Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Percentage Accurate: 99.8% → 99.9%

Time: 9.0s

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate--l+ 99.8%

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

   2. sub-neg 99.8%

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

   3. associate-+l+ 99.8%

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

   4. *-commutative 99.8%

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

   5. distribute-rgt-neg-in 99.8%

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

   6. fma-def 99.8%

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

   7. +-commutative 99.8%

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

   8. distribute-neg-in 99.8%

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

   9. unsub-neg 99.8%

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

   10. metadata-eval 99.8%

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

3. Simplified 99.8%

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

4. Final simplification 99.8%

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

Alternative 2: 68.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.3 \cdot 10^{-35}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y \leq 107000000:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{elif}\;y \leq 9.8 \cdot 10^{+86} \lor \neg \left(y \leq 5.2 \cdot 10^{+186}\right) \land y \leq 6.2 \cdot 10^{+208}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y - y \cdot \log y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 3.3e-35)
   (- x z)
   (if (<= y 107000000.0)
     (- (* (log y) -0.5) z)
     (if (or (<= y 9.8e+86) (and (not (<= y 5.2e+186)) (<= y 6.2e+208)))
       (- x z)
       (- y (* y (log y)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 3.3e-35) {
		tmp = x - z;
	} else if (y <= 107000000.0) {
		tmp = (log(y) * -0.5) - z;
	} else if ((y <= 9.8e+86) || (!(y <= 5.2e+186) && (y <= 6.2e+208))) {
		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) :: tmp
    if (y <= 3.3d-35) then
        tmp = x - z
    else if (y <= 107000000.0d0) then
        tmp = (log(y) * (-0.5d0)) - z
    else if ((y <= 9.8d+86) .or. (.not. (y <= 5.2d+186)) .and. (y <= 6.2d+208)) 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 tmp;
	if (y <= 3.3e-35) {
		tmp = x - z;
	} else if (y <= 107000000.0) {
		tmp = (Math.log(y) * -0.5) - z;
	} else if ((y <= 9.8e+86) || (!(y <= 5.2e+186) && (y <= 6.2e+208))) {
		tmp = x - z;
	} else {
		tmp = y - (y * Math.log(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 3.3e-35:
		tmp = x - z
	elif y <= 107000000.0:
		tmp = (math.log(y) * -0.5) - z
	elif (y <= 9.8e+86) or (not (y <= 5.2e+186) and (y <= 6.2e+208)):
		tmp = x - z
	else:
		tmp = y - (y * math.log(y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 3.3e-35)
		tmp = Float64(x - z);
	elseif (y <= 107000000.0)
		tmp = Float64(Float64(log(y) * -0.5) - z);
	elseif ((y <= 9.8e+86) || (!(y <= 5.2e+186) && (y <= 6.2e+208)))
		tmp = Float64(x - z);
	else
		tmp = Float64(y - Float64(y * log(y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 3.3e-35)
		tmp = x - z;
	elseif (y <= 107000000.0)
		tmp = (log(y) * -0.5) - z;
	elseif ((y <= 9.8e+86) || (~((y <= 5.2e+186)) && (y <= 6.2e+208)))
		tmp = x - z;
	else
		tmp = y - (y * log(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 3.3e-35], N[(x - z), $MachinePrecision], If[LessEqual[y, 107000000.0], N[(N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision] - z), $MachinePrecision], If[Or[LessEqual[y, 9.8e+86], And[N[Not[LessEqual[y, 5.2e+186]], $MachinePrecision], LessEqual[y, 6.2e+208]]], N[(x - z), $MachinePrecision], N[(y - N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 3.3e-35 or 1.07e8 < y < 9.7999999999999999e86 or 5.2000000000000001e186 < y < 6.19999999999999961e208

   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 0 100.0%

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

   3. Taylor expanded in x around inf 79.7%

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

   if 3.3e-35 < y < 1.07e8

   1. Initial program 99.5%

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

   2. Taylor expanded in y around 0 99.7%

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

   3. Taylor expanded in x around 0 94.9%

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

   4. Taylor expanded in y around 0 80.9%

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

   if 9.7999999999999999e86 < y < 5.2000000000000001e186 or 6.19999999999999961e208 < y

