Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Percentage Accurate: 99.8% → 99.9%

Time: 9.0s

Alternatives: 11

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   4. associate-+l+ 99.8%

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

   5. *-commutative 99.8%

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

   6. distribute-rgt-neg-in 99.8%

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

   7. fma-def 99.9%

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

   8. neg-sub0 99.9%

      \[\leadsto \mathsf{fma}\left(\log y, \color{blue}{0 - \left(y + 0.5\right)}, x + \left(y - z\right)\right) \]

   9. +-commutative 99.9%

      \[\leadsto \mathsf{fma}\left(\log y, 0 - \color{blue}{\left(0.5 + y\right)}, x + \left(y - z\right)\right) \]

   10. associate--r+ 99.9%

       \[\leadsto \mathsf{fma}\left(\log y, \color{blue}{\left(0 - 0.5\right) - y}, x + \left(y - z\right)\right) \]

   11. metadata-eval 99.9%

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

   12. associate-+r- 99.9%

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

   13. +-commutative 99.9%

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

   14. associate-+r- 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 2: 69.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y + x\right) - z\\ \mathbf{if}\;y \leq 6 \cdot 10^{-84}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{-21}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{+25}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 5.3 \cdot 10^{+45}:\\ \;\;\;\;y - y \cdot \log y\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{+135}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (+ y x) z)))
   (if (<= y 6e-84)
     t_0
     (if (<= y 2.4e-21)
       (- (* (log y) -0.5) z)
       (if (<= y 1.8e+25)
         t_0
         (if (<= y 5.3e+45)
           (- y (* y (log y)))
           (if (<= y 7.5e+135) (- x z) (* y (- 1.0 (log y))))))))))

C

double code(double x, double y, double z) {
	double t_0 = (y + x) - z;
	double tmp;
	if (y <= 6e-84) {
		tmp = t_0;
	} else if (y <= 2.4e-21) {
		tmp = (log(y) * -0.5) - z;
	} else if (y <= 1.8e+25) {
		tmp = t_0;
	} else if (y <= 5.3e+45) {
		tmp = y - (y * log(y));
	} else if (y <= 7.5e+135) {
		tmp = x - z;
	} else {
		tmp = y * (1.0 - log(y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (y + x) - z
    if (y <= 6d-84) then
        tmp = t_0
    else if (y <= 2.4d-21) then
        tmp = (log(y) * (-0.5d0)) - z
    else if (y <= 1.8d+25) then
        tmp = t_0
    else if (y <= 5.3d+45) then
        tmp = y - (y * log(y))
    else if (y <= 7.5d+135) then
        tmp = x - z
    else
        tmp = y * (1.0d0 - log(y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (y + x) - z;
	double tmp;
	if (y <= 6e-84) {
		tmp = t_0;
	} else if (y <= 2.4e-21) {
		tmp = (Math.log(y) * -0.5) - z;
	} else if (y <= 1.8e+25) {
		tmp = t_0;
	} else if (y <= 5.3e+45) {
		tmp = y - (y * Math.log(y));
	} else if (y <= 7.5e+135) {
		tmp = x - z;
	} else {
		tmp = y * (1.0 - Math.log(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (y + x) - z
	tmp = 0
	if y <= 6e-84:
		tmp = t_0
	elif y <= 2.4e-21:
		tmp = (math.log(y) * -0.5) - z
	elif y <= 1.8e+25:
		tmp = t_0
	elif y <= 5.3e+45:
		tmp = y - (y * math.log(y))
	elif y <= 7.5e+135:
		tmp = x - z
	else:
		tmp = y * (1.0 - math.log(y))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y + x) - z)
	tmp = 0.0
	if (y <= 6e-84)
		tmp = t_0;
	elseif (y <= 2.4e-21)
		tmp = Float64(Float64(log(y) * -0.5) - z);
	elseif (y <= 1.8e+25)
		tmp = t_0;
	elseif (y <= 5.3e+45)
		tmp = Float64(y - Float64(y * log(y)));
	elseif (y <= 7.5e+135)
		tmp = Float64(x - z);
	else
		tmp = Float64(y * Float64(1.0 - log(y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (y + x) - z;
	tmp = 0.0;
	if (y <= 6e-84)
		tmp = t_0;
	elseif (y <= 2.4e-21)
		tmp = (log(y) * -0.5) - z;
	elseif (y <= 1.8e+25)
		tmp = t_0;
	elseif (y <= 5.3e+45)
		tmp = y - (y * log(y));
	elseif (y <= 7.5e+135)
		tmp = x - z;
	else
		tmp = y * (1.0 - log(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y + x), $MachinePrecision] - z), $MachinePrecision]}, If[LessEqual[y, 6e-84], t$95$0, If[LessEqual[y, 2.4e-21], N[(N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[y, 1.8e+25], t$95$0, If[LessEqual[y, 5.3e+45], N[(y - N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7.5e+135], N[(x - z), $MachinePrecision], N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y < 6.0000000000000002e-84 or 2.3999999999999999e-21 < y < 1.80000000000000008e25

