Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Percentage Accurate: 99.8% → 99.9%

Time: 11.1s

Alternatives: 14

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate--l+ 99.8%

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

   2. sub-neg 99.8%

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

   3. associate-+l+ 99.8%

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

   4. associate-+r- 99.8%

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

   5. *-commutative 99.8%

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

   6. distribute-rgt-neg-in 99.8%

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

   7. fma-define 99.9%

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

   8. +-commutative 99.9%

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

   9. distribute-neg-in 99.9%

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

   10. unsub-neg 99.9%

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

   11. metadata-eval 99.9%

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

3. Simplified 99.9%

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

Alternative 2: 74.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x + y\right) - y \cdot \log y\\ t_1 := y \cdot \left(1 - \log y\right) - z\\ \mathbf{if}\;x \leq -6.7 \cdot 10^{+130}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-265}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{-112}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{elif}\;x \leq 2.45 \cdot 10^{+61}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (+ x y) (* y (log y)))) (t_1 (- (* y (- 1.0 (log y))) z)))
   (if (<= x -6.7e+130)
     t_0
     (if (<= x 1.4e-265)
       t_1
       (if (<= x 1.95e-112)
         (- (* (log y) -0.5) z)
         (if (<= x 2.45e+61) t_1 t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = (x + y) - (y * log(y));
	double t_1 = (y * (1.0 - log(y))) - z;
	double tmp;
	if (x <= -6.7e+130) {
		tmp = t_0;
	} else if (x <= 1.4e-265) {
		tmp = t_1;
	} else if (x <= 1.95e-112) {
		tmp = (log(y) * -0.5) - z;
	} else if (x <= 2.45e+61) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (x + y) - (y * log(y))
    t_1 = (y * (1.0d0 - log(y))) - z
    if (x <= (-6.7d+130)) then
        tmp = t_0
    else if (x <= 1.4d-265) then
        tmp = t_1
    else if (x <= 1.95d-112) then
        tmp = (log(y) * (-0.5d0)) - z
    else if (x <= 2.45d+61) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x + y) - (y * Math.log(y));
	double t_1 = (y * (1.0 - Math.log(y))) - z;
	double tmp;
	if (x <= -6.7e+130) {
		tmp = t_0;
	} else if (x <= 1.4e-265) {
		tmp = t_1;
	} else if (x <= 1.95e-112) {
		tmp = (Math.log(y) * -0.5) - z;
	} else if (x <= 2.45e+61) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x + y) - (y * math.log(y))
	t_1 = (y * (1.0 - math.log(y))) - z
	tmp = 0
	if x <= -6.7e+130:
		tmp = t_0
	elif x <= 1.4e-265:
		tmp = t_1
	elif x <= 1.95e-112:
		tmp = (math.log(y) * -0.5) - z
	elif x <= 2.45e+61:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x + y) - Float64(y * log(y)))
	t_1 = Float64(Float64(y * Float64(1.0 - log(y))) - z)
	tmp = 0.0
	if (x <= -6.7e+130)
		tmp = t_0;
	elseif (x <= 1.4e-265)
		tmp = t_1;
	elseif (x <= 1.95e-112)
		tmp = Float64(Float64(log(y) * -0.5) - z);
	elseif (x <= 2.45e+61)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x + y) - (y * log(y));
	t_1 = (y * (1.0 - log(y))) - z;
	tmp = 0.0;
	if (x <= -6.7e+130)
		tmp = t_0;
	elseif (x <= 1.4e-265)
		tmp = t_1;
	elseif (x <= 1.95e-112)
		tmp = (log(y) * -0.5) - z;
	elseif (x <= 2.45e+61)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x + y), $MachinePrecision] - N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]}, If[LessEqual[x, -6.7e+130], t$95$0, If[LessEqual[x, 1.4e-265], t$95$1, If[LessEqual[x, 1.95e-112], N[(N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[x, 2.45e+61], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.7000000000000001e130 or 2.45000000000000013e61 < x

   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. +-commutative 99.8%

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

      2. associate--l+ 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around inf 99.8%

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

      1. mul-1-neg 99.8%

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

      2. distribute-rgt-neg-in 99.8%

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

      3. log-rec 99.8%

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

      4. remove-double-neg 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in z around 0 89.7%

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

   if -6.7000000000000001e130 < x < 1.40000000000000012e-265 or 1.9500000000000001e-112 < x < 2.45000000000000013e61

