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

Percentage Accurate: 95.4% → 99.7%

Time: 12.5s

Alternatives: 12

Speedup: 3.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 95.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -17:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 9:\\ \;\;\;\;x + \frac{y}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.18806319451591877, 0.5641895835477563\right), 1.1283791670955126\right), 1.1283791670955126 - x \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -17.0)
   (+ x (/ -1.0 x))
   (if (<= z 9.0)
     (+
      x
      (/
       y
       (fma
        z
        (fma
         z
         (fma z 0.18806319451591877 0.5641895835477563)
         1.1283791670955126)
        (- 1.1283791670955126 (* x y)))))
     x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -17.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 9.0) {
		tmp = x + (y / fma(z, fma(z, fma(z, 0.18806319451591877, 0.5641895835477563), 1.1283791670955126), (1.1283791670955126 - (x * y))));
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -17.0)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (z <= 9.0)
		tmp = Float64(x + Float64(y / fma(z, fma(z, fma(z, 0.18806319451591877, 0.5641895835477563), 1.1283791670955126), Float64(1.1283791670955126 - Float64(x * y)))));
	else
		tmp = x;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -17.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 9.0], N[(x + N[(y / N[(z * N[(z * N[(z * 0.18806319451591877 + 0.5641895835477563), $MachinePrecision] + 1.1283791670955126), $MachinePrecision] + N[(1.1283791670955126 - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if z < -17

   1. Initial program 94.2%

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

   3. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{x}\right)\right)}\right) \]

      3. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(\frac{\mathsf{neg}\left(1\right)}{\color{blue}{x}}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(\frac{-1}{x}\right)\right) \]

      5. /-lowering-/.f64 100.0%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(-1, \color{blue}{x}\right)\right) \]

   5. Simplified 100.0%

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

   if -17 < z < 9

   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

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \color{blue}{\left(\left(\frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{10000000000000000} + \frac{5641895835477563}{30000000000000000} \cdot z\right)\right)\right) - x \cdot y\right)}\right)\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \left(\left(z \cdot \left(\frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{10000000000000000} + \frac{5641895835477563}{30000000000000000} \cdot z\right)\right) + \frac{5641895835477563}{5000000000000000}\right) - \color{blue}{x} \cdot y\right)\right)\right) \]

      2. associate--l+ N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \left(z \cdot \left(\frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{10000000000000000} + \frac{5641895835477563}{30000000000000000} \cdot z\right)\right) + \color{blue}{\left(\frac{5641895835477563}{5000000000000000} - x \cdot y\right)}\right)\right)\right) \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{fma.f64}\left(z, \color{blue}{\left(\frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{10000000000000000} + \frac{5641895835477563}{30000000000000000} \cdot z\right)\right)}, \left(\frac{5641895835477563}{5000000000000000} - x \cdot y\right)\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{fma.f64}\left(z, \left(z \cdot \left(\frac{5641895835477563}{10000000000000000} + \frac{5641895835477563}{30000000000000000} \cdot z\right) + \color{blue}{\frac{5641895835477563}{5000000000000000}}\right), \left(\frac{5641895835477563}{5000000000000000} - x \cdot y\right)\right)\right)\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{fma.f64}\left(z, \mathsf{fma.f64}\left(z, \color{blue}{\left(\frac{5641895835477563}{10000000000000000} + \frac{5641895835477563}{30000000000000000} \cdot z\right)}, \frac{5641895835477563}{5000000000000000}\right), \left(\frac{5641895835477563}{5000000000000000} - x \cdot y\right)\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{fma.f64}\left(z, \mathsf{fma.f64}\left(z, \left(\frac{5641895835477563}{30000000000000000} \cdot z + \color{blue}{\frac{5641895835477563}{10000000000000000}}\right), \frac{5641895835477563}{5000000000000000}\right), \left(\frac{5641895835477563}{5000000000000000} - x \cdot y\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{fma.f64}\left(z, \mathsf{fma.f64}\left(z, \left(z \cdot \frac{5641895835477563}{30000000000000000} + \frac{5641895835477563}{10000000000000000}\right), \frac{5641895835477563}{5000000000000000}\right), \left(\frac{5641895835477563}{5000000000000000} - x \cdot y\right)\right)\right)\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{fma.f64}\left(z, \mathsf{fma.f64}\left(z, \mathsf{fma.f64}\left(z, \color{blue}{\frac{5641895835477563}{30000000000000000}}, \frac{5641895835477563}{10000000000000000}\right), \frac{5641895835477563}{5000000000000000}\right), \left(\frac{5641895835477563}{5000000000000000} - x \cdot y\right)\right)\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{fma.f64}\left(z, \mathsf{fma.f64}\left(z, \mathsf{fma.f64}\left(z, \frac{5641895835477563}{30000000000000000}, \frac{5641895835477563}{10000000000000000}\right), \color{blue}{\frac{5641895835477563}{5000000000000000}}\right), \mathsf{\_.f64}\left(\frac{5641895835477563}{5000000000000000}, \left(x \cdot y\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{fma.f64}\left(z, \mathsf{fma.f64}\left(z, \mathsf{fma.f64}\left(z, \frac{5641895835477563}{30000000000000000}, \frac{5641895835477563}{10000000000000000}\right), \frac{5641895835477563}{5000000000000000}\right), \mathsf{\_.f64}\left(\frac{5641895835477563}{5000000000000000}, \left(y \cdot x\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 99.6%

