Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Percentage Accurate: 99.8% → 99.9%

Time: 11.9s

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

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

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 99.9%

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

   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: 74.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.15 \cdot 10^{-256}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y \leq 9 \cdot 10^{-154}:\\ \;\;\;\;x + \log y \cdot -0.5\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{+113}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \log \left(\frac{e}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2.15e-256)
   (- x z)
   (if (<= y 9e-154)
     (+ x (* (log y) -0.5))
     (if (<= y 7.8e+113) (- x z) (+ x (* y (log (/ E y))))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.15e-256) {
		tmp = x - z;
	} else if (y <= 9e-154) {
		tmp = x + (log(y) * -0.5);
	} else if (y <= 7.8e+113) {
		tmp = x - z;
	} else {
		tmp = x + (y * log((((double) M_E) / y)));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.15e-256) {
		tmp = x - z;
	} else if (y <= 9e-154) {
		tmp = x + (Math.log(y) * -0.5);
	} else if (y <= 7.8e+113) {
		tmp = x - z;
	} else {
		tmp = x + (y * Math.log((Math.E / y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 2.15e-256:
		tmp = x - z
	elif y <= 9e-154:
		tmp = x + (math.log(y) * -0.5)
	elif y <= 7.8e+113:
		tmp = x - z
	else:
		tmp = x + (y * math.log((math.e / y)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2.15e-256)
		tmp = Float64(x - z);
	elseif (y <= 9e-154)
		tmp = Float64(x + Float64(log(y) * -0.5));
	elseif (y <= 7.8e+113)
		tmp = Float64(x - z);
	else
		tmp = Float64(x + Float64(y * log(Float64(exp(1) / y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 2.15e-256)
		tmp = x - z;
	elseif (y <= 9e-154)
		tmp = x + (log(y) * -0.5);
	elseif (y <= 7.8e+113)
		tmp = x - z;
	else
		tmp = x + (y * log((2.71828182845904523536 / y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2.15e-256], N[(x - z), $MachinePrecision], If[LessEqual[y, 9e-154], N[(x + N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7.8e+113], N[(x - z), $MachinePrecision], N[(x + N[(y * N[Log[N[(E / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.1500000000000001e-256 or 8.9999999999999994e-154 < y < 7.80000000000000039e113

   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 91.2%

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

   4. Taylor expanded in x around inf 76.1%

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

   if 2.1500000000000001e-256 < y < 8.9999999999999994e-154

   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%

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

      2. sub-neg 100.0%

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

      3. associate-+l+ 100.0%

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

      4. associate-+r- 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-rgt-neg-in 100.0%