   1. Initial program 99.8%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 99.8%

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

      1. associate--l+ 99.8%

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

      2. mul-1-neg 99.8%

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

      3. log-rec 99.8%

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

      4. remove-double-neg 99.8%

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

      5. associate--r+ 99.8%

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

      6. associate-*r/ 99.8%

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

      7. log-rec 99.8%

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

      8. mul-1-neg 99.8%

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

      9. associate-*r* 99.8%

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

      10. metadata-eval 99.8%

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

      11. *-commutative 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in y around inf 82.6%

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

      1. log-rec 82.6%

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

      2. cancel-sign-sub-inv 82.6%

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

      3. metadata-eval 82.6%

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

      4. *-lft-identity 82.6%

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

      5. sub-neg 82.6%

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

   8. Simplified 82.6%

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

   if 1.40000000000000012e-265 < x < 1.9500000000000001e-112

   1. Initial program 99.9%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 99.9%

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

   4. Taylor expanded in y around 0 83.9%

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

      1. mul-1-neg 83.9%

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

      2. distribute-neg-in 83.9%

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

      3. distribute-lft-neg-in 83.9%

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

      4. metadata-eval 83.9%

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

   6. Simplified 83.9%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.7 \cdot 10^{+130}:\\ \;\;\;\;\left(x + y\right) - y \cdot \log y\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-265}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{-112}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{elif}\;x \leq 2.45 \cdot 10^{+61}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) - y \cdot \log y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 8.5 \cdot 10^{-208}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{+99}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 8.5e-208)
   (- (* (log y) -0.5) z)
   (if (<= y 9.5e+99) (- x z) (- (* y (- 1.0 (log y))) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 8.5e-208) {
		tmp = (log(y) * -0.5) - z;
	} else if (y <= 9.5e+99) {
		tmp = x - z;
	} else {
		tmp = (y * (1.0 - log(y))) - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 8.5d-208) then
        tmp = (log(y) * (-0.5d0)) - z
    else if (y <= 9.5d+99) then
        tmp = x - z
    else
        tmp = (y * (1.0d0 - log(y))) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 8.5e-208) {
		tmp = (Math.log(y) * -0.5) - z;
	} else if (y <= 9.5e+99) {
		tmp = x - z;
	} else {
		tmp = (y * (1.0 - Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 8.5e-208:
		tmp = (math.log(y) * -0.5) - z
	elif y <= 9.5e+99:
		tmp = x - z
	else:
		tmp = (y * (1.0 - math.log(y))) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 8.5e-208)
		tmp = Float64(Float64(log(y) * -0.5) - z);
	elseif (y <= 9.5e+99)
		tmp = Float64(x - z);
	else
		tmp = Float64(Float64(y * Float64(1.0 - log(y))) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 8.5e-208)
		tmp = (log(y) * -0.5) - z;
	elseif (y <= 9.5e+99)
		tmp = x - z;
	else
		tmp = (y * (1.0 - log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 8.5e-208], N[(N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[y, 9.5e+99], N[(x - z), $MachinePrecision], N[(N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 8.49999999999999997e-208

   1. Initial program 100.0%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 75.2%

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

   4. Taylor expanded in y around 0 75.2%

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

      1. mul-1-neg 75.2%

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

      2. distribute-neg-in 75.2%

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

      3. distribute-lft-neg-in 75.2%

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

      4. metadata-eval 75.2%

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

   6. Simplified 75.2%

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

   if 8.49999999999999997e-208 < y < 9.49999999999999908e99

   1. Initial program 100.0%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 86.9%

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

      1. associate--l+ 86.9%

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

      2. mul-1-neg 86.9%

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

      3. log-rec 86.9%

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

      4. remove-double-neg 86.9%

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

      5. associate--r+ 86.9%

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

      6. associate-*r/ 86.9%

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

      7. log-rec 86.9%

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

      8. mul-1-neg 86.9%

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

      9. associate-*r* 86.9%

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

      10. metadata-eval 86.9%

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

      11. *-commutative 86.9%

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

   5. Simplified 86.9%

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

   6. Taylor expanded in x around inf 80.0%

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

   if 9.49999999999999908e99 < y

   1. Initial program 99.6%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 99.8%

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

      1. associate--l+ 99.8%

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

      2. mul-1-neg 99.8%

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

      3. log-rec 99.8%

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

      4. remove-double-neg 99.8%

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

      5. associate--r+ 99.8%

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

      6. associate-*r/ 99.8%

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

      7. log-rec 99.8%

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

      8. mul-1-neg 99.8%

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

      9. associate-*r* 99.8%

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

      10. metadata-eval 99.8%

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

      11. *-commutative 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in y around inf 83.4%