   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 inf 81.1%

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

      1. mul-1-neg 81.1%

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

      2. distribute-rgt-neg-in 81.1%

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

      3. log-rec 81.1%

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

      4. remove-double-neg 81.1%

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

   4. Simplified 81.1%

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

      1. add-cube-cbrt 80.4%

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

      2. pow3 80.4%

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

   6. Applied egg-rr 80.4%

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

   7. Taylor expanded in y around 0 80.1%

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

      1. pow-base-1 80.1%

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

      2. *-lft-identity 80.1%

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

   9. Simplified 80.1%

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

   if 6.0000000000000002e-84 < y < 2.3999999999999999e-21

   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 0 100.0%

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

   3. Taylor expanded in x around 0 79.2%

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

      1. mul-1-neg 79.2%

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

      2. *-commutative 79.2%

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

      3. distribute-neg-in 79.2%

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

      4. unsub-neg 79.2%

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

      5. distribute-rgt-neg-in 79.2%

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

      6. metadata-eval 79.2%

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

   5. Simplified 79.2%

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

   if 1.80000000000000008e25 < y < 5.29999999999999991e45

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 69.4%

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

   3. Taylor expanded in z around 0 63.7%

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

   4. Taylor expanded in y around inf 63.7%

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

      1. mul-1-neg 63.7%

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

      2. distribute-rgt-neg-in 63.7%

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

      3. log-rec 63.7%

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

      4. remove-double-neg 63.7%

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

   6. Simplified 63.7%

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

   if 5.29999999999999991e45 < y < 7.49999999999999947e135

   1. Initial program 99.8%

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

   2. Taylor expanded in x around inf 75.9%

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

   if 7.49999999999999947e135 < y

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 91.5%

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

   3. Taylor expanded in z around 0 80.6%

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

   4. Taylor expanded in y around inf 80.8%

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

      1. mul-1-neg 80.8%

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

      2. log-rec 80.8%

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

      3. remove-double-neg 80.8%

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

   6. Simplified 80.8%

      \[\leadsto \color{blue}{y \cdot \left(1 - \log y\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6 \cdot 10^{-84}:\\ \;\;\;\;\left(y + x\right) - z\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{-21}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{+25}:\\ \;\;\;\;\left(y + x\right) - z\\ \mathbf{elif}\;y \leq 5.3 \cdot 10^{+45}:\\ \;\;\;\;y - y \cdot \log y\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{+135}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \end{array} \]

Alternative 3: 70.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.9 \cdot 10^{+25}:\\ \;\;\;\;\left(y + x\right) - z\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{+45} \lor \neg \left(y \leq 8 \cdot 10^{+135}\right):\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.9e+25)
   (- (+ y x) z)
   (if (or (<= y 4.5e+45) (not (<= y 8e+135))) (* y (- 1.0 (log y))) (- x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.9e+25) {
		tmp = (y + x) - z;
	} else if ((y <= 4.5e+45) || !(y <= 8e+135)) {
		tmp = 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 (y <= 1.9d+25) then
        tmp = (y + x) - z
    else if ((y <= 4.5d+45) .or. (.not. (y <= 8d+135))) then
        tmp = 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 (y <= 1.9e+25) {
		tmp = (y + x) - z;
	} else if ((y <= 4.5e+45) || !(y <= 8e+135)) {
		tmp = y * (1.0 - Math.log(y));
	} else {
		tmp = x - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1.9e+25:
		tmp = (y + x) - z
	elif (y <= 4.5e+45) or not (y <= 8e+135):
		tmp = y * (1.0 - math.log(y))
	else:
		tmp = x - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.9e+25)
		tmp = Float64(Float64(y + x) - z);
	elseif ((y <= 4.5e+45) || !(y <= 8e+135))
		tmp = 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 (y <= 1.9e+25)
		tmp = (y + x) - z;
	elseif ((y <= 4.5e+45) || ~((y <= 8e+135)))
		tmp = y * (1.0 - log(y));
	else
		tmp = x - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.9e+25], N[(N[(y + x), $MachinePrecision] - z), $MachinePrecision], If[Or[LessEqual[y, 4.5e+45], N[Not[LessEqual[y, 8e+135]], $MachinePrecision]], N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.9e25

