Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Percentage Accurate: 99.8% → 99.9%

Time: 8.6s

Alternatives: 12

Speedup: 0.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate--l+ 99.8%

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

   2. sub-neg 99.8%

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

   3. associate-+l+ 99.8%

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

   4. *-commutative 99.8%

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

   5. distribute-rgt-neg-in 99.8%

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

   6. fma-def 99.9%

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

   7. +-commutative 99.9%

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

   8. distribute-neg-in 99.9%

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

   9. unsub-neg 99.9%

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

   10. metadata-eval 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 2: 76.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{+63} \lor \neg \left(z \leq 2.8 \cdot 10^{+54}\right):\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(1 - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -2.8e+63) (not (<= z 2.8e+54)))
   (- x z)
   (+ x (* y (- 1.0 (log y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2.8e+63) || !(z <= 2.8e+54)) {
		tmp = x - z;
	} else {
		tmp = x + (y * (1.0 - log(y)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-2.8d+63)) .or. (.not. (z <= 2.8d+54))) then
        tmp = x - z
    else
        tmp = x + (y * (1.0d0 - log(y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2.8e+63) || !(z <= 2.8e+54)) {
		tmp = x - z;
	} else {
		tmp = x + (y * (1.0 - Math.log(y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -2.8e+63) or not (z <= 2.8e+54):
		tmp = x - z
	else:
		tmp = x + (y * (1.0 - math.log(y)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -2.8e+63) || !(z <= 2.8e+54))
		tmp = Float64(x - z);
	else
		tmp = Float64(x + Float64(y * Float64(1.0 - log(y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -2.8e+63) || ~((z <= 2.8e+54)))
		tmp = x - z;
	else
		tmp = x + (y * (1.0 - log(y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -2.8e+63], N[Not[LessEqual[z, 2.8e+54]], $MachinePrecision]], N[(x - z), $MachinePrecision], N[(x + N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.79999999999999987e63 or 2.80000000000000015e54 < 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%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around 0 87.3%

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

      1. metadata-eval 87.3%

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

      2. associate-*r* 87.3%

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

      3. mul-1-neg 87.3%

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

      4. log-rec 87.3%

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

      5. log-rec 87.3%

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

      6. mul-1-neg 87.3%

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

      7. associate-*r* 87.3%

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

      8. metadata-eval 87.3%

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

      9. *-commutative 87.3%

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

   6. Simplified 87.3%

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

   7. Taylor expanded in z around inf 87.3%

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

   if -2.79999999999999987e63 < z < 2.80000000000000015e54

   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 y around inf 76.3%

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

      1. log-rec 76.3%

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

      2. sub-neg 76.3%

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

   6. Simplified 76.3%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{+63} \lor \neg \left(z \leq 2.8 \cdot 10^{+54}\right):\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(1 - \log y\right)\\ \end{array} \]

Alternative 3: 89.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.35 \cdot 10^{+44}:\\ \;\;\;\;x - \left(z + \log y \cdot 0.5\right)\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{+117}:\\ \;\;\;\;x + y \cdot \left(1 - \log y\right)\\ \mathbf{else}:\\ \;\;\;\;y - \left(z + y \cdot \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2.35e+44)
   (- x (+ z (* (log y) 0.5)))
   (if (<= y 7.5e+117) (+ x (* y (- 1.0 (log y)))) (- y (+ z (* y (log y)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.35e+44) {
		tmp = x - (z + (log(y) * 0.5));
	} else if (y <= 7.5e+117) {
		tmp = x + (y * (1.0 - log(y)));
	} 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 <= 2.35d+44) then
        tmp = x - (z + (log(y) * 0.5d0))
    else if (y <= 7.5d+117) then
        tmp = x + (y * (1.0d0 - log(y)))
    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 <= 2.35e+44) {
		tmp = x - (z + (Math.log(y) * 0.5));
	} else if (y <= 7.5e+117) {
		tmp = x + (y * (1.0 - Math.log(y)));
	} else {
		tmp = y - (z + (y * Math.log(y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 2.35e+44:
		tmp = x - (z + (math.log(y) * 0.5))
	elif y <= 7.5e+117:
		tmp = x + (y * (1.0 - math.log(y)))
	else:
		tmp = y - (z + (y * math.log(y)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2.35e+44)
		tmp = Float64(x - Float64(z + Float64(log(y) * 0.5)));
	elseif (y <= 7.5e+117)
		tmp = Float64(x + Float64(y * Float64(1.0 - log(y))));
	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 <= 2.35e+44)
		tmp = x - (z + (log(y) * 0.5));
	elseif (y <= 7.5e+117)
		tmp = x + (y * (1.0 - log(y)));
	else
		tmp = y - (z + (y * log(y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2.35e+44], N[(x - N[(z + N[(N[Log[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7.5e+117], N[(x + N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $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 < 2.35000000000000009e44

