Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Percentage Accurate: 99.8% → 99.9%

Time: 12.3s

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

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

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. associate--l+ 99.8%

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

   2. sub-neg 99.8%

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

   3. associate-+l+ 99.8%

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

   4. associate-+r- 99.8%

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

   5. *-commutative 99.8%

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

   6. distribute-rgt-neg-in 99.8%

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

   7. fma-def 99.9%

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

   8. +-commutative 99.9%

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

   9. distribute-neg-in 99.9%

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

   10. unsub-neg 99.9%

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

   11. metadata-eval 99.9%

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

3. Simplified 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 75.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.4 \cdot 10^{-134}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{-57}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+103}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2.4e-134)
   (- x z)
   (if (<= y 3.8e-57)
     (- (* (log y) -0.5) z)
     (if (<= y 2.5e+103) (- x z) (- (* y (- 1.0 (log y))) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.4e-134) {
		tmp = x - z;
	} else if (y <= 3.8e-57) {
		tmp = (log(y) * -0.5) - z;
	} else if (y <= 2.5e+103) {
		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 <= 2.4d-134) then
        tmp = x - z
    else if (y <= 3.8d-57) then
        tmp = (log(y) * (-0.5d0)) - z
    else if (y <= 2.5d+103) 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 <= 2.4e-134) {
		tmp = x - z;
	} else if (y <= 3.8e-57) {
		tmp = (Math.log(y) * -0.5) - z;
	} else if (y <= 2.5e+103) {
		tmp = x - z;
	} else {
		tmp = (y * (1.0 - Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 2.4e-134:
		tmp = x - z
	elif y <= 3.8e-57:
		tmp = (math.log(y) * -0.5) - z
	elif y <= 2.5e+103:
		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 <= 2.4e-134)
		tmp = Float64(x - z);
	elseif (y <= 3.8e-57)
		tmp = Float64(Float64(log(y) * -0.5) - z);
	elseif (y <= 2.5e+103)
		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 <= 2.4e-134)
		tmp = x - z;
	elseif (y <= 3.8e-57)
		tmp = (log(y) * -0.5) - z;
	elseif (y <= 2.5e+103)
		tmp = x - z;
	else
		tmp = (y * (1.0 - log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2.4e-134], N[(x - z), $MachinePrecision], If[LessEqual[y, 3.8e-57], N[(N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[y, 2.5e+103], 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 < 2.4000000000000001e-134 or 3.7999999999999997e-57 < y < 2.5e103

   1. Initial program 99.9%

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

      1. flip-+ 99.9%

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

      2. associate-*l/ 99.9%

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

      3. fma-neg 99.9%

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

      4. metadata-eval 99.9%

         \[\leadsto \left(\left(x - \frac{\mathsf{fma}\left(y, y, -\color{blue}{0.25}\right) \cdot \log y}{y - 0.5}\right) + y\right) - z \]

      5. metadata-eval 99.9%

         \[\leadsto \left(\left(x - \frac{\mathsf{fma}\left(y, y, \color{blue}{-0.25}\right) \cdot \log y}{y - 0.5}\right) + y\right) - z \]

      6. sub-neg 99.9%

         \[\leadsto \left(\left(x - \frac{\mathsf{fma}\left(y, y, -0.25\right) \cdot \log y}{\color{blue}{y + \left(-0.5\right)}}\right) + y\right) - z \]

      7. metadata-eval 99.9%

         \[\leadsto \left(\left(x - \frac{\mathsf{fma}\left(y, y, -0.25\right) \cdot \log y}{y + \color{blue}{-0.5}}\right) + y\right) - z \]

   4. Applied egg-rr 99.9%

      \[\leadsto \left(\left(x - \color{blue}{\frac{\mathsf{fma}\left(y, y, -0.25\right) \cdot \log y}{y + -0.5}}\right) + y\right) - z \]

