Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Percentage Accurate: 99.8% → 99.9%

Time: 10.5s

Alternatives: 12

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 99.8%

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

   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. Add Preprocessing

Alternative 2: 74.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(1 - \log y\right)\\ t_1 := x - \log y \cdot 0.5\\ \mathbf{if}\;y \leq 2.6 \cdot 10^{-263}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.95 \cdot 10^{-233}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{-133}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 100000000:\\ \;\;\;\;\left(x + y\right) - z\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+112} \lor \neg \left(y \leq 1.9 \cdot 10^{+168}\right):\\ \;\;\;\;t\_0 - z\\ \mathbf{else}:\\ \;\;\;\;x + t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (- 1.0 (log y)))) (t_1 (- x (* (log y) 0.5))))
   (if (<= y 2.6e-263)
     t_1
     (if (<= y 1.95e-233)
       (- x z)
       (if (<= y 3.6e-133)
         t_1
         (if (<= y 100000000.0)
           (- (+ x y) z)
           (if (or (<= y 2e+112) (not (<= y 1.9e+168)))
             (- t_0 z)
             (+ x t_0))))))))

C

double code(double x, double y, double z) {
	double t_0 = y * (1.0 - log(y));
	double t_1 = x - (log(y) * 0.5);
	double tmp;
	if (y <= 2.6e-263) {
		tmp = t_1;
	} else if (y <= 1.95e-233) {
		tmp = x - z;
	} else if (y <= 3.6e-133) {
		tmp = t_1;
	} else if (y <= 100000000.0) {
		tmp = (x + y) - z;
	} else if ((y <= 2e+112) || !(y <= 1.9e+168)) {
		tmp = t_0 - z;
	} else {
		tmp = x + t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = y * (1.0d0 - log(y))
    t_1 = x - (log(y) * 0.5d0)
    if (y <= 2.6d-263) then
        tmp = t_1
    else if (y <= 1.95d-233) then
        tmp = x - z
    else if (y <= 3.6d-133) then
        tmp = t_1
    else if (y <= 100000000.0d0) then
        tmp = (x + y) - z
    else if ((y <= 2d+112) .or. (.not. (y <= 1.9d+168))) then
        tmp = t_0 - z
    else
        tmp = x + t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y * (1.0 - Math.log(y));
	double t_1 = x - (Math.log(y) * 0.5);
	double tmp;
	if (y <= 2.6e-263) {
		tmp = t_1;
	} else if (y <= 1.95e-233) {
		tmp = x - z;
	} else if (y <= 3.6e-133) {
		tmp = t_1;
	} else if (y <= 100000000.0) {
		tmp = (x + y) - z;
	} else if ((y <= 2e+112) || !(y <= 1.9e+168)) {
		tmp = t_0 - z;
	} else {
		tmp = x + t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * (1.0 - math.log(y))
	t_1 = x - (math.log(y) * 0.5)
	tmp = 0
	if y <= 2.6e-263:
		tmp = t_1
	elif y <= 1.95e-233:
		tmp = x - z
	elif y <= 3.6e-133:
		tmp = t_1
	elif y <= 100000000.0:
		tmp = (x + y) - z
	elif (y <= 2e+112) or not (y <= 1.9e+168):
		tmp = t_0 - z
	else:
		tmp = x + t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * Float64(1.0 - log(y)))
	t_1 = Float64(x - Float64(log(y) * 0.5))
	tmp = 0.0
	if (y <= 2.6e-263)
		tmp = t_1;
	elseif (y <= 1.95e-233)
		tmp = Float64(x - z);
	elseif (y <= 3.6e-133)
		tmp = t_1;
	elseif (y <= 100000000.0)
		tmp = Float64(Float64(x + y) - z);
	elseif ((y <= 2e+112) || !(y <= 1.9e+168))
		tmp = Float64(t_0 - z);
	else
		tmp = Float64(x + t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * (1.0 - log(y));
	t_1 = x - (log(y) * 0.5);
	tmp = 0.0;
	if (y <= 2.6e-263)
		tmp = t_1;
	elseif (y <= 1.95e-233)
		tmp = x - z;
	elseif (y <= 3.6e-133)
		tmp = t_1;
	elseif (y <= 100000000.0)
		tmp = (x + y) - z;
	elseif ((y <= 2e+112) || ~((y <= 1.9e+168)))
		tmp = t_0 - z;
	else
		tmp = x + t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x - N[(N[Log[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 2.6e-263], t$95$1, If[LessEqual[y, 1.95e-233], N[(x - z), $MachinePrecision], If[LessEqual[y, 3.6e-133], t$95$1, If[LessEqual[y, 100000000.0], N[(N[(x + y), $MachinePrecision] - z), $MachinePrecision], If[Or[LessEqual[y, 2e+112], N[Not[LessEqual[y, 1.9e+168]], $MachinePrecision]], N[(t$95$0 - z), $MachinePrecision], N[(x + t$95$0), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y < 2.6e-263 or 1.9500000000000001e-233 < y < 3.6000000000000004e-133

