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

Percentage Accurate: 95.9% → 99.3%

Time: 12.2s

Alternatives: 13

Speedup: 0.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 95.9% 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: 99.3% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (+ x (/ -1.0 x))
   (if (<= (exp z) 1.0)
     (- x (/ y (fma x y -1.1283791670955126)))
     (fma (exp (- z)) (* y 0.8862269254527579) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (exp(z) <= 0.0) {
		tmp = x + (-1.0 / x);
	} else if (exp(z) <= 1.0) {
		tmp = x - (y / fma(x, y, -1.1283791670955126));
	} else {
		tmp = fma(exp(-z), (y * 0.8862269254527579), x);
	}
	return tmp;
}

Julia

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

Wolfram

code[x_, y_, z_] := If[LessEqual[N[Exp[z], $MachinePrecision], 0.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Exp[z], $MachinePrecision], 1.0], N[(x - N[(y / N[(x * y + -1.1283791670955126), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Exp[(-z)], $MachinePrecision] * N[(y * 0.8862269254527579), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 81.8%

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

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

      2. lower-+.f64 N/A

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

      3. distribute-neg-frac N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 100.0

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

   5. Simplified 100.0%

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

   if 0.0 < (exp.f64 z) < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 99.9

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

   5. Simplified 99.9%

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

   6. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 99.8

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

   8. Simplified 99.8%

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

   9. Taylor expanded in z around 0

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

       1. mul-1-neg N/A

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

       2. unsub-neg N/A

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

       3. lower--.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. sub-neg N/A

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

       6. distribute-lft-in N/A

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

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

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

       8. *-commutative N/A

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

       9. associate-*l* N/A

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

       10. lft-mult-inverse N/A

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

       11. metadata-eval N/A

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

       12. metadata-eval N/A

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

       13. lower-fma.f64 99.9

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

   11. Simplified 99.9%

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

   if 1 < (exp.f64 z)

   1. Initial program 98.6%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-lft-identity N/A

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

      3. associate-*l/ N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

      8. rec-exp N/A

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

      9. neg-mul-1 N/A

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

      10. lower-exp.f64 N/A

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

      11. neg-mul-1 N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-*.f64 100.0

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

   5. Simplified 100.0%

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

4. Final simplification 99.9%

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

Alternative 2: 96.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;e^{z} \leq 1:\\ \;\;\;\;x - \frac{y}{\mathsf{fma}\left(x, y, -1.1283791670955126\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(5.317361552716548, \frac{y}{z \cdot \left(z \cdot z\right)}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (+ x (/ -1.0 x))
   (if (<= (exp z) 1.0)
     (- x (/ y (fma x y -1.1283791670955126)))
     (fma 5.317361552716548 (/ y (* z (* z z))) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (exp(z) <= 0.0) {
		tmp = x + (-1.0 / x);
	} else if (exp(z) <= 1.0) {
		tmp = x - (y / fma(x, y, -1.1283791670955126));
	} else {
		tmp = fma(5.317361552716548, (y / (z * (z * z))), x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (exp(z) <= 0.0)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (exp(z) <= 1.0)
		tmp = Float64(x - Float64(y / fma(x, y, -1.1283791670955126)));
	else
		tmp = fma(5.317361552716548, Float64(y / Float64(z * Float64(z * z))), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[Exp[z], $MachinePrecision], 0.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Exp[z], $MachinePrecision], 1.0], N[(x - N[(y / N[(x * y + -1.1283791670955126), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(5.317361552716548 * N[(y / N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 81.8%

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

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

      2. lower-+.f64 N/A

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

      3. distribute-neg-frac N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 100.0

