Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Percentage Accurate: 99.8% → 99.8%

Time: 8.7s

Alternatives: 8

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: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

Alternative 2: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{-27}:\\ \;\;\;\;\left(x - 0.5 \cdot \log y\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + \left(y \cdot \left(1 - \log y\right) - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 4e-27)
   (- (- x (* 0.5 (log y))) z)
   (+ x (- (* y (- 1.0 (log y))) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 4e-27) {
		tmp = (x - (0.5 * log(y))) - 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 <= 4d-27) then
        tmp = (x - (0.5d0 * log(y))) - 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 <= 4e-27) {
		tmp = (x - (0.5 * Math.log(y))) - z;
	} else {
		tmp = x + ((y * (1.0 - Math.log(y))) - z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 4e-27:
		tmp = (x - (0.5 * math.log(y))) - z
	else:
		tmp = x + ((y * (1.0 - math.log(y))) - z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 4e-27)
		tmp = Float64(Float64(x - Float64(0.5 * log(y))) - 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 <= 4e-27)
		tmp = (x - (0.5 * log(y))) - z;
	else
		tmp = x + ((y * (1.0 - log(y))) - z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 4e-27], N[(N[(x - N[(0.5 * N[Log[y], $MachinePrecision]), $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 < 4.0000000000000002e-27

   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 4.0000000000000002e-27 < 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 99.7%

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

      8. +-commutative 99.7%

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

      9. distribute-neg-in 99.7%

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

      10. unsub-neg 99.7%

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

      11. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around inf 99.5%

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

      1. log-rec 99.5%

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

      2. sub-neg 99.5%

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

   7. Simplified 99.5%

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

Alternative 3: 89.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.45 \cdot 10^{+51}:\\ \;\;\;\;\left(x - 0.5 \cdot \log y\right) - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2.45e+51) (- (- x (* 0.5 (log y))) z) (- (* y (- 1.0 (log y))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.45e+51) {
		tmp = (x - (0.5 * log(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 <= 2.45d+51) then
        tmp = (x - (0.5d0 * log(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 <= 2.45e+51) {
		tmp = (x - (0.5 * Math.log(y))) - z;
	} else {
		tmp = (y * (1.0 - Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 2.45e+51:
		tmp = (x - (0.5 * math.log(y))) - z
	else:
		tmp = (y * (1.0 - math.log(y))) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2.45e+51)
		tmp = Float64(Float64(x - Float64(0.5 * log(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 <= 2.45e+51)
		tmp = (x - (0.5 * log(y))) - z;
	else
		tmp = (y * (1.0 - log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2.45e+51], N[(N[(x - N[(0.5 * N[Log[y], $MachinePrecision]), $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 < 2.44999999999999992e51

   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 96.7%

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

   if 2.44999999999999992e51 < 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 98.2%

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

      2. pow3 98.2%

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

      3. sub-neg 98.2%

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

      4. *-commutative 98.2%

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

      5. distribute-rgt-neg-in 98.2%

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

      6. +-commutative 98.2%

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

      7. distribute-neg-in 98.2%

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

      8. metadata-eval 98.2%

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

      9. sub-neg 98.2%

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

   4. Applied egg-rr 98.2%

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

   5. Taylor expanded in y around inf 83.7%

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

      1. log-rec 83.7%

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

      2. sub-neg 83.7%

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

   7. Simplified 83.7%

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

Alternative 4: 77.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 10^{+51}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1e+51) (- x z) (- (* y (- 1.0 (log y))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1e+51) {
		tmp = x - z;
	} else {
		tmp = (y * (1.0 - log(y))) - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 1d+51) then
        tmp = x - z
    else
        tmp = (y * (1.0d0 - log(y))) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 1e+51) {
		tmp = x - z;
	} else {
		tmp = (y * (1.0 - Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1e+51:
		tmp = x - z
	else:
		tmp = (y * (1.0 - math.log(y))) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1e+51)
		tmp = Float64(x - z);
	else
		tmp = Float64(Float64(y * Float64(1.0 - log(y))) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1e+51)
		tmp = x - z;
	else
		tmp = (y * (1.0 - log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1e+51], N[(x - 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 < 1e51