   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. Add Preprocessing

   5. Taylor expanded in z around 0 83.7%

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

   6. Taylor expanded in y around 0 83.7%

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

   if 2.6e-263 < y < 1.9500000000000001e-233

   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. associate-+l+ 100.0%

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

      4. associate-+r- 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-rgt-neg-in 100.0%

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

      7. fma-define 100.0%

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

      8. +-commutative 100.0%

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

      9. distribute-neg-in 100.0%

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

      10. unsub-neg 100.0%

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

      11. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in z around inf 100.0%

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

      1. neg-mul-1 100.0%

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

   7. Simplified 100.0%

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

   if 3.6000000000000004e-133 < y < 1e8

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. mul-1-neg 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-rgt-in 99.9%

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

      5. +-commutative 99.9%

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

      6. distribute-neg-in 99.9%

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

      7. distribute-lft-neg-in 99.9%

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

      8. metadata-eval 99.9%

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

      9. *-commutative 99.9%

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

      10. *-commutative 99.9%

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

      11. distribute-rgt-neg-in 99.9%

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

      12. distribute-lft-in 100.0%

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

      13. sub-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf 80.6%

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

   if 1e8 < y < 1.9999999999999999e112 or 1.9000000000000001e168 < y

   1. Initial program 99.6%

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

      1. associate--l+ 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around inf 86.9%

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

      1. *-commutative 86.9%

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

      2. log-rec 86.9%

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

      3. distribute-lft-neg-in 86.9%

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

      4. distribute-rgt-neg-in 86.9%

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

   7. Simplified 86.9%

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

   8. Taylor expanded in y around 0 87.0%

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

      1. neg-mul-1 87.0%

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

      2. sub-neg 87.0%

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

   10. Simplified 87.0%

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

   if 1.9999999999999999e112 < y < 1.9000000000000001e168

   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. Add Preprocessing

   5. Taylor expanded in z around 0 87.2%

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

   6. Taylor expanded in y around inf 87.2%

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

      1. mul-1-neg 87.2%

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

      2. log-rec 87.2%

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

      3. distribute-rgt-neg-in 87.2%

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

      4. remove-double-neg 87.2%

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

   8. Simplified 87.2%

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

   9. Taylor expanded in y around 0 87.2%

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

       1. +-commutative 87.2%

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

   11. Simplified 87.2%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.6 \cdot 10^{-263}:\\ \;\;\;\;x - \log y \cdot 0.5\\ \mathbf{elif}\;y \leq 1.95 \cdot 10^{-233}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{-133}:\\ \;\;\;\;x - \log y \cdot 0.5\\ \mathbf{elif}\;y \leq 100000000:\\ \;\;\;\;\left(x + y\right) - z\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+112} \lor \neg \left(y \leq 1.9 \cdot 10^{+168}\right):\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(1 - \log y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \log y \cdot 0.5\\ \mathbf{if}\;y \leq 4.2 \cdot 10^{-263}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{-234}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{-133}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{+38}:\\ \;\;\;\;\left(x + y\right) - z\\ \mathbf{elif}\;y \leq 1.32 \cdot 10^{+79} \lor \neg \left(y \leq 1.7 \cdot 10^{+110}\right):\\ \;\;\;\;x + y \cdot \left(1 - \log y\right)\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- x (* (log y) 0.5))))
   (if (<= y 4.2e-263)
     t_0
     (if (<= y 8.2e-234)
       (- x z)
       (if (<= y 1.65e-133)
         t_0
         (if (<= y 4.4e+38)
           (- (+ x y) z)
           (if (or (<= y 1.32e+79) (not (<= y 1.7e+110)))
             (+ x (* y (- 1.0 (log y))))
             (- x z))))))))

