Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Percentage Accurate: 99.8% → 99.9%

Time: 14.1s

Alternatives: 11

Speedup: 0.5×

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: 46.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(1 - \log y\right)\\ \mathbf{if}\;x \leq -9.5 \cdot 10^{+60}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -1.15 \cdot 10^{-239}:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \leq -1.1 \cdot 10^{-303}:\\ \;\;\;\;\log y \cdot -0.5\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-222}:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-152}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 8.6 \cdot 10^{-86}:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \leq 0.051:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.42 \cdot 10^{+165}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (- 1.0 (log y)))))
   (if (<= x -9.5e+60)
     x
     (if (<= x -1.15e-239)
       (- z)
       (if (<= x -1.1e-303)
         (* (log y) -0.5)
         (if (<= x 5.2e-222)
           (- z)
           (if (<= x 5e-152)
             t_0
             (if (<= x 8.6e-86)
               (- z)
               (if (<= x 0.051) t_0 (if (<= x 1.42e+165) (- z) x))))))))))

C

double code(double x, double y, double z) {
	double t_0 = y * (1.0 - log(y));
	double tmp;
	if (x <= -9.5e+60) {
		tmp = x;
	} else if (x <= -1.15e-239) {
		tmp = -z;
	} else if (x <= -1.1e-303) {
		tmp = log(y) * -0.5;
	} else if (x <= 5.2e-222) {
		tmp = -z;
	} else if (x <= 5e-152) {
		tmp = t_0;
	} else if (x <= 8.6e-86) {
		tmp = -z;
	} else if (x <= 0.051) {
		tmp = t_0;
	} else if (x <= 1.42e+165) {
		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) :: t_0
    real(8) :: tmp
    t_0 = y * (1.0d0 - log(y))
    if (x <= (-9.5d+60)) then
        tmp = x
    else if (x <= (-1.15d-239)) then
        tmp = -z
    else if (x <= (-1.1d-303)) then
        tmp = log(y) * (-0.5d0)
    else if (x <= 5.2d-222) then
        tmp = -z
    else if (x <= 5d-152) then
        tmp = t_0
    else if (x <= 8.6d-86) then
        tmp = -z
    else if (x <= 0.051d0) then
        tmp = t_0
    else if (x <= 1.42d+165) then
        tmp = -z
    else
        tmp = x
    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 (x <= -9.5e+60) {
		tmp = x;
	} else if (x <= -1.15e-239) {
		tmp = -z;
	} else if (x <= -1.1e-303) {
		tmp = Math.log(y) * -0.5;
	} else if (x <= 5.2e-222) {
		tmp = -z;
	} else if (x <= 5e-152) {
		tmp = t_0;
	} else if (x <= 8.6e-86) {
		tmp = -z;
	} else if (x <= 0.051) {
		tmp = t_0;
	} else if (x <= 1.42e+165) {
		tmp = -z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * (1.0 - math.log(y))
	tmp = 0
	if x <= -9.5e+60:
		tmp = x
	elif x <= -1.15e-239:
		tmp = -z
	elif x <= -1.1e-303:
		tmp = math.log(y) * -0.5
	elif x <= 5.2e-222:
		tmp = -z
	elif x <= 5e-152:
		tmp = t_0
	elif x <= 8.6e-86:
		tmp = -z
	elif x <= 0.051:
		tmp = t_0
	elif x <= 1.42e+165:
		tmp = -z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * Float64(1.0 - log(y)))
	tmp = 0.0
	if (x <= -9.5e+60)
		tmp = x;
	elseif (x <= -1.15e-239)
		tmp = Float64(-z);
	elseif (x <= -1.1e-303)
		tmp = Float64(log(y) * -0.5);
	elseif (x <= 5.2e-222)
		tmp = Float64(-z);
	elseif (x <= 5e-152)
		tmp = t_0;
	elseif (x <= 8.6e-86)
		tmp = Float64(-z);
	elseif (x <= 0.051)
		tmp = t_0;
	elseif (x <= 1.42e+165)
		tmp = Float64(-z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * (1.0 - log(y));
	tmp = 0.0;
	if (x <= -9.5e+60)
		tmp = x;
	elseif (x <= -1.15e-239)
		tmp = -z;
	elseif (x <= -1.1e-303)
		tmp = log(y) * -0.5;
	elseif (x <= 5.2e-222)
		tmp = -z;
	elseif (x <= 5e-152)
		tmp = t_0;
	elseif (x <= 8.6e-86)
		tmp = -z;
	elseif (x <= 0.051)
		tmp = t_0;
	elseif (x <= 1.42e+165)
		tmp = -z;
	else
		tmp = x;
	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[x, -9.5e+60], x, If[LessEqual[x, -1.15e-239], (-z), If[LessEqual[x, -1.1e-303], N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision], If[LessEqual[x, 5.2e-222], (-z), If[LessEqual[x, 5e-152], t$95$0, If[LessEqual[x, 8.6e-86], (-z), If[LessEqual[x, 0.051], t$95$0, If[LessEqual[x, 1.42e+165], (-z), x]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -9.49999999999999988e60 or 1.42e165 < x

