Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Percentage Accurate: 99.8% → 99.9%

Time: 12.7s

Alternatives: 13

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

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

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-define 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: 74.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y - \log y \cdot \left(y + 0.5\right)\\ t_1 := x + y \cdot \left(1 - \log y\right)\\ \mathbf{if}\;z \leq -6.6 \cdot 10^{+107}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;z \leq -1.35 \cdot 10^{-133}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -2.65 \cdot 10^{-240}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq -9.5 \cdot 10^{-302}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-137}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{+48}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- y (* (log y) (+ y 0.5)))) (t_1 (+ x (* y (- 1.0 (log y))))))
   (if (<= z -6.6e+107)
     (- x z)
     (if (<= z -1.35e-133)
       t_1
       (if (<= z -2.65e-240)
         t_0
         (if (<= z -9.5e-302)
           t_1
           (if (<= z 6e-137) t_0 (if (<= z 2.7e+48) t_1 (- x z)))))))))

C

double code(double x, double y, double z) {
	double t_0 = y - (log(y) * (y + 0.5));
	double t_1 = x + (y * (1.0 - log(y)));
	double tmp;
	if (z <= -6.6e+107) {
		tmp = x - z;
	} else if (z <= -1.35e-133) {
		tmp = t_1;
	} else if (z <= -2.65e-240) {
		tmp = t_0;
	} else if (z <= -9.5e-302) {
		tmp = t_1;
	} else if (z <= 6e-137) {
		tmp = t_0;
	} else if (z <= 2.7e+48) {
		tmp = t_1;
	} else {
		tmp = x - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = y - (log(y) * (y + 0.5d0))
    t_1 = x + (y * (1.0d0 - log(y)))
    if (z <= (-6.6d+107)) then
        tmp = x - z
    else if (z <= (-1.35d-133)) then
        tmp = t_1
    else if (z <= (-2.65d-240)) then
        tmp = t_0
    else if (z <= (-9.5d-302)) then
        tmp = t_1
    else if (z <= 6d-137) then
        tmp = t_0
    else if (z <= 2.7d+48) then
        tmp = t_1
    else
        tmp = x - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y - (Math.log(y) * (y + 0.5));
	double t_1 = x + (y * (1.0 - Math.log(y)));
	double tmp;
	if (z <= -6.6e+107) {
		tmp = x - z;
	} else if (z <= -1.35e-133) {
		tmp = t_1;
	} else if (z <= -2.65e-240) {
		tmp = t_0;
	} else if (z <= -9.5e-302) {
		tmp = t_1;
	} else if (z <= 6e-137) {
		tmp = t_0;
	} else if (z <= 2.7e+48) {
		tmp = t_1;
	} else {
		tmp = x - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y - (math.log(y) * (y + 0.5))
	t_1 = x + (y * (1.0 - math.log(y)))
	tmp = 0
	if z <= -6.6e+107:
		tmp = x - z
	elif z <= -1.35e-133:
		tmp = t_1
	elif z <= -2.65e-240:
		tmp = t_0
	elif z <= -9.5e-302:
		tmp = t_1
	elif z <= 6e-137:
		tmp = t_0
	elif z <= 2.7e+48:
		tmp = t_1
	else:
		tmp = x - z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y - Float64(log(y) * Float64(y + 0.5)))
	t_1 = Float64(x + Float64(y * Float64(1.0 - log(y))))
	tmp = 0.0
	if (z <= -6.6e+107)
		tmp = Float64(x - z);
	elseif (z <= -1.35e-133)
		tmp = t_1;
	elseif (z <= -2.65e-240)
		tmp = t_0;
	elseif (z <= -9.5e-302)
		tmp = t_1;
	elseif (z <= 6e-137)
		tmp = t_0;
	elseif (z <= 2.7e+48)
		tmp = t_1;
	else
		tmp = Float64(x - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y - (log(y) * (y + 0.5));
	t_1 = x + (y * (1.0 - log(y)));
	tmp = 0.0;
	if (z <= -6.6e+107)
		tmp = x - z;
	elseif (z <= -1.35e-133)
		tmp = t_1;
	elseif (z <= -2.65e-240)
		tmp = t_0;
	elseif (z <= -9.5e-302)
		tmp = t_1;
	elseif (z <= 6e-137)
		tmp = t_0;
	elseif (z <= 2.7e+48)
		tmp = t_1;
	else
		tmp = x - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y - N[(N[Log[y], $MachinePrecision] * N[(y + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x + N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -6.6e+107], N[(x - z), $MachinePrecision], If[LessEqual[z, -1.35e-133], t$95$1, If[LessEqual[z, -2.65e-240], t$95$0, If[LessEqual[z, -9.5e-302], t$95$1, If[LessEqual[z, 6e-137], t$95$0, If[LessEqual[z, 2.7e+48], t$95$1, N[(x - z), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -6.60000000000000064e107 or 2.70000000000000004e48 < 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. add-cube-cbrt 99.8%

