Numeric.SpecFunctions:invErfc from math-functions-0.1.5.2, A

Percentage Accurate: 95.8% → 99.5%

Time: 6.7s

Alternatives: 10

Speedup: 0.5×

Specification

Math

\[\begin{array}{l} \\ x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+ x (/ y (- (* 1.1283791670955126 (exp z)) (* x y)))))

C

double code(double x, double y, double z) {
	return x + (y / ((1.1283791670955126 * exp(z)) - (x * 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 = x + (y / ((1.1283791670955126d0 * exp(z)) - (x * y)))
end function

Java

public static double code(double x, double y, double z) {
	return x + (y / ((1.1283791670955126 * Math.exp(z)) - (x * y)));
}

Python

def code(x, y, z):
	return x + (y / ((1.1283791670955126 * math.exp(z)) - (x * y)))

Julia

function code(x, y, z)
	return Float64(x + Float64(y / Float64(Float64(1.1283791670955126 * exp(z)) - Float64(x * y))))
end

MATLAB

function tmp = code(x, y, z)
	tmp = x + (y / ((1.1283791670955126 * exp(z)) - (x * y)));
end

Wolfram

code[x_, y_, z_] := N[(x + N[(y / N[(N[(1.1283791670955126 * N[Exp[z], $MachinePrecision]), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 95.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+ x (/ y (- (* 1.1283791670955126 (exp z)) (* x y)))))

C

double code(double x, double y, double z) {
	return x + (y / ((1.1283791670955126 * exp(z)) - (x * 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 = x + (y / ((1.1283791670955126d0 * exp(z)) - (x * y)))
end function

Java

public static double code(double x, double y, double z) {
	return x + (y / ((1.1283791670955126 * Math.exp(z)) - (x * y)));
}

Python

def code(x, y, z):
	return x + (y / ((1.1283791670955126 * math.exp(z)) - (x * y)))

Julia

function code(x, y, z)
	return Float64(x + Float64(y / Float64(Float64(1.1283791670955126 * exp(z)) - Float64(x * y))))
end

MATLAB

function tmp = code(x, y, z)
	tmp = x + (y / ((1.1283791670955126 * exp(z)) - (x * y)));
end

Wolfram

code[x_, y_, z_] := N[(x + N[(y / N[(N[(1.1283791670955126 * N[Exp[z], $MachinePrecision]), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;e^{z} \leq 1:\\ \;\;\;\;x + \frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(0.5641895835477563, z, 1.1283791670955126\right), z, 1.1283791670955126\right) - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.8862269254527579}{e^{z}}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (+ x (/ -1.0 x))
   (if (<= (exp z) 1.0)
     (+
      x
      (/
       y
       (-
        (fma
         (fma 0.5641895835477563 z 1.1283791670955126)
         z
         1.1283791670955126)
        (* x y))))
     (fma (/ 0.8862269254527579 (exp z)) y x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (exp(z) <= 0.0) {
		tmp = x + (-1.0 / x);
	} else if (exp(z) <= 1.0) {
		tmp = x + (y / (fma(fma(0.5641895835477563, z, 1.1283791670955126), z, 1.1283791670955126) - (x * y)));
	} else {
		tmp = fma((0.8862269254527579 / exp(z)), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (exp(z) <= 0.0)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (exp(z) <= 1.0)
		tmp = Float64(x + Float64(y / Float64(fma(fma(0.5641895835477563, z, 1.1283791670955126), z, 1.1283791670955126) - Float64(x * y))));
	else
		tmp = fma(Float64(0.8862269254527579 / exp(z)), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[Exp[z], $MachinePrecision], 0.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Exp[z], $MachinePrecision], 1.0], N[(x + N[(y / N[(N[(N[(0.5641895835477563 * z + 1.1283791670955126), $MachinePrecision] * z + 1.1283791670955126), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.8862269254527579 / N[Exp[z], $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 83.7%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto x + \color{blue}{\frac{-1}{x}} \]
   4. Step-by-step derivation

      1. lower-/.f64 100.0

         \[\leadsto x + \color{blue}{\frac{-1}{x}} \]

   5. Applied rewrites 100.0%

      \[\leadsto x + \color{blue}{\frac{-1}{x}} \]

   if 0.0 < (exp.f64 z) < 1

   1. Initial program 99.8%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto x + \frac{y}{\color{blue}{\left(\frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z\right)\right)} - x \cdot y} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x + \frac{y}{\color{blue}{\left(z \cdot \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z\right) + \frac{5641895835477563}{5000000000000000}\right)} - x \cdot y} \]

      2. *-commutative N/A

         \[\leadsto x + \frac{y}{\left(\color{blue}{\left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z\right) \cdot z} + \frac{5641895835477563}{5000000000000000}\right) - x \cdot y} \]

      3. lower-fma.f64 N/A

         \[\leadsto x + \frac{y}{\color{blue}{\mathsf{fma}\left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z, z, \frac{5641895835477563}{5000000000000000}\right)} - x \cdot y} \]

      4. +-commutative N/A

         \[\leadsto x + \frac{y}{\mathsf{fma}\left(\color{blue}{\frac{5641895835477563}{10000000000000000} \cdot z + \frac{5641895835477563}{5000000000000000}}, z, \frac{5641895835477563}{5000000000000000}\right) - x \cdot y} \]

      5. lower-fma.f64 99.8

         \[\leadsto x + \frac{y}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.5641895835477563, z, 1.1283791670955126\right)}, z, 1.1283791670955126\right) - x \cdot y} \]

   5. Applied rewrites 99.8%

      \[\leadsto x + \frac{y}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5641895835477563, z, 1.1283791670955126\right), z, 1.1283791670955126\right)} - x \cdot y} \]

   if 1 < (exp.f64 z)

   1. Initial program 96.9%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + \frac{5000000000000000}{5641895835477563} \cdot \frac{y}{e^{z}}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{5000000000000000}{5641895835477563} \cdot \frac{y}{e^{z}} + x} \]

      2. *-lft-identity N/A

         \[\leadsto \frac{5000000000000000}{5641895835477563} \cdot \frac{\color{blue}{1 \cdot y}}{e^{z}} + x \]

      3. associate-*l/ N/A

         \[\leadsto \frac{5000000000000000}{5641895835477563} \cdot \color{blue}{\left(\frac{1}{e^{z}} \cdot y\right)} + x \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\frac{5000000000000000}{5641895835477563} \cdot \frac{1}{e^{z}}\right) \cdot y} + x \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{5000000000000000}{5641895835477563} \cdot \frac{1}{e^{z}}, y, x\right)} \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{5000000000000000}{5641895835477563} \cdot 1}{e^{z}}}, y, x\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\frac{5000000000000000}{5641895835477563}}}{e^{z}}, y, x\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{5000000000000000}{5641895835477563}}{e^{z}}}, y, x\right) \]

