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

Percentage Accurate: 95.4% → 99.7%

Time: 12.1s

Alternatives: 9

Speedup: 0.9×

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: 95.4% 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: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;e^{z} \leq 2:\\ \;\;\;\;x + \frac{y}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.18806319451591877, 0.5641895835477563\right), 1.1283791670955126\right), 1.1283791670955126\right) - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(e^{-z}, y \cdot 0.8862269254527579, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (+ x (/ -1.0 x))
   (if (<= (exp z) 2.0)
     (+
      x
      (/
       y
       (-
        (fma
         z
         (fma
          z
          (fma z 0.18806319451591877 0.5641895835477563)
          1.1283791670955126)
         1.1283791670955126)
        (* x y))))
     (fma (exp (- z)) (* y 0.8862269254527579) 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) <= 2.0) {
		tmp = x + (y / (fma(z, fma(z, fma(z, 0.18806319451591877, 0.5641895835477563), 1.1283791670955126), 1.1283791670955126) - (x * y)));
	} else {
		tmp = fma(exp(-z), (y * 0.8862269254527579), 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) <= 2.0)
		tmp = Float64(x + Float64(y / Float64(fma(z, fma(z, fma(z, 0.18806319451591877, 0.5641895835477563), 1.1283791670955126), 1.1283791670955126) - Float64(x * y))));
	else
		tmp = fma(exp(Float64(-z)), Float64(y * 0.8862269254527579), 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], 2.0], N[(x + N[(y / N[(N[(z * N[(z * N[(z * 0.18806319451591877 + 0.5641895835477563), $MachinePrecision] + 1.1283791670955126), $MachinePrecision] + 1.1283791670955126), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Exp[(-z)], $MachinePrecision] * N[(y * 0.8862269254527579), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 83.0%

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

   3. Taylor expanded in y 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) < 2

   1. Initial program 99.7%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 99.8

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

   5. Applied rewrites 99.8%

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

   if 2 < (exp.f64 z)

   1. Initial program 88.0%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

      8. rec-exp N/A

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

      9. neg-mul-1 N/A

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

      10. lower-exp.f64 N/A

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

      11. neg-mul-1 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;e^{z} \leq 2:\\ \;\;\;\;x + \frac{y}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.18806319451591877, 0.5641895835477563\right), 1.1283791670955126\right), 1.1283791670955126\right) - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(e^{-z}, y \cdot 0.8862269254527579, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{-1}{x}\\ t_1 := x + \frac{y}{e^{z} \cdot 1.1283791670955126 - x \cdot y}\\ \mathbf{if}\;t\_1 \leq -50:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 1000:\\ \;\;\;\;x + y \cdot \mathsf{fma}\left(x \cdot y, 0.7853981633974483, 0.8862269254527579\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (/ -1.0 x)))
        (t_1 (+ x (/ y (- (* (exp z) 1.1283791670955126) (* x y))))))
   (if (<= t_1 -50.0)
     t_0
     (if (<= t_1 1000.0)
       (+ x (* y (fma (* x y) 0.7853981633974483 0.8862269254527579)))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x + (-1.0 / x);
	double t_1 = x + (y / ((exp(z) * 1.1283791670955126) - (x * y)));
	double tmp;
	if (t_1 <= -50.0) {
		tmp = t_0;
	} else if (t_1 <= 1000.0) {
		tmp = x + (y * fma((x * y), 0.7853981633974483, 0.8862269254527579));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x + Float64(-1.0 / x))
	t_1 = Float64(x + Float64(y / Float64(Float64(exp(z) * 1.1283791670955126) - Float64(x * y))))
	tmp = 0.0
	if (t_1 <= -50.0)
		tmp = t_0;
	elseif (t_1 <= 1000.0)
		tmp = Float64(x + Float64(y * fma(Float64(x * y), 0.7853981633974483, 0.8862269254527579)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x + N[(y / N[(N[(N[Exp[z], $MachinePrecision] * 1.1283791670955126), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -50.0], t$95$0, If[LessEqual[t$95$1, 1000.0], N[(x + N[(y * N[(N[(x * y), $MachinePrecision] * 0.7853981633974483 + 0.8862269254527579), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

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)))) < -50 or 1e3 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y))))