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

      7. fma-define 100.0%

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

      8. +-commutative 100.0%

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

      9. distribute-neg-in 100.0%

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

      10. unsub-neg 100.0%

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

      11. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in z around 0 77.2%

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

      1. associate-+r+ 77.2%

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

      2. associate-*r* 77.2%

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

      3. neg-mul-1 77.2%

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

      4. log-rec 77.2%

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

      5. distribute-lft-in 77.2%

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

      6. log-rec 77.2%

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

      7. distribute-lft-neg-in 77.2%

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

      8. distribute-rgt-neg-in 77.2%

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

      9. metadata-eval 77.2%

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

      10. log-rec 77.2%

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

      11. distribute-lft-neg-in 77.2%

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

      12. distribute-rgt-neg-in 77.2%

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

      13. distribute-lft-in 77.2%

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

      14. sub-neg 77.2%

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

      15. associate-+r+ 77.2%

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

      16. +-commutative 77.2%

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

      17. fma-define 77.2%

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

   7. Simplified 77.2%

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

   8. Taylor expanded in y around 0 77.2%

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

   if 7.80000000000000039e113 < y

   1. Initial program 99.4%

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

      1. associate--l+ 99.4%

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

      2. sub-neg 99.4%

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

      3. associate-+l+ 99.4%

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

      4. associate-+r- 99.4%

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

      5. *-commutative 99.4%

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

      6. distribute-rgt-neg-in 99.4%

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

      7. fma-define 99.6%

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

      8. +-commutative 99.6%

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

      9. distribute-neg-in 99.6%

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

      10. unsub-neg 99.6%

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

      11. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around inf 99.6%

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

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

      1. add-log-exp 99.6%

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

      2. exp-diff 99.6%

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

      3. add-exp-log 99.7%

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

   9. Applied egg-rr 99.7%

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

       1. *-un-lft-identity 99.7%

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

       2. exp-1-e 99.7%

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

   11. Applied egg-rr 99.7%

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

       1. *-lft-identity 99.7%

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

   13. Simplified 99.7%

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

   14. Taylor expanded in z around 0 85.7%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.15 \cdot 10^{-256}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y \leq 9 \cdot 10^{-154}:\\ \;\;\;\;x + \log y \cdot -0.5\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{+113}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \log \left(\frac{e}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.22 \cdot 10^{-256}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{-153}:\\ \;\;\;\;x + \log y \cdot -0.5\\ \mathbf{elif}\;y \leq 2.15 \cdot 10^{+64}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.22e-256)
   (- x z)
   (if (<= y 2.5e-153)
     (+ x (* (log y) -0.5))
     (if (<= y 2.15e+64) (- x z) (- (* y (- 1.0 (log y))) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.22e-256) {
		tmp = x - z;
	} else if (y <= 2.5e-153) {
		tmp = x + (log(y) * -0.5);
	} else if (y <= 2.15e+64) {
		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 <= 1.22d-256) then
        tmp = x - z
    else if (y <= 2.5d-153) then
        tmp = x + (log(y) * (-0.5d0))
    else if (y <= 2.15d+64) 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 <= 1.22e-256) {
		tmp = x - z;
	} else if (y <= 2.5e-153) {
		tmp = x + (Math.log(y) * -0.5);
	} else if (y <= 2.15e+64) {
		tmp = x - z;
	} else {
		tmp = (y * (1.0 - Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1.22e-256:
		tmp = x - z
	elif y <= 2.5e-153:
		tmp = x + (math.log(y) * -0.5)
	elif y <= 2.15e+64:
		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 <= 1.22e-256)
		tmp = Float64(x - z);
	elseif (y <= 2.5e-153)
		tmp = Float64(x + Float64(log(y) * -0.5));
	elseif (y <= 2.15e+64)
		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 <= 1.22e-256)
		tmp = x - z;
	elseif (y <= 2.5e-153)
		tmp = x + (log(y) * -0.5);
	elseif (y <= 2.15e+64)
		tmp = x - z;
	else
		tmp = (y * (1.0 - log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.22e-256], N[(x - z), $MachinePrecision], If[LessEqual[y, 2.5e-153], N[(x + N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.15e+64], N[(x - z), $MachinePrecision], N[(N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.2199999999999999e-256 or 2.50000000000000016e-153 < y < 2.1499999999999999e64

   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 95.0%

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

   4. Taylor expanded in x around inf 77.3%

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

   if 1.2199999999999999e-256 < y < 2.50000000000000016e-153

   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%

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

      2. sub-neg 100.0%

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

      3. associate-+l+ 100.0%

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

      4. associate-+r- 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-rgt-neg-in 100.0%

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

      7. fma-define 100.0%

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

      8. +-commutative 100.0%

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

      9. distribute-neg-in 100.0%

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

      10. unsub-neg 100.0%

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

      11. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in z around 0 77.2%

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

      1. associate-+r+ 77.2%

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

      2. associate-*r* 77.2%

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

      3. neg-mul-1 77.2%

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

      4. log-rec 77.2%

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

      5. distribute-lft-in 77.2%

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

      6. log-rec 77.2%

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

      7. distribute-lft-neg-in 77.2%

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

      8. distribute-rgt-neg-in 77.2%

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

      9. metadata-eval 77.2%

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

      10. log-rec 77.2%

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

      11. distribute-lft-neg-in 77.2%

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

      12. distribute-rgt-neg-in 77.2%

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

      13. distribute-lft-in 77.2%

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

      14. sub-neg 77.2%

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

      15. associate-+r+ 77.2%

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

      16. +-commutative 77.2%

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

      17. fma-define 77.2%

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

   7. Simplified 77.2%

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

   8. Taylor expanded in y around 0 77.2%

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

   if 2.1499999999999999e64 < 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. Taylor expanded in y around inf 89.2%

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

      1. *-commutative 89.2%

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

      2. log-rec 89.2%

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

      3. distribute-lft-neg-in 89.2%

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

      4. distribute-rgt-neg-in 89.2%

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

   5. Simplified 89.2%

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

      1. +-commutative 89.2%

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

      2. *-un-lft-identity 89.2%

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

      3. distribute-rgt-neg-out 89.2%

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

      4. distribute-lft-neg-in 89.2%

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

      5. distribute-rgt-in 89.3%

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

      6. sub-neg 89.3%

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

      7. *-commutative 89.3%

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

   7. Applied egg-rr 89.3%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.22 \cdot 10^{-256}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{-153}:\\ \;\;\;\;x + \log y \cdot -0.5\\ \mathbf{elif}\;y \leq 2.15 \cdot 10^{+64}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 69.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -11500000000000 \lor \neg \left(z \leq 145\right):\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;x + \log y \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -11500000000000.0) (not (<= z 145.0)))
   (- x z)
   (+ x (* (log y) -0.5))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -11500000000000.0) || !(z <= 145.0)) {
		tmp = x - z;
	} else {
		tmp = x + (log(y) * -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-11500000000000.0d0)) .or. (.not. (z <= 145.0d0))) then
        tmp = x - z
    else
        tmp = x + (log(y) * (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -11500000000000.0) || !(z <= 145.0)) {
		tmp = x - z;
	} else {
		tmp = x + (Math.log(y) * -0.5);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -11500000000000.0) or not (z <= 145.0):
		tmp = x - z
	else:
		tmp = x + (math.log(y) * -0.5)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -11500000000000.0) || !(z <= 145.0))
		tmp = Float64(x - z);
	else
		tmp = Float64(x + Float64(log(y) * -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -11500000000000.0) || ~((z <= 145.0)))
		tmp = x - z;
	else
		tmp = x + (log(y) * -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -11500000000000.0], N[Not[LessEqual[z, 145.0]], $MachinePrecision]], N[(x - z), $MachinePrecision], N[(x + N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.15e13 or 145 < z

