Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Percentage Accurate: 99.8% → 99.9%

Time: 13.0s

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate--l+ 99.8%

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

   2. sub-neg 99.8%

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

   3. associate-+l+ 99.8%

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

   4. associate-+r- 99.8%

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

   5. *-commutative 99.8%

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

   6. distribute-rgt-neg-in 99.8%

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

   7. fma-def 99.9%

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

   8. +-commutative 99.9%

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

   9. distribute-neg-in 99.9%

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

   10. unsub-neg 99.9%

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

   11. metadata-eval 99.9%

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

3. Simplified 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 77.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x + y\right) - z\\ \mathbf{if}\;y \leq 1.45 \cdot 10^{-68}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{-40}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{+69}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(1 - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (+ x y) z)))
   (if (<= y 1.45e-68)
     t_0
     (if (<= y 2.8e-40)
       (- (* (log y) -0.5) z)
       (if (<= y 4.8e+69) t_0 (+ x (* y (- 1.0 (log y)))))))))

C

double code(double x, double y, double z) {
	double t_0 = (x + y) - z;
	double tmp;
	if (y <= 1.45e-68) {
		tmp = t_0;
	} else if (y <= 2.8e-40) {
		tmp = (log(y) * -0.5) - z;
	} else if (y <= 4.8e+69) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = (x + y) - z
    if (y <= 1.45d-68) then
        tmp = t_0
    else if (y <= 2.8d-40) then
        tmp = (log(y) * (-0.5d0)) - z
    else if (y <= 4.8d+69) then
        tmp = t_0
    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 t_0 = (x + y) - z;
	double tmp;
	if (y <= 1.45e-68) {
		tmp = t_0;
	} else if (y <= 2.8e-40) {
		tmp = (Math.log(y) * -0.5) - z;
	} else if (y <= 4.8e+69) {
		tmp = t_0;
	} else {
		tmp = x + (y * (1.0 - Math.log(y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x + y) - z
	tmp = 0
	if y <= 1.45e-68:
		tmp = t_0
	elif y <= 2.8e-40:
		tmp = (math.log(y) * -0.5) - z
	elif y <= 4.8e+69:
		tmp = t_0
	else:
		tmp = x + (y * (1.0 - math.log(y)))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x + y) - z)
	tmp = 0.0
	if (y <= 1.45e-68)
		tmp = t_0;
	elseif (y <= 2.8e-40)
		tmp = Float64(Float64(log(y) * -0.5) - z);
	elseif (y <= 4.8e+69)
		tmp = t_0;
	else
		tmp = Float64(x + Float64(y * Float64(1.0 - log(y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x + y) - z;
	tmp = 0.0;
	if (y <= 1.45e-68)
		tmp = t_0;
	elseif (y <= 2.8e-40)
		tmp = (log(y) * -0.5) - z;
	elseif (y <= 4.8e+69)
		tmp = t_0;
	else
		tmp = x + (y * (1.0 - log(y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x + y), $MachinePrecision] - z), $MachinePrecision]}, If[LessEqual[y, 1.45e-68], t$95$0, If[LessEqual[y, 2.8e-40], N[(N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[y, 4.8e+69], t$95$0, N[(x + N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.45e-68 or 2.8e-40 < y < 4.8000000000000003e69

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. associate-+r- 99.9%

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

      3. *-commutative 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. add-cube-cbrt 99.5%

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

      2. pow3 99.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in y around inf 78.6%

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

      1. +-commutative 78.6%

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

   9. Simplified 78.6%

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

   if 1.45e-68 < y < 2.8e-40

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. flip-+ 100.0%

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

      3. associate-*r/ 100.0%

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

      4. fma-neg 100.0%

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

      5. metadata-eval 100.0%

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

      6. metadata-eval 100.0%

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

      7. sub-neg 100.0%

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

      8. metadata-eval 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-/l* 99.8%

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

      3. +-commutative 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in x around 0 94.2%

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

      1. sub-neg 94.2%

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

      2. metadata-eval 94.2%

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

      3. associate-/l* 94.2%

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

      4. unpow2 94.2%

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

      5. fma-neg 94.2%

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

      6. metadata-eval 94.2%

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

   9. Simplified 94.2%

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

   10. Taylor expanded in y around 0 94.2%

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

       1. *-commutative 94.2%

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

   12. Simplified 94.2%

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

   if 4.8000000000000003e69 < 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. sub-neg 99.5%