   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%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 96.6%

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

      1. metadata-eval 96.6%

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

      2. associate-*r* 96.6%

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

      3. mul-1-neg 96.6%

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

      4. log-rec 96.6%

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

      5. log-rec 96.6%

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

      6. mul-1-neg 96.6%

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

      7. associate-*r* 96.6%

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

      8. metadata-eval 96.6%

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

      9. *-commutative 96.6%

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

   6. Simplified 96.6%

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

   if 2.35000000000000009e44 < y < 7.5e117

   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. sub-neg 99.7%

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

      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. *-commutative 99.7%

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

      5. distribute-rgt-neg-in 99.7%

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

      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 y around inf 81.6%

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

      1. log-rec 81.6%

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

      2. sub-neg 81.6%

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

   6. Simplified 81.6%

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

   if 7.5e117 < 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%

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

   3. Simplified 99.6%

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

      1. add-cube-cbrt 98.6%

         \[\leadsto \left(x - \color{blue}{\left(\sqrt[3]{\left(y + 0.5\right) \cdot \log y} \cdot \sqrt[3]{\left(y + 0.5\right) \cdot \log y}\right) \cdot \sqrt[3]{\left(y + 0.5\right) \cdot \log y}}\right) + \left(y - z\right) \]

      2. pow3 98.6%

         \[\leadsto \left(x - \color{blue}{{\left(\sqrt[3]{\left(y + 0.5\right) \cdot \log y}\right)}^{3}}\right) + \left(y - z\right) \]

      3. *-commutative 98.6%

         \[\leadsto \left(x - {\left(\sqrt[3]{\color{blue}{\log y \cdot \left(y + 0.5\right)}}\right)}^{3}\right) + \left(y - z\right) \]

   5. Applied egg-rr 98.6%

      \[\leadsto \left(x - \color{blue}{{\left(\sqrt[3]{\log y \cdot \left(y + 0.5\right)}\right)}^{3}}\right) + \left(y - z\right) \]

   6. Taylor expanded in y around inf 98.6%

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

      1. mul-1-neg 98.6%

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

      2. distribute-rgt-neg-in 98.6%

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

      3. log-rec 98.6%

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

      4. remove-double-neg 98.6%

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

   8. Simplified 98.6%

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

   9. Taylor expanded in x around 0 87.8%

      \[\leadsto \color{blue}{y - \left(z + {1}^{0.3333333333333333} \cdot \left(y \cdot \log y\right)\right)} \]
   10. Step-by-step derivation

       1. +-commutative 87.8%

          \[\leadsto y - \color{blue}{\left({1}^{0.3333333333333333} \cdot \left(y \cdot \log y\right) + z\right)} \]

       2. pow-base-1 87.8%

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

       3. *-lft-identity 87.8%

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

   11. Simplified 87.8%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.35 \cdot 10^{+44}:\\ \;\;\;\;x - \left(z + \log y \cdot 0.5\right)\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{+117}:\\ \;\;\;\;x + y \cdot \left(1 - \log y\right)\\ \mathbf{else}:\\ \;\;\;\;y - \left(z + y \cdot \log y\right)\\ \end{array} \]

