Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Percentage Accurate: 99.8% → 99.9%

Time: 9.2s

Alternatives: 11

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

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

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 + fma(log(y), Float64(-0.5 - y), Float64(y - z)))
end

Wolfram

code[x_, y_, z_] := N[(x + N[(N[Log[y], $MachinePrecision] * N[(-0.5 - y), $MachinePrecision] + N[(y - z), $MachinePrecision]), $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. *-commutative 99.8%

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

   5. distribute-rgt-neg-in 99.8%

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

   6. fma-def 99.9%

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

   7. +-commutative 99.9%

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

   8. distribute-neg-in 99.9%

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

   9. unsub-neg 99.9%

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

   10. metadata-eval 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 2: 99.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_] := N[(x + N[(y - N[(N[Log[y], $MachinePrecision] * N[(y + 0.5), $MachinePrecision] + z), $MachinePrecision]), $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. *-commutative 99.8%

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

   5. distribute-rgt-neg-in 99.8%

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

   6. fma-def 99.9%

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

   7. +-commutative 99.9%

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

   8. distribute-neg-in 99.9%

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

   9. unsub-neg 99.9%

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

   10. metadata-eval 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in y around 0 99.9%

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

6. Simplified 99.8%

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

7. Final simplification 99.8%

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

Alternative 3: 75.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \log y \cdot 0.5\\ \mathbf{if}\;y \leq 6.5 \cdot 10^{-228}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.26 \cdot 10^{-53}:\\ \;\;\;\;\left(x + y\right) - z\\ \mathbf{elif}\;y \leq 6.1 \cdot 10^{-24}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 8.8 \cdot 10^{+18}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(1 - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- x (* (log y) 0.5))))
   (if (<= y 6.5e-228)
     t_0
     (if (<= y 1.26e-53)
       (- (+ x y) z)
       (if (<= y 6.1e-24)
         t_0
         (if (<= y 8.8e+18) (- x z) (+ x (* y (- 1.0 (log y))))))))))

C

double code(double x, double y, double z) {
	double t_0 = x - (log(y) * 0.5);
	double tmp;
	if (y <= 6.5e-228) {
		tmp = t_0;
	} else if (y <= 1.26e-53) {
		tmp = (x + y) - z;
	} else if (y <= 6.1e-24) {
		tmp = t_0;
	} else if (y <= 8.8e+18) {
		tmp = x - z;
	} else {
		tmp = x + (y * (1.0 - 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) :: t_0
    real(8) :: tmp
    t_0 = x - (log(y) * 0.5d0)
    if (y <= 6.5d-228) then
        tmp = t_0
    else if (y <= 1.26d-53) then
        tmp = (x + y) - z
    else if (y <= 6.1d-24) then
        tmp = t_0
    else if (y <= 8.8d+18) then
        tmp = x - z
    else
        tmp = x + (y * (1.0d0 - log(y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x - (Math.log(y) * 0.5);
	double tmp;
	if (y <= 6.5e-228) {
		tmp = t_0;
	} else if (y <= 1.26e-53) {
		tmp = (x + y) - z;
	} else if (y <= 6.1e-24) {
		tmp = t_0;
	} else if (y <= 8.8e+18) {
		tmp = x - z;
	} else {
		tmp = x + (y * (1.0 - Math.log(y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x - (math.log(y) * 0.5)
	tmp = 0
	if y <= 6.5e-228:
		tmp = t_0
	elif y <= 1.26e-53:
		tmp = (x + y) - z
	elif y <= 6.1e-24:
		tmp = t_0
	elif y <= 8.8e+18:
		tmp = x - z
	else:
		tmp = x + (y * (1.0 - math.log(y)))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x - Float64(log(y) * 0.5))
	tmp = 0.0
	if (y <= 6.5e-228)
		tmp = t_0;
	elseif (y <= 1.26e-53)
		tmp = Float64(Float64(x + y) - z);
	elseif (y <= 6.1e-24)
		tmp = t_0;
	elseif (y <= 8.8e+18)
		tmp = Float64(x - z);
	else
		tmp = Float64(x + Float64(y * Float64(1.0 - log(y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x - (log(y) * 0.5);
	tmp = 0.0;
	if (y <= 6.5e-228)
		tmp = t_0;
	elseif (y <= 1.26e-53)
		tmp = (x + y) - z;
	elseif (y <= 6.1e-24)
		tmp = t_0;
	elseif (y <= 8.8e+18)
		tmp = x - z;
	else
		tmp = x + (y * (1.0 - log(y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x - N[(N[Log[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 6.5e-228], t$95$0, If[LessEqual[y, 1.26e-53], N[(N[(x + y), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[y, 6.1e-24], t$95$0, If[LessEqual[y, 8.8e+18], N[(x - z), $MachinePrecision], N[(x + N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < 6.50000000000000029e-228 or 1.26e-53 < y < 6.10000000000000036e-24