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{fma.f64}\left(z, \mathsf{fma.f64}\left(z, \mathsf{fma.f64}\left(z, \frac{5641895835477563}{30000000000000000}, \frac{5641895835477563}{10000000000000000}\right), \frac{5641895835477563}{5000000000000000}\right), \mathsf{\_.f64}\left(\frac{5641895835477563}{5000000000000000}, \mathsf{*.f64}\left(y, x\right)\right)\right)\right)\right) \]

   5. Simplified 99.6%

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

   if 9 < z

   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

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

   5. Simplified 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -17:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 9:\\ \;\;\;\;x + \frac{y}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.18806319451591877, 0.5641895835477563\right), 1.1283791670955126\right), 1.1283791670955126 - x \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{y}{e^{z} \cdot 1.1283791670955126 - x \cdot y}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{+262}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;x + \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (/ y (- (* (exp z) 1.1283791670955126) (* x y))))))
   (if (<= t_0 5e+262) t_0 (+ x (/ -1.0 x)))))

C

double code(double x, double y, double z) {
	double t_0 = x + (y / ((exp(z) * 1.1283791670955126) - (x * y)));
	double tmp;
	if (t_0 <= 5e+262) {
		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 / ((exp(z) * 1.1283791670955126d0) - (x * y)))
    if (t_0 <= 5d+262) 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 / ((Math.exp(z) * 1.1283791670955126) - (x * y)));
	double tmp;
	if (t_0 <= 5e+262) {
		tmp = t_0;
	} else {
		tmp = x + (-1.0 / x);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x + (y / ((math.exp(z) * 1.1283791670955126) - (x * y)))
	tmp = 0
	if t_0 <= 5e+262:
		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(exp(z) * 1.1283791670955126) - Float64(x * y))))
	tmp = 0.0
	if (t_0 <= 5e+262)
		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 / ((exp(z) * 1.1283791670955126) - (x * y)));
	tmp = 0.0;
	if (t_0 <= 5e+262)
		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[(N[Exp[z], $MachinePrecision] * 1.1283791670955126), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e+262], 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)))) < 5.00000000000000008e262

   1. Initial program 99.4%

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

   if 5.00000000000000008e262 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y))))

   1. Initial program 65.3%

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

   3. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{x}\right)\right)}\right) \]

