Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Percentage Accurate: 99.8% → 99.9%

Time: 14.0s

Alternatives: 12

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.5× speedup

Math

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

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: 75.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6.6 \cdot 10^{-129}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y \leq 1.95 \cdot 10^{-23}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{+42}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{+109}:\\ \;\;\;\;x - y \cdot \left(\log y + -1\right)\\ \mathbf{else}:\\ \;\;\;\;y - \left(z + y \cdot \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 6.6e-129)
   (- x z)
   (if (<= y 1.95e-23)
     (- (* (log y) -0.5) z)
     (if (<= y 3.5e+42)
       (- x z)
       (if (<= y 2.3e+109)
         (- x (* y (+ (log y) -1.0)))
         (- y (+ z (* y (log y)))))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 6.6e-129) {
		tmp = x - z;
	} else if (y <= 1.95e-23) {
		tmp = (log(y) * -0.5) - z;
	} else if (y <= 3.5e+42) {
		tmp = x - z;
	} else if (y <= 2.3e+109) {
		tmp = x - (y * (log(y) + -1.0));
	} else {
		tmp = y - (z + (y * log(y)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 6.6d-129) then
        tmp = x - z
    else if (y <= 1.95d-23) then
        tmp = (log(y) * (-0.5d0)) - z
    else if (y <= 3.5d+42) then
        tmp = x - z
    else if (y <= 2.3d+109) then
        tmp = x - (y * (log(y) + (-1.0d0)))
    else
        tmp = y - (z + (y * log(y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 6.6e-129) {
		tmp = x - z;
	} else if (y <= 1.95e-23) {
		tmp = (Math.log(y) * -0.5) - z;
	} else if (y <= 3.5e+42) {
		tmp = x - z;
	} else if (y <= 2.3e+109) {
		tmp = x - (y * (Math.log(y) + -1.0));
	} else {
		tmp = y - (z + (y * Math.log(y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 6.6e-129:
		tmp = x - z
	elif y <= 1.95e-23:
		tmp = (math.log(y) * -0.5) - z
	elif y <= 3.5e+42:
		tmp = x - z
	elif y <= 2.3e+109:
		tmp = x - (y * (math.log(y) + -1.0))
	else:
		tmp = y - (z + (y * math.log(y)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 6.6e-129)
		tmp = Float64(x - z);
	elseif (y <= 1.95e-23)
		tmp = Float64(Float64(log(y) * -0.5) - z);
	elseif (y <= 3.5e+42)
		tmp = Float64(x - z);
	elseif (y <= 2.3e+109)
		tmp = Float64(x - Float64(y * Float64(log(y) + -1.0)));
	else
		tmp = Float64(y - Float64(z + Float64(y * log(y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 6.6e-129)
		tmp = x - z;
	elseif (y <= 1.95e-23)
		tmp = (log(y) * -0.5) - z;
	elseif (y <= 3.5e+42)
		tmp = x - z;
	elseif (y <= 2.3e+109)
		tmp = x - (y * (log(y) + -1.0));
	else
		tmp = y - (z + (y * log(y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 6.6e-129], N[(x - z), $MachinePrecision], If[LessEqual[y, 1.95e-23], N[(N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[y, 3.5e+42], N[(x - z), $MachinePrecision], If[LessEqual[y, 2.3e+109], N[(x - N[(y * N[(N[Log[y], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y - N[(z + N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if y < 6.59999999999999977e-129 or 1.95e-23 < y < 3.50000000000000023e42

   1. Initial program 100.0%

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

      1. associate--l+ 100.0%

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

      2. associate-+l- 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in z around inf 78.7%

