Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Percentage Accurate: 99.8% → 99.9%

Time: 10.6s

Alternatives: 11

Speedup: 0.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate--l+ 99.8%

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

   2. sub-neg 99.8%

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

   3. associate-+l+ 99.8%

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

   4. associate-+r- 99.8%

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

   5. *-commutative 99.8%

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

   6. distribute-rgt-neg-in 99.8%

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

   7. fma-define 99.8%

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

   8. +-commutative 99.8%

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

   9. distribute-neg-in 99.8%

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

   10. unsub-neg 99.8%

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

   11. metadata-eval 99.8%

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

3. Simplified 99.8%

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

Alternative 2: 71.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.35 \cdot 10^{-62}:\\ \;\;\;\;\left(x + y\right) - z\\ \mathbf{elif}\;y \leq 7 \cdot 10^{-29}:\\ \;\;\;\;\log y \cdot \left(-0.5\right) - z\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{+110}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y - y \cdot \log y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.35e-62)
   (- (+ x y) z)
   (if (<= y 7e-29)
     (- (* (log y) (- 0.5)) z)
     (if (<= y 1.05e+110) (- x z) (- y (* y (log y)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.35e-62) {
		tmp = (x + y) - z;
	} else if (y <= 7e-29) {
		tmp = (log(y) * -0.5) - z;
	} else if (y <= 1.05e+110) {
		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 <= 1.35d-62) then
        tmp = (x + y) - z
    else if (y <= 7d-29) then
        tmp = (log(y) * -0.5d0) - z
    else if (y <= 1.05d+110) 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 <= 1.35e-62) {
		tmp = (x + y) - z;
	} else if (y <= 7e-29) {
		tmp = (Math.log(y) * -0.5) - z;
	} else if (y <= 1.05e+110) {
		tmp = x - z;
	} else {
		tmp = y - (y * Math.log(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1.35e-62:
		tmp = (x + y) - z
	elif y <= 7e-29:
		tmp = (math.log(y) * -0.5) - z
	elif y <= 1.05e+110:
		tmp = x - z
	else:
		tmp = y - (y * math.log(y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.35e-62)
		tmp = Float64(Float64(x + y) - z);
	elseif (y <= 7e-29)
		tmp = Float64(Float64(log(y) * Float64(-0.5)) - z);
	elseif (y <= 1.05e+110)
		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 <= 1.35e-62)
		tmp = (x + y) - z;
	elseif (y <= 7e-29)
		tmp = (log(y) * -0.5) - z;
	elseif (y <= 1.05e+110)
		tmp = x - z;
	else
		tmp = y - (y * log(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.35e-62], N[(N[(x + y), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[y, 7e-29], N[(N[(N[Log[y], $MachinePrecision] * (-0.5)), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[y, 1.05e+110], N[(x - z), $MachinePrecision], N[(y - N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < 1.3500000000000001e-62