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

      2. pow3 99.8%

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

      3. *-commutative 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around inf 87.0%

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

   if -6.60000000000000064e107 < z < -1.3499999999999999e-133 or -2.6500000000000001e-240 < z < -9.49999999999999991e-302 or 5.9999999999999996e-137 < z < 2.70000000000000004e48

   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. Add Preprocessing

   5. Taylor expanded in z around 0 95.4%

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

      1. associate--l+ 95.4%

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

      2. +-commutative 95.4%

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

   7. Applied egg-rr 95.4%

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

   8. Taylor expanded in y around inf 82.2%

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

      1. mul-1-neg 82.2%

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

      2. log-rec 82.2%

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

      3. remove-double-neg 82.2%

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

   10. Simplified 82.2%

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

   if -1.3499999999999999e-133 < z < -2.6500000000000001e-240 or -9.49999999999999991e-302 < z < 5.9999999999999996e-137

   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. Add Preprocessing

   5. Taylor expanded in z around inf 62.0%

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

      1. +-commutative 62.0%

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

      2. associate-/l* 61.9%

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

      3. +-commutative 61.9%

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

   7. Simplified 61.9%

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

   8. Taylor expanded in y around inf 54.1%

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

   9. Taylor expanded in z around 0 80.8%

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

       1. +-commutative 80.8%

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

   11. Simplified 80.8%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.6 \cdot 10^{+107}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;z \leq -1.35 \cdot 10^{-133}:\\ \;\;\;\;x + y \cdot \left(1 - \log y\right)\\ \mathbf{elif}\;z \leq -2.65 \cdot 10^{-240}:\\ \;\;\;\;y - \log y \cdot \left(y + 0.5\right)\\ \mathbf{elif}\;z \leq -9.5 \cdot 10^{-302}:\\ \;\;\;\;x + y \cdot \left(1 - \log y\right)\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-137}:\\ \;\;\;\;y - \log y \cdot \left(y + 0.5\right)\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{+48}:\\ \;\;\;\;x + y \cdot \left(1 - \log y\right)\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y - \log y \cdot \left(y + 0.5\right)\\ t_1 := y \cdot \left(1 - \log y\right)\\ t_2 := x + t\_1\\ \mathbf{if}\;z \leq -2.4 \cdot 10^{+107}:\\ \;\;\;\;t\_1 - z\\ \mathbf{elif}\;z \leq -3.4 \cdot 10^{-137}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq -1.1 \cdot 10^{-242}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq -1.32 \cdot 10^{-298}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{-136}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{+48}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- y (* (log y) (+ y 0.5))))
        (t_1 (* y (- 1.0 (log y))))
        (t_2 (+ x t_1)))
   (if (<= z -2.4e+107)
     (- t_1 z)
     (if (<= z -3.4e-137)
       t_2
       (if (<= z -1.1e-242)
         t_0
         (if (<= z -1.32e-298)
           t_2
           (if (<= z 7.5e-136) t_0 (if (<= z 1.15e+48) t_2 (- x z)))))))))