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

      3. associate-+l+ 99.5%

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

      4. associate-+r- 99.5%

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

      5. *-commutative 99.5%

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

      6. distribute-rgt-neg-in 99.5%

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

      7. fma-def 99.7%

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

      8. +-commutative 99.7%

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

      9. distribute-neg-in 99.7%

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

      10. unsub-neg 99.7%

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

      11. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around inf 99.7%

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

      1. log-rec 99.7%

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

      2. sub-neg 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in z around 0 86.1%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.45 \cdot 10^{-68}:\\ \;\;\;\;\left(x + y\right) - z\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{-40}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{+69}:\\ \;\;\;\;\left(x + y\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(1 - \log y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(1 - \log y\right)\\ \mathbf{if}\;z \leq -1.3 \cdot 10^{+39} \lor \neg \left(z \leq 1.15 \cdot 10^{+91}\right):\\ \;\;\;\;t\_0 - z\\ \mathbf{else}:\\ \;\;\;\;x + t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (- 1.0 (log y)))))
   (if (or (<= z -1.3e+39) (not (<= z 1.15e+91))) (- t_0 z) (+ x t_0))))

C

double code(double x, double y, double z) {
	double t_0 = y * (1.0 - log(y));
	double tmp;
	if ((z <= -1.3e+39) || !(z <= 1.15e+91)) {
		tmp = t_0 - z;
	} else {
		tmp = x + t_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) :: t_0
    real(8) :: tmp
    t_0 = y * (1.0d0 - log(y))
    if ((z <= (-1.3d+39)) .or. (.not. (z <= 1.15d+91))) then
        tmp = t_0 - z
    else
        tmp = x + t_0
    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 ((z <= -1.3e+39) || !(z <= 1.15e+91)) {
		tmp = t_0 - z;
	} else {
		tmp = x + t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * (1.0 - math.log(y))
	tmp = 0
	if (z <= -1.3e+39) or not (z <= 1.15e+91):
		tmp = t_0 - z
	else:
		tmp = x + t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * Float64(1.0 - log(y)))
	tmp = 0.0
	if ((z <= -1.3e+39) || !(z <= 1.15e+91))
		tmp = Float64(t_0 - z);
	else
		tmp = Float64(x + t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * (1.0 - log(y));
	tmp = 0.0;
	if ((z <= -1.3e+39) || ~((z <= 1.15e+91)))
		tmp = t_0 - z;
	else
		tmp = x + t_0;
	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[Or[LessEqual[z, -1.3e+39], N[Not[LessEqual[z, 1.15e+91]], $MachinePrecision]], N[(t$95$0 - z), $MachinePrecision], N[(x + t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.3e39 or 1.14999999999999996e91 < z

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. associate-+r- 99.9%

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

      3. *-commutative 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in y around inf 90.3%