   1. Initial program 100.0%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]

   2. Taylor expanded in y around 0 100.0%

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

   3. Taylor expanded in z around 0 90.7%

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

   if 6.50000000000000029e-228 < y < 1.26e-53

   1. Initial program 100.0%

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

      1. expm1-log1p-u 68.6%

         \[\leadsto \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x - \left(y + 0.5\right) \cdot \log y\right)\right)} + y\right) - z \]

      2. sub-neg 68.6%

         \[\leadsto \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{x + \left(-\left(y + 0.5\right) \cdot \log y\right)}\right)\right) + y\right) - z \]

      3. distribute-lft-neg-in 68.6%

         \[\leadsto \left(\mathsf{expm1}\left(\mathsf{log1p}\left(x + \color{blue}{\left(-\left(y + 0.5\right)\right) \cdot \log y}\right)\right) + y\right) - z \]

      4. +-commutative 68.6%

         \[\leadsto \left(\mathsf{expm1}\left(\mathsf{log1p}\left(x + \left(-\color{blue}{\left(0.5 + y\right)}\right) \cdot \log y\right)\right) + y\right) - z \]

      5. distribute-neg-in 68.6%

         \[\leadsto \left(\mathsf{expm1}\left(\mathsf{log1p}\left(x + \color{blue}{\left(\left(-0.5\right) + \left(-y\right)\right)} \cdot \log y\right)\right) + y\right) - z \]

      6. metadata-eval 68.6%

         \[\leadsto \left(\mathsf{expm1}\left(\mathsf{log1p}\left(x + \left(\color{blue}{-0.5} + \left(-y\right)\right) \cdot \log y\right)\right) + y\right) - z \]

      7. sub-neg 68.6%

         \[\leadsto \left(\mathsf{expm1}\left(\mathsf{log1p}\left(x + \color{blue}{\left(-0.5 - y\right)} \cdot \log y\right)\right) + y\right) - z \]

      8. *-commutative 68.6%

         \[\leadsto \left(\mathsf{expm1}\left(\mathsf{log1p}\left(x + \color{blue}{\log y \cdot \left(-0.5 - y\right)}\right)\right) + y\right) - z \]

   3. Applied egg-rr 68.6%

      \[\leadsto \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x + \log y \cdot \left(-0.5 - y\right)\right)\right)} + y\right) - z \]

   4. Taylor expanded in x around inf 78.5%

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

   if 6.10000000000000036e-24 < y < 8.8e18

   1. Initial program 99.9%

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

      1. add-cube-cbrt 99.3%

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

      2. pow3 99.3%

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

      3. sub-neg 99.3%

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

      4. associate-+l+ 99.3%

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

      5. distribute-lft-neg-in 99.3%

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

      6. +-commutative 99.3%

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

      7. distribute-neg-in 99.3%

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

      8. metadata-eval 99.3%

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

      9. sub-neg 99.3%

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

      10. *-commutative 99.3%

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

   3. Applied egg-rr 99.3%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{x + \left(\log y \cdot \left(-0.5 - y\right) + y\right)}\right)}^{3}} - z \]
   4. Step-by-step derivation

      1. add-cube-cbrt 99.2%

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

      2. pow3 99.2%

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

      3. fma-def 99.2%

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

   5. Applied egg-rr 99.2%

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

   6. Taylor expanded in y around inf 81.8%

      \[\leadsto \color{blue}{{1}^{0.3333333333333333} \cdot x} - z \]
   7. Step-by-step derivation

      1. pow-base-1 81.8%

         \[\leadsto \color{blue}{1} \cdot x - z \]

