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

Percentage Accurate: 95.4% → 98.2%

Time: 12.7s

Alternatives: 11

Speedup: 3.7×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 95.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.2% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (+ x (/ -1.0 x))
   (+
    x
    (/
     y
     (*
      x
      (-
       (/
        (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 / (x * ((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(x * Float64(Float64(fma(z, fma(z, 0.5641895835477563, 1.1283791670955126), 1.1283791670955126) / 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[(x * N[(N[(N[(z * N[(z * 0.5641895835477563 + 1.1283791670955126), $MachinePrecision] + 1.1283791670955126), $MachinePrecision] / x), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 83.0%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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. Applied rewrites 100.0%

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

   if 0.0 < (exp.f64 z)

   1. Initial program 94.6%

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

   3. Taylor expanded in z around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 97.3

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

   5. Applied rewrites 97.3%

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

   6. Taylor expanded in x around -inf

      \[\leadsto x + \frac{y}{\color{blue}{-1 \cdot \left(x \cdot \left(y + -1 \cdot \frac{\frac{5641895835477563}{5000000000000000} + z \cdot \left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{10000000000000000} \cdot z\right)}{x}\right)\right)}} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 98.8

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

   8. Applied rewrites 98.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. remove-double-neg N/A

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

      7. lift-neg.f64 N/A

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

      8. distribute-frac-neg2 N/A

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

      9. lift-neg.f64 N/A

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

      10. remove-double-neg N/A

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

   10. Applied rewrites 98.8%

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

4. Final simplification 99.1%

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

Alternative 2: 84.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{-1}{x}\\ t_1 := x + \frac{y}{e^{z} \cdot 1.1283791670955126 - x \cdot y}\\ \mathbf{if}\;t\_1 \leq -50:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 1000:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(y, x \cdot 0.7853981633974483, 0.8862269254527579\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (/ -1.0 x)))
        (t_1 (+ x (/ y (- (* (exp z) 1.1283791670955126) (* x y))))))
   (if (<= t_1 -50.0)
     t_0
     (if (<= t_1 1000.0)
       (fma y (fma y (* x 0.7853981633974483) 0.8862269254527579) x)
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x + (-1.0 / x);
	double t_1 = x + (y / ((exp(z) * 1.1283791670955126) - (x * y)));
	double tmp;
	if (t_1 <= -50.0) {
		tmp = t_0;
	} else if (t_1 <= 1000.0) {
		tmp = fma(y, fma(y, (x * 0.7853981633974483), 0.8862269254527579), x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x + Float64(-1.0 / x))
	t_1 = Float64(x + Float64(y / Float64(Float64(exp(z) * 1.1283791670955126) - Float64(x * y))))
	tmp = 0.0
	if (t_1 <= -50.0)
		tmp = t_0;
	elseif (t_1 <= 1000.0)
		tmp = fma(y, fma(y, Float64(x * 0.7853981633974483), 0.8862269254527579), x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x + N[(y / N[(N[(N[Exp[z], $MachinePrecision] * 1.1283791670955126), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -50.0], t$95$0, If[LessEqual[t$95$1, 1000.0], N[(y * N[(y * N[(x * 0.7853981633974483), $MachinePrecision] + 0.8862269254527579), $MachinePrecision] + x), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y)))) < -50 or 1e3 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y))))

   1. Initial program 89.2%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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 90.3

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

   5. Applied rewrites 90.3%

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

   if -50 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y)))) < 1e3

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 92.0

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

   5. Applied rewrites 92.0%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 66.2%

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

   9. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + y \cdot \left(\frac{5000000000000000}{5641895835477563} + \frac{25000000000000000000000000000000}{31830988618379068626528276418969} \cdot \left(x \cdot y\right)\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{y \cdot \left(\frac{5000000000000000}{5641895835477563} + \frac{25000000000000000000000000000000}{31830988618379068626528276418969} \cdot \left(x \cdot y\right)\right) + x} \]