      9. lower-exp.f64 100.0

         \[\leadsto \mathsf{fma}\left(\frac{0.8862269254527579}{\color{blue}{e^{z}}}, y, x\right) \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.8862269254527579}{e^{z}}, y, x\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 86.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y}\\ \mathbf{if}\;t\_0 \leq -2 \lor \neg \left(t\_0 \leq 0.002\right):\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;-\left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (/ y (- (* 1.1283791670955126 (exp z)) (* x y))))))
   (if (or (<= t_0 -2.0) (not (<= t_0 0.002))) (+ x (/ -1.0 x)) (- (- x)))))

C

double code(double x, double y, double z) {
	double t_0 = x + (y / ((1.1283791670955126 * exp(z)) - (x * y)));
	double tmp;
	if ((t_0 <= -2.0) || !(t_0 <= 0.002)) {
		tmp = x + (-1.0 / x);
	} else {
		tmp = -(-x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x + (y / ((1.1283791670955126d0 * exp(z)) - (x * y)))
    if ((t_0 <= (-2.0d0)) .or. (.not. (t_0 <= 0.002d0))) then
        tmp = x + ((-1.0d0) / x)
    else
        tmp = -(-x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x + (y / ((1.1283791670955126 * Math.exp(z)) - (x * y)));
	double tmp;
	if ((t_0 <= -2.0) || !(t_0 <= 0.002)) {
		tmp = x + (-1.0 / x);
	} else {
		tmp = -(-x);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x + (y / ((1.1283791670955126 * math.exp(z)) - (x * y)))
	tmp = 0
	if (t_0 <= -2.0) or not (t_0 <= 0.002):
		tmp = x + (-1.0 / x)
	else:
		tmp = -(-x)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x + Float64(y / Float64(Float64(1.1283791670955126 * exp(z)) - Float64(x * y))))
	tmp = 0.0
	if ((t_0 <= -2.0) || !(t_0 <= 0.002))
		tmp = Float64(x + Float64(-1.0 / x));
	else
		tmp = Float64(-Float64(-x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x + (y / ((1.1283791670955126 * exp(z)) - (x * y)));
	tmp = 0.0;
	if ((t_0 <= -2.0) || ~((t_0 <= 0.002)))
		tmp = x + (-1.0 / x);
	else
		tmp = -(-x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x + N[(y / N[(N[(1.1283791670955126 * N[Exp[z], $MachinePrecision]), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -2.0], N[Not[LessEqual[t$95$0, 0.002]], $MachinePrecision]], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], (-(-x))]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y)))) < -2 or 2e-3 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y))))

   1. Initial program 92.7%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto x + \color{blue}{\frac{-1}{x}} \]
   4. Step-by-step derivation

      1. lower-/.f64 88.3

         \[\leadsto x + \color{blue}{\frac{-1}{x}} \]

   5. Applied rewrites 88.3%

      \[\leadsto x + \color{blue}{\frac{-1}{x}} \]

   if -2 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y)))) < 2e-3

   1. Initial program 99.9%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(1 - \frac{1}{{x}^{2}}\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto x \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{{x}^{2}}\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{{x}^{2}}\right)\right) + 1\right)} \]

      3. neg-sub0 N/A

         \[\leadsto x \cdot \left(\color{blue}{\left(0 - \frac{1}{{x}^{2}}\right)} + 1\right) \]

      4. associate-+l- N/A

         \[\leadsto x \cdot \color{blue}{\left(0 - \left(\frac{1}{{x}^{2}} - 1\right)\right)} \]

      5. neg-sub0 N/A

         \[\leadsto x \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{{x}^{2}} - 1\right)\right)\right)} \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(x \cdot \left(\frac{1}{{x}^{2}} - 1\right)\right)} \]

      7. lower-neg.f64 N/A

         \[\leadsto \color{blue}{-x \cdot \left(\frac{1}{{x}^{2}} - 1\right)} \]

      8. sub-neg N/A

         \[\leadsto -x \cdot \color{blue}{\left(\frac{1}{{x}^{2}} + \left(\mathsf{neg}\left(1\right)\right)\right)} \]

      9. metadata-eval N/A

         \[\leadsto -x \cdot \left(\frac{1}{{x}^{2}} + \color{blue}{-1}\right) \]

      10. distribute-rgt-in N/A

          \[\leadsto -\color{blue}{\left(\frac{1}{{x}^{2}} \cdot x + -1 \cdot x\right)} \]

      11. mul-1-neg N/A

          \[\leadsto -\left(\frac{1}{{x}^{2}} \cdot x + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right) \]

      12. unsub-neg N/A

          \[\leadsto -\color{blue}{\left(\frac{1}{{x}^{2}} \cdot x - x\right)} \]

      13. lower--.f64 N/A

          \[\leadsto -\color{blue}{\left(\frac{1}{{x}^{2}} \cdot x - x\right)} \]

      14. associate-*l/ N/A

          \[\leadsto -\left(\color{blue}{\frac{1 \cdot x}{{x}^{2}}} - x\right) \]

      15. *-lft-identity N/A

          \[\leadsto -\left(\frac{\color{blue}{x}}{{x}^{2}} - x\right) \]

      16. lower-/.f64 N/A

          \[\leadsto -\left(\color{blue}{\frac{x}{{x}^{2}}} - x\right) \]

      17. unpow2 N/A

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

      18. lower-*.f64 1.6

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

   5. Applied rewrites 1.6%

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

   6. Taylor expanded in x around inf

      \[\leadsto --1 \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 79.4%

      \[\leadsto -\left(-x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \leq -2 \lor \neg \left(x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \leq 0.002\right):\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;-\left(-x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;e^{z} \leq 1.03:\\ \;\;\;\;x + \frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.18806319451591877, z, 0.5641895835477563\right), z, 1.1283791670955126\right), z, 1.1283791670955126\right) - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;-\left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (+ x (/ -1.0 x))
   (if (<= (exp z) 1.03)
     (+
      x
      (/
       y
       (-
        (fma
         (fma
          (fma 0.18806319451591877 z 0.5641895835477563)
          z
          1.1283791670955126)
         z
         1.1283791670955126)
        (* x y))))
     (- (- x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (exp(z) <= 0.0) {
		tmp = x + (-1.0 / x);
	} else if (exp(z) <= 1.03) {
		tmp = x + (y / (fma(fma(fma(0.18806319451591877, z, 0.5641895835477563), z, 1.1283791670955126), z, 1.1283791670955126) - (x * y)));
	} else {
		tmp = -(-x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (exp(z) <= 0.0)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (exp(z) <= 1.03)
		tmp = Float64(x + Float64(y / Float64(fma(fma(fma(0.18806319451591877, z, 0.5641895835477563), z, 1.1283791670955126), z, 1.1283791670955126) - Float64(x * y))));
	else
		tmp = Float64(-Float64(-x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[Exp[z], $MachinePrecision], 0.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Exp[z], $MachinePrecision], 1.03], N[(x + N[(y / N[(N[(N[(N[(0.18806319451591877 * z + 0.5641895835477563), $MachinePrecision] * z + 1.1283791670955126), $MachinePrecision] * z + 1.1283791670955126), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-(-x))]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 83.7%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto x + \color{blue}{\frac{-1}{x}} \]
   4. Step-by-step derivation