      2. *-lft-identity 81.8%

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

   8. Simplified 81.8%

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

   if 8.8e18 < y

   1. Initial program 99.6%

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

      1. associate--l+ 99.6%

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

      2. sub-neg 99.6%

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

      3. associate-+l+ 99.6%

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

      4. *-commutative 99.6%

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

      5. distribute-rgt-neg-in 99.6%

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

      6. fma-def 99.7%

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

      7. +-commutative 99.7%

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

      8. distribute-neg-in 99.7%

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

      9. unsub-neg 99.7%

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

      10. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around inf 83.5%

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

      1. log-rec 83.5%

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

      2. sub-neg 83.5%

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

   6. Simplified 83.5%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.5 \cdot 10^{-228}:\\ \;\;\;\;x - \log y \cdot 0.5\\ \mathbf{elif}\;y \leq 1.26 \cdot 10^{-53}:\\ \;\;\;\;\left(x + y\right) - z\\ \mathbf{elif}\;y \leq 6.1 \cdot 10^{-24}:\\ \;\;\;\;x - \log y \cdot 0.5\\ \mathbf{elif}\;y \leq 8.8 \cdot 10^{+18}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(1 - \log y\right)\\ \end{array} \]

Alternative 4: 89.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.4 \cdot 10^{+22} \lor \neg \left(x \leq 8 \cdot 10^{+78}\right):\\ \;\;\;\;x + y \cdot \left(1 - \log y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y - \log y \cdot \left(y + 0.5\right)\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -7.4e+22) (not (<= x 8e+78)))
   (+ x (* y (- 1.0 (log y))))
   (- (- y (* (log y) (+ y 0.5))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -7.4e+22) || !(x <= 8e+78)) {
		tmp = x + (y * (1.0 - log(y)));
	} else {
		tmp = (y - (log(y) * (y + 0.5))) - 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 <= (-7.4d+22)) .or. (.not. (x <= 8d+78))) then
        tmp = x + (y * (1.0d0 - log(y)))
    else
        tmp = (y - (log(y) * (y + 0.5d0))) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -7.4e+22) || !(x <= 8e+78)) {
		tmp = x + (y * (1.0 - Math.log(y)));
	} else {
		tmp = (y - (Math.log(y) * (y + 0.5))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -7.4e+22) or not (x <= 8e+78):
		tmp = x + (y * (1.0 - math.log(y)))
	else:
		tmp = (y - (math.log(y) * (y + 0.5))) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -7.4e+22) || !(x <= 8e+78))
		tmp = Float64(x + Float64(y * Float64(1.0 - log(y))));
	else
		tmp = Float64(Float64(y - Float64(log(y) * Float64(y + 0.5))) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -7.4e+22) || ~((x <= 8e+78)))
		tmp = x + (y * (1.0 - log(y)));
	else
		tmp = (y - (log(y) * (y + 0.5))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -7.4e+22], N[Not[LessEqual[x, 8e+78]], $MachinePrecision]], N[(x + N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y - N[(N[Log[y], $MachinePrecision] * N[(y + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -7.3999999999999996e22 or 8.00000000000000007e78 < x

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-+l+ 100.0%

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

      4. *-commutative 100.0%

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

      5. distribute-rgt-neg-in 100.0%

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

      6. fma-def 100.0%

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

      7. +-commutative 100.0%

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

      8. distribute-neg-in 100.0%

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

      9. unsub-neg 100.0%

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

      10. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around inf 87.6%

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

      1. log-rec 87.6%

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

      2. sub-neg 87.6%

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

   6. Simplified 87.6%

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

   if -7.3999999999999996e22 < x < 8.00000000000000007e78

   1. Initial program 99.7%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]

   2. Taylor expanded in x around 0 98.3%

      \[\leadsto \color{blue}{\left(y - \log y \cdot \left(0.5 + y\right)\right)} - z \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.4 \cdot 10^{+22} \lor \neg \left(x \leq 8 \cdot 10^{+78}\right):\\ \;\;\;\;x + y \cdot \left(1 - \log y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y - \log y \cdot \left(y + 0.5\right)\right) - z\\ \end{array} \]

Alternative 5: 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. Final simplification 99.8%