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

   if 6.59999999999999977e-129 < y < 1.95e-23

   1. Initial program 100.0%

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

      1. associate--l+ 100.0%

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

      2. associate-+l- 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 89.1%

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

   5. Taylor expanded in y around 0 89.1%

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

      1. distribute-lft-in 89.1%

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

      2. mul-1-neg 89.1%

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

      3. unsub-neg 89.1%

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

      4. *-commutative 89.1%

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

      5. associate-*r* 89.1%

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

      6. *-commutative 89.1%

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

      7. associate-*l* 89.1%

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

      8. metadata-eval 89.1%

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

   7. Simplified 89.1%

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

   if 3.50000000000000023e42 < y < 2.3000000000000001e109

   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+ 99.7%

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

      2. associate-+l- 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around inf 86.2%

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

      1. sub-neg 86.2%

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

      2. mul-1-neg 86.2%

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

      3. log-rec 86.2%

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

      4. remove-double-neg 86.2%

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

      5. metadata-eval 86.2%

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

   6. Simplified 86.2%

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

   if 2.3000000000000001e109 < y

   1. Initial program 99.6%

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

      1. associate--l+ 99.6%

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

      2. associate-+l- 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around 0 89.7%

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

   5. Taylor expanded in y around inf 89.7%

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

      1. mul-1-neg 89.7%

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

      2. distribute-rgt-neg-in 89.7%

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

      3. log-rec 89.7%

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

      4. remove-double-neg 89.7%

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

   7. Simplified 89.7%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.6 \cdot 10^{-129}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y \leq 1.95 \cdot 10^{-23}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{+42}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{+109}:\\ \;\;\;\;x - y \cdot \left(\log y + -1\right)\\ \mathbf{else}:\\ \;\;\;\;y - \left(z + y \cdot \log y\right)\\ \end{array} \]

Alternative 3: 75.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7.6 \cdot 10^{-130}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{-23}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+43}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \left(\log y + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 7.6e-130)
   (- x z)
   (if (<= y 1.6e-23)
     (- (* (log y) -0.5) z)
     (if (<= y 1.5e+43) (- x z) (- x (* y (+ (log y) -1.0)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 7.6e-130) {
		tmp = x - z;
	} else if (y <= 1.6e-23) {
		tmp = (log(y) * -0.5) - z;
	} else if (y <= 1.5e+43) {
		tmp = x - z;
	} else {
		tmp = x - (y * (log(y) + -1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 7.6d-130) then
        tmp = x - z
    else if (y <= 1.6d-23) then
        tmp = (log(y) * (-0.5d0)) - z
    else if (y <= 1.5d+43) then
        tmp = x - z
    else
        tmp = x - (y * (log(y) + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 7.6e-130) {
		tmp = x - z;
	} else if (y <= 1.6e-23) {
		tmp = (Math.log(y) * -0.5) - z;
	} else if (y <= 1.5e+43) {
		tmp = x - z;
	} else {
		tmp = x - (y * (Math.log(y) + -1.0));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 7.6e-130:
		tmp = x - z
	elif y <= 1.6e-23:
		tmp = (math.log(y) * -0.5) - z
	elif y <= 1.5e+43:
		tmp = x - z
	else:
		tmp = x - (y * (math.log(y) + -1.0))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 7.6e-130)
		tmp = Float64(x - z);
	elseif (y <= 1.6e-23)
		tmp = Float64(Float64(log(y) * -0.5) - z);
	elseif (y <= 1.5e+43)
		tmp = Float64(x - z);
	else
		tmp = Float64(x - Float64(y * Float64(log(y) + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 7.6e-130)
		tmp = x - z;
	elseif (y <= 1.6e-23)
		tmp = (log(y) * -0.5) - z;
	elseif (y <= 1.5e+43)
		tmp = x - z;
	else
		tmp = x - (y * (log(y) + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 7.6e-130], N[(x - z), $MachinePrecision], If[LessEqual[y, 1.6e-23], N[(N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[y, 1.5e+43], N[(x - z), $MachinePrecision], N[(x - N[(y * N[(N[Log[y], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 7.5999999999999997e-130 or 1.59999999999999988e-23 < y < 1.50000000000000008e43

