Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Percentage Accurate: 99.8% → 99.9%

Time: 11.6s

Alternatives: 9

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 99.9%

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

   7. +-commutative 99.9%

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

   8. distribute-neg-in 99.9%

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

   9. unsub-neg 99.9%

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

   10. metadata-eval 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 2: 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 3: 77.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.15 \cdot 10^{+62}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(1 - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.15e+62) (- x z) (+ x (* y (- 1.0 (log y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.15e+62) {
		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 (y <= 1.15d+62) 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 (y <= 1.15e+62) {
		tmp = x - z;
	} else {
		tmp = x + (y * (1.0 - Math.log(y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1.15e+62:
		tmp = x - z
	else:
		tmp = x + (y * (1.0 - math.log(y)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.15e+62)
		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 (y <= 1.15e+62)
		tmp = x - z;
	else
		tmp = x + (y * (1.0 - log(y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.15e+62], 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 y < 1.14999999999999992e62

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 81.6%

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

   if 1.14999999999999992e62 < 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 83.4%

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

      1. log-rec 83.4%

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

      2. sub-neg 83.4%

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

   6. Simplified 83.4%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.15 \cdot 10^{+62}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(1 - \log y\right)\\ \end{array} \]

Alternative 4: 77.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.9 \cdot 10^{+61}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 4.9e+61) (- x z) (- (* y (- 1.0 (log y))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 4.9e+61) {
		tmp = x - z;
	} else {
		tmp = (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 <= 4.9d+61) then
        tmp = x - z
    else
        tmp = (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 <= 4.9e+61) {
		tmp = x - z;
	} else {
		tmp = (y * (1.0 - Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 4.9e+61:
		tmp = x - z
	else:
		tmp = (y * (1.0 - math.log(y))) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 4.9e+61)
		tmp = Float64(x - z);
	else
		tmp = 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 <= 4.9e+61)
		tmp = x - z;
	else
		tmp = (y * (1.0 - log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 4.9e+61], N[(x - z), $MachinePrecision], N[(N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.90000000000000025e61

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 81.4%

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

   if 4.90000000000000025e61 < 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. flip-+ 47.9%

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

      2. associate-*l/ 47.8%

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

      3. fma-neg 47.8%

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

      4. metadata-eval 47.8%

         \[\leadsto \left(\left(x - \frac{\mathsf{fma}\left(y, y, -\color{blue}{0.25}\right) \cdot \log y}{y - 0.5}\right) + y\right) - z \]

      5. metadata-eval 47.8%

         \[\leadsto \left(\left(x - \frac{\mathsf{fma}\left(y, y, \color{blue}{-0.25}\right) \cdot \log y}{y - 0.5}\right) + y\right) - z \]

      6. sub-neg 47.8%

         \[\leadsto \left(\left(x - \frac{\mathsf{fma}\left(y, y, -0.25\right) \cdot \log y}{\color{blue}{y + \left(-0.5\right)}}\right) + y\right) - z \]

      7. metadata-eval 47.8%

         \[\leadsto \left(\left(x - \frac{\mathsf{fma}\left(y, y, -0.25\right) \cdot \log y}{y + \color{blue}{-0.5}}\right) + y\right) - z \]

   3. Applied egg-rr 47.8%

      \[\leadsto \left(\left(x - \color{blue}{\frac{\mathsf{fma}\left(y, y, -0.25\right) \cdot \log y}{y + -0.5}}\right) + y\right) - z \]

   4. Taylor expanded in x around 0 41.3%

      \[\leadsto \color{blue}{\left(y - \frac{\log y \cdot \left({y}^{2} - 0.25\right)}{y - 0.5}\right)} - z \]
   5. Step-by-step derivation

      1. sub-neg 41.3%

         \[\leadsto \left(y - \frac{\log y \cdot \left({y}^{2} - 0.25\right)}{\color{blue}{y + \left(-0.5\right)}}\right) - z \]

      2. metadata-eval 41.3%

         \[\leadsto \left(y - \frac{\log y \cdot \left({y}^{2} - 0.25\right)}{y + \color{blue}{-0.5}}\right) - z \]

      3. *-commutative 41.3%

         \[\leadsto \left(y - \frac{\color{blue}{\left({y}^{2} - 0.25\right) \cdot \log y}}{y + -0.5}\right) - z \]

      4. unpow2 41.3%

         \[\leadsto \left(y - \frac{\left(\color{blue}{y \cdot y} - 0.25\right) \cdot \log y}{y + -0.5}\right) - z \]

