Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Percentage Accurate: 99.8% → 99.9%

Time: 9.0s

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: 99.9% 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. sub-neg 99.8%

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

   2. sub-neg 99.8%

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

   3. associate-+l+ 99.8%

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

   4. associate-+l+ 99.8%

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

   5. sub-neg 99.8%

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

   6. neg-sub0 99.8%

      \[\leadsto x + \left(\left(\color{blue}{\left(0 - \left(y + 0.5\right) \cdot \log y\right)} + y\right) - z\right) \]

   7. associate-+l- 99.8%

      \[\leadsto x + \left(\color{blue}{\left(0 - \left(\left(y + 0.5\right) \cdot \log y - y\right)\right)} - z\right) \]

   8. neg-sub0 99.8%

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

   9. neg-mul-1 99.8%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 2: 77.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(1 - \log y\right) - z\\ \mathbf{if}\;x \leq -3.6 \cdot 10^{+43}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;x \leq -3.3 \cdot 10^{-81}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -1.85 \cdot 10^{-138}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{elif}\;x \leq 9.8 \cdot 10^{+45}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (* y (- 1.0 (log y))) z)))
   (if (<= x -3.6e+43)
     (- x z)
     (if (<= x -3.3e-81)
       t_0
       (if (<= x -1.85e-138)
         (- (* (log y) -0.5) z)
         (if (<= x 9.8e+45) t_0 (- x z)))))))

C

double code(double x, double y, double z) {
	double t_0 = (y * (1.0 - log(y))) - z;
	double tmp;
	if (x <= -3.6e+43) {
		tmp = x - z;
	} else if (x <= -3.3e-81) {
		tmp = t_0;
	} else if (x <= -1.85e-138) {
		tmp = (log(y) * -0.5) - z;
	} else if (x <= 9.8e+45) {
		tmp = t_0;
	} else {
		tmp = x - 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))) - z
    if (x <= (-3.6d+43)) then
        tmp = x - z
    else if (x <= (-3.3d-81)) then
        tmp = t_0
    else if (x <= (-1.85d-138)) then
        tmp = (log(y) * (-0.5d0)) - z
    else if (x <= 9.8d+45) then
        tmp = t_0
    else
        tmp = x - 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))) - z;
	double tmp;
	if (x <= -3.6e+43) {
		tmp = x - z;
	} else if (x <= -3.3e-81) {
		tmp = t_0;
	} else if (x <= -1.85e-138) {
		tmp = (Math.log(y) * -0.5) - z;
	} else if (x <= 9.8e+45) {
		tmp = t_0;
	} else {
		tmp = x - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (y * (1.0 - math.log(y))) - z
	tmp = 0
	if x <= -3.6e+43:
		tmp = x - z
	elif x <= -3.3e-81:
		tmp = t_0
	elif x <= -1.85e-138:
		tmp = (math.log(y) * -0.5) - z
	elif x <= 9.8e+45:
		tmp = t_0
	else:
		tmp = x - z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y * Float64(1.0 - log(y))) - z)
	tmp = 0.0
	if (x <= -3.6e+43)
		tmp = Float64(x - z);
	elseif (x <= -3.3e-81)
		tmp = t_0;
	elseif (x <= -1.85e-138)
		tmp = Float64(Float64(log(y) * -0.5) - z);
	elseif (x <= 9.8e+45)
		tmp = t_0;
	else
		tmp = Float64(x - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (y * (1.0 - log(y))) - z;
	tmp = 0.0;
	if (x <= -3.6e+43)
		tmp = x - z;
	elseif (x <= -3.3e-81)
		tmp = t_0;
	elseif (x <= -1.85e-138)
		tmp = (log(y) * -0.5) - z;
	elseif (x <= 9.8e+45)
		tmp = t_0;
	else
		tmp = x - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]}, If[LessEqual[x, -3.6e+43], N[(x - z), $MachinePrecision], If[LessEqual[x, -3.3e-81], t$95$0, If[LessEqual[x, -1.85e-138], N[(N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[x, 9.8e+45], t$95$0, N[(x - z), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.6000000000000001e43 or 9.8000000000000004e45 < x

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf 90.6%

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

   if -3.6000000000000001e43 < x < -3.29999999999999987e-81 or -1.84999999999999995e-138 < x < 9.8000000000000004e45

   1. Initial program 99.8%

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

   2. Taylor expanded in y around inf 76.7%

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

      1. *-commutative 76.7%

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

      2. log-rec 76.7%

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

      3. distribute-lft-neg-in 76.7%

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

      4. distribute-rgt-neg-in 76.7%

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

   4. Simplified 76.7%

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

   5. Taylor expanded in y around 0 76.7%

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

      1. neg-mul-1 76.7%

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

      2. log-rec 76.7%

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

      3. log-rec 76.7%

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

      4. sub-neg 76.7%

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

   7. Simplified 76.7%

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

   if -3.29999999999999987e-81 < x < -1.84999999999999995e-138

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 99.9%

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

   3. Taylor expanded in y around 0 78.0%

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

      1. *-commutative 78.0%

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

   5. Simplified 78.0%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{+43}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;x \leq -3.3 \cdot 10^{-81}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \mathbf{elif}\;x \leq -1.85 \cdot 10^{-138}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{elif}\;x \leq 9.8 \cdot 10^{+45}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \]

