Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Percentage Accurate: 99.8% → 99.9%

Time: 8.2s

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 99.8%

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

   7. +-commutative 99.8%

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

   8. distribute-neg-in 99.8%

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

   9. unsub-neg 99.8%

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

   10. metadata-eval 99.8%

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

3. Simplified 99.8%

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

4. Final simplification 99.8%

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

Alternative 2: 76.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.5 \cdot 10^{-266}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{elif}\;y \leq 400000000:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(1 - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.5e-266)
   (- (* (log y) -0.5) z)
   (if (<= y 400000000.0) (- x z) (+ x (* y (- 1.0 (log y)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.5e-266) {
		tmp = (log(y) * -0.5) - z;
	} else if (y <= 400000000.0) {
		tmp = x - z;
	} else {
		tmp = x + (y * (1.0 - log(y)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 1.5d-266) then
        tmp = (log(y) * (-0.5d0)) - z
    else if (y <= 400000000.0d0) then
        tmp = x - z
    else
        tmp = x + (y * (1.0d0 - log(y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.5e-266) {
		tmp = (Math.log(y) * -0.5) - z;
	} else if (y <= 400000000.0) {
		tmp = x - z;
	} else {
		tmp = x + (y * (1.0 - Math.log(y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1.5e-266:
		tmp = (math.log(y) * -0.5) - z
	elif y <= 400000000.0:
		tmp = x - z
	else:
		tmp = x + (y * (1.0 - math.log(y)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.5e-266)
		tmp = Float64(Float64(log(y) * -0.5) - z);
	elseif (y <= 400000000.0)
		tmp = Float64(x - z);
	else
		tmp = Float64(x + Float64(y * Float64(1.0 - log(y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1.5e-266)
		tmp = (log(y) * -0.5) - z;
	elseif (y <= 400000000.0)
		tmp = x - z;
	else
		tmp = x + (y * (1.0 - log(y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.5e-266], N[(N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[y, 400000000.0], N[(x - z), $MachinePrecision], N[(x + N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.5e-266

   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%

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

      1. *-commutative 99.9%

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

   4. Simplified 99.9%

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

   5. Taylor expanded in x around 0 86.4%

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

      1. *-commutative 86.4%

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

   7. Simplified 86.4%

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

   if 1.5e-266 < y < 4e8

   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. +-commutative 100.0%

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

      2. add-sqr-sqrt 100.0%

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

      3. fma-def 100.0%

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

      4. sub-neg 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-rgt-neg-in 100.0%

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

      7. +-commutative 100.0%

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

      8. distribute-neg-in 100.0%

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

      9. metadata-eval 100.0%

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

      10. sub-neg 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in x around inf 73.7%

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

   if 4e8 < 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. sub-neg 99.6%

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

      3. associate-+l+ 99.6%

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

      4. *-commutative 99.6%

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

      5. distribute-rgt-neg-in 99.6%

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

      6. fma-def 99.6%

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

      7. +-commutative 99.6%

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

      8. distribute-neg-in 99.6%

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

      9. unsub-neg 99.6%

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

      10. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around inf 87.3%

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

      1. log-rec 87.3%

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

      2. sub-neg 87.3%

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

   6. Simplified 87.3%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.5 \cdot 10^{-266}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{elif}\;y \leq 400000000:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(1 - \log y\right)\\ \end{array} \]

Alternative 3: 75.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.4 \cdot 10^{-266}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{-9}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 4.4e-266)
   (- (* (log y) -0.5) z)
   (if (<= y 8.5e-9) (- x z) (- (* y (- 1.0 (log y))) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 4.4e-266) {
		tmp = (log(y) * -0.5) - z;
	} else if (y <= 8.5e-9) {
		tmp = x - z;
	} else {
		tmp = (y * (1.0 - log(y))) - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 4.4d-266) then
        tmp = (log(y) * (-0.5d0)) - z
    else if (y <= 8.5d-9) then
        tmp = x - z
    else
        tmp = (y * (1.0d0 - log(y))) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 4.4e-266) {
		tmp = (Math.log(y) * -0.5) - z;
	} else if (y <= 8.5e-9) {
		tmp = x - z;
	} else {
		tmp = (y * (1.0 - Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 4.4e-266:
		tmp = (math.log(y) * -0.5) - z
	elif y <= 8.5e-9:
		tmp = x - z
	else:
		tmp = (y * (1.0 - math.log(y))) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 4.4e-266)
		tmp = Float64(Float64(log(y) * -0.5) - z);
	elseif (y <= 8.5e-9)
		tmp = Float64(x - z);
	else
		tmp = Float64(Float64(y * Float64(1.0 - log(y))) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 4.4e-266)
		tmp = (log(y) * -0.5) - z;
	elseif (y <= 8.5e-9)
		tmp = x - z;
	else
		tmp = (y * (1.0 - log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 4.4e-266], N[(N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[y, 8.5e-9], N[(x - z), $MachinePrecision], N[(N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 4.3999999999999999e-266

