Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Percentage Accurate: 99.8% → 99.9%

Time: 11.1s

Alternatives: 9

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- (+ (- x (* (+ y 0.5) (log y))) y) z))

C

double code(double x, double y, double z) {
	return ((x - ((y + 0.5) * log(y))) + 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 - ((y + 0.5d0) * log(y))) + y) - z
end function

Java

public static double code(double x, double y, double z) {
	return ((x - ((y + 0.5) * Math.log(y))) + y) - z;
}

Python

def code(x, y, z):
	return ((x - ((y + 0.5) * math.log(y))) + y) - z

Julia

function code(x, y, z)
	return Float64(Float64(Float64(x - Float64(Float64(y + 0.5) * log(y))) + y) - z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((x - ((y + 0.5) * log(y))) + y) - z;
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(x - N[(N[(y + 0.5), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] - z), $MachinePrecision]

Initial Program: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- (+ (- x (* (+ y 0.5) (log y))) y) z))

C

double code(double x, double y, double z) {
	return ((x - ((y + 0.5) * log(y))) + 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 - ((y + 0.5d0) * log(y))) + y) - z
end function

Java

public static double code(double x, double y, double z) {
	return ((x - ((y + 0.5) * Math.log(y))) + y) - z;
}

Python

def code(x, y, z):
	return ((x - ((y + 0.5) * math.log(y))) + y) - z

Julia

function code(x, y, z)
	return Float64(Float64(Float64(x - Float64(Float64(y + 0.5) * log(y))) + y) - z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((x - ((y + 0.5) * log(y))) + y) - z;
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(x - N[(N[(y + 0.5), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] - z), $MachinePrecision]

Alternative 1: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ x + \left(\mathsf{fma}\left(\log y, -0.5 - y, y\right) - z\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ x (- (fma (log y) (- -0.5 y) y) z)))

C

double code(double x, double y, double z) {
	return x + (fma(log(y), (-0.5 - y), y) - z);
}

Julia

function code(x, y, z)
	return Float64(x + Float64(fma(log(y), Float64(-0.5 - y), y) - z))
end

Wolfram

code[x_, y_, z_] := N[(x + N[(N[(N[Log[y], $MachinePrecision] * N[(-0.5 - y), $MachinePrecision] + y), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
2. Step-by-step derivation

   1. associate--l+ 99.8%

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

   2. sub-neg 99.8%

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

   3. associate-+l+ 99.8%

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

   4. associate-+r- 99.8%

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

   5. *-commutative 99.8%

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

   6. distribute-rgt-neg-in 99.8%

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

   7. fma-define 99.9%

      \[\leadsto x + \left(\color{blue}{\mathsf{fma}\left(\log y, -\left(y + 0.5\right), y\right)} - z\right) \]

   8. +-commutative 99.9%

      \[\leadsto x + \left(\mathsf{fma}\left(\log y, -\color{blue}{\left(0.5 + y\right)}, y\right) - z\right) \]

   9. distribute-neg-in 99.9%

      \[\leadsto x + \left(\mathsf{fma}\left(\log y, \color{blue}{\left(-0.5\right) + \left(-y\right)}, y\right) - z\right) \]

   10. unsub-neg 99.9%

       \[\leadsto x + \left(\mathsf{fma}\left(\log y, \color{blue}{\left(-0.5\right) - y}, y\right) - z\right) \]

   11. metadata-eval 99.9%

       \[\leadsto x + \left(\mathsf{fma}\left(\log y, \color{blue}{-0.5} - y, y\right) - z\right) \]

3. Simplified 99.9%

   \[\leadsto \color{blue}{x + \left(\mathsf{fma}\left(\log y, -0.5 - y, y\right) - z\right)} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 2: 76.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.95 \cdot 10^{-138}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{-104}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{+76}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.95e-138)
   (- x z)
   (if (<= y 6.2e-104)
     (- (* (log y) -0.5) z)
     (if (<= y 1.45e+76) (- x z) (- (* y (- 1.0 (log y))) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.95e-138) {
		tmp = x - z;
	} else if (y <= 6.2e-104) {
		tmp = (log(y) * -0.5) - z;
	} else if (y <= 1.45e+76) {
		tmp = x - z;
	} else {
		tmp = (y * (1.0 - 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 <= 1.95d-138) then
        tmp = x - z
    else if (y <= 6.2d-104) then
        tmp = (log(y) * (-0.5d0)) - z
    else if (y <= 1.45d+76) then
        tmp = x - z
    else
        tmp = (y * (1.0d0 - log(y))) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.95e-138) {
		tmp = x - z;
	} else if (y <= 6.2e-104) {
		tmp = (Math.log(y) * -0.5) - z;
	} else if (y <= 1.45e+76) {
		tmp = x - z;
	} else {
		tmp = (y * (1.0 - Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1.95e-138:
		tmp = x - z
	elif y <= 6.2e-104:
		tmp = (math.log(y) * -0.5) - z
	elif y <= 1.45e+76:
		tmp = x - z
	else:
		tmp = (y * (1.0 - math.log(y))) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.95e-138)
		tmp = Float64(x - z);
	elseif (y <= 6.2e-104)
		tmp = Float64(Float64(log(y) * -0.5) - z);
	elseif (y <= 1.45e+76)
		tmp = Float64(x - z);
	else
		tmp = Float64(Float64(y * Float64(1.0 - log(y))) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1.95e-138)
		tmp = x - z;
	elseif (y <= 6.2e-104)
		tmp = (log(y) * -0.5) - z;
	elseif (y <= 1.45e+76)
		tmp = x - z;
	else
		tmp = (y * (1.0 - log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.95e-138], N[(x - z), $MachinePrecision], If[LessEqual[y, 6.2e-104], N[(N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[y, 1.45e+76], N[(x - z), $MachinePrecision], N[(N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.95e-138 or 6.19999999999999951e-104 < y < 1.4500000000000001e76