   5. Taylor expanded in x around inf 73.2%

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

   if 2.4000000000000001e-134 < y < 3.7999999999999997e-57

   1. Initial program 100.0%

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

      1. flip-+ 100.0%

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

      2. associate-*l/ 100.0%

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

      3. fma-neg 100.0%

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

      4. metadata-eval 100.0%

         \[\leadsto \left(\left(x - \frac{\mathsf{fma}\left(y, y, -\color{blue}{0.25}\right) \cdot \log y}{y - 0.5}\right) + y\right) - z \]

      5. metadata-eval 100.0%

         \[\leadsto \left(\left(x - \frac{\mathsf{fma}\left(y, y, \color{blue}{-0.25}\right) \cdot \log y}{y - 0.5}\right) + y\right) - z \]

      6. sub-neg 100.0%

         \[\leadsto \left(\left(x - \frac{\mathsf{fma}\left(y, y, -0.25\right) \cdot \log y}{\color{blue}{y + \left(-0.5\right)}}\right) + y\right) - z \]

      7. metadata-eval 100.0%

         \[\leadsto \left(\left(x - \frac{\mathsf{fma}\left(y, y, -0.25\right) \cdot \log y}{y + \color{blue}{-0.5}}\right) + y\right) - z \]

   4. Applied egg-rr 100.0%

      \[\leadsto \left(\left(x - \color{blue}{\frac{\mathsf{fma}\left(y, y, -0.25\right) \cdot \log y}{y + -0.5}}\right) + y\right) - z \]

   5. Taylor expanded in x around 0 82.6%

      \[\leadsto \color{blue}{\left(y - \frac{\log y \cdot \left({y}^{2} - 0.25\right)}{y - 0.5}\right)} - z \]
   6. Step-by-step derivation

      1. sub-neg 82.6%

         \[\leadsto \left(y - \frac{\log y \cdot \left({y}^{2} - 0.25\right)}{\color{blue}{y + \left(-0.5\right)}}\right) - z \]

      2. metadata-eval 82.6%

         \[\leadsto \left(y - \frac{\log y \cdot \left({y}^{2} - 0.25\right)}{y + \color{blue}{-0.5}}\right) - z \]

      3. associate-/l* 82.6%

         \[\leadsto \left(y - \color{blue}{\frac{\log y}{\frac{y + -0.5}{{y}^{2} - 0.25}}}\right) - z \]

      4. unpow2 82.6%

         \[\leadsto \left(y - \frac{\log y}{\frac{y + -0.5}{\color{blue}{y \cdot y} - 0.25}}\right) - z \]

      5. fma-neg 82.6%

         \[\leadsto \left(y - \frac{\log y}{\frac{y + -0.5}{\color{blue}{\mathsf{fma}\left(y, y, -0.25\right)}}}\right) - z \]

      6. metadata-eval 82.6%

         \[\leadsto \left(y - \frac{\log y}{\frac{y + -0.5}{\mathsf{fma}\left(y, y, \color{blue}{-0.25}\right)}}\right) - z \]

   7. Simplified 82.6%

      \[\leadsto \color{blue}{\left(y - \frac{\log y}{\frac{y + -0.5}{\mathsf{fma}\left(y, y, -0.25\right)}}\right)} - z \]

   8. Taylor expanded in y around 0 82.6%

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

      1. *-commutative 82.6%

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

   10. Simplified 82.6%

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

   if 2.5e103 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in y around inf 88.4%

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

      1. *-commutative 88.4%

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

      2. log-rec 88.4%

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

      3. distribute-lft-neg-in 88.4%

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

      4. distribute-rgt-neg-in 88.4%

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

   5. Simplified 88.4%

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

      1. +-commutative 88.4%

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

      2. *-un-lft-identity 88.4%

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

      3. distribute-rgt-neg-out 88.4%

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

      4. distribute-lft-neg-in 88.4%

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

      5. distribute-rgt-in 88.5%

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

      6. sub-neg 88.5%

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

      7. add-cube-cbrt 87.5%

         \[\leadsto y \cdot \color{blue}{\left(\left(\sqrt[3]{1 - \log y} \cdot \sqrt[3]{1 - \log y}\right) \cdot \sqrt[3]{1 - \log y}\right)} - z \]