       2. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{5000000000000000}{5641895835477563} + \frac{25000000000000000000000000000000}{31830988618379068626528276418969} \cdot \left(x \cdot y\right), x\right)} \]

       3. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{25000000000000000000000000000000}{31830988618379068626528276418969} \cdot \left(x \cdot y\right) + \frac{5000000000000000}{5641895835477563}}, x\right) \]

       4. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(y, \color{blue}{\left(\frac{25000000000000000000000000000000}{31830988618379068626528276418969} \cdot x\right) \cdot y} + \frac{5000000000000000}{5641895835477563}, x\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(y, \left(\color{blue}{\left(\mathsf{neg}\left(\frac{-25000000000000000000000000000000}{31830988618379068626528276418969}\right)\right)} \cdot x\right) \cdot y + \frac{5000000000000000}{5641895835477563}, x\right) \]

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

          \[\leadsto \mathsf{fma}\left(y, \color{blue}{\left(\mathsf{neg}\left(\frac{-25000000000000000000000000000000}{31830988618379068626528276418969} \cdot x\right)\right)} \cdot y + \frac{5000000000000000}{5641895835477563}, x\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(y, \color{blue}{y \cdot \left(\mathsf{neg}\left(\frac{-25000000000000000000000000000000}{31830988618379068626528276418969} \cdot x\right)\right)} + \frac{5000000000000000}{5641895835477563}, x\right) \]

       8. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, \mathsf{neg}\left(\frac{-25000000000000000000000000000000}{31830988618379068626528276418969} \cdot x\right), \frac{5000000000000000}{5641895835477563}\right)}, x\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{neg}\left(\color{blue}{x \cdot \frac{-25000000000000000000000000000000}{31830988618379068626528276418969}}\right), \frac{5000000000000000}{5641895835477563}\right), x\right) \]

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

           \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{x \cdot \left(\mathsf{neg}\left(\frac{-25000000000000000000000000000000}{31830988618379068626528276418969}\right)\right)}, \frac{5000000000000000}{5641895835477563}\right), x\right) \]

       11. metadata-eval N/A

           \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(y, x \cdot \color{blue}{\frac{25000000000000000000000000000000}{31830988618379068626528276418969}}, \frac{5000000000000000}{5641895835477563}\right), x\right) \]

       12. lower-*.f64 66.5

           \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{x \cdot 0.7853981633974483}, 0.8862269254527579\right), x\right) \]

   11. Applied rewrites 66.5%

       \[\leadsto \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, x \cdot 0.7853981633974483, 0.8862269254527579\right), x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{y}{e^{z} \cdot 1.1283791670955126 - x \cdot y} \leq -50:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;x + \frac{y}{e^{z} \cdot 1.1283791670955126 - x \cdot y} \leq 1000:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(y, x \cdot 0.7853981633974483, 0.8862269254527579\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{-1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{-1}{x}\\ t_1 := x + \frac{y}{e^{z} \cdot 1.1283791670955126 - x \cdot y}\\ \mathbf{if}\;t\_1 \leq -50:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 1000:\\ \;\;\;\;\mathsf{fma}\left(y, 0.8862269254527579, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (/ -1.0 x)))
        (t_1 (+ x (/ y (- (* (exp z) 1.1283791670955126) (* x y))))))
   (if (<= t_1 -50.0)
     t_0
     (if (<= t_1 1000.0) (fma y 0.8862269254527579 x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x + (-1.0 / x);
	double t_1 = x + (y / ((exp(z) * 1.1283791670955126) - (x * y)));
	double tmp;
	if (t_1 <= -50.0) {
		tmp = t_0;
	} else if (t_1 <= 1000.0) {
		tmp = fma(y, 0.8862269254527579, x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x + Float64(-1.0 / x))
	t_1 = Float64(x + Float64(y / Float64(Float64(exp(z) * 1.1283791670955126) - Float64(x * y))))
	tmp = 0.0
	if (t_1 <= -50.0)
		tmp = t_0;
	elseif (t_1 <= 1000.0)
		tmp = fma(y, 0.8862269254527579, x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x + N[(y / N[(N[(N[Exp[z], $MachinePrecision] * 1.1283791670955126), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -50.0], t$95$0, If[LessEqual[t$95$1, 1000.0], N[(y * 0.8862269254527579 + x), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y)))) < -50 or 1e3 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y))))

