Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Percentage Accurate: 99.8% → 99.9%

Time: 7.8s

Alternatives: 10

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.5× speedup

Math

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

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 + fma(log(y), Float64(-0.5 - y), Float64(y - z)))
end

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate--l+ 99.8%

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

   2. sub-neg 99.8%

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

   3. associate-+l+ 99.8%

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

   4. *-commutative 99.8%

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

   5. distribute-rgt-neg-in 99.8%

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

   6. fma-def 99.9%

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

   7. +-commutative 99.9%

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

   8. distribute-neg-in 99.9%

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

   9. unsub-neg 99.9%

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

   10. metadata-eval 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 2: 72.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - y \cdot \left(\log y + -1\right)\\ \mathbf{if}\;z \leq -9.6 \cdot 10^{+47}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;z \leq -9.4 \cdot 10^{-107}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{-235}:\\ \;\;\;\;y - \log y \cdot \left(y + 0.5\right)\\ \mathbf{elif}\;z \leq 1.75 \cdot 10^{+152}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- x (* y (+ (log y) -1.0)))))
   (if (<= z -9.6e+47)
     (- x z)
     (if (<= z -9.4e-107)
       t_0
       (if (<= z 8.5e-235)
         (- y (* (log y) (+ y 0.5)))
         (if (<= z 1.75e+152) t_0 (- x z)))))))

C

double code(double x, double y, double z) {
	double t_0 = x - (y * (log(y) + -1.0));
	double tmp;
	if (z <= -9.6e+47) {
		tmp = x - z;
	} else if (z <= -9.4e-107) {
		tmp = t_0;
	} else if (z <= 8.5e-235) {
		tmp = y - (log(y) * (y + 0.5));
	} else if (z <= 1.75e+152) {
		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 = x - (y * (log(y) + (-1.0d0)))
    if (z <= (-9.6d+47)) then
        tmp = x - z
    else if (z <= (-9.4d-107)) then
        tmp = t_0
    else if (z <= 8.5d-235) then
        tmp = y - (log(y) * (y + 0.5d0))
    else if (z <= 1.75d+152) 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 = x - (y * (Math.log(y) + -1.0));
	double tmp;
	if (z <= -9.6e+47) {
		tmp = x - z;
	} else if (z <= -9.4e-107) {
		tmp = t_0;
	} else if (z <= 8.5e-235) {
		tmp = y - (Math.log(y) * (y + 0.5));
	} else if (z <= 1.75e+152) {
		tmp = t_0;
	} else {
		tmp = x - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x - (y * (math.log(y) + -1.0))
	tmp = 0
	if z <= -9.6e+47:
		tmp = x - z
	elif z <= -9.4e-107:
		tmp = t_0
	elif z <= 8.5e-235:
		tmp = y - (math.log(y) * (y + 0.5))
	elif z <= 1.75e+152:
		tmp = t_0
	else:
		tmp = x - z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x - Float64(y * Float64(log(y) + -1.0)))
	tmp = 0.0
	if (z <= -9.6e+47)
		tmp = Float64(x - z);
	elseif (z <= -9.4e-107)
		tmp = t_0;
	elseif (z <= 8.5e-235)
		tmp = Float64(y - Float64(log(y) * Float64(y + 0.5)));
	elseif (z <= 1.75e+152)
		tmp = t_0;
	else
		tmp = Float64(x - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x - (y * (log(y) + -1.0));
	tmp = 0.0;
	if (z <= -9.6e+47)
		tmp = x - z;
	elseif (z <= -9.4e-107)
		tmp = t_0;
	elseif (z <= 8.5e-235)
		tmp = y - (log(y) * (y + 0.5));
	elseif (z <= 1.75e+152)
		tmp = t_0;
	else
		tmp = x - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x - N[(y * N[(N[Log[y], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -9.6e+47], N[(x - z), $MachinePrecision], If[LessEqual[z, -9.4e-107], t$95$0, If[LessEqual[z, 8.5e-235], N[(y - N[(N[Log[y], $MachinePrecision] * N[(y + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.75e+152], t$95$0, N[(x - z), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -9.60000000000000075e47 or 1.74999999999999991e152 < 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. associate-+l- 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in z around inf 84.9%

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

   if -9.60000000000000075e47 < z < -9.39999999999999995e-107 or 8.49999999999999964e-235 < z < 1.74999999999999991e152

   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. associate-+l- 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around inf 79.0%

