Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Percentage Accurate: 99.8% → 99.9%

Time: 16.0s

Alternatives: 11

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-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. Add Preprocessing

Alternative 2: 66.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log y \cdot -0.5 - z\\ t_1 := y \cdot \left(1 - \log y\right)\\ t_2 := \left(x + y\right) - z\\ \mathbf{if}\;y \leq 1.42 \cdot 10^{-164}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{-42}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{-18}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{+20}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{+74}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 9.2 \cdot 10^{+110}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{+183}:\\ \;\;\;\;y - y \cdot \log y\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{+183}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (* (log y) -0.5) z))
        (t_1 (* y (- 1.0 (log y))))
        (t_2 (- (+ x y) z)))
   (if (<= y 1.42e-164)
     t_0
     (if (<= y 4.8e-42)
       t_2
       (if (<= y 2.8e-18)
         t_0
         (if (<= y 1.85e+20)
           t_2
           (if (<= y 1.45e+74)
             t_1
             (if (<= y 9.2e+110)
               (- x z)
               (if (<= y 1.25e+183)
                 (- y (* y (log y)))
                 (if (<= y 1.3e+183) x t_1))))))))))

C

double code(double x, double y, double z) {
	double t_0 = (log(y) * -0.5) - z;
	double t_1 = y * (1.0 - log(y));
	double t_2 = (x + y) - z;
	double tmp;
	if (y <= 1.42e-164) {
		tmp = t_0;
	} else if (y <= 4.8e-42) {
		tmp = t_2;
	} else if (y <= 2.8e-18) {
		tmp = t_0;
	} else if (y <= 1.85e+20) {
		tmp = t_2;
	} else if (y <= 1.45e+74) {
		tmp = t_1;
	} else if (y <= 9.2e+110) {
		tmp = x - z;
	} else if (y <= 1.25e+183) {
		tmp = y - (y * log(y));
	} else if (y <= 1.3e+183) {
		tmp = x;
	} 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 = (log(y) * (-0.5d0)) - z
    t_1 = y * (1.0d0 - log(y))
    t_2 = (x + y) - z
    if (y <= 1.42d-164) then
        tmp = t_0
    else if (y <= 4.8d-42) then
        tmp = t_2
    else if (y <= 2.8d-18) then
        tmp = t_0
    else if (y <= 1.85d+20) then
        tmp = t_2
    else if (y <= 1.45d+74) then
        tmp = t_1
    else if (y <= 9.2d+110) then
        tmp = x - z
    else if (y <= 1.25d+183) then
        tmp = y - (y * log(y))
    else if (y <= 1.3d+183) then
        tmp = x
    else
        tmp = t_1
    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) - z;
	double t_1 = y * (1.0 - Math.log(y));
	double t_2 = (x + y) - z;
	double tmp;
	if (y <= 1.42e-164) {
		tmp = t_0;
	} else if (y <= 4.8e-42) {
		tmp = t_2;
	} else if (y <= 2.8e-18) {
		tmp = t_0;
	} else if (y <= 1.85e+20) {
		tmp = t_2;
	} else if (y <= 1.45e+74) {
		tmp = t_1;
	} else if (y <= 9.2e+110) {
		tmp = x - z;
	} else if (y <= 1.25e+183) {
		tmp = y - (y * Math.log(y));
	} else if (y <= 1.3e+183) {
		tmp = x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (math.log(y) * -0.5) - z
	t_1 = y * (1.0 - math.log(y))
	t_2 = (x + y) - z
	tmp = 0
	if y <= 1.42e-164:
		tmp = t_0
	elif y <= 4.8e-42:
		tmp = t_2
	elif y <= 2.8e-18:
		tmp = t_0
	elif y <= 1.85e+20:
		tmp = t_2
	elif y <= 1.45e+74:
		tmp = t_1
	elif y <= 9.2e+110:
		tmp = x - z
	elif y <= 1.25e+183:
		tmp = y - (y * math.log(y))
	elif y <= 1.3e+183:
		tmp = x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(log(y) * -0.5) - z)
	t_1 = Float64(y * Float64(1.0 - log(y)))
	t_2 = Float64(Float64(x + y) - z)
	tmp = 0.0
	if (y <= 1.42e-164)
		tmp = t_0;
	elseif (y <= 4.8e-42)
		tmp = t_2;
	elseif (y <= 2.8e-18)
		tmp = t_0;
	elseif (y <= 1.85e+20)
		tmp = t_2;
	elseif (y <= 1.45e+74)
		tmp = t_1;
	elseif (y <= 9.2e+110)
		tmp = Float64(x - z);
	elseif (y <= 1.25e+183)
		tmp = Float64(y - Float64(y * log(y)));
	elseif (y <= 1.3e+183)
		tmp = x;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (log(y) * -0.5) - z;
	t_1 = y * (1.0 - log(y));
	t_2 = (x + y) - z;
	tmp = 0.0;
	if (y <= 1.42e-164)
		tmp = t_0;
	elseif (y <= 4.8e-42)
		tmp = t_2;
	elseif (y <= 2.8e-18)
		tmp = t_0;
	elseif (y <= 1.85e+20)
		tmp = t_2;
	elseif (y <= 1.45e+74)
		tmp = t_1;
	elseif (y <= 9.2e+110)
		tmp = x - z;
	elseif (y <= 1.25e+183)
		tmp = y - (y * log(y));
	elseif (y <= 1.3e+183)
		tmp = x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision] - z), $MachinePrecision]}, Block[{t$95$1 = N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + y), $MachinePrecision] - z), $MachinePrecision]}, If[LessEqual[y, 1.42e-164], t$95$0, If[LessEqual[y, 4.8e-42], t$95$2, If[LessEqual[y, 2.8e-18], t$95$0, If[LessEqual[y, 1.85e+20], t$95$2, If[LessEqual[y, 1.45e+74], t$95$1, If[LessEqual[y, 9.2e+110], N[(x - z), $MachinePrecision], If[LessEqual[y, 1.25e+183], N[(y - N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.3e+183], x, t$95$1]]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if y < 1.4200000000000001e-164 or 4.80000000000000005e-42 < y < 2.80000000000000012e-18