   1. Initial program 99.5%

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

   2. Taylor expanded in y around inf 86.5%

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

      1. log-rec 86.5%

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

   4. Simplified 86.5%

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

   5. Taylor expanded in z around 0 86.5%

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

      1. neg-mul-1 86.5%

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

      2. associate-+r+ 86.5%

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

      3. sub-neg 86.5%

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

      4. neg-mul-1 86.5%

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

      5. sub-neg 86.5%

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

   7. Simplified 86.5%

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

   8. Taylor expanded in z around 0 79.0%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.3 \cdot 10^{-35}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y \leq 107000000:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{elif}\;y \leq 9.8 \cdot 10^{+86} \lor \neg \left(y \leq 5.2 \cdot 10^{+186}\right) \land y \leq 6.2 \cdot 10^{+208}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y - y \cdot \log y\\ \end{array} \]

Alternative 3: 88.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{+87}:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{+187} \lor \neg \left(y \leq 1.36 \cdot 10^{+208}\right):\\ \;\;\;\;x + y \cdot \left(1 - \log y\right)\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 5e+87)
   (- (- x (* (log y) 0.5)) z)
   (if (or (<= y 3.5e+187) (not (<= y 1.36e+208)))
     (+ x (* y (- 1.0 (log y))))
     (- x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 5e+87) {
		tmp = (x - (log(y) * 0.5)) - z;
	} else if ((y <= 3.5e+187) || !(y <= 1.36e+208)) {
		tmp = x + (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 <= 5d+87) then
        tmp = (x - (log(y) * 0.5d0)) - z
    else if ((y <= 3.5d+187) .or. (.not. (y <= 1.36d+208))) then
        tmp = x + (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 <= 5e+87) {
		tmp = (x - (Math.log(y) * 0.5)) - z;
	} else if ((y <= 3.5e+187) || !(y <= 1.36e+208)) {
		tmp = x + (y * (1.0 - Math.log(y)));
	} else {
		tmp = x - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 5e+87:
		tmp = (x - (math.log(y) * 0.5)) - z
	elif (y <= 3.5e+187) or not (y <= 1.36e+208):
		tmp = x + (y * (1.0 - math.log(y)))
	else:
		tmp = x - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 5e+87)
		tmp = Float64(Float64(x - Float64(log(y) * 0.5)) - z);
	elseif ((y <= 3.5e+187) || !(y <= 1.36e+208))
		tmp = Float64(x + 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 <= 5e+87)
		tmp = (x - (log(y) * 0.5)) - z;
	elseif ((y <= 3.5e+187) || ~((y <= 1.36e+208)))
		tmp = x + (y * (1.0 - log(y)));
	else
		tmp = x - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 5e+87], N[(N[(x - N[(N[Log[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], If[Or[LessEqual[y, 3.5e+187], N[Not[LessEqual[y, 1.36e+208]], $MachinePrecision]], N[(x + N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 4.9999999999999998e87

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 93.3%

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

   if 4.9999999999999998e87 < y < 3.4999999999999998e187 or 1.3599999999999999e208 < y

   1. Initial program 99.5%

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

      1. associate--l+ 99.5%

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

      2. sub-neg 99.5%

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

      3. associate-+l+ 99.5%

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

      4. *-commutative 99.5%

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

      5. distribute-rgt-neg-in 99.5%

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

      6. fma-def 99.7%

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

      7. +-commutative 99.7%

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

      8. distribute-neg-in 99.7%

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

      9. unsub-neg 99.7%

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

      10. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around inf 93.2%

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

      1. log-rec 93.2%

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

      2. sub-neg 93.2%

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

   6. Simplified 93.2%

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

   if 3.4999999999999998e187 < y < 1.3599999999999999e208

   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 0 100.0%

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

   3. Taylor expanded in x around inf 100.0%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{+87}:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{+187} \lor \neg \left(y \leq 1.36 \cdot 10^{+208}\right):\\ \;\;\;\;x + y \cdot \left(1 - \log y\right)\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \]