Alternative 4: 89.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(1 - \log y\right)\\ \mathbf{if}\;y \leq 8.4 \cdot 10^{+43}:\\ \;\;\;\;x - \left(z + \log y \cdot 0.5\right)\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{+120}:\\ \;\;\;\;x + t_0\\ \mathbf{else}:\\ \;\;\;\;t_0 - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (- 1.0 (log y)))))
   (if (<= y 8.4e+43)
     (- x (+ z (* (log y) 0.5)))
     (if (<= y 9.5e+120) (+ x t_0) (- t_0 z)))))

C

double code(double x, double y, double z) {
	double t_0 = y * (1.0 - log(y));
	double tmp;
	if (y <= 8.4e+43) {
		tmp = x - (z + (log(y) * 0.5));
	} else if (y <= 9.5e+120) {
		tmp = x + t_0;
	} else {
		tmp = t_0 - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y * (1.0d0 - log(y))
    if (y <= 8.4d+43) then
        tmp = x - (z + (log(y) * 0.5d0))
    else if (y <= 9.5d+120) then
        tmp = x + t_0
    else
        tmp = t_0 - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y * (1.0 - Math.log(y));
	double tmp;
	if (y <= 8.4e+43) {
		tmp = x - (z + (Math.log(y) * 0.5));
	} else if (y <= 9.5e+120) {
		tmp = x + t_0;
	} else {
		tmp = t_0 - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * (1.0 - math.log(y))
	tmp = 0
	if y <= 8.4e+43:
		tmp = x - (z + (math.log(y) * 0.5))
	elif y <= 9.5e+120:
		tmp = x + t_0
	else:
		tmp = t_0 - z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * Float64(1.0 - log(y)))
	tmp = 0.0
	if (y <= 8.4e+43)
		tmp = Float64(x - Float64(z + Float64(log(y) * 0.5)));
	elseif (y <= 9.5e+120)
		tmp = Float64(x + t_0);
	else
		tmp = Float64(t_0 - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * (1.0 - log(y));
	tmp = 0.0;
	if (y <= 8.4e+43)
		tmp = x - (z + (log(y) * 0.5));
	elseif (y <= 9.5e+120)
		tmp = x + t_0;
	else
		tmp = t_0 - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 8.4e+43], N[(x - N[(z + N[(N[Log[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 9.5e+120], N[(x + t$95$0), $MachinePrecision], N[(t$95$0 - z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 8.40000000000000007e43

   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%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 96.6%

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

      1. metadata-eval 96.6%

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

      2. associate-*r* 96.6%

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

      3. mul-1-neg 96.6%

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

      4. log-rec 96.6%

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

      5. log-rec 96.6%

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

      6. mul-1-neg 96.6%

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

      7. associate-*r* 96.6%

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

      8. metadata-eval 96.6%

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

      9. *-commutative 96.6%

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

   6. Simplified 96.6%

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

   if 8.40000000000000007e43 < y < 9.5e120

   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. sub-neg 99.7%

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

      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. *-commutative 99.7%

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

      5. distribute-rgt-neg-in 99.7%

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

      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 y around inf 81.6%

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

      1. log-rec 81.6%

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

      2. sub-neg 81.6%

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

   6. Simplified 81.6%

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

   if 9.5e120 < 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%

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

   3. Simplified 99.6%

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

      1. add-cube-cbrt 98.6%

         \[\leadsto \left(x - \color{blue}{\left(\sqrt[3]{\left(y + 0.5\right) \cdot \log y} \cdot \sqrt[3]{\left(y + 0.5\right) \cdot \log y}\right) \cdot \sqrt[3]{\left(y + 0.5\right) \cdot \log y}}\right) + \left(y - z\right) \]

      2. pow3 98.6%

         \[\leadsto \left(x - \color{blue}{{\left(\sqrt[3]{\left(y + 0.5\right) \cdot \log y}\right)}^{3}}\right) + \left(y - z\right) \]

      3. *-commutative 98.6%

         \[\leadsto \left(x - {\left(\sqrt[3]{\color{blue}{\log y \cdot \left(y + 0.5\right)}}\right)}^{3}\right) + \left(y - z\right) \]

   5. Applied egg-rr 98.6%

      \[\leadsto \left(x - \color{blue}{{\left(\sqrt[3]{\log y \cdot \left(y + 0.5\right)}\right)}^{3}}\right) + \left(y - z\right) \]