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

      1. mul-1-neg 90.3%

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

      2. log-rec 90.3%

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

      3. remove-double-neg 90.3%

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

   7. Simplified 90.3%

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

   if -1.3e39 < z < 1.14999999999999996e91

   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. associate-+r- 99.7%

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

      5. *-commutative 99.7%

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

      6. distribute-rgt-neg-in 99.7%

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

      7. fma-def 99.8%

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

      8. +-commutative 99.8%

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

      9. distribute-neg-in 99.8%

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

      10. unsub-neg 99.8%

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

      11. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around inf 79.4%

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

      1. log-rec 79.4%

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

      2. sub-neg 79.4%

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

   7. Simplified 79.4%

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

   8. Taylor expanded in z around 0 77.8%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{+39} \lor \neg \left(z \leq 1.15 \cdot 10^{+91}\right):\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(1 - \log y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(1 - \log y\right)\\ \mathbf{if}\;y \leq 5.5 \cdot 10^{+44}:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{+175}:\\ \;\;\;\;t\_0 - z\\ \mathbf{else}:\\ \;\;\;\;x + t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (- 1.0 (log y)))))
   (if (<= y 5.5e+44)
     (- (- x (* (log y) 0.5)) z)
     (if (<= y 2.8e+175) (- t_0 z) (+ x t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = y * (1.0 - log(y));
	double tmp;
	if (y <= 5.5e+44) {
		tmp = (x - (log(y) * 0.5)) - z;
	} else if (y <= 2.8e+175) {
		tmp = t_0 - z;
	} else {
		tmp = x + t_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) :: t_0
    real(8) :: tmp
    t_0 = y * (1.0d0 - log(y))
    if (y <= 5.5d+44) then
        tmp = (x - (log(y) * 0.5d0)) - z
    else if (y <= 2.8d+175) then
        tmp = t_0 - z
    else
        tmp = x + t_0
    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 <= 5.5e+44) {
		tmp = (x - (Math.log(y) * 0.5)) - z;
	} else if (y <= 2.8e+175) {
		tmp = t_0 - z;
	} else {
		tmp = x + t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * (1.0 - math.log(y))
	tmp = 0
	if y <= 5.5e+44:
		tmp = (x - (math.log(y) * 0.5)) - z
	elif y <= 2.8e+175:
		tmp = t_0 - z
	else:
		tmp = x + t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * Float64(1.0 - log(y)))
	tmp = 0.0
	if (y <= 5.5e+44)
		tmp = Float64(Float64(x - Float64(log(y) * 0.5)) - z);
	elseif (y <= 2.8e+175)
		tmp = Float64(t_0 - z);
	else
		tmp = Float64(x + t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * (1.0 - log(y));
	tmp = 0.0;
	if (y <= 5.5e+44)
		tmp = (x - (log(y) * 0.5)) - z;
	elseif (y <= 2.8e+175)
		tmp = t_0 - z;
	else
		tmp = x + t_0;
	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, 5.5e+44], N[(N[(x - N[(N[Log[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[y, 2.8e+175], N[(t$95$0 - z), $MachinePrecision], N[(x + t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 5.5000000000000001e44

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 98.2%

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

   if 5.5000000000000001e44 < y < 2.8000000000000001e175

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. associate-+r- 99.6%

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

      3. *-commutative 99.6%

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in y around inf 84.1%

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

      1. mul-1-neg 84.1%

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

      2. log-rec 84.1%

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

      3. remove-double-neg 84.1%

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

   7. Simplified 84.1%

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

   if 2.8000000000000001e175 < 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. sub-neg 99.5%

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

      3. associate-+l+ 99.5%

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

      4. associate-+r- 99.5%

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

      5. *-commutative 99.5%

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

      6. distribute-rgt-neg-in 99.5%

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

      7. fma-def 99.5%

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

      8. +-commutative 99.5%

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

      9. distribute-neg-in 99.5%

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

      10. unsub-neg 99.5%

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

      11. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in y around inf 99.5%

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

      1. log-rec 99.5%

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

      2. sub-neg 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in z around 0 92.8%

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

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.5 \cdot 10^{+44}:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{+175}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(1 - \log y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 70.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -230 \lor \neg \left(x \leq 2.1 \cdot 10^{+16}\right):\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -230.0) (not (<= x 2.1e+16))) (- x z) (- (* (log y) -0.5) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -230.0) || !(x <= 2.1e+16)) {
		tmp = x - z;
	} else {
		tmp = (log(y) * -0.5) - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-230.0d0)) .or. (.not. (x <= 2.1d+16))) then
        tmp = x - z
    else
        tmp = (log(y) * (-0.5d0)) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -230.0) || !(x <= 2.1e+16)) {
		tmp = x - z;
	} else {
		tmp = (Math.log(y) * -0.5) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -230.0) or not (x <= 2.1e+16):
		tmp = x - z
	else:
		tmp = (math.log(y) * -0.5) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -230.0) || !(x <= 2.1e+16))
		tmp = Float64(x - z);
	else
		tmp = Float64(Float64(log(y) * -0.5) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -230.0) || ~((x <= 2.1e+16)))
		tmp = x - z;
	else
		tmp = (log(y) * -0.5) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -230.0], N[Not[LessEqual[x, 2.1e+16]], $MachinePrecision]], N[(x - z), $MachinePrecision], N[(N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -230 or 2.1e16 < x

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      2. associate-+r- 99.9%

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

      3. *-commutative 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around inf 79.0%

