Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Percentage Accurate: 99.8% → 99.9%

Time: 11.0s

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. sub-neg 99.8%

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

   2. sub-neg 99.8%

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

   3. associate-+l+ 99.8%

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

   4. associate-+l+ 99.8%

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

   5. sub-neg 99.8%

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

   6. neg-sub0 99.8%

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

   7. associate-+l- 99.8%

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

   8. neg-sub0 99.8%

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

   9. neg-mul-1 99.8%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 2: 75.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \log y \cdot 0.5\\ \mathbf{if}\;y \leq 7.6 \cdot 10^{-281}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{-252}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.95 \cdot 10^{-178}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{-101}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{+92}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) - y \cdot \log y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- x (* (log y) 0.5))))
   (if (<= y 7.6e-281)
     (- x z)
     (if (<= y 2.3e-252)
       t_0
       (if (<= y 1.95e-178)
         (- x z)
         (if (<= y 7.2e-101)
           t_0
           (if (<= y 3.3e+92) (- x z) (- (+ x y) (* y (log y))))))))))

C

double code(double x, double y, double z) {
	double t_0 = x - (log(y) * 0.5);
	double tmp;
	if (y <= 7.6e-281) {
		tmp = x - z;
	} else if (y <= 2.3e-252) {
		tmp = t_0;
	} else if (y <= 1.95e-178) {
		tmp = x - z;
	} else if (y <= 7.2e-101) {
		tmp = t_0;
	} else if (y <= 3.3e+92) {
		tmp = x - z;
	} else {
		tmp = (x + y) - (y * log(y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x - (log(y) * 0.5d0)
    if (y <= 7.6d-281) then
        tmp = x - z
    else if (y <= 2.3d-252) then
        tmp = t_0
    else if (y <= 1.95d-178) then
        tmp = x - z
    else if (y <= 7.2d-101) then
        tmp = t_0
    else if (y <= 3.3d+92) then
        tmp = x - z
    else
        tmp = (x + y) - (y * log(y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x - (Math.log(y) * 0.5);
	double tmp;
	if (y <= 7.6e-281) {
		tmp = x - z;
	} else if (y <= 2.3e-252) {
		tmp = t_0;
	} else if (y <= 1.95e-178) {
		tmp = x - z;
	} else if (y <= 7.2e-101) {
		tmp = t_0;
	} else if (y <= 3.3e+92) {
		tmp = x - z;
	} else {
		tmp = (x + y) - (y * Math.log(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x - (math.log(y) * 0.5)
	tmp = 0
	if y <= 7.6e-281:
		tmp = x - z
	elif y <= 2.3e-252:
		tmp = t_0
	elif y <= 1.95e-178:
		tmp = x - z
	elif y <= 7.2e-101:
		tmp = t_0
	elif y <= 3.3e+92:
		tmp = x - z
	else:
		tmp = (x + y) - (y * math.log(y))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x - Float64(log(y) * 0.5))
	tmp = 0.0
	if (y <= 7.6e-281)
		tmp = Float64(x - z);
	elseif (y <= 2.3e-252)
		tmp = t_0;
	elseif (y <= 1.95e-178)
		tmp = Float64(x - z);
	elseif (y <= 7.2e-101)
		tmp = t_0;
	elseif (y <= 3.3e+92)
		tmp = Float64(x - z);
	else
		tmp = Float64(Float64(x + y) - Float64(y * log(y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x - (log(y) * 0.5);
	tmp = 0.0;
	if (y <= 7.6e-281)
		tmp = x - z;
	elseif (y <= 2.3e-252)
		tmp = t_0;
	elseif (y <= 1.95e-178)
		tmp = x - z;
	elseif (y <= 7.2e-101)
		tmp = t_0;
	elseif (y <= 3.3e+92)
		tmp = x - z;
	else
		tmp = (x + y) - (y * log(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x - N[(N[Log[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 7.6e-281], N[(x - z), $MachinePrecision], If[LessEqual[y, 2.3e-252], t$95$0, If[LessEqual[y, 1.95e-178], N[(x - z), $MachinePrecision], If[LessEqual[y, 7.2e-101], t$95$0, If[LessEqual[y, 3.3e+92], N[(x - z), $MachinePrecision], N[(N[(x + y), $MachinePrecision] - N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 7.59999999999999953e-281 or 2.2999999999999998e-252 < y < 1.95000000000000013e-178 or 7.19999999999999999e-101 < y < 3.29999999999999974e92

