Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Percentage Accurate: 99.8% → 99.9%

Time: 9.3s

Alternatives: 12

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

\[\begin{array}{l} \\ \left(\left(y \cdot \left(1 - \log y\right) + x\right) - \log y \cdot 0.5\right) - z \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (- (- (+ (* y (- 1.0 (log y))) x) (* (log y) 0.5)) z))

C

double code(double x, double y, double z) {
	return (((y * (1.0 - log(y))) + x) - (log(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 * (1.0d0 - log(y))) + x) - (log(y) * 0.5d0)) - z
end function

Java

public static double code(double x, double y, double z) {
	return (((y * (1.0 - Math.log(y))) + x) - (Math.log(y) * 0.5)) - z;
}

Python

def code(x, y, z):
	return (((y * (1.0 - math.log(y))) + x) - (math.log(y) * 0.5)) - z

Julia

function code(x, y, z)
	return Float64(Float64(Float64(Float64(y * Float64(1.0 - log(y))) + x) - Float64(log(y) * 0.5)) - z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = (((y * (1.0 - log(y))) + x) - (log(y) * 0.5)) - z;
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] - N[(N[Log[y], $MachinePrecision] * 0.5), $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. Taylor expanded in y around 0 99.8%

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

3. Final simplification 99.8%

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

Alternative 2: 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 99.8%

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

4. Final simplification 99.8%

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

Alternative 3: 76.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(1 - \log y\right)\\ t_1 := t_0 + x\\ t_2 := t_0 - z\\ \mathbf{if}\;z \leq -9.8 \cdot 10^{+210}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -1.88 \cdot 10^{+34}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;z \leq 1.08 \cdot 10^{-107}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{-6}:\\ \;\;\;\;x - \log y \cdot 0.5\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{+17}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (- 1.0 (log y)))) (t_1 (+ t_0 x)) (t_2 (- t_0 z)))
   (if (<= z -9.8e+210)
     t_2
     (if (<= z -1.88e+34)
       (- x z)
       (if (<= z 1.08e-107)
         t_1
         (if (<= z 2.5e-6)
           (- x (* (log y) 0.5))
           (if (<= z 1.8e+17) t_1 t_2)))))))

C

double code(double x, double y, double z) {
	double t_0 = y * (1.0 - log(y));
	double t_1 = t_0 + x;
	double t_2 = t_0 - z;
	double tmp;
	if (z <= -9.8e+210) {
		tmp = t_2;
	} else if (z <= -1.88e+34) {
		tmp = x - z;
	} else if (z <= 1.08e-107) {
		tmp = t_1;
	} else if (z <= 2.5e-6) {
		tmp = x - (log(y) * 0.5);
	} else if (z <= 1.8e+17) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = y * (1.0d0 - log(y))
    t_1 = t_0 + x
    t_2 = t_0 - z
    if (z <= (-9.8d+210)) then
        tmp = t_2
    else if (z <= (-1.88d+34)) then
        tmp = x - z
    else if (z <= 1.08d-107) then
        tmp = t_1
    else if (z <= 2.5d-6) then
        tmp = x - (log(y) * 0.5d0)
    else if (z <= 1.8d+17) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y * (1.0 - Math.log(y));
	double t_1 = t_0 + x;
	double t_2 = t_0 - z;
	double tmp;
	if (z <= -9.8e+210) {
		tmp = t_2;
	} else if (z <= -1.88e+34) {
		tmp = x - z;
	} else if (z <= 1.08e-107) {
		tmp = t_1;
	} else if (z <= 2.5e-6) {
		tmp = x - (Math.log(y) * 0.5);
	} else if (z <= 1.8e+17) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * (1.0 - math.log(y))
	t_1 = t_0 + x
	t_2 = t_0 - z
	tmp = 0
	if z <= -9.8e+210:
		tmp = t_2
	elif z <= -1.88e+34:
		tmp = x - z
	elif z <= 1.08e-107:
		tmp = t_1
	elif z <= 2.5e-6:
		tmp = x - (math.log(y) * 0.5)
	elif z <= 1.8e+17:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * Float64(1.0 - log(y)))
	t_1 = Float64(t_0 + x)
	t_2 = Float64(t_0 - z)
	tmp = 0.0
	if (z <= -9.8e+210)
		tmp = t_2;
	elseif (z <= -1.88e+34)
		tmp = Float64(x - z);
	elseif (z <= 1.08e-107)
		tmp = t_1;
	elseif (z <= 2.5e-6)
		tmp = Float64(x - Float64(log(y) * 0.5));
	elseif (z <= 1.8e+17)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * (1.0 - log(y));
	t_1 = t_0 + x;
	t_2 = t_0 - z;
	tmp = 0.0;
	if (z <= -9.8e+210)
		tmp = t_2;
	elseif (z <= -1.88e+34)
		tmp = x - z;
	elseif (z <= 1.08e-107)
		tmp = t_1;
	elseif (z <= 2.5e-6)
		tmp = x - (log(y) * 0.5);
	elseif (z <= 1.8e+17)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + x), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 - z), $MachinePrecision]}, If[LessEqual[z, -9.8e+210], t$95$2, If[LessEqual[z, -1.88e+34], N[(x - z), $MachinePrecision], If[LessEqual[z, 1.08e-107], t$95$1, If[LessEqual[z, 2.5e-6], N[(x - N[(N[Log[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.8e+17], t$95$1, t$95$2]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -9.80000000000000013e210 or 1.8e17 < 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. sub-neg 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-+l+ 99.9%

