Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Percentage Accurate: 99.8% → 99.9%

Time: 9.3s

Alternatives: 10

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: 89.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(1 - \log y\right)\\ \mathbf{if}\;y \leq 6.2 \cdot 10^{+37}:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{+136} \lor \neg \left(y \leq 8.2 \cdot 10^{+190}\right):\\ \;\;\;\;t_0 - z\\ \mathbf{else}:\\ \;\;\;\;x + t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (- 1.0 (log y)))))
   (if (<= y 6.2e+37)
     (- (- x (* (log y) 0.5)) z)
     (if (or (<= y 1.75e+136) (not (<= y 8.2e+190))) (- t_0 z) (+ x t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = y * (1.0 - log(y));
	double tmp;
	if (y <= 6.2e+37) {
		tmp = (x - (log(y) * 0.5)) - z;
	} else if ((y <= 1.75e+136) || !(y <= 8.2e+190)) {
		tmp = t_0 - z;
	} else {
		tmp = x + t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y * (1.0d0 - log(y))
    if (y <= 6.2d+37) then
        tmp = (x - (log(y) * 0.5d0)) - z
    else if ((y <= 1.75d+136) .or. (.not. (y <= 8.2d+190))) then
        tmp = t_0 - z
    else
        tmp = x + t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y * (1.0 - Math.log(y));
	double tmp;
	if (y <= 6.2e+37) {
		tmp = (x - (Math.log(y) * 0.5)) - z;
	} else if ((y <= 1.75e+136) || !(y <= 8.2e+190)) {
		tmp = t_0 - z;
	} else {
		tmp = x + t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * (1.0 - math.log(y))
	tmp = 0
	if y <= 6.2e+37:
		tmp = (x - (math.log(y) * 0.5)) - z
	elif (y <= 1.75e+136) or not (y <= 8.2e+190):
		tmp = t_0 - z
	else:
		tmp = x + t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * Float64(1.0 - log(y)))
	tmp = 0.0
	if (y <= 6.2e+37)
		tmp = Float64(Float64(x - Float64(log(y) * 0.5)) - z);
	elseif ((y <= 1.75e+136) || !(y <= 8.2e+190))
		tmp = Float64(t_0 - z);
	else
		tmp = Float64(x + t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * (1.0 - log(y));
	tmp = 0.0;
	if (y <= 6.2e+37)
		tmp = (x - (log(y) * 0.5)) - z;
	elseif ((y <= 1.75e+136) || ~((y <= 8.2e+190)))
		tmp = t_0 - z;
	else
		tmp = x + t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 6.2e+37], N[(N[(x - N[(N[Log[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], If[Or[LessEqual[y, 1.75e+136], N[Not[LessEqual[y, 8.2e+190]], $MachinePrecision]], N[(t$95$0 - z), $MachinePrecision], N[(x + t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 6.2000000000000004e37

   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 95.1%

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

      1. *-commutative 95.1%

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

   4. Simplified 95.1%

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

   if 6.2000000000000004e37 < y < 1.75000000000000001e136 or 8.2000000000000004e190 < 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. *-commutative 99.6%

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

      2. flip-+ 52.0%

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

      3. associate-*r/ 52.0%

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

      4. fma-neg 52.0%

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

      5. metadata-eval 52.0%

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

      6. metadata-eval 52.0%

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

      7. sub-neg 52.0%

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

      8. metadata-eval 52.0%

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

   3. Applied egg-rr 52.0%

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

      1. associate-/l* 52.0%

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

      2. associate-/r/ 52.0%

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

      3. +-commutative 52.0%

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

   5. Simplified 52.0%

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

   6. Taylor expanded in x around 0 44.4%

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

      1. *-commutative 44.4%

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

      2. unpow2 44.4%

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

      3. fma-neg 44.4%

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

      4. metadata-eval 44.4%

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

      5. sub-neg 44.4%

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

      6. metadata-eval 44.4%

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

      7. associate-*l/ 44.4%

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

   8. Simplified 44.4%

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

   9. Taylor expanded in y around inf 90.9%

      \[\leadsto \color{blue}{y \cdot \left(1 - -1 \cdot \log \left(\frac{1}{y}\right)\right)} - z \]
   10. 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%

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

   11. Simplified 90.9%

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

   if 1.75000000000000001e136 < y < 8.2000000000000004e190

   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. associate-+l- 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around inf 85.9%

