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

Percentage Accurate: 95.8% → 99.9%

Time: 8.6s

Alternatives: 13

Speedup: 2.8×

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x + (y / ((1.1283791670955126d0 * exp(z)) - (x * y)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 95.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x + (y / ((1.1283791670955126d0 * exp(z)) - (x * y)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4700:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\mathsf{fma}\left(-y, x, e^{z} \cdot 1.1283791670955126\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -4700.0)
   (+ x (/ -1.0 x))
   (+ x (/ y (fma (- y) x (* (exp z) 1.1283791670955126))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -4700.0) {
		tmp = x + (-1.0 / x);
	} else {
		tmp = x + (y / fma(-y, x, (exp(z) * 1.1283791670955126)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -4700.0)
		tmp = Float64(x + Float64(-1.0 / x));
	else
		tmp = Float64(x + Float64(y / fma(Float64(-y), x, Float64(exp(z) * 1.1283791670955126))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -4700.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], N[(x + N[(y / N[((-y) * x + N[(N[Exp[z], $MachinePrecision] * 1.1283791670955126), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4700

   1. Initial program 92.1%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -4700 < z

   1. Initial program 98.9%

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

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. +-commutative N/A

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

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

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

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

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-neg.f64 99.9

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

Alternative 2: 84.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y}\\ \mathbf{if}\;t\_0 \leq -100000 \lor \neg \left(t\_0 \leq 5 \cdot 10^{-5}\right):\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;x + 0.8862269254527579 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (/ y (- (* 1.1283791670955126 (exp z)) (* x y))))))
   (if (or (<= t_0 -100000.0) (not (<= t_0 5e-5)))
     (+ x (/ -1.0 x))
     (+ x (* 0.8862269254527579 y)))))

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 <= -100000.0) || !(t_0 <= 5e-5)) {
		tmp = x + (-1.0 / x);
	} else {
		tmp = x + (0.8862269254527579 * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x + (y / ((1.1283791670955126d0 * exp(z)) - (x * y)))
    if ((t_0 <= (-100000.0d0)) .or. (.not. (t_0 <= 5d-5))) then
        tmp = x + ((-1.0d0) / x)
    else
        tmp = x + (0.8862269254527579d0 * y)
    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 <= -100000.0) || !(t_0 <= 5e-5)) {
		tmp = x + (-1.0 / x);
	} else {
		tmp = x + (0.8862269254527579 * y);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x + (y / ((1.1283791670955126 * math.exp(z)) - (x * y)))
	tmp = 0
	if (t_0 <= -100000.0) or not (t_0 <= 5e-5):
		tmp = x + (-1.0 / x)
	else:
		tmp = x + (0.8862269254527579 * y)
	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 <= -100000.0) || !(t_0 <= 5e-5))
		tmp = Float64(x + Float64(-1.0 / x));
	else
		tmp = Float64(x + Float64(0.8862269254527579 * y));
	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 <= -100000.0) || ~((t_0 <= 5e-5)))
		tmp = x + (-1.0 / x);
	else
		tmp = x + (0.8862269254527579 * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x + N[(y / N[(N[(1.1283791670955126 * N[Exp[z], $MachinePrecision]), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -100000.0], N[Not[LessEqual[t$95$0, 5e-5]], $MachinePrecision]], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], N[(x + N[(0.8862269254527579 * y), $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)))) < -1e5 or 5.00000000000000024e-5 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y))))

   1. Initial program 96.4%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 93.0

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

   5. Applied rewrites 93.0%

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

   if -1e5 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y)))) < 5.00000000000000024e-5

   1. Initial program 99.9%

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

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

      2. unpow2 N/A

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

      3. associate-/r* N/A

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

      4. div-add-rev N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. metadata-eval N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 N/A

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

      14. lower--.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower--.f64 N/A

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

   5. Applied rewrites 62.6%

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

   6. Taylor expanded in x around 0

      \[\leadsto x + \frac{5000000000000000}{5641895835477563} \cdot \color{blue}{\left(y + -1 \cdot \left(y \cdot z\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 62.4%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 72.4%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \leq -100000 \lor \neg \left(x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \leq 5 \cdot 10^{-5}\right):\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;x + 0.8862269254527579 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (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 x around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 0.0 < (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z))