C

double code(double x, double y, double z) {
	double t_0 = y - (log(y) * (y + 0.5));
	double t_1 = y * (1.0 - log(y));
	double t_2 = x + t_1;
	double tmp;
	if (z <= -2.4e+107) {
		tmp = t_1 - z;
	} else if (z <= -3.4e-137) {
		tmp = t_2;
	} else if (z <= -1.1e-242) {
		tmp = t_0;
	} else if (z <= -1.32e-298) {
		tmp = t_2;
	} else if (z <= 7.5e-136) {
		tmp = t_0;
	} else if (z <= 1.15e+48) {
		tmp = t_2;
	} else {
		tmp = x - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = y - (log(y) * (y + 0.5d0))
    t_1 = y * (1.0d0 - log(y))
    t_2 = x + t_1
    if (z <= (-2.4d+107)) then
        tmp = t_1 - z
    else if (z <= (-3.4d-137)) then
        tmp = t_2
    else if (z <= (-1.1d-242)) then
        tmp = t_0
    else if (z <= (-1.32d-298)) then
        tmp = t_2
    else if (z <= 7.5d-136) then
        tmp = t_0
    else if (z <= 1.15d+48) then
        tmp = t_2
    else
        tmp = x - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y - (Math.log(y) * (y + 0.5));
	double t_1 = y * (1.0 - Math.log(y));
	double t_2 = x + t_1;
	double tmp;
	if (z <= -2.4e+107) {
		tmp = t_1 - z;
	} else if (z <= -3.4e-137) {
		tmp = t_2;
	} else if (z <= -1.1e-242) {
		tmp = t_0;
	} else if (z <= -1.32e-298) {
		tmp = t_2;
	} else if (z <= 7.5e-136) {
		tmp = t_0;
	} else if (z <= 1.15e+48) {
		tmp = t_2;
	} else {
		tmp = x - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y - (math.log(y) * (y + 0.5))
	t_1 = y * (1.0 - math.log(y))
	t_2 = x + t_1
	tmp = 0
	if z <= -2.4e+107:
		tmp = t_1 - z
	elif z <= -3.4e-137:
		tmp = t_2
	elif z <= -1.1e-242:
		tmp = t_0
	elif z <= -1.32e-298:
		tmp = t_2
	elif z <= 7.5e-136:
		tmp = t_0
	elif z <= 1.15e+48:
		tmp = t_2
	else:
		tmp = x - z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y - Float64(log(y) * Float64(y + 0.5)))
	t_1 = Float64(y * Float64(1.0 - log(y)))
	t_2 = Float64(x + t_1)
	tmp = 0.0
	if (z <= -2.4e+107)
		tmp = Float64(t_1 - z);
	elseif (z <= -3.4e-137)
		tmp = t_2;
	elseif (z <= -1.1e-242)
		tmp = t_0;
	elseif (z <= -1.32e-298)
		tmp = t_2;
	elseif (z <= 7.5e-136)
		tmp = t_0;
	elseif (z <= 1.15e+48)
		tmp = t_2;
	else
		tmp = Float64(x - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y - (log(y) * (y + 0.5));
	t_1 = y * (1.0 - log(y));
	t_2 = x + t_1;
	tmp = 0.0;
	if (z <= -2.4e+107)
		tmp = t_1 - z;
	elseif (z <= -3.4e-137)
		tmp = t_2;
	elseif (z <= -1.1e-242)
		tmp = t_0;
	elseif (z <= -1.32e-298)
		tmp = t_2;
	elseif (z <= 7.5e-136)
		tmp = t_0;
	elseif (z <= 1.15e+48)
		tmp = t_2;
	else
		tmp = x - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y - N[(N[Log[y], $MachinePrecision] * N[(y + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x + t$95$1), $MachinePrecision]}, If[LessEqual[z, -2.4e+107], N[(t$95$1 - z), $MachinePrecision], If[LessEqual[z, -3.4e-137], t$95$2, If[LessEqual[z, -1.1e-242], t$95$0, If[LessEqual[z, -1.32e-298], t$95$2, If[LessEqual[z, 7.5e-136], t$95$0, If[LessEqual[z, 1.15e+48], t$95$2, N[(x - z), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -2.4000000000000001e107

   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. add-cube-cbrt 99.8%

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

      2. pow3 99.8%

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

      3. *-commutative 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in y around inf 92.9%

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

      1. mul-1-neg 92.9%

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

      2. log-rec 92.9%

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

      3. remove-double-neg 92.9%

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

   7. Simplified 92.9%

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

   if -2.4000000000000001e107 < z < -3.40000000000000014e-137 or -1.10000000000000001e-242 < z < -1.3200000000000001e-298 or 7.5000000000000003e-136 < z < 1.15e48