   1. Initial program 100.0%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 79.7%

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

   if 1.95e-138 < y < 6.19999999999999951e-104

   1. Initial program 100.0%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-cube-cbrt 98.8%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\left(x - \left(y + 0.5\right) \cdot \log y\right) + y} \cdot \sqrt[3]{\left(x - \left(y + 0.5\right) \cdot \log y\right) + y}\right) \cdot \sqrt[3]{\left(x - \left(y + 0.5\right) \cdot \log y\right) + y}} - z \]

      2. pow3 98.9%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\left(x - \left(y + 0.5\right) \cdot \log y\right) + y}\right)}^{3}} - z \]

      3. sub-neg 98.9%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{\left(x + \left(-\left(y + 0.5\right) \cdot \log y\right)\right)} + y}\right)}^{3} - z \]

      4. associate-+l+ 98.9%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{x + \left(\left(-\left(y + 0.5\right) \cdot \log y\right) + y\right)}}\right)}^{3} - z \]

      5. *-commutative 98.9%

         \[\leadsto {\left(\sqrt[3]{x + \left(\left(-\color{blue}{\log y \cdot \left(y + 0.5\right)}\right) + y\right)}\right)}^{3} - z \]

      6. distribute-rgt-neg-in 98.9%

         \[\leadsto {\left(\sqrt[3]{x + \left(\color{blue}{\log y \cdot \left(-\left(y + 0.5\right)\right)} + y\right)}\right)}^{3} - z \]

      7. +-commutative 98.9%

         \[\leadsto {\left(\sqrt[3]{x + \left(\log y \cdot \left(-\color{blue}{\left(0.5 + y\right)}\right) + y\right)}\right)}^{3} - z \]

      8. distribute-neg-in 98.9%

         \[\leadsto {\left(\sqrt[3]{x + \left(\log y \cdot \color{blue}{\left(\left(-0.5\right) + \left(-y\right)\right)} + y\right)}\right)}^{3} - z \]

      9. metadata-eval 98.9%

         \[\leadsto {\left(\sqrt[3]{x + \left(\log y \cdot \left(\color{blue}{-0.5} + \left(-y\right)\right) + y\right)}\right)}^{3} - z \]

      10. sub-neg 98.9%

          \[\leadsto {\left(\sqrt[3]{x + \left(\log y \cdot \color{blue}{\left(-0.5 - y\right)} + y\right)}\right)}^{3} - z \]

      11. fma-undefine 98.9%

          \[\leadsto {\left(\sqrt[3]{x + \color{blue}{\mathsf{fma}\left(\log y, -0.5 - y, y\right)}}\right)}^{3} - z \]

   4. Applied egg-rr 98.9%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{x + \mathsf{fma}\left(\log y, -0.5 - y, y\right)}\right)}^{3}} - z \]

   5. Taylor expanded in x around 0 87.8%

      \[\leadsto \color{blue}{\left(y + -1 \cdot \left(\log y \cdot \left(0.5 + y\right)\right)\right)} - z \]
   6. Step-by-step derivation

      1. mul-1-neg 87.8%

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

      2. sub-neg 87.8%

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

      3. +-commutative 87.8%

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

   7. Simplified 87.8%

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

   8. Taylor expanded in y around 0 87.8%

      \[\leadsto \color{blue}{-0.5 \cdot \log y} - z \]
   9. Step-by-step derivation

      1. *-commutative 87.8%

         \[\leadsto \color{blue}{\log y \cdot -0.5} - z \]

   10. Simplified 87.8%

       \[\leadsto \color{blue}{\log y \cdot -0.5} - z \]

   if 1.4500000000000001e76 < y

   1. Initial program 99.6%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-cube-cbrt 98.4%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\left(x - \left(y + 0.5\right) \cdot \log y\right) + y} \cdot \sqrt[3]{\left(x - \left(y + 0.5\right) \cdot \log y\right) + y}\right) \cdot \sqrt[3]{\left(x - \left(y + 0.5\right) \cdot \log y\right) + y}} - z \]

      2. pow3 98.4%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\left(x - \left(y + 0.5\right) \cdot \log y\right) + y}\right)}^{3}} - z \]

      3. sub-neg 98.4%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{\left(x + \left(-\left(y + 0.5\right) \cdot \log y\right)\right)} + y}\right)}^{3} - z \]