Alternative 3: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.12:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + \left(y \cdot \left(1 - \log y\right) - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 0.12)
   (- (- 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 <= 0.12) {
		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 <= 0.12d0) 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 <= 0.12) {
		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 <= 0.12:
		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 <= 0.12)
		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 <= 0.12)
		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, 0.12], 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 < 0.12

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 99.5%

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

      1. *-commutative 99.5%

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

   4. Simplified 99.5%

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

   if 0.12 < 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. sub-neg 99.7%

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

      2. sub-neg 99.7%

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

      3. associate-+l+ 99.7%

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

      4. associate-+l+ 99.7%

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

      5. sub-neg 99.7%

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

      6. neg-sub0 99.7%

         \[\leadsto x + \left(\left(\color{blue}{\left(0 - \left(y + 0.5\right) \cdot \log y\right)} + y\right) - z\right) \]

      7. associate-+l- 99.7%

         \[\leadsto x + \left(\color{blue}{\left(0 - \left(\left(y + 0.5\right) \cdot \log y - y\right)\right)} - z\right) \]

      8. neg-sub0 99.7%

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

      9. neg-mul-1 99.7%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around inf 99.5%

      \[\leadsto x + \left(\color{blue}{y \cdot \left(1 + \log \left(\frac{1}{y}\right)\right)} - z\right) \]
   5. 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%

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

   6. 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. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.12:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + \left(y \cdot \left(1 - \log y\right) - z\right)\\ \end{array} \]

Alternative 4: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Final simplification 99.8%

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

Alternative 5: 69.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7.5 \cdot 10^{+35}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;z \leq 250:\\ \;\;\;\;x - \log y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -7.5e+35) (- x z) (if (<= z 250.0) (- x (* (log y) 0.5)) (- x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -7.5e+35) {
		tmp = x - z;
	} else if (z <= 250.0) {
		tmp = x - (log(y) * 0.5);
	} else {
		tmp = x - 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 (z <= (-7.5d+35)) then
        tmp = x - z
    else if (z <= 250.0d0) then
        tmp = x - (log(y) * 0.5d0)
    else
        tmp = x - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -7.5e+35) {
		tmp = x - z;
	} else if (z <= 250.0) {
		tmp = x - (Math.log(y) * 0.5);
	} else {
		tmp = x - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -7.5e+35:
		tmp = x - z
	elif z <= 250.0:
		tmp = x - (math.log(y) * 0.5)
	else:
		tmp = x - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -7.5e+35)
		tmp = Float64(x - z);
	elseif (z <= 250.0)
		tmp = Float64(x - Float64(log(y) * 0.5));
	else
		tmp = Float64(x - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -7.5e+35)
		tmp = x - z;
	elseif (z <= 250.0)
		tmp = x - (log(y) * 0.5);
	else
		tmp = x - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -7.5e+35], N[(x - z), $MachinePrecision], If[LessEqual[z, 250.0], N[(x - N[(N[Log[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], N[(x - z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -7.4999999999999999e35 or 250 < z

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf 78.6%

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

   if -7.4999999999999999e35 < z < 250

   1. Initial program 99.8%

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

   2. Taylor expanded in y around 0 70.8%

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

      1. *-commutative 70.8%

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

   4. Simplified 70.8%

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

   5. Taylor expanded in z around 0 70.8%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.5 \cdot 10^{+35}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;z \leq 250:\\ \;\;\;\;x - \log y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \]

Alternative 6: 89.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.15 \cdot 10^{+82}:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2.15e+82) (- (- 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 <= 2.15e+82) {
		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 <= 2.15d+82) 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 <= 2.15e+82) {
		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 <= 2.15e+82:
		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 <= 2.15e+82)
		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 <= 2.15e+82)
		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, 2.15e+82], 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 < 2.15000000000000007e82

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 95.4%

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

      1. *-commutative 95.4%

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

   4. Simplified 95.4%

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

   if 2.15000000000000007e82 < y

   1. Initial program 99.6%

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

   2. Taylor expanded in y around inf 80.8%

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

      1. *-commutative 80.8%

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

      2. log-rec 80.8%

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

      3. distribute-lft-neg-in 80.8%

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

      4. distribute-rgt-neg-in 80.8%

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

   4. Simplified 80.8%

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

   5. Taylor expanded in y around 0 81.0%

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

      1. neg-mul-1 81.0%

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

      2. log-rec 81.0%

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

      3. log-rec 81.0%

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

      4. sub-neg 81.0%

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

   7. Simplified 81.0%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.15 \cdot 10^{+82}:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \end{array} \]

Alternative 7: 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 99.8%

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

2. Taylor expanded in x around inf 59.8%

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

3. Final simplification 59.8%

   \[\leadsto x - z \]

Alternative 8: 30.2% accurate, 55.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := (-z)

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in y around inf 55.0%

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

   1. *-commutative 55.0%

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

   2. log-rec 55.0%

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

   3. distribute-lft-neg-in 55.0%

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

   4. distribute-rgt-neg-in 55.0%

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

4. Simplified 55.0%

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

5. Taylor expanded in y around 0 30.3%

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

   1. neg-mul-1 30.3%

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

7. Simplified 30.3%

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

8. Final simplification 30.3%

   \[\leadsto -z \]

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 2023178 
(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))