Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Percentage Accurate: 99.8% → 99.9%

Time: 9.8s

Alternatives: 9

Speedup: 1.0×

• could not determine a ground truth

  1. x = 5.899079023961052e+293
  2. y = 7.068242793033713e+76
  3. z = 5.03442329535394e+292

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate--l+ 99.8%

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

   2. sub-neg 99.8%

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

   3. associate-+l+ 99.8%

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

   4. associate-+r- 99.8%

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

   5. *-commutative 99.8%

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

   6. distribute-rgt-neg-in 99.8%

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

   7. fma-define 99.8%

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

   8. +-commutative 99.8%

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

   9. distribute-neg-in 99.8%

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

   10. unsub-neg 99.8%

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

   11. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 75.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7.8 \cdot 10^{-222}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{+44}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 7.8e-222)
   (- (* (log y) -0.5) z)
   (if (<= y 5.2e+44) (- x z) (- (* y (- 1.0 (log y))) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 7.8e-222) {
		tmp = (log(y) * -0.5) - z;
	} else if (y <= 5.2e+44) {
		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 <= 7.8d-222) then
        tmp = (log(y) * (-0.5d0)) - z
    else if (y <= 5.2d+44) 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 <= 7.8e-222) {
		tmp = (Math.log(y) * -0.5) - z;
	} else if (y <= 5.2e+44) {
		tmp = x - z;
	} else {
		tmp = (y * (1.0 - Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 7.8e-222:
		tmp = (math.log(y) * -0.5) - z
	elif y <= 5.2e+44:
		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 <= 7.8e-222)
		tmp = Float64(Float64(log(y) * -0.5) - z);
	elseif (y <= 5.2e+44)
		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 <= 7.8e-222)
		tmp = (log(y) * -0.5) - z;
	elseif (y <= 5.2e+44)
		tmp = x - z;
	else
		tmp = (y * (1.0 - log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 7.8e-222], N[(N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[y, 5.2e+44], 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 < 7.8000000000000002e-222

   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%

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

      1. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 76.6%

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

   if 7.8000000000000002e-222 < y < 5.1999999999999998e44

   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 82.8%

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

   if 5.1999999999999998e44 < 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 98.5%

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

      2. pow3 98.6%

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

      3. sub-neg 98.6%

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

      4. associate-+l+ 98.6%

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

      5. *-commutative 98.6%

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

      6. distribute-rgt-neg-in 98.6%

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

      7. +-commutative 98.6%

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

      8. distribute-neg-in 98.6%

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

      9. metadata-eval 98.6%

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

      10. sub-neg 98.6%

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

      11. fma-undefine 98.5%

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

   4. Applied egg-rr 98.5%

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

   5. Taylor expanded in y around inf 82.5%

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

      1. log-rec 82.5%

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

      2. sub-neg 82.5%

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

   7. Simplified 82.5%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7.8 \cdot 10^{-222}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{+44}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 69.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7 \cdot 10^{-222}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{elif}\;y \leq 4 \cdot 10^{+137}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y - y \cdot \log y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 7e-222)
   (- (* (log y) -0.5) z)
   (if (<= y 4e+137) (- x z) (- y (* y (log y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 7e-222) {
		tmp = (log(y) * -0.5) - z;
	} else if (y <= 4e+137) {
		tmp = x - z;
	} else {
		tmp = y - (y * log(y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 7d-222) then
        tmp = (log(y) * (-0.5d0)) - z
    else if (y <= 4d+137) then
        tmp = x - z
    else
        tmp = y - (y * log(y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 7e-222) {
		tmp = (Math.log(y) * -0.5) - z;
	} else if (y <= 4e+137) {
		tmp = x - z;
	} else {
		tmp = y - (y * Math.log(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 7e-222:
		tmp = (math.log(y) * -0.5) - z
	elif y <= 4e+137:
		tmp = x - z
	else:
		tmp = y - (y * math.log(y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 7e-222)
		tmp = Float64(Float64(log(y) * -0.5) - z);
	elseif (y <= 4e+137)
		tmp = Float64(x - z);
	else
		tmp = Float64(y - Float64(y * log(y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 7e-222)
		tmp = (log(y) * -0.5) - z;
	elseif (y <= 4e+137)
		tmp = x - z;
	else
		tmp = y - (y * log(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 7e-222], N[(N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[y, 4e+137], N[(x - z), $MachinePrecision], N[(y - N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 7.00000000000000049e-222

   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%

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

      1. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 76.6%

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

   if 7.00000000000000049e-222 < y < 4.0000000000000001e137

   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 x around inf 77.5%

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

   if 4.0000000000000001e137 < 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 90.7%