      1. lower-/.f64 100.0

         \[\leadsto x + \color{blue}{\frac{-1}{x}} \]

   5. Applied rewrites 100.0%

      \[\leadsto x + \color{blue}{\frac{-1}{x}} \]

   if 0.0 < (exp.f64 z) < 1.03000000000000003

   1. Initial program 99.8%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto x + \frac{y}{\color{blue}{\left(\frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{10000000000000000} + \frac{5641895835477563}{30000000000000000} \cdot z\right)\right)\right)} - x \cdot y} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x + \frac{y}{\color{blue}{\left(z \cdot \left(\frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{10000000000000000} + \frac{5641895835477563}{30000000000000000} \cdot z\right)\right) + \frac{5641895835477563}{5000000000000000}\right)} - x \cdot y} \]

      2. *-commutative N/A

         \[\leadsto x + \frac{y}{\left(\color{blue}{\left(\frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{10000000000000000} + \frac{5641895835477563}{30000000000000000} \cdot z\right)\right) \cdot z} + \frac{5641895835477563}{5000000000000000}\right) - x \cdot y} \]

      3. lower-fma.f64 N/A

         \[\leadsto x + \frac{y}{\color{blue}{\mathsf{fma}\left(\frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{10000000000000000} + \frac{5641895835477563}{30000000000000000} \cdot z\right), z, \frac{5641895835477563}{5000000000000000}\right)} - x \cdot y} \]

      4. +-commutative N/A

         \[\leadsto x + \frac{y}{\mathsf{fma}\left(\color{blue}{z \cdot \left(\frac{5641895835477563}{10000000000000000} + \frac{5641895835477563}{30000000000000000} \cdot z\right) + \frac{5641895835477563}{5000000000000000}}, z, \frac{5641895835477563}{5000000000000000}\right) - x \cdot y} \]

      5. *-commutative N/A

         \[\leadsto x + \frac{y}{\mathsf{fma}\left(\color{blue}{\left(\frac{5641895835477563}{10000000000000000} + \frac{5641895835477563}{30000000000000000} \cdot z\right) \cdot z} + \frac{5641895835477563}{5000000000000000}, z, \frac{5641895835477563}{5000000000000000}\right) - x \cdot y} \]

      6. lower-fma.f64 N/A

         \[\leadsto x + \frac{y}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{5641895835477563}{10000000000000000} + \frac{5641895835477563}{30000000000000000} \cdot z, z, \frac{5641895835477563}{5000000000000000}\right)}, z, \frac{5641895835477563}{5000000000000000}\right) - x \cdot y} \]

      7. +-commutative N/A

         \[\leadsto x + \frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{5641895835477563}{30000000000000000} \cdot z + \frac{5641895835477563}{10000000000000000}}, z, \frac{5641895835477563}{5000000000000000}\right), z, \frac{5641895835477563}{5000000000000000}\right) - x \cdot y} \]

      8. lower-fma.f64 99.5

         \[\leadsto x + \frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.18806319451591877, z, 0.5641895835477563\right)}, z, 1.1283791670955126\right), z, 1.1283791670955126\right) - x \cdot y} \]

   5. Applied rewrites 99.5%

      \[\leadsto x + \frac{y}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.18806319451591877, z, 0.5641895835477563\right), z, 1.1283791670955126\right), z, 1.1283791670955126\right)} - x \cdot y} \]

   if 1.03000000000000003 < (exp.f64 z)

   1. Initial program 96.9%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(1 - \frac{1}{{x}^{2}}\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto x \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{{x}^{2}}\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{{x}^{2}}\right)\right) + 1\right)} \]

      3. neg-sub0 N/A

         \[\leadsto x \cdot \left(\color{blue}{\left(0 - \frac{1}{{x}^{2}}\right)} + 1\right) \]

      4. associate-+l- N/A

         \[\leadsto x \cdot \color{blue}{\left(0 - \left(\frac{1}{{x}^{2}} - 1\right)\right)} \]

      5. neg-sub0 N/A

         \[\leadsto x \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{{x}^{2}} - 1\right)\right)\right)} \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(x \cdot \left(\frac{1}{{x}^{2}} - 1\right)\right)} \]

      7. lower-neg.f64 N/A

         \[\leadsto \color{blue}{-x \cdot \left(\frac{1}{{x}^{2}} - 1\right)} \]

      8. sub-neg N/A

         \[\leadsto -x \cdot \color{blue}{\left(\frac{1}{{x}^{2}} + \left(\mathsf{neg}\left(1\right)\right)\right)} \]

      9. metadata-eval N/A

         \[\leadsto -x \cdot \left(\frac{1}{{x}^{2}} + \color{blue}{-1}\right) \]

      10. distribute-rgt-in N/A

          \[\leadsto -\color{blue}{\left(\frac{1}{{x}^{2}} \cdot x + -1 \cdot x\right)} \]

      11. mul-1-neg N/A

          \[\leadsto -\left(\frac{1}{{x}^{2}} \cdot x + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right) \]

      12. unsub-neg N/A

          \[\leadsto -\color{blue}{\left(\frac{1}{{x}^{2}} \cdot x - x\right)} \]

      13. lower--.f64 N/A

          \[\leadsto -\color{blue}{\left(\frac{1}{{x}^{2}} \cdot x - x\right)} \]

      14. associate-*l/ N/A

          \[\leadsto -\left(\color{blue}{\frac{1 \cdot x}{{x}^{2}}} - x\right) \]

      15. *-lft-identity N/A

          \[\leadsto -\left(\frac{\color{blue}{x}}{{x}^{2}} - x\right) \]

      16. lower-/.f64 N/A

          \[\leadsto -\left(\color{blue}{\frac{x}{{x}^{2}}} - x\right) \]