   1. Initial program 98.9%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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 90.8%

      \[\leadsto x + \frac{y}{\color{blue}{1.1283791670955126} - x \cdot y} \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 90.8

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. fp-cancel-sub-sign-inv N/A

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

      8. lift-neg.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 90.9

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

   7. Applied rewrites 90.9%

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

   8. Taylor expanded in z around 0

      \[\leadsto \frac{y}{\color{blue}{\frac{5641895835477563}{5000000000000000} + \left(-1 \cdot \left(x \cdot y\right) + \frac{5641895835477563}{5000000000000000} \cdot z\right)}} + x \]
   9. Step-by-step derivation

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 93.0

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

   10. Applied rewrites 93.0%

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

Alternative 4: 93.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (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 x around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 0.0 < (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z))

   1. Initial program 98.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 93.0

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

   5. Applied rewrites 93.0%

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

Alternative 5: 98.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4700:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -4700.0)
   (+ x (/ -1.0 x))
   (+ x (/ y (- (* 1.1283791670955126 (exp z)) (* x y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -4700.0) {
		tmp = x + (-1.0 / x);
	} else {
		tmp = x + (y / ((1.1283791670955126 * exp(z)) - (x * y)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-4700.0d0)) then
        tmp = x + ((-1.0d0) / x)
    else
        tmp = x + (y / ((1.1283791670955126d0 * exp(z)) - (x * y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -4700.0) {
		tmp = x + (-1.0 / x);
	} else {
		tmp = x + (y / ((1.1283791670955126 * Math.exp(z)) - (x * y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -4700.0:
		tmp = x + (-1.0 / x)
	else:
		tmp = x + (y / ((1.1283791670955126 * math.exp(z)) - (x * y)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -4700.0)
		tmp = Float64(x + Float64(-1.0 / x));
	else
		tmp = Float64(x + Float64(y / Float64(Float64(1.1283791670955126 * exp(z)) - Float64(x * y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -4700.0)
		tmp = x + (-1.0 / x);
	else
		tmp = x + (y / ((1.1283791670955126 * exp(z)) - (x * y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -4700.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], N[(x + N[(y / N[(N[(1.1283791670955126 * N[Exp[z], $MachinePrecision]), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4700

   1. Initial program 92.1%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -4700 < z

   1. Initial program 98.9%

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

Alternative 6: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4700:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 52:\\ \;\;\;\;x + \frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.18806319451591877, z, 0.5641895835477563\right), z, 1.1283791670955126\right), z, 1.1283791670955126\right) - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.8862269254527579}{e^{z}}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -4700.0)
   (+ x (/ -1.0 x))
   (if (<= z 52.0)
     (+
      x
      (/
       y
       (-
        (fma
         (fma
          (fma 0.18806319451591877 z 0.5641895835477563)
          z
          1.1283791670955126)
         z
         1.1283791670955126)
        (* x y))))
     (fma (/ 0.8862269254527579 (exp z)) y x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -4700.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 52.0) {
		tmp = x + (y / (fma(fma(fma(0.18806319451591877, z, 0.5641895835477563), z, 1.1283791670955126), z, 1.1283791670955126) - (x * y)));
	} else {
		tmp = fma((0.8862269254527579 / exp(z)), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -4700.0)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (z <= 52.0)
		tmp = Float64(x + Float64(y / Float64(fma(fma(fma(0.18806319451591877, z, 0.5641895835477563), z, 1.1283791670955126), z, 1.1283791670955126) - Float64(x * y))));
	else
		tmp = fma(Float64(0.8862269254527579 / exp(z)), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -4700.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 52.0], N[(x + N[(y / N[(N[(N[(N[(0.18806319451591877 * z + 0.5641895835477563), $MachinePrecision] * z + 1.1283791670955126), $MachinePrecision] * z + 1.1283791670955126), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.8862269254527579 / N[Exp[z], $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -4700

