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

Percentage Accurate: 95.7% → 99.7%

Time: 10.8s

Alternatives: 11

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: 95.7% 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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;e^{z} \leq 5000000:\\ \;\;\;\;x + \frac{y}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.18806319451591877, 0.5641895835477563\right), 1.1283791670955126\right), 1.1283791670955126\right) - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(e^{-z}, y \cdot 0.8862269254527579, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (+ x (/ -1.0 x))
   (if (<= (exp z) 5000000.0)
     (+
      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) <= 5000000.0) {
		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) <= 5000000.0)
		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], 5000000.0], 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 92.3%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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) < 5e6

   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. 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 99.9

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

   5. Applied rewrites 99.9%

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

   if 5e6 < (exp.f64 z)

   1. Initial program 91.1%

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

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;e^{z} \leq 5000000:\\ \;\;\;\;x + \frac{y}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.18806319451591877, 0.5641895835477563\right), 1.1283791670955126\right), 1.1283791670955126\right) - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(e^{-z}, y \cdot 0.8862269254527579, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \leq 2 \cdot 10^{+268}:\\ \;\;\;\;x + \frac{y}{\mathsf{fma}\left(e^{z}, 1.1283791670955126, -x \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (+ x (/ y (- (* 1.1283791670955126 (exp z)) (* x y)))) 2e+268)
   (+ x (/ y (fma (exp z) 1.1283791670955126 (- (* x y)))))
   (+ x (/ -1.0 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x + (y / ((1.1283791670955126 * exp(z)) - (x * y)))) <= 2e+268) {
		tmp = x + (y / fma(exp(z), 1.1283791670955126, -(x * y)));
	} else {
		tmp = x + (-1.0 / x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(x + Float64(y / Float64(Float64(1.1283791670955126 * exp(z)) - Float64(x * y)))) <= 2e+268)
		tmp = Float64(x + Float64(y / fma(exp(z), 1.1283791670955126, Float64(-Float64(x * y)))));
	else
		tmp = Float64(x + Float64(-1.0 / x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(x + N[(y / N[(N[(1.1283791670955126 * N[Exp[z], $MachinePrecision]), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+268], N[(x + N[(y / N[(N[Exp[z], $MachinePrecision] * 1.1283791670955126 + (-N[(x * y), $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]

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)))) < 1.9999999999999999e268

   1. Initial program 98.3%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 98.3

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

   4. Applied rewrites 98.3%

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

   if 1.9999999999999999e268 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y))))

   1. Initial program 46.7%

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

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

Alternative 3: 98.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y}\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{+268}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;x + \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (/ y (- (* 1.1283791670955126 (exp z)) (* x y))))))
   (if (<= t_0 2e+268) t_0 (+ x (/ -1.0 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 <= 2e+268) {
		tmp = t_0;
	} else {
		tmp = x + (-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) :: t_0
    real(8) :: tmp
    t_0 = x + (y / ((1.1283791670955126d0 * exp(z)) - (x * y)))
    if (t_0 <= 2d+268) then
        tmp = t_0
    else
        tmp = x + ((-1.0d0) / 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 <= 2e+268) {
		tmp = t_0;
	} else {
		tmp = x + (-1.0 / 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 <= 2e+268:
		tmp = t_0
	else:
		tmp = x + (-1.0 / 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 <= 2e+268)
		tmp = t_0;
	else
		tmp = Float64(x + Float64(-1.0 / 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 <= 2e+268)
		tmp = t_0;
	else
		tmp = x + (-1.0 / 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[LessEqual[t$95$0, 2e+268], t$95$0, N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]]

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)))) < 1.9999999999999999e268

   1. Initial program 98.3%

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

   if 1.9999999999999999e268 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y))))

   1. Initial program 46.7%

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

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

Alternative 4: 92.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;e^{z} \leq 1.2:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 \cdot z - x \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (+ x (/ -1.0 x))
   (if (<= (exp z) 1.2)
     (+ x (/ y (- 1.1283791670955126 (* x y))))
     (+ x (/ y (- (* 1.1283791670955126 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 if (exp(z) <= 1.2) {
		tmp = x + (y / (1.1283791670955126 - (x * y)));
	} else {
		tmp = x + (y / ((1.1283791670955126 * 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 if (exp(z) <= 1.2d0) then
        tmp = x + (y / (1.1283791670955126d0 - (x * y)))
    else
        tmp = x + (y / ((1.1283791670955126d0 * 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 if (Math.exp(z) <= 1.2) {
		tmp = x + (y / (1.1283791670955126 - (x * y)));
	} else {
		tmp = x + (y / ((1.1283791670955126 * z) - (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) <= 1.2:
		tmp = x + (y / (1.1283791670955126 - (x * y)))
	else:
		tmp = x + (y / ((1.1283791670955126 * 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));
	elseif (exp(z) <= 1.2)
		tmp = Float64(x + Float64(y / Float64(1.1283791670955126 - Float64(x * y))));
	else
		tmp = Float64(x + Float64(y / Float64(Float64(1.1283791670955126 * 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);
	elseif (exp(z) <= 1.2)
		tmp = x + (y / (1.1283791670955126 - (x * y)));
	else
		tmp = x + (y / ((1.1283791670955126 * 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], If[LessEqual[N[Exp[z], $MachinePrecision], 1.2], N[(x + N[(y / N[(1.1283791670955126 - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(y / N[(N[(1.1283791670955126 * z), $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 92.3%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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) < 1.19999999999999996

