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

Percentage Accurate: 95.5% → 98.7%

Time: 7.8s

Alternatives: 8

Speedup: 3.7×

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: 95.5% 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: 98.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1.1283791670955126 - y \cdot x\\ \mathbf{if}\;z \leq -2.15 \cdot 10^{+29}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 9 \cdot 10^{-5}:\\ \;\;\;\;x + \frac{\frac{\left(-1.1283791670955126 \cdot z\right) \cdot y}{t\_0} + y}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- 1.1283791670955126 (* y x))))
   (if (<= z -2.15e+29)
     (+ x (/ -1.0 x))
     (if (<= z 9e-5)
       (+ x (/ (+ (/ (* (* -1.1283791670955126 z) y) t_0) y) t_0))
       (* 1.0 x)))))

C

double code(double x, double y, double z) {
	double t_0 = 1.1283791670955126 - (y * x);
	double tmp;
	if (z <= -2.15e+29) {
		tmp = x + (-1.0 / x);
	} else if (z <= 9e-5) {
		tmp = x + (((((-1.1283791670955126 * z) * y) / t_0) + y) / t_0);
	} else {
		tmp = 1.0 * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.1283791670955126d0 - (y * x)
    if (z <= (-2.15d+29)) then
        tmp = x + ((-1.0d0) / x)
    else if (z <= 9d-5) then
        tmp = x + ((((((-1.1283791670955126d0) * z) * y) / t_0) + y) / t_0)
    else
        tmp = 1.0d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = 1.1283791670955126 - (y * x);
	double tmp;
	if (z <= -2.15e+29) {
		tmp = x + (-1.0 / x);
	} else if (z <= 9e-5) {
		tmp = x + (((((-1.1283791670955126 * z) * y) / t_0) + y) / t_0);
	} else {
		tmp = 1.0 * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 1.1283791670955126 - (y * x)
	tmp = 0
	if z <= -2.15e+29:
		tmp = x + (-1.0 / x)
	elif z <= 9e-5:
		tmp = x + (((((-1.1283791670955126 * z) * y) / t_0) + y) / t_0)
	else:
		tmp = 1.0 * x
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(1.1283791670955126 - Float64(y * x))
	tmp = 0.0
	if (z <= -2.15e+29)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (z <= 9e-5)
		tmp = Float64(x + Float64(Float64(Float64(Float64(Float64(-1.1283791670955126 * z) * y) / t_0) + y) / t_0));
	else
		tmp = Float64(1.0 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 1.1283791670955126 - (y * x);
	tmp = 0.0;
	if (z <= -2.15e+29)
		tmp = x + (-1.0 / x);
	elseif (z <= 9e-5)
		tmp = x + (((((-1.1283791670955126 * z) * y) / t_0) + y) / t_0);
	else
		tmp = 1.0 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(1.1283791670955126 - N[(y * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2.15e+29], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 9e-5], N[(x + N[(N[(N[(N[(N[(-1.1283791670955126 * z), $MachinePrecision] * y), $MachinePrecision] / t$95$0), $MachinePrecision] + y), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], N[(1.0 * x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.1500000000000001e29

   1. Initial program 82.2%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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 -2.1500000000000001e29 < z < 9.00000000000000057e-5

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0

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

      1. 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 99.1%

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

   if 9.00000000000000057e-5 < z

   1. Initial program 88.1%

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

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 51.4

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

   5. Applied rewrites 51.4%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 100.0%

      \[\leadsto 1 \cdot x \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 86.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y}\\ \mathbf{if}\;t\_0 \leq -100000 \lor \neg \left(t\_0 \leq 5 \cdot 10^{-7}\right):\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \end{array} \]

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-7)))
     (+ x (/ -1.0 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 <= -100000.0) || !(t_0 <= 5e-7)) {
		tmp = x + (-1.0 / x);
	} else {
		tmp = 1.0 * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: 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-7))) then
        tmp = x + ((-1.0d0) / x)
    else
        tmp = 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 <= -100000.0) || !(t_0 <= 5e-7)) {
		tmp = x + (-1.0 / x);
	} else {
		tmp = 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 <= -100000.0) or not (t_0 <= 5e-7):
		tmp = x + (-1.0 / x)
	else:
		tmp = 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 <= -100000.0) || !(t_0 <= 5e-7))
		tmp = Float64(x + Float64(-1.0 / x));
	else
		tmp = 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 <= -100000.0) || ~((t_0 <= 5e-7)))
		tmp = x + (-1.0 / x);
	else
		tmp = 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[Or[LessEqual[t$95$0, -100000.0], N[Not[LessEqual[t$95$0, 5e-7]], $MachinePrecision]], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], N[(1.0 * x), $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 4.99999999999999977e-7 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y))))

   1. Initial program 91.2%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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 91.1

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

   5. Applied rewrites 91.1%

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

   if -1e5 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y)))) < 4.99999999999999977e-7