   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 0 100.0%

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

      1. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 78.9%

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

      1. *-commutative 78.9%

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

   8. Simplified 78.9%

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

   if 1.4200000000000001e-164 < y < 4.80000000000000005e-42 or 2.80000000000000012e-18 < y < 1.85e20

   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 x around -inf 99.9%

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

      1. mul-1-neg 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. +-commutative 99.9%

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

      5. metadata-eval 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in x around inf 83.1%

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

      1. neg-mul-1 83.1%

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

   8. Simplified 83.1%

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

   if 1.85e20 < y < 1.4500000000000001e74 or 1.3e183 < y

   1. Initial program 99.4%

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

   3. Taylor expanded in z around inf 58.6%

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

      1. associate--l+ 58.6%

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

      2. associate-/l* 58.7%

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

      3. +-commutative 58.7%

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

   5. Simplified 58.7%

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

   6. Taylor expanded in y around inf 58.6%

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

   7. Taylor expanded in y around inf 57.3%

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

      1. cancel-sign-sub-inv 57.3%

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

      2. metadata-eval 57.3%

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

      3. log-rec 57.3%

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

      4. distribute-neg-frac 57.3%

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

      5. *-lft-identity 57.3%

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

      6. sub-neg 57.3%

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

      7. div-sub 57.4%

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

   9. Simplified 57.4%

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

       1. clear-num 57.4%

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

       2. un-div-inv 57.3%

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

   11. Applied egg-rr 57.3%

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

       1. associate-/r/ 57.4%

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

   13. Simplified 57.4%

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

   14. Taylor expanded in x around 0 83.1%

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

   if 1.4500000000000001e74 < y < 9.2000000000000001e110

   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 x around inf 78.1%

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

   if 9.2000000000000001e110 < y < 1.25000000000000002e183

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 82.2%

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

   4. Taylor expanded in y around inf 82.2%

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

      1. mul-1-neg 82.2%

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

      2. distribute-rgt-neg-in 82.2%

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

      3. log-rec 82.2%

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

      4. remove-double-neg 82.2%

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

   6. Simplified 82.2%

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

   7. Taylor expanded in z around 0 60.1%

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

   if 1.25000000000000002e183 < y < 1.3e183

   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 z around inf 100.0%

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

      1. associate--l+ 100.0%

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

      2. associate-/l* 100.0%

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

      3. +-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{x} \]
3. Recombined 6 regimes into one program.