   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%

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

      1. *-commutative 99.9%

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

   4. Simplified 99.9%

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

   5. Taylor expanded in x around 0 86.4%

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

      1. *-commutative 86.4%

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

   7. Simplified 86.4%

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

   if 4.3999999999999999e-266 < y < 8.5e-9

   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. +-commutative 100.0%

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

      2. add-sqr-sqrt 100.0%

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

      3. fma-def 100.0%

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

      4. sub-neg 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-rgt-neg-in 100.0%

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

      7. +-commutative 100.0%

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

      8. distribute-neg-in 100.0%

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

      9. metadata-eval 100.0%

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

      10. sub-neg 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in x around inf 72.1%

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

   if 8.5e-9 < y

   1. Initial program 99.6%

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

   2. Taylor expanded in y around inf 88.7%

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

      1. *-commutative 88.7%

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

      2. log-rec 88.7%

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

      3. distribute-lft-neg-in 88.7%

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

      4. distribute-rgt-neg-in 88.7%

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

   4. Simplified 88.7%

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

   5. Taylor expanded in y around 0 88.8%

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

      1. mul-1-neg 88.8%

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

      2. log-rec 88.8%

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

      3. log-rec 88.8%

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

      4. sub-neg 88.8%

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

   7. Simplified 88.8%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.4 \cdot 10^{-266}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{-9}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \end{array} \]

Alternative 4: 89.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 8.5 \cdot 10^{-9}:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(y - \log y \cdot \left(y + 0.5\right)\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 8.5e-9)
   (- (- x (* (log y) 0.5)) z)
   (- (- y (* (log y) (+ y 0.5))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 8.5e-9) {
		tmp = (x - (log(y) * 0.5)) - z;
	} else {
		tmp = (y - (log(y) * (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 (y <= 8.5d-9) then
        tmp = (x - (log(y) * 0.5d0)) - z
    else
        tmp = (y - (log(y) * (y + 0.5d0))) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 8.5e-9) {
		tmp = (x - (Math.log(y) * 0.5)) - z;
	} else {
		tmp = (y - (Math.log(y) * (y + 0.5))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 8.5e-9:
		tmp = (x - (math.log(y) * 0.5)) - z
	else:
		tmp = (y - (math.log(y) * (y + 0.5))) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 8.5e-9)
		tmp = Float64(Float64(x - Float64(log(y) * 0.5)) - z);
	else
		tmp = Float64(Float64(y - Float64(log(y) * Float64(y + 0.5))) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 8.5e-9)
		tmp = (x - (log(y) * 0.5)) - z;
	else
		tmp = (y - (log(y) * (y + 0.5))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 8.5e-9], N[(N[(x - N[(N[Log[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(N[(y - N[(N[Log[y], $MachinePrecision] * N[(y + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 8.5e-9

   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 99.7%

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

      1. *-commutative 99.7%

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

   4. Simplified 99.7%

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

   if 8.5e-9 < y

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 89.1%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8.5 \cdot 10^{-9}:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(y - \log y \cdot \left(y + 0.5\right)\right) - z\\ \end{array} \]

Alternative 5: 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 6: 70.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -165:\\ \;\;\;\;x - z\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{-16}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -165.0) (- x z) (if (<= x 1.7e-16) (- (* (log y) -0.5) z) (- x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -165.0) {
		tmp = x - z;
	} else if (x <= 1.7e-16) {
		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 <= (-165.0d0)) then
        tmp = x - z
    else if (x <= 1.7d-16) 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 <= -165.0) {
		tmp = x - z;
	} else if (x <= 1.7e-16) {
		tmp = (Math.log(y) * -0.5) - z;
	} else {
		tmp = x - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -165.0:
		tmp = x - z
	elif x <= 1.7e-16:
		tmp = (math.log(y) * -0.5) - z
	else:
		tmp = x - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -165.0)
		tmp = Float64(x - z);
	elseif (x <= 1.7e-16)
		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 <= -165.0)
		tmp = x - z;
	elseif (x <= 1.7e-16)
		tmp = (log(y) * -0.5) - z;
	else
		tmp = x - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -165.0], N[(x - z), $MachinePrecision], If[LessEqual[x, 1.7e-16], 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 < -165 or 1.7e-16 < 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. +-commutative 99.9%

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

      2. add-sqr-sqrt 99.9%

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

      3. fma-def 99.9%

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

      4. sub-neg 99.9%

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

      5. *-commutative 99.9%

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

      6. distribute-rgt-neg-in 99.9%

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

      7. +-commutative 99.9%

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

      8. distribute-neg-in 99.9%

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

      9. metadata-eval 99.9%

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

      10. sub-neg 99.9%

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

   3. Applied egg-rr 99.9%

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

   4. Taylor expanded in x around inf 71.8%

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

   if -165 < x < 1.7e-16

   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 61.7%

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

      1. *-commutative 61.7%

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

   4. Simplified 61.7%

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

   5. Taylor expanded in x around 0 61.5%

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

      1. *-commutative 61.5%

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

   7. Simplified 61.5%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -165:\\ \;\;\;\;x - z\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{-16}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \]