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

      4. associate-+l+ 99.9%

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

      5. sub-neg 99.9%

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

      6. neg-sub0 99.9%

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

      7. associate-+l- 99.9%

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

      8. neg-sub0 99.9%

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

      9. neg-mul-1 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around inf 99.9%

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

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

   6. Simplified 99.9%

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

   7. Taylor expanded in x around 0 92.2%

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

   if -9.80000000000000013e210 < z < -1.87999999999999992e34

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 99.9%

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

   3. Taylor expanded in x around inf 85.8%

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

   if -1.87999999999999992e34 < z < 1.08000000000000002e-107 or 2.5000000000000002e-6 < z < 1.8e17

   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 99.7%

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

   4. Taylor expanded in y around inf 79.0%

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

      1. log-rec 79.0%

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

      2. sub-neg 79.0%

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

   6. Simplified 79.0%

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

   7. Taylor expanded in z around 0 77.9%

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

   if 1.08000000000000002e-107 < z < 2.5000000000000002e-6

   1. Initial program 99.7%

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

   2. Taylor expanded in y around 0 72.5%

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

   3. Taylor expanded in z around 0 71.0%

      \[\leadsto \color{blue}{x - 0.5 \cdot \log y} \]
3. Recombined 4 regimes into one program.

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9.8 \cdot 10^{+210}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \mathbf{elif}\;z \leq -1.88 \cdot 10^{+34}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;z \leq 1.08 \cdot 10^{-107}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) + x\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{-6}:\\ \;\;\;\;x - \log y \cdot 0.5\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{+17}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) + x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \end{array} \]

Alternative 4: 76.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(1 - \log y\right) + x\\ \mathbf{if}\;z \leq -6.2 \cdot 10^{+38}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{-108}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1.85 \cdot 10^{-6}:\\ \;\;\;\;x - \log y \cdot 0.5\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{+59}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (* y (- 1.0 (log y))) x)))
   (if (<= z -6.2e+38)
     (- x z)
     (if (<= z 1.45e-108)
       t_0
       (if (<= z 1.85e-6)
         (- x (* (log y) 0.5))
         (if (<= z 4.2e+59) t_0 (- x z)))))))