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.42 \cdot 10^{-164}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{-42}:\\ \;\;\;\;\left(x + y\right) - z\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{-18}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{+20}:\\ \;\;\;\;\left(x + y\right) - z\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{+74}:\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \mathbf{elif}\;y \leq 9.2 \cdot 10^{+110}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{+183}:\\ \;\;\;\;y - y \cdot \log y\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{+183}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log y \cdot -0.5 - z\\ t_1 := \left(x + y\right) - z\\ \mathbf{if}\;y \leq 1.4 \cdot 10^{-164}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-40}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{-17}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{+18}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.18 \cdot 10^{+70} \lor \neg \left(y \leq 5.7 \cdot 10^{+85}\right) \land y \leq 5.4 \cdot 10^{+119}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \log \left(\frac{e}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (* (log y) -0.5) z)) (t_1 (- (+ x y) z)))
   (if (<= y 1.4e-164)
     t_0
     (if (<= y 5.5e-40)
       t_1
       (if (<= y 1.2e-17)
         t_0
         (if (<= y 3.8e+18)
           t_1
           (if (or (<= y 1.18e+70) (and (not (<= y 5.7e+85)) (<= y 5.4e+119)))
             (- (* y (- 1.0 (log y))) z)
             (+ x (* y (log (/ E y)))))))))))

C

double code(double x, double y, double z) {
	double t_0 = (log(y) * -0.5) - z;
	double t_1 = (x + y) - z;
	double tmp;
	if (y <= 1.4e-164) {
		tmp = t_0;
	} else if (y <= 5.5e-40) {
		tmp = t_1;
	} else if (y <= 1.2e-17) {
		tmp = t_0;
	} else if (y <= 3.8e+18) {
		tmp = t_1;
	} else if ((y <= 1.18e+70) || (!(y <= 5.7e+85) && (y <= 5.4e+119))) {
		tmp = (y * (1.0 - log(y))) - z;
	} else {
		tmp = x + (y * log((((double) M_E) / y)));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = (Math.log(y) * -0.5) - z;
	double t_1 = (x + y) - z;
	double tmp;
	if (y <= 1.4e-164) {
		tmp = t_0;
	} else if (y <= 5.5e-40) {
		tmp = t_1;
	} else if (y <= 1.2e-17) {
		tmp = t_0;
	} else if (y <= 3.8e+18) {
		tmp = t_1;
	} else if ((y <= 1.18e+70) || (!(y <= 5.7e+85) && (y <= 5.4e+119))) {
		tmp = (y * (1.0 - Math.log(y))) - z;
	} else {
		tmp = x + (y * Math.log((Math.E / y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (math.log(y) * -0.5) - z
	t_1 = (x + y) - z
	tmp = 0
	if y <= 1.4e-164:
		tmp = t_0
	elif y <= 5.5e-40:
		tmp = t_1
	elif y <= 1.2e-17:
		tmp = t_0
	elif y <= 3.8e+18:
		tmp = t_1
	elif (y <= 1.18e+70) or (not (y <= 5.7e+85) and (y <= 5.4e+119)):
		tmp = (y * (1.0 - math.log(y))) - z
	else:
		tmp = x + (y * math.log((math.e / y)))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(log(y) * -0.5) - z)
	t_1 = Float64(Float64(x + y) - z)
	tmp = 0.0
	if (y <= 1.4e-164)
		tmp = t_0;
	elseif (y <= 5.5e-40)
		tmp = t_1;
	elseif (y <= 1.2e-17)
		tmp = t_0;
	elseif (y <= 3.8e+18)
		tmp = t_1;
	elseif ((y <= 1.18e+70) || (!(y <= 5.7e+85) && (y <= 5.4e+119)))
		tmp = Float64(Float64(y * Float64(1.0 - log(y))) - z);
	else
		tmp = Float64(x + Float64(y * log(Float64(exp(1) / y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (log(y) * -0.5) - z;
	t_1 = (x + y) - z;
	tmp = 0.0;
	if (y <= 1.4e-164)
		tmp = t_0;
	elseif (y <= 5.5e-40)
		tmp = t_1;
	elseif (y <= 1.2e-17)
		tmp = t_0;
	elseif (y <= 3.8e+18)
		tmp = t_1;
	elseif ((y <= 1.18e+70) || (~((y <= 5.7e+85)) && (y <= 5.4e+119)))
		tmp = (y * (1.0 - log(y))) - z;
	else
		tmp = x + (y * log((2.71828182845904523536 / y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision] - z), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x + y), $MachinePrecision] - z), $MachinePrecision]}, If[LessEqual[y, 1.4e-164], t$95$0, If[LessEqual[y, 5.5e-40], t$95$1, If[LessEqual[y, 1.2e-17], t$95$0, If[LessEqual[y, 3.8e+18], t$95$1, If[Or[LessEqual[y, 1.18e+70], And[N[Not[LessEqual[y, 5.7e+85]], $MachinePrecision], LessEqual[y, 5.4e+119]]], N[(N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(x + N[(y * N[Log[N[(E / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < 1.4000000000000001e-164 or 5.50000000000000002e-40 < y < 1.19999999999999993e-17