      17. unpow2 N/A

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

      18. lower-*.f64 46.1

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

   5. Applied rewrites 46.1%

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

   6. Taylor expanded in x around inf

      \[\leadsto --1 \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 100.0%

      \[\leadsto -\left(-x\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;e^{z} \leq 1.03:\\ \;\;\;\;x + \frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(0.5641895835477563, z, 1.1283791670955126\right), z, 1.1283791670955126\right) - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;-\left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (+ x (/ -1.0 x))
   (if (<= (exp z) 1.03)
     (+
      x
      (/
       y
       (-
        (fma
         (fma 0.5641895835477563 z 1.1283791670955126)
         z
         1.1283791670955126)
        (* x y))))
     (- (- x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (exp(z) <= 0.0) {
		tmp = x + (-1.0 / x);
	} else if (exp(z) <= 1.03) {
		tmp = x + (y / (fma(fma(0.5641895835477563, z, 1.1283791670955126), z, 1.1283791670955126) - (x * y)));
	} else {
		tmp = -(-x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (exp(z) <= 0.0)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (exp(z) <= 1.03)
		tmp = Float64(x + Float64(y / Float64(fma(fma(0.5641895835477563, z, 1.1283791670955126), z, 1.1283791670955126) - Float64(x * y))));
	else
		tmp = Float64(-Float64(-x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[Exp[z], $MachinePrecision], 0.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Exp[z], $MachinePrecision], 1.03], N[(x + N[(y / N[(N[(N[(0.5641895835477563 * z + 1.1283791670955126), $MachinePrecision] * z + 1.1283791670955126), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-(-x))]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 83.7%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto x + \color{blue}{\frac{-1}{x}} \]
   4. Step-by-step derivation

      1. lower-/.f64 100.0

         \[\leadsto x + \color{blue}{\frac{-1}{x}} \]

   5. Applied rewrites 100.0%

      \[\leadsto x + \color{blue}{\frac{-1}{x}} \]

   if 0.0 < (exp.f64 z) < 1.03000000000000003

   1. Initial program 99.8%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto x + \frac{y}{\color{blue}{\left(\frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z\right)\right)} - x \cdot y} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x + \frac{y}{\color{blue}{\left(z \cdot \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z\right) + \frac{5641895835477563}{5000000000000000}\right)} - x \cdot y} \]

      2. *-commutative N/A

         \[\leadsto x + \frac{y}{\left(\color{blue}{\left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z\right) \cdot z} + \frac{5641895835477563}{5000000000000000}\right) - x \cdot y} \]

      3. lower-fma.f64 N/A

         \[\leadsto x + \frac{y}{\color{blue}{\mathsf{fma}\left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z, z, \frac{5641895835477563}{5000000000000000}\right)} - x \cdot y} \]

      4. +-commutative N/A

         \[\leadsto x + \frac{y}{\mathsf{fma}\left(\color{blue}{\frac{5641895835477563}{10000000000000000} \cdot z + \frac{5641895835477563}{5000000000000000}}, z, \frac{5641895835477563}{5000000000000000}\right) - x \cdot y} \]

      5. lower-fma.f64 99.4

         \[\leadsto x + \frac{y}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.5641895835477563, z, 1.1283791670955126\right)}, z, 1.1283791670955126\right) - x \cdot y} \]

   5. Applied rewrites 99.4%

      \[\leadsto x + \frac{y}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5641895835477563, z, 1.1283791670955126\right), z, 1.1283791670955126\right)} - x \cdot y} \]

   if 1.03000000000000003 < (exp.f64 z)

   1. Initial program 96.9%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(1 - \frac{1}{{x}^{2}}\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto x \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{{x}^{2}}\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{{x}^{2}}\right)\right) + 1\right)} \]

      3. neg-sub0 N/A

         \[\leadsto x \cdot \left(\color{blue}{\left(0 - \frac{1}{{x}^{2}}\right)} + 1\right) \]

      4. associate-+l- N/A

         \[\leadsto x \cdot \color{blue}{\left(0 - \left(\frac{1}{{x}^{2}} - 1\right)\right)} \]

      5. neg-sub0 N/A

         \[\leadsto x \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{{x}^{2}} - 1\right)\right)\right)} \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(x \cdot \left(\frac{1}{{x}^{2}} - 1\right)\right)} \]

      7. lower-neg.f64 N/A

         \[\leadsto \color{blue}{-x \cdot \left(\frac{1}{{x}^{2}} - 1\right)} \]

      8. sub-neg N/A

         \[\leadsto -x \cdot \color{blue}{\left(\frac{1}{{x}^{2}} + \left(\mathsf{neg}\left(1\right)\right)\right)} \]

      9. metadata-eval N/A

         \[\leadsto -x \cdot \left(\frac{1}{{x}^{2}} + \color{blue}{-1}\right) \]

      10. distribute-rgt-in N/A

          \[\leadsto -\color{blue}{\left(\frac{1}{{x}^{2}} \cdot x + -1 \cdot x\right)} \]

      11. mul-1-neg N/A

          \[\leadsto -\left(\frac{1}{{x}^{2}} \cdot x + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right) \]

      12. unsub-neg N/A

          \[\leadsto -\color{blue}{\left(\frac{1}{{x}^{2}} \cdot x - x\right)} \]

      13. lower--.f64 N/A

          \[\leadsto -\color{blue}{\left(\frac{1}{{x}^{2}} \cdot x - x\right)} \]

      14. associate-*l/ N/A

          \[\leadsto -\left(\color{blue}{\frac{1 \cdot x}{{x}^{2}}} - x\right) \]

      15. *-lft-identity N/A

          \[\leadsto -\left(\frac{\color{blue}{x}}{{x}^{2}} - x\right) \]

      16. lower-/.f64 N/A

          \[\leadsto -\left(\color{blue}{\frac{x}{{x}^{2}}} - x\right) \]

      17. unpow2 N/A

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

      18. lower-*.f64 46.1

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

   5. Applied rewrites 46.1%

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

   6. Taylor expanded in x around inf

      \[\leadsto --1 \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 100.0%

      \[\leadsto -\left(-x\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;e^{z} \leq 1.03:\\ \;\;\;\;x + \frac{y}{\mathsf{fma}\left(1.1283791670955126, z, 1.1283791670955126\right) - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;-\left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (+ x (/ -1.0 x))
   (if (<= (exp z) 1.03)
     (+ x (/ y (- (fma 1.1283791670955126 z 1.1283791670955126) (* x y))))
     (- (- x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (exp(z) <= 0.0) {
		tmp = x + (-1.0 / x);
	} else if (exp(z) <= 1.03) {
		tmp = x + (y / (fma(1.1283791670955126, z, 1.1283791670955126) - (x * y)));
	} else {
		tmp = -(-x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (exp(z) <= 0.0)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (exp(z) <= 1.03)
		tmp = Float64(x + Float64(y / Float64(fma(1.1283791670955126, z, 1.1283791670955126) - Float64(x * y))));
	else
		tmp = Float64(-Float64(-x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[Exp[z], $MachinePrecision], 0.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Exp[z], $MachinePrecision], 1.03], N[(x + N[(y / N[(N[(1.1283791670955126 * z + 1.1283791670955126), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-(-x))]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 83.7%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto x + \color{blue}{\frac{-1}{x}} \]
   4. Step-by-step derivation