   1. Initial program 89.2%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 90.3

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

   5. Applied rewrites 90.3%

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

   if -50 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y)))) < 1e3

   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

      \[\leadsto x + \color{blue}{\left(\frac{-5641895835477563}{5000000000000000} \cdot \frac{y \cdot z}{{\left(\frac{5641895835477563}{5000000000000000} - x \cdot y\right)}^{2}} + \frac{y}{\frac{5641895835477563}{5000000000000000} - x \cdot y}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto x + \left(\color{blue}{\frac{y \cdot z}{{\left(\frac{5641895835477563}{5000000000000000} - x \cdot y\right)}^{2}} \cdot \frac{-5641895835477563}{5000000000000000}} + \frac{y}{\frac{5641895835477563}{5000000000000000} - x \cdot y}\right) \]

      2. associate-/l* N/A

         \[\leadsto x + \left(\color{blue}{\left(y \cdot \frac{z}{{\left(\frac{5641895835477563}{5000000000000000} - x \cdot y\right)}^{2}}\right)} \cdot \frac{-5641895835477563}{5000000000000000} + \frac{y}{\frac{5641895835477563}{5000000000000000} - x \cdot y}\right) \]

      3. associate-*l* N/A

         \[\leadsto x + \left(\color{blue}{y \cdot \left(\frac{z}{{\left(\frac{5641895835477563}{5000000000000000} - x \cdot y\right)}^{2}} \cdot \frac{-5641895835477563}{5000000000000000}\right)} + \frac{y}{\frac{5641895835477563}{5000000000000000} - x \cdot y}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto x + \color{blue}{\mathsf{fma}\left(y, \frac{z}{{\left(\frac{5641895835477563}{5000000000000000} - x \cdot y\right)}^{2}} \cdot \frac{-5641895835477563}{5000000000000000}, \frac{y}{\frac{5641895835477563}{5000000000000000} - x \cdot y}\right)} \]

      5. associate-*l/ N/A

         \[\leadsto x + \mathsf{fma}\left(y, \color{blue}{\frac{z \cdot \frac{-5641895835477563}{5000000000000000}}{{\left(\frac{5641895835477563}{5000000000000000} - x \cdot y\right)}^{2}}}, \frac{y}{\frac{5641895835477563}{5000000000000000} - x \cdot y}\right) \]

      6. lower-/.f64 N/A

         \[\leadsto x + \mathsf{fma}\left(y, \color{blue}{\frac{z \cdot \frac{-5641895835477563}{5000000000000000}}{{\left(\frac{5641895835477563}{5000000000000000} - x \cdot y\right)}^{2}}}, \frac{y}{\frac{5641895835477563}{5000000000000000} - x \cdot y}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto x + \mathsf{fma}\left(y, \frac{\color{blue}{z \cdot \frac{-5641895835477563}{5000000000000000}}}{{\left(\frac{5641895835477563}{5000000000000000} - x \cdot y\right)}^{2}}, \frac{y}{\frac{5641895835477563}{5000000000000000} - x \cdot y}\right) \]

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower--.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower--.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 52.9

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

   5. Applied rewrites 52.9%

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

   6. Taylor expanded in y around 0

      \[\leadsto x + y \cdot \color{blue}{\left(\frac{5000000000000000}{5641895835477563} + \left(\frac{-5000000000000000}{5641895835477563} \cdot z + y \cdot \left(\frac{-50000000000000000000000000000000}{31830988618379068626528276418969} \cdot \left(x \cdot z\right) - \frac{-25000000000000000000000000000000}{31830988618379068626528276418969} \cdot x\right)\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 52.9%

      \[\leadsto x + y \cdot \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(z, x \cdot -1.5707963267948966, x \cdot 0.7853981633974483\right), \mathsf{fma}\left(z, -0.8862269254527579, 0.8862269254527579\right)\right)} \]

