Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Percentage Accurate: 99.8% → 99.9%

Time: 8.9s

Alternatives: 8

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate--l+ 99.8%

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

   2. sub-neg 99.8%

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

   3. associate-+l+ 99.8%

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

   4. associate-+r- 99.8%

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

   5. *-commutative 99.8%

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

   6. distribute-rgt-neg-in 99.8%

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

   7. fma-def 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. Final simplification 99.9%

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

Alternative 2: 77.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{+49} \lor \neg \left(x \leq 5.2 \cdot 10^{+37}\right):\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.45e+49) (not (<= x 5.2e+37)))
   (- x z)
   (- (* y (- 1.0 (log y))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.45e+49) || !(x <= 5.2e+37)) {
		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 ((x <= (-1.45d+49)) .or. (.not. (x <= 5.2d+37))) 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 ((x <= -1.45e+49) || !(x <= 5.2e+37)) {
		tmp = x - z;
	} else {
		tmp = (y * (1.0 - Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.45e+49) or not (x <= 5.2e+37):
		tmp = x - z
	else:
		tmp = (y * (1.0 - math.log(y))) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.45e+49) || !(x <= 5.2e+37))
		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 ((x <= -1.45e+49) || ~((x <= 5.2e+37)))
		tmp = x - z;
	else
		tmp = (y * (1.0 - log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.45e+49], N[Not[LessEqual[x, 5.2e+37]], $MachinePrecision]], 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 x < -1.45e49 or 5.1999999999999998e37 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 90.1%

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

   if -1.45e49 < x < 5.1999999999999998e37

   1. Initial program 99.7%

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

      1. add-cube-cbrt 98.8%

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

      2. pow3 98.8%

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

      3. *-commutative 98.8%

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

   4. Applied egg-rr 98.8%

      \[\leadsto \left(\left(x - \color{blue}{{\left(\sqrt[3]{\log y \cdot \left(y + 0.5\right)}\right)}^{3}}\right) + y\right) - z \]
   5. Step-by-step derivation

      1. rem-cube-cbrt 99.7%

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

      2. flip-+ 77.5%

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

      3. fma-neg 77.5%

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

      4. metadata-eval 77.5%

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

      5. metadata-eval 77.5%

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

      6. sub-neg 77.5%

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

      7. metadata-eval 77.5%

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

      8. +-commutative 77.5%

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

      9. clear-num 77.5%

         \[\leadsto \left(\left(x - \log y \cdot \color{blue}{\frac{1}{\frac{-0.5 + y}{\mathsf{fma}\left(y, y, -0.25\right)}}}\right) + y\right) - z \]

      10. un-div-inv 77.5%

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

      11. clear-num 77.4%

          \[\leadsto \left(\left(x - \frac{\log y}{\color{blue}{\frac{1}{\frac{\mathsf{fma}\left(y, y, -0.25\right)}{-0.5 + y}}}}\right) + y\right) - z \]

      12. metadata-eval 77.4%

          \[\leadsto \left(\left(x - \frac{\log y}{\frac{1}{\frac{\mathsf{fma}\left(y, y, \color{blue}{-0.25}\right)}{-0.5 + y}}}\right) + y\right) - z \]

      13. metadata-eval 77.4%

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

      14. fma-neg 77.4%

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

      15. +-commutative 77.4%

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

      16. metadata-eval 77.4%

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

      17. sub-neg 77.4%

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

      18. flip-+ 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in y around inf 78.7%

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

      1. neg-mul-1 78.7%

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

      2. log-rec 78.8%

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

      3. remove-double-neg 78.8%

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

   9. Simplified 78.8%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{+49} \lor \neg \left(x \leq 5.2 \cdot 10^{+37}\right):\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.65 \cdot 10^{-15}:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + \left(y \cdot \left(1 - \log y\right) - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2.65e-15)
   (- (- x (* (log y) 0.5)) z)
   (+ x (- (* y (- 1.0 (log y))) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.65e-15) {
		tmp = (x - (log(y) * 0.5)) - z;
	} else {
		tmp = x + ((y * (1.0 - log(y))) - z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 2.65d-15) then
        tmp = (x - (log(y) * 0.5d0)) - z
    else
        tmp = x + ((y * (1.0d0 - log(y))) - z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.65e-15) {
		tmp = (x - (Math.log(y) * 0.5)) - z;
	} else {
		tmp = x + ((y * (1.0 - Math.log(y))) - z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 2.65e-15:
		tmp = (x - (math.log(y) * 0.5)) - z
	else:
		tmp = x + ((y * (1.0 - math.log(y))) - z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2.65e-15)
		tmp = Float64(Float64(x - Float64(log(y) * 0.5)) - z);
	else
		tmp = Float64(x + Float64(Float64(y * Float64(1.0 - log(y))) - z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 2.65e-15)
		tmp = (x - (log(y) * 0.5)) - z;
	else
		tmp = x + ((y * (1.0 - log(y))) - z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2.65e-15], N[(N[(x - N[(N[Log[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(x + N[(N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.6500000000000001e-15

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 100.0%

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

   if 2.6500000000000001e-15 < y

   1. Initial program 99.7%

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

      1. 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. associate-+r- 99.7%

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

      5. *-commutative 99.7%

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

      6. distribute-rgt-neg-in 99.7%

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

      7. fma-def 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 y around inf 99.5%

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

      1. log-rec 99.6%

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

      2. sub-neg 99.6%

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

   7. Simplified 99.6%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.65 \cdot 10^{-15}:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + \left(y \cdot \left(1 - \log y\right) - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Final simplification 99.8%