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-+l+ 100.0%

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

      4. associate-+l+ 100.0%

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

      5. sub-neg 100.0%

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

      6. neg-sub0 100.0%

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

      7. associate-+l- 100.0%

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

      8. neg-sub0 100.0%

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

      9. neg-mul-1 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in z around inf 78.4%

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

      1. neg-mul-1 78.4%

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

   6. Simplified 78.4%

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

   if 7.59999999999999953e-281 < y < 2.2999999999999998e-252 or 1.95000000000000013e-178 < y < 7.19999999999999999e-101

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-+l+ 100.0%

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

      4. associate-+l+ 100.0%

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

      5. sub-neg 100.0%

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

      6. neg-sub0 100.0%

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

      7. associate-+l- 100.0%

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

      8. neg-sub0 100.0%

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

      9. neg-mul-1 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in z around 0 82.5%

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

      1. +-commutative 82.5%

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

      2. mul-1-neg 82.5%

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

      3. +-commutative 82.5%

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

      4. *-commutative 82.5%

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

      5. sub-neg 82.5%

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

   6. Simplified 82.5%

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

   7. Taylor expanded in y around 0 82.5%

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

   if 3.29999999999999974e92 < 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. sub-neg 99.6%

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

      2. sub-neg 99.6%

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

      3. associate-+l+ 99.7%

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

      4. associate-+l+ 99.7%

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

      5. sub-neg 99.7%

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

      6. neg-sub0 99.7%

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

      7. associate-+l- 99.7%

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

      8. neg-sub0 99.7%

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

      9. neg-mul-1 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in z around 0 87.1%

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

      1. +-commutative 87.1%

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

      2. mul-1-neg 87.1%

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

      3. +-commutative 87.1%

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

      4. *-commutative 87.1%

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

      5. sub-neg 87.1%

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

   6. Simplified 87.1%

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

   7. Taylor expanded in y around inf 87.1%

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

      1. mul-1-neg 87.1%

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

      2. distribute-rgt-neg-in 87.1%

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

      3. log-rec 87.1%

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

      4. remove-double-neg 87.1%

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

   9. Simplified 87.1%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7.6 \cdot 10^{-281}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{-252}:\\ \;\;\;\;x - \log y \cdot 0.5\\ \mathbf{elif}\;y \leq 1.95 \cdot 10^{-178}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{-101}:\\ \;\;\;\;x - \log y \cdot 0.5\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{+92}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) - y \cdot \log y\\ \end{array} \]

Alternative 3: 70.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \log y \cdot 0.5\\ \mathbf{if}\;y \leq 6.5 \cdot 10^{-281}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y \leq 9.6 \cdot 10^{-252}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{-178}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{-101}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 6 \cdot 10^{+127}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y - y \cdot \log y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- x (* (log y) 0.5))))
   (if (<= y 6.5e-281)
     (- x z)
     (if (<= y 9.6e-252)
       t_0
       (if (<= y 1.8e-178)
         (- x z)
         (if (<= y 1.15e-101)
           t_0
           (if (<= y 6e+127) (- x z) (- y (* y (log y))))))))))

