Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Percentage Accurate: 99.8% → 99.9%

Time: 10.9s

Alternatives: 10

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: 89.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(1 - \log y\right)\\ \mathbf{if}\;y \leq 1.95 \cdot 10^{+14}:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{+115}:\\ \;\;\;\;x + t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_0 - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (- 1.0 (log y)))))
   (if (<= y 1.95e+14)
     (- (- x (* (log y) 0.5)) z)
     (if (<= y 4.6e+115) (+ x t_0) (- t_0 z)))))

C

double code(double x, double y, double z) {
	double t_0 = y * (1.0 - log(y));
	double tmp;
	if (y <= 1.95e+14) {
		tmp = (x - (log(y) * 0.5)) - z;
	} else if (y <= 4.6e+115) {
		tmp = x + t_0;
	} else {
		tmp = t_0 - 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) :: t_0
    real(8) :: tmp
    t_0 = y * (1.0d0 - log(y))
    if (y <= 1.95d+14) then
        tmp = (x - (log(y) * 0.5d0)) - z
    else if (y <= 4.6d+115) then
        tmp = x + t_0
    else
        tmp = t_0 - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y * (1.0 - Math.log(y));
	double tmp;
	if (y <= 1.95e+14) {
		tmp = (x - (Math.log(y) * 0.5)) - z;
	} else if (y <= 4.6e+115) {
		tmp = x + t_0;
	} else {
		tmp = t_0 - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * (1.0 - math.log(y))
	tmp = 0
	if y <= 1.95e+14:
		tmp = (x - (math.log(y) * 0.5)) - z
	elif y <= 4.6e+115:
		tmp = x + t_0
	else:
		tmp = t_0 - z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * Float64(1.0 - log(y)))
	tmp = 0.0
	if (y <= 1.95e+14)
		tmp = Float64(Float64(x - Float64(log(y) * 0.5)) - z);
	elseif (y <= 4.6e+115)
		tmp = Float64(x + t_0);
	else
		tmp = Float64(t_0 - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * (1.0 - log(y));
	tmp = 0.0;
	if (y <= 1.95e+14)
		tmp = (x - (log(y) * 0.5)) - z;
	elseif (y <= 4.6e+115)
		tmp = x + t_0;
	else
		tmp = t_0 - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 1.95e+14], N[(N[(x - N[(N[Log[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[y, 4.6e+115], N[(x + t$95$0), $MachinePrecision], N[(t$95$0 - z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.95e14

   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 98.9%

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

   if 1.95e14 < y < 4.60000000000000007e115

   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. Taylor expanded in y around inf 99.9%

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

      1. +-commutative 99.9%

         \[\leadsto x + y \cdot \left(\color{blue}{\left(\left(\log \left(\frac{1}{y}\right) + 0.5 \cdot \frac{\log \left(\frac{1}{y}\right)}{y}\right) + 1\right)} - \frac{z}{y}\right) \]

      2. associate--l+ 99.9%

         \[\leadsto x + y \cdot \color{blue}{\left(\left(\log \left(\frac{1}{y}\right) + 0.5 \cdot \frac{\log \left(\frac{1}{y}\right)}{y}\right) + \left(1 - \frac{z}{y}\right)\right)} \]

      3. +-commutative 99.9%

         \[\leadsto x + y \cdot \left(\color{blue}{\left(0.5 \cdot \frac{\log \left(\frac{1}{y}\right)}{y} + \log \left(\frac{1}{y}\right)\right)} + \left(1 - \frac{z}{y}\right)\right) \]

      4. log-rec 99.9%

         \[\leadsto x + y \cdot \left(\left(0.5 \cdot \frac{\log \left(\frac{1}{y}\right)}{y} + \color{blue}{\left(-\log y\right)}\right) + \left(1 - \frac{z}{y}\right)\right) \]

      5. unsub-neg 99.9%

         \[\leadsto x + y \cdot \left(\color{blue}{\left(0.5 \cdot \frac{\log \left(\frac{1}{y}\right)}{y} - \log y\right)} + \left(1 - \frac{z}{y}\right)\right) \]

      6. associate-*r/ 99.9%

         \[\leadsto x + y \cdot \left(\left(\color{blue}{\frac{0.5 \cdot \log \left(\frac{1}{y}\right)}{y}} - \log y\right) + \left(1 - \frac{z}{y}\right)\right) \]

      7. log-rec 99.9%

         \[\leadsto x + y \cdot \left(\left(\frac{0.5 \cdot \color{blue}{\left(-\log y\right)}}{y} - \log y\right) + \left(1 - \frac{z}{y}\right)\right) \]

      8. distribute-rgt-neg-in 99.9%

         \[\leadsto x + y \cdot \left(\left(\frac{\color{blue}{-0.5 \cdot \log y}}{y} - \log y\right) + \left(1 - \frac{z}{y}\right)\right) \]

      9. distribute-lft-neg-in 99.9%

         \[\leadsto x + y \cdot \left(\left(\frac{\color{blue}{\left(-0.5\right) \cdot \log y}}{y} - \log y\right) + \left(1 - \frac{z}{y}\right)\right) \]

      10. metadata-eval 99.9%

          \[\leadsto x + y \cdot \left(\left(\frac{\color{blue}{-0.5} \cdot \log y}{y} - \log y\right) + \left(1 - \frac{z}{y}\right)\right) \]

   7. Simplified 99.9%

      \[\leadsto x + \color{blue}{y \cdot \left(\left(\frac{-0.5 \cdot \log y}{y} - \log y\right) + \left(1 - \frac{z}{y}\right)\right)} \]