   6. Taylor expanded in y around inf 98.6%

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

      1. mul-1-neg 98.6%

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

      2. distribute-rgt-neg-in 98.6%

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

      3. log-rec 98.6%

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

      4. remove-double-neg 98.6%

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

   8. Simplified 98.6%

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

   9. Taylor expanded in x around 0 99.6%

      \[\leadsto \color{blue}{\left(x + y\right) - \left(z + {1}^{0.3333333333333333} \cdot \left(y \cdot \log y\right)\right)} \]
   10. Step-by-step derivation

       1. associate--l+ 99.6%

          \[\leadsto \color{blue}{x + \left(y - \left(z + {1}^{0.3333333333333333} \cdot \left(y \cdot \log y\right)\right)\right)} \]

       2. +-commutative 99.6%

          \[\leadsto x + \left(y - \color{blue}{\left({1}^{0.3333333333333333} \cdot \left(y \cdot \log y\right) + z\right)}\right) \]

       3. pow-base-1 99.6%

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

       4. *-lft-identity 99.6%

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

       5. associate--r+ 99.6%

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

       6. *-commutative 99.6%

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

       7. cancel-sign-sub-inv 99.6%

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

       8. *-lft-identity 99.6%

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

       9. distribute-rgt-in 99.7%

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

       10. sub-neg 99.7%

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

       11. associate--l+ 99.7%

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

       12. +-commutative 99.7%

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

       13. fma-udef 99.7%

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

   11. Simplified 99.7%

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

   12. Taylor expanded in y around inf 87.9%

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

       1. log-rec 87.9%

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

       2. mul-1-neg 87.9%

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

       3. remove-double-neg 87.9%

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

   14. Simplified 87.9%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8.4 \cdot 10^{+43}:\\ \;\;\;\;x - \left(z + \log y \cdot 0.5\right)\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{+120}:\\ \;\;\;\;x + y \cdot \left(1 - \log y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \end{array} \]

Alternative 5: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.1 \cdot 10^{-7}:\\ \;\;\;\;x - \left(z + \log y \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y - z\right) + \left(x - y \cdot \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 4.1e-7)
   (- x (+ z (* (log y) 0.5)))
   (+ (- y z) (- x (* y (log y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 4.1e-7) {
		tmp = x - (z + (log(y) * 0.5));
	} else {
		tmp = (y - z) + (x - (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 <= 4.1d-7) then
        tmp = x - (z + (log(y) * 0.5d0))
    else
        tmp = (y - z) + (x - (y * log(y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 4.1e-7) {
		tmp = x - (z + (Math.log(y) * 0.5));
	} else {
		tmp = (y - z) + (x - (y * Math.log(y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 4.1e-7:
		tmp = x - (z + (math.log(y) * 0.5))
	else:
		tmp = (y - z) + (x - (y * math.log(y)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 4.1e-7)
		tmp = Float64(x - Float64(z + Float64(log(y) * 0.5)));
	else
		tmp = Float64(Float64(y - z) + Float64(x - Float64(y * log(y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 4.1e-7)
		tmp = x - (z + (log(y) * 0.5));
	else
		tmp = (y - z) + (x - (y * log(y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 4.1e-7], N[(x - N[(z + N[(N[Log[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y - z), $MachinePrecision] + N[(x - N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.0999999999999999e-7

   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%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 99.7%

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

      1. metadata-eval 99.7%

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

      2. associate-*r* 99.7%

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

      3. mul-1-neg 99.7%

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

      4. log-rec 99.7%

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

      5. log-rec 99.7%

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

      6. mul-1-neg 99.7%

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

      7. associate-*r* 99.7%

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

      8. metadata-eval 99.7%

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

      9. *-commutative 99.7%

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

   6. Simplified 99.7%

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

   if 4.0999999999999999e-7 < 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+ 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around inf 98.9%