      1. lower-/.f64 100.0

         \[\leadsto x + \color{blue}{\frac{-1}{x}} \]

   5. Applied rewrites 100.0%

      \[\leadsto x + \color{blue}{\frac{-1}{x}} \]

   if 0.0 < (exp.f64 z) < 1.03000000000000003

   1. Initial program 99.8%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto x + \frac{y}{\color{blue}{\left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{5000000000000000} \cdot z\right)} - x \cdot y} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x + \frac{y}{\color{blue}{\left(\frac{5641895835477563}{5000000000000000} \cdot z + \frac{5641895835477563}{5000000000000000}\right)} - x \cdot y} \]

      2. lower-fma.f64 99.2

         \[\leadsto x + \frac{y}{\color{blue}{\mathsf{fma}\left(1.1283791670955126, z, 1.1283791670955126\right)} - x \cdot y} \]

   5. Applied rewrites 99.2%

      \[\leadsto x + \frac{y}{\color{blue}{\mathsf{fma}\left(1.1283791670955126, z, 1.1283791670955126\right)} - x \cdot y} \]

   if 1.03000000000000003 < (exp.f64 z)

   1. Initial program 96.9%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(1 - \frac{1}{{x}^{2}}\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto x \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{{x}^{2}}\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{{x}^{2}}\right)\right) + 1\right)} \]

      3. neg-sub0 N/A

         \[\leadsto x \cdot \left(\color{blue}{\left(0 - \frac{1}{{x}^{2}}\right)} + 1\right) \]

      4. associate-+l- N/A

         \[\leadsto x \cdot \color{blue}{\left(0 - \left(\frac{1}{{x}^{2}} - 1\right)\right)} \]

      5. neg-sub0 N/A

         \[\leadsto x \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{{x}^{2}} - 1\right)\right)\right)} \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(x \cdot \left(\frac{1}{{x}^{2}} - 1\right)\right)} \]

      7. lower-neg.f64 N/A

         \[\leadsto \color{blue}{-x \cdot \left(\frac{1}{{x}^{2}} - 1\right)} \]

      8. sub-neg N/A

         \[\leadsto -x \cdot \color{blue}{\left(\frac{1}{{x}^{2}} + \left(\mathsf{neg}\left(1\right)\right)\right)} \]

      9. metadata-eval N/A

         \[\leadsto -x \cdot \left(\frac{1}{{x}^{2}} + \color{blue}{-1}\right) \]

      10. distribute-rgt-in N/A

          \[\leadsto -\color{blue}{\left(\frac{1}{{x}^{2}} \cdot x + -1 \cdot x\right)} \]

      11. mul-1-neg N/A

          \[\leadsto -\left(\frac{1}{{x}^{2}} \cdot x + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right) \]

      12. unsub-neg N/A

          \[\leadsto -\color{blue}{\left(\frac{1}{{x}^{2}} \cdot x - x\right)} \]

      13. lower--.f64 N/A

          \[\leadsto -\color{blue}{\left(\frac{1}{{x}^{2}} \cdot x - x\right)} \]

      14. associate-*l/ N/A

          \[\leadsto -\left(\color{blue}{\frac{1 \cdot x}{{x}^{2}}} - x\right) \]

      15. *-lft-identity N/A

          \[\leadsto -\left(\frac{\color{blue}{x}}{{x}^{2}} - x\right) \]

      16. lower-/.f64 N/A

          \[\leadsto -\left(\color{blue}{\frac{x}{{x}^{2}}} - x\right) \]

      17. unpow2 N/A

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

      18. lower-*.f64 46.1

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

   5. Applied rewrites 46.1%

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

   6. Taylor expanded in x around inf

      \[\leadsto --1 \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 100.0%

      \[\leadsto -\left(-x\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;e^{z} \leq 1.03:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;-\left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (+ x (/ -1.0 x))
   (if (<= (exp z) 1.03)
     (+ x (/ y (- 1.1283791670955126 (* x y))))
     (- (- x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (exp(z) <= 0.0) {
		tmp = x + (-1.0 / x);
	} else if (exp(z) <= 1.03) {
		tmp = x + (y / (1.1283791670955126 - (x * y)));
	} 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 (exp(z) <= 0.0d0) then
        tmp = x + ((-1.0d0) / x)
    else if (exp(z) <= 1.03d0) then
        tmp = x + (y / (1.1283791670955126d0 - (x * y)))
    else
        tmp = -(-x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (Math.exp(z) <= 0.0) {
		tmp = x + (-1.0 / x);
	} else if (Math.exp(z) <= 1.03) {
		tmp = x + (y / (1.1283791670955126 - (x * y)));
	} else {
		tmp = -(-x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if math.exp(z) <= 0.0:
		tmp = x + (-1.0 / x)
	elif math.exp(z) <= 1.03:
		tmp = x + (y / (1.1283791670955126 - (x * y)))
	else:
		tmp = -(-x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (exp(z) <= 0.0)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (exp(z) <= 1.03)
		tmp = Float64(x + Float64(y / Float64(1.1283791670955126 - Float64(x * y))));
	else
		tmp = Float64(-Float64(-x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (exp(z) <= 0.0)
		tmp = x + (-1.0 / x);
	elseif (exp(z) <= 1.03)
		tmp = x + (y / (1.1283791670955126 - (x * y)));
	else
		tmp = -(-x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[Exp[z], $MachinePrecision], 0.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Exp[z], $MachinePrecision], 1.03], N[(x + N[(y / N[(1.1283791670955126 - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-(-x))]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 83.7%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto x + \color{blue}{\frac{-1}{x}} \]
   4. Step-by-step derivation

      1. lower-/.f64 100.0

         \[\leadsto x + \color{blue}{\frac{-1}{x}} \]

   5. Applied rewrites 100.0%

      \[\leadsto x + \color{blue}{\frac{-1}{x}} \]

   if 0.0 < (exp.f64 z) < 1.03000000000000003

   1. Initial program 99.8%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto x + \frac{y}{\color{blue}{\frac{5641895835477563}{5000000000000000}} - x \cdot y} \]
   4. Step-by-step derivation

   5. Applied rewrites 98.8%

      \[\leadsto x + \frac{y}{\color{blue}{1.1283791670955126} - x \cdot y} \]

   if 1.03000000000000003 < (exp.f64 z)

   1. Initial program 96.9%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(1 - \frac{1}{{x}^{2}}\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto x \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{{x}^{2}}\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{{x}^{2}}\right)\right) + 1\right)} \]

      3. neg-sub0 N/A

         \[\leadsto x \cdot \left(\color{blue}{\left(0 - \frac{1}{{x}^{2}}\right)} + 1\right) \]

      4. associate-+l- N/A

         \[\leadsto x \cdot \color{blue}{\left(0 - \left(\frac{1}{{x}^{2}} - 1\right)\right)} \]