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 1.5

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

   5. Applied rewrites 1.5%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 81.5%

      \[\leadsto 1 \cdot x \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \leq -100000 \lor \neg \left(x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \leq 5 \cdot 10^{-7}\right):\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{+253}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;x + \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (/ y (- (* 1.1283791670955126 (exp z)) (* x y))))))
   (if (<= t_0 5e+253) 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 <= 5e+253) {
		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 <= 5d+253) 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 <= 5e+253) {
		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 <= 5e+253:
		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 <= 5e+253)
		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 <= 5e+253)
		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, 5e+253], 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)))) < 4.9999999999999997e253

   1. Initial program 99.4%

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

   if 4.9999999999999997e253 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y))))

   1. Initial program 21.1%

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

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

Alternative 4: 98.6% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.15 \cdot 10^{+29}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 9 \cdot 10^{-5}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2.15e+29)
   (+ x (/ -1.0 x))
   (if (<= z 9e-5) (+ x (/ y (- 1.1283791670955126 (* x y)))) (* 1.0 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.15e+29) {
		tmp = x + (-1.0 / x);
	} else if (z <= 9e-5) {
		tmp = x + (y / (1.1283791670955126 - (x * y)));
	} else {
		tmp = 1.0 * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-2.15d+29)) then
        tmp = x + ((-1.0d0) / x)
    else if (z <= 9d-5) then
        tmp = x + (y / (1.1283791670955126d0 - (x * y)))
    else
        tmp = 1.0d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.15e+29) {
		tmp = x + (-1.0 / x);
	} else if (z <= 9e-5) {
		tmp = x + (y / (1.1283791670955126 - (x * y)));
	} else {
		tmp = 1.0 * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -2.15e+29:
		tmp = x + (-1.0 / x)
	elif z <= 9e-5:
		tmp = x + (y / (1.1283791670955126 - (x * y)))
	else:
		tmp = 1.0 * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2.15e+29)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (z <= 9e-5)
		tmp = Float64(x + Float64(y / Float64(1.1283791670955126 - Float64(x * y))));
	else
		tmp = Float64(1.0 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -2.15e+29)
		tmp = x + (-1.0 / x);
	elseif (z <= 9e-5)
		tmp = x + (y / (1.1283791670955126 - (x * y)));
	else
		tmp = 1.0 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2.15e+29], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 9e-5], N[(x + N[(y / N[(1.1283791670955126 - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.1500000000000001e29

   1. Initial program 82.2%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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 -2.1500000000000001e29 < z < 9.00000000000000057e-5

   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 99.1%

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

   if 9.00000000000000057e-5 < z

   1. Initial program 88.1%

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

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 51.4

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

   5. Applied rewrites 51.4%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 100.0%

      \[\leadsto 1 \cdot x \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 73.7% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.9 \cdot 10^{+123}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{elif}\;z \leq -7 \cdot 10^{-123} \lor \neg \left(z \leq 3 \cdot 10^{-60}\right):\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;x + \mathsf{fma}\left(-0.8862269254527579, z, 0.8862269254527579\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -4.9e+123)
   (/ -1.0 x)
   (if (or (<= z -7e-123) (not (<= z 3e-60)))
     (* 1.0 x)
     (+ x (* (fma -0.8862269254527579 z 0.8862269254527579) y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -4.9e+123) {
		tmp = -1.0 / x;
	} else if ((z <= -7e-123) || !(z <= 3e-60)) {
		tmp = 1.0 * x;
	} else {
		tmp = x + (fma(-0.8862269254527579, z, 0.8862269254527579) * y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -4.9e+123)
		tmp = Float64(-1.0 / x);
	elseif ((z <= -7e-123) || !(z <= 3e-60))
		tmp = Float64(1.0 * x);
	else
		tmp = Float64(x + Float64(fma(-0.8862269254527579, z, 0.8862269254527579) * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -4.9e+123], N[(-1.0 / x), $MachinePrecision], If[Or[LessEqual[z, -7e-123], N[Not[LessEqual[z, 3e-60]], $MachinePrecision]], N[(1.0 * x), $MachinePrecision], N[(x + N[(N[(-0.8862269254527579 * z + 0.8862269254527579), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.89999999999999976e123

   1. Initial program 75.6%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 76.2

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

   5. Applied rewrites 76.2%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 40.6%

      \[\leadsto 1 \cdot x \]

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 59.2%

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

   if -4.89999999999999976e123 < z < -6.9999999999999997e-123 or 3.00000000000000019e-60 < z