   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. Add Preprocessing

   5. Taylor expanded in z around 0 95.4%

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

      1. associate--l+ 95.4%

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

      2. +-commutative 95.4%

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

   7. Applied egg-rr 95.4%

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

   8. Taylor expanded in y around inf 82.2%

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

      1. mul-1-neg 82.2%

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

      2. log-rec 82.2%

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

      3. remove-double-neg 82.2%

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

   10. Simplified 82.2%

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

   if -3.40000000000000014e-137 < z < -1.10000000000000001e-242 or -1.3200000000000001e-298 < z < 7.5000000000000003e-136

   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. Add Preprocessing

   5. Taylor expanded in z around inf 62.0%

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

      1. +-commutative 62.0%

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

      2. associate-/l* 61.9%

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

      3. +-commutative 61.9%

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

   7. Simplified 61.9%

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

   8. Taylor expanded in y around inf 54.1%

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

   9. Taylor expanded in z around 0 80.8%

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

       1. +-commutative 80.8%

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

   11. Simplified 80.8%

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

   if 1.15e48 < 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. add-cube-cbrt 99.8%

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

      2. pow3 99.8%

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

      3. *-commutative 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around inf 87.1%

      \[\leadsto \color{blue}{x} - z \]
3. Recombined 4 regimes into one program.

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.4 \cdot 10^{+107}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \mathbf{elif}\;z \leq -3.4 \cdot 10^{-137}:\\ \;\;\;\;x + y \cdot \left(1 - \log y\right)\\ \mathbf{elif}\;z \leq -1.1 \cdot 10^{-242}:\\ \;\;\;\;y - \log y \cdot \left(y + 0.5\right)\\ \mathbf{elif}\;z \leq -1.32 \cdot 10^{-298}:\\ \;\;\;\;x + y \cdot \left(1 - \log y\right)\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{-136}:\\ \;\;\;\;y - \log y \cdot \left(y + 0.5\right)\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{+48}:\\ \;\;\;\;x + y \cdot \left(1 - \log y\right)\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(1 - \log y\right)\\ t_1 := t\_0 - z\\ t_2 := \left(x - \log y \cdot 0.5\right) - z\\ \mathbf{if}\;y \leq 6.8 \cdot 10^{+43}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{+81}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{+115}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{+151}:\\ \;\;\;\;x + t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (- 1.0 (log y))))
        (t_1 (- t_0 z))
        (t_2 (- (- x (* (log y) 0.5)) z)))
   (if (<= y 6.8e+43)
     t_2
     (if (<= y 1.05e+81)
       t_1
       (if (<= y 4.6e+115) t_2 (if (<= y 1.4e+151) (+ x t_0) t_1))))))

C

double code(double x, double y, double z) {
	double t_0 = y * (1.0 - log(y));
	double t_1 = t_0 - z;
	double t_2 = (x - (log(y) * 0.5)) - z;
	double tmp;
	if (y <= 6.8e+43) {
		tmp = t_2;
	} else if (y <= 1.05e+81) {
		tmp = t_1;
	} else if (y <= 4.6e+115) {
		tmp = t_2;
	} else if (y <= 1.4e+151) {
		tmp = x + t_0;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = y * (1.0d0 - log(y))
    t_1 = t_0 - z
    t_2 = (x - (log(y) * 0.5d0)) - z
    if (y <= 6.8d+43) then
        tmp = t_2
    else if (y <= 1.05d+81) then
        tmp = t_1
    else if (y <= 4.6d+115) then
        tmp = t_2
    else if (y <= 1.4d+151) then
        tmp = x + t_0
    else
        tmp = t_1
    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 t_1 = t_0 - z;
	double t_2 = (x - (Math.log(y) * 0.5)) - z;
	double tmp;
	if (y <= 6.8e+43) {
		tmp = t_2;
	} else if (y <= 1.05e+81) {
		tmp = t_1;
	} else if (y <= 4.6e+115) {
		tmp = t_2;
	} else if (y <= 1.4e+151) {
		tmp = x + t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * (1.0 - math.log(y))
	t_1 = t_0 - z
	t_2 = (x - (math.log(y) * 0.5)) - z
	tmp = 0
	if y <= 6.8e+43:
		tmp = t_2
	elif y <= 1.05e+81:
		tmp = t_1
	elif y <= 4.6e+115:
		tmp = t_2
	elif y <= 1.4e+151:
		tmp = x + t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * Float64(1.0 - log(y)))
	t_1 = Float64(t_0 - z)
	t_2 = Float64(Float64(x - Float64(log(y) * 0.5)) - z)
	tmp = 0.0
	if (y <= 6.8e+43)
		tmp = t_2;
	elseif (y <= 1.05e+81)
		tmp = t_1;
	elseif (y <= 4.6e+115)
		tmp = t_2;
	elseif (y <= 1.4e+151)
		tmp = Float64(x + t_0);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * (1.0 - log(y));
	t_1 = t_0 - z;
	t_2 = (x - (log(y) * 0.5)) - z;
	tmp = 0.0;
	if (y <= 6.8e+43)
		tmp = t_2;
	elseif (y <= 1.05e+81)
		tmp = t_1;
	elseif (y <= 4.6e+115)
		tmp = t_2;
	elseif (y <= 1.4e+151)
		tmp = x + t_0;
	else
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(t$95$0 - z), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x - N[(N[Log[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]}, If[LessEqual[y, 6.8e+43], t$95$2, If[LessEqual[y, 1.05e+81], t$95$1, If[LessEqual[y, 4.6e+115], t$95$2, If[LessEqual[y, 1.4e+151], N[(x + t$95$0), $MachinePrecision], t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 6.80000000000000024e43 or 1.0499999999999999e81 < y < 4.60000000000000007e115