   8. Taylor expanded in y around 0 99.9%

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

      1. mul-1-neg 99.9%

         \[\leadsto x + y \cdot \left(\frac{\color{blue}{\left(-y \cdot \log y\right)} + -0.5 \cdot \log y}{y} + \left(1 - \frac{z}{y}\right)\right) \]

      2. metadata-eval 99.9%

         \[\leadsto x + y \cdot \left(\frac{\left(-y \cdot \log y\right) + \color{blue}{\left(-0.5\right)} \cdot \log y}{y} + \left(1 - \frac{z}{y}\right)\right) \]

      3. distribute-lft-neg-in 99.9%

         \[\leadsto x + y \cdot \left(\frac{\left(-y \cdot \log y\right) + \color{blue}{\left(-0.5 \cdot \log y\right)}}{y} + \left(1 - \frac{z}{y}\right)\right) \]

      4. distribute-neg-in 99.9%

         \[\leadsto x + y \cdot \left(\frac{\color{blue}{-\left(y \cdot \log y + 0.5 \cdot \log y\right)}}{y} + \left(1 - \frac{z}{y}\right)\right) \]

      5. distribute-rgt-in 99.9%

         \[\leadsto x + y \cdot \left(\frac{-\color{blue}{\log y \cdot \left(y + 0.5\right)}}{y} + \left(1 - \frac{z}{y}\right)\right) \]

      6. distribute-rgt-neg-in 99.9%

         \[\leadsto x + y \cdot \left(\frac{\color{blue}{\log y \cdot \left(-\left(y + 0.5\right)\right)}}{y} + \left(1 - \frac{z}{y}\right)\right) \]

      7. +-commutative 99.9%

         \[\leadsto x + y \cdot \left(\frac{\log y \cdot \left(-\color{blue}{\left(0.5 + y\right)}\right)}{y} + \left(1 - \frac{z}{y}\right)\right) \]

      8. distribute-neg-in 99.9%

         \[\leadsto x + y \cdot \left(\frac{\log y \cdot \color{blue}{\left(\left(-0.5\right) + \left(-y\right)\right)}}{y} + \left(1 - \frac{z}{y}\right)\right) \]

      9. metadata-eval 99.9%

         \[\leadsto x + y \cdot \left(\frac{\log y \cdot \left(\color{blue}{-0.5} + \left(-y\right)\right)}{y} + \left(1 - \frac{z}{y}\right)\right) \]

      10. sub-neg 99.9%

          \[\leadsto x + y \cdot \left(\frac{\log y \cdot \color{blue}{\left(-0.5 - y\right)}}{y} + \left(1 - \frac{z}{y}\right)\right) \]

   10. Simplified 99.9%

       \[\leadsto x + y \cdot \left(\color{blue}{\frac{\log y \cdot \left(-0.5 - y\right)}{y}} + \left(1 - \frac{z}{y}\right)\right) \]

   11. Taylor expanded in y around inf 85.1%

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

       1. log-rec 85.1%

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

       2. sub-neg 85.1%

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

   13. Simplified 85.1%

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

   if 4.60000000000000007e115 < 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 98.5%

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

      2. pow3 98.3%

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

      3. sub-neg 98.3%

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

      4. *-commutative 98.3%

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

      5. distribute-rgt-neg-in 98.3%

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

      6. +-commutative 98.3%

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

      7. distribute-neg-in 98.3%

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

      8. metadata-eval 98.3%

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

      9. sub-neg 98.3%

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

   4. Applied egg-rr 98.3%

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

   5. Taylor expanded in y around inf 88.3%

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

      1. log-rec 88.3%

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

      2. sub-neg 88.3%

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

   7. Simplified 88.3%

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

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.95 \cdot 10^{+14}:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{+115}:\\ \;\;\;\;x + y \cdot \left(1 - \log y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(1 - \log y\right)\\ \mathbf{if}\;y \leq 1.4 \cdot 10^{+16}:\\ \;\;\;\;\left(x + y\right) - z\\ \mathbf{elif}\;y \leq 2.55 \cdot 10^{+117}:\\ \;\;\;\;x + t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_0 - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (- 1.0 (log y)))))
   (if (<= y 1.4e+16)
     (- (+ x y) z)
     (if (<= y 2.55e+117) (+ x t_0) (- t_0 z)))))

C

double code(double x, double y, double z) {
	double t_0 = y * (1.0 - log(y));
	double tmp;
	if (y <= 1.4e+16) {
		tmp = (x + y) - z;
	} else if (y <= 2.55e+117) {
		tmp = x + t_0;
	} else {
		tmp = t_0 - 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) :: t_0
    real(8) :: tmp
    t_0 = y * (1.0d0 - log(y))
    if (y <= 1.4d+16) then
        tmp = (x + y) - z
    else if (y <= 2.55d+117) then
        tmp = x + t_0
    else
        tmp = t_0 - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y * (1.0 - Math.log(y));
	double tmp;
	if (y <= 1.4e+16) {
		tmp = (x + y) - z;
	} else if (y <= 2.55e+117) {
		tmp = x + t_0;
	} else {
		tmp = t_0 - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * (1.0 - math.log(y))
	tmp = 0
	if y <= 1.4e+16:
		tmp = (x + y) - z
	elif y <= 2.55e+117:
		tmp = x + t_0
	else:
		tmp = t_0 - z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * Float64(1.0 - log(y)))
	tmp = 0.0
	if (y <= 1.4e+16)
		tmp = Float64(Float64(x + y) - z);
	elseif (y <= 2.55e+117)
		tmp = Float64(x + t_0);
	else
		tmp = Float64(t_0 - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * (1.0 - log(y));
	tmp = 0.0;
	if (y <= 1.4e+16)
		tmp = (x + y) - z;
	elseif (y <= 2.55e+117)
		tmp = x + t_0;
	else
		tmp = t_0 - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 1.4e+16], N[(N[(x + y), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[y, 2.55e+117], N[(x + t$95$0), $MachinePrecision], N[(t$95$0 - z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.4e16