      3. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(\frac{\mathsf{neg}\left(1\right)}{\color{blue}{x}}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(\frac{-1}{x}\right)\right) \]

      5. /-lowering-/.f64 100.0%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(-1, \color{blue}{x}\right)\right) \]

   5. Simplified 100.0%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{y}{e^{z} \cdot 1.1283791670955126 - x \cdot y} \leq 5 \cdot 10^{+262}:\\ \;\;\;\;x + \frac{y}{e^{z} \cdot 1.1283791670955126 - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{-1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{-1}{\mathsf{fma}\left(x, y, e^{z} \cdot -1.1283791670955126\right)}, y, x\right) \end{array} \]

FPCore

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

C

double code(double x, double y, double z) {
	return fma((-1.0 / fma(x, y, (exp(z) * -1.1283791670955126))), y, x);
}

Julia

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

Wolfram

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

Derivation

1. Initial program 97.1%

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

   1. +-commutative N/A

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

   2. *-lft-identity N/A

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

   3. associate-*l/ N/A

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

   4. flip-- N/A

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

   5. clear-num N/A

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

   6. accelerator-lowering-fma.f64 N/A

      \[\leadsto \mathsf{fma.f64}\left(\left(\frac{\frac{5641895835477563}{5000000000000000} \cdot e^{z} + x \cdot y}{\left(\frac{5641895835477563}{5000000000000000} \cdot e^{z}\right) \cdot \left(\frac{5641895835477563}{5000000000000000} \cdot e^{z}\right) - \left(x \cdot y\right) \cdot \left(x \cdot y\right)}\right), \color{blue}{y}, x\right) \]

4. Applied egg-rr 98.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{\mathsf{fma}\left(x, y, e^{z} \cdot -1.1283791670955126\right)}, y, x\right)} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -350:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 9:\\ \;\;\;\;x + \frac{y}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5641895835477563, 1.1283791670955126\right), 1.1283791670955126\right) - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -350.0)
   (+ x (/ -1.0 x))
   (if (<= z 9.0)
     (+
      x
      (/
       y
       (-
        (fma
         z
         (fma z 0.5641895835477563 1.1283791670955126)
         1.1283791670955126)
        (* x y))))
     x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -350.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 9.0) {
		tmp = x + (y / (fma(z, fma(z, 0.5641895835477563, 1.1283791670955126), 1.1283791670955126) - (x * y)));
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -350.0)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (z <= 9.0)
		tmp = Float64(x + Float64(y / Float64(fma(z, fma(z, 0.5641895835477563, 1.1283791670955126), 1.1283791670955126) - Float64(x * y))));
	else
		tmp = x;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -350.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 9.0], N[(x + N[(y / N[(N[(z * N[(z * 0.5641895835477563 + 1.1283791670955126), $MachinePrecision] + 1.1283791670955126), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if z < -350

   1. Initial program 94.2%

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

   3. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{x}\right)\right)}\right) \]

      3. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(\frac{\mathsf{neg}\left(1\right)}{\color{blue}{x}}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(\frac{-1}{x}\right)\right) \]

      5. /-lowering-/.f64 100.0%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(-1, \color{blue}{x}\right)\right) \]

   5. Simplified 100.0%

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

   if -350 < z < 9

   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

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(\color{blue}{\left(\frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z\right)\right)}, \mathsf{*.f64}\left(x, y\right)\right)\right)\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(\left(z \cdot \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z\right) + \frac{5641895835477563}{5000000000000000}\right), \mathsf{*.f64}\left(\color{blue}{x}, y\right)\right)\right)\right) \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(z, \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z\right), \frac{5641895835477563}{5000000000000000}\right), \mathsf{*.f64}\left(\color{blue}{x}, y\right)\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(z, \left(\frac{5641895835477563}{10000000000000000} \cdot z + \frac{5641895835477563}{5000000000000000}\right), \frac{5641895835477563}{5000000000000000}\right), \mathsf{*.f64}\left(x, y\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(z, \left(z \cdot \frac{5641895835477563}{10000000000000000} + \frac{5641895835477563}{5000000000000000}\right), \frac{5641895835477563}{5000000000000000}\right), \mathsf{*.f64}\left(x, y\right)\right)\right)\right) \]