      4. associate-+l+ 98.4%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{x + \left(\left(-\left(y + 0.5\right) \cdot \log y\right) + y\right)}}\right)}^{3} - z \]

      5. *-commutative 98.4%

         \[\leadsto {\left(\sqrt[3]{x + \left(\left(-\color{blue}{\log y \cdot \left(y + 0.5\right)}\right) + y\right)}\right)}^{3} - z \]

      6. distribute-rgt-neg-in 98.4%

         \[\leadsto {\left(\sqrt[3]{x + \left(\color{blue}{\log y \cdot \left(-\left(y + 0.5\right)\right)} + y\right)}\right)}^{3} - z \]

      7. +-commutative 98.4%

         \[\leadsto {\left(\sqrt[3]{x + \left(\log y \cdot \left(-\color{blue}{\left(0.5 + y\right)}\right) + y\right)}\right)}^{3} - z \]

      8. distribute-neg-in 98.4%

         \[\leadsto {\left(\sqrt[3]{x + \left(\log y \cdot \color{blue}{\left(\left(-0.5\right) + \left(-y\right)\right)} + y\right)}\right)}^{3} - z \]

      9. metadata-eval 98.4%

         \[\leadsto {\left(\sqrt[3]{x + \left(\log y \cdot \left(\color{blue}{-0.5} + \left(-y\right)\right) + y\right)}\right)}^{3} - z \]

      10. sub-neg 98.4%

          \[\leadsto {\left(\sqrt[3]{x + \left(\log y \cdot \color{blue}{\left(-0.5 - y\right)} + y\right)}\right)}^{3} - z \]

      11. fma-undefine 98.3%

          \[\leadsto {\left(\sqrt[3]{x + \color{blue}{\mathsf{fma}\left(\log y, -0.5 - y, y\right)}}\right)}^{3} - z \]

   4. Applied egg-rr 98.3%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{x + \mathsf{fma}\left(\log y, -0.5 - y, y\right)}\right)}^{3}} - z \]

   5. Taylor expanded in y around inf 87.2%

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

      1. log-rec 87.2%

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

      2. sub-neg 87.2%

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

   7. Simplified 87.2%

      \[\leadsto \color{blue}{y \cdot \left(1 - \log y\right)} - z \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 71.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.35 \cdot 10^{-137}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y \leq 6.6 \cdot 10^{-104}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{elif}\;y \leq 5.4 \cdot 10^{+116}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y - y \cdot \log y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.35e-137)
   (- x z)
   (if (<= y 6.6e-104)
     (- (* (log y) -0.5) z)
     (if (<= y 5.4e+116) (- x z) (- y (* y (log y)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.35e-137) {
		tmp = x - z;
	} else if (y <= 6.6e-104) {
		tmp = (log(y) * -0.5) - z;
	} else if (y <= 5.4e+116) {
		tmp = x - z;
	} else {
		tmp = y - (y * log(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 (y <= 1.35d-137) then
        tmp = x - z
    else if (y <= 6.6d-104) then
        tmp = (log(y) * (-0.5d0)) - z
    else if (y <= 5.4d+116) then
        tmp = x - z
    else
        tmp = y - (y * log(y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.35e-137) {
		tmp = x - z;
	} else if (y <= 6.6e-104) {
		tmp = (Math.log(y) * -0.5) - z;
	} else if (y <= 5.4e+116) {
		tmp = x - z;
	} else {
		tmp = y - (y * Math.log(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1.35e-137:
		tmp = x - z
	elif y <= 6.6e-104:
		tmp = (math.log(y) * -0.5) - z
	elif y <= 5.4e+116:
		tmp = x - z
	else:
		tmp = y - (y * math.log(y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.35e-137)
		tmp = Float64(x - z);
	elseif (y <= 6.6e-104)
		tmp = Float64(Float64(log(y) * -0.5) - z);
	elseif (y <= 5.4e+116)
		tmp = Float64(x - z);
	else
		tmp = Float64(y - Float64(y * log(y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1.35e-137)
		tmp = x - z;
	elseif (y <= 6.6e-104)
		tmp = (log(y) * -0.5) - z;
	elseif (y <= 5.4e+116)
		tmp = x - z;
	else
		tmp = y - (y * log(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.35e-137], N[(x - z), $MachinePrecision], If[LessEqual[y, 6.6e-104], N[(N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[y, 5.4e+116], N[(x - z), $MachinePrecision], N[(y - N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.34999999999999996e-137 or 6.60000000000000004e-104 < y < 5.3999999999999999e116

   1. Initial program 100.0%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 79.2%

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

   if 1.34999999999999996e-137 < y < 6.60000000000000004e-104

   1. Initial program 100.0%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-cube-cbrt 98.8%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\left(x - \left(y + 0.5\right) \cdot \log y\right) + y} \cdot \sqrt[3]{\left(x - \left(y + 0.5\right) \cdot \log y\right) + y}\right) \cdot \sqrt[3]{\left(x - \left(y + 0.5\right) \cdot \log y\right) + y}} - z \]

      2. pow3 98.9%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\left(x - \left(y + 0.5\right) \cdot \log y\right) + y}\right)}^{3}} - z \]