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

   5. Simplified 100.0%

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

   if 0.0 < (exp.f64 z) < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 99.9

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

   5. Simplified 99.9%

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

   6. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 99.8

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

   8. Simplified 99.8%

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

   9. Taylor expanded in z around 0

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

       1. mul-1-neg N/A

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

       2. unsub-neg N/A

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

       3. lower--.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. sub-neg N/A

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

       6. distribute-lft-in N/A

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

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

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

       8. *-commutative N/A

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

       9. associate-*l* N/A

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

       10. lft-mult-inverse N/A

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

       11. metadata-eval N/A

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

       12. metadata-eval N/A

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

       13. lower-fma.f64 99.9

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

   11. Simplified 99.9%

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

   if 1 < (exp.f64 z)

   1. Initial program 98.6%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

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

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

      13. mul-1-neg N/A

          \[\leadsto x + \frac{y}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{5641895835477563}{30000000000000000}, \frac{5641895835477563}{10000000000000000}\right), \frac{5641895835477563}{5000000000000000}\right), y \cdot \color{blue}{\left(-1 \cdot x\right)} + \frac{5641895835477563}{5000000000000000}\right)} \]

      14. lower-fma.f64 N/A

          \[\leadsto x + \frac{y}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{5641895835477563}{30000000000000000}, \frac{5641895835477563}{10000000000000000}\right), \frac{5641895835477563}{5000000000000000}\right), \color{blue}{\mathsf{fma}\left(y, -1 \cdot x, \frac{5641895835477563}{5000000000000000}\right)}\right)} \]

      15. mul-1-neg N/A

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

      16. lower-neg.f64 94.0

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

   5. Simplified 94.0%

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

   6. Taylor expanded in z around inf

      \[\leadsto \color{blue}{x + \frac{30000000000000000}{5641895835477563} \cdot \frac{y}{{z}^{3}}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{30000000000000000}{5641895835477563} \cdot \frac{y}{{z}^{3}} + x} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{30000000000000000}{5641895835477563}, \frac{y}{{z}^{3}}, x\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{30000000000000000}{5641895835477563}, \color{blue}{\frac{y}{{z}^{3}}}, x\right) \]

      4. cube-mult N/A

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

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{30000000000000000}{5641895835477563}, \frac{y}{z \cdot \color{blue}{{z}^{2}}}, x\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{30000000000000000}{5641895835477563}, \frac{y}{\color{blue}{z \cdot {z}^{2}}}, x\right) \]

      7. unpow2 N/A

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

      8. lower-*.f64 94.0

         \[\leadsto \mathsf{fma}\left(5.317361552716548, \frac{y}{z \cdot \color{blue}{\left(z \cdot z\right)}}, x\right) \]

   8. Simplified 94.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(5.317361552716548, \frac{y}{z \cdot \left(z \cdot z\right)}, x\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 98.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (exp(z) <= 0.0d0) then
        tmp = x + ((-1.0d0) / x)
    else
        tmp = x + (y / ((exp(z) * 1.1283791670955126d0) - (x * y)))
    end if
    code = tmp
end function

Java

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

Python

def code(x, y, z):
	tmp = 0
	if math.exp(z) <= 0.0:
		tmp = x + (-1.0 / x)
	else:
		tmp = x + (y / ((math.exp(z) * 1.1283791670955126) - (x * y)))
	return tmp

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 81.8%

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

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

      2. lower-+.f64 N/A

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

      3. distribute-neg-frac N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 100.0

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

   5. Simplified 100.0%

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

   if 0.0 < (exp.f64 z)

   1. Initial program 99.4%

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

4. Final simplification 99.5%

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

Alternative 4: 97.1% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (+ x (/ -1.0 x))
   (+
    x
    (/
     y
     (fma
      z
      (fma z (fma z 0.18806319451591877 0.5641895835477563) 1.1283791670955126)
      (fma y (- x) 1.1283791670955126))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (exp(z) <= 0.0) {
		tmp = x + (-1.0 / x);
	} else {
		tmp = x + (y / fma(z, fma(z, fma(z, 0.18806319451591877, 0.5641895835477563), 1.1283791670955126), fma(y, -x, 1.1283791670955126)));
	}
	return tmp;
}