   1. Initial program 100.0%

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

      1. associate--l+ 100.0%

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

      2. associate-+l- 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in z around inf 78.7%

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

   if 7.5999999999999997e-130 < y < 1.59999999999999988e-23

   1. Initial program 100.0%

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

      1. associate--l+ 100.0%

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

      2. associate-+l- 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 89.1%

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

   5. Taylor expanded in y around 0 89.1%

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

      1. distribute-lft-in 89.1%

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

      2. mul-1-neg 89.1%

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

      3. unsub-neg 89.1%

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

      4. *-commutative 89.1%

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

      5. associate-*r* 89.1%

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

      6. *-commutative 89.1%

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

      7. associate-*l* 89.1%

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

      8. metadata-eval 89.1%

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

   7. Simplified 89.1%

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

   if 1.50000000000000008e43 < y

   1. Initial program 99.6%

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

      1. associate--l+ 99.6%

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

      2. associate-+l- 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around inf 82.0%

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

      1. sub-neg 82.0%

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

      2. mul-1-neg 82.0%

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

      3. log-rec 82.0%

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

      4. remove-double-neg 82.0%

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

      5. metadata-eval 82.0%

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

   6. Simplified 82.0%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7.6 \cdot 10^{-130}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{-23}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+43}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \left(\log y + -1\right)\\ \end{array} \]

Alternative 4: 88.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{+98} \lor \neg \left(x \leq 8.2 \cdot 10^{+30}\right):\\ \;\;\;\;x - y \cdot \left(\log y + -1\right)\\ \mathbf{else}:\\ \;\;\;\;y - \left(z + \log y \cdot \left(y + 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -9e+98) (not (<= x 8.2e+30)))
   (- x (* y (+ (log y) -1.0)))
   (- y (+ z (* (log y) (+ y 0.5))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -9e+98) || !(x <= 8.2e+30)) {
		tmp = x - (y * (log(y) + -1.0));
	} else {
		tmp = y - (z + (log(y) * (y + 0.5)));
	}
	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 <= (-9d+98)) .or. (.not. (x <= 8.2d+30))) then
        tmp = x - (y * (log(y) + (-1.0d0)))
    else
        tmp = y - (z + (log(y) * (y + 0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -9e+98) || !(x <= 8.2e+30)) {
		tmp = x - (y * (Math.log(y) + -1.0));
	} else {
		tmp = y - (z + (Math.log(y) * (y + 0.5)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -9e+98) or not (x <= 8.2e+30):
		tmp = x - (y * (math.log(y) + -1.0))
	else:
		tmp = y - (z + (math.log(y) * (y + 0.5)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -9e+98) || !(x <= 8.2e+30))
		tmp = Float64(x - Float64(y * Float64(log(y) + -1.0)));
	else
		tmp = Float64(y - Float64(z + Float64(log(y) * Float64(y + 0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -9e+98) || ~((x <= 8.2e+30)))
		tmp = x - (y * (log(y) + -1.0));
	else
		tmp = y - (z + (log(y) * (y + 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -9e+98], N[Not[LessEqual[x, 8.2e+30]], $MachinePrecision]], N[(x - N[(y * N[(N[Log[y], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y - N[(z + N[(N[Log[y], $MachinePrecision] * N[(y + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -9.0000000000000004e98 or 8.20000000000000011e30 < x

   1. Initial program 99.8%

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

      1. associate--l+ 99.8%

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

      2. associate-+l- 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around inf 86.3%

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

      1. sub-neg 86.3%

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

      2. mul-1-neg 86.3%

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

      3. log-rec 86.3%

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

      4. remove-double-neg 86.3%

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

      5. metadata-eval 86.3%

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

   6. Simplified 86.3%

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

   if -9.0000000000000004e98 < x < 8.20000000000000011e30

   1. Initial program 99.8%

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

      1. associate--l+ 99.8%

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

      2. associate-+l- 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 97.9%

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

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{+98} \lor \neg \left(x \leq 8.2 \cdot 10^{+30}\right):\\ \;\;\;\;x - y \cdot \left(\log y + -1\right)\\ \mathbf{else}:\\ \;\;\;\;y - \left(z + \log y \cdot \left(y + 0.5\right)\right)\\ \end{array} \]