   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 inf 77.0%

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

      1. mul-1-neg 77.0%

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

      2. distribute-rgt-neg-in 77.0%

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

      3. log-rec 77.0%

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

      4. remove-double-neg 77.0%

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

   4. Simplified 77.0%

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

      1. add-cube-cbrt 76.4%

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

      2. pow3 76.4%

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

   6. Applied egg-rr 76.4%

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

   7. Taylor expanded in y around 0 76.2%

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

      1. pow-base-1 76.2%

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

      2. *-lft-identity 76.2%

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

   9. Simplified 76.2%

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

   if 1.9e25 < y < 4.4999999999999998e45 or 7.99999999999999969e135 < y

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 87.7%

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

   3. Taylor expanded in z around 0 77.7%

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

   4. Taylor expanded in y around inf 77.8%

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

      1. mul-1-neg 77.8%

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

      2. log-rec 77.8%

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

      3. remove-double-neg 77.8%

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

   6. Simplified 77.8%

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

   if 4.4999999999999998e45 < y < 7.99999999999999969e135

   1. Initial program 99.8%

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

   2. Taylor expanded in x around inf 75.9%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.9 \cdot 10^{+25}:\\ \;\;\;\;\left(y + x\right) - z\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{+45} \lor \neg \left(y \leq 8 \cdot 10^{+135}\right):\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \]

Alternative 4: 70.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.15 \cdot 10^{+24}:\\ \;\;\;\;\left(y + x\right) - z\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{+45}:\\ \;\;\;\;y - y \cdot \log y\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{+134}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.15e+24)
   (- (+ y x) z)
   (if (<= y 7.8e+45)
     (- y (* y (log y)))
     (if (<= y 1.85e+134) (- x z) (* y (- 1.0 (log y)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.15e+24) {
		tmp = (y + x) - z;
	} else if (y <= 7.8e+45) {
		tmp = y - (y * log(y));
	} else if (y <= 1.85e+134) {
		tmp = x - z;
	} else {
		tmp = y * (1.0 - log(y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 1.15d+24) then
        tmp = (y + x) - z
    else if (y <= 7.8d+45) then
        tmp = y - (y * log(y))
    else if (y <= 1.85d+134) then
        tmp = x - z
    else
        tmp = y * (1.0d0 - log(y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.15e+24) {
		tmp = (y + x) - z;
	} else if (y <= 7.8e+45) {
		tmp = y - (y * Math.log(y));
	} else if (y <= 1.85e+134) {
		tmp = x - z;
	} else {
		tmp = y * (1.0 - Math.log(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1.15e+24:
		tmp = (y + x) - z
	elif y <= 7.8e+45:
		tmp = y - (y * math.log(y))
	elif y <= 1.85e+134:
		tmp = x - z
	else:
		tmp = y * (1.0 - math.log(y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.15e+24)
		tmp = Float64(Float64(y + x) - z);
	elseif (y <= 7.8e+45)
		tmp = Float64(y - Float64(y * log(y)));
	elseif (y <= 1.85e+134)
		tmp = Float64(x - z);
	else
		tmp = Float64(y * Float64(1.0 - log(y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1.15e+24)
		tmp = (y + x) - z;
	elseif (y <= 7.8e+45)
		tmp = y - (y * log(y));
	elseif (y <= 1.85e+134)
		tmp = x - z;
	else
		tmp = y * (1.0 - log(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.15e+24], N[(N[(y + x), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[y, 7.8e+45], N[(y - N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.85e+134], N[(x - z), $MachinePrecision], N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < 1.15e24