   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%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around inf 80.1%

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

   if -9.49999999999999988e60 < x < -1.1499999999999999e-239 or -1.10000000000000007e-303 < x < 5.1999999999999997e-222 or 4.9999999999999997e-152 < x < 8.60000000000000026e-86 or 0.0509999999999999967 < x < 1.42e165

   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%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in z around inf 48.8%

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

      1. mul-1-neg 48.8%

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

   6. Simplified 48.8%

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

   if -1.1499999999999999e-239 < x < -1.10000000000000007e-303

   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%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around 0 99.7%

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

      1. fma-def 99.7%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in z around 0 88.0%

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

   8. Taylor expanded in y around 0 53.2%

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

      1. *-commutative 53.2%

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

   10. Simplified 53.2%

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

   if 5.1999999999999997e-222 < x < 4.9999999999999997e-152 or 8.60000000000000026e-86 < x < 0.0509999999999999967

   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%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around inf 55.4%

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

      1. *-commutative 55.4%

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

      2. log-rec 55.4%

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

      3. cancel-sign-sub 55.4%

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

      4. *-commutative 55.4%

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

      5. neg-mul-1 55.4%

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

      6. sub-neg 55.4%

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

   6. Simplified 55.4%

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

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{+60}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -1.15 \cdot 10^{-239}:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \leq -1.1 \cdot 10^{-303}:\\ \;\;\;\;\log y \cdot -0.5\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-222}:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-152}:\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \mathbf{elif}\;x \leq 8.6 \cdot 10^{-86}:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \leq 0.051:\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \mathbf{elif}\;x \leq 1.42 \cdot 10^{+165}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3: 60.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \log y \cdot 0.5\\ t_1 := y - \log y \cdot \left(y + 0.5\right)\\ \mathbf{if}\;z \leq -1.9 \cdot 10^{+128}:\\ \;\;\;\;y - z\\ \mathbf{elif}\;z \leq -2.4 \cdot 10^{+71}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -2.1 \cdot 10^{-36}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -9.5 \cdot 10^{-205}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{-53}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 9 \cdot 10^{-14}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 7.9 \cdot 10^{+34}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- x (* (log y) 0.5))) (t_1 (- y (* (log y) (+ y 0.5)))))
   (if (<= z -1.9e+128)
     (- y z)
     (if (<= z -2.4e+71)
       t_1
       (if (<= z -2.1e-36)
         t_0
         (if (<= z -9.5e-205)
           t_1
           (if (<= z 1.4e-53)
             t_0
             (if (<= z 9e-14) t_1 (if (<= z 7.9e+34) x (- z))))))))))