Alternative 5: 65.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{-115}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{-44}:\\ \;\;\;\;\log y \cdot -0.5\\ \mathbf{elif}\;y \leq 7 \cdot 10^{+179}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 5e-115)
   (- x z)
   (if (<= y 1.65e-44)
     (* (log y) -0.5)
     (if (<= y 7e+179) (- x z) (* y (- 1.0 (log y)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 5e-115) {
		tmp = x - z;
	} else if (y <= 1.65e-44) {
		tmp = log(y) * -0.5;
	} else if (y <= 7e+179) {
		tmp = x - z;
	} else {
		tmp = 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 <= 5d-115) then
        tmp = x - z
    else if (y <= 1.65d-44) then
        tmp = log(y) * (-0.5d0)
    else if (y <= 7d+179) then
        tmp = x - z
    else
        tmp = 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 <= 5e-115) {
		tmp = x - z;
	} else if (y <= 1.65e-44) {
		tmp = Math.log(y) * -0.5;
	} else if (y <= 7e+179) {
		tmp = x - z;
	} else {
		tmp = y * (1.0 - Math.log(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 5e-115:
		tmp = x - z
	elif y <= 1.65e-44:
		tmp = math.log(y) * -0.5
	elif y <= 7e+179:
		tmp = x - z
	else:
		tmp = y * (1.0 - math.log(y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 5e-115)
		tmp = Float64(x - z);
	elseif (y <= 1.65e-44)
		tmp = Float64(log(y) * -0.5);
	elseif (y <= 7e+179)
		tmp = Float64(x - z);
	else
		tmp = Float64(y * Float64(1.0 - log(y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 5e-115)
		tmp = x - z;
	elseif (y <= 1.65e-44)
		tmp = log(y) * -0.5;
	elseif (y <= 7e+179)
		tmp = x - z;
	else
		tmp = y * (1.0 - log(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 5e-115], N[(x - z), $MachinePrecision], If[LessEqual[y, 1.65e-44], N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision], If[LessEqual[y, 7e+179], N[(x - z), $MachinePrecision], N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 5.0000000000000003e-115 or 1.65000000000000003e-44 < y < 7.0000000000000003e179

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. associate-+l- 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in z around inf 73.0%

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

   if 5.0000000000000003e-115 < y < 1.65000000000000003e-44

   1. Initial program 100.0%

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

      1. associate--l+ 100.0%

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

      2. associate-+l- 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 89.3%

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

   5. Taylor expanded in z around 0 59.7%

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

   6. Taylor expanded in y around 0 59.7%

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

   if 7.0000000000000003e179 < y

   1. Initial program 99.5%

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

      1. associate--l+ 99.5%

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

      2. associate-+l- 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around 0 96.1%

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

   5. Taylor expanded in y around inf 83.0%

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

      1. mul-1-neg 83.0%

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

      2. log-rec 83.0%

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

      3. remove-double-neg 83.0%

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

   7. Simplified 83.0%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{-115}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{-44}:\\ \;\;\;\;\log y \cdot -0.5\\ \mathbf{elif}\;y \leq 7 \cdot 10^{+179}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \end{array} \]