      8. unpow3 87.5%

         \[\leadsto y \cdot \color{blue}{{\left(\sqrt[3]{1 - \log y}\right)}^{3}} - z \]

      9. *-commutative 87.5%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{1 - \log y}\right)}^{3} \cdot y} - z \]

      10. unpow3 87.5%

          \[\leadsto \color{blue}{\left(\left(\sqrt[3]{1 - \log y} \cdot \sqrt[3]{1 - \log y}\right) \cdot \sqrt[3]{1 - \log y}\right)} \cdot y - z \]

      11. add-cube-cbrt 88.5%

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

   7. Applied egg-rr 88.5%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.4 \cdot 10^{-134}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{-57}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+103}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 68.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.3 \cdot 10^{+43}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;x \leq -8.5 \cdot 10^{-50}:\\ \;\;\;\;y - y \cdot \log y\\ \mathbf{elif}\;x \leq 300:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -4.3e+43)
   (- x z)
   (if (<= x -8.5e-50)
     (- y (* y (log y)))
     (if (<= x 300.0) (- (* (log y) -0.5) z) (- x z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -4.3e+43) {
		tmp = x - z;
	} else if (x <= -8.5e-50) {
		tmp = y - (y * log(y));
	} else if (x <= 300.0) {
		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 <= (-4.3d+43)) then
        tmp = x - z
    else if (x <= (-8.5d-50)) then
        tmp = y - (y * log(y))
    else if (x <= 300.0d0) 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 <= -4.3e+43) {
		tmp = x - z;
	} else if (x <= -8.5e-50) {
		tmp = y - (y * Math.log(y));
	} else if (x <= 300.0) {
		tmp = (Math.log(y) * -0.5) - z;
	} else {
		tmp = x - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -4.3e+43:
		tmp = x - z
	elif x <= -8.5e-50:
		tmp = y - (y * math.log(y))
	elif x <= 300.0:
		tmp = (math.log(y) * -0.5) - z
	else:
		tmp = x - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -4.3e+43)
		tmp = Float64(x - z);
	elseif (x <= -8.5e-50)
		tmp = Float64(y - Float64(y * log(y)));
	elseif (x <= 300.0)
		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 <= -4.3e+43)
		tmp = x - z;
	elseif (x <= -8.5e-50)
		tmp = y - (y * log(y));
	elseif (x <= 300.0)
		tmp = (log(y) * -0.5) - z;
	else
		tmp = x - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -4.3e+43], N[(x - z), $MachinePrecision], If[LessEqual[x, -8.5e-50], N[(y - N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 300.0], N[(N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision] - z), $MachinePrecision], N[(x - z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.3e43 or 300 < x

   1. Initial program 99.9%

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

      1. flip-+ 82.0%

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

      2. associate-*l/ 82.0%

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

      3. fma-neg 82.0%

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

      4. metadata-eval 82.0%

         \[\leadsto \left(\left(x - \frac{\mathsf{fma}\left(y, y, -\color{blue}{0.25}\right) \cdot \log y}{y - 0.5}\right) + y\right) - z \]

      5. metadata-eval 82.0%

         \[\leadsto \left(\left(x - \frac{\mathsf{fma}\left(y, y, \color{blue}{-0.25}\right) \cdot \log y}{y - 0.5}\right) + y\right) - z \]

      6. sub-neg 82.0%

         \[\leadsto \left(\left(x - \frac{\mathsf{fma}\left(y, y, -0.25\right) \cdot \log y}{\color{blue}{y + \left(-0.5\right)}}\right) + y\right) - z \]

      7. metadata-eval 82.0%

         \[\leadsto \left(\left(x - \frac{\mathsf{fma}\left(y, y, -0.25\right) \cdot \log y}{y + \color{blue}{-0.5}}\right) + y\right) - z \]

   4. Applied egg-rr 82.0%

      \[\leadsto \left(\left(x - \color{blue}{\frac{\mathsf{fma}\left(y, y, -0.25\right) \cdot \log y}{y + -0.5}}\right) + y\right) - z \]