C

double code(double x, double y, double z) {
	double t_0 = x - (log(y) * 0.5);
	double tmp;
	if (y <= 4.2e-263) {
		tmp = t_0;
	} else if (y <= 8.2e-234) {
		tmp = x - z;
	} else if (y <= 1.65e-133) {
		tmp = t_0;
	} else if (y <= 4.4e+38) {
		tmp = (x + y) - z;
	} else if ((y <= 1.32e+79) || !(y <= 1.7e+110)) {
		tmp = x + (y * (1.0 - log(y)));
	} else {
		tmp = x - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x - (log(y) * 0.5d0)
    if (y <= 4.2d-263) then
        tmp = t_0
    else if (y <= 8.2d-234) then
        tmp = x - z
    else if (y <= 1.65d-133) then
        tmp = t_0
    else if (y <= 4.4d+38) then
        tmp = (x + y) - z
    else if ((y <= 1.32d+79) .or. (.not. (y <= 1.7d+110))) then
        tmp = x + (y * (1.0d0 - log(y)))
    else
        tmp = x - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x - (Math.log(y) * 0.5);
	double tmp;
	if (y <= 4.2e-263) {
		tmp = t_0;
	} else if (y <= 8.2e-234) {
		tmp = x - z;
	} else if (y <= 1.65e-133) {
		tmp = t_0;
	} else if (y <= 4.4e+38) {
		tmp = (x + y) - z;
	} else if ((y <= 1.32e+79) || !(y <= 1.7e+110)) {
		tmp = x + (y * (1.0 - Math.log(y)));
	} else {
		tmp = x - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x - (math.log(y) * 0.5)
	tmp = 0
	if y <= 4.2e-263:
		tmp = t_0
	elif y <= 8.2e-234:
		tmp = x - z
	elif y <= 1.65e-133:
		tmp = t_0
	elif y <= 4.4e+38:
		tmp = (x + y) - z
	elif (y <= 1.32e+79) or not (y <= 1.7e+110):
		tmp = x + (y * (1.0 - math.log(y)))
	else:
		tmp = x - z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x - Float64(log(y) * 0.5))
	tmp = 0.0
	if (y <= 4.2e-263)
		tmp = t_0;
	elseif (y <= 8.2e-234)
		tmp = Float64(x - z);
	elseif (y <= 1.65e-133)
		tmp = t_0;
	elseif (y <= 4.4e+38)
		tmp = Float64(Float64(x + y) - z);
	elseif ((y <= 1.32e+79) || !(y <= 1.7e+110))
		tmp = Float64(x + Float64(y * Float64(1.0 - log(y))));
	else
		tmp = Float64(x - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x - (log(y) * 0.5);
	tmp = 0.0;
	if (y <= 4.2e-263)
		tmp = t_0;
	elseif (y <= 8.2e-234)
		tmp = x - z;
	elseif (y <= 1.65e-133)
		tmp = t_0;
	elseif (y <= 4.4e+38)
		tmp = (x + y) - z;
	elseif ((y <= 1.32e+79) || ~((y <= 1.7e+110)))
		tmp = x + (y * (1.0 - log(y)));
	else
		tmp = x - 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]}, If[LessEqual[y, 4.2e-263], t$95$0, If[LessEqual[y, 8.2e-234], N[(x - z), $MachinePrecision], If[LessEqual[y, 1.65e-133], t$95$0, If[LessEqual[y, 4.4e+38], N[(N[(x + y), $MachinePrecision] - z), $MachinePrecision], If[Or[LessEqual[y, 1.32e+79], N[Not[LessEqual[y, 1.7e+110]], $MachinePrecision]], N[(x + N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - z), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < 4.20000000000000005e-263 or 8.20000000000000021e-234 < y < 1.65000000000000005e-133