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

      1. log-rec 83.4%

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

      2. cancel-sign-sub-inv 83.4%

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

      3. metadata-eval 83.4%

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

      4. *-lft-identity 83.4%

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

      5. sub-neg 83.4%

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

   8. Simplified 83.4%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8.5 \cdot 10^{-208}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{+99}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.6 \cdot 10^{-208}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+99}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y - \left(z + y \cdot \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 3.6e-208)
   (- (* (log y) -0.5) z)
   (if (<= y 2.5e+99) (- x z) (- y (+ z (* y (log y)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 3.6e-208) {
		tmp = (log(y) * -0.5) - z;
	} else if (y <= 2.5e+99) {
		tmp = x - 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 <= 3.6d-208) then
        tmp = (log(y) * (-0.5d0)) - z
    else if (y <= 2.5d+99) then
        tmp = x - 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 <= 3.6e-208) {
		tmp = (Math.log(y) * -0.5) - z;
	} else if (y <= 2.5e+99) {
		tmp = x - z;
	} else {
		tmp = y - (z + (y * Math.log(y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 3.6e-208:
		tmp = (math.log(y) * -0.5) - z
	elif y <= 2.5e+99:
		tmp = x - z
	else:
		tmp = y - (z + (y * math.log(y)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 3.6e-208)
		tmp = Float64(Float64(log(y) * -0.5) - z);
	elseif (y <= 2.5e+99)
		tmp = Float64(x - z);
	else
		tmp = Float64(y - Float64(z + Float64(y * log(y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 3.6e-208)
		tmp = (log(y) * -0.5) - z;
	elseif (y <= 2.5e+99)
		tmp = x - z;
	else
		tmp = y - (z + (y * log(y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 3.6e-208], N[(N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[y, 2.5e+99], N[(x - z), $MachinePrecision], N[(y - N[(z + N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 3.5999999999999998e-208

   1. Initial program 100.0%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 75.2%

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

   4. Taylor expanded in y around 0 75.2%

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

      1. mul-1-neg 75.2%

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

      2. distribute-neg-in 75.2%

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

      3. distribute-lft-neg-in 75.2%

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

      4. metadata-eval 75.2%

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

   6. Simplified 75.2%

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

   if 3.5999999999999998e-208 < y < 2.50000000000000004e99

   1. Initial program 100.0%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 86.9%

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

      1. associate--l+ 86.9%

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

      2. mul-1-neg 86.9%

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

      3. log-rec 86.9%

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

      4. remove-double-neg 86.9%

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

      5. associate--r+ 86.9%

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

      6. associate-*r/ 86.9%

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

      7. log-rec 86.9%

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

      8. mul-1-neg 86.9%

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

      9. associate-*r* 86.9%

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

      10. metadata-eval 86.9%

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

      11. *-commutative 86.9%

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

   5. Simplified 86.9%

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

   6. Taylor expanded in x around inf 80.0%

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

   if 2.50000000000000004e99 < y

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. associate--l+ 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around inf 99.6%

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

      1. mul-1-neg 99.6%

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

      2. distribute-rgt-neg-in 99.6%

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

      3. log-rec 99.6%

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

      4. remove-double-neg 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in x around 0 83.3%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.6 \cdot 10^{-208}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+99}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y - \left(z + y \cdot \log y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 69.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.8 \cdot 10^{-208}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{+153}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y - \log y \cdot \left(y + 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 4.8e-208)
   (- (* (log y) -0.5) z)
   (if (<= y 4.6e+153) (- x z) (- y (* (log y) (+ y 0.5))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 4.8e-208) {
		tmp = (log(y) * -0.5) - z;
	} else if (y <= 4.6e+153) {
		tmp = x - 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 <= 4.8d-208) then
        tmp = (log(y) * (-0.5d0)) - z
    else if (y <= 4.6d+153) then
        tmp = x - 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 <= 4.8e-208) {
		tmp = (Math.log(y) * -0.5) - z;
	} else if (y <= 4.6e+153) {
		tmp = x - z;
	} else {
		tmp = y - (Math.log(y) * (y + 0.5));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 4.8e-208:
		tmp = (math.log(y) * -0.5) - z
	elif y <= 4.6e+153:
		tmp = x - z
	else:
		tmp = y - (math.log(y) * (y + 0.5))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 4.8e-208)
		tmp = Float64(Float64(log(y) * -0.5) - z);
	elseif (y <= 4.6e+153)
		tmp = Float64(x - 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 <= 4.8e-208)
		tmp = (log(y) * -0.5) - z;
	elseif (y <= 4.6e+153)
		tmp = x - z;
	else
		tmp = y - (log(y) * (y + 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 4.8e-208], N[(N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[y, 4.6e+153], N[(x - z), $MachinePrecision], N[(y - N[(N[Log[y], $MachinePrecision] * N[(y + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 4.7999999999999998e-208