   1. Initial program 89.2%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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 90.3

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

   5. Applied rewrites 90.3%

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

   if -50 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y)))) < 1e3

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 92.0

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

   5. Applied rewrites 92.0%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 66.2%

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

   9. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 66.4

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

   11. Applied rewrites 66.4%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{y}{e^{z} \cdot 1.1283791670955126 - x \cdot y} \leq -50:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;x + \frac{y}{e^{z} \cdot 1.1283791670955126 - x \cdot y} \leq 1000:\\ \;\;\;\;\mathsf{fma}\left(y, 0.8862269254527579, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{-1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.7% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (+ x (/ -1.0 x))
   (+
    x
    (/
     y
     (fma
      y
      (- x)
      (fma
       z
       (fma z 0.5641895835477563 1.1283791670955126)
       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(y, -x, fma(z, fma(z, 0.5641895835477563, 1.1283791670955126), 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(y, Float64(-x), fma(z, fma(z, 0.5641895835477563, 1.1283791670955126), 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[(y * (-x) + N[(z * N[(z * 0.5641895835477563 + 1.1283791670955126), $MachinePrecision] + 1.1283791670955126), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 83.0%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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. Applied rewrites 100.0%

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

   if 0.0 < (exp.f64 z)

   1. Initial program 94.6%

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

   3. Taylor expanded in z around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 97.3

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

   5. Applied rewrites 97.3%

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

Alternative 5: 97.4% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -220000000:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{+23}:\\ \;\;\;\;x + \frac{y}{\mathsf{fma}\left(y, -x, \mathsf{fma}\left(z, 1.1283791670955126, 1.1283791670955126\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z \cdot \left(z \cdot z\right)}, 5.317361552716548, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -220000000.0)
   (+ x (/ -1.0 x))
   (if (<= z 9.5e+23)
     (+ x (/ y (fma y (- x) (fma z 1.1283791670955126 1.1283791670955126))))
     (fma (/ y (* z (* z z))) 5.317361552716548 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -220000000.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 9.5e+23) {
		tmp = x + (y / fma(y, -x, fma(z, 1.1283791670955126, 1.1283791670955126)));
	} else {
		tmp = fma((y / (z * (z * z))), 5.317361552716548, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -220000000.0)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (z <= 9.5e+23)
		tmp = Float64(x + Float64(y / fma(y, Float64(-x), fma(z, 1.1283791670955126, 1.1283791670955126))));
	else
		tmp = fma(Float64(y / Float64(z * Float64(z * z))), 5.317361552716548, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -220000000.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 9.5e+23], N[(x + N[(y / N[(y * (-x) + N[(z * 1.1283791670955126 + 1.1283791670955126), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y / N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 5.317361552716548 + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.2e8

   1. Initial program 82.2%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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 100.0

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

   5. Applied rewrites 100.0%

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

   if -2.2e8 < z < 9.50000000000000038e23

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0

      \[\leadsto x + \frac{y}{\color{blue}{\left(\frac{5641895835477563}{5000000000000000} + \frac{5641895835477563}{5000000000000000} \cdot z\right) - x \cdot y}} \]
   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 98.7

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

   5. Applied rewrites 98.7%

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

   if 9.50000000000000038e23 < z

   1. Initial program 87.2%

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

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

      10. *-commutative N/A

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

      11. lower-*.f64 86.0

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

   5. Applied rewrites 86.0%

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

   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. cube-mult N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 96.2