      3. sub-neg 98.9%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{\left(x + \left(-\left(y + 0.5\right) \cdot \log y\right)\right)} + y}\right)}^{3} - z \]

      4. associate-+l+ 98.9%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{x + \left(\left(-\left(y + 0.5\right) \cdot \log y\right) + y\right)}}\right)}^{3} - z \]

      5. *-commutative 98.9%

         \[\leadsto {\left(\sqrt[3]{x + \left(\left(-\color{blue}{\log y \cdot \left(y + 0.5\right)}\right) + y\right)}\right)}^{3} - z \]

      6. distribute-rgt-neg-in 98.9%

         \[\leadsto {\left(\sqrt[3]{x + \left(\color{blue}{\log y \cdot \left(-\left(y + 0.5\right)\right)} + y\right)}\right)}^{3} - z \]

      7. +-commutative 98.9%

         \[\leadsto {\left(\sqrt[3]{x + \left(\log y \cdot \left(-\color{blue}{\left(0.5 + y\right)}\right) + y\right)}\right)}^{3} - z \]

      8. distribute-neg-in 98.9%

         \[\leadsto {\left(\sqrt[3]{x + \left(\log y \cdot \color{blue}{\left(\left(-0.5\right) + \left(-y\right)\right)} + y\right)}\right)}^{3} - z \]

      9. metadata-eval 98.9%

         \[\leadsto {\left(\sqrt[3]{x + \left(\log y \cdot \left(\color{blue}{-0.5} + \left(-y\right)\right) + y\right)}\right)}^{3} - z \]

      10. sub-neg 98.9%

          \[\leadsto {\left(\sqrt[3]{x + \left(\log y \cdot \color{blue}{\left(-0.5 - y\right)} + y\right)}\right)}^{3} - z \]

      11. fma-undefine 98.9%

          \[\leadsto {\left(\sqrt[3]{x + \color{blue}{\mathsf{fma}\left(\log y, -0.5 - y, y\right)}}\right)}^{3} - z \]

   4. Applied egg-rr 98.9%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{x + \mathsf{fma}\left(\log y, -0.5 - y, y\right)}\right)}^{3}} - z \]

   5. Taylor expanded in x around 0 87.8%

      \[\leadsto \color{blue}{\left(y + -1 \cdot \left(\log y \cdot \left(0.5 + y\right)\right)\right)} - z \]
   6. Step-by-step derivation

      1. mul-1-neg 87.8%

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

      2. sub-neg 87.8%

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

      3. +-commutative 87.8%

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

   7. Simplified 87.8%

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

   8. Taylor expanded in y around 0 87.8%

      \[\leadsto \color{blue}{-0.5 \cdot \log y} - z \]
   9. Step-by-step derivation

      1. *-commutative 87.8%

         \[\leadsto \color{blue}{\log y \cdot -0.5} - z \]

   10. Simplified 87.8%

       \[\leadsto \color{blue}{\log y \cdot -0.5} - z \]

   if 5.3999999999999999e116 < y

   1. Initial program 99.5%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-cube-cbrt 98.2%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\left(x - \left(y + 0.5\right) \cdot \log y\right) + y} \cdot \sqrt[3]{\left(x - \left(y + 0.5\right) \cdot \log y\right) + y}\right) \cdot \sqrt[3]{\left(x - \left(y + 0.5\right) \cdot \log y\right) + y}} - z \]

      2. pow3 98.2%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\left(x - \left(y + 0.5\right) \cdot \log y\right) + y}\right)}^{3}} - z \]

      3. sub-neg 98.2%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{\left(x + \left(-\left(y + 0.5\right) \cdot \log y\right)\right)} + y}\right)}^{3} - z \]

      4. associate-+l+ 98.2%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{x + \left(\left(-\left(y + 0.5\right) \cdot \log y\right) + y\right)}}\right)}^{3} - z \]

      5. *-commutative 98.2%

         \[\leadsto {\left(\sqrt[3]{x + \left(\left(-\color{blue}{\log y \cdot \left(y + 0.5\right)}\right) + y\right)}\right)}^{3} - z \]

      6. distribute-rgt-neg-in 98.2%

         \[\leadsto {\left(\sqrt[3]{x + \left(\color{blue}{\log y \cdot \left(-\left(y + 0.5\right)\right)} + y\right)}\right)}^{3} - z \]

      7. +-commutative 98.2%

         \[\leadsto {\left(\sqrt[3]{x + \left(\log y \cdot \left(-\color{blue}{\left(0.5 + y\right)}\right) + y\right)}\right)}^{3} - z \]

      8. distribute-neg-in 98.2%

         \[\leadsto {\left(\sqrt[3]{x + \left(\log y \cdot \color{blue}{\left(\left(-0.5\right) + \left(-y\right)\right)} + y\right)}\right)}^{3} - z \]

      9. metadata-eval 98.2%

         \[\leadsto {\left(\sqrt[3]{x + \left(\log y \cdot \left(\color{blue}{-0.5} + \left(-y\right)\right) + y\right)}\right)}^{3} - z \]