   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 98.9%

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

      2. pow2 98.9%

         \[\leadsto \left(\color{blue}{{\left(\sqrt[3]{x - \left(y + 0.5\right) \cdot \log y}\right)}^{2}} \cdot \sqrt[3]{x - \left(y + 0.5\right) \cdot \log y} + y\right) - z \]

      3. sub-neg 98.9%

         \[\leadsto \left({\left(\sqrt[3]{\color{blue}{x + \left(-\left(y + 0.5\right) \cdot \log y\right)}}\right)}^{2} \cdot \sqrt[3]{x - \left(y + 0.5\right) \cdot \log y} + y\right) - z \]

      4. *-commutative 98.9%

         \[\leadsto \left({\left(\sqrt[3]{x + \left(-\color{blue}{\log y \cdot \left(y + 0.5\right)}\right)}\right)}^{2} \cdot \sqrt[3]{x - \left(y + 0.5\right) \cdot \log y} + y\right) - z \]

      5. distribute-rgt-neg-in 98.9%

         \[\leadsto \left({\left(\sqrt[3]{x + \color{blue}{\log y \cdot \left(-\left(y + 0.5\right)\right)}}\right)}^{2} \cdot \sqrt[3]{x - \left(y + 0.5\right) \cdot \log y} + y\right) - z \]

      6. +-commutative 98.9%

         \[\leadsto \left({\left(\sqrt[3]{x + \log y \cdot \left(-\color{blue}{\left(0.5 + y\right)}\right)}\right)}^{2} \cdot \sqrt[3]{x - \left(y + 0.5\right) \cdot \log y} + y\right) - z \]

      7. distribute-neg-in 98.9%

         \[\leadsto \left({\left(\sqrt[3]{x + \log y \cdot \color{blue}{\left(\left(-0.5\right) + \left(-y\right)\right)}}\right)}^{2} \cdot \sqrt[3]{x - \left(y + 0.5\right) \cdot \log y} + y\right) - z \]

      8. metadata-eval 98.9%

         \[\leadsto \left({\left(\sqrt[3]{x + \log y \cdot \left(\color{blue}{-0.5} + \left(-y\right)\right)}\right)}^{2} \cdot \sqrt[3]{x - \left(y + 0.5\right) \cdot \log y} + y\right) - z \]

      9. sub-neg 98.9%

         \[\leadsto \left({\left(\sqrt[3]{x + \log y \cdot \color{blue}{\left(-0.5 - y\right)}}\right)}^{2} \cdot \sqrt[3]{x - \left(y + 0.5\right) \cdot \log y} + y\right) - z \]

      10. sub-neg 98.9%

          \[\leadsto \left({\left(\sqrt[3]{x + \log y \cdot \left(-0.5 - y\right)}\right)}^{2} \cdot \sqrt[3]{\color{blue}{x + \left(-\left(y + 0.5\right) \cdot \log y\right)}} + y\right) - z \]

      11. *-commutative 98.9%

          \[\leadsto \left({\left(\sqrt[3]{x + \log y \cdot \left(-0.5 - y\right)}\right)}^{2} \cdot \sqrt[3]{x + \left(-\color{blue}{\log y \cdot \left(y + 0.5\right)}\right)} + y\right) - z \]

      12. distribute-rgt-neg-in 98.9%

          \[\leadsto \left({\left(\sqrt[3]{x + \log y \cdot \left(-0.5 - y\right)}\right)}^{2} \cdot \sqrt[3]{x + \color{blue}{\log y \cdot \left(-\left(y + 0.5\right)\right)}} + y\right) - z \]

      13. +-commutative 98.9%

          \[\leadsto \left({\left(\sqrt[3]{x + \log y \cdot \left(-0.5 - y\right)}\right)}^{2} \cdot \sqrt[3]{x + \log y \cdot \left(-\color{blue}{\left(0.5 + y\right)}\right)} + y\right) - z \]

      14. distribute-neg-in 98.9%

          \[\leadsto \left({\left(\sqrt[3]{x + \log y \cdot \left(-0.5 - y\right)}\right)}^{2} \cdot \sqrt[3]{x + \log y \cdot \color{blue}{\left(\left(-0.5\right) + \left(-y\right)\right)}} + y\right) - z \]

      15. metadata-eval 98.9%

          \[\leadsto \left({\left(\sqrt[3]{x + \log y \cdot \left(-0.5 - y\right)}\right)}^{2} \cdot \sqrt[3]{x + \log y \cdot \left(\color{blue}{-0.5} + \left(-y\right)\right)} + y\right) - z \]

      16. sub-neg 98.9%

          \[\leadsto \left({\left(\sqrt[3]{x + \log y \cdot \left(-0.5 - y\right)}\right)}^{2} \cdot \sqrt[3]{x + \log y \cdot \color{blue}{\left(-0.5 - y\right)}} + y\right) - z \]

   4. Applied egg-rr 98.9%

      \[\leadsto \left(\color{blue}{{\left(\sqrt[3]{x + \log y \cdot \left(-0.5 - y\right)}\right)}^{2} \cdot \sqrt[3]{x + \log y \cdot \left(-0.5 - y\right)}} + y\right) - z \]

   5. Taylor expanded in x around inf 72.9%

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

   if 1.4e16 < y < 2.5499999999999998e117

   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. Taylor expanded in y around inf 99.9%

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

      1. +-commutative 99.9%

         \[\leadsto x + y \cdot \left(\color{blue}{\left(\left(\log \left(\frac{1}{y}\right) + 0.5 \cdot \frac{\log \left(\frac{1}{y}\right)}{y}\right) + 1\right)} - \frac{z}{y}\right) \]