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

   if -230 < x < 2.1e16

   1. Initial program 99.7%

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

      1. *-commutative 99.7%

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

      2. flip-+ 80.0%

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

      3. associate-*r/ 80.0%

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

      4. fma-neg 80.0%

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

      5. metadata-eval 80.0%

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

      6. metadata-eval 80.0%

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

      7. sub-neg 80.0%

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

      8. metadata-eval 80.0%

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

   4. Applied egg-rr 80.0%

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

      1. *-commutative 80.0%

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

      2. associate-/l* 80.0%

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

      3. +-commutative 80.0%

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

   6. Simplified 80.0%

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

   7. Taylor expanded in x around 0 80.0%

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

      1. sub-neg 80.0%

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

      2. metadata-eval 80.0%

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

      3. associate-/l* 80.0%

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

      4. unpow2 80.0%

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

      5. fma-neg 80.0%

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

      6. metadata-eval 80.0%

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

   9. Simplified 80.0%

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

   10. Taylor expanded in y around 0 64.6%

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

       1. *-commutative 64.6%

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

   12. Simplified 64.6%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -230 \lor \neg \left(x \leq 2.1 \cdot 10^{+16}\right):\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.95 \cdot 10^{-10}:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + \left(y \cdot \left(1 - \log y\right) - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 3.95e-10)
   (- (- x (* (log y) 0.5)) z)
   (+ x (- (* y (- 1.0 (log y))) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 3.95e-10) {
		tmp = (x - (log(y) * 0.5)) - z;
	} else {
		tmp = x + ((y * (1.0 - log(y))) - z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 3.95d-10) then
        tmp = (x - (log(y) * 0.5d0)) - z
    else
        tmp = x + ((y * (1.0d0 - log(y))) - z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 3.95e-10) {
		tmp = (x - (Math.log(y) * 0.5)) - z;
	} else {
		tmp = x + ((y * (1.0 - Math.log(y))) - z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 3.95e-10:
		tmp = (x - (math.log(y) * 0.5)) - z
	else:
		tmp = x + ((y * (1.0 - math.log(y))) - z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 3.95e-10)
		tmp = Float64(Float64(x - Float64(log(y) * 0.5)) - z);
	else
		tmp = Float64(x + Float64(Float64(y * Float64(1.0 - log(y))) - z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 3.95e-10)
		tmp = (x - (log(y) * 0.5)) - z;
	else
		tmp = x + ((y * (1.0 - log(y))) - z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 3.95e-10], N[(N[(x - N[(N[Log[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(x + N[(N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.9499999999999998e-10

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 99.8%

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

   if 3.9499999999999998e-10 < 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. associate-+r- 99.6%

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

      5. *-commutative 99.6%

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

      6. distribute-rgt-neg-in 99.6%

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

      7. fma-def 99.8%

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

      8. +-commutative 99.8%

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

      9. distribute-neg-in 99.8%

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

      10. unsub-neg 99.8%

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

      11. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around inf 99.5%

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

      1. log-rec 99.5%

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

      2. sub-neg 99.5%

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

   7. Simplified 99.5%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.95 \cdot 10^{-10}:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + \left(y \cdot \left(1 - \log y\right) - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(x + y\right) - \frac{\log y}{\frac{1}{y + 0.5}}\right) - z \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (- (- (+ x y) (/ (log y) (/ 1.0 (+ y 0.5)))) z))

C

double code(double x, double y, double z) {
	return ((x + y) - (log(y) / (1.0 / (y + 0.5)))) - z;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = ((x + y) - (log(y) / (1.0d0 / (y + 0.5d0)))) - z
end function

Java

public static double code(double x, double y, double z) {
	return ((x + y) - (Math.log(y) / (1.0 / (y + 0.5)))) - z;
}

Python

def code(x, y, z):
	return ((x + y) - (math.log(y) / (1.0 / (y + 0.5)))) - z

Julia

function code(x, y, z)
	return Float64(Float64(Float64(x + y) - Float64(log(y) / Float64(1.0 / Float64(y + 0.5)))) - z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((x + y) - (log(y) / (1.0 / (y + 0.5)))) - z;
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(x + y), $MachinePrecision] - N[(N[Log[y], $MachinePrecision] / N[(1.0 / N[(y + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

   1. +-commutative 99.8%

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

   2. associate-+r- 99.8%

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

   3. *-commutative 99.8%

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

4. Applied egg-rr 99.8%

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

   1. add-cube-cbrt 99.2%

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

   2. pow3 99.2%

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

6. Applied egg-rr 99.2%

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

   1. rem-cube-cbrt 99.8%

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

   2. metadata-eval 99.8%

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

   3. sub-neg 99.8%

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

   4. flip-- 79.1%

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

   5. fma-neg 79.1%

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

   6. metadata-eval 79.1%

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

   7. metadata-eval 79.1%

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

   8. clear-num 79.1%

      \[\leadsto \left(\left(y + x\right) - \log y \cdot \color{blue}{\frac{1}{\frac{y + -0.5}{\mathsf{fma}\left(y, y, -0.25\right)}}}\right) - z \]