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

      1. sub-neg 79.0%

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

      2. mul-1-neg 79.0%

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

      3. log-rec 79.0%

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

      4. remove-double-neg 79.0%

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

      5. metadata-eval 79.0%

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

   6. Simplified 79.0%

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

   if -9.39999999999999995e-107 < z < 8.49999999999999964e-235

   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 0 74.0%

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

   3. Taylor expanded in z around 0 74.0%

      \[\leadsto \color{blue}{y - \left(0.5 + y\right) \cdot \log y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9.6 \cdot 10^{+47}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;z \leq -9.4 \cdot 10^{-107}:\\ \;\;\;\;x - y \cdot \left(\log y + -1\right)\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{-235}:\\ \;\;\;\;y - \log y \cdot \left(y + 0.5\right)\\ \mathbf{elif}\;z \leq 1.75 \cdot 10^{+152}:\\ \;\;\;\;x - y \cdot \left(\log y + -1\right)\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \]

Alternative 3: 74.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -27000000000000:\\ \;\;\;\;x - z\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-152}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{+51}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \left(\log y + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -27000000000000.0)
   (- x z)
   (if (<= x 1.4e-152)
     (- (* y (- 1.0 (log y))) z)
     (if (<= x 1.95e+51)
       (- (* (log y) -0.5) z)
       (- x (* y (+ (log y) -1.0)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -27000000000000.0) {
		tmp = x - z;
	} else if (x <= 1.4e-152) {
		tmp = (y * (1.0 - log(y))) - z;
	} else if (x <= 1.95e+51) {
		tmp = (log(y) * -0.5) - z;
	} else {
		tmp = x - (y * (log(y) + -1.0));
	}
	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 <= (-27000000000000.0d0)) then
        tmp = x - z
    else if (x <= 1.4d-152) then
        tmp = (y * (1.0d0 - log(y))) - z
    else if (x <= 1.95d+51) then
        tmp = (log(y) * (-0.5d0)) - z
    else
        tmp = x - (y * (log(y) + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -27000000000000.0) {
		tmp = x - z;
	} else if (x <= 1.4e-152) {
		tmp = (y * (1.0 - Math.log(y))) - z;
	} else if (x <= 1.95e+51) {
		tmp = (Math.log(y) * -0.5) - z;
	} else {
		tmp = x - (y * (Math.log(y) + -1.0));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -27000000000000.0:
		tmp = x - z
	elif x <= 1.4e-152:
		tmp = (y * (1.0 - math.log(y))) - z
	elif x <= 1.95e+51:
		tmp = (math.log(y) * -0.5) - z
	else:
		tmp = x - (y * (math.log(y) + -1.0))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -27000000000000.0)
		tmp = Float64(x - z);
	elseif (x <= 1.4e-152)
		tmp = Float64(Float64(y * Float64(1.0 - log(y))) - z);
	elseif (x <= 1.95e+51)
		tmp = Float64(Float64(log(y) * -0.5) - z);
	else
		tmp = Float64(x - Float64(y * Float64(log(y) + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -27000000000000.0)
		tmp = x - z;
	elseif (x <= 1.4e-152)
		tmp = (y * (1.0 - log(y))) - z;
	elseif (x <= 1.95e+51)
		tmp = (log(y) * -0.5) - z;
	else
		tmp = x - (y * (log(y) + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -27000000000000.0], N[(x - z), $MachinePrecision], If[LessEqual[x, 1.4e-152], N[(N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[x, 1.95e+51], N[(N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision] - z), $MachinePrecision], N[(x - N[(y * N[(N[Log[y], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2.7e13

   1. Initial program 100.0%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   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. associate-+l- 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in z around inf 87.9%

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

   if -2.7e13 < x < 1.39999999999999992e-152

   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 78.7%

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

   3. Taylor expanded in y around 0 78.7%

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

   if 1.39999999999999992e-152 < x < 1.94999999999999992e51

   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 0 95.5%

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

   3. Taylor expanded in y around 0 76.6%

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

      1. *-commutative 76.6%

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

   5. Simplified 76.6%

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

   if 1.94999999999999992e51 < x

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. associate-+l- 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around inf 87.5%

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

      1. sub-neg 87.5%

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

      2. mul-1-neg 87.5%

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

      3. log-rec 87.5%

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

      4. remove-double-neg 87.5%

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

      5. metadata-eval 87.5%

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

   6. Simplified 87.5%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -27000000000000:\\ \;\;\;\;x - z\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-152}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{+51}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \left(\log y + -1\right)\\ \end{array} \]