   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 99.3%

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

   if 1.19999999999999996 < (exp.f64 z)

   1. Initial program 91.5%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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 78.8

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

   5. Applied rewrites 78.8%

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

   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 78.8%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;e^{z} \leq 1.2:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 \cdot z - x \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.6% 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 92.3%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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 97.1%

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

   3. Taylor expanded in z around 0

      \[\leadsto x + \frac{y}{\color{blue}{\left(\frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{10000000000000000} + \frac{5641895835477563}{30000000000000000} \cdot z\right)\right)\right)} - x \cdot y} \]
   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 95.6

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

   5. Applied rewrites 95.6%

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

Alternative 6: 96.3% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (+ x (/ -1.0 x))
   (+
    x
    (/
     y
     (- (fma z (* 0.18806319451591877 (* z 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 {
		tmp = x + (y / (fma(z, (0.18806319451591877 * (z * 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));
	else
		tmp = Float64(x + Float64(y / Float64(fma(z, Float64(0.18806319451591877 * Float64(z * z)), 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[(0.18806319451591877 * N[(z * z), $MachinePrecision]), $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 92.3%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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 97.1%

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

   3. Taylor expanded in z around 0

      \[\leadsto x + \frac{y}{\color{blue}{\left(\frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{10000000000000000} + \frac{5641895835477563}{30000000000000000} \cdot z\right)\right)\right)} - x \cdot y} \]
   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 95.6

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

   5. Applied rewrites 95.6%

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

   6. Taylor expanded in z around inf

      \[\leadsto x + \frac{y}{\mathsf{fma}\left(z, \frac{5641895835477563}{30000000000000000} \cdot \color{blue}{{z}^{2}}, \frac{5641895835477563}{5000000000000000}\right) - x \cdot y} \]
   7. Step-by-step derivation

   8. Applied rewrites 95.2%

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

Alternative 7: 95.6% 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 92.3%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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 97.1%

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

   3. Taylor expanded in z around 0

      \[\leadsto x + \frac{y}{\color{blue}{\left(\frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z\right)\right)} - x \cdot y} \]
   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 95.0

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

   5. Applied rewrites 95.0%

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

Alternative 8: 93.0% 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 92.3%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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 97.1%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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 92.8

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

   5. Applied rewrites 92.8%

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

Alternative 9: 89.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 3.8 \cdot 10^{-110}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 - x \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 3.8e-110)
   (+ x (/ -1.0 x))
   (+ x (/ y (- 1.1283791670955126 (* x y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (exp(z) <= 3.8e-110) {
		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) <= 3.8d-110) 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) <= 3.8e-110) {
		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) <= 3.8e-110:
		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) <= 3.8e-110)
		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) <= 3.8e-110)
		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], 3.8e-110], 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) < 3.7999999999999998e-110

   1. Initial program 92.3%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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 3.7999999999999998e-110 < (exp.f64 z)

   1. Initial program 97.1%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 90.3%

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

Alternative 10: 68.2% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ x + \frac{-1}{x} \end{array} \]

FPCore

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

C

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

Java

public static double code(double x, double y, double z) {
	return x + (-1.0 / x);
}

Python

def code(x, y, z):
	return x + (-1.0 / x)

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.7%

   \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
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 72.1

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

5. Applied rewrites 72.1%

   \[\leadsto x + \color{blue}{\frac{-1}{x}} \]
6. Add Preprocessing

Alternative 11: 19.2% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{x} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ -1.0 x))

C

double code(double x, double y, double z) {
	return -1.0 / 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 = (-1.0d0) / x
end function

Java

public static double code(double x, double y, double z) {
	return -1.0 / x;
}

Python

def code(x, y, z):
	return -1.0 / x

Julia

function code(x, y, z)
	return Float64(-1.0 / x)
end

MATLAB

function tmp = code(x, y, z)
	tmp = -1.0 / x;
end

Wolfram

code[x_, y_, z_] := N[(-1.0 / x), $MachinePrecision]

Derivation

1. Initial program 95.7%

   \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
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. distribute-rgt-in N/A

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

   3. *-lft-identity N/A

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

   4. distribute-lft-neg-out N/A

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

   5. distribute-rgt-neg-out N/A

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

   6. mul-1-neg N/A

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

   7. lower-+.f64 N/A

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

   8. mul-1-neg N/A

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

   9. distribute-rgt-neg-out N/A

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

   10. distribute-lft-neg-out N/A

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

   11. distribute-neg-frac2 N/A

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

   12. associate-*l/ N/A

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

   13. *-lft-identity N/A

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

   14. lower-/.f64 N/A

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

   15. unpow2 N/A

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

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

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

   17. mul-1-neg N/A

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

   18. lower-*.f64 N/A

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

   19. mul-1-neg N/A

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

   20. lower-neg.f64 65.2

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

5. Applied rewrites 65.2%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 20.6%

   \[\leadsto \frac{-1}{\color{blue}{x}} \]
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 2024220 
(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)))))