Alternative 4: 69.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.76 \cdot 10^{+87} \lor \neg \left(y \leq 3.5 \cdot 10^{+187}\right) \land y \leq 6.2 \cdot 10^{+208}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y - y \cdot \log y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y 1.76e+87) (and (not (<= y 3.5e+187)) (<= y 6.2e+208)))
   (- x z)
   (- y (* y (log y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= 1.76e+87) || (!(y <= 3.5e+187) && (y <= 6.2e+208))) {
		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) :: tmp
    if ((y <= 1.76d+87) .or. (.not. (y <= 3.5d+187)) .and. (y <= 6.2d+208)) 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 tmp;
	if ((y <= 1.76e+87) || (!(y <= 3.5e+187) && (y <= 6.2e+208))) {
		tmp = x - z;
	} else {
		tmp = y - (y * Math.log(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= 1.76e+87) or (not (y <= 3.5e+187) and (y <= 6.2e+208)):
		tmp = x - z
	else:
		tmp = y - (y * math.log(y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= 1.76e+87) || (!(y <= 3.5e+187) && (y <= 6.2e+208)))
		tmp = Float64(x - z);
	else
		tmp = Float64(y - Float64(y * log(y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= 1.76e+87) || (~((y <= 3.5e+187)) && (y <= 6.2e+208)))
		tmp = x - z;
	else
		tmp = y - (y * log(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, 1.76e+87], And[N[Not[LessEqual[y, 3.5e+187]], $MachinePrecision], LessEqual[y, 6.2e+208]]], N[(x - z), $MachinePrecision], N[(y - N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.76000000000000003e87 or 3.4999999999999998e187 < y < 6.19999999999999961e208

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 99.9%

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

   3. Taylor expanded in x around inf 76.8%

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

   if 1.76000000000000003e87 < y < 3.4999999999999998e187 or 6.19999999999999961e208 < y

   1. Initial program 99.5%

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

   2. Taylor expanded in y around inf 86.5%

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

      1. log-rec 86.5%

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

   4. Simplified 86.5%

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

   5. Taylor expanded in z around 0 86.5%

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

      1. neg-mul-1 86.5%

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

      2. associate-+r+ 86.5%

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

      3. sub-neg 86.5%

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

      4. neg-mul-1 86.5%

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

      5. sub-neg 86.5%

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

   7. Simplified 86.5%

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

   8. Taylor expanded in z around 0 79.0%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.76 \cdot 10^{+87} \lor \neg \left(y \leq 3.5 \cdot 10^{+187}\right) \land y \leq 6.2 \cdot 10^{+208}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y - y \cdot \log y\\ \end{array} \]

Alternative 5: 76.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.2 \cdot 10^{+42} \lor \neg \left(z \leq 3.8 \cdot 10^{+107}\right):\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(1 - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -3.2e+42) (not (<= z 3.8e+107)))
   (- x z)
   (+ x (* y (- 1.0 (log y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -3.2e+42) || !(z <= 3.8e+107)) {
		tmp = x - z;
	} else {
		tmp = x + (y * (1.0 - log(y)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-3.2d+42)) .or. (.not. (z <= 3.8d+107))) then
        tmp = x - z
    else
        tmp = x + (y * (1.0d0 - log(y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -3.2e+42) || !(z <= 3.8e+107)) {
		tmp = x - z;
	} else {
		tmp = x + (y * (1.0 - Math.log(y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -3.2e+42) or not (z <= 3.8e+107):
		tmp = x - z
	else:
		tmp = x + (y * (1.0 - math.log(y)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -3.2e+42) || !(z <= 3.8e+107))
		tmp = Float64(x - z);
	else
		tmp = Float64(x + Float64(y * Float64(1.0 - log(y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -3.2e+42) || ~((z <= 3.8e+107)))
		tmp = x - z;
	else
		tmp = x + (y * (1.0 - log(y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -3.2e+42], N[Not[LessEqual[z, 3.8e+107]], $MachinePrecision]], N[(x - z), $MachinePrecision], N[(x + N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.20000000000000002e42 or 3.7999999999999998e107 < z