C

double code(double x, double y, double z) {
	double t_0 = (y * (1.0 - log(y))) + x;
	double tmp;
	if (z <= -6.2e+38) {
		tmp = x - z;
	} else if (z <= 1.45e-108) {
		tmp = t_0;
	} else if (z <= 1.85e-6) {
		tmp = x - (log(y) * 0.5);
	} else if (z <= 4.2e+59) {
		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))) + x
    if (z <= (-6.2d+38)) then
        tmp = x - z
    else if (z <= 1.45d-108) then
        tmp = t_0
    else if (z <= 1.85d-6) then
        tmp = x - (log(y) * 0.5d0)
    else if (z <= 4.2d+59) 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))) + x;
	double tmp;
	if (z <= -6.2e+38) {
		tmp = x - z;
	} else if (z <= 1.45e-108) {
		tmp = t_0;
	} else if (z <= 1.85e-6) {
		tmp = x - (Math.log(y) * 0.5);
	} else if (z <= 4.2e+59) {
		tmp = t_0;
	} else {
		tmp = x - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (y * (1.0 - math.log(y))) + x
	tmp = 0
	if z <= -6.2e+38:
		tmp = x - z
	elif z <= 1.45e-108:
		tmp = t_0
	elif z <= 1.85e-6:
		tmp = x - (math.log(y) * 0.5)
	elif z <= 4.2e+59:
		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))) + x)
	tmp = 0.0
	if (z <= -6.2e+38)
		tmp = Float64(x - z);
	elseif (z <= 1.45e-108)
		tmp = t_0;
	elseif (z <= 1.85e-6)
		tmp = Float64(x - Float64(log(y) * 0.5));
	elseif (z <= 4.2e+59)
		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))) + x;
	tmp = 0.0;
	if (z <= -6.2e+38)
		tmp = x - z;
	elseif (z <= 1.45e-108)
		tmp = t_0;
	elseif (z <= 1.85e-6)
		tmp = x - (log(y) * 0.5);
	elseif (z <= 4.2e+59)
		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] + x), $MachinePrecision]}, If[LessEqual[z, -6.2e+38], N[(x - z), $MachinePrecision], If[LessEqual[z, 1.45e-108], t$95$0, If[LessEqual[z, 1.85e-6], N[(x - N[(N[Log[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 4.2e+59], t$95$0, N[(x - z), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -6.20000000000000035e38 or 4.19999999999999968e59 < 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 y around 0 99.9%

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

   3. Taylor expanded in x around inf 84.9%

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

   if -6.20000000000000035e38 < z < 1.45e-108 or 1.8500000000000001e-6 < z < 4.19999999999999968e59

   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 99.7%

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

   4. Taylor expanded in y around inf 80.0%

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

      1. log-rec 80.0%

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

      2. sub-neg 80.0%

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

   6. Simplified 80.0%

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

   7. Taylor expanded in z around 0 77.3%

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

   if 1.45e-108 < z < 1.8500000000000001e-6

   1. Initial program 99.7%

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

   2. Taylor expanded in y around 0 72.5%

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

   3. Taylor expanded in z around 0 71.0%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.2 \cdot 10^{+38}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{-108}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) + x\\ \mathbf{elif}\;z \leq 1.85 \cdot 10^{-6}:\\ \;\;\;\;x - \log y \cdot 0.5\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{+59}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) + x\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \]

Alternative 5: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.0062:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + \left(y \cdot \left(1 - \log y\right) - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 0.0062)
   (- (- 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 <= 0.0062) {
		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 <= 0.0062d0) 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 <= 0.0062) {
		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 <= 0.0062:
		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 <= 0.0062)
		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 <= 0.0062)
		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, 0.0062], 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 < 0.00619999999999999978

   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 99.4%

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

   if 0.00619999999999999978 < y

   1. Initial program 99.6%

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

      1. sub-neg 99.6%

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

      2. sub-neg 99.6%

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

      3. associate-+l+ 99.6%

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

      4. associate-+l+ 99.6%

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

      5. sub-neg 99.6%

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

      6. neg-sub0 99.6%

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

      7. associate-+l- 99.6%

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

      8. neg-sub0 99.6%

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

      9. neg-mul-1 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around inf 99.0%