   1. Initial program 100.0%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 75.2%

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

   4. Taylor expanded in y around 0 75.2%

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

      1. mul-1-neg 75.2%

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

      2. distribute-neg-in 75.2%

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

      3. distribute-lft-neg-in 75.2%

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

      4. metadata-eval 75.2%

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

   6. Simplified 75.2%

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

   if 4.7999999999999998e-208 < y < 4.6000000000000003e153

   1. Initial program 99.9%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 89.5%

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

      1. associate--l+ 89.5%

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

      2. mul-1-neg 89.5%

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

      3. log-rec 89.5%

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

      4. remove-double-neg 89.5%

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

      5. associate--r+ 89.5%

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

      6. associate-*r/ 89.5%

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

      7. log-rec 89.5%

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

      8. mul-1-neg 89.5%

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

      9. associate-*r* 89.5%

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

      10. metadata-eval 89.5%

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

      11. *-commutative 89.5%

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

   5. Simplified 89.5%

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

   6. Taylor expanded in x around inf 76.3%

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

   if 4.6000000000000003e153 < y

   1. Initial program 99.6%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 87.7%

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

   4. Taylor expanded in z around 0 77.8%

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

      1. +-commutative 77.8%

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

   6. Simplified 77.8%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.8 \cdot 10^{-208}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{+153}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y - \log y \cdot \left(y + 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 69.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.5 \cdot 10^{-208}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+153}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y - y \cdot \log y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 3.5e-208)
   (- (* (log y) -0.5) z)
   (if (<= y 5e+153) (- x z) (- y (* y (log y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 3.5e-208) {
		tmp = (log(y) * -0.5) - z;
	} else if (y <= 5e+153) {
		tmp = x - z;
	} else {
		tmp = y - (y * log(y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 3.5d-208) then
        tmp = (log(y) * (-0.5d0)) - z
    else if (y <= 5d+153) then
        tmp = x - z
    else
        tmp = y - (y * log(y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 3.5e-208) {
		tmp = (Math.log(y) * -0.5) - z;
	} else if (y <= 5e+153) {
		tmp = x - z;
	} else {
		tmp = y - (y * Math.log(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 3.5e-208:
		tmp = (math.log(y) * -0.5) - z
	elif y <= 5e+153:
		tmp = x - z
	else:
		tmp = y - (y * math.log(y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 3.5e-208)
		tmp = Float64(Float64(log(y) * -0.5) - z);
	elseif (y <= 5e+153)
		tmp = Float64(x - z);
	else
		tmp = Float64(y - Float64(y * log(y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 3.5e-208)
		tmp = (log(y) * -0.5) - z;
	elseif (y <= 5e+153)
		tmp = x - z;
	else
		tmp = y - (y * log(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 3.5e-208], N[(N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[y, 5e+153], N[(x - z), $MachinePrecision], N[(y - N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 3.49999999999999991e-208