Julia

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

Wolfram

code[x_, y_, z_] := If[LessEqual[N[Exp[z], $MachinePrecision], 0.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], N[(x + N[(y / N[(z * N[(z * N[(z * 0.18806319451591877 + 0.5641895835477563), $MachinePrecision] + 1.1283791670955126), $MachinePrecision] + N[(y * (-x) + 1.1283791670955126), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 81.8%

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

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

      2. lower-+.f64 N/A

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

      3. distribute-neg-frac N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 100.0

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

   5. Simplified 100.0%

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

   if 0.0 < (exp.f64 z)

   1. Initial program 99.4%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

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

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

      13. mul-1-neg N/A

          \[\leadsto x + \frac{y}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{5641895835477563}{30000000000000000}, \frac{5641895835477563}{10000000000000000}\right), \frac{5641895835477563}{5000000000000000}\right), y \cdot \color{blue}{\left(-1 \cdot x\right)} + \frac{5641895835477563}{5000000000000000}\right)} \]

      14. lower-fma.f64 N/A

          \[\leadsto x + \frac{y}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{5641895835477563}{30000000000000000}, \frac{5641895835477563}{10000000000000000}\right), \frac{5641895835477563}{5000000000000000}\right), \color{blue}{\mathsf{fma}\left(y, -1 \cdot x, \frac{5641895835477563}{5000000000000000}\right)}\right)} \]

      15. mul-1-neg N/A

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

      16. lower-neg.f64 97.7

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

   5. Simplified 97.7%

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

Alternative 5: 96.0% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (+ x (/ -1.0 x))
   (+
    x
    (/
     y
     (-
      (fma z (fma z 0.5641895835477563 1.1283791670955126) 1.1283791670955126)
      (* x y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (exp(z) <= 0.0) {
		tmp = x + (-1.0 / x);
	} else {
		tmp = x + (y / (fma(z, fma(z, 0.5641895835477563, 1.1283791670955126), 1.1283791670955126) - (x * y)));
	}
	return tmp;
}

Julia

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

Wolfram

code[x_, y_, z_] := If[LessEqual[N[Exp[z], $MachinePrecision], 0.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], N[(x + N[(y / N[(N[(z * N[(z * 0.5641895835477563 + 1.1283791670955126), $MachinePrecision] + 1.1283791670955126), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 81.8%

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

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

      2. lower-+.f64 N/A

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

      3. distribute-neg-frac N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 100.0

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

   5. Simplified 100.0%

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

   if 0.0 < (exp.f64 z)

   1. Initial program 99.4%

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

   3. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 97.0

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

   5. Simplified 97.0%

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

4. Final simplification 97.7%

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

Alternative 6: 93.0% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -140000:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 4.7 \cdot 10^{+48}:\\ \;\;\;\;x - \frac{y}{\mathsf{fma}\left(x, y, -1.1283791670955126\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, 0.8862269254527579, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -140000.0)
   (+ x (/ -1.0 x))
   (if (<= z 4.7e+48)
     (- x (/ y (fma x y -1.1283791670955126)))
     (fma (/ y z) 0.8862269254527579 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -140000.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 4.7e+48) {
		tmp = x - (y / fma(x, y, -1.1283791670955126));
	} else {
		tmp = fma((y / z), 0.8862269254527579, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -140000.0)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (z <= 4.7e+48)
		tmp = Float64(x - Float64(y / fma(x, y, -1.1283791670955126)));
	else
		tmp = fma(Float64(y / z), 0.8862269254527579, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -140000.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 4.7e+48], N[(x - N[(y / N[(x * y + -1.1283791670955126), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y / z), $MachinePrecision] * 0.8862269254527579 + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.4e5

   1. Initial program 81.5%

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

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

      2. lower-+.f64 N/A

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

      3. distribute-neg-frac N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 100.0