   5. Taylor expanded in x around inf 87.5%

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

   if -4.3e43 < x < -8.50000000000000012e-50

   1. Initial program 99.8%

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

      1. flip-+ 57.3%

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

      2. associate-*l/ 57.4%

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

      3. fma-neg 57.4%

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

      4. metadata-eval 57.4%

         \[\leadsto \left(\left(x - \frac{\mathsf{fma}\left(y, y, -\color{blue}{0.25}\right) \cdot \log y}{y - 0.5}\right) + y\right) - z \]

      5. metadata-eval 57.4%

         \[\leadsto \left(\left(x - \frac{\mathsf{fma}\left(y, y, \color{blue}{-0.25}\right) \cdot \log y}{y - 0.5}\right) + y\right) - z \]

      6. sub-neg 57.4%

         \[\leadsto \left(\left(x - \frac{\mathsf{fma}\left(y, y, -0.25\right) \cdot \log y}{\color{blue}{y + \left(-0.5\right)}}\right) + y\right) - z \]

      7. metadata-eval 57.4%

         \[\leadsto \left(\left(x - \frac{\mathsf{fma}\left(y, y, -0.25\right) \cdot \log y}{y + \color{blue}{-0.5}}\right) + y\right) - z \]

   4. Applied egg-rr 57.4%

      \[\leadsto \left(\left(x - \color{blue}{\frac{\mathsf{fma}\left(y, y, -0.25\right) \cdot \log y}{y + -0.5}}\right) + y\right) - z \]

   5. Taylor expanded in x around 0 50.2%

      \[\leadsto \color{blue}{\left(y - \frac{\log y \cdot \left({y}^{2} - 0.25\right)}{y - 0.5}\right)} - z \]
   6. Step-by-step derivation

      1. sub-neg 50.2%

         \[\leadsto \left(y - \frac{\log y \cdot \left({y}^{2} - 0.25\right)}{\color{blue}{y + \left(-0.5\right)}}\right) - z \]

      2. metadata-eval 50.2%

         \[\leadsto \left(y - \frac{\log y \cdot \left({y}^{2} - 0.25\right)}{y + \color{blue}{-0.5}}\right) - z \]

      3. associate-/l* 50.0%

         \[\leadsto \left(y - \color{blue}{\frac{\log y}{\frac{y + -0.5}{{y}^{2} - 0.25}}}\right) - z \]

      4. unpow2 50.0%

         \[\leadsto \left(y - \frac{\log y}{\frac{y + -0.5}{\color{blue}{y \cdot y} - 0.25}}\right) - z \]

      5. fma-neg 50.0%

         \[\leadsto \left(y - \frac{\log y}{\frac{y + -0.5}{\color{blue}{\mathsf{fma}\left(y, y, -0.25\right)}}}\right) - z \]

      6. metadata-eval 50.0%

         \[\leadsto \left(y - \frac{\log y}{\frac{y + -0.5}{\mathsf{fma}\left(y, y, \color{blue}{-0.25}\right)}}\right) - z \]

   7. Simplified 50.0%

      \[\leadsto \color{blue}{\left(y - \frac{\log y}{\frac{y + -0.5}{\mathsf{fma}\left(y, y, -0.25\right)}}\right)} - z \]

   8. Taylor expanded in y around inf 77.0%

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

      1. mul-1-neg 77.0%

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

      2. log-rec 77.0%

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

      3. distribute-rgt-neg-in 77.0%

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

      4. remove-double-neg 77.0%

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

   10. Simplified 77.0%

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

   11. Taylor expanded in z around 0 62.6%

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

   if -8.50000000000000012e-50 < x < 300

   1. Initial program 99.8%

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

      1. flip-+ 81.5%

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

      2. associate-*l/ 81.5%

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

      3. fma-neg 81.5%

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

      4. metadata-eval 81.5%

         \[\leadsto \left(\left(x - \frac{\mathsf{fma}\left(y, y, -\color{blue}{0.25}\right) \cdot \log y}{y - 0.5}\right) + y\right) - z \]