      2. associate--l+ 99.9%

         \[\leadsto x + y \cdot \color{blue}{\left(\left(\log \left(\frac{1}{y}\right) + 0.5 \cdot \frac{\log \left(\frac{1}{y}\right)}{y}\right) + \left(1 - \frac{z}{y}\right)\right)} \]

      3. +-commutative 99.9%

         \[\leadsto x + y \cdot \left(\color{blue}{\left(0.5 \cdot \frac{\log \left(\frac{1}{y}\right)}{y} + \log \left(\frac{1}{y}\right)\right)} + \left(1 - \frac{z}{y}\right)\right) \]

      4. log-rec 99.9%

         \[\leadsto x + y \cdot \left(\left(0.5 \cdot \frac{\log \left(\frac{1}{y}\right)}{y} + \color{blue}{\left(-\log y\right)}\right) + \left(1 - \frac{z}{y}\right)\right) \]

      5. unsub-neg 99.9%

         \[\leadsto x + y \cdot \left(\color{blue}{\left(0.5 \cdot \frac{\log \left(\frac{1}{y}\right)}{y} - \log y\right)} + \left(1 - \frac{z}{y}\right)\right) \]

      6. associate-*r/ 99.9%

         \[\leadsto x + y \cdot \left(\left(\color{blue}{\frac{0.5 \cdot \log \left(\frac{1}{y}\right)}{y}} - \log y\right) + \left(1 - \frac{z}{y}\right)\right) \]

      7. log-rec 99.9%

         \[\leadsto x + y \cdot \left(\left(\frac{0.5 \cdot \color{blue}{\left(-\log y\right)}}{y} - \log y\right) + \left(1 - \frac{z}{y}\right)\right) \]

      8. distribute-rgt-neg-in 99.9%

         \[\leadsto x + y \cdot \left(\left(\frac{\color{blue}{-0.5 \cdot \log y}}{y} - \log y\right) + \left(1 - \frac{z}{y}\right)\right) \]

      9. distribute-lft-neg-in 99.9%

         \[\leadsto x + y \cdot \left(\left(\frac{\color{blue}{\left(-0.5\right) \cdot \log y}}{y} - \log y\right) + \left(1 - \frac{z}{y}\right)\right) \]

      10. metadata-eval 99.9%

          \[\leadsto x + y \cdot \left(\left(\frac{\color{blue}{-0.5} \cdot \log y}{y} - \log y\right) + \left(1 - \frac{z}{y}\right)\right) \]

   7. Simplified 99.9%

      \[\leadsto x + \color{blue}{y \cdot \left(\left(\frac{-0.5 \cdot \log y}{y} - \log y\right) + \left(1 - \frac{z}{y}\right)\right)} \]

   8. Taylor expanded in y around 0 99.9%

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

      1. mul-1-neg 99.9%

         \[\leadsto x + y \cdot \left(\frac{\color{blue}{\left(-y \cdot \log y\right)} + -0.5 \cdot \log y}{y} + \left(1 - \frac{z}{y}\right)\right) \]

      2. metadata-eval 99.9%

         \[\leadsto x + y \cdot \left(\frac{\left(-y \cdot \log y\right) + \color{blue}{\left(-0.5\right)} \cdot \log y}{y} + \left(1 - \frac{z}{y}\right)\right) \]

      3. distribute-lft-neg-in 99.9%

         \[\leadsto x + y \cdot \left(\frac{\left(-y \cdot \log y\right) + \color{blue}{\left(-0.5 \cdot \log y\right)}}{y} + \left(1 - \frac{z}{y}\right)\right) \]

      4. distribute-neg-in 99.9%

         \[\leadsto x + y \cdot \left(\frac{\color{blue}{-\left(y \cdot \log y + 0.5 \cdot \log y\right)}}{y} + \left(1 - \frac{z}{y}\right)\right) \]

      5. distribute-rgt-in 99.9%

         \[\leadsto x + y \cdot \left(\frac{-\color{blue}{\log y \cdot \left(y + 0.5\right)}}{y} + \left(1 - \frac{z}{y}\right)\right) \]

      6. distribute-rgt-neg-in 99.9%

         \[\leadsto x + y \cdot \left(\frac{\color{blue}{\log y \cdot \left(-\left(y + 0.5\right)\right)}}{y} + \left(1 - \frac{z}{y}\right)\right) \]

      7. +-commutative 99.9%

         \[\leadsto x + y \cdot \left(\frac{\log y \cdot \left(-\color{blue}{\left(0.5 + y\right)}\right)}{y} + \left(1 - \frac{z}{y}\right)\right) \]

      8. distribute-neg-in 99.9%

         \[\leadsto x + y \cdot \left(\frac{\log y \cdot \color{blue}{\left(\left(-0.5\right) + \left(-y\right)\right)}}{y} + \left(1 - \frac{z}{y}\right)\right) \]

      9. metadata-eval 99.9%

         \[\leadsto x + y \cdot \left(\frac{\log y \cdot \left(\color{blue}{-0.5} + \left(-y\right)\right)}{y} + \left(1 - \frac{z}{y}\right)\right) \]

      10. sub-neg 99.9%

          \[\leadsto x + y \cdot \left(\frac{\log y \cdot \color{blue}{\left(-0.5 - y\right)}}{y} + \left(1 - \frac{z}{y}\right)\right) \]

   10. Simplified 99.9%

       \[\leadsto x + y \cdot \left(\color{blue}{\frac{\log y \cdot \left(-0.5 - y\right)}{y}} + \left(1 - \frac{z}{y}\right)\right) \]