Alternative 4: 75.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.3 \cdot 10^{+48} \lor \neg \left(z \leq 1.75 \cdot 10^{+151}\right):\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \left(\log y + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -3.3e+48) (not (<= z 1.75e+151)))
   (- x z)
   (- x (* y (+ (log y) -1.0)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -3.3e+48) || !(z <= 1.75e+151)) {
		tmp = x - z;
	} else {
		tmp = x - (y * (log(y) + -1.0));
	}
	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 <= (-3.3d+48)) .or. (.not. (z <= 1.75d+151))) then
        tmp = x - z
    else
        tmp = x - (y * (log(y) + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -3.3e+48) || !(z <= 1.75e+151)) {
		tmp = x - z;
	} else {
		tmp = x - (y * (Math.log(y) + -1.0));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -3.3e+48) or not (z <= 1.75e+151):
		tmp = x - z
	else:
		tmp = x - (y * (math.log(y) + -1.0))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -3.3e+48) || !(z <= 1.75e+151))
		tmp = Float64(x - z);
	else
		tmp = Float64(x - Float64(y * Float64(log(y) + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -3.3e+48) || ~((z <= 1.75e+151)))
		tmp = x - z;
	else
		tmp = x - (y * (log(y) + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -3.3e+48], N[Not[LessEqual[z, 1.75e+151]], $MachinePrecision]], N[(x - z), $MachinePrecision], N[(x - N[(y * N[(N[Log[y], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.30000000000000023e48 or 1.7500000000000001e151 < 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. associate-+l- 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in z around inf 84.9%

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

   if -3.30000000000000023e48 < z < 1.7500000000000001e151

   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. associate-+l- 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around inf 72.6%

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

      1. sub-neg 72.6%

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

      2. mul-1-neg 72.6%

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

      3. log-rec 72.6%

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

      4. remove-double-neg 72.6%

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

      5. metadata-eval 72.6%

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

   6. Simplified 72.6%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.3 \cdot 10^{+48} \lor \neg \left(z \leq 1.75 \cdot 10^{+151}\right):\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \left(\log y + -1\right)\\ \end{array} \]

Alternative 5: 89.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.8 \cdot 10^{+27}:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{+217}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \left(\log y + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2.8e+27)
   (- (- x (* (log y) 0.5)) z)
   (if (<= y 3.2e+217)
     (- (* y (- 1.0 (log y))) z)
     (- x (* y (+ (log y) -1.0))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.8e+27) {
		tmp = (x - (log(y) * 0.5)) - z;
	} else if (y <= 3.2e+217) {
		tmp = (y * (1.0 - log(y))) - z;
	} else {
		tmp = x - (y * (log(y) + -1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 2.8d+27) then
        tmp = (x - (log(y) * 0.5d0)) - z
    else if (y <= 3.2d+217) then
        tmp = (y * (1.0d0 - log(y))) - z
    else
        tmp = x - (y * (log(y) + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.8e+27) {
		tmp = (x - (Math.log(y) * 0.5)) - z;
	} else if (y <= 3.2e+217) {
		tmp = (y * (1.0 - Math.log(y))) - z;
	} else {
		tmp = x - (y * (Math.log(y) + -1.0));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 2.8e+27:
		tmp = (x - (math.log(y) * 0.5)) - z
	elif y <= 3.2e+217:
		tmp = (y * (1.0 - math.log(y))) - z
	else:
		tmp = x - (y * (math.log(y) + -1.0))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2.8e+27)
		tmp = Float64(Float64(x - Float64(log(y) * 0.5)) - z);
	elseif (y <= 3.2e+217)
		tmp = Float64(Float64(y * Float64(1.0 - log(y))) - z);
	else
		tmp = Float64(x - Float64(y * Float64(log(y) + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 2.8e+27)
		tmp = (x - (log(y) * 0.5)) - z;
	elseif (y <= 3.2e+217)
		tmp = (y * (1.0 - log(y))) - z;
	else
		tmp = x - (y * (log(y) + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2.8e+27], N[(N[(x - N[(N[Log[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[y, 3.2e+217], N[(N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(x - N[(y * N[(N[Log[y], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.7999999999999999e27

   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 96.8%

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

      1. *-commutative 96.8%

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

   4. Simplified 96.8%

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

   if 2.7999999999999999e27 < y < 3.2000000000000001e217

   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 inf 84.7%

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

   3. Taylor expanded in y around 0 84.7%

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

   if 3.2000000000000001e217 < 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. associate--l+ 99.5%