Alternative 6: 68.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 9.5 \cdot 10^{-131}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-23}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{elif}\;y \leq 7 \cdot 10^{+179}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 9.5e-131)
   (- x z)
   (if (<= y 2e-23)
     (- (* (log y) -0.5) z)
     (if (<= y 7e+179) (- x z) (* y (- 1.0 (log y)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 9.5e-131) {
		tmp = x - z;
	} else if (y <= 2e-23) {
		tmp = (log(y) * -0.5) - z;
	} else if (y <= 7e+179) {
		tmp = x - z;
	} else {
		tmp = 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 <= 9.5d-131) then
        tmp = x - z
    else if (y <= 2d-23) then
        tmp = (log(y) * (-0.5d0)) - z
    else if (y <= 7d+179) then
        tmp = x - z
    else
        tmp = 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 <= 9.5e-131) {
		tmp = x - z;
	} else if (y <= 2e-23) {
		tmp = (Math.log(y) * -0.5) - z;
	} else if (y <= 7e+179) {
		tmp = x - z;
	} else {
		tmp = y * (1.0 - Math.log(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 9.5e-131:
		tmp = x - z
	elif y <= 2e-23:
		tmp = (math.log(y) * -0.5) - z
	elif y <= 7e+179:
		tmp = x - z
	else:
		tmp = y * (1.0 - math.log(y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 9.5e-131)
		tmp = Float64(x - z);
	elseif (y <= 2e-23)
		tmp = Float64(Float64(log(y) * -0.5) - z);
	elseif (y <= 7e+179)
		tmp = Float64(x - z);
	else
		tmp = Float64(y * Float64(1.0 - log(y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 9.5e-131)
		tmp = x - z;
	elseif (y <= 2e-23)
		tmp = (log(y) * -0.5) - z;
	elseif (y <= 7e+179)
		tmp = x - z;
	else
		tmp = y * (1.0 - log(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 9.5e-131], N[(x - z), $MachinePrecision], If[LessEqual[y, 2e-23], N[(N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[y, 7e+179], N[(x - z), $MachinePrecision], N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 9.4999999999999996e-131 or 1.99999999999999992e-23 < y < 7.0000000000000003e179

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. associate-+l- 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in z around inf 72.9%

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

   if 9.4999999999999996e-131 < y < 1.99999999999999992e-23

   1. Initial program 100.0%

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

      1. associate--l+ 100.0%

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

      2. associate-+l- 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 89.1%

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

   5. Taylor expanded in y around 0 89.1%

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

      1. distribute-lft-in 89.1%

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

      2. mul-1-neg 89.1%

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

      3. unsub-neg 89.1%

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

      4. *-commutative 89.1%

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

      5. associate-*r* 89.1%

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

      6. *-commutative 89.1%

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

      7. associate-*l* 89.1%

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

      8. metadata-eval 89.1%

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

   7. Simplified 89.1%

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

   if 7.0000000000000003e179 < y

   1. Initial program 99.5%

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

      1. associate--l+ 99.5%

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

      2. associate-+l- 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around 0 96.1%

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

   5. Taylor expanded in y around inf 83.0%

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

      1. mul-1-neg 83.0%

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

      2. log-rec 83.0%

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

      3. remove-double-neg 83.0%

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

   7. Simplified 83.0%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9.5 \cdot 10^{-131}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-23}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{elif}\;y \leq 7 \cdot 10^{+179}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \end{array} \]