      10. sub-neg 98.2%

          \[\leadsto {\left(\sqrt[3]{x + \left(\log y \cdot \color{blue}{\left(-0.5 - y\right)} + y\right)}\right)}^{3} - z \]

      11. fma-undefine 98.2%

          \[\leadsto {\left(\sqrt[3]{x + \color{blue}{\mathsf{fma}\left(\log y, -0.5 - y, y\right)}}\right)}^{3} - z \]

   4. Applied egg-rr 98.2%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{x + \mathsf{fma}\left(\log y, -0.5 - y, y\right)}\right)}^{3}} - z \]

   5. Taylor expanded in x around 0 88.7%

      \[\leadsto \color{blue}{\left(y + -1 \cdot \left(\log y \cdot \left(0.5 + y\right)\right)\right)} - z \]
   6. Step-by-step derivation

      1. mul-1-neg 88.7%

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

      2. sub-neg 88.7%

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

      3. +-commutative 88.7%

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

   7. Simplified 88.7%

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

   8. Taylor expanded in z around 0 75.8%

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

      1. cancel-sign-sub-inv 75.8%

         \[\leadsto \color{blue}{y + \left(-\log y\right) \cdot \left(0.5 + y\right)} \]

      2. +-commutative 75.8%

         \[\leadsto y + \left(-\log y\right) \cdot \color{blue}{\left(y + 0.5\right)} \]

      3. cancel-sign-sub-inv 75.8%

         \[\leadsto \color{blue}{y - \log y \cdot \left(y + 0.5\right)} \]

   10. Simplified 75.8%

       \[\leadsto \color{blue}{y - \log y \cdot \left(y + 0.5\right)} \]

   11. Taylor expanded in y around inf 75.8%

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

       1. mul-1-neg 75.8%

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

       2. log-rec 75.8%

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

       3. distribute-rgt-neg-in 75.8%

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

       4. remove-double-neg 75.8%

          \[\leadsto y - y \cdot \color{blue}{\log y} \]

   13. Simplified 75.8%

       \[\leadsto y - \color{blue}{y \cdot \log y} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.28:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + \left(y \cdot \left(1 - \log y\right) - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 0.28)
   (- (- x (* (log y) 0.5)) z)
   (+ x (- (* y (- 1.0 (log y))) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 0.28) {
		tmp = (x - (log(y) * 0.5)) - z;
	} else {
		tmp = x + ((y * (1.0 - 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 <= 0.28d0) then
        tmp = (x - (log(y) * 0.5d0)) - z
    else
        tmp = x + ((y * (1.0d0 - log(y))) - z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 0.28) {
		tmp = (x - (Math.log(y) * 0.5)) - z;
	} else {
		tmp = x + ((y * (1.0 - Math.log(y))) - z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 0.28:
		tmp = (x - (math.log(y) * 0.5)) - z
	else:
		tmp = x + ((y * (1.0 - math.log(y))) - z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 0.28)
		tmp = Float64(Float64(x - Float64(log(y) * 0.5)) - z);
	else
		tmp = Float64(x + Float64(Float64(y * Float64(1.0 - log(y))) - z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 0.28)
		tmp = (x - (log(y) * 0.5)) - z;
	else
		tmp = x + ((y * (1.0 - log(y))) - z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 0.28], N[(N[(x - N[(N[Log[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(x + N[(N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 0.28000000000000003

   1. Initial program 100.0%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 99.4%

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

      1. *-commutative 99.4%

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

   5. Simplified 99.4%

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

   if 0.28000000000000003 < y

   1. Initial program 99.7%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Step-by-step derivation

      1. associate--l+ 99.7%

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

      2. sub-neg 99.7%

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

      3. associate-+l+ 99.7%

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

      4. associate-+r- 99.7%

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

      5. *-commutative 99.7%

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

      6. distribute-rgt-neg-in 99.7%

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

      7. fma-define 99.8%

         \[\leadsto x + \left(\color{blue}{\mathsf{fma}\left(\log y, -\left(y + 0.5\right), y\right)} - z\right) \]

      8. +-commutative 99.8%

         \[\leadsto x + \left(\mathsf{fma}\left(\log y, -\color{blue}{\left(0.5 + y\right)}, y\right) - z\right) \]

      9. distribute-neg-in 99.8%

         \[\leadsto x + \left(\mathsf{fma}\left(\log y, \color{blue}{\left(-0.5\right) + \left(-y\right)}, y\right) - z\right) \]

      10. unsub-neg 99.8%

          \[\leadsto x + \left(\mathsf{fma}\left(\log y, \color{blue}{\left(-0.5\right) - y}, y\right) - z\right) \]

      11. metadata-eval 99.8%

          \[\leadsto x + \left(\mathsf{fma}\left(\log y, \color{blue}{-0.5} - y, y\right) - z\right) \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{x + \left(\mathsf{fma}\left(\log y, -0.5 - y, y\right) - z\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 99.3%