   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. Add Preprocessing

   5. Taylor expanded in z around 0 83.7%

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

   6. Taylor expanded in y around 0 83.7%

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

   if 4.20000000000000005e-263 < y < 8.20000000000000021e-234 or 1.32e79 < y < 1.7000000000000001e110

   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. associate-+l+ 99.9%

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

      4. associate-+r- 99.9%

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

      5. *-commutative 99.9%

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

      6. distribute-rgt-neg-in 99.9%

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

      7. fma-define 100.0%

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

      8. +-commutative 100.0%

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

      9. distribute-neg-in 100.0%

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

      10. unsub-neg 100.0%

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

      11. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in z around inf 92.1%

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

      1. neg-mul-1 92.1%

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

   7. Simplified 92.1%

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

   if 1.65000000000000005e-133 < y < 4.40000000000000013e38

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 98.5%

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

      1. associate-*r/ 98.5%

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

      2. mul-1-neg 98.5%

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

      3. +-commutative 98.5%

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

      4. distribute-rgt-in 98.4%

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

      5. +-commutative 98.4%

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

      6. distribute-neg-in 98.4%

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

      7. distribute-lft-neg-in 98.4%

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

      8. metadata-eval 98.4%

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

      9. *-commutative 98.4%

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

      10. *-commutative 98.4%

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

      11. distribute-rgt-neg-in 98.4%

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

      12. distribute-lft-in 98.5%

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

      13. sub-neg 98.5%

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

   5. Simplified 98.5%

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

   6. Taylor expanded in x around inf 77.1%

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

   if 4.40000000000000013e38 < y < 1.32e79 or 1.7000000000000001e110 < y

   1. Initial program 99.6%

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

      1. associate--l+ 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in z around 0 80.5%

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

   6. Taylor expanded in y around inf 80.5%

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

      1. mul-1-neg 80.5%

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

      2. log-rec 80.5%

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

      3. distribute-rgt-neg-in 80.5%

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

      4. remove-double-neg 80.5%

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

   8. Simplified 80.5%

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

   9. Taylor expanded in y around 0 80.5%

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

       1. +-commutative 80.5%

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

   11. Simplified 80.5%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.2 \cdot 10^{-263}:\\ \;\;\;\;x - \log y \cdot 0.5\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{-234}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{-133}:\\ \;\;\;\;x - \log y \cdot 0.5\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{+38}:\\ \;\;\;\;\left(x + y\right) - z\\ \mathbf{elif}\;y \leq 1.32 \cdot 10^{+79} \lor \neg \left(y \leq 1.7 \cdot 10^{+110}\right):\\ \;\;\;\;x + y \cdot \left(1 - \log y\right)\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 69.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \log y \cdot 0.5\\ \mathbf{if}\;y \leq 6.6 \cdot 10^{-263}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-234}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y \leq 4 \cdot 10^{-132}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 5.3 \cdot 10^{+118}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- x (* (log y) 0.5))))
   (if (<= y 6.6e-263)
     t_0
     (if (<= y 4.2e-234)
       (- x z)
       (if (<= y 4e-132)
         t_0
         (if (<= y 5.3e+118) (- x z) (* y (- 1.0 (log y)))))))))