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

      1. mul-1-neg 98.9%

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

      2. distribute-rgt-neg-in 98.9%

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

      3. log-rec 98.9%

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

      4. remove-double-neg 98.9%

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

   6. Simplified 98.9%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.1 \cdot 10^{-7}:\\ \;\;\;\;x - \left(z + \log y \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y - z\right) + \left(x - y \cdot \log y\right)\\ \end{array} \]

Alternative 6: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.1 \cdot 10^{-7}:\\ \;\;\;\;x - \left(z + \log y \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + y \cdot \left(1 - \log y\right)\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 4.1e-7)
   (- x (+ z (* (log y) 0.5)))
   (- (+ x (* y (- 1.0 (log y)))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 4.1e-7) {
		tmp = x - (z + (log(y) * 0.5));
	} 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 <= 4.1d-7) then
        tmp = x - (z + (log(y) * 0.5d0))
    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 <= 4.1e-7) {
		tmp = x - (z + (Math.log(y) * 0.5));
	} else {
		tmp = (x + (y * (1.0 - Math.log(y)))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 4.1e-7:
		tmp = x - (z + (math.log(y) * 0.5))
	else:
		tmp = (x + (y * (1.0 - math.log(y)))) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 4.1e-7)
		tmp = Float64(x - Float64(z + Float64(log(y) * 0.5)));
	else
		tmp = Float64(Float64(x + Float64(y * Float64(1.0 - log(y)))) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 4.1e-7)
		tmp = x - (z + (log(y) * 0.5));
	else
		tmp = (x + (y * (1.0 - log(y)))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 4.1e-7], N[(x - N[(z + N[(N[Log[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x + N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.0999999999999999e-7

   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%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 99.7%

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

      1. metadata-eval 99.7%

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

      2. associate-*r* 99.7%

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

      3. mul-1-neg 99.7%

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

      4. log-rec 99.7%

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

      5. log-rec 99.7%

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

      6. mul-1-neg 99.7%

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

      7. associate-*r* 99.7%

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

      8. metadata-eval 99.7%

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

      9. *-commutative 99.7%

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

   6. Simplified 99.7%

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

   if 4.0999999999999999e-7 < 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+ 99.7%

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

   3. Simplified 99.7%

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

      1. add-cube-cbrt 99.0%

         \[\leadsto \left(x - \color{blue}{\left(\sqrt[3]{\left(y + 0.5\right) \cdot \log y} \cdot \sqrt[3]{\left(y + 0.5\right) \cdot \log y}\right) \cdot \sqrt[3]{\left(y + 0.5\right) \cdot \log y}}\right) + \left(y - z\right) \]

      2. pow3 98.9%

         \[\leadsto \left(x - \color{blue}{{\left(\sqrt[3]{\left(y + 0.5\right) \cdot \log y}\right)}^{3}}\right) + \left(y - z\right) \]

      3. *-commutative 98.9%

         \[\leadsto \left(x - {\left(\sqrt[3]{\color{blue}{\log y \cdot \left(y + 0.5\right)}}\right)}^{3}\right) + \left(y - z\right) \]

   5. Applied egg-rr 98.9%

      \[\leadsto \left(x - \color{blue}{{\left(\sqrt[3]{\log y \cdot \left(y + 0.5\right)}\right)}^{3}}\right) + \left(y - z\right) \]

   6. Taylor expanded in y around inf 98.1%

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

      1. mul-1-neg 98.1%

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

      2. distribute-rgt-neg-in 98.1%

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

      3. log-rec 98.1%

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

      4. remove-double-neg 98.1%

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

   8. Simplified 98.1%

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

   9. Taylor expanded in x around 0 98.8%

      \[\leadsto \color{blue}{\left(x + y\right) - \left(z + {1}^{0.3333333333333333} \cdot \left(y \cdot \log y\right)\right)} \]
   10. Step-by-step derivation

       1. associate--l+ 98.8%

          \[\leadsto \color{blue}{x + \left(y - \left(z + {1}^{0.3333333333333333} \cdot \left(y \cdot \log y\right)\right)\right)} \]

       2. +-commutative 98.8%

          \[\leadsto x + \left(y - \color{blue}{\left({1}^{0.3333333333333333} \cdot \left(y \cdot \log y\right) + z\right)}\right) \]