   1. Initial program 92.1%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -4700 < z < 52

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 99.4

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

   5. Applied rewrites 99.4%

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

   if 52 < z

   1. Initial program 96.7%

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

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

      3. associate-*l/ N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. lower-fma.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

Alternative 7: 96.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4700:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.18806319451591877, z, 0.5641895835477563\right), z, 1.1283791670955126\right), z, 1.1283791670955126\right) - x \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -4700.0)
   (+ x (/ -1.0 x))
   (+
    x
    (/
     y
     (-
      (fma
       (fma
        (fma 0.18806319451591877 z 0.5641895835477563)
        z
        1.1283791670955126)
       z
       1.1283791670955126)
      (* x y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -4700.0) {
		tmp = x + (-1.0 / x);
	} else {
		tmp = x + (y / (fma(fma(fma(0.18806319451591877, z, 0.5641895835477563), z, 1.1283791670955126), z, 1.1283791670955126) - (x * y)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -4700.0)
		tmp = Float64(x + Float64(-1.0 / x));
	else
		tmp = Float64(x + Float64(y / Float64(fma(fma(fma(0.18806319451591877, z, 0.5641895835477563), z, 1.1283791670955126), z, 1.1283791670955126) - Float64(x * y))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -4700.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], N[(x + N[(y / N[(N[(N[(N[(0.18806319451591877 * z + 0.5641895835477563), $MachinePrecision] * z + 1.1283791670955126), $MachinePrecision] * z + 1.1283791670955126), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4700

   1. Initial program 92.1%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -4700 < z

   1. Initial program 98.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 97.0

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

   5. Applied rewrites 97.0%

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

Alternative 8: 96.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4700:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\mathsf{fma}\left(\left(z \cdot z\right) \cdot 0.18806319451591877, z, 1.1283791670955126\right) - x \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -4700.0)
   (+ x (/ -1.0 x))
   (+
    x
    (/
     y
     (- (fma (* (* z z) 0.18806319451591877) z 1.1283791670955126) (* x y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -4700.0) {
		tmp = x + (-1.0 / x);
	} else {
		tmp = x + (y / (fma(((z * z) * 0.18806319451591877), z, 1.1283791670955126) - (x * y)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -4700.0)
		tmp = Float64(x + Float64(-1.0 / x));
	else
		tmp = Float64(x + Float64(y / Float64(fma(Float64(Float64(z * z) * 0.18806319451591877), z, 1.1283791670955126) - Float64(x * y))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -4700.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], N[(x + N[(y / N[(N[(N[(N[(z * z), $MachinePrecision] * 0.18806319451591877), $MachinePrecision] * z + 1.1283791670955126), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4700

   1. Initial program 92.1%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -4700 < z

   1. Initial program 98.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 97.0

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

   5. Applied rewrites 97.0%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 96.5%

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

Alternative 9: 95.6% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4700:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(0.5641895835477563, z, 1.1283791670955126\right), z, 1.1283791670955126\right) - x \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -4700.0)
   (+ x (/ -1.0 x))
   (+
    x
    (/
     y
     (-
      (fma (fma 0.5641895835477563 z 1.1283791670955126) z 1.1283791670955126)
      (* x y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -4700.0) {
		tmp = x + (-1.0 / x);
	} else {
		tmp = x + (y / (fma(fma(0.5641895835477563, z, 1.1283791670955126), z, 1.1283791670955126) - (x * y)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -4700.0)
		tmp = Float64(x + Float64(-1.0 / x));
	else
		tmp = Float64(x + Float64(y / Float64(fma(fma(0.5641895835477563, z, 1.1283791670955126), z, 1.1283791670955126) - Float64(x * y))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -4700.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], N[(x + N[(y / N[(N[(N[(0.5641895835477563 * z + 1.1283791670955126), $MachinePrecision] * z + 1.1283791670955126), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4700