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

Alternative 6: 70.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -330000000000 \lor \neg \left(z \leq 202\right):\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;x - \log y \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -330000000000.0) (not (<= z 202.0)))
   (- x z)
   (- x (* (log y) 0.5))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -330000000000.0) || !(z <= 202.0)) {
		tmp = x - z;
	} else {
		tmp = x - (log(y) * 0.5);
	}
	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 <= (-330000000000.0d0)) .or. (.not. (z <= 202.0d0))) then
        tmp = x - z
    else
        tmp = x - (log(y) * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -330000000000.0) || !(z <= 202.0)) {
		tmp = x - z;
	} else {
		tmp = x - (Math.log(y) * 0.5);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -330000000000.0) or not (z <= 202.0):
		tmp = x - z
	else:
		tmp = x - (math.log(y) * 0.5)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -330000000000.0) || !(z <= 202.0))
		tmp = Float64(x - z);
	else
		tmp = Float64(x - Float64(log(y) * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -330000000000.0) || ~((z <= 202.0)))
		tmp = x - z;
	else
		tmp = x - (log(y) * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -330000000000.0], N[Not[LessEqual[z, 202.0]], $MachinePrecision]], N[(x - z), $MachinePrecision], N[(x - N[(N[Log[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.3e11 or 202 < z

   1. Initial program 99.9%

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

      1. add-cube-cbrt 99.0%

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

      2. pow3 99.0%

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

      3. sub-neg 99.0%

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

      4. associate-+l+ 99.0%

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

      5. distribute-lft-neg-in 99.0%

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

      6. +-commutative 99.0%

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

      7. distribute-neg-in 99.0%

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

      8. metadata-eval 99.0%

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

      9. sub-neg 99.0%

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

      10. *-commutative 99.0%

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

   3. Applied egg-rr 99.0%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{x + \left(\log y \cdot \left(-0.5 - y\right) + y\right)}\right)}^{3}} - z \]
   4. Step-by-step derivation

      1. add-cube-cbrt 99.0%

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

      2. pow3 99.0%

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

      3. fma-def 98.9%

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

   5. Applied egg-rr 98.9%

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

   6. Taylor expanded in y around inf 71.3%

      \[\leadsto \color{blue}{{1}^{0.3333333333333333} \cdot x} - z \]
   7. Step-by-step derivation

      1. pow-base-1 71.3%

         \[\leadsto \color{blue}{1} \cdot x - z \]

      2. *-lft-identity 71.3%

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

   8. Simplified 71.3%

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

   if -3.3e11 < z < 202

   1. Initial program 99.7%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]

   2. Taylor expanded in y around 0 57.5%

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

   3. Taylor expanded in z around 0 57.5%

      \[\leadsto \color{blue}{x - 0.5 \cdot \log y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -330000000000 \lor \neg \left(z \leq 202\right):\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;x - \log y \cdot 0.5\\ \end{array} \]

Alternative 7: 69.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -130 \lor \neg \left(x \leq 4.55 \cdot 10^{+36}\right):\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -130.0) (not (<= x 4.55e+36))) (- x z) (- (* (log y) -0.5) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -130.0) || !(x <= 4.55e+36)) {
		tmp = x - z;
	} else {
		tmp = (log(y) * -0.5) - 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 <= (-130.0d0)) .or. (.not. (x <= 4.55d+36))) then
        tmp = x - z
    else
        tmp = (log(y) * (-0.5d0)) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -130.0) || !(x <= 4.55e+36)) {
		tmp = x - z;
	} else {
		tmp = (Math.log(y) * -0.5) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -130.0) or not (x <= 4.55e+36):
		tmp = x - z
	else:
		tmp = (math.log(y) * -0.5) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -130.0) || !(x <= 4.55e+36))
		tmp = Float64(x - z);
	else
		tmp = Float64(Float64(log(y) * -0.5) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -130.0) || ~((x <= 4.55e+36)))
		tmp = x - z;
	else
		tmp = (log(y) * -0.5) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -130.0], N[Not[LessEqual[x, 4.55e+36]], $MachinePrecision]], N[(x - z), $MachinePrecision], N[(N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -130 or 4.54999999999999995e36 < x