   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 95.5%

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

      1. *-commutative 95.5%

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

   5. Simplified 95.5%

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

   if 6.80000000000000024e43 < y < 1.0499999999999999e81 or 1.39999999999999994e151 < y

   1. Initial program 99.6%

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

      1. add-cube-cbrt 98.6%

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

      2. pow3 98.7%

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

      3. *-commutative 98.7%

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

   4. Applied egg-rr 98.7%

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

   5. Taylor expanded in y around inf 88.1%

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

      1. mul-1-neg 88.1%

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

      2. log-rec 88.1%

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

      3. remove-double-neg 88.1%

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

   7. Simplified 88.1%

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

   if 4.60000000000000007e115 < y < 1.39999999999999994e151

   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+ 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in z around 0 87.6%

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

      1. associate--l+ 87.7%

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

      2. +-commutative 87.7%

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

   7. Applied egg-rr 87.7%

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

   8. Taylor expanded in y around inf 87.6%

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

      1. mul-1-neg 87.6%

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

      2. log-rec 87.6%

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

      3. remove-double-neg 87.6%

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

   10. Simplified 87.6%

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

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.8 \cdot 10^{+43}:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{+81}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{+115}:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{+151}:\\ \;\;\;\;x + y \cdot \left(1 - \log y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 55.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log y \cdot -0.5\\ \mathbf{if}\;z \leq -1.6 \cdot 10^{-135}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;z \leq -3.7 \cdot 10^{-240}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq -1.36 \cdot 10^{-292}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 4.05 \cdot 10^{-137}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (log y) -0.5)))
   (if (<= z -1.6e-135)
     (- x z)
     (if (<= z -3.7e-240)
       t_0
       (if (<= z -1.36e-292) x (if (<= z 4.05e-137) t_0 (- x z)))))))