C

double code(double x, double y, double z) {
	double t_0 = x - (log(y) * 0.5);
	double t_1 = y - (log(y) * (y + 0.5));
	double tmp;
	if (z <= -1.9e+128) {
		tmp = y - z;
	} else if (z <= -2.4e+71) {
		tmp = t_1;
	} else if (z <= -2.1e-36) {
		tmp = t_0;
	} else if (z <= -9.5e-205) {
		tmp = t_1;
	} else if (z <= 1.4e-53) {
		tmp = t_0;
	} else if (z <= 9e-14) {
		tmp = t_1;
	} else if (z <= 7.9e+34) {
		tmp = x;
	} else {
		tmp = -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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x - (log(y) * 0.5d0)
    t_1 = y - (log(y) * (y + 0.5d0))
    if (z <= (-1.9d+128)) then
        tmp = y - z
    else if (z <= (-2.4d+71)) then
        tmp = t_1
    else if (z <= (-2.1d-36)) then
        tmp = t_0
    else if (z <= (-9.5d-205)) then
        tmp = t_1
    else if (z <= 1.4d-53) then
        tmp = t_0
    else if (z <= 9d-14) then
        tmp = t_1
    else if (z <= 7.9d+34) then
        tmp = x
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x - (Math.log(y) * 0.5);
	double t_1 = y - (Math.log(y) * (y + 0.5));
	double tmp;
	if (z <= -1.9e+128) {
		tmp = y - z;
	} else if (z <= -2.4e+71) {
		tmp = t_1;
	} else if (z <= -2.1e-36) {
		tmp = t_0;
	} else if (z <= -9.5e-205) {
		tmp = t_1;
	} else if (z <= 1.4e-53) {
		tmp = t_0;
	} else if (z <= 9e-14) {
		tmp = t_1;
	} else if (z <= 7.9e+34) {
		tmp = x;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x - (math.log(y) * 0.5)
	t_1 = y - (math.log(y) * (y + 0.5))
	tmp = 0
	if z <= -1.9e+128:
		tmp = y - z
	elif z <= -2.4e+71:
		tmp = t_1
	elif z <= -2.1e-36:
		tmp = t_0
	elif z <= -9.5e-205:
		tmp = t_1
	elif z <= 1.4e-53:
		tmp = t_0
	elif z <= 9e-14:
		tmp = t_1
	elif z <= 7.9e+34:
		tmp = x
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x - Float64(log(y) * 0.5))
	t_1 = Float64(y - Float64(log(y) * Float64(y + 0.5)))
	tmp = 0.0
	if (z <= -1.9e+128)
		tmp = Float64(y - z);
	elseif (z <= -2.4e+71)
		tmp = t_1;
	elseif (z <= -2.1e-36)
		tmp = t_0;
	elseif (z <= -9.5e-205)
		tmp = t_1;
	elseif (z <= 1.4e-53)
		tmp = t_0;
	elseif (z <= 9e-14)
		tmp = t_1;
	elseif (z <= 7.9e+34)
		tmp = x;
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x - (log(y) * 0.5);
	t_1 = y - (log(y) * (y + 0.5));
	tmp = 0.0;
	if (z <= -1.9e+128)
		tmp = y - z;
	elseif (z <= -2.4e+71)
		tmp = t_1;
	elseif (z <= -2.1e-36)
		tmp = t_0;
	elseif (z <= -9.5e-205)
		tmp = t_1;
	elseif (z <= 1.4e-53)
		tmp = t_0;
	elseif (z <= 9e-14)
		tmp = t_1;
	elseif (z <= 7.9e+34)
		tmp = x;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x - N[(N[Log[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y - N[(N[Log[y], $MachinePrecision] * N[(y + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.9e+128], N[(y - z), $MachinePrecision], If[LessEqual[z, -2.4e+71], t$95$1, If[LessEqual[z, -2.1e-36], t$95$0, If[LessEqual[z, -9.5e-205], t$95$1, If[LessEqual[z, 1.4e-53], t$95$0, If[LessEqual[z, 9e-14], t$95$1, If[LessEqual[z, 7.9e+34], x, (-z)]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if z < -1.89999999999999995e128