      5. accelerator-lowering-fma.f64 99.5%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(z, \mathsf{fma.f64}\left(z, \frac{5641895835477563}{10000000000000000}, \frac{5641895835477563}{5000000000000000}\right), \frac{5641895835477563}{5000000000000000}\right), \mathsf{*.f64}\left(x, y\right)\right)\right)\right) \]

   5. Simplified 99.5%

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

   if 9 < z

   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

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

   5. Simplified 100.0%

      \[\leadsto \color{blue}{x} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 99.6% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -225:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 9:\\ \;\;\;\;x + \frac{y}{\mathsf{fma}\left(y, 0 - x, \mathsf{fma}\left(z, 1.1283791670955126, 1.1283791670955126\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -225.0)
   (+ x (/ -1.0 x))
   (if (<= z 9.0)
     (+
      x
      (/ y (fma y (- 0.0 x) (fma z 1.1283791670955126 1.1283791670955126))))
     x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -225.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 9.0) {
		tmp = x + (y / fma(y, (0.0 - x), fma(z, 1.1283791670955126, 1.1283791670955126)));
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -225.0)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (z <= 9.0)
		tmp = Float64(x + Float64(y / fma(y, Float64(0.0 - x), fma(z, 1.1283791670955126, 1.1283791670955126))));
	else
		tmp = x;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -225.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 9.0], N[(x + N[(y / N[(y * N[(0.0 - x), $MachinePrecision] + N[(z * 1.1283791670955126 + 1.1283791670955126), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if z < -225

   1. Initial program 94.2%

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

   3. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{x}\right)\right)}\right) \]

      3. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(\frac{\mathsf{neg}\left(1\right)}{\color{blue}{x}}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(\frac{-1}{x}\right)\right) \]

      5. /-lowering-/.f64 100.0%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(-1, \color{blue}{x}\right)\right) \]

   5. Simplified 100.0%

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

   if -225 < z < 9

   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

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \color{blue}{\left(\left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{5000000000000000} \cdot z\right) - x \cdot y\right)}\right)\right) \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \left(\left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{5000000000000000} \cdot z\right) + \color{blue}{\left(\mathsf{neg}\left(x \cdot y\right)\right)}\right)\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \left(\left(\mathsf{neg}\left(x \cdot y\right)\right) + \color{blue}{\left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{5000000000000000} \cdot z\right)}\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \left(\left(\mathsf{neg}\left(y \cdot x\right)\right) + \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{5000000000000000} \cdot z\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \left(y \cdot \left(\mathsf{neg}\left(x\right)\right) + \left(\color{blue}{\frac{5641895835477563}{5000000000000000}} + \frac{5641895835477563}{5000000000000000} \cdot z\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \left(y \cdot \left(-1 \cdot x\right) + \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{5000000000000000} \cdot z\right)\right)\right)\right) \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{fma.f64}\left(y, \color{blue}{\left(-1 \cdot x\right)}, \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{5000000000000000} \cdot z\right)\right)\right)\right) \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{fma.f64}\left(y, \left(\mathsf{neg}\left(x\right)\right), \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{5000000000000000} \cdot z\right)\right)\right)\right) \]

      8. neg-sub0 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{fma.f64}\left(y, \left(0 - \color{blue}{x}\right), \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{5000000000000000} \cdot z\right)\right)\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{fma.f64}\left(y, \mathsf{\_.f64}\left(0, \color{blue}{x}\right), \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{5000000000000000} \cdot z\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{fma.f64}\left(y, \mathsf{\_.f64}\left(0, x\right), \left(\frac{5641895835477563}{5000000000000000} \cdot z + \frac{5641895835477563}{5000000000000000}\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{fma.f64}\left(y, \mathsf{\_.f64}\left(0, x\right), \left(z \cdot \frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{5000000000000000}\right)\right)\right)\right) \]