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

      1. log-rec 99.0%

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

      2. sub-neg 99.0%

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

   6. Simplified 99.0%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.0062:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + \left(y \cdot \left(1 - \log y\right) - z\right)\\ \end{array} \]

Alternative 6: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Final simplification 99.8%

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

Alternative 7: 69.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3000000:\\ \;\;\;\;x - z\\ \mathbf{elif}\;z \leq 1.95:\\ \;\;\;\;x - \log y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -3000000.0) (- x z) (if (<= z 1.95) (- x (* (log y) 0.5)) (- x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -3000000.0) {
		tmp = x - z;
	} else if (z <= 1.95) {
		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 <= (-3000000.0d0)) then
        tmp = x - z
    else if (z <= 1.95d0) 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 <= -3000000.0) {
		tmp = x - z;
	} else if (z <= 1.95) {
		tmp = x - (Math.log(y) * 0.5);
	} else {
		tmp = x - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -3000000.0:
		tmp = x - z
	elif z <= 1.95:
		tmp = x - (math.log(y) * 0.5)
	else:
		tmp = x - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -3000000.0)
		tmp = Float64(x - z);
	elseif (z <= 1.95)
		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 <= -3000000.0)
		tmp = x - z;
	elseif (z <= 1.95)
		tmp = x - (log(y) * 0.5);
	else
		tmp = x - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -3000000.0], N[(x - z), $MachinePrecision], If[LessEqual[z, 1.95], 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 < -3e6 or 1.94999999999999996 < 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 y around 0 99.9%

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

   3. Taylor expanded in x around inf 79.6%

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

   if -3e6 < z < 1.94999999999999996

   1. Initial program 99.7%

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

   2. Taylor expanded in y around 0 61.7%

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

   3. Taylor expanded in z around 0 61.0%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3000000:\\ \;\;\;\;x - z\\ \mathbf{elif}\;z \leq 1.95:\\ \;\;\;\;x - \log y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \]

Alternative 8: 69.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -80:\\ \;\;\;\;x - z\\ \mathbf{elif}\;x \leq 72000000000:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -80.0)
   (- x z)
   (if (<= x 72000000000.0) (- (* (log y) -0.5) z) (- x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -80.0) {
		tmp = x - z;
	} else if (x <= 72000000000.0) {
		tmp = (log(y) * -0.5) - z;
	} 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 (x <= (-80.0d0)) then
        tmp = x - z
    else if (x <= 72000000000.0d0) then
        tmp = (log(y) * (-0.5d0)) - z
    else
        tmp = x - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -80.0) {
		tmp = x - z;
	} else if (x <= 72000000000.0) {
		tmp = (Math.log(y) * -0.5) - z;
	} else {
		tmp = x - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -80.0:
		tmp = x - z
	elif x <= 72000000000.0:
		tmp = (math.log(y) * -0.5) - z
	else:
		tmp = x - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -80.0)
		tmp = Float64(x - z);
	elseif (x <= 72000000000.0)
		tmp = Float64(Float64(log(y) * -0.5) - z);
	else
		tmp = Float64(x - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -80.0)
		tmp = x - z;
	elseif (x <= 72000000000.0)
		tmp = (log(y) * -0.5) - z;
	else
		tmp = x - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -80.0], N[(x - z), $MachinePrecision], If[LessEqual[x, 72000000000.0], N[(N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision] - z), $MachinePrecision], N[(x - z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -80 or 7.2e10 < x

   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 99.8%

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

   3. Taylor expanded in x around inf 78.3%

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

   if -80 < x < 7.2e10

   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 63.5%

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

   3. Taylor expanded in x around 0 63.4%

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

      1. distribute-lft-in 63.4%

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

      2. neg-mul-1 63.4%

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

      3. unsub-neg 63.4%

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

      4. neg-mul-1 63.4%

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

      5. distribute-lft-neg-in 63.4%

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

      6. metadata-eval 63.4%

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

   5. Simplified 63.4%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -80:\\ \;\;\;\;x - z\\ \mathbf{elif}\;x \leq 72000000000:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \]