   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 0 100.0%

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

      1. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 78.9%

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

      1. *-commutative 78.9%

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

   8. Simplified 78.9%

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

   if 1.4000000000000001e-164 < y < 5.50000000000000002e-40 or 1.19999999999999993e-17 < y < 3.8e18

   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 x around -inf 99.9%

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

      1. mul-1-neg 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. +-commutative 99.9%

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

      5. metadata-eval 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in x around inf 83.1%

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

      1. neg-mul-1 83.1%

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

   8. Simplified 83.1%

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

   if 3.8e18 < y < 1.18000000000000001e70 or 5.7000000000000002e85 < y < 5.3999999999999997e119

   1. Initial program 99.6%

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

   3. Taylor expanded in x around -inf 79.4%

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

      1. mul-1-neg 79.4%

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

      2. sub-neg 79.4%

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

      3. associate-/l* 79.6%

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

      4. +-commutative 79.6%

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

      5. metadata-eval 79.6%

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

   5. Simplified 79.6%

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

   6. Taylor expanded in y around inf 89.9%

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

      1. mul-1-neg 89.9%

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

      2. log-rec 89.9%

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

      3. remove-double-neg 89.9%

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

   8. Simplified 89.9%

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

   if 1.18000000000000001e70 < y < 5.7000000000000002e85 or 5.3999999999999997e119 < y

   1. Initial program 99.5%

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

   3. Taylor expanded in z around inf 61.1%

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

      1. associate--l+ 61.1%

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

      2. associate-/l* 61.1%

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

      3. +-commutative 61.1%

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

   5. Simplified 61.1%

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

   6. Taylor expanded in y around inf 61.0%

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

   7. Taylor expanded in y around inf 57.9%

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

      1. cancel-sign-sub-inv 57.9%

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

      2. metadata-eval 57.9%

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

      3. log-rec 57.9%

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

      4. distribute-neg-frac 57.9%

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

      5. *-lft-identity 57.9%

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

      6. sub-neg 57.9%

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

      7. div-sub 58.0%

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

   9. Simplified 58.0%

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

   10. Taylor expanded in z around 0 91.4%

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

       1. +-commutative 91.4%

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

   12. Simplified 91.4%

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

       1. add-log-exp 99.6%

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

       2. exp-diff 99.6%

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

       3. add-exp-log 99.6%

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

   14. Applied egg-rr 91.4%

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

       1. exp-1-e 99.6%

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

   16. Simplified 91.4%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.4 \cdot 10^{-164}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-40}:\\ \;\;\;\;\left(x + y\right) - z\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{-17}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{+18}:\\ \;\;\;\;\left(x + y\right) - z\\ \mathbf{elif}\;y \leq 1.18 \cdot 10^{+70} \lor \neg \left(y \leq 5.7 \cdot 10^{+85}\right) \land y \leq 5.4 \cdot 10^{+119}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \log \left(\frac{e}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log y \cdot -0.5 - z\\ \mathbf{if}\;y \leq 1.5 \cdot 10^{-164}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{-40}:\\ \;\;\;\;\left(x + y\right) - z\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{-17}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \log \left(\frac{e}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (* (log y) -0.5) z)))
   (if (<= y 1.5e-164)
     t_0
     (if (<= y 8.2e-40)
       (- (+ x y) z)
       (if (<= y 3.2e-17) t_0 (+ x (* y (log (/ E y)))))))))