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

      2. associate-+l- 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around inf 97.2%

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

      1. sub-neg 97.2%

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

      2. mul-1-neg 97.2%

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

      3. log-rec 97.2%

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

      4. remove-double-neg 97.2%

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

      5. metadata-eval 97.2%

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

   6. Simplified 97.2%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.8 \cdot 10^{+27}:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{+217}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \left(\log y + -1\right)\\ \end{array} \]

Alternative 6: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.2 \cdot 10^{-7}:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(y - z\right) - y \cdot \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 5.2e-7)
   (- (- x (* (log y) 0.5)) z)
   (+ x (- (- y z) (* y (log y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 5.2e-7) {
		tmp = (x - (log(y) * 0.5)) - z;
	} else {
		tmp = x + ((y - z) - (y * log(y)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 5.2d-7) then
        tmp = (x - (log(y) * 0.5d0)) - z
    else
        tmp = x + ((y - z) - (y * log(y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 5.2e-7) {
		tmp = (x - (Math.log(y) * 0.5)) - z;
	} else {
		tmp = x + ((y - z) - (y * Math.log(y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 5.2e-7:
		tmp = (x - (math.log(y) * 0.5)) - z
	else:
		tmp = x + ((y - z) - (y * math.log(y)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 5.2e-7)
		tmp = Float64(Float64(x - Float64(log(y) * 0.5)) - z);
	else
		tmp = Float64(x + Float64(Float64(y - z) - Float64(y * log(y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 5.2e-7)
		tmp = (x - (log(y) * 0.5)) - z;
	else
		tmp = x + ((y - z) - (y * log(y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 5.2e-7], N[(N[(x - N[(N[Log[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(x + N[(N[(y - z), $MachinePrecision] - N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.19999999999999998e-7

   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 99.2%

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

      1. *-commutative 99.2%

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

   4. Simplified 99.2%

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

   if 5.19999999999999998e-7 < 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. associate--l+ 99.6%

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

      2. associate-+l- 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around inf 98.0%

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

      1. mul-1-neg 98.0%

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

      2. distribute-rgt-neg-in 98.0%

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

      3. log-rec 98.0%

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

      4. remove-double-neg 98.0%

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

   6. Simplified 98.0%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.2 \cdot 10^{-7}:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(y - z\right) - y \cdot \log y\right)\\ \end{array} \]

Alternative 7: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 - z) - (log(y) * (y + 0.5d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate--l+ 99.8%

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

   2. associate-+l- 99.8%

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

3. Simplified 99.8%

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

4. Final simplification 99.8%

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

Alternative 8: 70.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -90 \lor \neg \left(x \leq 650\right):\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -90.0) (not (<= x 650.0))) (- x z) (- (* (log y) -0.5) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -90.0) || !(x <= 650.0)) {
		tmp = x - z;
	} else {
		tmp = (log(y) * -0.5) - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-90.0d0)) .or. (.not. (x <= 650.0d0))) then
        tmp = x - z
    else
        tmp = (log(y) * (-0.5d0)) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -90.0) || !(x <= 650.0)) {
		tmp = x - z;
	} else {
		tmp = (Math.log(y) * -0.5) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -90.0) or not (x <= 650.0):
		tmp = x - z
	else:
		tmp = (math.log(y) * -0.5) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -90.0) || !(x <= 650.0))
		tmp = Float64(x - z);
	else
		tmp = Float64(Float64(log(y) * -0.5) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -90.0) || ~((x <= 650.0)))
		tmp = x - z;
	else
		tmp = (log(y) * -0.5) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -90.0], N[Not[LessEqual[x, 650.0]], $MachinePrecision]], N[(x - z), $MachinePrecision], N[(N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -90 or 650 < x

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. associate-+l- 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in z around inf 84.2%

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

   if -90 < x < 650

   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 0 99.8%

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

   3. Taylor expanded in y around 0 61.7%

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

      1. *-commutative 61.7%

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

   5. Simplified 61.7%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -90 \lor \neg \left(x \leq 650\right):\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \end{array} \]

Alternative 9: 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. 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. associate-+l- 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in z around inf 59.3%

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

5. Final simplification 59.3%

   \[\leadsto x - z \]

Alternative 10: 29.7% 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 x around 0 70.8%

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

3. Taylor expanded in z around inf 30.4%

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

   1. neg-mul-1 30.4%

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

5. Simplified 30.4%

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

6. Final simplification 30.4%

   \[\leadsto -z \]

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 2023268 
(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))