C

double code(double x, double y, double z) {
	double t_0 = x - (log(y) * 0.5);
	double tmp;
	if (y <= 6.5e-281) {
		tmp = x - z;
	} else if (y <= 9.6e-252) {
		tmp = t_0;
	} else if (y <= 1.8e-178) {
		tmp = x - z;
	} else if (y <= 1.15e-101) {
		tmp = t_0;
	} else if (y <= 6e+127) {
		tmp = x - z;
	} else {
		tmp = y - (y * log(y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x - (log(y) * 0.5d0)
    if (y <= 6.5d-281) then
        tmp = x - z
    else if (y <= 9.6d-252) then
        tmp = t_0
    else if (y <= 1.8d-178) then
        tmp = x - z
    else if (y <= 1.15d-101) then
        tmp = t_0
    else if (y <= 6d+127) then
        tmp = x - z
    else
        tmp = y - (y * log(y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x - (Math.log(y) * 0.5);
	double tmp;
	if (y <= 6.5e-281) {
		tmp = x - z;
	} else if (y <= 9.6e-252) {
		tmp = t_0;
	} else if (y <= 1.8e-178) {
		tmp = x - z;
	} else if (y <= 1.15e-101) {
		tmp = t_0;
	} else if (y <= 6e+127) {
		tmp = x - z;
	} else {
		tmp = y - (y * Math.log(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x - (math.log(y) * 0.5)
	tmp = 0
	if y <= 6.5e-281:
		tmp = x - z
	elif y <= 9.6e-252:
		tmp = t_0
	elif y <= 1.8e-178:
		tmp = x - z
	elif y <= 1.15e-101:
		tmp = t_0
	elif y <= 6e+127:
		tmp = x - z
	else:
		tmp = y - (y * math.log(y))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x - Float64(log(y) * 0.5))
	tmp = 0.0
	if (y <= 6.5e-281)
		tmp = Float64(x - z);
	elseif (y <= 9.6e-252)
		tmp = t_0;
	elseif (y <= 1.8e-178)
		tmp = Float64(x - z);
	elseif (y <= 1.15e-101)
		tmp = t_0;
	elseif (y <= 6e+127)
		tmp = Float64(x - z);
	else
		tmp = Float64(y - Float64(y * log(y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x - (log(y) * 0.5);
	tmp = 0.0;
	if (y <= 6.5e-281)
		tmp = x - z;
	elseif (y <= 9.6e-252)
		tmp = t_0;
	elseif (y <= 1.8e-178)
		tmp = x - z;
	elseif (y <= 1.15e-101)
		tmp = t_0;
	elseif (y <= 6e+127)
		tmp = x - z;
	else
		tmp = y - (y * log(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x - N[(N[Log[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 6.5e-281], N[(x - z), $MachinePrecision], If[LessEqual[y, 9.6e-252], t$95$0, If[LessEqual[y, 1.8e-178], N[(x - z), $MachinePrecision], If[LessEqual[y, 1.15e-101], t$95$0, If[LessEqual[y, 6e+127], N[(x - z), $MachinePrecision], N[(y - N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 6.5e-281 or 9.6000000000000006e-252 < y < 1.79999999999999997e-178 or 1.15e-101 < y < 6.0000000000000005e127

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-+l+ 99.9%

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

      4. associate-+l+ 99.9%

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

      5. sub-neg 99.9%

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

      6. neg-sub0 99.9%

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

      7. associate-+l- 99.9%

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

      8. neg-sub0 99.9%

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

      9. neg-mul-1 99.9%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in z around inf 75.1%

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

      1. neg-mul-1 75.1%

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

   6. Simplified 75.1%

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

   if 6.5e-281 < y < 9.6000000000000006e-252 or 1.79999999999999997e-178 < y < 1.15e-101

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-+l+ 100.0%

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

      4. associate-+l+ 100.0%

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

      5. sub-neg 100.0%

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

      6. neg-sub0 100.0%

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

      7. associate-+l- 100.0%

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

      8. neg-sub0 100.0%

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

      9. neg-mul-1 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in z around 0 82.5%

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

      1. +-commutative 82.5%

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

      2. mul-1-neg 82.5%

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

      3. +-commutative 82.5%

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

      4. *-commutative 82.5%

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

      5. sub-neg 82.5%

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

   6. Simplified 82.5%

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

   7. Taylor expanded in y around 0 82.5%

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

   if 6.0000000000000005e127 < 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. sub-neg 99.6%

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

      2. sub-neg 99.6%

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

      3. associate-+l+ 99.7%

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

      4. associate-+l+ 99.7%

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

      5. sub-neg 99.7%

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

      6. neg-sub0 99.7%

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

      7. associate-+l- 99.7%

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

      8. neg-sub0 99.7%

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

      9. neg-mul-1 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in z around 0 91.1%