   1. Initial program 100.0%

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

      1. associate--l+ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 85.7%

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

      1. fma-def 85.8%

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

   6. Simplified 85.8%

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

   7. Taylor expanded in z around inf 78.3%

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

   if -1.89999999999999995e128 < z < -2.39999999999999981e71 or -2.09999999999999991e-36 < z < -9.49999999999999957e-205 or 1.39999999999999993e-53 < z < 8.9999999999999995e-14

   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%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around 0 86.9%

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

      1. fma-def 86.9%

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

   6. Simplified 86.9%

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

   7. Taylor expanded in z around 0 85.1%

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

   if -2.39999999999999981e71 < z < -2.09999999999999991e-36 or -9.49999999999999957e-205 < z < 1.39999999999999993e-53

   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%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in z around 0 94.6%

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

   5. Taylor expanded in y around 0 70.5%

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

   if 8.9999999999999995e-14 < z < 7.89999999999999997e34

   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%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around inf 59.4%

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

   if 7.89999999999999997e34 < 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%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in z around inf 67.9%

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

      1. mul-1-neg 67.9%

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

   6. Simplified 67.9%

      \[\leadsto \color{blue}{-z} \]
3. Recombined 5 regimes into one program.

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{+128}:\\ \;\;\;\;y - z\\ \mathbf{elif}\;z \leq -2.4 \cdot 10^{+71}:\\ \;\;\;\;y - \log y \cdot \left(y + 0.5\right)\\ \mathbf{elif}\;z \leq -2.1 \cdot 10^{-36}:\\ \;\;\;\;x - \log y \cdot 0.5\\ \mathbf{elif}\;z \leq -9.5 \cdot 10^{-205}:\\ \;\;\;\;y - \log y \cdot \left(y + 0.5\right)\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{-53}:\\ \;\;\;\;x - \log y \cdot 0.5\\ \mathbf{elif}\;z \leq 9 \cdot 10^{-14}:\\ \;\;\;\;y - \log y \cdot \left(y + 0.5\right)\\ \mathbf{elif}\;z \leq 7.9 \cdot 10^{+34}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 4: 45.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log y \cdot -0.5\\ \mathbf{if}\;x \leq -2.15 \cdot 10^{+61}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -1.25 \cdot 10^{-239}:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \leq -8.2 \cdot 10^{-306}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-82}:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \leq 8.8 \cdot 10^{-41}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.28 \cdot 10^{+165}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (log y) -0.5)))
   (if (<= x -2.15e+61)
     x
     (if (<= x -1.25e-239)
       (- z)
       (if (<= x -8.2e-306)
         t_0
         (if (<= x 2.5e-82)
           (- z)
           (if (<= x 8.8e-41) t_0 (if (<= x 1.28e+165) (- z) x))))))))