   1. Initial program 100.0%

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

      1. add-cube-cbrt 98.9%

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

      2. pow3 98.8%

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

      3. sub-neg 98.8%

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

      4. *-commutative 98.8%

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

      5. distribute-rgt-neg-in 98.8%

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

      6. +-commutative 98.8%

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

      7. distribute-neg-in 98.8%

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

      8. metadata-eval 98.8%

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

      9. sub-neg 98.8%

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

   4. Applied egg-rr 98.8%

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

   5. Taylor expanded in x around inf 74.6%

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

   if 1.3500000000000001e-62 < y < 6.9999999999999995e-29

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 73.3%

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

   4. Taylor expanded in y around 0 73.3%

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

      1. mul-1-neg 73.3%

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

      2. *-commutative 73.3%

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

   6. Simplified 73.3%

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

   if 6.9999999999999995e-29 < y < 1.05000000000000007e110

   1. Initial program 99.8%

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

      1. associate--l+ 99.8%

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

      2. sub-neg 99.8%

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

      3. associate-+l+ 99.8%

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

      4. associate-+r- 99.8%

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

      5. *-commutative 99.8%

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

      6. distribute-rgt-neg-in 99.8%

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

      7. fma-define 99.9%

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

      8. +-commutative 99.9%

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

      9. distribute-neg-in 99.9%

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

      10. unsub-neg 99.9%

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

      11. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in z around inf 67.1%

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

      1. neg-mul-1 67.1%

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

   7. Simplified 67.1%

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

   if 1.05000000000000007e110 < y

   1. Initial program 99.4%

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

   3. Taylor expanded in x around 0 97.5%

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

   4. Taylor expanded in z around 0 85.3%

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

   5. Taylor expanded in y around inf 85.3%

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

      1. mul-1-neg 85.3%

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

      2. log-rec 85.3%

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

      3. distribute-rgt-neg-in 85.3%

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

      4. remove-double-neg 85.3%

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

   7. Simplified 85.3%

      \[\leadsto y - \color{blue}{y \cdot \log y} \]
3. Recombined 4 regimes into one program.

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.35 \cdot 10^{-62}:\\ \;\;\;\;\left(x + y\right) - z\\ \mathbf{elif}\;y \leq 7 \cdot 10^{-29}:\\ \;\;\;\;\log y \cdot \left(-0.5\right) - z\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{+110}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y - y \cdot \log y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -30000000 \lor \neg \left(z \leq 4.5 \cdot 10^{+32}\right):\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(1 - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -30000000.0) (not (<= z 4.5e+32)))
   (- x z)
   (+ x (* y (- 1.0 (log y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -30000000.0) || !(z <= 4.5e+32)) {
		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 <= (-30000000.0d0)) .or. (.not. (z <= 4.5d+32))) 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 <= -30000000.0) || !(z <= 4.5e+32)) {
		tmp = x - z;
	} else {
		tmp = x + (y * (1.0 - Math.log(y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -30000000.0) or not (z <= 4.5e+32):
		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 <= -30000000.0) || !(z <= 4.5e+32))
		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 <= -30000000.0) || ~((z <= 4.5e+32)))
		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, -30000000.0], N[Not[LessEqual[z, 4.5e+32]], $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 < -3e7 or 4.5000000000000003e32 < z

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-+l+ 99.9%

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

      4. associate-+r- 99.9%

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

      5. *-commutative 99.9%

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

      6. distribute-rgt-neg-in 99.9%

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

      7. fma-define 99.9%

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

      8. +-commutative 99.9%

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

      9. distribute-neg-in 99.9%

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

      10. unsub-neg 99.9%

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

      11. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in z around inf 82.2%

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

      1. neg-mul-1 82.2%

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

   7. Simplified 82.2%

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

   if -3e7 < z < 4.5000000000000003e32

   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%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in z around 0 98.2%

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

      1. sub-neg 98.2%

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

      2. distribute-rgt-in 98.3%

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

      3. distribute-neg-in 98.3%

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

      4. distribute-lft-neg-in 98.3%

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

      5. metadata-eval 98.3%

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

      6. distribute-lft-neg-in 98.3%

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

      7. distribute-rgt-in 98.2%

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

      8. sub-neg 98.2%

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

      9. associate-+r+ 98.2%

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

      10. +-commutative 98.2%

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

      11. fma-define 98.3%

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

   7. Simplified 98.3%

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

   8. Taylor expanded in y around inf 75.0%

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

      1. log-rec 76.2%

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

      2. sub-neg 76.2%

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

   10. Simplified 75.0%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -30000000 \lor \neg \left(z \leq 4.5 \cdot 10^{+32}\right):\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(1 - \log y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(1 - \log y\right)\\ \mathbf{if}\;y \leq 3.1 \cdot 10^{+36}:\\ \;\;\;\;x - \left(z + \log y \cdot 0.5\right)\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{+65}:\\ \;\;\;\;x + t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_0 - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (- 1.0 (log y)))))
   (if (<= y 3.1e+36)
     (- x (+ z (* (log y) 0.5)))
     (if (<= y 3.2e+65) (+ x t_0) (- t_0 z)))))