   1. Initial program 92.1%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -4700 < z

   1. Initial program 98.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 96.1

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

   5. Applied rewrites 96.1%

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

Alternative 10: 95.4% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4700:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\mathsf{fma}\left(0.5641895835477563 \cdot z, z, 1.1283791670955126\right) - x \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -4700.0)
   (+ x (/ -1.0 x))
   (+
    x
    (/ y (- (fma (* 0.5641895835477563 z) z 1.1283791670955126) (* x y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -4700.0) {
		tmp = x + (-1.0 / x);
	} else {
		tmp = x + (y / (fma((0.5641895835477563 * z), z, 1.1283791670955126) - (x * y)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -4700.0)
		tmp = Float64(x + Float64(-1.0 / x));
	else
		tmp = Float64(x + Float64(y / Float64(fma(Float64(0.5641895835477563 * z), z, 1.1283791670955126) - Float64(x * y))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -4700.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], N[(x + N[(y / N[(N[(N[(0.5641895835477563 * z), $MachinePrecision] * z + 1.1283791670955126), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4700

   1. Initial program 92.1%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -4700 < z

   1. Initial program 98.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 96.1

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

   5. Applied rewrites 96.1%

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

   6. Taylor expanded in z around inf

      \[\leadsto x + \frac{y}{\mathsf{fma}\left(\frac{5641895835477563}{10000000000000000} \cdot z, z, \frac{5641895835477563}{5000000000000000}\right) - x \cdot y} \]
   7. Step-by-step derivation

   8. Applied rewrites 95.6%

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

Alternative 11: 90.7% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4700:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(-y, x, 1.1283791670955126\right)} + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -4700.0)
   (+ x (/ -1.0 x))
   (+ (/ y (fma (- y) x 1.1283791670955126)) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -4700.0) {
		tmp = x + (-1.0 / x);
	} else {
		tmp = (y / fma(-y, x, 1.1283791670955126)) + x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -4700.0)
		tmp = Float64(x + Float64(-1.0 / x));
	else
		tmp = Float64(Float64(y / fma(Float64(-y), x, 1.1283791670955126)) + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -4700.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], N[(N[(y / N[((-y) * x + 1.1283791670955126), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4700

   1. Initial program 92.1%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -4700 < z

   1. Initial program 98.9%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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 90.9%

      \[\leadsto x + \frac{y}{\color{blue}{1.1283791670955126} - x \cdot y} \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 90.9

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. fp-cancel-sub-sign-inv N/A

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

      8. lift-neg.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 90.9

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

   7. Applied rewrites 90.9%

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

Alternative 12: 90.7% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4700:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 - x \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -4700.0)
   (+ x (/ -1.0 x))
   (+ x (/ y (- 1.1283791670955126 (* x y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -4700.0) {
		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 (z <= (-4700.0d0)) 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 (z <= -4700.0) {
		tmp = x + (-1.0 / x);
	} else {
		tmp = x + (y / (1.1283791670955126 - (x * y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -4700.0:
		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 (z <= -4700.0)
		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 (z <= -4700.0)
		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[z, -4700.0], 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 z < -4700

   1. Initial program 92.1%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -4700 < z

   1. Initial program 98.9%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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 90.9%

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

Alternative 13: 60.0% accurate, 14.2× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return x + (0.8862269254527579 * 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 + (0.8862269254527579d0 * y)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.3%

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

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

   2. unpow2 N/A

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

   3. associate-/r* N/A

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

   4. div-add-rev N/A

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

   5. lower-/.f64 N/A

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

   6. lower-+.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. *-commutative N/A

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

   9. associate-*r* N/A

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

   10. metadata-eval N/A

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

   11. lower-*.f64 N/A

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

   12. metadata-eval N/A

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

   13. lower-*.f64 N/A

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

   14. lower--.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

   17. lower--.f64 N/A

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

5. Applied rewrites 71.0%

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

6. Taylor expanded in x around 0

   \[\leadsto x + \frac{5000000000000000}{5641895835477563} \cdot \color{blue}{\left(y + -1 \cdot \left(y \cdot z\right)\right)} \]
7. Step-by-step derivation

8. Applied rewrites 53.5%

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

9. Taylor expanded in z around 0

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

11. Applied rewrites 61.1%

    \[\leadsto x + 0.8862269254527579 \cdot y \]
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 2024339 
(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)))))