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

      1. log-rec 99.3%

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

      2. sub-neg 99.3%

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

   7. Simplified 99.3%

      \[\leadsto x + \left(\color{blue}{y \cdot \left(1 - \log y\right)} - z\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 89.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.45 \cdot 10^{+76}:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.45e+76) (- (- x (* (log y) 0.5)) z) (- (* y (- 1.0 (log y))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.45e+76) {
		tmp = (x - (log(y) * 0.5)) - z;
	} else {
		tmp = (y * (1.0 - 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 <= 1.45d+76) then
        tmp = (x - (log(y) * 0.5d0)) - z
    else
        tmp = (y * (1.0d0 - log(y))) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.45e+76) {
		tmp = (x - (Math.log(y) * 0.5)) - z;
	} else {
		tmp = (y * (1.0 - Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1.45e+76:
		tmp = (x - (math.log(y) * 0.5)) - z
	else:
		tmp = (y * (1.0 - math.log(y))) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.45e+76)
		tmp = Float64(Float64(x - Float64(log(y) * 0.5)) - z);
	else
		tmp = Float64(Float64(y * Float64(1.0 - log(y))) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1.45e+76)
		tmp = (x - (log(y) * 0.5)) - z;
	else
		tmp = (y * (1.0 - log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.45e+76], N[(N[(x - N[(N[Log[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.4500000000000001e76

   1. Initial program 100.0%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 94.8%

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

      1. *-commutative 94.8%

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

   5. Simplified 94.8%

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

   if 1.4500000000000001e76 < y

   1. Initial program 99.6%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-cube-cbrt 98.4%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\left(x - \left(y + 0.5\right) \cdot \log y\right) + y} \cdot \sqrt[3]{\left(x - \left(y + 0.5\right) \cdot \log y\right) + y}\right) \cdot \sqrt[3]{\left(x - \left(y + 0.5\right) \cdot \log y\right) + y}} - z \]

      2. pow3 98.4%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\left(x - \left(y + 0.5\right) \cdot \log y\right) + y}\right)}^{3}} - z \]

      3. sub-neg 98.4%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{\left(x + \left(-\left(y + 0.5\right) \cdot \log y\right)\right)} + y}\right)}^{3} - z \]

      4. associate-+l+ 98.4%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{x + \left(\left(-\left(y + 0.5\right) \cdot \log y\right) + y\right)}}\right)}^{3} - z \]

      5. *-commutative 98.4%

         \[\leadsto {\left(\sqrt[3]{x + \left(\left(-\color{blue}{\log y \cdot \left(y + 0.5\right)}\right) + y\right)}\right)}^{3} - z \]

      6. distribute-rgt-neg-in 98.4%

         \[\leadsto {\left(\sqrt[3]{x + \left(\color{blue}{\log y \cdot \left(-\left(y + 0.5\right)\right)} + y\right)}\right)}^{3} - z \]

      7. +-commutative 98.4%

         \[\leadsto {\left(\sqrt[3]{x + \left(\log y \cdot \left(-\color{blue}{\left(0.5 + y\right)}\right) + y\right)}\right)}^{3} - z \]

      8. distribute-neg-in 98.4%

         \[\leadsto {\left(\sqrt[3]{x + \left(\log y \cdot \color{blue}{\left(\left(-0.5\right) + \left(-y\right)\right)} + y\right)}\right)}^{3} - z \]

      9. metadata-eval 98.4%

         \[\leadsto {\left(\sqrt[3]{x + \left(\log y \cdot \left(\color{blue}{-0.5} + \left(-y\right)\right) + y\right)}\right)}^{3} - z \]

      10. sub-neg 98.4%

          \[\leadsto {\left(\sqrt[3]{x + \left(\log y \cdot \color{blue}{\left(-0.5 - y\right)} + y\right)}\right)}^{3} - z \]

      11. fma-undefine 98.3%

          \[\leadsto {\left(\sqrt[3]{x + \color{blue}{\mathsf{fma}\left(\log y, -0.5 - y, y\right)}}\right)}^{3} - z \]

   4. Applied egg-rr 98.3%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{x + \mathsf{fma}\left(\log y, -0.5 - y, y\right)}\right)}^{3}} - z \]

   5. Taylor expanded in y around inf 87.2%

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

      1. log-rec 87.2%

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

      2. sub-neg 87.2%

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

   7. Simplified 87.2%

      \[\leadsto \color{blue}{y \cdot \left(1 - \log y\right)} - z \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(y + \left(x - \log y \cdot \left(y + 0.5\right)\right)\right) - z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- (+ y (- x (* (log y) (+ y 0.5)))) z))

C

double code(double x, double y, double z) {
	return (y + (x - (log(y) * (y + 0.5)))) - 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 = (y + (x - (log(y) * (y + 0.5d0)))) - z
end function

Java

public static double code(double x, double y, double z) {
	return (y + (x - (Math.log(y) * (y + 0.5)))) - z;
}

Python

def code(x, y, z):
	return (y + (x - (math.log(y) * (y + 0.5)))) - z

Julia

function code(x, y, z)
	return Float64(Float64(y + Float64(x - Float64(log(y) * Float64(y + 0.5)))) - z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = (y + (x - (log(y) * (y + 0.5)))) - z;
end

Wolfram

code[x_, y_, z_] := N[(N[(y + N[(x - N[(N[Log[y], $MachinePrecision] * N[(y + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
2. Add Preprocessing