   11. Taylor expanded in y around inf 85.1%

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

       1. log-rec 85.1%

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

       2. sub-neg 85.1%

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

   13. Simplified 85.1%

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

   if 2.5499999999999998e117 < 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 98.5%

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

      2. pow3 98.3%

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

      3. sub-neg 98.3%

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

      4. *-commutative 98.3%

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

      5. distribute-rgt-neg-in 98.3%

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

      6. +-commutative 98.3%

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

      7. distribute-neg-in 98.3%

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

      8. metadata-eval 98.3%

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

      9. sub-neg 98.3%

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

   4. Applied egg-rr 98.3%

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

   5. Taylor expanded in y around inf 88.3%

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

      1. log-rec 88.3%

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

      2. sub-neg 88.3%

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

   7. Simplified 88.3%

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

Alternative 4: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.3:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + \left(y \cdot \left(1 - \log y\right) - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 0.3) (- (- 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 <= 0.3) {
		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 <= 0.3d0) 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 <= 0.3) {
		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 <= 0.3:
		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 <= 0.3)
		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 <= 0.3)
		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, 0.3], 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 < 0.299999999999999989

   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%

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

   if 0.299999999999999989 < 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 99.1%

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

      1. log-rec 99.1%

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

      2. sub-neg 99.1%

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

   7. Simplified 99.1%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.3:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + \left(y \cdot \left(1 - \log y\right) - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.8 \cdot 10^{+14}:\\ \;\;\;\;\left(x + y\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(1 - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2.8e+14) (- (+ x y) z) (+ x (* y (- 1.0 (log y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.8e+14) {
		tmp = (x + y) - 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 <= 2.8d+14) then
        tmp = (x + y) - 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 <= 2.8e+14) {
		tmp = (x + y) - z;
	} else {
		tmp = x + (y * (1.0 - Math.log(y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 2.8e+14:
		tmp = (x + y) - z
	else:
		tmp = x + (y * (1.0 - math.log(y)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2.8e+14)
		tmp = Float64(Float64(x + y) - 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 <= 2.8e+14)
		tmp = (x + y) - z;
	else
		tmp = x + (y * (1.0 - log(y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2.8e+14], N[(N[(x + y), $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 < 2.8e14

   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 98.9%

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

      2. pow2 98.9%

         \[\leadsto \left(\color{blue}{{\left(\sqrt[3]{x - \left(y + 0.5\right) \cdot \log y}\right)}^{2}} \cdot \sqrt[3]{x - \left(y + 0.5\right) \cdot \log y} + y\right) - z \]

      3. sub-neg 98.9%

         \[\leadsto \left({\left(\sqrt[3]{\color{blue}{x + \left(-\left(y + 0.5\right) \cdot \log y\right)}}\right)}^{2} \cdot \sqrt[3]{x - \left(y + 0.5\right) \cdot \log y} + y\right) - z \]

      4. *-commutative 98.9%

         \[\leadsto \left({\left(\sqrt[3]{x + \left(-\color{blue}{\log y \cdot \left(y + 0.5\right)}\right)}\right)}^{2} \cdot \sqrt[3]{x - \left(y + 0.5\right) \cdot \log y} + y\right) - z \]

      5. distribute-rgt-neg-in 98.9%

         \[\leadsto \left({\left(\sqrt[3]{x + \color{blue}{\log y \cdot \left(-\left(y + 0.5\right)\right)}}\right)}^{2} \cdot \sqrt[3]{x - \left(y + 0.5\right) \cdot \log y} + y\right) - z \]

      6. +-commutative 98.9%

         \[\leadsto \left({\left(\sqrt[3]{x + \log y \cdot \left(-\color{blue}{\left(0.5 + y\right)}\right)}\right)}^{2} \cdot \sqrt[3]{x - \left(y + 0.5\right) \cdot \log y} + y\right) - z \]

      7. distribute-neg-in 98.9%

         \[\leadsto \left({\left(\sqrt[3]{x + \log y \cdot \color{blue}{\left(\left(-0.5\right) + \left(-y\right)\right)}}\right)}^{2} \cdot \sqrt[3]{x - \left(y + 0.5\right) \cdot \log y} + y\right) - z \]

      8. metadata-eval 98.9%

         \[\leadsto \left({\left(\sqrt[3]{x + \log y \cdot \left(\color{blue}{-0.5} + \left(-y\right)\right)}\right)}^{2} \cdot \sqrt[3]{x - \left(y + 0.5\right) \cdot \log y} + y\right) - z \]

      9. sub-neg 98.9%

         \[\leadsto \left({\left(\sqrt[3]{x + \log y \cdot \color{blue}{\left(-0.5 - y\right)}}\right)}^{2} \cdot \sqrt[3]{x - \left(y + 0.5\right) \cdot \log y} + y\right) - z \]

      10. sub-neg 98.9%

          \[\leadsto \left({\left(\sqrt[3]{x + \log y \cdot \left(-0.5 - y\right)}\right)}^{2} \cdot \sqrt[3]{\color{blue}{x + \left(-\left(y + 0.5\right) \cdot \log y\right)}} + y\right) - z \]

      11. *-commutative 98.9%

          \[\leadsto \left({\left(\sqrt[3]{x + \log y \cdot \left(-0.5 - y\right)}\right)}^{2} \cdot \sqrt[3]{x + \left(-\color{blue}{\log y \cdot \left(y + 0.5\right)}\right)} + y\right) - z \]

      12. distribute-rgt-neg-in 98.9%

          \[\leadsto \left({\left(\sqrt[3]{x + \log y \cdot \left(-0.5 - y\right)}\right)}^{2} \cdot \sqrt[3]{x + \color{blue}{\log y \cdot \left(-\left(y + 0.5\right)\right)}} + y\right) - z \]