   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 inf 77.0%

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

      1. mul-1-neg 77.0%

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

      2. distribute-rgt-neg-in 77.0%

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

      3. log-rec 77.0%

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

      4. remove-double-neg 77.0%

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

   4. Simplified 77.0%

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

      1. add-cube-cbrt 76.4%

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

      2. pow3 76.4%

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

   6. Applied egg-rr 76.4%

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

   7. Taylor expanded in y around 0 76.2%

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

      1. pow-base-1 76.2%

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

      2. *-lft-identity 76.2%

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

   9. Simplified 76.2%

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

   if 1.15e24 < y < 7.7999999999999999e45

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 69.4%

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

   3. Taylor expanded in z around 0 63.7%

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

   4. Taylor expanded in y around inf 63.7%

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

      1. mul-1-neg 63.7%

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

      2. distribute-rgt-neg-in 63.7%

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

      3. log-rec 63.7%

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

      4. remove-double-neg 63.7%

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

   6. Simplified 63.7%

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

   if 7.7999999999999999e45 < y < 1.85000000000000007e134

   1. Initial program 99.8%

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

   2. Taylor expanded in x around inf 75.9%

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

   if 1.85000000000000007e134 < y

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 91.5%

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

   3. Taylor expanded in z around 0 80.6%

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

   4. Taylor expanded in y around inf 80.8%

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

      1. mul-1-neg 80.8%

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

      2. log-rec 80.8%

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

      3. remove-double-neg 80.8%

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

   6. Simplified 80.8%

      \[\leadsto \color{blue}{y \cdot \left(1 - \log y\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.15 \cdot 10^{+24}:\\ \;\;\;\;\left(y + x\right) - z\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{+45}:\\ \;\;\;\;y - y \cdot \log y\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{+134}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \end{array} \]

Alternative 5: 76.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{+50}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{+35}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -4.4e+50)
   (- x z)
   (if (<= x 2.3e+35) (- (* y (- 1.0 (log y))) z) (- x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -4.4e+50) {
		tmp = x - z;
	} else if (x <= 2.3e+35) {
		tmp = (y * (1.0 - log(y))) - z;
	} 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 (x <= (-4.4d+50)) then
        tmp = x - z
    else if (x <= 2.3d+35) then
        tmp = (y * (1.0d0 - log(y))) - z
    else
        tmp = x - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -4.4e+50) {
		tmp = x - z;
	} else if (x <= 2.3e+35) {
		tmp = (y * (1.0 - Math.log(y))) - z;
	} else {
		tmp = x - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -4.4e+50:
		tmp = x - z
	elif x <= 2.3e+35:
		tmp = (y * (1.0 - math.log(y))) - z
	else:
		tmp = x - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -4.4e+50)
		tmp = Float64(x - z);
	elseif (x <= 2.3e+35)
		tmp = Float64(Float64(y * Float64(1.0 - log(y))) - z);
	else
		tmp = Float64(x - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -4.4e+50)
		tmp = x - z;
	elseif (x <= 2.3e+35)
		tmp = (y * (1.0 - log(y))) - z;
	else
		tmp = x - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -4.4e+50], N[(x - z), $MachinePrecision], If[LessEqual[x, 2.3e+35], N[(N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(x - z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.40000000000000034e50 or 2.2999999999999998e35 < 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 x around inf 79.3%

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

   if -4.40000000000000034e50 < x < 2.2999999999999998e35

   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 inf 80.0%

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

      1. cancel-sign-sub-inv 80.0%

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

      2. metadata-eval 80.0%

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

      3. *-lft-identity 80.0%

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

      4. log-rec 80.0%

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

      5. sub-neg 80.0%

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

   4. Simplified 80.0%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{+50}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{+35}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \]

Alternative 6: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7.2 \cdot 10^{-18}:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(y + \left(x - y \cdot \log y\right)\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 7.2e-18)
   (- (- x (* (log y) 0.5)) z)
   (- (+ y (- x (* y (log y)))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 7.2e-18) {
		tmp = (x - (log(y) * 0.5)) - z;
	} else {
		tmp = (y + (x - (y * 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 <= 7.2d-18) then
        tmp = (x - (log(y) * 0.5d0)) - z
    else
        tmp = (y + (x - (y * log(y)))) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 7.2e-18) {
		tmp = (x - (Math.log(y) * 0.5)) - z;
	} else {
		tmp = (y + (x - (y * Math.log(y)))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 7.2e-18:
		tmp = (x - (math.log(y) * 0.5)) - z
	else:
		tmp = (y + (x - (y * math.log(y)))) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 7.2e-18)
		tmp = Float64(Float64(x - Float64(log(y) * 0.5)) - z);
	else
		tmp = Float64(Float64(y + Float64(x - Float64(y * log(y)))) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 7.2e-18)
		tmp = (x - (log(y) * 0.5)) - z;
	else
		tmp = (y + (x - (y * log(y)))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 7.2e-18], N[(N[(x - N[(N[Log[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(N[(y + N[(x - N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 7.20000000000000021e-18