C

double code(double x, double y, double z) {
	double t_0 = x - (log(y) * 0.5);
	double tmp;
	if (y <= 6.6e-263) {
		tmp = t_0;
	} else if (y <= 4.2e-234) {
		tmp = x - z;
	} else if (y <= 4e-132) {
		tmp = t_0;
	} else if (y <= 5.3e+118) {
		tmp = x - z;
	} else {
		tmp = y * (1.0 - log(y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x - (log(y) * 0.5d0)
    if (y <= 6.6d-263) then
        tmp = t_0
    else if (y <= 4.2d-234) then
        tmp = x - z
    else if (y <= 4d-132) then
        tmp = t_0
    else if (y <= 5.3d+118) then
        tmp = x - z
    else
        tmp = y * (1.0d0 - log(y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x - (Math.log(y) * 0.5);
	double tmp;
	if (y <= 6.6e-263) {
		tmp = t_0;
	} else if (y <= 4.2e-234) {
		tmp = x - z;
	} else if (y <= 4e-132) {
		tmp = t_0;
	} else if (y <= 5.3e+118) {
		tmp = x - z;
	} else {
		tmp = y * (1.0 - Math.log(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x - (math.log(y) * 0.5)
	tmp = 0
	if y <= 6.6e-263:
		tmp = t_0
	elif y <= 4.2e-234:
		tmp = x - z
	elif y <= 4e-132:
		tmp = t_0
	elif y <= 5.3e+118:
		tmp = x - z
	else:
		tmp = y * (1.0 - math.log(y))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x - Float64(log(y) * 0.5))
	tmp = 0.0
	if (y <= 6.6e-263)
		tmp = t_0;
	elseif (y <= 4.2e-234)
		tmp = Float64(x - z);
	elseif (y <= 4e-132)
		tmp = t_0;
	elseif (y <= 5.3e+118)
		tmp = Float64(x - z);
	else
		tmp = Float64(y * Float64(1.0 - log(y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x - (log(y) * 0.5);
	tmp = 0.0;
	if (y <= 6.6e-263)
		tmp = t_0;
	elseif (y <= 4.2e-234)
		tmp = x - z;
	elseif (y <= 4e-132)
		tmp = t_0;
	elseif (y <= 5.3e+118)
		tmp = x - z;
	else
		tmp = y * (1.0 - log(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x - N[(N[Log[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 6.6e-263], t$95$0, If[LessEqual[y, 4.2e-234], N[(x - z), $MachinePrecision], If[LessEqual[y, 4e-132], t$95$0, If[LessEqual[y, 5.3e+118], N[(x - z), $MachinePrecision], N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 6.5999999999999994e-263 or 4.19999999999999982e-234 < y < 3.9999999999999999e-132

   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. Add Preprocessing

   5. Taylor expanded in z around 0 83.7%

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

   6. Taylor expanded in y around 0 83.7%

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

   if 6.5999999999999994e-263 < y < 4.19999999999999982e-234 or 3.9999999999999999e-132 < y < 5.2999999999999997e118

   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. associate-+l+ 99.9%

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

      4. associate-+r- 99.9%

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

      5. *-commutative 99.9%

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

      6. distribute-rgt-neg-in 99.9%

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

      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. Taylor expanded in z around inf 73.7%