      13. +-commutative 98.9%

          \[\leadsto \left({\left(\sqrt[3]{x + \log y \cdot \left(-0.5 - y\right)}\right)}^{2} \cdot \sqrt[3]{x + \log y \cdot \left(-\color{blue}{\left(0.5 + y\right)}\right)} + y\right) - z \]

      14. distribute-neg-in 98.9%

          \[\leadsto \left({\left(\sqrt[3]{x + \log y \cdot \left(-0.5 - y\right)}\right)}^{2} \cdot \sqrt[3]{x + \log y \cdot \color{blue}{\left(\left(-0.5\right) + \left(-y\right)\right)}} + y\right) - z \]

      15. metadata-eval 98.9%

          \[\leadsto \left({\left(\sqrt[3]{x + \log y \cdot \left(-0.5 - y\right)}\right)}^{2} \cdot \sqrt[3]{x + \log y \cdot \left(\color{blue}{-0.5} + \left(-y\right)\right)} + y\right) - z \]

      16. sub-neg 98.9%

          \[\leadsto \left({\left(\sqrt[3]{x + \log y \cdot \left(-0.5 - y\right)}\right)}^{2} \cdot \sqrt[3]{x + \log y \cdot \color{blue}{\left(-0.5 - y\right)}} + y\right) - z \]

   4. Applied egg-rr 98.9%

      \[\leadsto \left(\color{blue}{{\left(\sqrt[3]{x + \log y \cdot \left(-0.5 - y\right)}\right)}^{2} \cdot \sqrt[3]{x + \log y \cdot \left(-0.5 - y\right)}} + y\right) - z \]

   5. Taylor expanded in x around inf 72.9%

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

   if 2.8e14 < 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 99.7%

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

      1. +-commutative 99.7%

         \[\leadsto x + y \cdot \left(\color{blue}{\left(\left(\log \left(\frac{1}{y}\right) + 0.5 \cdot \frac{\log \left(\frac{1}{y}\right)}{y}\right) + 1\right)} - \frac{z}{y}\right) \]

      2. associate--l+ 99.7%

         \[\leadsto x + y \cdot \color{blue}{\left(\left(\log \left(\frac{1}{y}\right) + 0.5 \cdot \frac{\log \left(\frac{1}{y}\right)}{y}\right) + \left(1 - \frac{z}{y}\right)\right)} \]

      3. +-commutative 99.7%

         \[\leadsto x + y \cdot \left(\color{blue}{\left(0.5 \cdot \frac{\log \left(\frac{1}{y}\right)}{y} + \log \left(\frac{1}{y}\right)\right)} + \left(1 - \frac{z}{y}\right)\right) \]

      4. log-rec 99.7%

         \[\leadsto x + y \cdot \left(\left(0.5 \cdot \frac{\log \left(\frac{1}{y}\right)}{y} + \color{blue}{\left(-\log y\right)}\right) + \left(1 - \frac{z}{y}\right)\right) \]

      5. unsub-neg 99.7%

         \[\leadsto x + y \cdot \left(\color{blue}{\left(0.5 \cdot \frac{\log \left(\frac{1}{y}\right)}{y} - \log y\right)} + \left(1 - \frac{z}{y}\right)\right) \]

      6. associate-*r/ 99.7%

         \[\leadsto x + y \cdot \left(\left(\color{blue}{\frac{0.5 \cdot \log \left(\frac{1}{y}\right)}{y}} - \log y\right) + \left(1 - \frac{z}{y}\right)\right) \]

      7. log-rec 99.7%

         \[\leadsto x + y \cdot \left(\left(\frac{0.5 \cdot \color{blue}{\left(-\log y\right)}}{y} - \log y\right) + \left(1 - \frac{z}{y}\right)\right) \]

      8. distribute-rgt-neg-in 99.7%

         \[\leadsto x + y \cdot \left(\left(\frac{\color{blue}{-0.5 \cdot \log y}}{y} - \log y\right) + \left(1 - \frac{z}{y}\right)\right) \]

      9. distribute-lft-neg-in 99.7%

         \[\leadsto x + y \cdot \left(\left(\frac{\color{blue}{\left(-0.5\right) \cdot \log y}}{y} - \log y\right) + \left(1 - \frac{z}{y}\right)\right) \]

      10. metadata-eval 99.7%

          \[\leadsto x + y \cdot \left(\left(\frac{\color{blue}{-0.5} \cdot \log y}{y} - \log y\right) + \left(1 - \frac{z}{y}\right)\right) \]

   7. Simplified 99.7%

      \[\leadsto x + \color{blue}{y \cdot \left(\left(\frac{-0.5 \cdot \log y}{y} - \log y\right) + \left(1 - \frac{z}{y}\right)\right)} \]

   8. Taylor expanded in y around 0 99.7%

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

      1. mul-1-neg 99.7%

         \[\leadsto x + y \cdot \left(\frac{\color{blue}{\left(-y \cdot \log y\right)} + -0.5 \cdot \log y}{y} + \left(1 - \frac{z}{y}\right)\right) \]

      2. metadata-eval 99.7%

         \[\leadsto x + y \cdot \left(\frac{\left(-y \cdot \log y\right) + \color{blue}{\left(-0.5\right)} \cdot \log y}{y} + \left(1 - \frac{z}{y}\right)\right) \]

      3. distribute-lft-neg-in 99.7%

         \[\leadsto x + y \cdot \left(\frac{\left(-y \cdot \log y\right) + \color{blue}{\left(-0.5 \cdot \log y\right)}}{y} + \left(1 - \frac{z}{y}\right)\right) \]