3. Final simplification 99.8%

   \[\leadsto \left(y + \left(x - \log y \cdot \left(y + 0.5\right)\right)\right) - z \]
4. Add Preprocessing

Alternative 7: 71.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 9 \cdot 10^{+116}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y - y \cdot \log y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 9e+116) (- x z) (- y (* y (log y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 9e+116) {
		tmp = x - z;
	} else {
		tmp = y - (y * log(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 (y <= 9d+116) then
        tmp = x - z
    else
        tmp = y - (y * log(y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 9e+116) {
		tmp = x - z;
	} else {
		tmp = y - (y * Math.log(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 9e+116:
		tmp = x - z
	else:
		tmp = y - (y * math.log(y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 9e+116)
		tmp = Float64(x - z);
	else
		tmp = Float64(y - Float64(y * log(y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 9e+116)
		tmp = x - z;
	else
		tmp = y - (y * log(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 9e+116], N[(x - z), $MachinePrecision], N[(y - N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 9.00000000000000032e116

   1. Initial program 100.0%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 77.5%

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

   if 9.00000000000000032e116 < y

   1. Initial program 99.5%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-cube-cbrt 98.2%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\left(x - \left(y + 0.5\right) \cdot \log y\right) + y} \cdot \sqrt[3]{\left(x - \left(y + 0.5\right) \cdot \log y\right) + y}\right) \cdot \sqrt[3]{\left(x - \left(y + 0.5\right) \cdot \log y\right) + y}} - z \]

      2. pow3 98.2%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\left(x - \left(y + 0.5\right) \cdot \log y\right) + y}\right)}^{3}} - z \]

      3. sub-neg 98.2%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{\left(x + \left(-\left(y + 0.5\right) \cdot \log y\right)\right)} + y}\right)}^{3} - z \]

      4. associate-+l+ 98.2%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{x + \left(\left(-\left(y + 0.5\right) \cdot \log y\right) + y\right)}}\right)}^{3} - z \]

      5. *-commutative 98.2%

         \[\leadsto {\left(\sqrt[3]{x + \left(\left(-\color{blue}{\log y \cdot \left(y + 0.5\right)}\right) + y\right)}\right)}^{3} - z \]

      6. distribute-rgt-neg-in 98.2%

         \[\leadsto {\left(\sqrt[3]{x + \left(\color{blue}{\log y \cdot \left(-\left(y + 0.5\right)\right)} + y\right)}\right)}^{3} - z \]

      7. +-commutative 98.2%

         \[\leadsto {\left(\sqrt[3]{x + \left(\log y \cdot \left(-\color{blue}{\left(0.5 + y\right)}\right) + y\right)}\right)}^{3} - z \]

      8. distribute-neg-in 98.2%

         \[\leadsto {\left(\sqrt[3]{x + \left(\log y \cdot \color{blue}{\left(\left(-0.5\right) + \left(-y\right)\right)} + y\right)}\right)}^{3} - z \]

      9. metadata-eval 98.2%

         \[\leadsto {\left(\sqrt[3]{x + \left(\log y \cdot \left(\color{blue}{-0.5} + \left(-y\right)\right) + y\right)}\right)}^{3} - z \]

      10. sub-neg 98.2%

          \[\leadsto {\left(\sqrt[3]{x + \left(\log y \cdot \color{blue}{\left(-0.5 - y\right)} + y\right)}\right)}^{3} - z \]

      11. fma-undefine 98.2%

          \[\leadsto {\left(\sqrt[3]{x + \color{blue}{\mathsf{fma}\left(\log y, -0.5 - y, y\right)}}\right)}^{3} - z \]

   4. Applied egg-rr 98.2%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{x + \mathsf{fma}\left(\log y, -0.5 - y, y\right)}\right)}^{3}} - z \]

   5. Taylor expanded in x around 0 88.7%

      \[\leadsto \color{blue}{\left(y + -1 \cdot \left(\log y \cdot \left(0.5 + y\right)\right)\right)} - z \]
   6. Step-by-step derivation

      1. mul-1-neg 88.7%

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

      2. sub-neg 88.7%

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

      3. +-commutative 88.7%

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

   7. Simplified 88.7%

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

   8. Taylor expanded in z around 0 75.8%

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

      1. cancel-sign-sub-inv 75.8%

         \[\leadsto \color{blue}{y + \left(-\log y\right) \cdot \left(0.5 + y\right)} \]

      2. +-commutative 75.8%

         \[\leadsto y + \left(-\log y\right) \cdot \color{blue}{\left(y + 0.5\right)} \]

      3. cancel-sign-sub-inv 75.8%

         \[\leadsto \color{blue}{y - \log y \cdot \left(y + 0.5\right)} \]

   10. Simplified 75.8%

       \[\leadsto \color{blue}{y - \log y \cdot \left(y + 0.5\right)} \]

   11. Taylor expanded in y around inf 75.8%

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

       1. mul-1-neg 75.8%

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

       2. log-rec 75.8%

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

       3. distribute-rgt-neg-in 75.8%

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

       4. remove-double-neg 75.8%

          \[\leadsto y - y \cdot \color{blue}{\log y} \]

