Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Percentage Accurate: 99.8% → 99.9%

Time: 12.5s

Alternatives: 9

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. associate--l+ 99.8%

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

   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- 99.9%

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

   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-def 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. Final simplification 99.9%

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

Alternative 2: 70.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -19500 \lor \neg \left(x \leq 8000000000000\right):\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -19500.0) (not (<= x 8000000000000.0)))
   (- x z)
   (- (* (log y) -0.5) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -19500.0) || !(x <= 8000000000000.0)) {
		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 <= (-19500.0d0)) .or. (.not. (x <= 8000000000000.0d0))) 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 <= -19500.0) || !(x <= 8000000000000.0)) {
		tmp = x - z;
	} else {
		tmp = (Math.log(y) * -0.5) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -19500.0) or not (x <= 8000000000000.0):
		tmp = x - z
	else:
		tmp = (math.log(y) * -0.5) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -19500.0) || !(x <= 8000000000000.0))
		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 <= -19500.0) || ~((x <= 8000000000000.0)))
		tmp = x - z;
	else
		tmp = (log(y) * -0.5) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -19500.0], N[Not[LessEqual[x, 8000000000000.0]], $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 < -19500 or 8e12 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 77.6%

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

   4. Taylor expanded in x around inf 77.1%

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

   if -19500 < x < 8e12

   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 0 72.0%

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

   4. Taylor expanded in x around 0 71.1%

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

      1. *-commutative 71.1%

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

   6. Simplified 71.1%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -19500 \lor \neg \left(x \leq 8000000000000\right):\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7 \cdot 10^{-20}:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + \left(y \cdot \left(1 - \log y\right) - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 7e-20)
   (- (- 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 <= 7e-20) {
		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 <= 7d-20) 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 <= 7e-20) {
		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 <= 7e-20:
		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 <= 7e-20)
		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 <= 7e-20)
		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, 7e-20], 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 < 7.00000000000000007e-20

   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 0 100.0%

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

   if 7.00000000000000007e-20 < 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-def 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%

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

      1. log-rec 99.9%

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

      2. sub-neg 99.9%

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

   7. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7 \cdot 10^{-20}:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + \left(y \cdot \left(1 - \log y\right) - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.32 \cdot 10^{+23}:\\ \;\;\;\;\left(x + y\right) - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.32e+23) (- (+ x y) z) (- (* y (- 1.0 (log y))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.32e+23) {
		tmp = (x + y) - z;
	} else {
		tmp = (y * (1.0 - log(y))) - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 1.32d+23) then
        tmp = (x + y) - z
    else
        tmp = (y * (1.0d0 - log(y))) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.32e+23) {
		tmp = (x + y) - z;
	} else {
		tmp = (y * (1.0 - Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1.32e+23:
		tmp = (x + y) - z
	else:
		tmp = (y * (1.0 - math.log(y))) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.32e+23)
		tmp = Float64(Float64(x + y) - z);
	else
		tmp = Float64(Float64(y * Float64(1.0 - log(y))) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1.32e+23)
		tmp = (x + y) - z;
	else
		tmp = (y * (1.0 - log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.32e+23], N[(N[(x + y), $MachinePrecision] - z), $MachinePrecision], N[(N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.3199999999999999e23