      5. metadata-eval 81.5%

         \[\leadsto \left(\left(x - \frac{\mathsf{fma}\left(y, y, \color{blue}{-0.25}\right) \cdot \log y}{y - 0.5}\right) + y\right) - z \]

      6. sub-neg 81.5%

         \[\leadsto \left(\left(x - \frac{\mathsf{fma}\left(y, y, -0.25\right) \cdot \log y}{\color{blue}{y + \left(-0.5\right)}}\right) + y\right) - z \]

      7. metadata-eval 81.5%

         \[\leadsto \left(\left(x - \frac{\mathsf{fma}\left(y, y, -0.25\right) \cdot \log y}{y + \color{blue}{-0.5}}\right) + y\right) - z \]

   4. Applied egg-rr 81.5%

      \[\leadsto \left(\left(x - \color{blue}{\frac{\mathsf{fma}\left(y, y, -0.25\right) \cdot \log y}{y + -0.5}}\right) + y\right) - z \]

   5. Taylor expanded in x around 0 80.2%

      \[\leadsto \color{blue}{\left(y - \frac{\log y \cdot \left({y}^{2} - 0.25\right)}{y - 0.5}\right)} - z \]
   6. Step-by-step derivation

      1. sub-neg 80.2%

         \[\leadsto \left(y - \frac{\log y \cdot \left({y}^{2} - 0.25\right)}{\color{blue}{y + \left(-0.5\right)}}\right) - z \]

      2. metadata-eval 80.2%

         \[\leadsto \left(y - \frac{\log y \cdot \left({y}^{2} - 0.25\right)}{y + \color{blue}{-0.5}}\right) - z \]

      3. associate-/l* 80.2%

         \[\leadsto \left(y - \color{blue}{\frac{\log y}{\frac{y + -0.5}{{y}^{2} - 0.25}}}\right) - z \]

      4. unpow2 80.2%

         \[\leadsto \left(y - \frac{\log y}{\frac{y + -0.5}{\color{blue}{y \cdot y} - 0.25}}\right) - z \]

      5. fma-neg 80.2%

         \[\leadsto \left(y - \frac{\log y}{\frac{y + -0.5}{\color{blue}{\mathsf{fma}\left(y, y, -0.25\right)}}}\right) - z \]

      6. metadata-eval 80.2%

         \[\leadsto \left(y - \frac{\log y}{\frac{y + -0.5}{\mathsf{fma}\left(y, y, \color{blue}{-0.25}\right)}}\right) - z \]

   7. Simplified 80.2%

      \[\leadsto \color{blue}{\left(y - \frac{\log y}{\frac{y + -0.5}{\mathsf{fma}\left(y, y, -0.25\right)}}\right)} - z \]

   8. Taylor expanded in y around 0 68.1%

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

      1. *-commutative 68.1%

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

   10. Simplified 68.1%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.3 \cdot 10^{+43}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;x \leq -8.5 \cdot 10^{-50}:\\ \;\;\;\;y - y \cdot \log y\\ \mathbf{elif}\;x \leq 300:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.029:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + \left(y \cdot \left(1 - \log y\right) - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 0.029)
   (- (- 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 <= 0.029) {
		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 <= 0.029d0) 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 <= 0.029) {
		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 <= 0.029:
		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 <= 0.029)
		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 <= 0.029)
		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, 0.029], 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 < 0.0290000000000000015

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 99.5%

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

   if 0.0290000000000000015 < y

   1. Initial program 99.7%

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

      1. associate--l+ 99.6%

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

      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+ 99.7%

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

      4. associate-+r- 99.7%

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

      5. *-commutative 99.7%

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

      6. distribute-rgt-neg-in 99.7%

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

      7. fma-def 99.8%

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

      8. +-commutative 99.8%

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

      9. distribute-neg-in 99.8%

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

      10. unsub-neg 99.8%

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

      11. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around inf 98.1%

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

      1. log-rec 98.1%

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

      2. sub-neg 98.1%

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

   7. Simplified 98.1%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.029:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + \left(y \cdot \left(1 - \log y\right) - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.65 \cdot 10^{+105}:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.65e+105) (- (- 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 <= 1.65e+105) {
		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 <= 1.65d+105) 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 <= 1.65e+105) {
		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 <= 1.65e+105:
		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 <= 1.65e+105)
		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 <= 1.65e+105)
		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, 1.65e+105], 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 < 1.64999999999999999e105