   13. Simplified 75.8%

       \[\leadsto y - \color{blue}{y \cdot \log y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 58.0% accurate, 37.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
2. Add Preprocessing

3. Taylor expanded in x around inf 60.1%

   \[\leadsto \color{blue}{x} - z \]
4. Add Preprocessing

Alternative 9: 30.4% accurate, 55.5× 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 99.8%

   \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
2. Add Preprocessing
3. Step-by-step derivation

   1. add-cube-cbrt 98.7%

      \[\leadsto \color{blue}{\left(\sqrt[3]{\left(x - \left(y + 0.5\right) \cdot \log y\right) + y} \cdot \sqrt[3]{\left(x - \left(y + 0.5\right) \cdot \log y\right) + y}\right) \cdot \sqrt[3]{\left(x - \left(y + 0.5\right) \cdot \log y\right) + y}} - z \]

   2. pow3 98.7%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\left(x - \left(y + 0.5\right) \cdot \log y\right) + y}\right)}^{3}} - z \]

   3. sub-neg 98.7%

      \[\leadsto {\left(\sqrt[3]{\color{blue}{\left(x + \left(-\left(y + 0.5\right) \cdot \log y\right)\right)} + y}\right)}^{3} - z \]

   4. associate-+l+ 98.7%

      \[\leadsto {\left(\sqrt[3]{\color{blue}{x + \left(\left(-\left(y + 0.5\right) \cdot \log y\right) + y\right)}}\right)}^{3} - z \]

   5. *-commutative 98.7%

      \[\leadsto {\left(\sqrt[3]{x + \left(\left(-\color{blue}{\log y \cdot \left(y + 0.5\right)}\right) + y\right)}\right)}^{3} - z \]

   6. distribute-rgt-neg-in 98.7%

      \[\leadsto {\left(\sqrt[3]{x + \left(\color{blue}{\log y \cdot \left(-\left(y + 0.5\right)\right)} + y\right)}\right)}^{3} - z \]

   7. +-commutative 98.7%

      \[\leadsto {\left(\sqrt[3]{x + \left(\log y \cdot \left(-\color{blue}{\left(0.5 + y\right)}\right) + y\right)}\right)}^{3} - z \]

   8. distribute-neg-in 98.7%

      \[\leadsto {\left(\sqrt[3]{x + \left(\log y \cdot \color{blue}{\left(\left(-0.5\right) + \left(-y\right)\right)} + y\right)}\right)}^{3} - z \]

   9. metadata-eval 98.7%

      \[\leadsto {\left(\sqrt[3]{x + \left(\log y \cdot \left(\color{blue}{-0.5} + \left(-y\right)\right) + y\right)}\right)}^{3} - z \]

   10. sub-neg 98.7%

       \[\leadsto {\left(\sqrt[3]{x + \left(\log y \cdot \color{blue}{\left(-0.5 - y\right)} + y\right)}\right)}^{3} - z \]

   11. fma-undefine 98.7%

       \[\leadsto {\left(\sqrt[3]{x + \color{blue}{\mathsf{fma}\left(\log y, -0.5 - y, y\right)}}\right)}^{3} - z \]

4. Applied egg-rr 98.7%

   \[\leadsto \color{blue}{{\left(\sqrt[3]{x + \mathsf{fma}\left(\log y, -0.5 - y, y\right)}\right)}^{3}} - z \]

5. Taylor expanded in x around 0 75.5%

   \[\leadsto \color{blue}{\left(y + -1 \cdot \left(\log y \cdot \left(0.5 + y\right)\right)\right)} - z \]
6. Step-by-step derivation

   1. mul-1-neg 75.5%

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

   2. sub-neg 75.5%

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

   3. +-commutative 75.5%

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

7. Simplified 75.5%

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

8. Taylor expanded in z around inf 36.2%

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

   1. neg-mul-1 36.2%

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

10. Simplified 36.2%

    \[\leadsto \color{blue}{-z} \]
11. Add Preprocessing

Developer target: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(y + x\right) - z\right) - \left(y + 0.5\right) \cdot \log y \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- (- (+ y x) z) (* (+ y 0.5) (log y))))

C

double code(double x, double y, double z) {
	return ((y + x) - z) - ((y + 0.5) * log(y));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = ((y + x) - z) - ((y + 0.5d0) * log(y))
end function

Java

public static double code(double x, double y, double z) {
	return ((y + x) - z) - ((y + 0.5) * Math.log(y));
}

Python

def code(x, y, z):
	return ((y + x) - z) - ((y + 0.5) * math.log(y))

Julia

function code(x, y, z)
	return Float64(Float64(Float64(y + x) - z) - Float64(Float64(y + 0.5) * log(y)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((y + x) - z) - ((y + 0.5) * log(y));
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(y + x), $MachinePrecision] - z), $MachinePrecision] - N[(N[(y + 0.5), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024087 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2"
  :precision binary64

  :alt
  (- (- (+ y x) z) (* (+ y 0.5) (log y)))

  (- (+ (- x (* (+ y 0.5) (log y))) y) z))