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

      1. +-commutative 91.1%

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

      2. mul-1-neg 91.1%

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

      3. +-commutative 91.1%

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

      4. *-commutative 91.1%

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

      5. sub-neg 91.1%

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

   6. Simplified 91.1%

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

   7. Taylor expanded in x around 0 83.2%

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

   8. Taylor expanded in y around inf 83.2%

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

      1. mul-1-neg 83.2%

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

      2. log-rec 83.2%

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

      3. distribute-rgt-neg-out 83.2%

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

      4. remove-double-neg 83.2%

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

   10. Simplified 83.2%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.5 \cdot 10^{-281}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y \leq 9.6 \cdot 10^{-252}:\\ \;\;\;\;x - \log y \cdot 0.5\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{-178}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{-101}:\\ \;\;\;\;x - \log y \cdot 0.5\\ \mathbf{elif}\;y \leq 6 \cdot 10^{+127}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y - y \cdot \log y\\ \end{array} \]

Alternative 4: 99.3% 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. Taylor expanded in y around 0 99.3%

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

   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. sub-neg 99.7%

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

      2. sub-neg 99.7%

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

      3. associate-+l+ 99.7%

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

      4. associate-+l+ 99.7%

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

      5. sub-neg 99.7%

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

      6. neg-sub0 99.7%

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

      7. associate-+l- 99.7%

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

      8. neg-sub0 99.7%

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

      9. neg-mul-1 99.7%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around inf 99.2%

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

      1. log-rec 99.2%

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

      2. sub-neg 99.2%

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

   6. Simplified 99.2%

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

4. Final simplification 99.3%

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

Alternative 5: 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. Final simplification 99.8%

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

Alternative 6: 70.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -14000000000000:\\ \;\;\;\;x - z\\ \mathbf{elif}\;z \leq 185:\\ \;\;\;\;x - \log y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -14000000000000.0)
   (- x z)
   (if (<= z 185.0) (- x (* (log y) 0.5)) (- x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -14000000000000.0) {
		tmp = x - z;
	} else if (z <= 185.0) {
		tmp = x - (log(y) * 0.5);
	} else {
		tmp = x - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-14000000000000.0d0)) then
        tmp = x - z
    else if (z <= 185.0d0) then
        tmp = x - (log(y) * 0.5d0)
    else
        tmp = x - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -14000000000000.0) {
		tmp = x - z;
	} else if (z <= 185.0) {
		tmp = x - (Math.log(y) * 0.5);
	} else {
		tmp = x - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -14000000000000.0:
		tmp = x - z
	elif z <= 185.0:
		tmp = x - (math.log(y) * 0.5)
	else:
		tmp = x - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -14000000000000.0)
		tmp = Float64(x - z);
	elseif (z <= 185.0)
		tmp = Float64(x - Float64(log(y) * 0.5));
	else
		tmp = Float64(x - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -14000000000000.0)
		tmp = x - z;
	elseif (z <= 185.0)
		tmp = x - (log(y) * 0.5);
	else
		tmp = x - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -14000000000000.0], N[(x - z), $MachinePrecision], If[LessEqual[z, 185.0], N[(x - N[(N[Log[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], N[(x - z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.4e13 or 185 < z

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-+l+ 99.9%

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

      4. associate-+l+ 99.9%

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

      5. sub-neg 99.9%

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

      6. neg-sub0 99.9%

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

      7. associate-+l- 99.9%

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

      8. neg-sub0 99.9%

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

      9. neg-mul-1 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in z around inf 77.7%

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

      1. neg-mul-1 77.7%

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

   6. Simplified 77.7%

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

   if -1.4e13 < z < 185

   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. sub-neg 99.8%

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

      2. sub-neg 99.8%

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

      3. associate-+l+ 99.8%

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

      4. associate-+l+ 99.8%

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

      5. sub-neg 99.8%

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

      6. neg-sub0 99.8%

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

      7. associate-+l- 99.8%

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

      8. neg-sub0 99.8%

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

      9. neg-mul-1 99.8%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in z around 0 99.7%

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

      1. +-commutative 99.7%

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

      2. mul-1-neg 99.7%

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

      3. +-commutative 99.7%

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

      4. *-commutative 99.7%

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

      5. sub-neg 99.7%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in y around 0 58.8%