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

Alternative 5: 70.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -13500 \lor \neg \left(x \leq 0.00345\right):\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -13500.0) (not (<= x 0.00345))) (- x z) (- (* (log y) -0.5) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -13500.0) || !(x <= 0.00345)) {
		tmp = x - z;
	} else {
		tmp = (log(y) * -0.5) - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-13500.0d0)) .or. (.not. (x <= 0.00345d0))) then
        tmp = x - z
    else
        tmp = (log(y) * (-0.5d0)) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -13500.0) || !(x <= 0.00345)) {
		tmp = x - z;
	} else {
		tmp = (Math.log(y) * -0.5) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -13500.0) or not (x <= 0.00345):
		tmp = x - z
	else:
		tmp = (math.log(y) * -0.5) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -13500.0) || !(x <= 0.00345))
		tmp = Float64(x - z);
	else
		tmp = Float64(Float64(log(y) * -0.5) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -13500.0) || ~((x <= 0.00345)))
		tmp = x - z;
	else
		tmp = (log(y) * -0.5) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -13500.0], N[Not[LessEqual[x, 0.00345]], $MachinePrecision]], N[(x - z), $MachinePrecision], N[(N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -13500 or 0.0034499999999999999 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 85.2%

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

   if -13500 < x < 0.0034499999999999999

   1. Initial program 99.7%

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

      1. add-cube-cbrt 98.8%

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

      2. pow3 98.8%

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

      3. *-commutative 98.8%

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

   4. Applied egg-rr 98.8%

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

   5. Taylor expanded in x around 0 99.1%

      \[\leadsto \color{blue}{\left(y - {1}^{0.3333333333333333} \cdot \left(\log y \cdot \left(0.5 + y\right)\right)\right)} - z \]
   6. Step-by-step derivation

      1. pow-base-1 99.1%

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

      2. associate-*r* 99.1%

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

      3. +-commutative 99.1%

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

      4. *-lft-identity 99.1%

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

   7. Simplified 99.1%

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

   8. Taylor expanded in y around 0 61.1%

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

      1. *-commutative 61.1%

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

   10. Simplified 61.1%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -13500 \lor \neg \left(x \leq 0.00345\right):\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 88.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.1 \cdot 10^{+107}:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.1e+107) (- (- 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 <= 1.1e+107) {
		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 <= 1.1d+107) 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 <= 1.1e+107) {
		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 <= 1.1e+107:
		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 <= 1.1e+107)
		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 <= 1.1e+107)
		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, 1.1e+107], 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 < 1.1e107

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 93.4%

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

   if 1.1e107 < y

   1. Initial program 99.6%

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

      1. add-cube-cbrt 98.6%

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

      2. pow3 98.6%

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

      3. *-commutative 98.6%

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

   4. Applied egg-rr 98.6%

      \[\leadsto \left(\left(x - \color{blue}{{\left(\sqrt[3]{\log y \cdot \left(y + 0.5\right)}\right)}^{3}}\right) + y\right) - z \]
   5. Step-by-step derivation

      1. rem-cube-cbrt 99.6%

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

      2. flip-+ 37.7%

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

      3. fma-neg 37.7%

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

      4. metadata-eval 37.7%

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

      5. metadata-eval 37.7%

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

      6. sub-neg 37.7%

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

      7. metadata-eval 37.7%

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

      8. +-commutative 37.7%

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

      9. clear-num 37.7%

         \[\leadsto \left(\left(x - \log y \cdot \color{blue}{\frac{1}{\frac{-0.5 + y}{\mathsf{fma}\left(y, y, -0.25\right)}}}\right) + y\right) - z \]

      10. un-div-inv 37.7%

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

      11. clear-num 37.7%

          \[\leadsto \left(\left(x - \frac{\log y}{\color{blue}{\frac{1}{\frac{\mathsf{fma}\left(y, y, -0.25\right)}{-0.5 + y}}}}\right) + y\right) - z \]

      12. metadata-eval 37.7%

          \[\leadsto \left(\left(x - \frac{\log y}{\frac{1}{\frac{\mathsf{fma}\left(y, y, \color{blue}{-0.25}\right)}{-0.5 + y}}}\right) + y\right) - z \]

      13. metadata-eval 37.7%

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

      14. fma-neg 37.7%

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

      15. +-commutative 37.7%

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

      16. metadata-eval 37.7%

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

      17. sub-neg 37.7%

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

      18. flip-+ 99.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in y around inf 86.2%

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

      1. neg-mul-1 86.2%

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

      2. log-rec 86.2%

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

      3. remove-double-neg 86.2%

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

   9. Simplified 86.2%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.1 \cdot 10^{+107}:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 58.2% 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. Add Preprocessing

3. Taylor expanded in x around inf 62.6%

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

4. Final simplification 62.6%

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

Alternative 8: 30.5% 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. Add Preprocessing
3. Step-by-step derivation

   1. add-cube-cbrt 99.3%

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

   2. pow3 99.3%

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

   3. *-commutative 99.3%

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

4. Applied egg-rr 99.3%

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

5. Taylor expanded in x around 0 66.6%

   \[\leadsto \color{blue}{\left(y - {1}^{0.3333333333333333} \cdot \left(\log y \cdot \left(0.5 + y\right)\right)\right)} - z \]
6. Step-by-step derivation

   1. pow-base-1 66.6%

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

   2. associate-*r* 66.6%

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

   3. +-commutative 66.6%

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

   4. *-lft-identity 66.6%

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

7. Simplified 66.6%

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

8. Taylor expanded in z around inf 29.9%

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

   1. neg-mul-1 29.9%

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

10. Simplified 29.9%

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

11. Final simplification 29.9%

    \[\leadsto -z \]
12. 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 2024011 
(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))