      12. accelerator-lowering-fma.f64 99.4%

          \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{fma.f64}\left(y, \mathsf{\_.f64}\left(0, x\right), \mathsf{fma.f64}\left(z, \frac{5641895835477563}{5000000000000000}, \frac{5641895835477563}{5000000000000000}\right)\right)\right)\right) \]

   5. Simplified 99.4%

      \[\leadsto x + \frac{y}{\color{blue}{\mathsf{fma}\left(y, 0 - x, \mathsf{fma}\left(z, 1.1283791670955126, 1.1283791670955126\right)\right)}} \]

   if 9 < z

   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

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

   5. Simplified 100.0%

      \[\leadsto \color{blue}{x} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 99.6% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -150:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 9:\\ \;\;\;\;x + \frac{y}{\mathsf{fma}\left(z, 1.1283791670955126, 1.1283791670955126\right) - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -150.0)
   (+ x (/ -1.0 x))
   (if (<= z 9.0)
     (+ x (/ y (- (fma z 1.1283791670955126 1.1283791670955126) (* x y))))
     x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -150.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 9.0) {
		tmp = x + (y / (fma(z, 1.1283791670955126, 1.1283791670955126) - (x * y)));
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -150.0)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (z <= 9.0)
		tmp = Float64(x + Float64(y / Float64(fma(z, 1.1283791670955126, 1.1283791670955126) - Float64(x * y))));
	else
		tmp = x;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -150.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 9.0], N[(x + N[(y / N[(N[(z * 1.1283791670955126 + 1.1283791670955126), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if z < -150

   1. Initial program 94.2%

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

   3. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{x}\right)\right)}\right) \]

      3. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(\frac{\mathsf{neg}\left(1\right)}{\color{blue}{x}}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(\frac{-1}{x}\right)\right) \]

      5. /-lowering-/.f64 100.0%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(-1, \color{blue}{x}\right)\right) \]

   5. Simplified 100.0%

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

   if -150 < z < 9

   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

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(\color{blue}{\left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{5000000000000000} \cdot z\right)}, \mathsf{*.f64}\left(x, y\right)\right)\right)\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(\left(\frac{5641895835477563}{5000000000000000} \cdot z + \frac{5641895835477563}{5000000000000000}\right), \mathsf{*.f64}\left(\color{blue}{x}, y\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(\left(z \cdot \frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{5000000000000000}\right), \mathsf{*.f64}\left(x, y\right)\right)\right)\right) \]

      3. accelerator-lowering-fma.f64 99.4%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{fma.f64}\left(z, \frac{5641895835477563}{5000000000000000}, \frac{5641895835477563}{5000000000000000}\right), \mathsf{*.f64}\left(\color{blue}{x}, y\right)\right)\right)\right) \]

   5. Simplified 99.4%

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

   if 9 < z

   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

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

   5. Simplified 100.0%

      \[\leadsto \color{blue}{x} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 99.5% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -95:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 9:\\ \;\;\;\;x - \frac{y}{\mathsf{fma}\left(y, x, -1.1283791670955126\right)}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -95.0)
   (+ x (/ -1.0 x))
   (if (<= z 9.0) (- x (/ y (fma y x -1.1283791670955126))) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -95.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 9.0) {
		tmp = x - (y / fma(y, x, -1.1283791670955126));
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -95.0)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (z <= 9.0)
		tmp = Float64(x - Float64(y / fma(y, x, -1.1283791670955126)));
	else
		tmp = x;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -95.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 9.0], N[(x - N[(y / N[(y * x + -1.1283791670955126), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if z < -95