   1. Initial program 100.0%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 75.2%

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

   4. Taylor expanded in y around 0 75.2%

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

      1. mul-1-neg 75.2%

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

      2. distribute-neg-in 75.2%

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

      3. distribute-lft-neg-in 75.2%

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

      4. metadata-eval 75.2%

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

   6. Simplified 75.2%

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

   if 3.49999999999999991e-208 < y < 5.00000000000000018e153

   1. Initial program 99.9%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 89.5%

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

      1. associate--l+ 89.5%

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

      2. mul-1-neg 89.5%

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

      3. log-rec 89.5%

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

      4. remove-double-neg 89.5%

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

      5. associate--r+ 89.5%

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

      6. associate-*r/ 89.5%

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

      7. log-rec 89.5%

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

      8. mul-1-neg 89.5%

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

      9. associate-*r* 89.5%

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

      10. metadata-eval 89.5%

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

      11. *-commutative 89.5%

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

   5. Simplified 89.5%

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

   6. Taylor expanded in x around inf 76.3%

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

   if 5.00000000000000018e153 < y

   1. Initial program 99.6%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 87.7%

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

   4. Taylor expanded in y around inf 87.7%

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

      1. mul-1-neg 87.7%

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

      2. distribute-rgt-neg-in 87.7%

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

      3. log-rec 87.7%

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

      4. remove-double-neg 87.7%

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

   6. Simplified 87.7%

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

   7. Taylor expanded in z around 0 77.8%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.5 \cdot 10^{-208}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+153}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y - y \cdot \log y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.3 \cdot 10^{-12}:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + \left(y \cdot \left(1 - \log y\right) - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 3.3e-12)
   (- (- x (* (log y) 0.5)) z)
   (+ x (- (* y (- 1.0 (log y))) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 3.3e-12) {
		tmp = (x - (log(y) * 0.5)) - z;
	} else {
		tmp = x + ((y * (1.0 - log(y))) - z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 3.3d-12) then
        tmp = (x - (log(y) * 0.5d0)) - z
    else
        tmp = x + ((y * (1.0d0 - log(y))) - z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 3.3e-12) {
		tmp = (x - (Math.log(y) * 0.5)) - z;
	} else {
		tmp = x + ((y * (1.0 - Math.log(y))) - z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 3.3e-12:
		tmp = (x - (math.log(y) * 0.5)) - z
	else:
		tmp = x + ((y * (1.0 - math.log(y))) - z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 3.3e-12)
		tmp = Float64(Float64(x - Float64(log(y) * 0.5)) - z);
	else
		tmp = Float64(x + Float64(Float64(y * Float64(1.0 - log(y))) - z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 3.3e-12)
		tmp = (x - (log(y) * 0.5)) - z;
	else
		tmp = x + ((y * (1.0 - log(y))) - z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 3.3e-12], N[(N[(x - N[(N[Log[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(x + N[(N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.3000000000000001e-12

   1. Initial program 100.0%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 100.0%

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

   if 3.3000000000000001e-12 < y

   1. Initial program 99.7%

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

      1. associate--l+ 99.7%

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

      2. sub-neg 99.7%

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

      3. associate-+l+ 99.7%

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

      4. associate-+r- 99.7%

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

      5. *-commutative 99.7%

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

      6. distribute-rgt-neg-in 99.7%

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

      7. fma-define 99.8%

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

      8. +-commutative 99.8%

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

      9. distribute-neg-in 99.8%

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

      10. unsub-neg 99.8%

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

      11. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around inf 99.8%

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

      1. log-rec 99.8%

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

      2. sub-neg 99.8%

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

   7. Simplified 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.3 \cdot 10^{-12}:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + \left(y \cdot \left(1 - \log y\right) - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 89.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7.8 \cdot 10^{+111}:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) - y \cdot \log y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 7.8e+111) (- (- x (* (log y) 0.5)) z) (- (+ x y) (* y (log y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 7.8e+111) {
		tmp = (x - (log(y) * 0.5)) - z;
	} else {
		tmp = (x + y) - (y * log(y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 7.8d+111) then
        tmp = (x - (log(y) * 0.5d0)) - z
    else
        tmp = (x + y) - (y * log(y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 7.8e+111) {
		tmp = (x - (Math.log(y) * 0.5)) - z;
	} else {
		tmp = (x + y) - (y * Math.log(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 7.8e+111:
		tmp = (x - (math.log(y) * 0.5)) - z
	else:
		tmp = (x + y) - (y * math.log(y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 7.8e+111)
		tmp = Float64(Float64(x - Float64(log(y) * 0.5)) - z);
	else
		tmp = Float64(Float64(x + y) - Float64(y * log(y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 7.8e+111)
		tmp = (x - (log(y) * 0.5)) - z;
	else
		tmp = (x + y) - (y * log(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 7.8e+111], N[(N[(x - N[(N[Log[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(N[(x + y), $MachinePrecision] - N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 7.79999999999999958e111

   1. Initial program 100.0%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 95.1%

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

   if 7.79999999999999958e111 < y

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. associate--l+ 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around inf 99.6%

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

      1. mul-1-neg 99.6%

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

      2. distribute-rgt-neg-in 99.6%

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

      3. log-rec 99.6%

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

      4. remove-double-neg 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in z around 0 85.5%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7.8 \cdot 10^{+111}:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) - y \cdot \log y\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
2. Add Preprocessing

3. Final simplification 99.8%

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

Alternative 10: 99.8% accurate, 1.0× speedup

Math

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

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(y + Float64(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[(y + N[(N[(x - N[(N[Log[y], $MachinePrecision] * N[(y + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