      5. fma-neg 41.3%

         \[\leadsto \left(y - \frac{\color{blue}{\mathsf{fma}\left(y, y, -0.25\right)} \cdot \log y}{y + -0.5}\right) - z \]

      6. metadata-eval 41.3%

         \[\leadsto \left(y - \frac{\mathsf{fma}\left(y, y, \color{blue}{-0.25}\right) \cdot \log y}{y + -0.5}\right) - z \]

      7. associate-*r/ 41.4%

         \[\leadsto \left(y - \color{blue}{\mathsf{fma}\left(y, y, -0.25\right) \cdot \frac{\log y}{y + -0.5}}\right) - z \]

      8. +-commutative 41.4%

         \[\leadsto \left(y - \mathsf{fma}\left(y, y, -0.25\right) \cdot \frac{\log y}{\color{blue}{-0.5 + y}}\right) - z \]

   6. Simplified 41.4%

      \[\leadsto \color{blue}{\left(y - \mathsf{fma}\left(y, y, -0.25\right) \cdot \frac{\log y}{-0.5 + y}\right)} - z \]

   7. Taylor expanded in y around inf 90.9%

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

      1. mul-1-neg 90.9%

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

      2. log-rec 90.9%

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

      3. remove-double-neg 90.9%

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

   9. Simplified 90.9%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.9 \cdot 10^{+61}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \end{array} \]

Alternative 5: 90.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6.8 \cdot 10^{+61}:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 6.8e+61) (- (- x (* (log y) 0.5)) z) (- (* y (- 1.0 (log y))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 6.8e+61) {
		tmp = (x - (log(y) * 0.5)) - z;
	} else {
		tmp = (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 <= 6.8d+61) then
        tmp = (x - (log(y) * 0.5d0)) - z
    else
        tmp = (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 <= 6.8e+61) {
		tmp = (x - (Math.log(y) * 0.5)) - z;
	} else {
		tmp = (y * (1.0 - Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 6.8e+61:
		tmp = (x - (math.log(y) * 0.5)) - z
	else:
		tmp = (y * (1.0 - math.log(y))) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 6.8e+61)
		tmp = Float64(Float64(x - Float64(log(y) * 0.5)) - z);
	else
		tmp = 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 <= 6.8e+61)
		tmp = (x - (log(y) * 0.5)) - z;
	else
		tmp = (y * (1.0 - log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 6.8e+61], N[(N[(x - N[(N[Log[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.80000000000000051e61

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 96.4%

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

   if 6.80000000000000051e61 < 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. flip-+ 47.9%

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

      2. associate-*l/ 47.8%

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

      3. fma-neg 47.8%

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

      4. metadata-eval 47.8%

         \[\leadsto \left(\left(x - \frac{\mathsf{fma}\left(y, y, -\color{blue}{0.25}\right) \cdot \log y}{y - 0.5}\right) + y\right) - z \]

      5. metadata-eval 47.8%

         \[\leadsto \left(\left(x - \frac{\mathsf{fma}\left(y, y, \color{blue}{-0.25}\right) \cdot \log y}{y - 0.5}\right) + y\right) - z \]

      6. sub-neg 47.8%

         \[\leadsto \left(\left(x - \frac{\mathsf{fma}\left(y, y, -0.25\right) \cdot \log y}{\color{blue}{y + \left(-0.5\right)}}\right) + y\right) - z \]

      7. metadata-eval 47.8%

         \[\leadsto \left(\left(x - \frac{\mathsf{fma}\left(y, y, -0.25\right) \cdot \log y}{y + \color{blue}{-0.5}}\right) + y\right) - z \]

   3. Applied egg-rr 47.8%

      \[\leadsto \left(\left(x - \color{blue}{\frac{\mathsf{fma}\left(y, y, -0.25\right) \cdot \log y}{y + -0.5}}\right) + y\right) - z \]

   4. Taylor expanded in x around 0 41.3%

      \[\leadsto \color{blue}{\left(y - \frac{\log y \cdot \left({y}^{2} - 0.25\right)}{y - 0.5}\right)} - z \]
   5. Step-by-step derivation

      1. sub-neg 41.3%

         \[\leadsto \left(y - \frac{\log y \cdot \left({y}^{2} - 0.25\right)}{\color{blue}{y + \left(-0.5\right)}}\right) - z \]

      2. metadata-eval 41.3%

         \[\leadsto \left(y - \frac{\log y \cdot \left({y}^{2} - 0.25\right)}{y + \color{blue}{-0.5}}\right) - z \]

      3. *-commutative 41.3%

         \[\leadsto \left(y - \frac{\color{blue}{\left({y}^{2} - 0.25\right) \cdot \log y}}{y + -0.5}\right) - z \]