      5. neg-sub0 N/A

         \[\leadsto x \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{{x}^{2}} - 1\right)\right)\right)} \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(x \cdot \left(\frac{1}{{x}^{2}} - 1\right)\right)} \]

      7. lower-neg.f64 N/A

         \[\leadsto \color{blue}{-x \cdot \left(\frac{1}{{x}^{2}} - 1\right)} \]

      8. sub-neg N/A

         \[\leadsto -x \cdot \color{blue}{\left(\frac{1}{{x}^{2}} + \left(\mathsf{neg}\left(1\right)\right)\right)} \]

      9. metadata-eval N/A

         \[\leadsto -x \cdot \left(\frac{1}{{x}^{2}} + \color{blue}{-1}\right) \]

      10. distribute-rgt-in N/A

          \[\leadsto -\color{blue}{\left(\frac{1}{{x}^{2}} \cdot x + -1 \cdot x\right)} \]

      11. mul-1-neg N/A

          \[\leadsto -\left(\frac{1}{{x}^{2}} \cdot x + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right) \]

      12. unsub-neg N/A

          \[\leadsto -\color{blue}{\left(\frac{1}{{x}^{2}} \cdot x - x\right)} \]

      13. lower--.f64 N/A

          \[\leadsto -\color{blue}{\left(\frac{1}{{x}^{2}} \cdot x - x\right)} \]

      14. associate-*l/ N/A

          \[\leadsto -\left(\color{blue}{\frac{1 \cdot x}{{x}^{2}}} - x\right) \]

      15. *-lft-identity N/A

          \[\leadsto -\left(\frac{\color{blue}{x}}{{x}^{2}} - x\right) \]

      16. lower-/.f64 N/A

          \[\leadsto -\left(\color{blue}{\frac{x}{{x}^{2}}} - x\right) \]

      17. unpow2 N/A

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

      18. lower-*.f64 46.1

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

   5. Applied rewrites 46.1%

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

   6. Taylor expanded in x around inf

      \[\leadsto --1 \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 100.0%

      \[\leadsto -\left(-x\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 98.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (+ x (/ -1.0 x))
   (+ x (/ y (- (* 1.1283791670955126 (exp z)) (* x y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (exp(z) <= 0.0) {
		tmp = x + (-1.0 / x);
	} else {
		tmp = x + (y / ((1.1283791670955126 * exp(z)) - (x * 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 (exp(z) <= 0.0d0) then
        tmp = x + ((-1.0d0) / x)
    else
        tmp = x + (y / ((1.1283791670955126d0 * exp(z)) - (x * y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (Math.exp(z) <= 0.0) {
		tmp = x + (-1.0 / x);
	} else {
		tmp = x + (y / ((1.1283791670955126 * Math.exp(z)) - (x * y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if math.exp(z) <= 0.0:
		tmp = x + (-1.0 / x)
	else:
		tmp = x + (y / ((1.1283791670955126 * math.exp(z)) - (x * y)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (exp(z) <= 0.0)
		tmp = Float64(x + Float64(-1.0 / x));
	else
		tmp = Float64(x + Float64(y / Float64(Float64(1.1283791670955126 * exp(z)) - Float64(x * y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (exp(z) <= 0.0)
		tmp = x + (-1.0 / x);
	else
		tmp = x + (y / ((1.1283791670955126 * exp(z)) - (x * y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[Exp[z], $MachinePrecision], 0.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], N[(x + N[(y / N[(N[(1.1283791670955126 * N[Exp[z], $MachinePrecision]), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 83.7%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto x + \color{blue}{\frac{-1}{x}} \]
   4. Step-by-step derivation

      1. lower-/.f64 100.0

         \[\leadsto x + \color{blue}{\frac{-1}{x}} \]

   5. Applied rewrites 100.0%

      \[\leadsto x + \color{blue}{\frac{-1}{x}} \]

   if 0.0 < (exp.f64 z)

   1. Initial program 98.8%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 68.3% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.45 \cdot 10^{+200} \lor \neg \left(z \leq -1.65 \cdot 10^{+60}\right):\\ \;\;\;\;-\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -2.45e+200) (not (<= z -1.65e+60))) (- (- x)) (/ -1.0 x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2.45e+200) || !(z <= -1.65e+60)) {
		tmp = -(-x);
	} else {
		tmp = -1.0 / 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 ((z <= (-2.45d+200)) .or. (.not. (z <= (-1.65d+60)))) then
        tmp = -(-x)
    else
        tmp = (-1.0d0) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2.45e+200) || !(z <= -1.65e+60)) {
		tmp = -(-x);
	} else {
		tmp = -1.0 / x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -2.45e+200) or not (z <= -1.65e+60):
		tmp = -(-x)
	else:
		tmp = -1.0 / x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -2.45e+200) || !(z <= -1.65e+60))
		tmp = Float64(-Float64(-x));
	else
		tmp = Float64(-1.0 / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -2.45e+200) || ~((z <= -1.65e+60)))
		tmp = -(-x);
	else
		tmp = -1.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -2.45e+200], N[Not[LessEqual[z, -1.65e+60]], $MachinePrecision]], (-(-x)), N[(-1.0 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.44999999999999991e200 or -1.6499999999999999e60 < z

   1. Initial program 95.6%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(1 - \frac{1}{{x}^{2}}\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto x \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{{x}^{2}}\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{{x}^{2}}\right)\right) + 1\right)} \]

      3. neg-sub0 N/A

         \[\leadsto x \cdot \left(\color{blue}{\left(0 - \frac{1}{{x}^{2}}\right)} + 1\right) \]

      4. associate-+l- N/A

         \[\leadsto x \cdot \color{blue}{\left(0 - \left(\frac{1}{{x}^{2}} - 1\right)\right)} \]

      5. neg-sub0 N/A

         \[\leadsto x \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{{x}^{2}} - 1\right)\right)\right)} \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(x \cdot \left(\frac{1}{{x}^{2}} - 1\right)\right)} \]

      7. lower-neg.f64 N/A

         \[\leadsto \color{blue}{-x \cdot \left(\frac{1}{{x}^{2}} - 1\right)} \]

      8. sub-neg N/A

         \[\leadsto -x \cdot \color{blue}{\left(\frac{1}{{x}^{2}} + \left(\mathsf{neg}\left(1\right)\right)\right)} \]

      9. metadata-eval N/A

         \[\leadsto -x \cdot \left(\frac{1}{{x}^{2}} + \color{blue}{-1}\right) \]

      10. distribute-rgt-in N/A

          \[\leadsto -\color{blue}{\left(\frac{1}{{x}^{2}} \cdot x + -1 \cdot x\right)} \]

      11. mul-1-neg N/A

          \[\leadsto -\left(\frac{1}{{x}^{2}} \cdot x + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right) \]