C

double code(double x, double y, double z) {
	double t_0 = y * (1.0 - log(y));
	double tmp;
	if (y <= 3.1e+36) {
		tmp = x - (z + (log(y) * 0.5));
	} else if (y <= 3.2e+65) {
		tmp = x + t_0;
	} else {
		tmp = t_0 - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y * (1.0d0 - log(y))
    if (y <= 3.1d+36) then
        tmp = x - (z + (log(y) * 0.5d0))
    else if (y <= 3.2d+65) then
        tmp = x + t_0
    else
        tmp = t_0 - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y * (1.0 - Math.log(y));
	double tmp;
	if (y <= 3.1e+36) {
		tmp = x - (z + (Math.log(y) * 0.5));
	} else if (y <= 3.2e+65) {
		tmp = x + t_0;
	} else {
		tmp = t_0 - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * (1.0 - math.log(y))
	tmp = 0
	if y <= 3.1e+36:
		tmp = x - (z + (math.log(y) * 0.5))
	elif y <= 3.2e+65:
		tmp = x + t_0
	else:
		tmp = t_0 - z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * Float64(1.0 - log(y)))
	tmp = 0.0
	if (y <= 3.1e+36)
		tmp = Float64(x - Float64(z + Float64(log(y) * 0.5)));
	elseif (y <= 3.2e+65)
		tmp = Float64(x + t_0);
	else
		tmp = Float64(t_0 - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * (1.0 - log(y));
	tmp = 0.0;
	if (y <= 3.1e+36)
		tmp = x - (z + (log(y) * 0.5));
	elseif (y <= 3.2e+65)
		tmp = x + t_0;
	else
		tmp = t_0 - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 3.1e+36], N[(x - N[(z + N[(N[Log[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.2e+65], N[(x + t$95$0), $MachinePrecision], N[(t$95$0 - z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 3.0999999999999999e36

   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. associate--l+ 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 98.1%

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

      1. *-commutative 98.1%

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

   7. Simplified 98.1%

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

   if 3.0999999999999999e36 < y < 3.20000000000000007e65

   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%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in z around 0 99.6%

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

      1. sub-neg 99.6%

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

      2. distribute-rgt-in 99.6%

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

      3. distribute-neg-in 99.6%

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

      4. distribute-lft-neg-in 99.6%

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

      5. metadata-eval 99.6%

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

      6. distribute-lft-neg-in 99.6%

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

      7. distribute-rgt-in 99.6%

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

      8. sub-neg 99.6%

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

      9. associate-+r+ 99.6%

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

      10. +-commutative 99.6%

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

      11. fma-define 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in y around inf 99.8%