       3. pow-base-1 98.8%

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

       4. *-lft-identity 98.8%

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

       5. associate--r+ 98.8%

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

       6. *-commutative 98.8%

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

       7. cancel-sign-sub-inv 98.8%

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

       8. *-lft-identity 98.8%

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

       9. distribute-rgt-in 99.0%

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

       10. sub-neg 99.0%

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

       11. associate--l+ 99.0%

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

       12. +-commutative 99.0%

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

       13. fma-udef 99.0%

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

   11. Simplified 99.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(y, 1 - \log y, x\right) - z} \]
   12. Step-by-step derivation

       1. fma-udef 99.0%

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

   13. Applied egg-rr 99.0%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.1 \cdot 10^{-7}:\\ \;\;\;\;x - \left(z + \log y \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + y \cdot \left(1 - \log y\right)\right) - z\\ \end{array} \]

Alternative 7: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

3. Simplified 99.8%

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

4. Final simplification 99.8%

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

Alternative 8: 90.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.35 \cdot 10^{+43}:\\ \;\;\;\;x - \left(z + \log y \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(1 - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.35e+43) (- x (+ z (* (log y) 0.5))) (+ x (* y (- 1.0 (log y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.35e+43) {
		tmp = x - (z + (log(y) * 0.5));
	} else {
		tmp = x + (y * (1.0 - log(y)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 1.35d+43) then
        tmp = x - (z + (log(y) * 0.5d0))
    else
        tmp = x + (y * (1.0d0 - log(y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.35e+43) {
		tmp = x - (z + (Math.log(y) * 0.5));
	} else {
		tmp = x + (y * (1.0 - Math.log(y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1.35e+43:
		tmp = x - (z + (math.log(y) * 0.5))
	else:
		tmp = x + (y * (1.0 - math.log(y)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.35e+43)
		tmp = Float64(x - Float64(z + Float64(log(y) * 0.5)));
	else
		tmp = Float64(x + Float64(y * Float64(1.0 - log(y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1.35e+43)
		tmp = x - (z + (log(y) * 0.5));
	else
		tmp = x + (y * (1.0 - log(y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.35e+43], N[(x - N[(z + N[(N[Log[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.3500000000000001e43

   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%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 96.6%

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

      1. metadata-eval 96.6%

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

      2. associate-*r* 96.6%

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

      3. mul-1-neg 96.6%

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

      4. log-rec 96.6%

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

      5. log-rec 96.6%

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

      6. mul-1-neg 96.6%

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

      7. associate-*r* 96.6%

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

      8. metadata-eval 96.6%

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

      9. *-commutative 96.6%

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

   6. Simplified 96.6%

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

   if 1.3500000000000001e43 < y

   1. Initial program 99.6%

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

      1. associate--l+ 99.6%

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

      2. sub-neg 99.6%

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

      3. associate-+l+ 99.6%

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

      4. *-commutative 99.6%

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

      5. distribute-rgt-neg-in 99.6%

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

      6. fma-def 99.8%

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

      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 y around inf 81.2%

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

      1. log-rec 81.2%

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

      2. sub-neg 81.2%

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

   6. Simplified 81.2%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.35 \cdot 10^{+43}:\\ \;\;\;\;x - \left(z + \log y \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(1 - \log y\right)\\ \end{array} \]

Alternative 9: 70.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.65 \cdot 10^{+191}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.65e+191) (- x z) (* y (- 1.0 (log y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.65e+191) {
		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 <= 1.65d+191) 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 <= 1.65e+191) {
		tmp = x - z;
	} else {
		tmp = y * (1.0 - Math.log(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1.65e+191:
		tmp = x - z
	else:
		tmp = y * (1.0 - math.log(y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.65e+191)
		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 <= 1.65e+191)
		tmp = x - z;
	else
		tmp = y * (1.0 - log(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.65e+191], N[(x - z), $MachinePrecision], N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.6499999999999999e191

   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%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around 0 85.7%