   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 82.1%

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

   4. Taylor expanded in x around inf 81.6%

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

   if -1.15e13 < z < 145

   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-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 0 98.5%

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

      1. associate-+r+ 98.5%

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

      2. associate-*r* 98.5%

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

      3. neg-mul-1 98.5%

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

      4. log-rec 98.5%

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

      5. distribute-lft-in 98.5%

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

      6. log-rec 98.5%

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

      7. distribute-lft-neg-in 98.5%

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

      8. distribute-rgt-neg-in 98.5%

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

      9. metadata-eval 98.5%

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

      10. log-rec 98.5%

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

      11. distribute-lft-neg-in 98.5%

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

      12. distribute-rgt-neg-in 98.5%

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

      13. distribute-lft-in 98.5%

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

      14. sub-neg 98.5%

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

      15. associate-+r+ 98.5%

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

      16. +-commutative 98.5%

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

      17. fma-define 98.6%

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

   7. Simplified 98.6%

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

   8. Taylor expanded in y around 0 61.8%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -11500000000000 \lor \neg \left(z \leq 145\right):\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;x + \log y \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.0007:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + \left(y \cdot \log \left(\frac{e}{y}\right) - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 0.0007)
   (- (- x (* (log y) 0.5)) z)
   (+ x (- (* y (log (/ E y))) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 0.0007) {
		tmp = (x - (log(y) * 0.5)) - z;
	} else {
		tmp = x + ((y * log((((double) M_E) / y))) - z);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 0.0007) {
		tmp = (x - (Math.log(y) * 0.5)) - z;
	} else {
		tmp = x + ((y * Math.log((Math.E / y))) - z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 0.0007:
		tmp = (x - (math.log(y) * 0.5)) - z
	else:
		tmp = x + ((y * math.log((math.e / y))) - z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 0.0007)
		tmp = Float64(Float64(x - Float64(log(y) * 0.5)) - z);
	else
		tmp = Float64(x + Float64(Float64(y * log(Float64(exp(1) / y))) - z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 0.0007)
		tmp = (x - (log(y) * 0.5)) - z;
	else
		tmp = x + ((y * log((2.71828182845904523536 / y))) - z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 0.0007], N[(N[(x - N[(N[Log[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(x + N[(N[(y * N[Log[N[(E / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.99999999999999993e-4

   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 99.1%

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

   if 6.99999999999999993e-4 < 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 98.8%

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

      1. log-rec 98.8%

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

      2. sub-neg 98.8%

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

   7. Simplified 98.8%

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

      1. add-log-exp 98.8%

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

      2. exp-diff 98.8%

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

      3. add-exp-log 98.8%

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

   9. Applied egg-rr 98.8%

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

       1. *-un-lft-identity 98.8%

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

       2. exp-1-e 98.8%

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

   11. Applied egg-rr 98.8%

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

       1. *-lft-identity 98.8%

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

   13. Simplified 98.8%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.0007:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + \left(y \cdot \log \left(\frac{e}{y}\right) - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.7 \cdot 10^{+64}:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 4.7e+64) (- (- 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 <= 4.7e+64) {
		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 <= 4.7d+64) 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 <= 4.7e+64) {
		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 <= 4.7e+64:
		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 <= 4.7e+64)
		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 <= 4.7e+64)
		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, 4.7e+64], 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 < 4.70000000000000029e64

   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 96.4%

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

   if 4.70000000000000029e64 < 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. Taylor expanded in y around inf 89.2%

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

      1. *-commutative 89.2%

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

      2. log-rec 89.2%

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

      3. distribute-lft-neg-in 89.2%

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

      4. distribute-rgt-neg-in 89.2%

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

   5. Simplified 89.2%

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

      1. +-commutative 89.2%

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

      2. *-un-lft-identity 89.2%

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

      3. distribute-rgt-neg-out 89.2%

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

      4. distribute-lft-neg-in 89.2%

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

      5. distribute-rgt-in 89.3%

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

      6. sub-neg 89.3%

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

      7. *-commutative 89.3%

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

   7. Applied egg-rr 89.3%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.7 \cdot 10^{+64}:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 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 8: 57.7% 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. Add Preprocessing

3. Taylor expanded in y around 0 72.1%

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

4. Taylor expanded in x around inf 56.7%

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

5. Final simplification 56.7%

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

Alternative 9: 29.7% accurate, 111.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 99.8%

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

   1. associate--l+ 99.8%

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

   2. sub-neg 99.8%

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

   3. associate-+l+ 99.8%

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

   4. 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 x around inf 26.1%

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

6. Final simplification 26.1%

   \[\leadsto x \]
7. 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 2024043 
(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))