C

double code(double x, double y, double z) {
	double t_0 = log(y) * -0.5;
	double tmp;
	if (z <= -1.6e-135) {
		tmp = x - z;
	} else if (z <= -3.7e-240) {
		tmp = t_0;
	} else if (z <= -1.36e-292) {
		tmp = x;
	} else if (z <= 4.05e-137) {
		tmp = t_0;
	} else {
		tmp = x - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = log(y) * (-0.5d0)
    if (z <= (-1.6d-135)) then
        tmp = x - z
    else if (z <= (-3.7d-240)) then
        tmp = t_0
    else if (z <= (-1.36d-292)) then
        tmp = x
    else if (z <= 4.05d-137) then
        tmp = t_0
    else
        tmp = x - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.log(y) * -0.5;
	double tmp;
	if (z <= -1.6e-135) {
		tmp = x - z;
	} else if (z <= -3.7e-240) {
		tmp = t_0;
	} else if (z <= -1.36e-292) {
		tmp = x;
	} else if (z <= 4.05e-137) {
		tmp = t_0;
	} else {
		tmp = x - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.log(y) * -0.5
	tmp = 0
	if z <= -1.6e-135:
		tmp = x - z
	elif z <= -3.7e-240:
		tmp = t_0
	elif z <= -1.36e-292:
		tmp = x
	elif z <= 4.05e-137:
		tmp = t_0
	else:
		tmp = x - z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(log(y) * -0.5)
	tmp = 0.0
	if (z <= -1.6e-135)
		tmp = Float64(x - z);
	elseif (z <= -3.7e-240)
		tmp = t_0;
	elseif (z <= -1.36e-292)
		tmp = x;
	elseif (z <= 4.05e-137)
		tmp = t_0;
	else
		tmp = Float64(x - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = log(y) * -0.5;
	tmp = 0.0;
	if (z <= -1.6e-135)
		tmp = x - z;
	elseif (z <= -3.7e-240)
		tmp = t_0;
	elseif (z <= -1.36e-292)
		tmp = x;
	elseif (z <= 4.05e-137)
		tmp = t_0;
	else
		tmp = x - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision]}, If[LessEqual[z, -1.6e-135], N[(x - z), $MachinePrecision], If[LessEqual[z, -3.7e-240], t$95$0, If[LessEqual[z, -1.36e-292], x, If[LessEqual[z, 4.05e-137], t$95$0, N[(x - z), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.6e-135 or 4.0500000000000001e-137 < z

   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. add-cube-cbrt 99.4%

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

      2. pow3 99.4%

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

      3. *-commutative 99.4%

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

   4. Applied egg-rr 99.4%

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

   5. Taylor expanded in x around inf 70.7%

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

   if -1.6e-135 < z < -3.7000000000000002e-240 or -1.36e-292 < z < 4.0500000000000001e-137

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0 66.9%

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

      1. *-commutative 66.9%

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

   5. Simplified 66.9%

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

   6. Taylor expanded in x around 0 48.1%

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

      1. *-commutative 48.1%

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

   8. Simplified 48.1%

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

   9. Taylor expanded in z around 0 48.1%

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

       1. *-commutative 48.1%

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

   11. Simplified 48.1%

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

   if -3.7000000000000002e-240 < z < -1.36e-292

   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. Add Preprocessing

   5. Taylor expanded in x around inf 58.4%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.6 \cdot 10^{-135}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;z \leq -3.7 \cdot 10^{-240}:\\ \;\;\;\;\log y \cdot -0.5\\ \mathbf{elif}\;z \leq -1.36 \cdot 10^{-292}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 4.05 \cdot 10^{-137}:\\ \;\;\;\;\log y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.8 \cdot 10^{+98}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{+47}:\\ \;\;\;\;x + \left(y - \log y \cdot \left(y + 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -6.8e+98)
   (- (* y (- 1.0 (log y))) z)
   (if (<= z 3.2e+47)
     (+ x (- y (* (log y) (+ y 0.5))))
     (- (- x (* (log y) 0.5)) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -6.8e+98) {
		tmp = (y * (1.0 - log(y))) - z;
	} else if (z <= 3.2e+47) {
		tmp = x + (y - (log(y) * (y + 0.5)));
	} else {
		tmp = (x - (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 (z <= (-6.8d+98)) then
        tmp = (y * (1.0d0 - log(y))) - z
    else if (z <= 3.2d+47) then
        tmp = x + (y - (log(y) * (y + 0.5d0)))
    else
        tmp = (x - (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 (z <= -6.8e+98) {
		tmp = (y * (1.0 - Math.log(y))) - z;
	} else if (z <= 3.2e+47) {
		tmp = x + (y - (Math.log(y) * (y + 0.5)));
	} else {
		tmp = (x - (Math.log(y) * 0.5)) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -6.8e+98:
		tmp = (y * (1.0 - math.log(y))) - z
	elif z <= 3.2e+47:
		tmp = x + (y - (math.log(y) * (y + 0.5)))
	else:
		tmp = (x - (math.log(y) * 0.5)) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -6.8e+98)
		tmp = Float64(Float64(y * Float64(1.0 - log(y))) - z);
	elseif (z <= 3.2e+47)
		tmp = Float64(x + Float64(y - Float64(log(y) * Float64(y + 0.5))));
	else
		tmp = Float64(Float64(x - Float64(log(y) * 0.5)) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -6.8e+98)
		tmp = (y * (1.0 - log(y))) - z;
	elseif (z <= 3.2e+47)
		tmp = x + (y - (log(y) * (y + 0.5)));
	else
		tmp = (x - (log(y) * 0.5)) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -6.8e+98], N[(N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[z, 3.2e+47], N[(x + N[(y - N[(N[Log[y], $MachinePrecision] * N[(y + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x - N[(N[Log[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -6.79999999999999944e98