      12. unsub-neg N/A

          \[\leadsto -\color{blue}{\left(\frac{1}{{x}^{2}} \cdot x - x\right)} \]

      13. lower--.f64 N/A

          \[\leadsto -\color{blue}{\left(\frac{1}{{x}^{2}} \cdot x - x\right)} \]

      14. associate-*l/ N/A

          \[\leadsto -\left(\color{blue}{\frac{1 \cdot x}{{x}^{2}}} - x\right) \]

      15. *-lft-identity N/A

          \[\leadsto -\left(\frac{\color{blue}{x}}{{x}^{2}} - x\right) \]

      16. lower-/.f64 N/A

          \[\leadsto -\left(\color{blue}{\frac{x}{{x}^{2}}} - x\right) \]

      17. unpow2 N/A

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

      18. lower-*.f64 55.9

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

   5. Applied rewrites 55.9%

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

   6. Taylor expanded in x around inf

      \[\leadsto --1 \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 71.9%

      \[\leadsto -\left(-x\right) \]

   if -2.44999999999999991e200 < z < -1.6499999999999999e60

   1. Initial program 87.9%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(1 - \frac{1}{{x}^{2}}\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto x \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{{x}^{2}}\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{{x}^{2}}\right)\right) + 1\right)} \]

      3. neg-sub0 N/A

         \[\leadsto x \cdot \left(\color{blue}{\left(0 - \frac{1}{{x}^{2}}\right)} + 1\right) \]

      4. associate-+l- N/A

         \[\leadsto x \cdot \color{blue}{\left(0 - \left(\frac{1}{{x}^{2}} - 1\right)\right)} \]

      5. neg-sub0 N/A

         \[\leadsto x \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{{x}^{2}} - 1\right)\right)\right)} \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(x \cdot \left(\frac{1}{{x}^{2}} - 1\right)\right)} \]

      7. lower-neg.f64 N/A

         \[\leadsto \color{blue}{-x \cdot \left(\frac{1}{{x}^{2}} - 1\right)} \]

      8. sub-neg N/A

         \[\leadsto -x \cdot \color{blue}{\left(\frac{1}{{x}^{2}} + \left(\mathsf{neg}\left(1\right)\right)\right)} \]

      9. metadata-eval N/A

         \[\leadsto -x \cdot \left(\frac{1}{{x}^{2}} + \color{blue}{-1}\right) \]

      10. distribute-rgt-in N/A

          \[\leadsto -\color{blue}{\left(\frac{1}{{x}^{2}} \cdot x + -1 \cdot x\right)} \]

      11. mul-1-neg N/A

          \[\leadsto -\left(\frac{1}{{x}^{2}} \cdot x + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right) \]

      12. unsub-neg N/A

          \[\leadsto -\color{blue}{\left(\frac{1}{{x}^{2}} \cdot x - x\right)} \]

      13. lower--.f64 N/A

          \[\leadsto -\color{blue}{\left(\frac{1}{{x}^{2}} \cdot x - x\right)} \]

      14. associate-*l/ N/A

          \[\leadsto -\left(\color{blue}{\frac{1 \cdot x}{{x}^{2}}} - x\right) \]

      15. *-lft-identity N/A

          \[\leadsto -\left(\frac{\color{blue}{x}}{{x}^{2}} - x\right) \]

      16. lower-/.f64 N/A

          \[\leadsto -\left(\color{blue}{\frac{x}{{x}^{2}}} - x\right) \]

      17. unpow2 N/A

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

      18. lower-*.f64 63.7

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

   5. Applied rewrites 63.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 75.2%

      \[\leadsto \frac{-1}{\color{blue}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.45 \cdot 10^{+200} \lor \neg \left(z \leq -1.65 \cdot 10^{+60}\right):\\ \;\;\;\;-\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 69.4% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.55 \cdot 10^{-234} \lor \neg \left(x \leq 1.3 \cdot 10^{-136}\right):\\ \;\;\;\;-\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;0.8862269254527579 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.55e-234) (not (<= x 1.3e-136)))
   (- (- x))
   (* 0.8862269254527579 y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.55e-234) || !(x <= 1.3e-136)) {
		tmp = -(-x);
	} else {
		tmp = 0.8862269254527579 * 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 ((x <= (-1.55d-234)) .or. (.not. (x <= 1.3d-136))) then
        tmp = -(-x)
    else
        tmp = 0.8862269254527579d0 * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.55e-234) || !(x <= 1.3e-136)) {
		tmp = -(-x);
	} else {
		tmp = 0.8862269254527579 * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.55e-234) or not (x <= 1.3e-136):
		tmp = -(-x)
	else:
		tmp = 0.8862269254527579 * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.55e-234) || !(x <= 1.3e-136))
		tmp = Float64(-Float64(-x));
	else
		tmp = Float64(0.8862269254527579 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -1.55e-234) || ~((x <= 1.3e-136)))
		tmp = -(-x);
	else
		tmp = 0.8862269254527579 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.55e-234], N[Not[LessEqual[x, 1.3e-136]], $MachinePrecision]], (-(-x)), N[(0.8862269254527579 * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.5500000000000001e-234 or 1.29999999999999998e-136 < x

   1. Initial program 96.6%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(1 - \frac{1}{{x}^{2}}\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto x \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{{x}^{2}}\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{{x}^{2}}\right)\right) + 1\right)} \]

      3. neg-sub0 N/A

         \[\leadsto x \cdot \left(\color{blue}{\left(0 - \frac{1}{{x}^{2}}\right)} + 1\right) \]

      4. associate-+l- N/A

         \[\leadsto x \cdot \color{blue}{\left(0 - \left(\frac{1}{{x}^{2}} - 1\right)\right)} \]

      5. neg-sub0 N/A

         \[\leadsto x \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{{x}^{2}} - 1\right)\right)\right)} \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(x \cdot \left(\frac{1}{{x}^{2}} - 1\right)\right)} \]

      7. lower-neg.f64 N/A

         \[\leadsto \color{blue}{-x \cdot \left(\frac{1}{{x}^{2}} - 1\right)} \]

      8. sub-neg N/A

         \[\leadsto -x \cdot \color{blue}{\left(\frac{1}{{x}^{2}} + \left(\mathsf{neg}\left(1\right)\right)\right)} \]

      9. metadata-eval N/A

         \[\leadsto -x \cdot \left(\frac{1}{{x}^{2}} + \color{blue}{-1}\right) \]

      10. distribute-rgt-in N/A

          \[\leadsto -\color{blue}{\left(\frac{1}{{x}^{2}} \cdot x + -1 \cdot x\right)} \]

      11. mul-1-neg N/A

          \[\leadsto -\left(\frac{1}{{x}^{2}} \cdot x + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right) \]

      12. unsub-neg N/A

          \[\leadsto -\color{blue}{\left(\frac{1}{{x}^{2}} \cdot x - x\right)} \]