Alternative 9: 89.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.65 \cdot 10^{+45}:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.65e+45) (- (- x (* (log y) 0.5)) z) (+ (* y (- 1.0 (log y))) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.65e+45) {
		tmp = (x - (log(y) * 0.5)) - z;
	} else {
		tmp = (y * (1.0 - log(y))) + 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 (y <= 1.65d+45) then
        tmp = (x - (log(y) * 0.5d0)) - z
    else
        tmp = (y * (1.0d0 - log(y))) + x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.65e+45) {
		tmp = (x - (Math.log(y) * 0.5)) - z;
	} else {
		tmp = (y * (1.0 - Math.log(y))) + x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1.65e+45:
		tmp = (x - (math.log(y) * 0.5)) - z
	else:
		tmp = (y * (1.0 - math.log(y))) + x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.65e+45)
		tmp = Float64(Float64(x - Float64(log(y) * 0.5)) - z);
	else
		tmp = Float64(Float64(y * Float64(1.0 - log(y))) + x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1.65e+45)
		tmp = (x - (log(y) * 0.5)) - z;
	else
		tmp = (y * (1.0 - log(y))) + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.65e+45], 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] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.65e45

   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 97.1%

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

   if 1.65e45 < y

   1. Initial program 99.5%

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

      1. sub-neg 99.5%

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

      2. sub-neg 99.5%

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

      3. associate-+l+ 99.6%

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

      4. associate-+l+ 99.6%

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

      5. sub-neg 99.6%

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

      6. neg-sub0 99.6%

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

      7. associate-+l- 99.6%

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

      8. neg-sub0 99.6%

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

      9. neg-mul-1 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around inf 99.6%

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

      1. log-rec 99.6%

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

      2. sub-neg 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in z around 0 82.7%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.65 \cdot 10^{+45}:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) + x\\ \end{array} \]

Alternative 10: 47.9% accurate, 18.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{+50}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 2.25 \cdot 10^{+19}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.4e+50) (- z) (if (<= z 2.25e+19) x (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.4e+50) {
		tmp = -z;
	} else if (z <= 2.25e+19) {
		tmp = x;
	} else {
		tmp = -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 <= (-1.4d+50)) then
        tmp = -z
    else if (z <= 2.25d+19) then
        tmp = x
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.4e+50) {
		tmp = -z;
	} else if (z <= 2.25e+19) {
		tmp = x;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.4e+50:
		tmp = -z
	elif z <= 2.25e+19:
		tmp = x
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.4e+50)
		tmp = Float64(-z);
	elseif (z <= 2.25e+19)
		tmp = x;
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.4e+50)
		tmp = -z;
	elseif (z <= 2.25e+19)
		tmp = x;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.4e+50], (-z), If[LessEqual[z, 2.25e+19], x, (-z)]]

Derivation

1. Split input into 2 regimes

2. if z < -1.3999999999999999e50 or 2.25e19 < 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. sub-neg 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-+l+ 99.9%

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

      4. associate-+l+ 99.9%

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

      5. sub-neg 99.9%

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

      6. neg-sub0 99.9%

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

      7. associate-+l- 99.9%

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

      8. neg-sub0 99.9%

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

      9. neg-mul-1 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in z around inf 67.7%

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

      1. neg-mul-1 67.7%

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

   6. Simplified 67.7%

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

   if -1.3999999999999999e50 < z < 2.25e19

   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 99.7%

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

   4. Taylor expanded in x around inf 35.9%

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

4. Final simplification 49.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{+50}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 2.25 \cdot 10^{+19}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 11: 57.4% 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 y around 0 99.8%

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

3. Taylor expanded in x around inf 56.4%

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

4. Final simplification 56.4%

   \[\leadsto x - z \]

Alternative 12: 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. 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 99.8%

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

4. Taylor expanded in x around inf 27.2%

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

5. Final simplification 27.2%

   \[\leadsto x \]

Developer target: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023171 
(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))