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 98.2%

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

   4. Taylor expanded in x around inf 77.7%

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

   if 1e51 < 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 98.2%

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

      2. pow3 98.2%

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

      3. sub-neg 98.2%

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

      4. *-commutative 98.2%

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

      5. distribute-rgt-neg-in 98.2%

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

      6. +-commutative 98.2%

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

      7. distribute-neg-in 98.2%

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

      8. metadata-eval 98.2%

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

      9. sub-neg 98.2%

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

   4. Applied egg-rr 98.2%

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

   5. Taylor expanded in y around inf 83.7%

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

      1. log-rec 83.7%

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

      2. sub-neg 83.7%

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

   7. Simplified 83.7%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 10^{+51}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 71.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.8 \cdot 10^{+163}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.8e+163) (- x z) (* y (- 1.0 (log y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.8e+163) {
		tmp = x - z;
	} else {
		tmp = 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 <= 1.8d+163) then
        tmp = x - z
    else
        tmp = 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 <= 1.8e+163) {
		tmp = x - z;
	} else {
		tmp = y * (1.0 - Math.log(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1.8e+163:
		tmp = x - z
	else:
		tmp = y * (1.0 - math.log(y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.8e+163)
		tmp = Float64(x - z);
	else
		tmp = Float64(y * Float64(1.0 - log(y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1.8e+163)
		tmp = x - z;
	else
		tmp = y * (1.0 - log(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.8e+163], N[(x - z), $MachinePrecision], N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.79999999999999989e163

   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 x around inf 97.0%

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

   4. Taylor expanded in x around inf 74.6%

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

   if 1.79999999999999989e163 < y

   1. Initial program 99.5%

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

      1. add-cube-cbrt 97.9%

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

      2. pow3 97.9%

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

      3. sub-neg 97.9%

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

      4. *-commutative 97.9%

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

      5. distribute-rgt-neg-in 97.9%

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

      6. +-commutative 97.9%

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

      7. distribute-neg-in 97.9%

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

      8. metadata-eval 97.9%

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

      9. sub-neg 97.9%

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

   4. Applied egg-rr 97.9%

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

   5. Taylor expanded in y around inf 94.8%

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

      1. log-rec 94.8%

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

      2. distribute-rgt-neg-out 94.8%

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

      3. distribute-lft-neg-out 94.8%

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

      4. *-commutative 94.8%

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

   7. Simplified 94.8%

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

   8. Taylor expanded in z around 0 81.6%

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

   9. Taylor expanded in y around 0 81.7%

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

       1. mul-1-neg 81.7%

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

       2. unsub-neg 81.7%

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

   11. Simplified 81.7%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.8 \cdot 10^{+163}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 48.2% accurate, 9.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.75 \cdot 10^{+52} \lor \neg \left(z \leq 9.5 \cdot 10^{+69}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.75e+52) (not (<= z 9.5e+69))) (- z) x))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.75e+52) || !(z <= 9.5e+69)) {
		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 ((z <= (-1.75d+52)) .or. (.not. (z <= 9.5d+69))) 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 ((z <= -1.75e+52) || !(z <= 9.5e+69)) {
		tmp = -z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.75e+52) or not (z <= 9.5e+69):
		tmp = -z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.75e+52) || !(z <= 9.5e+69))
		tmp = Float64(-z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.75e+52) || ~((z <= 9.5e+69)))
		tmp = -z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.75e+52], N[Not[LessEqual[z, 9.5e+69]], $MachinePrecision]], (-z), x]

Derivation

1. Split input into 2 regimes

2. if z < -1.75e52 or 9.4999999999999995e69 < z

   1. Initial program 99.9%

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

      1. 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 z around inf 73.6%

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

      1. neg-mul-1 73.6%

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

   7. Simplified 73.6%

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

   if -1.75e52 < z < 9.4999999999999995e69

   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 99.8%

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

      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 x around inf 42.0%

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

4. Final simplification 55.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.75 \cdot 10^{+52} \lor \neg \left(z \leq 9.5 \cdot 10^{+69}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 57.8% accurate, 37.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in x around inf 89.7%

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

4. Taylor expanded in x around inf 63.1%

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

5. Final simplification 63.1%

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

Alternative 8: 29.9% 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 99.8%

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

   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 x around inf 30.5%

   \[\leadsto \color{blue}{x} \]
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 2024101 
(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))