   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. add-cube-cbrt 99.7%

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

      2. pow3 99.8%

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

      3. *-commutative 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in y around inf 93.0%

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

      1. mul-1-neg 93.0%

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

      2. log-rec 93.0%

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

      3. remove-double-neg 93.0%

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

   7. Simplified 93.0%

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

   if -6.79999999999999944e98 < z < 3.2e47

   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. Add Preprocessing

   5. Taylor expanded in z around 0 97.2%

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

      1. associate--l+ 97.2%

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

      2. +-commutative 97.2%

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

   7. Applied egg-rr 97.2%

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

   if 3.2e47 < 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. Taylor expanded in y around 0 87.1%

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

      1. *-commutative 87.1%

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

   5. Simplified 87.1%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.8 \cdot 10^{+98}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{+47}:\\ \;\;\;\;x + \left(y - \log y \cdot \left(y + 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{+107} \lor \neg \left(z \leq 1.38 \cdot 10^{+51}\right):\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(1 - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.25e+107) (not (<= z 1.38e+51)))
   (- x z)
   (+ x (* y (- 1.0 (log y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.25e+107) || !(z <= 1.38e+51)) {
		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 <= (-1.25d+107)) .or. (.not. (z <= 1.38d+51))) 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 <= -1.25e+107) || !(z <= 1.38e+51)) {
		tmp = x - z;
	} else {
		tmp = x + (y * (1.0 - Math.log(y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.25e+107) or not (z <= 1.38e+51):
		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 <= -1.25e+107) || !(z <= 1.38e+51))
		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 <= -1.25e+107) || ~((z <= 1.38e+51)))
		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, -1.25e+107], N[Not[LessEqual[z, 1.38e+51]], $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 < -1.25e107 or 1.38000000000000006e51 < 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. add-cube-cbrt 99.8%

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

      2. pow3 99.8%

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

      3. *-commutative 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around inf 87.0%

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

   if -1.25e107 < z < 1.38000000000000006e51

   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. Add Preprocessing

   5. Taylor expanded in z around 0 97.2%

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

      1. associate--l+ 97.2%

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

      2. +-commutative 97.2%

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

   7. Applied egg-rr 97.2%

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

   8. Taylor expanded in y around inf 70.2%

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

      1. mul-1-neg 70.2%

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

      2. log-rec 70.2%

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

      3. remove-double-neg 70.2%

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

   10. Simplified 70.2%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{+107} \lor \neg \left(z \leq 1.38 \cdot 10^{+51}\right):\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(1 - \log y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 69.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -49 \lor \neg \left(x \leq 920\right):\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -49.0) (not (<= x 920.0))) (- x z) (- (* (log y) -0.5) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -49.0) || !(x <= 920.0)) {
		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 <= (-49.0d0)) .or. (.not. (x <= 920.0d0))) 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 <= -49.0) || !(x <= 920.0)) {
		tmp = x - z;
	} else {
		tmp = (Math.log(y) * -0.5) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -49.0) or not (x <= 920.0):
		tmp = x - z
	else:
		tmp = (math.log(y) * -0.5) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -49.0) || !(x <= 920.0))
		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 <= -49.0) || ~((x <= 920.0)))
		tmp = x - z;
	else
		tmp = (log(y) * -0.5) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -49.0], N[Not[LessEqual[x, 920.0]], $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 < -49 or 920 < x