   9. Taylor expanded in z around 0

      \[\leadsto x + y \cdot \left(\frac{5000000000000000}{5641895835477563} + \frac{25000000000000000000000000000000}{31830988618379068626528276418969} \cdot \color{blue}{\left(x \cdot y\right)}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 66.4%

       \[\leadsto x + y \cdot \mathsf{fma}\left(x \cdot y, 0.7853981633974483, 0.8862269254527579\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{y}{e^{z} \cdot 1.1283791670955126 - x \cdot y} \leq -50:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;x + \frac{y}{e^{z} \cdot 1.1283791670955126 - x \cdot y} \leq 1000:\\ \;\;\;\;x + y \cdot \mathsf{fma}\left(x \cdot y, 0.7853981633974483, 0.8862269254527579\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{-1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{-1}{x}\\ t_1 := x + \frac{y}{e^{z} \cdot 1.1283791670955126 - x \cdot y}\\ \mathbf{if}\;t\_1 \leq -50:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 1000:\\ \;\;\;\;x + y \cdot 0.8862269254527579\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (/ -1.0 x)))
        (t_1 (+ x (/ y (- (* (exp z) 1.1283791670955126) (* x y))))))
   (if (<= t_1 -50.0)
     t_0
     (if (<= t_1 1000.0) (+ x (* y 0.8862269254527579)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x + (-1.0 / x);
	double t_1 = x + (y / ((exp(z) * 1.1283791670955126) - (x * y)));
	double tmp;
	if (t_1 <= -50.0) {
		tmp = t_0;
	} else if (t_1 <= 1000.0) {
		tmp = x + (y * 0.8862269254527579);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x + ((-1.0d0) / x)
    t_1 = x + (y / ((exp(z) * 1.1283791670955126d0) - (x * y)))
    if (t_1 <= (-50.0d0)) then
        tmp = t_0
    else if (t_1 <= 1000.0d0) then
        tmp = x + (y * 0.8862269254527579d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x + (-1.0 / x);
	double t_1 = x + (y / ((Math.exp(z) * 1.1283791670955126) - (x * y)));
	double tmp;
	if (t_1 <= -50.0) {
		tmp = t_0;
	} else if (t_1 <= 1000.0) {
		tmp = x + (y * 0.8862269254527579);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x + (-1.0 / x)
	t_1 = x + (y / ((math.exp(z) * 1.1283791670955126) - (x * y)))
	tmp = 0
	if t_1 <= -50.0:
		tmp = t_0
	elif t_1 <= 1000.0:
		tmp = x + (y * 0.8862269254527579)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x + Float64(-1.0 / x))
	t_1 = Float64(x + Float64(y / Float64(Float64(exp(z) * 1.1283791670955126) - Float64(x * y))))
	tmp = 0.0
	if (t_1 <= -50.0)
		tmp = t_0;
	elseif (t_1 <= 1000.0)
		tmp = Float64(x + Float64(y * 0.8862269254527579));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x + (-1.0 / x);
	t_1 = x + (y / ((exp(z) * 1.1283791670955126) - (x * y)));
	tmp = 0.0;
	if (t_1 <= -50.0)
		tmp = t_0;
	elseif (t_1 <= 1000.0)
		tmp = x + (y * 0.8862269254527579);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x + N[(y / N[(N[(N[Exp[z], $MachinePrecision] * 1.1283791670955126), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -50.0], t$95$0, If[LessEqual[t$95$1, 1000.0], N[(x + N[(y * 0.8862269254527579), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

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)))) < -50 or 1e3 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y))))

   1. Initial program 89.2%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 90.3

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

   5. Applied rewrites 90.3%

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

   if -50 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y)))) < 1e3

   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

      \[\leadsto x + \color{blue}{\left(\frac{-5641895835477563}{5000000000000000} \cdot \frac{y \cdot z}{{\left(\frac{5641895835477563}{5000000000000000} - x \cdot y\right)}^{2}} + \frac{y}{\frac{5641895835477563}{5000000000000000} - x \cdot y}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto x + \left(\color{blue}{\frac{y \cdot z}{{\left(\frac{5641895835477563}{5000000000000000} - x \cdot y\right)}^{2}} \cdot \frac{-5641895835477563}{5000000000000000}} + \frac{y}{\frac{5641895835477563}{5000000000000000} - x \cdot y}\right) \]

      2. associate-/l* N/A

         \[\leadsto x + \left(\color{blue}{\left(y \cdot \frac{z}{{\left(\frac{5641895835477563}{5000000000000000} - x \cdot y\right)}^{2}}\right)} \cdot \frac{-5641895835477563}{5000000000000000} + \frac{y}{\frac{5641895835477563}{5000000000000000} - x \cdot y}\right) \]