   1. Initial program 94.2%

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

   3. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{x}\right)\right)}\right) \]

      3. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(\frac{\mathsf{neg}\left(1\right)}{\color{blue}{x}}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(\frac{-1}{x}\right)\right) \]

      5. /-lowering-/.f64 100.0%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(-1, \color{blue}{x}\right)\right) \]

   5. Simplified 100.0%

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

   if -95 < z < 9

   1. Initial program 99.8%

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

      1. +-commutative N/A

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

      2. *-lft-identity N/A

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

      3. associate-*l/ N/A

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

      4. flip-- N/A

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

      5. clear-num N/A

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

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(\frac{\frac{5641895835477563}{5000000000000000} \cdot e^{z} + x \cdot y}{\left(\frac{5641895835477563}{5000000000000000} \cdot e^{z}\right) \cdot \left(\frac{5641895835477563}{5000000000000000} \cdot e^{z}\right) - \left(x \cdot y\right) \cdot \left(x \cdot y\right)}\right), \color{blue}{y}, x\right) \]

   4. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{\mathsf{fma}\left(x, y, e^{z} \cdot -1.1283791670955126\right)}, y, x\right)} \]

   5. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + -1 \cdot \frac{y}{x \cdot y - \frac{5641895835477563}{5000000000000000}}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \color{blue}{\left(\frac{y}{x \cdot y - \frac{5641895835477563}{5000000000000000}}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(y, \color{blue}{\left(x \cdot y - \frac{5641895835477563}{5000000000000000}\right)}\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(y, \left(x \cdot y + \color{blue}{\left(\mathsf{neg}\left(\frac{5641895835477563}{5000000000000000}\right)\right)}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(y, \left(y \cdot x + \left(\mathsf{neg}\left(\color{blue}{\frac{5641895835477563}{5000000000000000}}\right)\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(y, \left(y \cdot x + \frac{-5641895835477563}{5000000000000000}\right)\right)\right) \]

      8. accelerator-lowering-fma.f64 99.0%

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{fma.f64}\left(y, \color{blue}{x}, \frac{-5641895835477563}{5000000000000000}\right)\right)\right) \]

   7. Simplified 99.0%

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

   if 9 < z

   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

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

   5. Simplified 100.0%

      \[\leadsto \color{blue}{x} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 71.4% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.3 \cdot 10^{-21}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{-37}:\\ \;\;\;\;\mathsf{fma}\left(0.8862269254527579, y, x\right)\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-8}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -4.3e-21)
   x
   (if (<= x 2.9e-37)
     (fma 0.8862269254527579 y x)
     (if (<= x 4.2e-8) (/ -1.0 x) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -4.3e-21) {
		tmp = x;
	} else if (x <= 2.9e-37) {
		tmp = fma(0.8862269254527579, y, x);
	} else if (x <= 4.2e-8) {
		tmp = -1.0 / x;
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -4.3e-21)
		tmp = x;
	elseif (x <= 2.9e-37)
		tmp = fma(0.8862269254527579, y, x);
	elseif (x <= 4.2e-8)
		tmp = Float64(-1.0 / x);
	else
		tmp = x;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -4.3e-21], x, If[LessEqual[x, 2.9e-37], N[(0.8862269254527579 * y + x), $MachinePrecision], If[LessEqual[x, 4.2e-8], N[(-1.0 / x), $MachinePrecision], x]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.2999999999999998e-21 or 4.19999999999999989e-8 < x

   1. Initial program 97.1%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 98.0%

      \[\leadsto \color{blue}{x} \]

   if -4.2999999999999998e-21 < x < 2.90000000000000005e-37

   1. Initial program 97.0%

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

   3. Taylor expanded in z around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(\frac{y}{\frac{5641895835477563}{5000000000000000} - x \cdot y}\right)}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \color{blue}{\left(\frac{5641895835477563}{5000000000000000} - x \cdot y\right)}\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(\frac{5641895835477563}{5000000000000000}, \color{blue}{\left(x \cdot y\right)}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(\frac{5641895835477563}{5000000000000000}, \left(y \cdot \color{blue}{x}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 73.4%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(\frac{5641895835477563}{5000000000000000}, \mathsf{*.f64}\left(y, \color{blue}{x}\right)\right)\right)\right) \]