   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 99.9%

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

   if 7.20000000000000021e-18 < y

   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 inf 99.2%

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

      1. mul-1-neg 99.2%

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

      2. distribute-rgt-neg-in 99.2%

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

      3. log-rec 99.2%

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

      4. remove-double-neg 99.2%

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

   4. Simplified 99.2%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7.2 \cdot 10^{-18}:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(y + \left(x - y \cdot \log y\right)\right) - z\\ \end{array} \]

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. Final simplification 99.8%

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

Alternative 8: 89.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.3 \cdot 10^{+24}:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.3e+24) (- (- x (* (log y) 0.5)) z) (- (* y (- 1.0 (log y))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.3e+24) {
		tmp = (x - (log(y) * 0.5)) - z;
	} else {
		tmp = (y * (1.0 - log(y))) - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 1.3d+24) then
        tmp = (x - (log(y) * 0.5d0)) - z
    else
        tmp = (y * (1.0d0 - log(y))) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.3e+24) {
		tmp = (x - (Math.log(y) * 0.5)) - z;
	} else {
		tmp = (y * (1.0 - Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1.3e+24:
		tmp = (x - (math.log(y) * 0.5)) - z
	else:
		tmp = (y * (1.0 - math.log(y))) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.3e+24)
		tmp = Float64(Float64(x - Float64(log(y) * 0.5)) - z);
	else
		tmp = Float64(Float64(y * Float64(1.0 - log(y))) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1.3e+24)
		tmp = (x - (log(y) * 0.5)) - z;
	else
		tmp = (y * (1.0 - log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.3e+24], N[(N[(x - N[(N[Log[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.2999999999999999e24

   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 98.6%

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

   if 1.2999999999999999e24 < y

   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 inf 81.8%

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

      1. cancel-sign-sub-inv 81.8%

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

      2. metadata-eval 81.8%

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

      3. *-lft-identity 81.8%

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

      4. log-rec 81.8%

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

      5. sub-neg 81.8%

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

   4. Simplified 81.8%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.3 \cdot 10^{+24}:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \end{array} \]

Alternative 9: 47.5% accurate, 18.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+51}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.75 \cdot 10^{+134}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1e+51) x (if (<= x 2.75e+134) (- z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1e+51) {
		tmp = x;
	} else if (x <= 2.75e+134) {
		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 <= (-1d+51)) then
        tmp = x
    else if (x <= 2.75d+134) 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 <= -1e+51) {
		tmp = x;
	} else if (x <= 2.75e+134) {
		tmp = -z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1e+51:
		tmp = x
	elif x <= 2.75e+134:
		tmp = -z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1e+51)
		tmp = x;
	elseif (x <= 2.75e+134)
		tmp = Float64(-z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1e+51)
		tmp = x;
	elseif (x <= 2.75e+134)
		tmp = -z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1e+51], x, If[LessEqual[x, 2.75e+134], (-z), x]]

Derivation

1. Split input into 2 regimes

2. if x < -1e51 or 2.7499999999999999e134 < 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 80.6%

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

   3. Taylor expanded in x around inf 66.1%

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

   if -1e51 < x < 2.7499999999999999e134

   1. Initial program 99.8%

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

   2. Taylor expanded in z around inf 40.5%

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

      1. neg-mul-1 40.5%

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

   4. Simplified 40.5%

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

4. Final simplification 48.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+51}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.75 \cdot 10^{+134}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 10: 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. Taylor expanded in x around inf 56.1%

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

3. Final simplification 56.1%

   \[\leadsto x - z \]

Alternative 11: 30.3% 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 67.4%

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

3. Taylor expanded in x around inf 25.3%

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

4. Final simplification 25.3%

   \[\leadsto x \]

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 2023200 
(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))