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

      1. sub-neg 85.9%

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

      2. mul-1-neg 85.9%

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

      3. log-rec 85.9%

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

      4. remove-double-neg 85.9%

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

      5. metadata-eval 85.9%

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

   6. Simplified 85.9%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.2 \cdot 10^{+37}:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{+136} \lor \neg \left(y \leq 8.2 \cdot 10^{+190}\right):\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(1 - \log y\right)\\ \end{array} \]

Alternative 3: 78.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+32}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;z \leq 4.3 \cdot 10^{+42}:\\ \;\;\;\;x + y \cdot \left(1 - \log y\right)\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1e+32)
   (- x z)
   (if (<= z 4.3e+42) (+ x (* y (- 1.0 (log y)))) (- x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1e+32) {
		tmp = x - z;
	} else if (z <= 4.3e+42) {
		tmp = x + (y * (1.0 - log(y)));
	} else {
		tmp = x - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-1d+32)) then
        tmp = x - z
    else if (z <= 4.3d+42) then
        tmp = x + (y * (1.0d0 - log(y)))
    else
        tmp = x - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1e+32) {
		tmp = x - z;
	} else if (z <= 4.3e+42) {
		tmp = x + (y * (1.0 - Math.log(y)));
	} else {
		tmp = x - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1e+32:
		tmp = x - z
	elif z <= 4.3e+42:
		tmp = x + (y * (1.0 - math.log(y)))
	else:
		tmp = x - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1e+32)
		tmp = Float64(x - z);
	elseif (z <= 4.3e+42)
		tmp = Float64(x + Float64(y * Float64(1.0 - log(y))));
	else
		tmp = Float64(x - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1e+32)
		tmp = x - z;
	elseif (z <= 4.3e+42)
		tmp = x + (y * (1.0 - log(y)));
	else
		tmp = x - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1e+32], N[(x - z), $MachinePrecision], If[LessEqual[z, 4.3e+42], N[(x + N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.00000000000000005e32 or 4.2999999999999998e42 < z

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. associate-+l- 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in z around inf 88.5%

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

   if -1.00000000000000005e32 < z < 4.2999999999999998e42

   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. associate-+l- 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around inf 77.0%

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

      1. sub-neg 77.0%

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

      2. mul-1-neg 77.0%

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

      3. log-rec 77.0%

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

      4. remove-double-neg 77.0%

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

      5. metadata-eval 77.0%

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

   6. Simplified 77.0%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+32}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;z \leq 4.3 \cdot 10^{+42}:\\ \;\;\;\;x + y \cdot \left(1 - \log y\right)\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \]

Alternative 4: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.13:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(y - z\right) - y \cdot \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 0.13) (- (- x (* (log y) 0.5)) z) (+ x (- (- y z) (* y (log y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 0.13) {
		tmp = (x - (log(y) * 0.5)) - z;
	} else {
		tmp = x + ((y - z) - (y * log(y)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 0.13d0) then
        tmp = (x - (log(y) * 0.5d0)) - z
    else
        tmp = x + ((y - z) - (y * log(y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 0.13) {
		tmp = (x - (Math.log(y) * 0.5)) - z;
	} else {
		tmp = x + ((y - z) - (y * Math.log(y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 0.13:
		tmp = (x - (math.log(y) * 0.5)) - z
	else:
		tmp = x + ((y - z) - (y * math.log(y)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 0.13)
		tmp = Float64(Float64(x - Float64(log(y) * 0.5)) - z);
	else
		tmp = Float64(x + Float64(Float64(y - z) - Float64(y * log(y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 0.13)
		tmp = (x - (log(y) * 0.5)) - z;
	else
		tmp = x + ((y - z) - (y * log(y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 0.13], N[(N[(x - N[(N[Log[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(x + N[(N[(y - z), $MachinePrecision] - N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 0.13

   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 99.3%

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

      1. *-commutative 99.3%

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

   4. Simplified 99.3%

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

   if 0.13 < 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. associate-+l- 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around inf 98.5%

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

      1. mul-1-neg 98.5%

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

      2. distribute-rgt-neg-in 98.5%

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

      3. log-rec 98.5%

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

      4. remove-double-neg 98.5%

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

   6. Simplified 98.5%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.13:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(y - z\right) - y \cdot \log y\right)\\ \end{array} \]