Alternative 7: 90.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6 \cdot 10^{+43}:\\ \;\;\;\;\left(x + \log y \cdot -0.5\right) - z\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{+108}:\\ \;\;\;\;x - y \cdot \left(\log y + -1\right)\\ \mathbf{else}:\\ \;\;\;\;y - \left(z + y \cdot \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 6e+43)
   (- (+ x (* (log y) -0.5)) z)
   (if (<= y 7.5e+108)
     (- x (* y (+ (log y) -1.0)))
     (- y (+ z (* y (log y)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 6e+43) {
		tmp = (x + (log(y) * -0.5)) - z;
	} else if (y <= 7.5e+108) {
		tmp = x - (y * (log(y) + -1.0));
	} else {
		tmp = y - (z + (y * log(y)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 6d+43) then
        tmp = (x + (log(y) * (-0.5d0))) - z
    else if (y <= 7.5d+108) then
        tmp = x - (y * (log(y) + (-1.0d0)))
    else
        tmp = y - (z + (y * log(y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 6e+43) {
		tmp = (x + (Math.log(y) * -0.5)) - z;
	} else if (y <= 7.5e+108) {
		tmp = x - (y * (Math.log(y) + -1.0));
	} else {
		tmp = y - (z + (y * Math.log(y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 6e+43:
		tmp = (x + (math.log(y) * -0.5)) - z
	elif y <= 7.5e+108:
		tmp = x - (y * (math.log(y) + -1.0))
	else:
		tmp = y - (z + (y * math.log(y)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 6e+43)
		tmp = Float64(Float64(x + Float64(log(y) * -0.5)) - z);
	elseif (y <= 7.5e+108)
		tmp = Float64(x - Float64(y * Float64(log(y) + -1.0)));
	else
		tmp = Float64(y - Float64(z + Float64(y * log(y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 6e+43)
		tmp = (x + (log(y) * -0.5)) - z;
	elseif (y <= 7.5e+108)
		tmp = x - (y * (log(y) + -1.0));
	else
		tmp = y - (z + (y * log(y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 6e+43], N[(N[(x + N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[y, 7.5e+108], N[(x - N[(y * N[(N[Log[y], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y - N[(z + N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 6.00000000000000033e43

   1. Initial program 100.0%

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

      1. associate--l+ 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-+l+ 100.0%

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

      4. *-commutative 100.0%

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

      5. distribute-rgt-neg-in 100.0%

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

      6. fma-def 100.0%

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

      7. +-commutative 100.0%

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

      8. distribute-neg-in 100.0%

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

      9. unsub-neg 100.0%

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

      10. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 97.4%

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

   if 6.00000000000000033e43 < y < 7.50000000000000039e108

   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+ 99.7%

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

      2. associate-+l- 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around inf 86.2%

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

      1. sub-neg 86.2%

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

      2. mul-1-neg 86.2%

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

      3. log-rec 86.2%

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

      4. remove-double-neg 86.2%

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

      5. metadata-eval 86.2%

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

   6. Simplified 86.2%

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

   if 7.50000000000000039e108 < y

   1. Initial program 99.6%

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

      1. associate--l+ 99.6%

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

      2. associate-+l- 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around 0 89.7%

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

   5. Taylor expanded in y around inf 89.7%

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

      1. mul-1-neg 89.7%

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

      2. distribute-rgt-neg-in 89.7%

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

      3. log-rec 89.7%

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

      4. remove-double-neg 89.7%

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

   7. Simplified 89.7%

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

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6 \cdot 10^{+43}:\\ \;\;\;\;\left(x + \log y \cdot -0.5\right) - z\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{+108}:\\ \;\;\;\;x - y \cdot \left(\log y + -1\right)\\ \mathbf{else}:\\ \;\;\;\;y - \left(z + y \cdot \log y\right)\\ \end{array} \]

Alternative 8: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate--l+ 99.8%

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

   2. associate-+l- 99.8%

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

3. Simplified 99.8%

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

4. Final simplification 99.8%

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

Alternative 9: 54.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.4 \cdot 10^{-194}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{-40}:\\ \;\;\;\;\log y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2.4e-194) (- x z) (if (<= z 1.4e-40) (* (log y) -0.5) (- x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.4e-194) {
		tmp = x - z;
	} else if (z <= 1.4e-40) {
		tmp = log(y) * -0.5;
	} else {
		tmp = x - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-2.4d-194)) then
        tmp = x - z
    else if (z <= 1.4d-40) then
        tmp = log(y) * (-0.5d0)
    else
        tmp = x - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.4e-194) {
		tmp = x - z;
	} else if (z <= 1.4e-40) {
		tmp = Math.log(y) * -0.5;
	} else {
		tmp = x - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -2.4e-194:
		tmp = x - z
	elif z <= 1.4e-40:
		tmp = math.log(y) * -0.5
	else:
		tmp = x - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2.4e-194)
		tmp = Float64(x - z);
	elseif (z <= 1.4e-40)
		tmp = Float64(log(y) * -0.5);
	else
		tmp = Float64(x - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -2.4e-194)
		tmp = x - z;
	elseif (z <= 1.4e-40)
		tmp = log(y) * -0.5;
	else
		tmp = x - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2.4e-194], N[(x - z), $MachinePrecision], If[LessEqual[z, 1.4e-40], N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision], N[(x - z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.4e-194 or 1.4e-40 < z