      4. unpow2 41.3%

         \[\leadsto \left(y - \frac{\left(\color{blue}{y \cdot y} - 0.25\right) \cdot \log y}{y + -0.5}\right) - z \]

      5. fma-neg 41.3%

         \[\leadsto \left(y - \frac{\color{blue}{\mathsf{fma}\left(y, y, -0.25\right)} \cdot \log y}{y + -0.5}\right) - z \]

      6. metadata-eval 41.3%

         \[\leadsto \left(y - \frac{\mathsf{fma}\left(y, y, \color{blue}{-0.25}\right) \cdot \log y}{y + -0.5}\right) - z \]

      7. associate-*r/ 41.4%

         \[\leadsto \left(y - \color{blue}{\mathsf{fma}\left(y, y, -0.25\right) \cdot \frac{\log y}{y + -0.5}}\right) - z \]

      8. +-commutative 41.4%

         \[\leadsto \left(y - \mathsf{fma}\left(y, y, -0.25\right) \cdot \frac{\log y}{\color{blue}{-0.5 + y}}\right) - z \]

   6. Simplified 41.4%

      \[\leadsto \color{blue}{\left(y - \mathsf{fma}\left(y, y, -0.25\right) \cdot \frac{\log y}{-0.5 + y}\right)} - z \]

   7. Taylor expanded in y around inf 90.9%

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

      1. mul-1-neg 90.9%

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

      2. log-rec 90.9%

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

      3. remove-double-neg 90.9%

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

   9. Simplified 90.9%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.8 \cdot 10^{+61}:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \end{array} \]

Alternative 6: 72.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6.8 \cdot 10^{+115}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 6.8e+115) (- x z) (* y (- 1.0 (log y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 6.8e+115) {
		tmp = x - z;
	} else {
		tmp = 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 <= 6.8d+115) then
        tmp = x - z
    else
        tmp = 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 <= 6.8e+115) {
		tmp = x - z;
	} else {
		tmp = y * (1.0 - Math.log(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 6.8e+115:
		tmp = x - z
	else:
		tmp = y * (1.0 - math.log(y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 6.8e+115)
		tmp = Float64(x - z);
	else
		tmp = Float64(y * Float64(1.0 - log(y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 6.8e+115)
		tmp = x - z;
	else
		tmp = y * (1.0 - log(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 6.8e+115], N[(x - z), $MachinePrecision], N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.8000000000000001e115

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf 78.4%

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

   if 6.8000000000000001e115 < 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 x around 0 90.9%

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

   3. Taylor expanded in z around 0 79.1%

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

   4. Taylor expanded in y around inf 79.1%

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

      1. mul-1-neg 79.1%

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

      2. distribute-rgt-neg-in 79.1%

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

      3. log-rec 79.1%

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

      4. remove-double-neg 79.1%

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

   6. Simplified 79.1%

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

      1. *-commutative 79.1%

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

      2. cancel-sign-sub-inv 79.1%

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

      3. *-un-lft-identity 79.1%

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

      4. distribute-rgt-in 79.3%

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

      5. sub-neg 79.3%

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

      6. *-commutative 79.3%

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

   8. Applied egg-rr 79.3%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.8 \cdot 10^{+115}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \end{array} \]

Alternative 7: 57.2% accurate, 37.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in x around inf 58.9%

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

3. Final simplification 58.9%

   \[\leadsto x - z \]

Alternative 8: 29.6% accurate, 55.5× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return -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 = -z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := (-z)

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 x around 0 72.7%

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

3. Taylor expanded in z around inf 31.8%

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

   1. neg-mul-1 31.8%

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

5. Simplified 31.8%

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

6. Final simplification 31.8%

   \[\leadsto -z \]

Alternative 9: 2.4% accurate, 111.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := y

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 x around 0 72.7%

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

3. Taylor expanded in z around 0 42.7%

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

   1. add-sqr-sqrt 32.7%

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

   2. pow2 32.7%

      \[\leadsto y - \color{blue}{{\left(\sqrt{\log y \cdot \left(0.5 + y\right)}\right)}^{2}} \]

   3. +-commutative 32.7%

      \[\leadsto y - {\left(\sqrt{\log y \cdot \color{blue}{\left(y + 0.5\right)}}\right)}^{2} \]

5. Applied egg-rr 32.7%

   \[\leadsto y - \color{blue}{{\left(\sqrt{\log y \cdot \left(y + 0.5\right)}\right)}^{2}} \]

6. Taylor expanded in y around inf 2.3%

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

7. Final simplification 2.3%

   \[\leadsto y \]

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 2023308 
(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))