   1. Initial program 99.9%

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

      1. add-cube-cbrt 98.3%

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

      2. pow3 98.3%

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

      3. sub-neg 98.3%

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

      4. associate-+l+ 98.3%

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

      5. distribute-lft-neg-in 98.3%

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

      6. +-commutative 98.3%

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

      7. distribute-neg-in 98.3%

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

      8. metadata-eval 98.3%

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

      9. sub-neg 98.3%

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

      10. *-commutative 98.3%

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

   3. Applied egg-rr 98.3%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{x + \left(\log y \cdot \left(-0.5 - y\right) + y\right)}\right)}^{3}} - z \]
   4. Step-by-step derivation

      1. add-cube-cbrt 98.2%

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

      2. pow3 98.3%

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

      3. fma-def 98.3%

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

   5. Applied egg-rr 98.3%

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

   6. Taylor expanded in y around inf 74.7%

      \[\leadsto \color{blue}{{1}^{0.3333333333333333} \cdot x} - z \]
   7. Step-by-step derivation

      1. pow-base-1 74.7%

         \[\leadsto \color{blue}{1} \cdot x - z \]

      2. *-lft-identity 74.7%

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

   8. Simplified 74.7%

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

   if -130 < x < 4.54999999999999995e36

   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. flip-+ 72.2%

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

      2. associate-*l/ 72.2%

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

      3. fma-neg 72.2%

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

      4. metadata-eval 72.2%

         \[\leadsto \left(\left(x - \frac{\mathsf{fma}\left(y, y, -\color{blue}{0.25}\right) \cdot \log y}{y - 0.5}\right) + y\right) - z \]

      5. metadata-eval 72.2%

         \[\leadsto \left(\left(x - \frac{\mathsf{fma}\left(y, y, \color{blue}{-0.25}\right) \cdot \log y}{y - 0.5}\right) + y\right) - z \]

      6. sub-neg 72.2%

         \[\leadsto \left(\left(x - \frac{\mathsf{fma}\left(y, y, -0.25\right) \cdot \log y}{\color{blue}{y + \left(-0.5\right)}}\right) + y\right) - z \]

      7. metadata-eval 72.2%

         \[\leadsto \left(\left(x - \frac{\mathsf{fma}\left(y, y, -0.25\right) \cdot \log y}{y + \color{blue}{-0.5}}\right) + y\right) - z \]

   3. Applied egg-rr 72.2%

      \[\leadsto \left(\left(x - \color{blue}{\frac{\mathsf{fma}\left(y, y, -0.25\right) \cdot \log y}{y + -0.5}}\right) + y\right) - z \]

   4. Taylor expanded in x around 0 72.0%

      \[\leadsto \color{blue}{\left(y - \frac{\log y \cdot \left({y}^{2} - 0.25\right)}{y - 0.5}\right)} - z \]

   5. Taylor expanded in y around 0 55.8%

      \[\leadsto \color{blue}{-0.5 \cdot \log y} - z \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -130 \lor \neg \left(x \leq 4.55 \cdot 10^{+36}\right):\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \end{array} \]

Alternative 8: 89.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 65000000000000:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(1 - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 65000000000000.0)
   (- (- x (* (log y) 0.5)) z)
   (+ x (* y (- 1.0 (log y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 65000000000000.0) {
		tmp = (x - (log(y) * 0.5)) - z;
	} else {
		tmp = x + (y * (1.0 - 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 <= 65000000000000.0d0) then
        tmp = (x - (log(y) * 0.5d0)) - z
    else
        tmp = x + (y * (1.0d0 - log(y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 65000000000000.0) {
		tmp = (x - (Math.log(y) * 0.5)) - z;
	} else {
		tmp = x + (y * (1.0 - Math.log(y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 65000000000000.0:
		tmp = (x - (math.log(y) * 0.5)) - z
	else:
		tmp = x + (y * (1.0 - math.log(y)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 65000000000000.0)
		tmp = Float64(Float64(x - Float64(log(y) * 0.5)) - z);
	else
		tmp = Float64(x + Float64(y * Float64(1.0 - log(y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 65000000000000.0)
		tmp = (x - (log(y) * 0.5)) - z;
	else
		tmp = x + (y * (1.0 - log(y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 65000000000000.0], N[(N[(x - N[(N[Log[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(x + N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.5e13