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

   8. Applied rewrites 96.2%

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

Alternative 6: 97.2% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -220000000:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{+23}:\\ \;\;\;\;x + \frac{y}{\mathsf{fma}\left(y, -x, 1.1283791670955126\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z \cdot \left(z \cdot z\right)}, 5.317361552716548, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -220000000.0)
   (+ x (/ -1.0 x))
   (if (<= z 9.5e+23)
     (+ x (/ y (fma y (- x) 1.1283791670955126)))
     (fma (/ y (* z (* z z))) 5.317361552716548 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -220000000.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 9.5e+23) {
		tmp = x + (y / fma(y, -x, 1.1283791670955126));
	} else {
		tmp = fma((y / (z * (z * z))), 5.317361552716548, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -220000000.0)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (z <= 9.5e+23)
		tmp = Float64(x + Float64(y / fma(y, Float64(-x), 1.1283791670955126)));
	else
		tmp = fma(Float64(y / Float64(z * Float64(z * z))), 5.317361552716548, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -220000000.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 9.5e+23], N[(x + N[(y / N[(y * (-x) + 1.1283791670955126), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y / N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 5.317361552716548 + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.2e8

   1. Initial program 82.2%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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 100.0

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

   5. Applied rewrites 100.0%

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

   if -2.2e8 < z < 9.50000000000000038e23

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 98.9

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 97.9%

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

   if 9.50000000000000038e23 < z

   1. Initial program 87.2%

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

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

      10. *-commutative N/A

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

      11. lower-*.f64 86.0

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

   5. Applied rewrites 86.0%

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

   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. cube-mult N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 96.2

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

   8. Applied rewrites 96.2%

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

Alternative 7: 95.7% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -220000000:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 8.2 \cdot 10^{+101}:\\ \;\;\;\;x + \frac{y}{\mathsf{fma}\left(y, -x, 1.1283791670955126\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1.7724538509055159, \frac{y}{z \cdot z}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -220000000.0)
   (+ x (/ -1.0 x))
   (if (<= z 8.2e+101)
     (+ x (/ y (fma y (- x) 1.1283791670955126)))
     (fma 1.7724538509055159 (/ y (* z z)) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -220000000.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 8.2e+101) {
		tmp = x + (y / fma(y, -x, 1.1283791670955126));
	} else {
		tmp = fma(1.7724538509055159, (y / (z * z)), x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -220000000.0)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (z <= 8.2e+101)
		tmp = Float64(x + Float64(y / fma(y, Float64(-x), 1.1283791670955126)));
	else
		tmp = fma(1.7724538509055159, Float64(y / Float64(z * z)), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -220000000.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 8.2e+101], N[(x + N[(y / N[(y * (-x) + 1.1283791670955126), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.7724538509055159 * N[(y / N[(z * z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.2e8

   1. Initial program 82.2%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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 100.0

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

   5. Applied rewrites 100.0%

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

   if -2.2e8 < z < 8.1999999999999999e101

   1. Initial program 99.0%

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

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 96.8

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

   5. Applied rewrites 96.8%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 95.9%

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

   if 8.1999999999999999e101 < z

   1. Initial program 85.7%

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

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 98.5

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

   5. Applied rewrites 98.5%

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

   6. Taylor expanded in z around inf

      \[\leadsto \color{blue}{x + \frac{10000000000000000}{5641895835477563} \cdot \frac{y}{{z}^{2}}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{10000000000000000}{5641895835477563} \cdot \frac{y}{{z}^{2}} + x} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{10000000000000000}{5641895835477563}, \frac{y}{{z}^{2}}, x\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{10000000000000000}{5641895835477563}, \color{blue}{\frac{y}{{z}^{2}}}, x\right) \]

      4. unpow2 N/A

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

      5. lower-*.f64 98.5

         \[\leadsto \mathsf{fma}\left(1.7724538509055159, \frac{y}{\color{blue}{z \cdot z}}, x\right) \]