   1. Initial program 92.2%

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

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 65.0

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

   5. Applied rewrites 65.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 85.1%

      \[\leadsto 1 \cdot x \]

   if -6.9999999999999997e-123 < z < 3.00000000000000019e-60

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0

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

      1. 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 99.8%

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

   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 73.2%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.9 \cdot 10^{+123}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{elif}\;z \leq -7 \cdot 10^{-123} \lor \neg \left(z \leq 3 \cdot 10^{-60}\right):\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;x + \mathsf{fma}\left(-0.8862269254527579, z, 0.8862269254527579\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.7% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.9 \cdot 10^{+123}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{elif}\;z \leq -7 \cdot 10^{-123} \lor \neg \left(z \leq 3 \cdot 10^{-60}\right):\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;x + 0.8862269254527579 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -4.9e+123)
   (/ -1.0 x)
   (if (or (<= z -7e-123) (not (<= z 3e-60)))
     (* 1.0 x)
     (+ x (* 0.8862269254527579 y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -4.9e+123) {
		tmp = -1.0 / x;
	} else if ((z <= -7e-123) || !(z <= 3e-60)) {
		tmp = 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) :: tmp
    if (z <= (-4.9d+123)) then
        tmp = (-1.0d0) / x
    else if ((z <= (-7d-123)) .or. (.not. (z <= 3d-60))) then
        tmp = 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 tmp;
	if (z <= -4.9e+123) {
		tmp = -1.0 / x;
	} else if ((z <= -7e-123) || !(z <= 3e-60)) {
		tmp = 1.0 * x;
	} else {
		tmp = x + (0.8862269254527579 * y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -4.9e+123:
		tmp = -1.0 / x
	elif (z <= -7e-123) or not (z <= 3e-60):
		tmp = 1.0 * x
	else:
		tmp = x + (0.8862269254527579 * y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -4.9e+123)
		tmp = Float64(-1.0 / x);
	elseif ((z <= -7e-123) || !(z <= 3e-60))
		tmp = Float64(1.0 * x);
	else
		tmp = Float64(x + Float64(0.8862269254527579 * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -4.9e+123)
		tmp = -1.0 / x;
	elseif ((z <= -7e-123) || ~((z <= 3e-60)))
		tmp = 1.0 * x;
	else
		tmp = x + (0.8862269254527579 * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -4.9e+123], N[(-1.0 / x), $MachinePrecision], If[Or[LessEqual[z, -7e-123], N[Not[LessEqual[z, 3e-60]], $MachinePrecision]], N[(1.0 * x), $MachinePrecision], N[(x + N[(0.8862269254527579 * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.89999999999999976e123

   1. Initial program 75.6%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 76.2

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

   5. Applied rewrites 76.2%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 40.6%

      \[\leadsto 1 \cdot x \]

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 59.2%

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

   if -4.89999999999999976e123 < z < -6.9999999999999997e-123 or 3.00000000000000019e-60 < z

   1. Initial program 92.2%

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

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 65.0

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

   5. Applied rewrites 65.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 85.1%

      \[\leadsto 1 \cdot x \]

   if -6.9999999999999997e-123 < z < 3.00000000000000019e-60

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0

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

      1. 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 99.8%

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

   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 73.2%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 73.2%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.9 \cdot 10^{+123}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{elif}\;z \leq -7 \cdot 10^{-123} \lor \neg \left(z \leq 3 \cdot 10^{-60}\right):\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;x + 0.8862269254527579 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.2% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{-123} \lor \neg \left(z \leq 3 \cdot 10^{-60}\right):\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;x + 0.8862269254527579 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -7e-123) (not (<= z 3e-60)))
   (* 1.0 x)
   (+ x (* 0.8862269254527579 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -7e-123) || !(z <= 3e-60)) {
		tmp = 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) :: tmp
    if ((z <= (-7d-123)) .or. (.not. (z <= 3d-60))) then
        tmp = 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 tmp;
	if ((z <= -7e-123) || !(z <= 3e-60)) {
		tmp = 1.0 * x;
	} else {
		tmp = x + (0.8862269254527579 * y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -7e-123) or not (z <= 3e-60):
		tmp = 1.0 * x
	else:
		tmp = x + (0.8862269254527579 * y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -7e-123) || !(z <= 3e-60))
		tmp = Float64(1.0 * x);
	else
		tmp = Float64(x + Float64(0.8862269254527579 * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -7e-123) || ~((z <= 3e-60)))
		tmp = 1.0 * x;
	else
		tmp = x + (0.8862269254527579 * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -7e-123], N[Not[LessEqual[z, 3e-60]], $MachinePrecision]], N[(1.0 * x), $MachinePrecision], N[(x + N[(0.8862269254527579 * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -6.9999999999999997e-123 or 3.00000000000000019e-60 < z

   1. Initial program 88.6%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 67.4

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

   5. Applied rewrites 67.4%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 75.5%

      \[\leadsto 1 \cdot x \]

   if -6.9999999999999997e-123 < z < 3.00000000000000019e-60

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0

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

      1. 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 99.8%

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

   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 73.2%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 73.2%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{-123} \lor \neg \left(z \leq 3 \cdot 10^{-60}\right):\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;x + 0.8862269254527579 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 68.9% accurate, 21.3× speedup

Math

\[\begin{array}{l} \\ 1 \cdot x \end{array} \]

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 93.3%

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

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

   2. lower-*.f64 N/A

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

   3. lower--.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 61.7

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

5. Applied rewrites 61.7%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 67.5%

   \[\leadsto 1 \cdot x \]
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 2024338 
(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)))))