      4. distribute-neg-in 99.7%

         \[\leadsto x + y \cdot \left(\frac{\color{blue}{-\left(y \cdot \log y + 0.5 \cdot \log y\right)}}{y} + \left(1 - \frac{z}{y}\right)\right) \]

      5. distribute-rgt-in 99.7%

         \[\leadsto x + y \cdot \left(\frac{-\color{blue}{\log y \cdot \left(y + 0.5\right)}}{y} + \left(1 - \frac{z}{y}\right)\right) \]

      6. distribute-rgt-neg-in 99.7%

         \[\leadsto x + y \cdot \left(\frac{\color{blue}{\log y \cdot \left(-\left(y + 0.5\right)\right)}}{y} + \left(1 - \frac{z}{y}\right)\right) \]

      7. +-commutative 99.7%

         \[\leadsto x + y \cdot \left(\frac{\log y \cdot \left(-\color{blue}{\left(0.5 + y\right)}\right)}{y} + \left(1 - \frac{z}{y}\right)\right) \]

      8. distribute-neg-in 99.7%

         \[\leadsto x + y \cdot \left(\frac{\log y \cdot \color{blue}{\left(\left(-0.5\right) + \left(-y\right)\right)}}{y} + \left(1 - \frac{z}{y}\right)\right) \]

      9. metadata-eval 99.7%

         \[\leadsto x + y \cdot \left(\frac{\log y \cdot \left(\color{blue}{-0.5} + \left(-y\right)\right)}{y} + \left(1 - \frac{z}{y}\right)\right) \]

      10. sub-neg 99.7%

          \[\leadsto x + y \cdot \left(\frac{\log y \cdot \color{blue}{\left(-0.5 - y\right)}}{y} + \left(1 - \frac{z}{y}\right)\right) \]

   10. Simplified 99.7%

       \[\leadsto x + y \cdot \left(\color{blue}{\frac{\log y \cdot \left(-0.5 - y\right)}{y}} + \left(1 - \frac{z}{y}\right)\right) \]

   11. Taylor expanded in y around inf 82.7%

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

       1. log-rec 82.7%

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

       2. sub-neg 82.7%

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

   13. Simplified 82.7%

       \[\leadsto x + \color{blue}{y \cdot \left(1 - \log y\right)} \]
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: 72.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 8.5 \cdot 10^{+118}:\\ \;\;\;\;\left(x + y\right) - z\\ \mathbf{else}:\\ \;\;\;\;y - y \cdot \log y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 8.5e+118) (- (+ x y) z) (- y (* y (log y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 8.5e+118) {
		tmp = (x + y) - 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 <= 8.5d+118) then
        tmp = (x + y) - 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 <= 8.5e+118) {
		tmp = (x + y) - z;
	} else {
		tmp = y - (y * Math.log(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 8.5e+118:
		tmp = (x + y) - z
	else:
		tmp = y - (y * math.log(y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 8.5e+118)
		tmp = Float64(Float64(x + y) - 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 <= 8.5e+118)
		tmp = (x + y) - z;
	else
		tmp = y - (y * log(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 8.5e+118], N[(N[(x + y), $MachinePrecision] - z), $MachinePrecision], N[(y - N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 8.50000000000000033e118

   1. Initial program 99.9%

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

      1. add-cube-cbrt 98.8%

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

      2. pow2 98.8%

         \[\leadsto \left(\color{blue}{{\left(\sqrt[3]{x - \left(y + 0.5\right) \cdot \log y}\right)}^{2}} \cdot \sqrt[3]{x - \left(y + 0.5\right) \cdot \log y} + y\right) - z \]

      3. sub-neg 98.8%

         \[\leadsto \left({\left(\sqrt[3]{\color{blue}{x + \left(-\left(y + 0.5\right) \cdot \log y\right)}}\right)}^{2} \cdot \sqrt[3]{x - \left(y + 0.5\right) \cdot \log y} + y\right) - z \]

      4. *-commutative 98.8%

         \[\leadsto \left({\left(\sqrt[3]{x + \left(-\color{blue}{\log y \cdot \left(y + 0.5\right)}\right)}\right)}^{2} \cdot \sqrt[3]{x - \left(y + 0.5\right) \cdot \log y} + y\right) - z \]

      5. distribute-rgt-neg-in 98.8%

         \[\leadsto \left({\left(\sqrt[3]{x + \color{blue}{\log y \cdot \left(-\left(y + 0.5\right)\right)}}\right)}^{2} \cdot \sqrt[3]{x - \left(y + 0.5\right) \cdot \log y} + y\right) - z \]

      6. +-commutative 98.8%

         \[\leadsto \left({\left(\sqrt[3]{x + \log y \cdot \left(-\color{blue}{\left(0.5 + y\right)}\right)}\right)}^{2} \cdot \sqrt[3]{x - \left(y + 0.5\right) \cdot \log y} + y\right) - z \]

      7. distribute-neg-in 98.8%

         \[\leadsto \left({\left(\sqrt[3]{x + \log y \cdot \color{blue}{\left(\left(-0.5\right) + \left(-y\right)\right)}}\right)}^{2} \cdot \sqrt[3]{x - \left(y + 0.5\right) \cdot \log y} + y\right) - z \]

      8. metadata-eval 98.8%

         \[\leadsto \left({\left(\sqrt[3]{x + \log y \cdot \left(\color{blue}{-0.5} + \left(-y\right)\right)}\right)}^{2} \cdot \sqrt[3]{x - \left(y + 0.5\right) \cdot \log y} + y\right) - z \]

      9. sub-neg 98.8%

         \[\leadsto \left({\left(\sqrt[3]{x + \log y \cdot \color{blue}{\left(-0.5 - y\right)}}\right)}^{2} \cdot \sqrt[3]{x - \left(y + 0.5\right) \cdot \log y} + y\right) - z \]