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

      1. neg-mul-1 73.7%

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

   7. Simplified 73.7%

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

   if 5.2999999999999997e118 < y

   1. Initial program 99.6%

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

      1. associate--l+ 99.6%

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

      2. sub-neg 99.6%

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

      3. associate-+l+ 99.6%

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

      4. associate-+r- 99.6%

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

      5. *-commutative 99.6%

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

      6. distribute-rgt-neg-in 99.6%

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

      7. fma-define 99.7%

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

      8. +-commutative 99.7%

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

      9. distribute-neg-in 99.7%

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

      10. unsub-neg 99.7%

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

      11. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

      1. add-cube-cbrt 98.0%

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

      2. pow3 98.1%

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

      3. +-commutative 98.1%

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

      4. associate-+l- 98.1%

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

   6. Applied egg-rr 98.1%

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

   7. Taylor expanded in y around inf 72.4%

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

      1. log-rec 72.4%

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

      2. sub-neg 72.4%

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

   9. Simplified 72.4%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.6 \cdot 10^{-263}:\\ \;\;\;\;x - \log y \cdot 0.5\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-234}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y \leq 4 \cdot 10^{-132}:\\ \;\;\;\;x - \log y \cdot 0.5\\ \mathbf{elif}\;y \leq 5.3 \cdot 10^{+118}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(1 - \log y\right)\\ \mathbf{if}\;y \leq 700000000:\\ \;\;\;\;\left(x + \log y \cdot -0.5\right) - z\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{+111} \lor \neg \left(y \leq 1.75 \cdot 10^{+168}\right):\\ \;\;\;\;t\_0 - z\\ \mathbf{else}:\\ \;\;\;\;x + t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (- 1.0 (log y)))))
   (if (<= y 700000000.0)
     (- (+ x (* (log y) -0.5)) z)
     (if (or (<= y 5.8e+111) (not (<= y 1.75e+168))) (- t_0 z) (+ x t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = y * (1.0 - log(y));
	double tmp;
	if (y <= 700000000.0) {
		tmp = (x + (log(y) * -0.5)) - z;
	} else if ((y <= 5.8e+111) || !(y <= 1.75e+168)) {
		tmp = t_0 - z;
	} else {
		tmp = x + t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y * (1.0d0 - log(y))
    if (y <= 700000000.0d0) then
        tmp = (x + (log(y) * (-0.5d0))) - z
    else if ((y <= 5.8d+111) .or. (.not. (y <= 1.75d+168))) then
        tmp = t_0 - z
    else
        tmp = x + t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y * (1.0 - Math.log(y));
	double tmp;
	if (y <= 700000000.0) {
		tmp = (x + (Math.log(y) * -0.5)) - z;
	} else if ((y <= 5.8e+111) || !(y <= 1.75e+168)) {
		tmp = t_0 - z;
	} else {
		tmp = x + t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * (1.0 - math.log(y))
	tmp = 0
	if y <= 700000000.0:
		tmp = (x + (math.log(y) * -0.5)) - z
	elif (y <= 5.8e+111) or not (y <= 1.75e+168):
		tmp = t_0 - z
	else:
		tmp = x + t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * Float64(1.0 - log(y)))
	tmp = 0.0
	if (y <= 700000000.0)
		tmp = Float64(Float64(x + Float64(log(y) * -0.5)) - z);
	elseif ((y <= 5.8e+111) || !(y <= 1.75e+168))
		tmp = Float64(t_0 - z);
	else
		tmp = Float64(x + t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * (1.0 - log(y));
	tmp = 0.0;
	if (y <= 700000000.0)
		tmp = (x + (log(y) * -0.5)) - z;
	elseif ((y <= 5.8e+111) || ~((y <= 1.75e+168)))
		tmp = t_0 - z;
	else
		tmp = x + t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 700000000.0], N[(N[(x + N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], If[Or[LessEqual[y, 5.8e+111], N[Not[LessEqual[y, 1.75e+168]], $MachinePrecision]], N[(t$95$0 - z), $MachinePrecision], N[(x + t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 7e8

   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. associate-+l+ 100.0%

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

      4. associate-+r- 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-rgt-neg-in 100.0%

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

      7. fma-define 100.0%

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

      8. +-commutative 100.0%

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

      9. distribute-neg-in 100.0%

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

      10. unsub-neg 100.0%

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

      11. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 98.2%

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

   if 7e8 < y < 5.7999999999999999e111 or 1.7500000000000001e168 < y

   1. Initial program 99.6%

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

      1. associate--l+ 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around inf 86.9%