   9. div-inv 79.1%

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

   10. clear-num 79.1%

       \[\leadsto \left(\left(y + x\right) - \frac{\log y}{\color{blue}{\frac{1}{\frac{\mathsf{fma}\left(y, y, -0.25\right)}{y + -0.5}}}}\right) - z \]

   11. metadata-eval 79.1%

       \[\leadsto \left(\left(y + x\right) - \frac{\log y}{\frac{1}{\frac{\mathsf{fma}\left(y, y, \color{blue}{-0.25}\right)}{y + -0.5}}}\right) - z \]

   12. metadata-eval 79.1%

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

   13. fma-neg 79.1%

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

   14. flip-- 99.8%

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

   15. sub-neg 99.8%

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

   16. metadata-eval 99.8%

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

8. Applied egg-rr 99.8%

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

9. Final simplification 99.8%

   \[\leadsto \left(\left(x + y\right) - \frac{\log y}{\frac{1}{y + 0.5}}\right) - z \]
10. Add Preprocessing

Alternative 8: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Final simplification 99.8%

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

Alternative 9: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. +-commutative 99.8%

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

   2. associate-+r- 99.8%

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

   3. *-commutative 99.8%

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

4. Applied egg-rr 99.8%

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

5. Final simplification 99.8%

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

Alternative 10: 48.0% accurate, 9.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.2 \cdot 10^{+37} \lor \neg \left(z \leq 1.08 \cdot 10^{+89}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -2.2e+37) (not (<= z 1.08e+89))) (- z) x))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2.2e+37) || !(z <= 1.08e+89)) {
		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 <= (-2.2d+37)) .or. (.not. (z <= 1.08d+89))) 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 <= -2.2e+37) || !(z <= 1.08e+89)) {
		tmp = -z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -2.2e+37) or not (z <= 1.08e+89):
		tmp = -z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -2.2e+37) || !(z <= 1.08e+89))
		tmp = Float64(-z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -2.2e+37) || ~((z <= 1.08e+89)))
		tmp = -z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -2.2e+37], N[Not[LessEqual[z, 1.08e+89]], $MachinePrecision]], (-z), x]

Derivation

1. Split input into 2 regimes

2. if z < -2.2000000000000001e37 or 1.08000000000000006e89 < z

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-+l+ 99.9%

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

      4. associate-+r- 99.9%

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

      5. *-commutative 99.9%

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

      6. distribute-rgt-neg-in 99.9%

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

      7. fma-def 99.9%

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

      8. +-commutative 99.9%

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

      9. distribute-neg-in 99.9%

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

      10. unsub-neg 99.9%

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

      11. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in z around inf 73.1%

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

      1. neg-mul-1 73.1%

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

   7. Simplified 73.1%

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

   if -2.2000000000000001e37 < z < 1.08000000000000006e89

   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. associate-+r- 99.7%

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

      5. *-commutative 99.7%

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

      6. distribute-rgt-neg-in 99.7%

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

      7. fma-def 99.8%

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

      8. +-commutative 99.8%

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

      9. distribute-neg-in 99.8%

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

      10. unsub-neg 99.8%

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

      11. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around inf 39.1%

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

4. Final simplification 53.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.2 \cdot 10^{+37} \lor \neg \left(z \leq 1.08 \cdot 10^{+89}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 58.4% accurate, 37.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. +-commutative 99.8%

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

   2. associate-+r- 99.8%

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

   3. *-commutative 99.8%

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

4. Applied egg-rr 99.8%

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

5. Taylor expanded in x around inf 58.8%

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

6. Final simplification 58.8%

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

Alternative 12: 30.8% accurate, 111.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 99.8%

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

   1. associate--l+ 99.8%

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

   2. sub-neg 99.8%

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

   3. associate-+l+ 99.8%

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

   4. associate-+r- 99.8%

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

   5. *-commutative 99.8%

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

   6. distribute-rgt-neg-in 99.8%

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

   7. fma-def 99.9%

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

   8. +-commutative 99.9%

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

   9. distribute-neg-in 99.9%

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

   10. unsub-neg 99.9%

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

   11. metadata-eval 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in x around inf 27.4%

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

6. Final simplification 27.4%

   \[\leadsto x \]
7. Add Preprocessing

Developer target: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024027 
(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))