   1. +-commutative 99.8%

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

   2. associate--l+ 99.8%

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

3. Simplified 99.8%

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

5. Final simplification 99.8%

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

Alternative 11: 71.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6 \cdot 10^{+153}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y - y \cdot \log y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 6e+153) (- x z) (- y (* y (log y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 6e+153) {
		tmp = x - z;
	} else {
		tmp = y - (y * log(y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 6d+153) then
        tmp = x - z
    else
        tmp = y - (y * log(y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 6e+153) {
		tmp = x - z;
	} else {
		tmp = y - (y * Math.log(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 6e+153:
		tmp = x - z
	else:
		tmp = y - (y * math.log(y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 6e+153)
		tmp = Float64(x - z);
	else
		tmp = Float64(y - Float64(y * log(y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 6e+153)
		tmp = x - z;
	else
		tmp = y - (y * log(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 6e+153], N[(x - z), $MachinePrecision], N[(y - N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.00000000000000037e153

   1. Initial program 100.0%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 86.8%

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

      1. associate--l+ 86.8%

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

      2. mul-1-neg 86.8%

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

      3. log-rec 86.8%

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

      4. remove-double-neg 86.8%

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

      5. associate--r+ 86.8%

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

      6. associate-*r/ 86.8%

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

      7. log-rec 86.8%

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

      8. mul-1-neg 86.8%

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

      9. associate-*r* 86.8%

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

      10. metadata-eval 86.8%

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

      11. *-commutative 86.8%

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

   5. Simplified 86.8%

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

   6. Taylor expanded in x around inf 73.5%

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

   if 6.00000000000000037e153 < y

   1. Initial program 99.6%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 87.7%

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

   4. Taylor expanded in y around inf 87.7%

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

      1. mul-1-neg 87.7%

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

      2. distribute-rgt-neg-in 87.7%

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

      3. log-rec 87.7%

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

      4. remove-double-neg 87.7%

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

   6. Simplified 87.7%

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

   7. Taylor expanded in z around 0 77.8%

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

Alternative 12: 47.1% accurate, 9.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{+130}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{+70}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -4.5e+130) x (if (<= x 2.05e+70) (- z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -4.5e+130) {
		tmp = x;
	} else if (x <= 2.05e+70) {
		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 <= (-4.5d+130)) then
        tmp = x
    else if (x <= 2.05d+70) 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 <= -4.5e+130) {
		tmp = x;
	} else if (x <= 2.05e+70) {
		tmp = -z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -4.5e+130:
		tmp = x
	elif x <= 2.05e+70:
		tmp = -z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -4.5e+130)
		tmp = x;
	elseif (x <= 2.05e+70)
		tmp = Float64(-z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -4.5e+130)
		tmp = x;
	elseif (x <= 2.05e+70)
		tmp = -z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -4.5e+130], x, If[LessEqual[x, 2.05e+70], (-z), x]]

Derivation

1. Split input into 2 regimes

2. if x < -4.50000000000000039e130 or 2.0500000000000001e70 < 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. +-commutative 99.9%

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

      2. associate--l+ 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around inf 72.1%

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

   if -4.50000000000000039e130 < x < 2.0500000000000001e70

   1. Initial program 99.8%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 98.0%

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

   4. Taylor expanded in z around inf 42.0%

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

      1. neg-mul-1 42.0%

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

   6. Simplified 42.0%

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

Alternative 13: 57.6% accurate, 37.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
2. Add Preprocessing

3. Taylor expanded in y around inf 90.7%

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

   1. associate--l+ 90.7%

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

   2. mul-1-neg 90.7%

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

   3. log-rec 90.7%

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

   4. remove-double-neg 90.7%

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

   5. associate--r+ 90.7%

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

   6. associate-*r/ 90.7%

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

   7. log-rec 90.7%

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

   8. mul-1-neg 90.7%

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

   9. associate-*r* 90.7%

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

   10. metadata-eval 90.7%

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

   11. *-commutative 90.7%

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

5. Simplified 90.7%

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

6. Taylor expanded in x around inf 57.4%

   \[\leadsto \color{blue}{x} - z \]
7. Add Preprocessing

Alternative 14: 30.4% 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. +-commutative 99.8%

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

   2. associate--l+ 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in x around inf 28.4%

   \[\leadsto \color{blue}{x} \]
6. Add Preprocessing

Developer target: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024086 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2"
  :precision binary64

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

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