   5. Simplified 73.4%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 54.8%

         \[\leadsto \mathsf{fma.f64}\left(\frac{5000000000000000}{5641895835477563}, \color{blue}{y}, x\right) \]

   8. Simplified 54.8%

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

   if 2.90000000000000005e-37 < x < 4.19999999999999989e-8

   1. Initial program 99.7%

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

   3. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{x}\right)\right)}\right) \]

      3. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(\frac{\mathsf{neg}\left(1\right)}{\color{blue}{x}}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(\frac{-1}{x}\right)\right) \]

      5. /-lowering-/.f64 100.0%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(-1, \color{blue}{x}\right)\right) \]

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(-1, \color{blue}{x}\right) \]

   8. Simplified 100.0%

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

Alternative 9: 86.4% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.004:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(0.8862269254527579, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -0.004)
   (+ x (/ -1.0 x))
   (if (<= z 1.7e-13) (fma 0.8862269254527579 y x) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -0.004) {
		tmp = x + (-1.0 / x);
	} else if (z <= 1.7e-13) {
		tmp = fma(0.8862269254527579, y, x);
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -0.004)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (z <= 1.7e-13)
		tmp = fma(0.8862269254527579, y, x);
	else
		tmp = x;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -0.004], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.7e-13], N[(0.8862269254527579 * y + x), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if z < -0.0040000000000000001

   1. Initial program 94.2%

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

   3. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{x}\right)\right)}\right) \]

      3. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(\frac{\mathsf{neg}\left(1\right)}{\color{blue}{x}}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(\frac{-1}{x}\right)\right) \]

      5. /-lowering-/.f64 100.0%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(-1, \color{blue}{x}\right)\right) \]

   5. Simplified 100.0%

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

   if -0.0040000000000000001 < z < 1.70000000000000008e-13

   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

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(\frac{y}{\frac{5641895835477563}{5000000000000000} - x \cdot y}\right)}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \color{blue}{\left(\frac{5641895835477563}{5000000000000000} - x \cdot y\right)}\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(\frac{5641895835477563}{5000000000000000}, \color{blue}{\left(x \cdot y\right)}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(\frac{5641895835477563}{5000000000000000}, \left(y \cdot \color{blue}{x}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 98.9%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(\frac{5641895835477563}{5000000000000000}, \mathsf{*.f64}\left(y, \color{blue}{x}\right)\right)\right)\right) \]

   5. Simplified 98.9%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 75.9%

         \[\leadsto \mathsf{fma.f64}\left(\frac{5000000000000000}{5641895835477563}, \color{blue}{y}, x\right) \]

   8. Simplified 75.9%

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

   if 1.70000000000000008e-13 < z

   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

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

   5. Simplified 100.0%

      \[\leadsto \color{blue}{x} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 71.5% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{-21}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 8.4 \cdot 10^{-16}:\\ \;\;\;\;\mathsf{fma}\left(0.8862269254527579, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.75e-21) x (if (<= x 8.4e-16) (fma 0.8862269254527579 y x) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.75e-21) {
		tmp = x;
	} else if (x <= 8.4e-16) {
		tmp = fma(0.8862269254527579, y, x);
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.75e-21)
		tmp = x;
	elseif (x <= 8.4e-16)
		tmp = fma(0.8862269254527579, y, x);
	else
		tmp = x;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.75e-21], x, If[LessEqual[x, 8.4e-16], N[(0.8862269254527579 * y + x), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -1.7500000000000002e-21 or 8.4000000000000004e-16 < x