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

   5. Simplified 100.0%

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

   if -1.4e5 < z < 4.70000000000000012e48

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 97.5

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

   5. Simplified 97.5%

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

   6. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 98.2

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

   8. Simplified 98.2%

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

   9. Taylor expanded in z around 0

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

       1. mul-1-neg N/A

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

       2. unsub-neg N/A

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

       3. lower--.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. sub-neg N/A

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

       6. distribute-lft-in N/A

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

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

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

       8. *-commutative N/A

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

       9. associate-*l* N/A

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

       10. lft-mult-inverse N/A

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

       11. metadata-eval N/A

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

       12. metadata-eval N/A

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

       13. lower-fma.f64 97.5

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

   11. Simplified 97.5%

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

   if 4.70000000000000012e48 < z

   1. Initial program 98.4%

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

   3. Taylor expanded in z around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 84.6

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

   5. Simplified 84.6%

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

   6. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 81.5

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

   8. Simplified 81.5%

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

Alternative 7: 79.3% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{-83}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-9}:\\ \;\;\;\;x - \frac{y}{-1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, 0.8862269254527579, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2.8e-83)
   (+ x (/ -1.0 x))
   (if (<= z 5e-9)
     (- x (/ y -1.1283791670955126))
     (fma (/ y z) 0.8862269254527579 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.8e-83) {
		tmp = x + (-1.0 / x);
	} else if (z <= 5e-9) {
		tmp = x - (y / -1.1283791670955126);
	} else {
		tmp = fma((y / z), 0.8862269254527579, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2.8e-83)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (z <= 5e-9)
		tmp = Float64(x - Float64(y / -1.1283791670955126));
	else
		tmp = fma(Float64(y / z), 0.8862269254527579, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2.8e-83], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 5e-9], N[(x - N[(y / -1.1283791670955126), $MachinePrecision]), $MachinePrecision], N[(N[(y / z), $MachinePrecision] * 0.8862269254527579 + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.8000000000000001e-83

   1. Initial program 85.2%

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

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

      2. lower-+.f64 N/A

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

      3. distribute-neg-frac N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 98.8

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

   5. Simplified 98.8%

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

   if -2.8000000000000001e-83 < z < 5.0000000000000001e-9

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 99.9

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

   5. Simplified 99.9%

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

   6. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 99.8

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

   8. Simplified 99.8%

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

   9. Taylor expanded in z around 0

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

       1. mul-1-neg N/A

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

       2. unsub-neg N/A

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

       3. lower--.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. sub-neg N/A

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

       6. distribute-lft-in N/A

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

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

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

       8. *-commutative N/A

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

       9. associate-*l* N/A

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

       10. lft-mult-inverse N/A

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

       11. metadata-eval N/A

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

       12. metadata-eval N/A

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

       13. lower-fma.f64 99.9

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

   11. Simplified 99.9%

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

   12. Taylor expanded in x around 0

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

   14. Simplified 84.0%

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

   if 5.0000000000000001e-9 < z

   1. Initial program 98.6%

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

   3. Taylor expanded in z around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 82.1

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

   5. Simplified 82.1%

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

   6. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 78.7

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

   8. Simplified 78.7%

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

Alternative 8: 79.5% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{-83}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\mathsf{fma}\left(1.1283791670955126, z, 1.1283791670955126\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2.8e-83)
   (+ x (/ -1.0 x))
   (+ x (/ y (fma 1.1283791670955126 z 1.1283791670955126)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.8e-83) {
		tmp = x + (-1.0 / x);
	} else {
		tmp = x + (y / fma(1.1283791670955126, z, 1.1283791670955126));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2.8e-83)
		tmp = Float64(x + Float64(-1.0 / x));
	else
		tmp = Float64(x + Float64(y / fma(1.1283791670955126, z, 1.1283791670955126)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2.8e-83], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], N[(x + N[(y / N[(1.1283791670955126 * z + 1.1283791670955126), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.8000000000000001e-83

   1. Initial program 85.2%

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

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

      2. lower-+.f64 N/A

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

      3. distribute-neg-frac N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 98.8