      13. lower--.f64 N/A

          \[\leadsto -\color{blue}{\left(\frac{1}{{x}^{2}} \cdot x - x\right)} \]

      14. associate-*l/ N/A

          \[\leadsto -\left(\color{blue}{\frac{1 \cdot x}{{x}^{2}}} - x\right) \]

      15. *-lft-identity N/A

          \[\leadsto -\left(\frac{\color{blue}{x}}{{x}^{2}} - x\right) \]

      16. lower-/.f64 N/A

          \[\leadsto -\left(\color{blue}{\frac{x}{{x}^{2}}} - x\right) \]

      17. unpow2 N/A

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

      18. lower-*.f64 68.8

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

   5. Applied rewrites 68.8%

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

   6. Taylor expanded in x around inf

      \[\leadsto --1 \cdot x \]
   7. Step-by-step derivation

   8. Applied rewrites 77.6%

      \[\leadsto -\left(-x\right) \]

   if -1.5500000000000001e-234 < x < 1.29999999999999998e-136

   1. Initial program 87.5%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{5000000000000000}{5641895835477563} \cdot \frac{y}{e^{z}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y}{e^{z}} \cdot \frac{5000000000000000}{5641895835477563}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{e^{z}} \cdot \frac{5000000000000000}{5641895835477563}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{e^{z}}} \cdot \frac{5000000000000000}{5641895835477563} \]

      4. lower-exp.f64 50.7

         \[\leadsto \frac{y}{\color{blue}{e^{z}}} \cdot 0.8862269254527579 \]

   5. Applied rewrites 50.7%

      \[\leadsto \color{blue}{\frac{y}{e^{z}} \cdot 0.8862269254527579} \]

   6. Taylor expanded in z around 0

      \[\leadsto \frac{5000000000000000}{5641895835477563} \cdot \color{blue}{y} \]
   7. Step-by-step derivation

   8. Applied rewrites 47.1%

      \[\leadsto 0.8862269254527579 \cdot \color{blue}{y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.55 \cdot 10^{-234} \lor \neg \left(x \leq 1.3 \cdot 10^{-136}\right):\\ \;\;\;\;-\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;0.8862269254527579 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 67.9% accurate, 25.6× speedup

Math

\[\begin{array}{l} \\ -\left(-x\right) \end{array} \]

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 Float64(-Float64(-x))
end

MATLAB

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

Wolfram

code[x_, y_, z_] := (-(-x))

Derivation

1. Initial program 94.7%

   \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
2. Add Preprocessing

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{x \cdot \left(1 - \frac{1}{{x}^{2}}\right)} \]
4. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto x \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(\frac{1}{{x}^{2}}\right)\right)\right)} \]

   2. +-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{{x}^{2}}\right)\right) + 1\right)} \]

   3. neg-sub0 N/A

      \[\leadsto x \cdot \left(\color{blue}{\left(0 - \frac{1}{{x}^{2}}\right)} + 1\right) \]

   4. associate-+l- N/A

      \[\leadsto x \cdot \color{blue}{\left(0 - \left(\frac{1}{{x}^{2}} - 1\right)\right)} \]

   5. neg-sub0 N/A

      \[\leadsto x \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{{x}^{2}} - 1\right)\right)\right)} \]

   6. distribute-rgt-neg-in N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(x \cdot \left(\frac{1}{{x}^{2}} - 1\right)\right)} \]

   7. lower-neg.f64 N/A

      \[\leadsto \color{blue}{-x \cdot \left(\frac{1}{{x}^{2}} - 1\right)} \]

   8. sub-neg N/A

      \[\leadsto -x \cdot \color{blue}{\left(\frac{1}{{x}^{2}} + \left(\mathsf{neg}\left(1\right)\right)\right)} \]

   9. metadata-eval N/A

      \[\leadsto -x \cdot \left(\frac{1}{{x}^{2}} + \color{blue}{-1}\right) \]

   10. distribute-rgt-in N/A

       \[\leadsto -\color{blue}{\left(\frac{1}{{x}^{2}} \cdot x + -1 \cdot x\right)} \]

   11. mul-1-neg N/A

       \[\leadsto -\left(\frac{1}{{x}^{2}} \cdot x + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right) \]

   12. unsub-neg N/A

       \[\leadsto -\color{blue}{\left(\frac{1}{{x}^{2}} \cdot x - x\right)} \]

   13. lower--.f64 N/A

       \[\leadsto -\color{blue}{\left(\frac{1}{{x}^{2}} \cdot x - x\right)} \]

   14. associate-*l/ N/A

       \[\leadsto -\left(\color{blue}{\frac{1 \cdot x}{{x}^{2}}} - x\right) \]

   15. *-lft-identity N/A

       \[\leadsto -\left(\frac{\color{blue}{x}}{{x}^{2}} - x\right) \]

   16. lower-/.f64 N/A

       \[\leadsto -\left(\color{blue}{\frac{x}{{x}^{2}}} - x\right) \]

   17. unpow2 N/A

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

   18. lower-*.f64 56.7

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

5. Applied rewrites 56.7%

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

6. Taylor expanded in x around inf

   \[\leadsto --1 \cdot x \]
7. Step-by-step derivation

8. Applied rewrites 66.9%

   \[\leadsto -\left(-x\right) \]
9. Add Preprocessing

Developer Target 1: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \frac{1}{\frac{1.1283791670955126}{y} \cdot e^{z} - x} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+ x (/ 1.0 (- (* (/ 1.1283791670955126 y) (exp z)) x))))

C

double code(double x, double y, double z) {
	return x + (1.0 / (((1.1283791670955126 / y) * exp(z)) - 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 + (1.0d0 / (((1.1283791670955126d0 / y) * exp(z)) - x))
end function

Java

public static double code(double x, double y, double z) {
	return x + (1.0 / (((1.1283791670955126 / y) * Math.exp(z)) - x));
}

Python

def code(x, y, z):
	return x + (1.0 / (((1.1283791670955126 / y) * math.exp(z)) - x))

Julia

function code(x, y, z)
	return Float64(x + Float64(1.0 / Float64(Float64(Float64(1.1283791670955126 / y) * exp(z)) - x)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = x + (1.0 / (((1.1283791670955126 / y) * exp(z)) - x));
end

Wolfram

code[x_, y_, z_] := N[(x + N[(1.0 / N[(N[(N[(1.1283791670955126 / y), $MachinePrecision] * N[Exp[z], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024340 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:invErfc from math-functions-0.1.5.2, A"
  :precision binary64

  :alt
  (! :herbie-platform default (+ x (/ 1 (- (* (/ 5641895835477563/5000000000000000 y) (exp z)) x))))

  (+ x (/ y (- (* 1.1283791670955126 (exp z)) (* x y)))))