   1. Initial program 100.0%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]

   2. Taylor expanded in y around 0 98.8%

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

   if 6.5e13 < y

   1. Initial program 99.6%

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

      1. associate--l+ 99.6%

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

      2. sub-neg 99.6%

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

      3. associate-+l+ 99.6%

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

      4. *-commutative 99.6%

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

      5. distribute-rgt-neg-in 99.6%

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

      6. fma-def 99.7%

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

      7. +-commutative 99.7%

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

      8. distribute-neg-in 99.7%

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

      9. unsub-neg 99.7%

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

      10. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around inf 83.5%

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

      1. log-rec 83.5%

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

      2. sub-neg 83.5%

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

   6. Simplified 83.5%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 65000000000000:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(1 - \log y\right)\\ \end{array} \]

Alternative 9: 48.5% accurate, 18.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{+24}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{+38}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -3e+24) x (if (<= x 1.65e+38) (- z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -3e+24) {
		tmp = x;
	} else if (x <= 1.65e+38) {
		tmp = -z;
	} else {
		tmp = 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 (x <= (-3d+24)) then
        tmp = x
    else if (x <= 1.65d+38) then
        tmp = -z
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -3e+24) {
		tmp = x;
	} else if (x <= 1.65e+38) {
		tmp = -z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -3e+24:
		tmp = x
	elif x <= 1.65e+38:
		tmp = -z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -3e+24)
		tmp = x;
	elseif (x <= 1.65e+38)
		tmp = Float64(-z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -3e+24)
		tmp = x;
	elseif (x <= 1.65e+38)
		tmp = -z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -3e+24], x, If[LessEqual[x, 1.65e+38], (-z), x]]

Derivation

1. Split input into 2 regimes

2. if x < -2.99999999999999995e24 or 1.65e38 < x

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-+l+ 99.9%

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

      4. *-commutative 99.9%

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

      5. distribute-rgt-neg-in 99.9%

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

      6. fma-def 100.0%

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

      7. +-commutative 100.0%

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

      8. distribute-neg-in 100.0%

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

      9. unsub-neg 100.0%

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

      10. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 63.1%

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

   if -2.99999999999999995e24 < x < 1.65e38

   1. Initial program 99.7%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]

   2. Taylor expanded in y around 0 55.6%

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

   3. Taylor expanded in z around inf 34.1%

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

      1. neg-mul-1 34.1%

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

   5. Simplified 34.1%

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

4. Final simplification 46.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{+24}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{+38}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 10: 58.5% 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. Step-by-step derivation

   1. add-cube-cbrt 98.5%

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

   2. pow3 98.5%

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

   3. sub-neg 98.5%

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

   4. associate-+l+ 98.5%

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

   5. distribute-lft-neg-in 98.5%

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

   6. +-commutative 98.5%

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

   7. distribute-neg-in 98.5%

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

   8. metadata-eval 98.5%

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

   9. sub-neg 98.5%

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

   10. *-commutative 98.5%

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

3. Applied egg-rr 98.5%

   \[\leadsto \color{blue}{{\left(\sqrt[3]{x + \left(\log y \cdot \left(-0.5 - y\right) + y\right)}\right)}^{3}} - z \]
4. Step-by-step derivation

   1. add-cube-cbrt 98.4%

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

   2. pow3 98.4%

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

   3. fma-def 98.4%

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

5. Applied egg-rr 98.4%

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

6. Taylor expanded in y around inf 51.4%

   \[\leadsto \color{blue}{{1}^{0.3333333333333333} \cdot x} - z \]
7. Step-by-step derivation

   1. pow-base-1 51.4%

      \[\leadsto \color{blue}{1} \cdot x - z \]

   2. *-lft-identity 51.4%

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

8. Simplified 51.4%

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

9. Final simplification 51.4%

   \[\leadsto x - z \]

Alternative 11: 30.1% accurate, 111.0× speedup

Math

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

FPCore

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

C

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

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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := x

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. *-commutative 99.8%

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

   5. distribute-rgt-neg-in 99.8%

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

   6. fma-def 99.9%

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

   7. +-commutative 99.9%

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

   8. distribute-neg-in 99.9%

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

   9. unsub-neg 99.9%

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

   10. metadata-eval 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in x around inf 27.3%

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

5. Final simplification 27.3%

   \[\leadsto x \]

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 2023318 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (- (- (+ y x) z) (* (+ y 0.5) (log y)))

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