      3. associate-*l* N/A

         \[\leadsto x + \left(\color{blue}{y \cdot \left(\frac{z}{{\left(\frac{5641895835477563}{5000000000000000} - x \cdot y\right)}^{2}} \cdot \frac{-5641895835477563}{5000000000000000}\right)} + \frac{y}{\frac{5641895835477563}{5000000000000000} - x \cdot y}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto x + \color{blue}{\mathsf{fma}\left(y, \frac{z}{{\left(\frac{5641895835477563}{5000000000000000} - x \cdot y\right)}^{2}} \cdot \frac{-5641895835477563}{5000000000000000}, \frac{y}{\frac{5641895835477563}{5000000000000000} - x \cdot y}\right)} \]

      5. associate-*l/ N/A

         \[\leadsto x + \mathsf{fma}\left(y, \color{blue}{\frac{z \cdot \frac{-5641895835477563}{5000000000000000}}{{\left(\frac{5641895835477563}{5000000000000000} - x \cdot y\right)}^{2}}}, \frac{y}{\frac{5641895835477563}{5000000000000000} - x \cdot y}\right) \]

      6. lower-/.f64 N/A

         \[\leadsto x + \mathsf{fma}\left(y, \color{blue}{\frac{z \cdot \frac{-5641895835477563}{5000000000000000}}{{\left(\frac{5641895835477563}{5000000000000000} - x \cdot y\right)}^{2}}}, \frac{y}{\frac{5641895835477563}{5000000000000000} - x \cdot y}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto x + \mathsf{fma}\left(y, \frac{\color{blue}{z \cdot \frac{-5641895835477563}{5000000000000000}}}{{\left(\frac{5641895835477563}{5000000000000000} - x \cdot y\right)}^{2}}, \frac{y}{\frac{5641895835477563}{5000000000000000} - x \cdot y}\right) \]

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower--.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower--.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 52.9

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

   5. Applied rewrites 52.9%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 53.0%

      \[\leadsto x + y \cdot \color{blue}{\mathsf{fma}\left(z, -0.8862269254527579, 0.8862269254527579\right)} \]

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 66.4%

       \[\leadsto x + y \cdot 0.8862269254527579 \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{y}{e^{z} \cdot 1.1283791670955126 - x \cdot y} \leq -50:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;x + \frac{y}{e^{z} \cdot 1.1283791670955126 - x \cdot y} \leq 1000:\\ \;\;\;\;x + y \cdot 0.8862269254527579\\ \mathbf{else}:\\ \;\;\;\;x + \frac{-1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;e^{z} \leq 1.05:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{z \cdot 1.1283791670955126 - x \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (+ x (/ -1.0 x))
   (if (<= (exp z) 1.05)
     (+ x (/ y (- 1.1283791670955126 (* x y))))
     (+ x (/ y (- (* z 1.1283791670955126) (* x y)))))))

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) <= 1.05) {
		tmp = x + (y / (1.1283791670955126 - (x * y)));
	} else {
		tmp = x + (y / ((z * 1.1283791670955126) - (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 if (exp(z) <= 1.05d0) then
        tmp = x + (y / (1.1283791670955126d0 - (x * y)))
    else
        tmp = x + (y / ((z * 1.1283791670955126d0) - (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 if (Math.exp(z) <= 1.05) {
		tmp = x + (y / (1.1283791670955126 - (x * y)));
	} else {
		tmp = x + (y / ((z * 1.1283791670955126) - (x * y)));
	}
	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) <= 1.05:
		tmp = x + (y / (1.1283791670955126 - (x * y)))
	else:
		tmp = x + (y / ((z * 1.1283791670955126) - (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));
	elseif (exp(z) <= 1.05)
		tmp = Float64(x + Float64(y / Float64(1.1283791670955126 - Float64(x * y))));
	else
		tmp = Float64(x + Float64(y / Float64(Float64(z * 1.1283791670955126) - 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);
	elseif (exp(z) <= 1.05)
		tmp = x + (y / (1.1283791670955126 - (x * y)));
	else
		tmp = x + (y / ((z * 1.1283791670955126) - (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], If[LessEqual[N[Exp[z], $MachinePrecision], 1.05], N[(x + N[(y / N[(1.1283791670955126 - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(y / N[(N[(z * 1.1283791670955126), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 83.0%