C

double code(double x, double y, double z) {
	double t_0 = (log(y) * -0.5) - z;
	double tmp;
	if (y <= 1.5e-164) {
		tmp = t_0;
	} else if (y <= 8.2e-40) {
		tmp = (x + y) - z;
	} else if (y <= 3.2e-17) {
		tmp = t_0;
	} else {
		tmp = x + (y * log((((double) M_E) / y)));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = (Math.log(y) * -0.5) - z;
	double tmp;
	if (y <= 1.5e-164) {
		tmp = t_0;
	} else if (y <= 8.2e-40) {
		tmp = (x + y) - z;
	} else if (y <= 3.2e-17) {
		tmp = t_0;
	} else {
		tmp = x + (y * Math.log((Math.E / y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (math.log(y) * -0.5) - z
	tmp = 0
	if y <= 1.5e-164:
		tmp = t_0
	elif y <= 8.2e-40:
		tmp = (x + y) - z
	elif y <= 3.2e-17:
		tmp = t_0
	else:
		tmp = x + (y * math.log((math.e / y)))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(log(y) * -0.5) - z)
	tmp = 0.0
	if (y <= 1.5e-164)
		tmp = t_0;
	elseif (y <= 8.2e-40)
		tmp = Float64(Float64(x + y) - z);
	elseif (y <= 3.2e-17)
		tmp = t_0;
	else
		tmp = Float64(x + Float64(y * log(Float64(exp(1) / y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (log(y) * -0.5) - z;
	tmp = 0.0;
	if (y <= 1.5e-164)
		tmp = t_0;
	elseif (y <= 8.2e-40)
		tmp = (x + y) - z;
	elseif (y <= 3.2e-17)
		tmp = t_0;
	else
		tmp = x + (y * log((2.71828182845904523536 / y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision] - z), $MachinePrecision]}, If[LessEqual[y, 1.5e-164], t$95$0, If[LessEqual[y, 8.2e-40], N[(N[(x + y), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[y, 3.2e-17], t$95$0, N[(x + N[(y * N[Log[N[(E / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.5e-164 or 8.19999999999999926e-40 < y < 3.2000000000000002e-17

   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 0 100.0%

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

      1. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 78.9%

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

      1. *-commutative 78.9%

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

   8. Simplified 78.9%

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

   if 1.5e-164 < y < 8.19999999999999926e-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. Taylor expanded in x around -inf 99.9%

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

      1. mul-1-neg 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.9%

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

      4. +-commutative 99.9%

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

      5. metadata-eval 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in x around inf 83.6%

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

      1. neg-mul-1 83.6%

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

   8. Simplified 83.6%

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

   if 3.2000000000000002e-17 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in z around inf 64.9%