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

      1. *-commutative 86.9%

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

      2. log-rec 86.9%

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

      3. distribute-lft-neg-in 86.9%

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

      4. distribute-rgt-neg-in 86.9%

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

   7. Simplified 86.9%

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

   8. Taylor expanded in y around 0 87.0%

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

      1. neg-mul-1 87.0%

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

      2. sub-neg 87.0%

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

   10. Simplified 87.0%

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

   if 5.7999999999999999e111 < y < 1.7500000000000001e168

   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. Add Preprocessing

   5. Taylor expanded in z around 0 87.2%

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

   6. Taylor expanded in y around inf 87.2%

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

      1. mul-1-neg 87.2%

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

      2. log-rec 87.2%

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

      3. distribute-rgt-neg-in 87.2%

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

      4. remove-double-neg 87.2%

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

   8. Simplified 87.2%

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

   9. Taylor expanded in y around 0 87.2%

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

       1. +-commutative 87.2%

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

   11. Simplified 87.2%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 700000000:\\ \;\;\;\;\left(x + \log y \cdot -0.5\right) - z\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{+111} \lor \neg \left(y \leq 1.75 \cdot 10^{+168}\right):\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(1 - \log y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.9 \cdot 10^{-7}:\\ \;\;\;\;\left(x + \log y \cdot -0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + \left(y \cdot \left(1 - \log y\right) - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2.9e-7)
   (- (+ 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 <= 2.9e-7) {
		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 <= 2.9d-7) 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 <= 2.9e-7) {
		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 <= 2.9e-7:
		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 <= 2.9e-7)
		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 <= 2.9e-7)
		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, 2.9e-7], 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 < 2.8999999999999998e-7

   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. associate-+l+ 100.0%

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

      4. associate-+r- 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-rgt-neg-in 100.0%

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

      7. fma-define 100.0%

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

      8. +-commutative 100.0%

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

      9. distribute-neg-in 100.0%

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

      10. unsub-neg 100.0%

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

      11. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 99.6%

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

   if 2.8999999999999998e-7 < 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+ 99.6%

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

      2. sub-neg 99.6%

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

      3. associate-+l+ 99.6%

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

      4. associate-+r- 99.7%

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

      5. *-commutative 99.7%

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

      6. distribute-rgt-neg-in 99.7%

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

      7. fma-define 99.7%

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

      8. +-commutative 99.7%

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

      9. distribute-neg-in 99.7%

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

      10. unsub-neg 99.7%

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

      11. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around inf 98.4%

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

      1. log-rec 98.4%

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

      2. sub-neg 98.4%

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

   7. Simplified 98.4%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.9 \cdot 10^{-7}:\\ \;\;\;\;\left(x + \log y \cdot -0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + \left(y \cdot \left(1 - \log y\right) - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
2. 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 8: 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. Add Preprocessing

5. Final simplification 99.8%

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

Alternative 9: 72.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.25 \cdot 10^{+121}:\\ \;\;\;\;\left(x + y\right) - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.25e+121) (- (+ x y) z) (* y (- 1.0 (log y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.25e+121) {
		tmp = (x + y) - z;
	} else {
		tmp = y * (1.0 - log(y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 1.25d+121) then
        tmp = (x + y) - z
    else
        tmp = y * (1.0d0 - log(y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.25e+121) {
		tmp = (x + y) - z;
	} else {
		tmp = y * (1.0 - Math.log(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1.25e+121:
		tmp = (x + y) - z
	else:
		tmp = y * (1.0 - math.log(y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.25e+121)
		tmp = Float64(Float64(x + y) - z);
	else
		tmp = Float64(y * Float64(1.0 - log(y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1.25e+121)
		tmp = (x + y) - z;
	else
		tmp = y * (1.0 - log(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.25e+121], N[(N[(x + y), $MachinePrecision] - z), $MachinePrecision], N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.25000000000000002e121