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 93.3%

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

   if 1.64999999999999999e105 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in y around inf 88.4%

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

      1. *-commutative 88.4%

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

      2. log-rec 88.4%

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

      3. distribute-lft-neg-in 88.4%

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

      4. distribute-rgt-neg-in 88.4%

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

   5. Simplified 88.4%

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

      1. +-commutative 88.4%

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

      2. *-un-lft-identity 88.4%

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

      3. distribute-rgt-neg-out 88.4%

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

      4. distribute-lft-neg-in 88.4%

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

      5. distribute-rgt-in 88.5%

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

      6. sub-neg 88.5%

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

      7. add-cube-cbrt 87.5%

         \[\leadsto y \cdot \color{blue}{\left(\left(\sqrt[3]{1 - \log y} \cdot \sqrt[3]{1 - \log y}\right) \cdot \sqrt[3]{1 - \log y}\right)} - z \]

      8. unpow3 87.5%

         \[\leadsto y \cdot \color{blue}{{\left(\sqrt[3]{1 - \log y}\right)}^{3}} - z \]

      9. *-commutative 87.5%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{1 - \log y}\right)}^{3} \cdot y} - z \]

      10. unpow3 87.5%

          \[\leadsto \color{blue}{\left(\left(\sqrt[3]{1 - \log y} \cdot \sqrt[3]{1 - \log y}\right) \cdot \sqrt[3]{1 - \log y}\right)} \cdot y - z \]

      11. add-cube-cbrt 88.5%

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

   7. Applied egg-rr 88.5%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.65 \cdot 10^{+105}:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \end{array} \]
5. Add Preprocessing

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. Add Preprocessing

3. Final simplification 99.8%

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

Alternative 7: 71.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.02 \cdot 10^{+108}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y - y \cdot \log y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.02e+108) (- x z) (- y (* y (log y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.02e+108) {
		tmp = x - z;
	} else {
		tmp = y - (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 <= 1.02d+108) then
        tmp = x - z
    else
        tmp = y - (y * log(y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.02e+108) {
		tmp = x - z;
	} else {
		tmp = y - (y * Math.log(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1.02e+108:
		tmp = x - z
	else:
		tmp = y - (y * math.log(y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.02e+108)
		tmp = Float64(x - z);
	else
		tmp = Float64(y - Float64(y * log(y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1.02e+108)
		tmp = x - z;
	else
		tmp = y - (y * log(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.02e+108], N[(x - z), $MachinePrecision], N[(y - N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.02e108

   1. Initial program 100.0%

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

      1. flip-+ 99.9%

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

      2. associate-*l/ 99.9%

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

      3. fma-neg 99.9%

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

      4. metadata-eval 99.9%

         \[\leadsto \left(\left(x - \frac{\mathsf{fma}\left(y, y, -\color{blue}{0.25}\right) \cdot \log y}{y - 0.5}\right) + y\right) - z \]

      5. metadata-eval 99.9%

         \[\leadsto \left(\left(x - \frac{\mathsf{fma}\left(y, y, \color{blue}{-0.25}\right) \cdot \log y}{y - 0.5}\right) + y\right) - z \]

      6. sub-neg 99.9%

         \[\leadsto \left(\left(x - \frac{\mathsf{fma}\left(y, y, -0.25\right) \cdot \log y}{\color{blue}{y + \left(-0.5\right)}}\right) + y\right) - z \]

      7. metadata-eval 99.9%

         \[\leadsto \left(\left(x - \frac{\mathsf{fma}\left(y, y, -0.25\right) \cdot \log y}{y + \color{blue}{-0.5}}\right) + y\right) - z \]