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

      1. metadata-eval 85.7%

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

      2. associate-*r* 85.7%

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

      3. mul-1-neg 85.7%

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

      4. log-rec 85.7%

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

      5. log-rec 85.7%

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

      6. mul-1-neg 85.7%

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

      7. associate-*r* 85.7%

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

      8. metadata-eval 85.7%

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

      9. *-commutative 85.7%

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

   6. Simplified 85.7%

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

   7. Taylor expanded in z around inf 70.8%

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

   if 1.6499999999999999e191 < 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%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around 0 93.8%

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

   5. Taylor expanded in y around inf 84.1%

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

      1. mul-1-neg 84.1%

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

      2. log-rec 84.1%

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

      3. remove-double-neg 84.1%

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

   7. Simplified 84.1%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.65 \cdot 10^{+191}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \end{array} \]

Alternative 10: 47.1% accurate, 18.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.35 \cdot 10^{+162} \lor \neg \left(z \leq 1.4 \cdot 10^{+95}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -2.35e+162) (not (<= z 1.4e+95))) (- z) x))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2.35e+162) || !(z <= 1.4e+95)) {
		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 <= (-2.35d+162)) .or. (.not. (z <= 1.4d+95))) 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 <= -2.35e+162) || !(z <= 1.4e+95)) {
		tmp = -z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -2.35e+162) or not (z <= 1.4e+95):
		tmp = -z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -2.35e+162) || !(z <= 1.4e+95))
		tmp = Float64(-z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -2.35e+162) || ~((z <= 1.4e+95)))
		tmp = -z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -2.35e+162], N[Not[LessEqual[z, 1.4e+95]], $MachinePrecision]], (-z), x]

Derivation

1. Split input into 2 regimes

2. if z < -2.35000000000000001e162 or 1.3999999999999999e95 < 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%

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

   3. Simplified 99.9%

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

      1. add-cube-cbrt 99.8%

         \[\leadsto \left(x - \color{blue}{\left(\sqrt[3]{\left(y + 0.5\right) \cdot \log y} \cdot \sqrt[3]{\left(y + 0.5\right) \cdot \log y}\right) \cdot \sqrt[3]{\left(y + 0.5\right) \cdot \log y}}\right) + \left(y - z\right) \]

      2. pow3 99.8%

         \[\leadsto \left(x - \color{blue}{{\left(\sqrt[3]{\left(y + 0.5\right) \cdot \log y}\right)}^{3}}\right) + \left(y - z\right) \]

      3. *-commutative 99.8%

         \[\leadsto \left(x - {\left(\sqrt[3]{\color{blue}{\log y \cdot \left(y + 0.5\right)}}\right)}^{3}\right) + \left(y - z\right) \]

   5. Applied egg-rr 99.8%

      \[\leadsto \left(x - \color{blue}{{\left(\sqrt[3]{\log y \cdot \left(y + 0.5\right)}\right)}^{3}}\right) + \left(y - z\right) \]

   6. Taylor expanded in z around inf 81.0%

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

      1. neg-mul-1 81.0%

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

   8. Simplified 81.0%

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

   if -2.35000000000000001e162 < z < 1.3999999999999999e95

   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%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around inf 39.4%

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

4. Final simplification 52.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.35 \cdot 10^{+162} \lor \neg \left(z \leq 1.4 \cdot 10^{+95}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 11: 58.0% 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%

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

3. Simplified 99.8%

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

4. Taylor expanded in y around 0 72.0%

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

   1. metadata-eval 72.0%

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

   2. associate-*r* 72.0%

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

   3. mul-1-neg 72.0%

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

   4. log-rec 72.0%

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

   5. log-rec 72.0%

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

   6. mul-1-neg 72.0%

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

   7. associate-*r* 72.0%

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

   8. metadata-eval 72.0%

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

   9. *-commutative 72.0%

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

6. Simplified 72.0%

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

7. Taylor expanded in z around inf 60.0%

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

8. Final simplification 60.0%

   \[\leadsto x - z \]

Alternative 12: 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%

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

3. Simplified 99.8%

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

4. Taylor expanded in x around inf 30.0%

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

5. Final simplification 30.0%

   \[\leadsto x \]

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 2023336 
(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))