Alternative 7: 88.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 8.5 \cdot 10^{-9}:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 8.5e-9) (- (- x (* (log y) 0.5)) z) (- (* y (- 1.0 (log y))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 8.5e-9) {
		tmp = (x - (log(y) * 0.5)) - z;
	} else {
		tmp = (y * (1.0 - log(y))) - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 8.5d-9) then
        tmp = (x - (log(y) * 0.5d0)) - z
    else
        tmp = (y * (1.0d0 - log(y))) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 8.5e-9) {
		tmp = (x - (Math.log(y) * 0.5)) - z;
	} else {
		tmp = (y * (1.0 - Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 8.5e-9:
		tmp = (x - (math.log(y) * 0.5)) - z
	else:
		tmp = (y * (1.0 - math.log(y))) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 8.5e-9)
		tmp = Float64(Float64(x - Float64(log(y) * 0.5)) - z);
	else
		tmp = Float64(Float64(y * Float64(1.0 - log(y))) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 8.5e-9)
		tmp = (x - (log(y) * 0.5)) - z;
	else
		tmp = (y * (1.0 - log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 8.5e-9], N[(N[(x - N[(N[Log[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 8.5e-9

   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 99.7%

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

      1. *-commutative 99.7%

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

   4. Simplified 99.7%

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

   if 8.5e-9 < y

   1. Initial program 99.6%

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

   2. Taylor expanded in y around inf 88.7%

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

      1. *-commutative 88.7%

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

      2. log-rec 88.7%

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

      3. distribute-lft-neg-in 88.7%

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

      4. distribute-rgt-neg-in 88.7%

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

   4. Simplified 88.7%

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

   5. Taylor expanded in y around 0 88.8%

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

      1. mul-1-neg 88.8%

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

      2. log-rec 88.8%

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

      3. log-rec 88.8%

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

      4. sub-neg 88.8%

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

   7. Simplified 88.8%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8.5 \cdot 10^{-9}:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \end{array} \]

Alternative 8: 48.5% accurate, 18.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.85 \cdot 10^{+14}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1600000000:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.85e+14) x (if (<= x 1600000000.0) (- z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.85e+14) {
		tmp = x;
	} else if (x <= 1600000000.0) {
		tmp = -z;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-1.85d+14)) then
        tmp = x
    else if (x <= 1600000000.0d0) then
        tmp = -z
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.85e+14) {
		tmp = x;
	} else if (x <= 1600000000.0) {
		tmp = -z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.85e+14:
		tmp = x
	elif x <= 1600000000.0:
		tmp = -z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.85e+14)
		tmp = x;
	elseif (x <= 1600000000.0)
		tmp = Float64(-z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.85e+14)
		tmp = x;
	elseif (x <= 1600000000.0)
		tmp = -z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.85e+14], x, If[LessEqual[x, 1600000000.0], (-z), x]]

Derivation

1. Split input into 2 regimes

2. if x < -1.85e14 or 1.6e9 < x

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-+l+ 99.9%

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

      4. *-commutative 99.9%

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

      5. distribute-rgt-neg-in 99.9%

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

      6. fma-def 99.8%

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

      7. +-commutative 99.8%

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

      8. distribute-neg-in 99.8%

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

      9. unsub-neg 99.8%

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

      10. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around inf 60.4%

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

   if -1.85e14 < x < 1.6e9

   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 62.5%

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

      1. *-commutative 62.5%

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

   4. Simplified 62.5%

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

   5. Taylor expanded in z around inf 33.9%

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

      1. neg-mul-1 33.9%

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

   7. Simplified 33.9%

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

4. Final simplification 44.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.85 \cdot 10^{+14}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1600000000:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9: 58.6% 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. +-commutative 99.8%

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

   2. add-sqr-sqrt 99.8%

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

   3. fma-def 99.8%

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

   4. sub-neg 99.8%

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

   5. *-commutative 99.8%

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

   6. distribute-rgt-neg-in 99.8%

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

   7. +-commutative 99.8%

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

   8. distribute-neg-in 99.8%

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

   9. metadata-eval 99.8%

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

   10. sub-neg 99.8%

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

3. Applied egg-rr 99.8%

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

4. Taylor expanded in x around inf 49.6%

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

5. Final simplification 49.6%

   \[\leadsto x - z \]

Alternative 10: 30.2% accurate, 111.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 99.8%

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

   1. associate--l+ 99.8%

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

   2. sub-neg 99.8%

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

   3. associate-+l+ 99.8%

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

   4. *-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 99.8%

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

   7. +-commutative 99.8%

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

   8. distribute-neg-in 99.8%

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

   9. unsub-neg 99.8%

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

   10. metadata-eval 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in x around inf 26.3%

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

5. Final simplification 26.3%

   \[\leadsto x \]

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 2023271 
(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))