C

double code(double x, double y, double z) {
	double t_0 = log(y) * -0.5;
	double tmp;
	if (x <= -2.15e+61) {
		tmp = x;
	} else if (x <= -1.25e-239) {
		tmp = -z;
	} else if (x <= -8.2e-306) {
		tmp = t_0;
	} else if (x <= 2.5e-82) {
		tmp = -z;
	} else if (x <= 8.8e-41) {
		tmp = t_0;
	} else if (x <= 1.28e+165) {
		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) :: t_0
    real(8) :: tmp
    t_0 = log(y) * (-0.5d0)
    if (x <= (-2.15d+61)) then
        tmp = x
    else if (x <= (-1.25d-239)) then
        tmp = -z
    else if (x <= (-8.2d-306)) then
        tmp = t_0
    else if (x <= 2.5d-82) then
        tmp = -z
    else if (x <= 8.8d-41) then
        tmp = t_0
    else if (x <= 1.28d+165) then
        tmp = -z
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.log(y) * -0.5;
	double tmp;
	if (x <= -2.15e+61) {
		tmp = x;
	} else if (x <= -1.25e-239) {
		tmp = -z;
	} else if (x <= -8.2e-306) {
		tmp = t_0;
	} else if (x <= 2.5e-82) {
		tmp = -z;
	} else if (x <= 8.8e-41) {
		tmp = t_0;
	} else if (x <= 1.28e+165) {
		tmp = -z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.log(y) * -0.5
	tmp = 0
	if x <= -2.15e+61:
		tmp = x
	elif x <= -1.25e-239:
		tmp = -z
	elif x <= -8.2e-306:
		tmp = t_0
	elif x <= 2.5e-82:
		tmp = -z
	elif x <= 8.8e-41:
		tmp = t_0
	elif x <= 1.28e+165:
		tmp = -z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(log(y) * -0.5)
	tmp = 0.0
	if (x <= -2.15e+61)
		tmp = x;
	elseif (x <= -1.25e-239)
		tmp = Float64(-z);
	elseif (x <= -8.2e-306)
		tmp = t_0;
	elseif (x <= 2.5e-82)
		tmp = Float64(-z);
	elseif (x <= 8.8e-41)
		tmp = t_0;
	elseif (x <= 1.28e+165)
		tmp = Float64(-z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = log(y) * -0.5;
	tmp = 0.0;
	if (x <= -2.15e+61)
		tmp = x;
	elseif (x <= -1.25e-239)
		tmp = -z;
	elseif (x <= -8.2e-306)
		tmp = t_0;
	elseif (x <= 2.5e-82)
		tmp = -z;
	elseif (x <= 8.8e-41)
		tmp = t_0;
	elseif (x <= 1.28e+165)
		tmp = -z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision]}, If[LessEqual[x, -2.15e+61], x, If[LessEqual[x, -1.25e-239], (-z), If[LessEqual[x, -8.2e-306], t$95$0, If[LessEqual[x, 2.5e-82], (-z), If[LessEqual[x, 8.8e-41], t$95$0, If[LessEqual[x, 1.28e+165], (-z), x]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.1500000000000001e61 or 1.2799999999999999e165 < x

   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%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around inf 80.1%

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

   if -2.1500000000000001e61 < x < -1.25e-239 or -8.19999999999999969e-306 < x < 2.4999999999999999e-82 or 8.7999999999999999e-41 < x < 1.2799999999999999e165

   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%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in z around inf 44.9%

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

      1. mul-1-neg 44.9%

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

   6. Simplified 44.9%

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

   if -1.25e-239 < x < -8.19999999999999969e-306 or 2.4999999999999999e-82 < x < 8.7999999999999999e-41

   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%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around 0 99.7%

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

      1. fma-def 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in z around 0 89.5%

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

   8. Taylor expanded in y around 0 52.3%

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

      1. *-commutative 52.3%

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

   10. Simplified 52.3%

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

4. Final simplification 55.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.15 \cdot 10^{+61}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -1.25 \cdot 10^{-239}:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \leq -8.2 \cdot 10^{-306}:\\ \;\;\;\;\log y \cdot -0.5\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-82}:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \leq 8.8 \cdot 10^{-41}:\\ \;\;\;\;\log y \cdot -0.5\\ \mathbf{elif}\;x \leq 1.28 \cdot 10^{+165}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5: 89.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.8 \cdot 10^{+127}:\\ \;\;\;\;\left(x + \log y \cdot -0.5\right) - z\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{+65}:\\ \;\;\;\;\left(y + x\right) - \log y \cdot \left(y + 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y - z\right) - y \cdot \log y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.8e+127)
   (- (+ x (* (log y) -0.5)) z)
   (if (<= z 7.2e+65)
     (- (+ y x) (* (log y) (+ y 0.5)))
     (- (- y z) (* y (log y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.8e+127) {
		tmp = (x + (log(y) * -0.5)) - z;
	} else if (z <= 7.2e+65) {
		tmp = (y + x) - (log(y) * (y + 0.5));
	} else {
		tmp = (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 (z <= (-1.8d+127)) then
        tmp = (x + (log(y) * (-0.5d0))) - z
    else if (z <= 7.2d+65) then
        tmp = (y + x) - (log(y) * (y + 0.5d0))
    else
        tmp = (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 (z <= -1.8e+127) {
		tmp = (x + (Math.log(y) * -0.5)) - z;
	} else if (z <= 7.2e+65) {
		tmp = (y + x) - (Math.log(y) * (y + 0.5));
	} else {
		tmp = (y - z) - (y * Math.log(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.8e+127:
		tmp = (x + (math.log(y) * -0.5)) - z
	elif z <= 7.2e+65:
		tmp = (y + x) - (math.log(y) * (y + 0.5))
	else:
		tmp = (y - z) - (y * math.log(y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.8e+127)
		tmp = Float64(Float64(x + Float64(log(y) * -0.5)) - z);
	elseif (z <= 7.2e+65)
		tmp = Float64(Float64(y + x) - Float64(log(y) * Float64(y + 0.5)));
	else
		tmp = Float64(Float64(y - z) - Float64(y * log(y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.8e+127)
		tmp = (x + (log(y) * -0.5)) - z;
	elseif (z <= 7.2e+65)
		tmp = (y + x) - (log(y) * (y + 0.5));
	else
		tmp = (y - z) - (y * log(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.8e+127], N[(N[(x + N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[z, 7.2e+65], N[(N[(y + x), $MachinePrecision] - N[(N[Log[y], $MachinePrecision] * N[(y + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y - z), $MachinePrecision] - N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.79999999999999989e127