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 95.0%

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

      1. associate-*r/ 95.0%

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

      2. mul-1-neg 95.0%

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

      3. +-commutative 95.0%

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

      4. distribute-rgt-in 95.0%

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

      5. +-commutative 95.0%

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

      6. distribute-neg-in 95.0%

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

      7. distribute-lft-neg-in 95.0%

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

      8. metadata-eval 95.0%

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

      9. *-commutative 95.0%

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

      10. *-commutative 95.0%

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

      11. distribute-rgt-neg-in 95.0%

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

      12. distribute-lft-in 95.0%

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

      13. sub-neg 95.0%

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

   5. Simplified 95.0%

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

   6. Taylor expanded in x around inf 65.4%

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

   if 1.25000000000000002e121 < y

   1. Initial program 99.6%

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

      1. associate--l+ 99.6%

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

      2. sub-neg 99.6%

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

      3. associate-+l+ 99.6%

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

      4. associate-+r- 99.6%

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

      5. *-commutative 99.6%

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

      6. distribute-rgt-neg-in 99.6%

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

      7. fma-define 99.7%

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

      8. +-commutative 99.7%

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

      9. distribute-neg-in 99.7%

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

      10. unsub-neg 99.7%

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

      11. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

      1. add-cube-cbrt 98.0%

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

      2. pow3 98.1%

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

      3. +-commutative 98.1%

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

      4. associate-+l- 98.1%

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

   6. Applied egg-rr 98.1%

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

   7. Taylor expanded in y around inf 72.4%

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

      1. log-rec 72.4%

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

      2. sub-neg 72.4%

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

   9. Simplified 72.4%

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

Alternative 10: 48.2% accurate, 9.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.4 \cdot 10^{+34}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{+32}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -7.4e+34) x (if (<= x 3.2e+32) (- z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -7.4e+34) {
		tmp = x;
	} else if (x <= 3.2e+32) {
		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 <= (-7.4d+34)) then
        tmp = x
    else if (x <= 3.2d+32) 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 <= -7.4e+34) {
		tmp = x;
	} else if (x <= 3.2e+32) {
		tmp = -z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -7.4e+34:
		tmp = x
	elif x <= 3.2e+32:
		tmp = -z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -7.4e+34)
		tmp = x;
	elseif (x <= 3.2e+32)
		tmp = Float64(-z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -7.4e+34)
		tmp = x;
	elseif (x <= 3.2e+32)
		tmp = -z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -7.4e+34], x, If[LessEqual[x, 3.2e+32], (-z), x]]

Derivation

1. Split input into 2 regimes

2. if x < -7.40000000000000017e34 or 3.1999999999999999e32 < 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. 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. Taylor expanded in x around inf 61.5%

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

   if -7.40000000000000017e34 < x < 3.1999999999999999e32

   1. Initial program 99.8%

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

      1. associate--l+ 99.8%

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

      2. sub-neg 99.8%

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

      3. associate-+l+ 99.8%

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

      4. associate-+r- 99.8%

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

      5. *-commutative 99.8%

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

      6. distribute-rgt-neg-in 99.8%

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

      7. fma-define 99.8%

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

      8. +-commutative 99.8%

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

      9. distribute-neg-in 99.8%

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

      10. unsub-neg 99.8%

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

      11. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

      1. add-cube-cbrt 98.0%

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

      2. pow3 98.0%

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

      3. +-commutative 98.0%

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

      4. associate-+l- 98.0%

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

   6. Applied egg-rr 98.0%

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

   7. Taylor expanded in z around inf 33.9%

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

      1. mul-1-neg 33.9%

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

   9. Simplified 33.9%

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

Alternative 11: 57.9% accurate, 37.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate--l+ 99.8%

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

   2. sub-neg 99.8%

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

   3. associate-+l+ 99.8%

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

   4. associate-+r- 99.8%

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

   5. *-commutative 99.8%

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

   6. distribute-rgt-neg-in 99.8%

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

   7. fma-define 99.8%

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

   8. +-commutative 99.8%

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

   9. distribute-neg-in 99.8%

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

   10. unsub-neg 99.8%

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

   11. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in z around inf 52.7%

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

   1. neg-mul-1 52.7%

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

7. Simplified 52.7%

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

8. Final simplification 52.7%

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

Alternative 12: 29.9% 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%

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

   2. sub-neg 99.8%

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

   3. associate-+l+ 99.8%

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

   4. associate-+r- 99.8%

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

   5. *-commutative 99.8%

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

   6. distribute-rgt-neg-in 99.8%

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

   7. fma-define 99.8%

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

   8. +-commutative 99.8%

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

   9. distribute-neg-in 99.8%

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

   10. unsub-neg 99.8%

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

   11. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in x around inf 26.8%

   \[\leadsto \color{blue}{x} \]
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 2024108 
(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))