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 99.9%

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

   3. Taylor expanded in x around inf 84.2%

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

   if -3.20000000000000002e42 < z < 3.7999999999999998e107

   1. Initial program 99.7%

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

      1. associate--l+ 99.7%

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

      2. sub-neg 99.7%

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

      3. associate-+l+ 99.7%

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

      4. *-commutative 99.7%

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

      5. distribute-rgt-neg-in 99.7%

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

      6. fma-def 99.8%

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

      7. +-commutative 99.8%

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

      8. distribute-neg-in 99.8%

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

      9. unsub-neg 99.8%

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

      10. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around inf 77.9%

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

      1. log-rec 77.9%

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

      2. sub-neg 77.9%

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

   6. Simplified 77.9%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.2 \cdot 10^{+42} \lor \neg \left(z \leq 3.8 \cdot 10^{+107}\right):\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(1 - \log y\right)\\ \end{array} \]

Alternative 6: 89.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{+88}:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{elif}\;y \leq 6.7 \cdot 10^{+187}:\\ \;\;\;\;x + y \cdot \left(1 - \log y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y - z\right) - y \cdot \log y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2e+88)
   (- (- x (* (log y) 0.5)) z)
   (if (<= y 6.7e+187) (+ x (* y (- 1.0 (log y)))) (- (- y z) (* y (log y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2e+88) {
		tmp = (x - (log(y) * 0.5)) - z;
	} else if (y <= 6.7e+187) {
		tmp = x + (y * (1.0 - log(y)));
	} else {
		tmp = (y - z) - (y * log(y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 2d+88) then
        tmp = (x - (log(y) * 0.5d0)) - z
    else if (y <= 6.7d+187) then
        tmp = x + (y * (1.0d0 - log(y)))
    else
        tmp = (y - z) - (y * log(y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 2e+88) {
		tmp = (x - (Math.log(y) * 0.5)) - z;
	} else if (y <= 6.7e+187) {
		tmp = x + (y * (1.0 - Math.log(y)));
	} else {
		tmp = (y - z) - (y * Math.log(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 2e+88:
		tmp = (x - (math.log(y) * 0.5)) - z
	elif y <= 6.7e+187:
		tmp = x + (y * (1.0 - math.log(y)))
	else:
		tmp = (y - z) - (y * math.log(y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2e+88)
		tmp = Float64(Float64(x - Float64(log(y) * 0.5)) - z);
	elseif (y <= 6.7e+187)
		tmp = Float64(x + Float64(y * Float64(1.0 - log(y))));
	else
		tmp = Float64(Float64(y - z) - Float64(y * log(y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 2e+88)
		tmp = (x - (log(y) * 0.5)) - z;
	elseif (y <= 6.7e+187)
		tmp = x + (y * (1.0 - log(y)));
	else
		tmp = (y - z) - (y * log(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2e+88], N[(N[(x - N[(N[Log[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[y, 6.7e+187], N[(x + N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y - z), $MachinePrecision] - N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.99999999999999992e88

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 93.3%

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

   if 1.99999999999999992e88 < y < 6.7000000000000001e187

   1. Initial program 99.6%

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

      1. associate--l+ 99.6%

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

      2. sub-neg 99.6%

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

      3. associate-+l+ 99.7%

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

      4. *-commutative 99.7%

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

      5. distribute-rgt-neg-in 99.7%

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

      6. fma-def 99.8%

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

      7. +-commutative 99.8%

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

      8. distribute-neg-in 99.8%

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

      9. unsub-neg 99.8%

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

      10. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around inf 94.8%

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

      1. log-rec 94.8%

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

      2. sub-neg 94.8%

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

   6. Simplified 94.8%

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

   if 6.7000000000000001e187 < y

   1. Initial program 99.5%

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

   2. Taylor expanded in y around inf 91.3%

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

      1. log-rec 91.3%

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

   4. Simplified 91.3%

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

   5. Taylor expanded in z around 0 91.3%

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

      1. neg-mul-1 91.3%

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

      2. associate-+r+ 91.3%

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

      3. sub-neg 91.3%

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

      4. neg-mul-1 91.3%

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

      5. sub-neg 91.3%

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

   7. Simplified 91.3%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{+88}:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{elif}\;y \leq 6.7 \cdot 10^{+187}:\\ \;\;\;\;x + y \cdot \left(1 - \log y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y - z\right) - y \cdot \log y\\ \end{array} \]