Alternative 5: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 - z) - (log(y) * (y + 0.5d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(x + N[(N[(y - z), $MachinePrecision] - N[(N[Log[y], $MachinePrecision] * N[(y + 0.5), $MachinePrecision]), $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. associate-+l- 99.8%

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

3. Simplified 99.8%

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

4. Final simplification 99.8%

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

Alternative 6: 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 7: 70.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+30}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;z \leq 175:\\ \;\;\;\;x - \log y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1e+30) (- x z) (if (<= z 175.0) (- x (* (log y) 0.5)) (- x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1e+30) {
		tmp = x - z;
	} else if (z <= 175.0) {
		tmp = x - (log(y) * 0.5);
	} else {
		tmp = x - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-1d+30)) then
        tmp = x - z
    else if (z <= 175.0d0) then
        tmp = x - (log(y) * 0.5d0)
    else
        tmp = x - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1e+30) {
		tmp = x - z;
	} else if (z <= 175.0) {
		tmp = x - (Math.log(y) * 0.5);
	} else {
		tmp = x - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1e+30:
		tmp = x - z
	elif z <= 175.0:
		tmp = x - (math.log(y) * 0.5)
	else:
		tmp = x - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1e+30)
		tmp = Float64(x - z);
	elseif (z <= 175.0)
		tmp = Float64(x - Float64(log(y) * 0.5));
	else
		tmp = Float64(x - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1e+30)
		tmp = x - z;
	elseif (z <= 175.0)
		tmp = x - (log(y) * 0.5);
	else
		tmp = x - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1e+30], N[(x - z), $MachinePrecision], If[LessEqual[z, 175.0], N[(x - N[(N[Log[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], N[(x - z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -1e30 or 175 < z

   1. Initial program 99.9%

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

      1. associate--l+ 99.9%

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

      2. associate-+l- 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in z around inf 86.2%

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

   if -1e30 < z < 175

   1. Initial program 99.7%

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

   2. Taylor expanded in y around 0 64.5%

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

      1. *-commutative 64.5%

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

   4. Simplified 64.5%

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

   5. Taylor expanded in z around 0 64.6%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+30}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;z \leq 175:\\ \;\;\;\;x - \log y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \]

Alternative 8: 49.7% accurate, 18.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.15 \cdot 10^{+32}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 9.2 \cdot 10^{+39}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.15e+32) x (if (<= x 9.2e+39) (- z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.15e+32) {
		tmp = x;
	} else if (x <= 9.2e+39) {
		tmp = -z;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-1.15d+32)) then
        tmp = x
    else if (x <= 9.2d+39) then
        tmp = -z
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.15e+32) {
		tmp = x;
	} else if (x <= 9.2e+39) {
		tmp = -z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.15e+32:
		tmp = x
	elif x <= 9.2e+39:
		tmp = -z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.15e+32)
		tmp = x;
	elseif (x <= 9.2e+39)
		tmp = Float64(-z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.15e+32)
		tmp = x;
	elseif (x <= 9.2e+39)
		tmp = -z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.15e+32], x, If[LessEqual[x, 9.2e+39], (-z), x]]

Derivation

1. Split input into 2 regimes

2. if x < -1.15e32 or 9.20000000000000047e39 < x

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 86.3%

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

      1. *-commutative 86.3%

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

   4. Simplified 86.3%

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

   5. Taylor expanded in x around inf 71.0%

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

   if -1.15e32 < x < 9.20000000000000047e39

   1. Initial program 99.7%

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

   2. Taylor expanded in y around 0 67.6%

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

      1. *-commutative 67.6%

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

   4. Simplified 67.6%

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

   5. Taylor expanded in z around inf 46.7%

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

      1. neg-mul-1 46.7%

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

   7. Simplified 46.7%

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

4. Final simplification 57.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.15 \cdot 10^{+32}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 9.2 \cdot 10^{+39}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9: 59.4% accurate, 37.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate--l+ 99.8%

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

   2. associate-+l- 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in z around inf 64.4%

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

5. Final simplification 64.4%

   \[\leadsto x - z \]

Alternative 10: 31.2% 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. Taylor expanded in y around 0 75.9%

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

   1. *-commutative 75.9%

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

4. Simplified 75.9%

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

5. Taylor expanded in x around inf 33.1%

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

6. Final simplification 33.1%

   \[\leadsto x \]

Developer target: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023279 
(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))