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

   5. Simplified 98.8%

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

   if -2.8000000000000001e-83 < z

   1. Initial program 99.4%

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

   3. Taylor expanded in z around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 92.9

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

   5. Simplified 92.9%

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

   6. Taylor expanded in y around 0

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

      1. lower-+.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 81.9

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

   8. Simplified 81.9%

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

Alternative 9: 74.1% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{-83}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{-1.1283791670955126}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2.8e-83) (+ x (/ -1.0 x)) (- x (/ y -1.1283791670955126))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.8e-83) {
		tmp = x + (-1.0 / x);
	} else {
		tmp = x - (y / -1.1283791670955126);
	}
	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 <= (-2.8d-83)) then
        tmp = x + ((-1.0d0) / x)
    else
        tmp = x - (y / (-1.1283791670955126d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.8e-83) {
		tmp = x + (-1.0 / x);
	} else {
		tmp = x - (y / -1.1283791670955126);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -2.8e-83:
		tmp = x + (-1.0 / x)
	else:
		tmp = x - (y / -1.1283791670955126)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2.8e-83)
		tmp = Float64(x + Float64(-1.0 / x));
	else
		tmp = Float64(x - Float64(y / -1.1283791670955126));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -2.8e-83)
		tmp = x + (-1.0 / x);
	else
		tmp = x - (y / -1.1283791670955126);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2.8e-83], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], N[(x - N[(y / -1.1283791670955126), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.8000000000000001e-83

   1. Initial program 85.2%

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

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

      2. lower-+.f64 N/A

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

      3. distribute-neg-frac N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 98.8

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

   5. Simplified 98.8%

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

   if -2.8000000000000001e-83 < z

   1. Initial program 99.4%

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

   3. Taylor expanded in z around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 92.9

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

   5. Simplified 92.9%

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

   6. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 96.7

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

   8. Simplified 96.7%

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

   9. Taylor expanded in z around 0

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

       1. mul-1-neg N/A

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

       2. unsub-neg N/A

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

       3. lower--.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. sub-neg N/A

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

       6. distribute-lft-in N/A

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

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

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

       8. *-commutative N/A

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

       9. associate-*l* N/A

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

       10. lft-mult-inverse N/A

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

       11. metadata-eval N/A

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

       12. metadata-eval N/A

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

       13. lower-fma.f64 87.5

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

   11. Simplified 87.5%

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

   12. Taylor expanded in x around 0

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

   14. Simplified 75.8%

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

Alternative 10: 74.1% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{-83}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.8862269254527579, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2.8e-83) (+ x (/ -1.0 x)) (fma 0.8862269254527579 y x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.8e-83) {
		tmp = x + (-1.0 / x);
	} else {
		tmp = fma(0.8862269254527579, y, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2.8e-83)
		tmp = Float64(x + Float64(-1.0 / x));
	else
		tmp = fma(0.8862269254527579, y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2.8e-83], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], N[(0.8862269254527579 * y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.8000000000000001e-83

   1. Initial program 85.2%

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

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

      2. lower-+.f64 N/A

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

      3. distribute-neg-frac N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 98.8

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

   5. Simplified 98.8%

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

   if -2.8000000000000001e-83 < z

   1. Initial program 99.4%

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

   3. Taylor expanded in z around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 92.9

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

   5. Simplified 92.9%

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

   6. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 96.7

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

   8. Simplified 96.7%

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

   9. Taylor expanded in z around 0

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

       1. mul-1-neg N/A

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

       2. unsub-neg N/A

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

       3. lower--.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. sub-neg N/A

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

       6. distribute-lft-in N/A

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

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

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

       8. *-commutative N/A

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

       9. associate-*l* N/A

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

       10. lft-mult-inverse N/A

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

       11. metadata-eval N/A

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

       12. metadata-eval N/A

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

       13. lower-fma.f64 87.5

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

   11. Simplified 87.5%

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

   12. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 75.8

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

   14. Simplified 75.8%

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

Alternative 11: 62.5% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.55 \cdot 10^{+64}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.8862269254527579, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -4.55e+64) (/ -1.0 x) (fma 0.8862269254527579 y x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -4.55e+64) {
		tmp = -1.0 / x;
	} else {
		tmp = fma(0.8862269254527579, y, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -4.55e+64)
		tmp = Float64(-1.0 / x);
	else
		tmp = fma(0.8862269254527579, y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -4.55e+64], N[(-1.0 / x), $MachinePrecision], N[(0.8862269254527579 * y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.5500000000000002e64