   1. Initial program 100.0%

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

      1. associate--l+ 100.0%

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

      2. sub-neg 100.0%

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

      3. +-commutative 100.0%

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

      4. associate-+l+ 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-rgt-neg-in 100.0%

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

      7. fma-def 100.0%

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

      8. neg-sub0 100.0%

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

      9. +-commutative 100.0%

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

      10. associate--r+ 100.0%

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

      11. metadata-eval 100.0%

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

      12. associate-+r- 100.0%

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

      13. +-commutative 100.0%

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

      14. associate-+r- 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 92.5%

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

   if -1.79999999999999989e127 < z < 7.19999999999999957e65

   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%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in z around 0 94.4%

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

   if 7.19999999999999957e65 < 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+ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around inf 87.9%

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

      1. *-commutative 87.9%

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

      2. log-rec 87.9%

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

      3. distribute-lft-neg-in 87.9%

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

      4. distribute-rgt-neg-in 87.9%

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

   6. Simplified 87.9%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.8 \cdot 10^{+127}:\\ \;\;\;\;\left(x + \log y \cdot -0.5\right) - z\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{+65}:\\ \;\;\;\;\left(y + x\right) - \log y \cdot \left(y + 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y - z\right) - y \cdot \log y\\ \end{array} \]

Alternative 6: 60.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.8 \cdot 10^{+133}:\\ \;\;\;\;y - z\\ \mathbf{elif}\;z \leq -7.3 \cdot 10^{+71}:\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \mathbf{elif}\;z \leq 10^{+71}:\\ \;\;\;\;x - \log y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -3.8e+133)
   (- y z)
   (if (<= z -7.3e+71)
     (* y (- 1.0 (log y)))
     (if (<= z 1e+71) (- x (* (log y) 0.5)) (- z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -3.8e+133) {
		tmp = y - z;
	} else if (z <= -7.3e+71) {
		tmp = y * (1.0 - log(y));
	} else if (z <= 1e+71) {
		tmp = x - (log(y) * 0.5);
	} else {
		tmp = -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 <= (-3.8d+133)) then
        tmp = y - z
    else if (z <= (-7.3d+71)) then
        tmp = y * (1.0d0 - log(y))
    else if (z <= 1d+71) then
        tmp = x - (log(y) * 0.5d0)
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -3.8e+133) {
		tmp = y - z;
	} else if (z <= -7.3e+71) {
		tmp = y * (1.0 - Math.log(y));
	} else if (z <= 1e+71) {
		tmp = x - (Math.log(y) * 0.5);
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -3.8e+133:
		tmp = y - z
	elif z <= -7.3e+71:
		tmp = y * (1.0 - math.log(y))
	elif z <= 1e+71:
		tmp = x - (math.log(y) * 0.5)
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -3.8e+133)
		tmp = Float64(y - z);
	elseif (z <= -7.3e+71)
		tmp = Float64(y * Float64(1.0 - log(y)));
	elseif (z <= 1e+71)
		tmp = Float64(x - Float64(log(y) * 0.5));
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -3.8e+133)
		tmp = y - z;
	elseif (z <= -7.3e+71)
		tmp = y * (1.0 - log(y));
	elseif (z <= 1e+71)
		tmp = x - (log(y) * 0.5);
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -3.8e+133], N[(y - z), $MachinePrecision], If[LessEqual[z, -7.3e+71], N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1e+71], N[(x - N[(N[Log[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], (-z)]]]