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

      1. associate--l+ 64.9%

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

      2. associate-/l* 64.9%

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

      3. +-commutative 64.9%

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

   5. Simplified 64.9%

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

   6. Taylor expanded in y around inf 64.8%

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

   7. Taylor expanded in y around inf 51.1%

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

      1. cancel-sign-sub-inv 51.1%

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

      2. metadata-eval 51.1%

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

      3. log-rec 51.1%

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

      4. distribute-neg-frac 51.1%

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

      5. *-lft-identity 51.1%

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

      6. sub-neg 51.1%

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

      7. div-sub 51.1%

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

   9. Simplified 51.1%

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

   10. Taylor expanded in z around 0 82.2%

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

       1. +-commutative 82.2%

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

   12. Simplified 82.2%

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

       1. add-log-exp 97.8%

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

       2. exp-diff 97.8%

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

       3. add-exp-log 97.8%

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

   14. Applied egg-rr 82.2%

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

       1. exp-1-e 97.8%

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

   16. Simplified 82.2%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.5 \cdot 10^{-164}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{-40}:\\ \;\;\;\;\left(x + y\right) - z\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{-17}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \log \left(\frac{e}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 69.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.85 \cdot 10^{+20}:\\ \;\;\;\;\left(x + y\right) - z\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+74} \lor \neg \left(y \leq 1.02 \cdot 10^{+111}\right):\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.85e+20)
   (- (+ x y) z)
   (if (or (<= y 1.9e+74) (not (<= y 1.02e+111)))
     (* y (- 1.0 (log y)))
     (- x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.85e+20) {
		tmp = (x + y) - z;
	} else if ((y <= 1.9e+74) || !(y <= 1.02e+111)) {
		tmp = y * (1.0 - log(y));
	} else {
		tmp = x - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 1.85d+20) then
        tmp = (x + y) - z
    else if ((y <= 1.9d+74) .or. (.not. (y <= 1.02d+111))) then
        tmp = y * (1.0d0 - log(y))
    else
        tmp = x - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.85e+20) {
		tmp = (x + y) - z;
	} else if ((y <= 1.9e+74) || !(y <= 1.02e+111)) {
		tmp = y * (1.0 - Math.log(y));
	} else {
		tmp = x - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1.85e+20:
		tmp = (x + y) - z
	elif (y <= 1.9e+74) or not (y <= 1.02e+111):
		tmp = y * (1.0 - math.log(y))
	else:
		tmp = x - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.85e+20)
		tmp = Float64(Float64(x + y) - z);
	elseif ((y <= 1.9e+74) || !(y <= 1.02e+111))
		tmp = Float64(y * Float64(1.0 - log(y)));
	else
		tmp = Float64(x - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1.85e+20)
		tmp = (x + y) - z;
	elseif ((y <= 1.9e+74) || ~((y <= 1.02e+111)))
		tmp = y * (1.0 - log(y));
	else
		tmp = x - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.85e+20], N[(N[(x + y), $MachinePrecision] - z), $MachinePrecision], If[Or[LessEqual[y, 1.9e+74], N[Not[LessEqual[y, 1.02e+111]], $MachinePrecision]], N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.85e20

   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 x around -inf 99.9%

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

      1. mul-1-neg 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-/l* 99.8%

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

      4. +-commutative 99.8%

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

      5. metadata-eval 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in x around inf 75.1%