Alternative 7: 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 8: 70.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4100000000 \lor \neg \left(z \leq 8.6 \cdot 10^{-11}\right):\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;x - \log y \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -4100000000.0) (not (<= z 8.6e-11)))
   (- x z)
   (- x (* (log y) 0.5))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -4100000000.0) || !(z <= 8.6e-11)) {
		tmp = x - z;
	} else {
		tmp = x - (log(y) * 0.5);
	}
	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 <= (-4100000000.0d0)) .or. (.not. (z <= 8.6d-11))) then
        tmp = x - z
    else
        tmp = x - (log(y) * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -4100000000.0) || !(z <= 8.6e-11)) {
		tmp = x - z;
	} else {
		tmp = x - (Math.log(y) * 0.5);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -4100000000.0) or not (z <= 8.6e-11):
		tmp = x - z
	else:
		tmp = x - (math.log(y) * 0.5)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -4100000000.0) || !(z <= 8.6e-11))
		tmp = Float64(x - z);
	else
		tmp = Float64(x - Float64(log(y) * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -4100000000.0) || ~((z <= 8.6e-11)))
		tmp = x - z;
	else
		tmp = x - (log(y) * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -4100000000.0], N[Not[LessEqual[z, 8.6e-11]], $MachinePrecision]], N[(x - z), $MachinePrecision], N[(x - N[(N[Log[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.1e9 or 8.60000000000000003e-11 < z

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 99.9%

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

   3. Taylor expanded in x around inf 77.8%

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

   if -4.1e9 < z < 8.60000000000000003e-11

   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 0 58.6%

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

   3. Taylor expanded in z around 0 58.2%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4100000000 \lor \neg \left(z \leq 8.6 \cdot 10^{-11}\right):\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;x - \log y \cdot 0.5\\ \end{array} \]

Alternative 9: 48.3% accurate, 18.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.2 \cdot 10^{+44} \lor \neg \left(z \leq 1.2 \cdot 10^{+50}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -6.2e+44) (not (<= z 1.2e+50))) (- z) x))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -6.2e+44) || !(z <= 1.2e+50)) {
		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 <= (-6.2d+44)) .or. (.not. (z <= 1.2d+50))) 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 <= -6.2e+44) || !(z <= 1.2e+50)) {
		tmp = -z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -6.2e+44) or not (z <= 1.2e+50):
		tmp = -z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -6.2e+44) || !(z <= 1.2e+50))
		tmp = Float64(-z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -6.2e+44) || ~((z <= 1.2e+50)))
		tmp = -z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -6.2e+44], N[Not[LessEqual[z, 1.2e+50]], $MachinePrecision]], (-z), x]

Derivation

1. Split input into 2 regimes

2. if z < -6.19999999999999991e44 or 1.2000000000000001e50 < z

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 99.9%

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

   3. Taylor expanded in z around inf 67.7%

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

      1. neg-mul-1 67.7%

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

   5. Simplified 67.7%

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

   if -6.19999999999999991e44 < z < 1.2000000000000001e50

   1. Initial program 99.7%

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

      1. associate--l+ 99.7%

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

      2. sub-neg 99.7%

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

      3. associate-+l+ 99.7%

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

      4. *-commutative 99.7%

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

      5. distribute-rgt-neg-in 99.7%

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

      6. fma-def 99.8%

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

      7. +-commutative 99.8%

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

      8. distribute-neg-in 99.8%

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

      9. unsub-neg 99.8%

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

      10. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around inf 37.7%

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

4. Final simplification 50.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.2 \cdot 10^{+44} \lor \neg \left(z \leq 1.2 \cdot 10^{+50}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 10: 58.0% accurate, 37.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in y around 0 99.8%

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

3. Taylor expanded in x around inf 57.4%

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

4. Final simplification 57.4%

   \[\leadsto x - z \]

Alternative 11: 30.7% 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. Step-by-step derivation

   1. associate--l+ 99.8%

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

   2. sub-neg 99.8%

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

   3. associate-+l+ 99.8%

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

   4. *-commutative 99.8%

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

   5. distribute-rgt-neg-in 99.8%

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

   6. fma-def 99.8%

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

   7. +-commutative 99.8%

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

   8. distribute-neg-in 99.8%

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

   9. unsub-neg 99.8%

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

   10. metadata-eval 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in x around inf 28.1%

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

5. Final simplification 28.1%

   \[\leadsto x \]

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 2023310 
(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))