Derivation

1. Split input into 4 regimes

2. if z < -3.8000000000000002e133

   1. Initial program 100.0%

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

      1. associate--l+ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 85.7%

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

      1. fma-def 85.8%

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

   6. Simplified 85.8%

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

   7. Taylor expanded in z around inf 78.3%

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

   if -3.8000000000000002e133 < z < -7.29999999999999996e71

   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%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around inf 85.9%

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

      1. *-commutative 85.9%

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

      2. log-rec 85.9%

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

      3. cancel-sign-sub 85.9%

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

      4. *-commutative 85.9%

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

      5. neg-mul-1 85.9%

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

      6. sub-neg 85.9%

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

   6. Simplified 85.9%

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

   if -7.29999999999999996e71 < z < 1e71

   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%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in z around 0 94.8%

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

   5. Taylor expanded in y around 0 64.3%

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

   if 1e71 < 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+ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in z around inf 71.6%

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

      1. mul-1-neg 71.6%

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

   6. Simplified 71.6%

      \[\leadsto \color{blue}{-z} \]
3. Recombined 4 regimes into one program.

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.8 \cdot 10^{+133}:\\ \;\;\;\;y - z\\ \mathbf{elif}\;z \leq -7.3 \cdot 10^{+71}:\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \mathbf{elif}\;z \leq 10^{+71}:\\ \;\;\;\;x - \log y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 7: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return (x - (log(y) * (y + 0.5))) + (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 - (log(y) * (y + 0.5d0))) + (y - z)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

3. Simplified 99.8%

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

4. Final simplification 99.8%

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

Alternative 8: 84.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.22 \cdot 10^{+148}:\\ \;\;\;\;\left(x + \log y \cdot -0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;y - \log y \cdot \left(y + 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.22e+148)
   (- (+ x (* (log y) -0.5)) z)
   (- y (* (log y) (+ y 0.5)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.22e+148) {
		tmp = (x + (log(y) * -0.5)) - z;
	} else {
		tmp = y - (log(y) * (y + 0.5));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 1.22d+148) then
        tmp = (x + (log(y) * (-0.5d0))) - z
    else
        tmp = y - (log(y) * (y + 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.22e+148) {
		tmp = (x + (Math.log(y) * -0.5)) - z;
	} else {
		tmp = y - (Math.log(y) * (y + 0.5));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1.22e+148:
		tmp = (x + (math.log(y) * -0.5)) - z
	else:
		tmp = y - (math.log(y) * (y + 0.5))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.22e+148)
		tmp = Float64(Float64(x + Float64(log(y) * -0.5)) - z);
	else
		tmp = Float64(y - Float64(log(y) * Float64(y + 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1.22e+148)
		tmp = (x + (log(y) * -0.5)) - z;
	else
		tmp = y - (log(y) * (y + 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.22e+148], N[(N[(x + N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(y - N[(N[Log[y], $MachinePrecision] * N[(y + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.22000000000000007e148

   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. sub-neg 99.9%

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

      3. +-commutative 99.9%

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

      4. associate-+l+ 99.9%

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

      5. *-commutative 99.9%

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

      6. distribute-rgt-neg-in 99.9%

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

      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. Taylor expanded in y around 0 89.6%

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

   if 1.22000000000000007e148 < 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+ 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around 0 86.8%