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

      1. neg-mul-1 75.1%

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

   8. Simplified 75.1%

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

   if 1.85e20 < y < 1.8999999999999999e74 or 1.02e111 < y

   1. Initial program 99.4%

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

   3. Taylor expanded in z around inf 61.5%

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

      1. associate--l+ 61.5%

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

      2. associate-/l* 61.5%

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

      3. +-commutative 61.5%

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

   5. Simplified 61.5%

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

   6. Taylor expanded in y around inf 61.4%

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

   7. Taylor expanded in y around inf 55.4%

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

      1. cancel-sign-sub-inv 55.4%

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

      2. metadata-eval 55.4%

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

      3. log-rec 55.4%

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

      4. distribute-neg-frac 55.4%

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

      5. *-lft-identity 55.4%

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

      6. sub-neg 55.4%

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

      7. div-sub 55.4%

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

   9. Simplified 55.4%

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

       1. clear-num 55.4%

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

       2. un-div-inv 55.4%

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

   11. Applied egg-rr 55.4%

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

       1. associate-/r/ 55.4%

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

   13. Simplified 55.4%

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

   14. Taylor expanded in x around 0 76.5%

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

   if 1.8999999999999999e74 < y < 1.02e111

   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 x around inf 78.1%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.85 \cdot 10^{+20}:\\ \;\;\;\;\left(x + y\right) - z\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+74} \lor \neg \left(y \leq 1.02 \cdot 10^{+111}\right):\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.15 \cdot 10^{+20}:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{+128}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \log \left(\frac{e}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.15e+20)
   (- (- x (* (log y) 0.5)) z)
   (if (<= y 3.5e+128) (- (* y (- 1.0 (log y))) z) (+ x (* y (log (/ E y)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.15e+20) {
		tmp = (x - (log(y) * 0.5)) - z;
	} else if (y <= 3.5e+128) {
		tmp = (y * (1.0 - log(y))) - z;
	} else {
		tmp = x + (y * log((((double) M_E) / y)));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.15e+20) {
		tmp = (x - (Math.log(y) * 0.5)) - z;
	} else if (y <= 3.5e+128) {
		tmp = (y * (1.0 - Math.log(y))) - z;
	} else {
		tmp = x + (y * Math.log((Math.E / y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1.15e+20:
		tmp = (x - (math.log(y) * 0.5)) - z
	elif y <= 3.5e+128:
		tmp = (y * (1.0 - math.log(y))) - z
	else:
		tmp = x + (y * math.log((math.e / y)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.15e+20)
		tmp = Float64(Float64(x - Float64(log(y) * 0.5)) - z);
	elseif (y <= 3.5e+128)
		tmp = Float64(Float64(y * Float64(1.0 - log(y))) - z);
	else
		tmp = Float64(x + Float64(y * log(Float64(exp(1) / y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1.15e+20)
		tmp = (x - (log(y) * 0.5)) - z;
	elseif (y <= 3.5e+128)
		tmp = (y * (1.0 - log(y))) - z;
	else
		tmp = x + (y * log((2.71828182845904523536 / y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.15e+20], N[(N[(x - N[(N[Log[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[y, 3.5e+128], N[(N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(x + N[(y * N[Log[N[(E / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.15e20

   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 99.3%

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

      1. *-commutative 99.3%

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

   5. Simplified 99.3%

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

   if 1.15e20 < y < 3.49999999999999969e128

   1. Initial program 99.6%

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

   3. Taylor expanded in x around -inf 83.7%

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

      1. mul-1-neg 83.7%

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

      2. sub-neg 83.7%

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

      3. associate-/l* 83.7%

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

      4. +-commutative 83.7%

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

      5. metadata-eval 83.7%

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

   5. Simplified 83.7%

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

   6. Taylor expanded in y around inf 84.1%

      \[\leadsto \color{blue}{y \cdot \left(1 - -1 \cdot \log \left(\frac{1}{y}\right)\right)} - z \]
   7. 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%

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

   8. Simplified 84.1%

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

   if 3.49999999999999969e128 < y

   1. Initial program 99.5%

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

   3. Taylor expanded in z around inf 58.0%

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

      1. associate--l+ 58.0%

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

      2. associate-/l* 58.0%

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

      3. +-commutative 58.0%

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

   5. Simplified 58.0%

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

   6. Taylor expanded in y around inf 57.9%

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

   7. Taylor expanded in y around inf 56.8%

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

      1. cancel-sign-sub-inv 56.8%

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

      2. metadata-eval 56.8%

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

      3. log-rec 56.8%

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

      4. distribute-neg-frac 56.8%

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

      5. *-lft-identity 56.8%

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

      6. sub-neg 56.8%

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

      7. div-sub 56.9%

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

   9. Simplified 56.9%

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

   10. Taylor expanded in z around 0 92.9%

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

       1. +-commutative 92.9%

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

   12. Simplified 92.9%

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

       1. add-log-exp 99.6%

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

       2. exp-diff 99.6%

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

       3. add-exp-log 99.6%

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

   14. Applied egg-rr 92.9%

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

       1. exp-1-e 99.6%

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

   16. Simplified 92.9%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.15 \cdot 10^{+20}:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{+128}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \log \left(\frac{e}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.1 \cdot 10^{-5}:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + \left(y \cdot \log \left(\frac{e}{y}\right) - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.1e-5)
   (- (- x (* (log y) 0.5)) z)
   (+ x (- (* y (log (/ E y))) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.1e-5) {
		tmp = (x - (log(y) * 0.5)) - z;
	} else {
		tmp = x + ((y * log((((double) M_E) / y))) - z);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.1e-5) {
		tmp = (x - (Math.log(y) * 0.5)) - z;
	} else {
		tmp = x + ((y * Math.log((Math.E / y))) - z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1.1e-5:
		tmp = (x - (math.log(y) * 0.5)) - z
	else:
		tmp = x + ((y * math.log((math.e / y))) - z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.1e-5)
		tmp = Float64(Float64(x - Float64(log(y) * 0.5)) - z);
	else
		tmp = Float64(x + Float64(Float64(y * log(Float64(exp(1) / y))) - z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1.1e-5)
		tmp = (x - (log(y) * 0.5)) - z;
	else
		tmp = x + ((y * log((2.71828182845904523536 / y))) - z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.1e-5], N[(N[(x - N[(N[Log[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(x + N[(N[(y * N[Log[N[(E / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.1e-5