   4. Applied egg-rr 99.9%

      \[\leadsto \left(\left(x - \color{blue}{\frac{\mathsf{fma}\left(y, y, -0.25\right) \cdot \log y}{y + -0.5}}\right) + y\right) - z \]

   5. Taylor expanded in x around inf 69.5%

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

   if 1.02e108 < y

   1. Initial program 99.6%

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

      1. flip-+ 29.3%

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

      2. associate-*l/ 29.3%

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

      3. fma-neg 29.3%

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

      4. metadata-eval 29.3%

         \[\leadsto \left(\left(x - \frac{\mathsf{fma}\left(y, y, -\color{blue}{0.25}\right) \cdot \log y}{y - 0.5}\right) + y\right) - z \]

      5. metadata-eval 29.3%

         \[\leadsto \left(\left(x - \frac{\mathsf{fma}\left(y, y, \color{blue}{-0.25}\right) \cdot \log y}{y - 0.5}\right) + y\right) - z \]

      6. sub-neg 29.3%

         \[\leadsto \left(\left(x - \frac{\mathsf{fma}\left(y, y, -0.25\right) \cdot \log y}{\color{blue}{y + \left(-0.5\right)}}\right) + y\right) - z \]

      7. metadata-eval 29.3%

         \[\leadsto \left(\left(x - \frac{\mathsf{fma}\left(y, y, -0.25\right) \cdot \log y}{y + \color{blue}{-0.5}}\right) + y\right) - z \]

   4. Applied egg-rr 29.3%

      \[\leadsto \left(\left(x - \color{blue}{\frac{\mathsf{fma}\left(y, y, -0.25\right) \cdot \log y}{y + -0.5}}\right) + y\right) - z \]

   5. Taylor expanded in x around 0 23.2%

      \[\leadsto \color{blue}{\left(y - \frac{\log y \cdot \left({y}^{2} - 0.25\right)}{y - 0.5}\right)} - z \]
   6. Step-by-step derivation

      1. sub-neg 23.2%

         \[\leadsto \left(y - \frac{\log y \cdot \left({y}^{2} - 0.25\right)}{\color{blue}{y + \left(-0.5\right)}}\right) - z \]

      2. metadata-eval 23.2%

         \[\leadsto \left(y - \frac{\log y \cdot \left({y}^{2} - 0.25\right)}{y + \color{blue}{-0.5}}\right) - z \]

      3. associate-/l* 23.2%

         \[\leadsto \left(y - \color{blue}{\frac{\log y}{\frac{y + -0.5}{{y}^{2} - 0.25}}}\right) - z \]

      4. unpow2 23.2%

         \[\leadsto \left(y - \frac{\log y}{\frac{y + -0.5}{\color{blue}{y \cdot y} - 0.25}}\right) - z \]

      5. fma-neg 23.2%

         \[\leadsto \left(y - \frac{\log y}{\frac{y + -0.5}{\color{blue}{\mathsf{fma}\left(y, y, -0.25\right)}}}\right) - z \]

      6. metadata-eval 23.2%

         \[\leadsto \left(y - \frac{\log y}{\frac{y + -0.5}{\mathsf{fma}\left(y, y, \color{blue}{-0.25}\right)}}\right) - z \]

   7. Simplified 23.2%

      \[\leadsto \color{blue}{\left(y - \frac{\log y}{\frac{y + -0.5}{\mathsf{fma}\left(y, y, -0.25\right)}}\right)} - z \]