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

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -14000000000000:\\ \;\;\;\;x - z\\ \mathbf{elif}\;z \leq 185:\\ \;\;\;\;x - \log y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \]

Alternative 7: 89.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.3 \cdot 10^{+94}:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) - y \cdot \log y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2.3e+94) (- (- x (* (log y) 0.5)) z) (- (+ x y) (* y (log y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.3e+94) {
		tmp = (x - (log(y) * 0.5)) - z;
	} else {
		tmp = (x + y) - (y * log(y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 2.3d+94) then
        tmp = (x - (log(y) * 0.5d0)) - z
    else
        tmp = (x + y) - (y * log(y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.3e+94) {
		tmp = (x - (Math.log(y) * 0.5)) - z;
	} else {
		tmp = (x + y) - (y * Math.log(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 2.3e+94:
		tmp = (x - (math.log(y) * 0.5)) - z
	else:
		tmp = (x + y) - (y * math.log(y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2.3e+94)
		tmp = Float64(Float64(x - Float64(log(y) * 0.5)) - z);
	else
		tmp = Float64(Float64(x + y) - Float64(y * log(y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 2.3e+94)
		tmp = (x - (log(y) * 0.5)) - z;
	else
		tmp = (x + y) - (y * log(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2.3e+94], N[(N[(x - N[(N[Log[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(N[(x + y), $MachinePrecision] - N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.3e94

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 95.2%

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

   if 2.3e94 < 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. sub-neg 99.6%

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

      2. sub-neg 99.6%

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

      3. associate-+l+ 99.7%

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

      4. associate-+l+ 99.7%

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

      5. sub-neg 99.7%

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

      6. neg-sub0 99.7%

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

      7. associate-+l- 99.7%

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

      8. neg-sub0 99.7%

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

      9. neg-mul-1 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in z around 0 87.1%

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

      1. +-commutative 87.1%

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

      2. mul-1-neg 87.1%

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

      3. +-commutative 87.1%

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

      4. *-commutative 87.1%

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

      5. sub-neg 87.1%

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

   6. Simplified 87.1%

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

   7. Taylor expanded in y around inf 87.1%

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

      1. mul-1-neg 87.1%

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

      2. distribute-rgt-neg-in 87.1%

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

      3. log-rec 87.1%

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

      4. remove-double-neg 87.1%

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

   9. Simplified 87.1%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.3 \cdot 10^{+94}:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) - y \cdot \log y\\ \end{array} \]

Alternative 8: 89.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6 \cdot 10^{+65}:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(y - y \cdot \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 6e+65) (- (- x (* (log y) 0.5)) z) (- (- y (* y (log y))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 6e+65) {
		tmp = (x - (log(y) * 0.5)) - z;
	} else {
		tmp = (y - (y * 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 <= 6d+65) then
        tmp = (x - (log(y) * 0.5d0)) - z
    else
        tmp = (y - (y * log(y))) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 6e+65) {
		tmp = (x - (Math.log(y) * 0.5)) - z;
	} else {
		tmp = (y - (y * Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 6e+65:
		tmp = (x - (math.log(y) * 0.5)) - z
	else:
		tmp = (y - (y * math.log(y))) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 6e+65)
		tmp = Float64(Float64(x - Float64(log(y) * 0.5)) - z);
	else
		tmp = Float64(Float64(y - Float64(y * log(y))) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 6e+65)
		tmp = (x - (log(y) * 0.5)) - z;
	else
		tmp = (y - (y * log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 6e+65], N[(N[(x - N[(N[Log[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(N[(y - N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.0000000000000004e65

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 96.3%

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

   if 6.0000000000000004e65 < y

   1. Initial program 99.7%

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

   2. Taylor expanded in y around inf 87.8%

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

      1. *-commutative 87.8%

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

      2. log-rec 87.8%

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

      3. distribute-lft-neg-in 87.8%

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

      4. distribute-rgt-neg-in 87.8%

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

   4. Simplified 87.8%

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

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6 \cdot 10^{+65}:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(y - y \cdot \log y\right) - z\\ \end{array} \]