   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 99.9%

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

      1. *-commutative 99.9%

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

   5. Simplified 99.9%

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

   if 1.1e-5 < 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+ 99.5%

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

      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- 99.6%

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

      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-define 99.6%

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

      8. +-commutative 99.6%

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

      9. distribute-neg-in 99.6%

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

      10. unsub-neg 99.6%

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

      11. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

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

      1. add-log-exp 99.4%

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

      2. exp-diff 99.4%

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

      3. add-exp-log 99.4%

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

   9. Applied egg-rr 99.4%

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

       1. exp-1-e 99.4%

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

   11. Simplified 99.4%

       \[\leadsto x + \left(y \cdot \color{blue}{\log \left(\frac{e}{y}\right)} - z\right) \]
3. Recombined 2 regimes into one program.
4. 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: 47.9% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{+88} \lor \neg \left(z \leq 1.22 \cdot 10^{+39}\right) \land \left(z \leq 7 \cdot 10^{+66} \lor \neg \left(z \leq 1.4 \cdot 10^{+70}\right)\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.1e+88)
         (and (not (<= z 1.22e+39)) (or (<= z 7e+66) (not (<= z 1.4e+70)))))
   (- z)
   x))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.1e+88) || (!(z <= 1.22e+39) && ((z <= 7e+66) || !(z <= 1.4e+70)))) {
		tmp = -z;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-1.1d+88)) .or. (.not. (z <= 1.22d+39)) .and. (z <= 7d+66) .or. (.not. (z <= 1.4d+70))) then
        tmp = -z
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.1e+88) || (!(z <= 1.22e+39) && ((z <= 7e+66) || !(z <= 1.4e+70)))) {
		tmp = -z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.1e+88) or (not (z <= 1.22e+39) and ((z <= 7e+66) or not (z <= 1.4e+70))):
		tmp = -z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.1e+88) || (!(z <= 1.22e+39) && ((z <= 7e+66) || !(z <= 1.4e+70))))
		tmp = Float64(-z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.1e+88) || (~((z <= 1.22e+39)) && ((z <= 7e+66) || ~((z <= 1.4e+70)))))
		tmp = -z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.1e+88], And[N[Not[LessEqual[z, 1.22e+39]], $MachinePrecision], Or[LessEqual[z, 7e+66], N[Not[LessEqual[z, 1.4e+70]], $MachinePrecision]]]], (-z), x]

Derivation

1. Split input into 2 regimes

2. if z < -1.10000000000000004e88 or 1.22e39 < z < 6.9999999999999994e66 or 1.39999999999999995e70 < 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 x around 0 83.0%

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

   4. Taylor expanded in z around inf 66.1%

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

      1. neg-mul-1 66.1%

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

   6. Simplified 66.1%

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

   if -1.10000000000000004e88 < z < 1.22e39 or 6.9999999999999994e66 < z < 1.39999999999999995e70

   1. Initial program 99.7%

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

   3. Taylor expanded in z around inf 62.8%

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

      1. associate--l+ 62.8%

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

      2. associate-/l* 62.7%

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

      3. +-commutative 62.7%

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

   5. Simplified 62.7%

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

   6. Taylor expanded in x around inf 34.2%

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

4. Final simplification 47.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{+88} \lor \neg \left(z \leq 1.22 \cdot 10^{+39}\right) \land \left(z \leq 7 \cdot 10^{+66} \lor \neg \left(z \leq 1.4 \cdot 10^{+70}\right)\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 57.8% 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. Taylor expanded in x around inf 57.2%

   \[\leadsto \color{blue}{x} - z \]
4. Add Preprocessing

Alternative 11: 29.7% 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. Add Preprocessing

3. Taylor expanded in z around inf 77.7%

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

   1. associate--l+ 77.7%

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

   2. associate-/l* 77.6%

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

   3. +-commutative 77.6%

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

5. Simplified 77.6%

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

6. Taylor expanded in x around inf 27.8%

   \[\leadsto \color{blue}{x} \]
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 2024107 
(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))