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

      1. *-commutative 90.7%

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

      2. log-rec 90.7%

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

      3. distribute-lft-neg-in 90.7%

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

      4. distribute-rgt-neg-in 90.7%

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

   5. Simplified 90.7%

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

   6. Taylor expanded in z around 0 90.7%

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

      1. neg-mul-1 90.7%

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

      2. associate-+r+ 90.7%

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

      3. sub-neg 90.7%

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

      4. neg-mul-1 90.7%

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

      5. sub-neg 90.7%

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

   8. Simplified 90.7%

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

   9. Taylor expanded in z around 0 77.4%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7 \cdot 10^{-222}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{elif}\;y \leq 4 \cdot 10^{+137}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y - y \cdot \log y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7 \cdot 10^{-21}:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + \left(y \cdot \left(1 - \log y\right) - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 7e-21)
   (- (- 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 <= 7e-21) {
		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 <= 7d-21) 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 <= 7e-21) {
		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 <= 7e-21:
		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 <= 7e-21)
		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 <= 7e-21)
		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, 7e-21], 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 < 7.0000000000000007e-21

   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%

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

      1. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   if 7.0000000000000007e-21 < 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-define 99.7%

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

      8. +-commutative 99.7%

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

      9. distribute-neg-in 99.7%

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

      10. unsub-neg 99.7%

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

      11. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around inf 99.7%

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

      1. log-rec 99.7%

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

      2. sub-neg 99.7%

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

   7. Simplified 99.7%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7 \cdot 10^{-21}:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + \left(y \cdot \left(1 - \log y\right) - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.8 \cdot 10^{+44}:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 4.8e+44) (- (- 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 <= 4.8e+44) {
		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 <= 4.8d+44) 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 <= 4.8e+44) {
		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 <= 4.8e+44:
		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 <= 4.8e+44)
		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 <= 4.8e+44)
		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, 4.8e+44], 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 < 4.80000000000000026e44

   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 99.3%

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

      1. *-commutative 99.3%

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

   5. Simplified 99.3%

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

   if 4.80000000000000026e44 < 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 98.5%

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

      2. pow3 98.6%

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

      3. sub-neg 98.6%

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

      4. associate-+l+ 98.6%

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

      5. *-commutative 98.6%

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

      6. distribute-rgt-neg-in 98.6%

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

      7. +-commutative 98.6%

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

      8. distribute-neg-in 98.6%

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

      9. metadata-eval 98.6%

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

      10. sub-neg 98.6%

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

      11. fma-undefine 98.5%

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

   4. Applied egg-rr 98.5%

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

   5. Taylor expanded in y around inf 82.5%

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

      1. log-rec 82.5%

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

      2. sub-neg 82.5%

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

   7. Simplified 82.5%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.8 \cdot 10^{+44}:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 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. 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 7: 71.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.1 \cdot 10^{+137}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y - y \cdot \log y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 4.1e+137) (- x z) (- y (* y (log y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 4.1e+137) {
		tmp = x - z;
	} else {
		tmp = y - (y * log(y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 4.1d+137) then
        tmp = x - z
    else
        tmp = y - (y * log(y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 4.1e+137) {
		tmp = x - z;
	} else {
		tmp = y - (y * Math.log(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 4.1e+137:
		tmp = x - z
	else:
		tmp = y - (y * math.log(y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 4.1e+137)
		tmp = Float64(x - z);
	else
		tmp = Float64(y - Float64(y * log(y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 4.1e+137)
		tmp = x - z;
	else
		tmp = y - (y * log(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 4.1e+137], N[(x - z), $MachinePrecision], N[(y - N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.09999999999999997e137

   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 74.8%

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

   if 4.09999999999999997e137 < 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 90.7%

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

      1. *-commutative 90.7%

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

      2. log-rec 90.7%

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

      3. distribute-lft-neg-in 90.7%

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

      4. distribute-rgt-neg-in 90.7%

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

   5. Simplified 90.7%

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

   6. Taylor expanded in z around 0 90.7%

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

      1. neg-mul-1 90.7%

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

      2. associate-+r+ 90.7%

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

      3. sub-neg 90.7%

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

      4. neg-mul-1 90.7%

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

      5. sub-neg 90.7%

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

   8. Simplified 90.7%

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

   9. Taylor expanded in z around 0 77.4%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.1 \cdot 10^{+137}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y - y \cdot \log y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 58.3% 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 59.3%

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

4. Final simplification 59.3%

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

Alternative 9: 30.7% 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. Taylor expanded in y around inf 58.6%

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

   1. *-commutative 58.6%

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

   2. log-rec 58.6%

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

   3. distribute-lft-neg-in 58.6%

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

   4. distribute-rgt-neg-in 58.6%

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

5. Simplified 58.6%

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

6. Taylor expanded in y around 0 30.3%

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

   1. neg-mul-1 30.3%

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

8. Simplified 30.3%

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

9. Final simplification 30.3%

   \[\leadsto -z \]
10. 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 2024074 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2"
  :precision binary64

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

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