   1. Initial program 100.0%

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

      1. add-cube-cbrt 98.8%

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

      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 70.9%

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

   if 1.3199999999999999e23 < y

   1. Initial program 99.6%

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

      1. add-cube-cbrt 98.4%

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

      2. pow2 98.4%

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

      3. sub-neg 98.4%

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

      4. *-commutative 98.4%

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

      5. distribute-rgt-neg-in 98.4%

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

      6. +-commutative 98.4%

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

      7. distribute-neg-in 98.4%

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

      8. metadata-eval 98.4%

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

      9. sub-neg 98.4%

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

      10. sub-neg 98.4%

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

      11. *-commutative 98.4%

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

      12. distribute-rgt-neg-in 98.4%

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

      13. +-commutative 98.4%

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

      14. distribute-neg-in 98.4%

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

      15. metadata-eval 98.4%

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

      16. sub-neg 98.4%

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

   4. Applied egg-rr 98.4%

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

   5. Taylor expanded in y around inf 83.8%

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

      1. log-rec 83.8%

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

      2. sub-neg 83.8%

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

   7. Simplified 83.8%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.32 \cdot 10^{+23}:\\ \;\;\;\;\left(x + y\right) - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6.2 \cdot 10^{+22}:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 6.2e+22) (- (- x (* (log y) 0.5)) z) (- (* y (- 1.0 (log y))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 6.2e+22) {
		tmp = (x - (log(y) * 0.5)) - z;
	} else {
		tmp = (y * (1.0 - log(y))) - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 6.2d+22) then
        tmp = (x - (log(y) * 0.5d0)) - z
    else
        tmp = (y * (1.0d0 - log(y))) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 6.2e+22) {
		tmp = (x - (Math.log(y) * 0.5)) - z;
	} else {
		tmp = (y * (1.0 - Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 6.2e+22:
		tmp = (x - (math.log(y) * 0.5)) - z
	else:
		tmp = (y * (1.0 - math.log(y))) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 6.2e+22)
		tmp = Float64(Float64(x - Float64(log(y) * 0.5)) - z);
	else
		tmp = Float64(Float64(y * Float64(1.0 - log(y))) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 6.2e+22)
		tmp = (x - (log(y) * 0.5)) - z;
	else
		tmp = (y * (1.0 - log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 6.2e+22], N[(N[(x - N[(N[Log[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.2000000000000004e22

   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 6.2000000000000004e22 < y

   1. Initial program 99.6%

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

      1. add-cube-cbrt 98.4%

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

      2. pow2 98.4%

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

      3. sub-neg 98.4%

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

      4. *-commutative 98.4%

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

      5. distribute-rgt-neg-in 98.4%

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

      6. +-commutative 98.4%

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

      7. distribute-neg-in 98.4%

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

      8. metadata-eval 98.4%

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

      9. sub-neg 98.4%

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

      10. sub-neg 98.4%

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

      11. *-commutative 98.4%

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

      12. distribute-rgt-neg-in 98.4%

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

      13. +-commutative 98.4%

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

      14. distribute-neg-in 98.4%

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

      15. metadata-eval 98.4%

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

      16. sub-neg 98.4%

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

   4. Applied egg-rr 98.4%

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

   5. Taylor expanded in y around inf 83.8%

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

      1. log-rec 83.8%

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

      2. sub-neg 83.8%

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

   7. Simplified 83.8%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.2 \cdot 10^{+22}:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \end{array} \]
5. 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 99.9%

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

3. Final simplification 99.9%

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

Alternative 7: 46.9% accurate, 9.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{+119}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{+141}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -9.5e+119) x (if (<= x 4.2e+141) (- z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -9.5e+119) {
		tmp = x;
	} else if (x <= 4.2e+141) {
		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 <= (-9.5d+119)) then
        tmp = x
    else if (x <= 4.2d+141) 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 <= -9.5e+119) {
		tmp = x;
	} else if (x <= 4.2e+141) {
		tmp = -z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -9.5e+119:
		tmp = x
	elif x <= 4.2e+141:
		tmp = -z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -9.5e+119)
		tmp = x;
	elseif (x <= 4.2e+141)
		tmp = Float64(-z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -9.5e+119)
		tmp = x;
	elseif (x <= 4.2e+141)
		tmp = -z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -9.5e+119], x, If[LessEqual[x, 4.2e+141], (-z), x]]

Derivation

1. Split input into 2 regimes

2. if x < -9.4999999999999994e119 or 4.1999999999999997e141 < 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. associate-+r- 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-rgt-neg-in 100.0%

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

      7. fma-def 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 76.4%

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

   if -9.4999999999999994e119 < x < 4.1999999999999997e141

   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-def 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 40.0%

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

      1. mul-1-neg 40.0%

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

   7. Simplified 40.0%

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

4. Final simplification 49.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{+119}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{+141}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 58.1% 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 99.9%

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

3. Taylor expanded in y around 0 74.6%

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

4. Taylor expanded in x around inf 57.4%

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

5. Final simplification 57.4%

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

Alternative 9: 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 99.9%

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

   1. associate--l+ 99.8%

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

   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- 99.9%

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

   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-def 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 27.6%

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

6. Final simplification 27.6%

   \[\leadsto x \]
7. 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 2024036 
(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))