28 \cdot 10^{-93}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1400000.0) (not (<= z 1.28e-93))) (- z) (* x (log (/ x y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1400000.0) || !(z <= 1.28e-93)) {
		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 <= (-1400000.0d0)) .or. (.not. (z <= 1.28d-93))) 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 <= -1400000.0) || !(z <= 1.28e-93)) {
		tmp = -z;
	} else {
		tmp = x * Math.log((x / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1400000.0) or not (z <= 1.28e-93):
		tmp = -z
	else:
		tmp = x * math.log((x / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1400000.0) || !(z <= 1.28e-93))
		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 <= -1400000.0) || ~((z <= 1.28e-93)))
		tmp = -z;
	else
		tmp = x * log((x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1400000.0], N[Not[LessEqual[z, 1.28e-93]], $MachinePrecision]], (-z), N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.4e6 or 1.27999999999999999e-93 < z

   1. Initial program 79.2%

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

      1. remove-double-neg 79.2%

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

      2. sub-neg 79.2%

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

      3. distribute-neg-in 79.2%

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

      4. distribute-rgt-neg-in 79.2%

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

      5. remove-double-neg 79.2%

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

      6. fma-udef 79.2%

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

      7. log-div 50.5%

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

      8. sub-neg 50.5%

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

      9. distribute-neg-in 50.5%

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

      10. remove-double-neg 50.5%

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

      11. +-commutative 50.5%

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

      12. sub-neg 50.5%

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

      13. log-div 76.2%

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

   3. Simplified 76.2%

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

   5. Taylor expanded in x around 0 70.7%

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

   if -1.4e6 < z < 1.27999999999999999e-93

   1. Initial program 78.1%

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

   3. Taylor expanded in z around 0 67.9%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1400000 \lor \neg \left(z \leq 1.28 \cdot 10^{-93}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 50.8% accurate, 53.5× speedup

Math

\[\begin{array}{l} \\ -z \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := (-z)

Derivation

1. Initial program 78.7%

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

   1. remove-double-neg 78.7%

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

   2. sub-neg 78.7%

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

   3. distribute-neg-in 78.7%

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

   4. distribute-rgt-neg-in 78.7%

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

   5. remove-double-neg 78.7%

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

   6. fma-udef 78.7%

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

   7. log-div 50.1%

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

   8. sub-neg 50.1%

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

   9. distribute-neg-in 50.1%

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

   10. remove-double-neg 50.1%

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

   11. +-commutative 50.1%

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

   12. sub-neg 50.1%

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

   13. log-div 77.2%

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

3. Simplified 77.2%

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

5. Taylor expanded in x around 0 47.1%

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

6. Final simplification 47.1%

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

Developer target: 88.8% 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 2024020 
(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))