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

   3. Taylor expanded in y 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) < 1.05000000000000004

   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 98.6%

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

   if 1.05000000000000004 < (exp.f64 z)

   1. Initial program 88.1%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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. *-commutative N/A

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

      3. lower-fma.f64 83.9

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

   5. Applied rewrites 83.9%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 83.9%

      \[\leadsto x + \frac{y}{z \cdot \color{blue}{1.1283791670955126} - x \cdot y} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 96.8% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (+ x (/ -1.0 x))
   (+
    x
    (/
     y
     (-
      (fma
       z
       (fma
        z
        (fma z 0.18806319451591877 0.5641895835477563)
        1.1283791670955126)
       1.1283791670955126)
      (* 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 / (fma(z, fma(z, fma(z, 0.18806319451591877, 0.5641895835477563), 1.1283791670955126), 1.1283791670955126) - (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(fma(z, fma(z, fma(z, 0.18806319451591877, 0.5641895835477563), 1.1283791670955126), 1.1283791670955126) - Float64(x * y))));
	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], N[(x + N[(y / N[(N[(z * N[(z * N[(z * 0.18806319451591877 + 0.5641895835477563), $MachinePrecision] + 1.1283791670955126), $MachinePrecision] + 1.1283791670955126), $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 83.0%

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

   3. Taylor expanded in y 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 94.6%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 93.1

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

   5. Applied rewrites 93.1%

      \[\leadsto x + \frac{y}{\color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.18806319451591877, 0.5641895835477563\right), 1.1283791670955126\right), 1.1283791670955126\right)} - x \cdot y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 96.1% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (+ x (/ -1.0 x))
   (+
    x
    (/
     y
     (-
      (fma z (fma z 0.5641895835477563 1.1283791670955126) 1.1283791670955126)
      (* 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 / (fma(z, fma(z, 0.5641895835477563, 1.1283791670955126), 1.1283791670955126) - (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(fma(z, fma(z, 0.5641895835477563, 1.1283791670955126), 1.1283791670955126) - Float64(x * y))));
	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], N[(x + N[(y / N[(N[(z * N[(z * 0.5641895835477563 + 1.1283791670955126), $MachinePrecision] + 1.1283791670955126), $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 83.0%

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

   3. Taylor expanded in y 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 94.6%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 93.1

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

   5. Applied rewrites 93.1%

      \[\leadsto x + \frac{y}{\color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5641895835477563, 1.1283791670955126\right), 1.1283791670955126\right)} - x \cdot y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 93.7% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (+ x (/ -1.0 x))
   (+ x (/ y (- (fma z 1.1283791670955126 1.1283791670955126) (* 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 / (fma(z, 1.1283791670955126, 1.1283791670955126) - (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(fma(z, 1.1283791670955126, 1.1283791670955126) - Float64(x * y))));
	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], N[(x + N[(y / N[(N[(z * 1.1283791670955126 + 1.1283791670955126), $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 83.0%

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

   3. Taylor expanded in y 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 94.6%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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. *-commutative N/A

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

      3. lower-fma.f64 92.7

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

   5. Applied rewrites 92.7%

      \[\leadsto x + \frac{y}{\color{blue}{\mathsf{fma}\left(z, 1.1283791670955126, 1.1283791670955126\right)} - x \cdot y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 90.7% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (+ x (/ -1.0 x))
   (+ x (/ y (- 1.1283791670955126 (* 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 - (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 - (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 - (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 - (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(1.1283791670955126 - 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 - (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[(1.1283791670955126 - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 83.0%

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

   3. Taylor expanded in y 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 94.6%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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 88.6%

      \[\leadsto x + \frac{y}{\color{blue}{1.1283791670955126} - x \cdot y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 59.4% accurate, 14.2× speedup

Math

\[\begin{array}{l} \\ x + y \cdot 0.8862269254527579 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ x (* y 0.8862269254527579)))

C

double code(double x, double y, double z) {
	return x + (y * 0.8862269254527579);
}