   1. Initial program 81.4%

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

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

      2. lower-+.f64 N/A

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

      3. distribute-neg-frac N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 100.0

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

   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. lower-/.f64 69.2

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

   8. Simplified 69.2%

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

   if -4.5500000000000002e64 < z

   1. Initial program 98.5%

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

   3. Taylor expanded in z around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 92.0

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

   5. Simplified 92.0%

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

   6. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 95.3

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

   8. Simplified 95.3%

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

   9. Taylor expanded in z around 0

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

       1. mul-1-neg N/A

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

       2. unsub-neg N/A

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

       3. lower--.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. sub-neg N/A

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

       6. distribute-lft-in N/A

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

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

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

       8. *-commutative N/A

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

       9. associate-*l* N/A

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

       10. lft-mult-inverse N/A

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

       11. metadata-eval N/A

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

       12. metadata-eval N/A

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

       13. lower-fma.f64 87.2

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

   11. Simplified 87.2%

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

   12. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 73.4

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

   14. Simplified 73.4%

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

Alternative 12: 59.9% accurate, 18.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(0.8862269254527579, y, x\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma 0.8862269254527579 y x))

C

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

Julia

function code(x, y, z)
	return fma(0.8862269254527579, y, x)
end

Wolfram

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

Derivation

1. Initial program 95.3%

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

3. Taylor expanded in z around 0

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

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

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

   5. mul-1-neg N/A

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

   6. lower-fma.f64 N/A

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

   7. mul-1-neg N/A

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

   8. lower-neg.f64 N/A

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

   9. +-commutative N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f64 81.2

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

5. Simplified 81.2%

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

6. Taylor expanded in x around -inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. mul-1-neg N/A

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

   5. unsub-neg N/A

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

   6. lower--.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-fma.f64 83.8

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

8. Simplified 83.8%

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

9. Taylor expanded in z around 0

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

    1. mul-1-neg N/A

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

    2. unsub-neg N/A

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

    3. lower--.f64 N/A

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

    4. lower-/.f64 N/A

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

    5. sub-neg N/A

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

    6. distribute-lft-in N/A

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

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

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

    8. *-commutative N/A

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

    9. associate-*l* N/A

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

    10. lft-mult-inverse N/A

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

    11. metadata-eval N/A

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

    12. metadata-eval N/A

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

    13. lower-fma.f64 80.8

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

11. Simplified 80.8%

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

12. Taylor expanded in y around 0

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

    1. +-commutative N/A

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

    2. lower-fma.f64 65.2

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

14. Simplified 65.2%

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

Alternative 13: 14.8% accurate, 21.3× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = y * 0.8862269254527579d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.3%

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

3. Taylor expanded in x around 0

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

   1. *-lft-identity N/A

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

   2. associate-*l/ N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. lower-*.f64 N/A

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

   7. rec-exp N/A

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

   8. neg-mul-1 N/A

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

   9. lower-exp.f64 N/A

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

   10. neg-mul-1 N/A

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

   11. lower-neg.f64 N/A

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

   12. lower-*.f64 16.7

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

5. Simplified 16.7%

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

6. Taylor expanded in z around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 16.1

      \[\leadsto \color{blue}{y \cdot 0.8862269254527579} \]

8. Simplified 16.1%

   \[\leadsto \color{blue}{y \cdot 0.8862269254527579} \]
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 2024208 
(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)))))