   1. Initial program 99.8%

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

      1. associate--l+ 99.8%

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

      2. associate-+l- 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in z around inf 69.0%

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

   if -2.4e-194 < z < 1.4e-40

   1. Initial program 99.8%

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

      1. associate--l+ 99.8%

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

      2. associate-+l- 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 76.3%

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

   5. Taylor expanded in z around 0 76.3%

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

   6. Taylor expanded in y around 0 42.0%

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

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.4 \cdot 10^{-194}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{-40}:\\ \;\;\;\;\log y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \]

Alternative 10: 48.0% accurate, 18.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{+100}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{+37}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -2.9e+100) x (if (<= x 1.05e+37) (- z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.9e+100) {
		tmp = x;
	} else if (x <= 1.05e+37) {
		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 <= (-2.9d+100)) then
        tmp = x
    else if (x <= 1.05d+37) 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 <= -2.9e+100) {
		tmp = x;
	} else if (x <= 1.05e+37) {
		tmp = -z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -2.9e+100:
		tmp = x
	elif x <= 1.05e+37:
		tmp = -z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -2.9e+100)
		tmp = x;
	elseif (x <= 1.05e+37)
		tmp = Float64(-z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -2.9e+100)
		tmp = x;
	elseif (x <= 1.05e+37)
		tmp = -z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -2.9e+100], x, If[LessEqual[x, 1.05e+37], (-z), x]]

Derivation

1. Split input into 2 regimes

2. if x < -2.9e100 or 1.0500000000000001e37 < x

   1. Initial program 99.8%

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

      1. associate--l+ 99.8%

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

      2. sub-neg 99.8%

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

      3. associate-+l+ 99.8%

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

      4. *-commutative 99.8%

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

      5. distribute-rgt-neg-in 99.8%

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

      6. fma-def 99.9%

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

      7. +-commutative 99.9%

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

      8. distribute-neg-in 99.9%

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

      9. unsub-neg 99.9%

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

      10. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around inf 64.7%

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

   if -2.9e100 < x < 1.0500000000000001e37

   1. Initial program 99.8%

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

      1. associate--l+ 99.8%

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

      2. associate-+l- 99.8%

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

   3. Simplified 99.8%

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

      1. add-sqr-sqrt 99.2%

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

      2. associate-*l* 99.3%

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

      3. fma-neg 99.3%

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

   5. Applied egg-rr 99.3%

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

      1. *-commutative 99.3%

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

      2. neg-sub0 99.3%

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

      3. metadata-eval 99.3%

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

      4. sub-neg 99.3%

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

      5. +-commutative 99.3%

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

      6. associate--l- 99.3%

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

      7. metadata-eval 99.3%

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

      8. neg-sub0 99.3%

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

      9. remove-double-neg 99.3%

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

   7. Simplified 99.3%

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

   8. Taylor expanded in z around inf 41.1%

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

      1. mul-1-neg 41.1%

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

   10. Simplified 41.1%

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

4. Final simplification 49.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{+100}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{+37}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 11: 58.3% accurate, 37.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate--l+ 99.8%

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

   2. associate-+l- 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in z around inf 55.3%

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

5. Final simplification 55.3%

   \[\leadsto x - z \]

Alternative 12: 30.1% 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 25.5%

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

5. Final simplification 25.5%

   \[\leadsto x \]

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 2023274 
(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))