   8. Taylor expanded in y around inf 88.4%

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

      1. mul-1-neg 88.4%

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

      2. log-rec 88.4%

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

      3. distribute-rgt-neg-in 88.4%

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

      4. remove-double-neg 88.4%

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

   10. Simplified 88.4%

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

   11. Taylor expanded in z around 0 71.4%

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

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.02 \cdot 10^{+108}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y - y \cdot \log y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 48.0% accurate, 9.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{+16} \lor \neg \left(z \leq 2.6 \cdot 10^{+128}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.25e+16) (not (<= z 2.6e+128))) (- z) x))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.25e+16) || !(z <= 2.6e+128)) {
		tmp = -z;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-1.25d+16)) .or. (.not. (z <= 2.6d+128))) then
        tmp = -z
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.25e+16) || !(z <= 2.6e+128)) {
		tmp = -z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.25e+16) or not (z <= 2.6e+128):
		tmp = -z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.25e+16) || !(z <= 2.6e+128))
		tmp = Float64(-z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.25e+16) || ~((z <= 2.6e+128)))
		tmp = -z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.25e+16], N[Not[LessEqual[z, 2.6e+128]], $MachinePrecision]], (-z), x]

Derivation

1. Split input into 2 regimes

2. if z < -1.25e16 or 2.6e128 < z

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-+l+ 99.9%

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

      4. associate-+r- 99.9%

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

      5. *-commutative 99.9%

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

      6. distribute-rgt-neg-in 99.9%

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

      7. fma-def 99.9%

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

      8. +-commutative 99.9%

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

      9. distribute-neg-in 99.9%

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

      10. unsub-neg 99.9%

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

      11. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in z around inf 75.8%

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

      1. neg-mul-1 75.8%

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

   7. Simplified 75.8%

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

   if -1.25e16 < z < 2.6e128

   1. Initial program 99.8%

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

      1. associate--l+ 99.8%

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

      2. sub-neg 99.8%

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

      3. associate-+l+ 99.8%

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

      4. associate-+r- 99.8%

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

      5. *-commutative 99.8%

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

      6. distribute-rgt-neg-in 99.8%

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

      7. fma-def 99.9%

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

      8. +-commutative 99.9%

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

      9. distribute-neg-in 99.9%

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

      10. unsub-neg 99.9%

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

      11. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around inf 37.4%

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

4. Final simplification 50.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{+16} \lor \neg \left(z \leq 2.6 \cdot 10^{+128}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 58.5% accurate, 37.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. flip-+ 79.8%

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

   2. associate-*l/ 79.8%

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

   3. fma-neg 79.8%

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

   4. metadata-eval 79.8%

      \[\leadsto \left(\left(x - \frac{\mathsf{fma}\left(y, y, -\color{blue}{0.25}\right) \cdot \log y}{y - 0.5}\right) + y\right) - z \]

   5. metadata-eval 79.8%

      \[\leadsto \left(\left(x - \frac{\mathsf{fma}\left(y, y, \color{blue}{-0.25}\right) \cdot \log y}{y - 0.5}\right) + y\right) - z \]

   6. sub-neg 79.8%

      \[\leadsto \left(\left(x - \frac{\mathsf{fma}\left(y, y, -0.25\right) \cdot \log y}{\color{blue}{y + \left(-0.5\right)}}\right) + y\right) - z \]

   7. metadata-eval 79.8%

      \[\leadsto \left(\left(x - \frac{\mathsf{fma}\left(y, y, -0.25\right) \cdot \log y}{y + \color{blue}{-0.5}}\right) + y\right) - z \]

4. Applied egg-rr 79.8%

   \[\leadsto \left(\left(x - \color{blue}{\frac{\mathsf{fma}\left(y, y, -0.25\right) \cdot \log y}{y + -0.5}}\right) + y\right) - z \]

5. Taylor expanded in x around inf 57.6%

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

6. Final simplification 57.6%

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

Alternative 10: 30.3% accurate, 111.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 99.8%

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

   1. associate--l+ 99.8%

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

   2. sub-neg 99.8%

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

   3. associate-+l+ 99.8%

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

   4. associate-+r- 99.8%

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

   5. *-commutative 99.8%

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

   6. distribute-rgt-neg-in 99.8%

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

   7. fma-def 99.9%

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

   8. +-commutative 99.9%

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

   9. distribute-neg-in 99.9%

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

   10. unsub-neg 99.9%

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

   11. metadata-eval 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in x around inf 29.4%

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

6. Final simplification 29.4%

   \[\leadsto x \]
7. Add Preprocessing

Developer target: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024019 
(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))