      10. sub-neg 98.8%

          \[\leadsto \left({\left(\sqrt[3]{x + \log y \cdot \left(-0.5 - y\right)}\right)}^{2} \cdot \sqrt[3]{\color{blue}{x + \left(-\left(y + 0.5\right) \cdot \log y\right)}} + y\right) - z \]

      11. *-commutative 98.8%

          \[\leadsto \left({\left(\sqrt[3]{x + \log y \cdot \left(-0.5 - y\right)}\right)}^{2} \cdot \sqrt[3]{x + \left(-\color{blue}{\log y \cdot \left(y + 0.5\right)}\right)} + y\right) - z \]

      12. distribute-rgt-neg-in 98.8%

          \[\leadsto \left({\left(\sqrt[3]{x + \log y \cdot \left(-0.5 - y\right)}\right)}^{2} \cdot \sqrt[3]{x + \color{blue}{\log y \cdot \left(-\left(y + 0.5\right)\right)}} + y\right) - z \]

      13. +-commutative 98.8%

          \[\leadsto \left({\left(\sqrt[3]{x + \log y \cdot \left(-0.5 - y\right)}\right)}^{2} \cdot \sqrt[3]{x + \log y \cdot \left(-\color{blue}{\left(0.5 + y\right)}\right)} + y\right) - z \]

      14. distribute-neg-in 98.8%

          \[\leadsto \left({\left(\sqrt[3]{x + \log y \cdot \left(-0.5 - y\right)}\right)}^{2} \cdot \sqrt[3]{x + \log y \cdot \color{blue}{\left(\left(-0.5\right) + \left(-y\right)\right)}} + y\right) - z \]

      15. metadata-eval 98.8%

          \[\leadsto \left({\left(\sqrt[3]{x + \log y \cdot \left(-0.5 - y\right)}\right)}^{2} \cdot \sqrt[3]{x + \log y \cdot \left(\color{blue}{-0.5} + \left(-y\right)\right)} + y\right) - z \]

      16. sub-neg 98.8%

          \[\leadsto \left({\left(\sqrt[3]{x + \log y \cdot \left(-0.5 - y\right)}\right)}^{2} \cdot \sqrt[3]{x + \log y \cdot \color{blue}{\left(-0.5 - y\right)}} + y\right) - z \]

   4. Applied egg-rr 98.8%

      \[\leadsto \left(\color{blue}{{\left(\sqrt[3]{x + \log y \cdot \left(-0.5 - y\right)}\right)}^{2} \cdot \sqrt[3]{x + \log y \cdot \left(-0.5 - y\right)}} + y\right) - z \]

   5. Taylor expanded in x around inf 70.6%

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

   if 8.50000000000000033e118 < 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. Taylor expanded in y around inf 88.1%

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

      1. *-commutative 88.1%

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

      2. log-rec 88.1%

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

      3. distribute-lft-neg-in 88.1%

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

      4. distribute-rgt-neg-in 88.1%

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

   5. Simplified 88.1%

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

   6. Taylor expanded in z around inf 51.4%

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

      1. sub-neg 51.4%

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

      2. metadata-eval 51.4%

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

      3. +-commutative 51.4%

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

      4. +-commutative 51.4%

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

      5. mul-1-neg 51.4%

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

      6. unsub-neg 51.4%

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

      7. associate-/l* 51.4%

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

   8. Simplified 51.4%

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

   9. Taylor expanded in z around 0 71.2%

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

Alternative 8: 49.4% accurate, 9.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{+38}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 9.8 \cdot 10^{+86}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.25e+38) x (if (<= x 9.8e+86) (- z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.25e+38) {
		tmp = x;
	} else if (x <= 9.8e+86) {
		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 <= (-1.25d+38)) then
        tmp = x
    else if (x <= 9.8d+86) 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 <= -1.25e+38) {
		tmp = x;
	} else if (x <= 9.8e+86) {
		tmp = -z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.25e+38:
		tmp = x
	elif x <= 9.8e+86:
		tmp = -z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.25e+38)
		tmp = x;
	elseif (x <= 9.8e+86)
		tmp = Float64(-z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.25e+38)
		tmp = x;
	elseif (x <= 9.8e+86)
		tmp = -z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.25e+38], x, If[LessEqual[x, 9.8e+86], (-z), x]]

Derivation

1. Split input into 2 regimes

2. if x < -1.24999999999999992e38 or 9.7999999999999999e86 < 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. associate-+r- 99.9%

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

      5. *-commutative 99.9%

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

      6. distribute-rgt-neg-in 99.9%

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

      7. fma-define 100.0%

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

      8. +-commutative 100.0%

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

      9. distribute-neg-in 100.0%

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

      10. unsub-neg 100.0%

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

      11. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 66.4%

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

   if -1.24999999999999992e38 < x < 9.7999999999999999e86

   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 y around inf 72.7%

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

      1. *-commutative 72.7%

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

      2. log-rec 72.7%

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

      3. distribute-lft-neg-in 72.7%

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

      4. distribute-rgt-neg-in 72.7%

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

   5. Simplified 72.7%

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

   6. Taylor expanded in y around 0 38.2%

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

      1. neg-mul-1 38.2%

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

   8. Simplified 38.2%

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

Alternative 9: 59.0% 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. 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. Taylor expanded in z around inf 58.1%

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

   1. neg-mul-1 58.1%

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

7. Simplified 58.1%

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

8. Final simplification 58.1%

   \[\leadsto x - z \]
9. Add Preprocessing

Alternative 10: 30.2% 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. 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. Taylor expanded in x around inf 28.3%

   \[\leadsto \color{blue}{x} \]
6. 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 2024103 
(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))