Alternative 9: 58.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{-20}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;x \leq -6.5 \cdot 10^{-87}:\\ \;\;\;\;\log y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.4e-20) (- x z) (if (<= x -6.5e-87) (* (log y) -0.5) (- x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.4e-20) {
		tmp = x - z;
	} else if (x <= -6.5e-87) {
		tmp = log(y) * -0.5;
	} else {
		tmp = x - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-1.4d-20)) then
        tmp = x - z
    else if (x <= (-6.5d-87)) then
        tmp = log(y) * (-0.5d0)
    else
        tmp = x - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.4e-20) {
		tmp = x - z;
	} else if (x <= -6.5e-87) {
		tmp = Math.log(y) * -0.5;
	} else {
		tmp = x - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.4e-20:
		tmp = x - z
	elif x <= -6.5e-87:
		tmp = math.log(y) * -0.5
	else:
		tmp = x - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.4e-20)
		tmp = Float64(x - z);
	elseif (x <= -6.5e-87)
		tmp = Float64(log(y) * -0.5);
	else
		tmp = Float64(x - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.4e-20)
		tmp = x - z;
	elseif (x <= -6.5e-87)
		tmp = log(y) * -0.5;
	else
		tmp = x - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.4e-20], N[(x - z), $MachinePrecision], If[LessEqual[x, -6.5e-87], N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision], N[(x - z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.4000000000000001e-20 or -6.5000000000000003e-87 < x

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. sub-neg 99.8%

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

      3. associate-+l+ 99.8%

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

      4. associate-+l+ 99.8%

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

      5. sub-neg 99.8%

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

      6. neg-sub0 99.8%

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

      7. associate-+l- 99.8%

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

      8. neg-sub0 99.8%

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

      9. neg-mul-1 99.8%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in z around inf 55.9%

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

      1. neg-mul-1 55.9%

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

   6. Simplified 55.9%

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

   if -1.4000000000000001e-20 < x < -6.5000000000000003e-87

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-+l+ 100.0%

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

      4. associate-+l+ 100.0%

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

      5. sub-neg 100.0%

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

      6. neg-sub0 100.0%

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

      7. associate-+l- 100.0%

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

      8. neg-sub0 100.0%

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

      9. neg-mul-1 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in z around 0 77.1%

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

      1. +-commutative 77.1%

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

      2. mul-1-neg 77.1%

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

      3. +-commutative 77.1%

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

      4. *-commutative 77.1%

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

      5. sub-neg 77.1%

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

   6. Simplified 77.1%

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

   7. Taylor expanded in y around 0 60.6%

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

   8. Taylor expanded in x around 0 60.6%

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

      1. *-commutative 60.6%

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

   10. Simplified 60.6%

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

4. Final simplification 56.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{-20}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;x \leq -6.5 \cdot 10^{-87}:\\ \;\;\;\;\log y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \]

Alternative 10: 58.6% accurate, 37.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. sub-neg 99.8%

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

   2. sub-neg 99.8%

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

   3. associate-+l+ 99.8%

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

   4. associate-+l+ 99.8%

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

   5. sub-neg 99.8%

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

   6. neg-sub0 99.8%

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

   7. associate-+l- 99.8%

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

   8. neg-sub0 99.8%

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

   9. neg-mul-1 99.8%

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

3. Simplified 99.9%

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

4. Taylor expanded in z around inf 53.9%

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

   1. neg-mul-1 53.9%

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

6. Simplified 53.9%

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

7. Final simplification 53.9%

   \[\leadsto x - z \]

Alternative 11: 29.9% 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. sub-neg 99.8%

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

   2. sub-neg 99.8%

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

   3. associate-+l+ 99.8%

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

   4. associate-+l+ 99.8%

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

   5. sub-neg 99.8%

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

   6. neg-sub0 99.8%

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

   7. associate-+l- 99.8%

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

   8. neg-sub0 99.8%

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

   9. neg-mul-1 99.8%

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

3. Simplified 99.9%

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

4. Taylor expanded in x around inf 28.8%

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

5. Final simplification 28.8%

   \[\leadsto x \]

Developer target: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023214 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (- (- (+ y x) z) (* (+ y 0.5) (log y)))

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