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 * 0.8862269254527579d0)
end function

Java

public static double code(double x, double y, double z) {
	return x + (y * 0.8862269254527579);
}

Python

def code(x, y, z):
	return x + (y * 0.8862269254527579)

Julia

function code(x, y, z)
	return Float64(x + Float64(y * 0.8862269254527579))
end

MATLAB

function tmp = code(x, y, z)
	tmp = x + (y * 0.8862269254527579);
end

Wolfram

code[x_, y_, z_] := N[(x + N[(y * 0.8862269254527579), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.7%

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

3. Taylor expanded in z around 0

   \[\leadsto x + \color{blue}{\left(\frac{-5641895835477563}{5000000000000000} \cdot \frac{y \cdot z}{{\left(\frac{5641895835477563}{5000000000000000} - x \cdot y\right)}^{2}} + \frac{y}{\frac{5641895835477563}{5000000000000000} - x \cdot y}\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto x + \left(\color{blue}{\frac{y \cdot z}{{\left(\frac{5641895835477563}{5000000000000000} - x \cdot y\right)}^{2}} \cdot \frac{-5641895835477563}{5000000000000000}} + \frac{y}{\frac{5641895835477563}{5000000000000000} - x \cdot y}\right) \]

   2. associate-/l* N/A

      \[\leadsto x + \left(\color{blue}{\left(y \cdot \frac{z}{{\left(\frac{5641895835477563}{5000000000000000} - x \cdot y\right)}^{2}}\right)} \cdot \frac{-5641895835477563}{5000000000000000} + \frac{y}{\frac{5641895835477563}{5000000000000000} - x \cdot y}\right) \]

   3. associate-*l* N/A

      \[\leadsto x + \left(\color{blue}{y \cdot \left(\frac{z}{{\left(\frac{5641895835477563}{5000000000000000} - x \cdot y\right)}^{2}} \cdot \frac{-5641895835477563}{5000000000000000}\right)} + \frac{y}{\frac{5641895835477563}{5000000000000000} - x \cdot y}\right) \]

   4. lower-fma.f64 N/A

      \[\leadsto x + \color{blue}{\mathsf{fma}\left(y, \frac{z}{{\left(\frac{5641895835477563}{5000000000000000} - x \cdot y\right)}^{2}} \cdot \frac{-5641895835477563}{5000000000000000}, \frac{y}{\frac{5641895835477563}{5000000000000000} - x \cdot y}\right)} \]

   5. associate-*l/ N/A

      \[\leadsto x + \mathsf{fma}\left(y, \color{blue}{\frac{z \cdot \frac{-5641895835477563}{5000000000000000}}{{\left(\frac{5641895835477563}{5000000000000000} - x \cdot y\right)}^{2}}}, \frac{y}{\frac{5641895835477563}{5000000000000000} - x \cdot y}\right) \]

   6. lower-/.f64 N/A

      \[\leadsto x + \mathsf{fma}\left(y, \color{blue}{\frac{z \cdot \frac{-5641895835477563}{5000000000000000}}{{\left(\frac{5641895835477563}{5000000000000000} - x \cdot y\right)}^{2}}}, \frac{y}{\frac{5641895835477563}{5000000000000000} - x \cdot y}\right) \]

   7. lower-*.f64 N/A

      \[\leadsto x + \mathsf{fma}\left(y, \frac{\color{blue}{z \cdot \frac{-5641895835477563}{5000000000000000}}}{{\left(\frac{5641895835477563}{5000000000000000} - x \cdot y\right)}^{2}}, \frac{y}{\frac{5641895835477563}{5000000000000000} - x \cdot y}\right) \]

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. lower--.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower--.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

   16. lower-/.f64 N/A

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

   17. lower--.f64 N/A

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

   18. *-commutative N/A

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

   19. lower-*.f64 74.4

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

5. Applied rewrites 74.4%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 45.7%

   \[\leadsto x + y \cdot \color{blue}{\mathsf{fma}\left(z, -0.8862269254527579, 0.8862269254527579\right)} \]

9. Taylor expanded in z around 0

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

11. Applied rewrites 57.8%

    \[\leadsto x + y \cdot 0.8862269254527579 \]
12. 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 2024219 
(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)))))