   8. Applied rewrites 98.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1.7724538509055159, \frac{y}{z \cdot z}, x\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 92.6% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -220000000:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{+131}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, 0.8862269254527579, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -220000000.0)
   (+ x (/ -1.0 x))
   (if (<= z 1.8e+131)
     (+ x (/ y (- 1.1283791670955126 (* x y))))
     (fma (/ y z) 0.8862269254527579 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -220000000.0) {
		tmp = x + (-1.0 / x);
	} else if (z <= 1.8e+131) {
		tmp = x + (y / (1.1283791670955126 - (x * y)));
	} else {
		tmp = fma((y / z), 0.8862269254527579, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -220000000.0)
		tmp = Float64(x + Float64(-1.0 / x));
	elseif (z <= 1.8e+131)
		tmp = Float64(x + Float64(y / Float64(1.1283791670955126 - Float64(x * y))));
	else
		tmp = fma(Float64(y / z), 0.8862269254527579, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -220000000.0], N[(x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.8e+131], N[(x + N[(y / N[(1.1283791670955126 - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y / z), $MachinePrecision] * 0.8862269254527579 + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.2e8

   1. Initial program 82.2%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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 100.0

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

   5. Applied rewrites 100.0%

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

   if -2.2e8 < z < 1.80000000000000016e131

   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

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

      1. lower-+.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 95.5

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

   5. Applied rewrites 95.5%

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

   if 1.80000000000000016e131 < z

   1. Initial program 84.6%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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 83.3

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

   5. Applied rewrites 83.3%

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

   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 81.4

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

   8. Applied rewrites 81.4%

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

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -220000000:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{+131}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, 0.8862269254527579, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 62.2% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -220000000:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 0.8862269254527579, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -220000000.0) (/ -1.0 x) (fma y 0.8862269254527579 x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -220000000.0) {
		tmp = -1.0 / x;
	} else {
		tmp = fma(y, 0.8862269254527579, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -220000000.0)
		tmp = Float64(-1.0 / x);
	else
		tmp = fma(y, 0.8862269254527579, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -220000000.0], N[(-1.0 / x), $MachinePrecision], N[(y * 0.8862269254527579 + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.2e8

   1. Initial program 82.2%

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
   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 100.0

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

   5. Applied rewrites 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 62.9

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

   8. Applied rewrites 62.9%

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

   if -2.2e8 < z

   1. Initial program 94.7%

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

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 97.3

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

   5. Applied rewrites 97.3%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 88.8%

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

   9. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 66.1

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

   11. Applied rewrites 66.1%

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

Alternative 10: 59.4% accurate, 18.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 91.7%

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

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

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

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

   5. mul-1-neg N/A

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

   6. lower-fma.f64 N/A

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

   7. mul-1-neg N/A

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

   8. lower-neg.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. +-commutative N/A

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

   12. *-commutative N/A

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

   13. lower-fma.f64 83.8

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

5. Applied rewrites 83.8%

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

6. Taylor expanded in z around 0

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

8. Applied rewrites 79.8%

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

9. Taylor expanded in y around 0

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

    1. +-commutative N/A

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

    2. *-commutative N/A

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

    3. lower-fma.f64 57.8

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

11. Applied rewrites 57.8%

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

Alternative 11: 14.3% 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 91.7%

   \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
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 80.4

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

5. Applied rewrites 80.4%

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

6. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-fma.f64 14.0

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

8. Applied rewrites 14.0%

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

9. Taylor expanded in z around 0

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

    1. *-commutative N/A

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

    2. lower-*.f64 13.8

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

11. Applied rewrites 13.8%

    \[\leadsto \color{blue}{y \cdot 0.8862269254527579} \]
12. Add Preprocessing

Developer Target 1: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \frac{1}{\frac{1.1283791670955126}{y} \cdot e^{z} - x} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x + (1.0d0 / (((1.1283791670955126d0 / y) * exp(z)) - x))
end function

Java

public static double code(double x, double y, double z) {
	return x + (1.0 / (((1.1283791670955126 / y) * Math.exp(z)) - x));
}

Python

def code(x, y, z):
	return x + (1.0 / (((1.1283791670955126 / y) * math.exp(z)) - x))

Julia

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

MATLAB

function tmp = code(x, y, z)
	tmp = x + (1.0 / (((1.1283791670955126 / y) * exp(z)) - x));
end

Wolfram

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

Reproduce

herbie shell --seed 2024219 
(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)))))