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

      1. log-rec 99.8%

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

      2. sub-neg 99.8%

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

   10. Simplified 99.8%

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

   if 3.20000000000000007e65 < y

   1. Initial program 99.5%

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

      1. add-cube-cbrt 98.5%

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

      2. pow3 98.5%

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

      3. sub-neg 98.5%

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

      4. *-commutative 98.5%

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

      5. distribute-rgt-neg-in 98.5%

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

      6. +-commutative 98.5%

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

      7. distribute-neg-in 98.5%

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

      8. metadata-eval 98.5%

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

      9. sub-neg 98.5%

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

   4. Applied egg-rr 98.5%

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

   5. Taylor expanded in y around inf 91.4%

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

      1. log-rec 91.4%

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

      2. sub-neg 91.4%

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

   7. Simplified 91.4%

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

Alternative 5: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.6 \cdot 10^{-8}:\\ \;\;\;\;x - \left(z + \log y \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(y \cdot \left(1 - \log y\right) - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 3.6e-8)
   (- x (+ z (* (log y) 0.5)))
   (+ x (- (* y (- 1.0 (log y))) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 3.6e-8) {
		tmp = x - (z + (log(y) * 0.5));
	} else {
		tmp = x + ((y * (1.0 - log(y))) - z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 3.6d-8) then
        tmp = x - (z + (log(y) * 0.5d0))
    else
        tmp = x + ((y * (1.0d0 - log(y))) - z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 3.6e-8) {
		tmp = x - (z + (Math.log(y) * 0.5));
	} else {
		tmp = x + ((y * (1.0 - Math.log(y))) - z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 3.6e-8:
		tmp = x - (z + (math.log(y) * 0.5))
	else:
		tmp = x + ((y * (1.0 - math.log(y))) - z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 3.6e-8)
		tmp = Float64(x - Float64(z + Float64(log(y) * 0.5)));
	else
		tmp = Float64(x + Float64(Float64(y * Float64(1.0 - log(y))) - z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 3.6e-8)
		tmp = x - (z + (log(y) * 0.5));
	else
		tmp = x + ((y * (1.0 - log(y))) - z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 3.6e-8], N[(x - N[(z + N[(N[Log[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.59999999999999981e-8

   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. associate--l+ 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 99.9%

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

      1. *-commutative 99.9%

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

   7. Simplified 99.9%

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

   if 3.59999999999999981e-8 < y

   1. Initial program 99.6%

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

      1. associate--l+ 99.6%

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

      2. sub-neg 99.6%

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

      3. associate-+l+ 99.6%

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

      4. associate-+r- 99.6%

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

      5. *-commutative 99.6%

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

      6. distribute-rgt-neg-in 99.6%

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

      7. fma-define 99.7%

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

      8. +-commutative 99.7%

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

      9. distribute-neg-in 99.7%

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

      10. unsub-neg 99.7%

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

      11. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around inf 99.3%

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

      1. log-rec 99.3%

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

      2. sub-neg 99.3%

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

   7. Simplified 99.3%

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

Alternative 6: 89.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.1 \cdot 10^{+37}:\\ \;\;\;\;x - \left(z + \log y \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(1 - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.1e+37) (- x (+ z (* (log y) 0.5))) (+ x (* y (- 1.0 (log y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.1e+37) {
		tmp = x - (z + (log(y) * 0.5));
	} 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 (y <= 1.1d+37) then
        tmp = x - (z + (log(y) * 0.5d0))
    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 (y <= 1.1e+37) {
		tmp = x - (z + (Math.log(y) * 0.5));
	} else {
		tmp = x + (y * (1.0 - Math.log(y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1.1e+37:
		tmp = x - (z + (math.log(y) * 0.5))
	else:
		tmp = x + (y * (1.0 - math.log(y)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.1e+37)
		tmp = Float64(x - Float64(z + Float64(log(y) * 0.5)));
	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 (y <= 1.1e+37)
		tmp = x - (z + (log(y) * 0.5));
	else
		tmp = x + (y * (1.0 - log(y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.1e+37], N[(x - N[(z + N[(N[Log[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.1e37

   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. associate--l+ 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 98.1%

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

      1. *-commutative 98.1%

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

   7. Simplified 98.1%

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

   if 1.1e37 < 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%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in z around 0 84.4%