   1. Initial program 97.1%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 97.3%

      \[\leadsto \color{blue}{x} \]

   if -1.7500000000000002e-21 < x < 8.4000000000000004e-16

   1. Initial program 97.2%

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

   3. Taylor expanded in z around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(\frac{y}{\frac{5641895835477563}{5000000000000000} - x \cdot y}\right)}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \color{blue}{\left(\frac{5641895835477563}{5000000000000000} - x \cdot y\right)}\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(\frac{5641895835477563}{5000000000000000}, \color{blue}{\left(x \cdot y\right)}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(\frac{5641895835477563}{5000000000000000}, \left(y \cdot \color{blue}{x}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 72.8%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(\frac{5641895835477563}{5000000000000000}, \mathsf{*.f64}\left(y, \color{blue}{x}\right)\right)\right)\right) \]

   5. Simplified 72.8%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 52.6%

         \[\leadsto \mathsf{fma.f64}\left(\frac{5000000000000000}{5641895835477563}, \color{blue}{y}, x\right) \]

   8. Simplified 52.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.8862269254527579, y, x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 70.3% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{-193}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-248}:\\ \;\;\;\;y \cdot 0.8862269254527579\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -8.5e-193) x (if (<= x 4.5e-248) (* y 0.8862269254527579) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -8.5e-193) {
		tmp = x;
	} else if (x <= 4.5e-248) {
		tmp = y * 0.8862269254527579;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-8.5d-193)) then
        tmp = x
    else if (x <= 4.5d-248) then
        tmp = y * 0.8862269254527579d0
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -8.5e-193) {
		tmp = x;
	} else if (x <= 4.5e-248) {
		tmp = y * 0.8862269254527579;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -8.5e-193:
		tmp = x
	elif x <= 4.5e-248:
		tmp = y * 0.8862269254527579
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -8.5e-193)
		tmp = x;
	elseif (x <= 4.5e-248)
		tmp = Float64(y * 0.8862269254527579);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -8.5e-193)
		tmp = x;
	elseif (x <= 4.5e-248)
		tmp = y * 0.8862269254527579;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -8.5e-193], x, If[LessEqual[x, 4.5e-248], N[(y * 0.8862269254527579), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -8.50000000000000004e-193 or 4.4999999999999996e-248 < x

   1. Initial program 97.6%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 77.9%

      \[\leadsto \color{blue}{x} \]

   if -8.50000000000000004e-193 < x < 4.4999999999999996e-248

   1. Initial program 95.2%

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

   3. Taylor expanded in z around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(\frac{y}{\frac{5641895835477563}{5000000000000000} - x \cdot y}\right)}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \color{blue}{\left(\frac{5641895835477563}{5000000000000000} - x \cdot y\right)}\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(\frac{5641895835477563}{5000000000000000}, \color{blue}{\left(x \cdot y\right)}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(\frac{5641895835477563}{5000000000000000}, \left(y \cdot \color{blue}{x}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 66.3%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(\frac{5641895835477563}{5000000000000000}, \mathsf{*.f64}\left(y, \color{blue}{x}\right)\right)\right)\right) \]

   5. Simplified 66.3%

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

   6. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 56.6%

         \[\leadsto \mathsf{*.f64}\left(\frac{5000000000000000}{5641895835477563}, \color{blue}{y}\right) \]

   8. Simplified 56.6%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{-193}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-248}:\\ \;\;\;\;y \cdot 0.8862269254527579\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 69.3% accurate, 128.0× speedup

Math

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

FPCore

(FPCore (x y z) :precision binary64 x)

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x
end function

Java

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

Python

def code(x, y, z):
	return x

Julia

function code(x, y, z)
	return x
end

MATLAB

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

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 97.1%

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

3. Taylor expanded in x around inf

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

5. Simplified 67.8%

   \[\leadsto \color{blue}{x} \]
6. 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 2024193 
(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)))))