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

      1. fma-def 86.8%

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

   6. Simplified 86.8%

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

   7. Taylor expanded in z around 0 68.4%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.22 \cdot 10^{+148}:\\ \;\;\;\;\left(x + \log y \cdot -0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;y - \log y \cdot \left(y + 0.5\right)\\ \end{array} \]

Alternative 9: 89.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6.8 \cdot 10^{+52}:\\ \;\;\;\;\left(x + \log y \cdot -0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(y - z\right) - y \cdot \log y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 6.8e+52) (- (+ x (* (log y) -0.5)) z) (- (- y z) (* y (log y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 6.8e+52) {
		tmp = (x + (log(y) * -0.5)) - z;
	} else {
		tmp = (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 <= 6.8d+52) then
        tmp = (x + (log(y) * (-0.5d0))) - z
    else
        tmp = (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 <= 6.8e+52) {
		tmp = (x + (Math.log(y) * -0.5)) - z;
	} else {
		tmp = (y - z) - (y * Math.log(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 6.8e+52:
		tmp = (x + (math.log(y) * -0.5)) - z
	else:
		tmp = (y - z) - (y * math.log(y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 6.8e+52)
		tmp = Float64(Float64(x + Float64(log(y) * -0.5)) - z);
	else
		tmp = 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 <= 6.8e+52)
		tmp = (x + (log(y) * -0.5)) - z;
	else
		tmp = (y - z) - (y * log(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 6.8e+52], N[(N[(x + N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(N[(y - z), $MachinePrecision] - N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.8e52

   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. sub-neg 99.9%

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

      3. +-commutative 99.9%

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

      4. associate-+l+ 99.9%

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

      5. *-commutative 99.9%

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

      6. distribute-rgt-neg-in 99.9%

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

      7. fma-def 100.0%

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

      8. neg-sub0 100.0%

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

      9. +-commutative 100.0%

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

      10. associate--r+ 100.0%

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

      11. metadata-eval 100.0%

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

      12. associate-+r- 100.0%

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

      13. +-commutative 100.0%

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

      14. associate-+r- 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 96.5%

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

   if 6.8e52 < 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%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around inf 80.5%

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

      1. *-commutative 80.5%

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

      2. log-rec 80.5%

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

      3. distribute-lft-neg-in 80.5%

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

      4. distribute-rgt-neg-in 80.5%

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

   6. Simplified 80.5%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.8 \cdot 10^{+52}:\\ \;\;\;\;\left(x + \log y \cdot -0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(y - z\right) - y \cdot \log y\\ \end{array} \]

Alternative 10: 46.8% accurate, 18.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.95 \cdot 10^{+61}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.28 \cdot 10^{+165}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.95e+61) x (if (<= x 1.28e+165) (- z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.95e+61) {
		tmp = x;
	} else if (x <= 1.28e+165) {
		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 <= (-1.95d+61)) then
        tmp = x
    else if (x <= 1.28d+165) 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 <= -1.95e+61) {
		tmp = x;
	} else if (x <= 1.28e+165) {
		tmp = -z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.95e+61:
		tmp = x
	elif x <= 1.28e+165:
		tmp = -z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.95e+61)
		tmp = x;
	elseif (x <= 1.28e+165)
		tmp = Float64(-z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.95e+61)
		tmp = x;
	elseif (x <= 1.28e+165)
		tmp = -z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.95e+61], x, If[LessEqual[x, 1.28e+165], (-z), x]]

Derivation

1. Split input into 2 regimes

2. if x < -1.94999999999999994e61 or 1.2799999999999999e165 < x

   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%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around inf 80.1%

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

   if -1.94999999999999994e61 < x < 1.2799999999999999e165

   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%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in z around inf 40.2%

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

      1. mul-1-neg 40.2%

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

   6. Simplified 40.2%

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

4. Final simplification 51.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.95 \cdot 10^{+61}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.28 \cdot 10^{+165}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 11: 29.7% accurate, 111.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 99.8%

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

   1. associate--l+ 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in x around inf 29.4%

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

5. Final simplification 29.4%

   \[\leadsto x \]

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 2023223 
(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))