   1. Initial program 99.9%

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

      1. add-cube-cbrt 99.5%

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

      2. pow3 99.5%

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

      3. *-commutative 99.5%

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

   4. Applied egg-rr 99.5%

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

   5. Taylor expanded in x around inf 75.3%

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

   if -49 < x < 920

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0 72.7%

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

      1. *-commutative 72.7%

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

   5. Simplified 72.7%

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

   6. Taylor expanded in x around 0 72.0%

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

      1. *-commutative 72.0%

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

   8. Simplified 72.0%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -49 \lor \neg \left(x \leq 920\right):\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.28:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + \left(y \cdot \left(1 - \log y\right) - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 0.28)
   (- (- x (* (log y) 0.5)) z)
   (+ x (- (* y (- 1.0 (log y))) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 0.28) {
		tmp = (x - (log(y) * 0.5)) - z;
	} else {
		tmp = x + ((y * (1.0 - log(y))) - z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 0.28d0) then
        tmp = (x - (log(y) * 0.5d0)) - z
    else
        tmp = x + ((y * (1.0d0 - log(y))) - z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 0.28) {
		tmp = (x - (Math.log(y) * 0.5)) - z;
	} else {
		tmp = x + ((y * (1.0 - Math.log(y))) - z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 0.28:
		tmp = (x - (math.log(y) * 0.5)) - z
	else:
		tmp = x + ((y * (1.0 - math.log(y))) - z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 0.28)
		tmp = Float64(Float64(x - Float64(log(y) * 0.5)) - z);
	else
		tmp = Float64(x + Float64(Float64(y * Float64(1.0 - log(y))) - z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 0.28)
		tmp = (x - (log(y) * 0.5)) - z;
	else
		tmp = x + ((y * (1.0 - log(y))) - z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 0.28], N[(N[(x - N[(N[Log[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(x + N[(N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 0.28000000000000003

   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 99.4%

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

      1. *-commutative 99.4%

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

   5. Simplified 99.4%

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

   if 0.28000000000000003 < 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%

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

      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-define 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 99.4%

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

      1. log-rec 99.4%

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

      2. sub-neg 99.4%

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

   7. Simplified 99.4%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.28:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + \left(y \cdot \left(1 - \log y\right) - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

3. Simplified 99.8%

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

5. Final simplification 99.8%

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

Alternative 11: 47.6% accurate, 9.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8.2 \cdot 10^{+106} \lor \neg \left(z \leq 2.75 \cdot 10^{+35}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -8.2e+106) (not (<= z 2.75e+35))) (- z) x))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -8.2e+106) || !(z <= 2.75e+35)) {
		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 <= (-8.2d+106)) .or. (.not. (z <= 2.75d+35))) 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 <= -8.2e+106) || !(z <= 2.75e+35)) {
		tmp = -z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -8.2e+106) or not (z <= 2.75e+35):
		tmp = -z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -8.2e+106) || !(z <= 2.75e+35))
		tmp = Float64(-z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -8.2e+106) || ~((z <= 2.75e+35)))
		tmp = -z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -8.2e+106], N[Not[LessEqual[z, 2.75e+35]], $MachinePrecision]], (-z), x]

Derivation

1. Split input into 2 regimes

2. if z < -8.2000000000000005e106 or 2.75000000000000001e35 < 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. Taylor expanded in y around 0 85.8%

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

      1. *-commutative 85.8%

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

   5. Simplified 85.8%

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

   6. Taylor expanded in z around inf 73.2%

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

      1. neg-mul-1 73.2%

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

   8. Simplified 73.2%

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

   if -8.2000000000000005e106 < z < 2.75000000000000001e35

   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. Add Preprocessing

   5. Taylor expanded in x around inf 36.0%

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

4. Final simplification 52.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8.2 \cdot 10^{+106} \lor \neg \left(z \leq 2.75 \cdot 10^{+35}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 57.2% 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. add-cube-cbrt 99.2%

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

   2. pow3 99.2%

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

   3. *-commutative 99.2%

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

4. Applied egg-rr 99.2%

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

5. Taylor expanded in x around inf 58.6%

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

6. Final simplification 58.6%

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

Alternative 13: 29.4% 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. Add Preprocessing

5. Taylor expanded in x around inf 26.4%

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

6. Final simplification 26.4%

   \[\leadsto x \]
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 2024096 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2"
  :precision binary64

  :alt
  (- (- (+ y x) z) (* (+ y 0.5) (log y)))

  (- (+ (- x (* (+ y 0.5) (log y))) y) z))