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

      1. sub-neg 84.4%

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

      2. distribute-rgt-in 84.4%

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

      3. distribute-neg-in 84.4%

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

      4. distribute-lft-neg-in 84.4%

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

      5. metadata-eval 84.4%

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

      6. distribute-lft-neg-in 84.4%

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

      7. distribute-rgt-in 84.4%

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

      8. sub-neg 84.4%

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

      9. associate-+r+ 84.4%

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

      10. +-commutative 84.4%

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

      11. fma-define 84.5%

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

   7. Simplified 84.5%

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

   8. Taylor expanded in y around inf 84.5%

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

      1. log-rec 99.7%

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

      2. sub-neg 99.7%

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

   10. Simplified 84.5%

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

Alternative 7: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate--l+ 99.8%

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

3. Simplified 99.8%

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

5. Final simplification 99.8%

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

Alternative 8: 72.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6.2 \cdot 10^{+109}:\\ \;\;\;\;\left(x + y\right) - z\\ \mathbf{else}:\\ \;\;\;\;y - y \cdot \log y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 6.2e+109) (- (+ x y) z) (- y (* y (log y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 6.2e+109) {
		tmp = (x + y) - 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 <= 6.2d+109) then
        tmp = (x + y) - 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 <= 6.2e+109) {
		tmp = (x + y) - z;
	} else {
		tmp = y - (y * Math.log(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 6.2e+109:
		tmp = (x + y) - z
	else:
		tmp = y - (y * math.log(y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 6.2e+109)
		tmp = Float64(Float64(x + y) - 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 <= 6.2e+109)
		tmp = (x + y) - z;
	else
		tmp = y - (y * log(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 6.2e+109], N[(N[(x + y), $MachinePrecision] - z), $MachinePrecision], N[(y - N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.19999999999999985e109

   1. Initial program 99.9%

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

      1. add-cube-cbrt 98.8%

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

      2. pow3 98.7%

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

      3. sub-neg 98.7%

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

      4. *-commutative 98.7%

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

      5. distribute-rgt-neg-in 98.7%

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

      6. +-commutative 98.7%

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

      7. distribute-neg-in 98.7%

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

      8. metadata-eval 98.7%

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

      9. sub-neg 98.7%

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

   4. Applied egg-rr 98.7%

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

   5. Taylor expanded in x around inf 69.1%

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

   if 6.19999999999999985e109 < y

   1. Initial program 99.4%

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

   3. Taylor expanded in x around 0 97.5%

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

   4. Taylor expanded in z around 0 85.3%

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

   5. Taylor expanded in y around inf 85.3%

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

      1. mul-1-neg 85.3%

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

      2. log-rec 85.3%

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

      3. distribute-rgt-neg-in 85.3%

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

      4. remove-double-neg 85.3%

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

   7. Simplified 85.3%

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

Alternative 9: 49.2% accurate, 9.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -21000000 \lor \neg \left(z \leq 4.7 \cdot 10^{+17}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -21000000.0) (not (<= z 4.7e+17))) (- z) x))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -21000000.0) || !(z <= 4.7e+17)) {
		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 <= (-21000000.0d0)) .or. (.not. (z <= 4.7d+17))) 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 <= -21000000.0) || !(z <= 4.7e+17)) {
		tmp = -z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -21000000.0) or not (z <= 4.7e+17):
		tmp = -z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -21000000.0) || !(z <= 4.7e+17))
		tmp = Float64(-z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -21000000.0) || ~((z <= 4.7e+17)))
		tmp = -z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -21000000.0], N[Not[LessEqual[z, 4.7e+17]], $MachinePrecision]], (-z), x]

Derivation

1. Split input into 2 regimes

2. if z < -2.1e7 or 4.7e17 < z

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in y around inf 83.1%

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

      1. *-commutative 83.1%

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

      2. log-rec 83.1%

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

      3. distribute-lft-neg-in 83.1%

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

      4. distribute-rgt-neg-in 83.1%

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

   7. Simplified 83.1%

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

   8. Taylor expanded in y around 0 65.0%

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

      1. neg-mul-1 65.0%

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

   10. Simplified 65.0%

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

   if -2.1e7 < z < 4.7e17

   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%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around inf 34.5%

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

4. Final simplification 47.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -21000000 \lor \neg \left(z \leq 4.7 \cdot 10^{+17}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 58.7% 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. associate--l+ 99.8%

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

   2. sub-neg 99.8%

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

   3. associate-+l+ 99.8%

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

   4. associate-+r- 99.8%

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

   5. *-commutative 99.8%

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

   6. distribute-rgt-neg-in 99.8%

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

   7. fma-define 99.8%

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

   8. +-commutative 99.8%

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

   9. distribute-neg-in 99.8%

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

   10. unsub-neg 99.8%

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

   11. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in z around inf 53.9%

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

   1. neg-mul-1 53.9%

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

7. Simplified 53.9%

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

8. Final simplification 53.9%

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

Alternative 11: 30.8% accurate, 111.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 99.8%

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

   1. associate--l+ 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in x around inf 27.1%

   \[\leadsto \color{blue}{x} \]
6. Add Preprocessing

Developer target: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024102 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2"
  :precision binary64

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

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