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

Percentage Accurate: 95.7% → 98.7%

Time: 9.7s

Alternatives: 13

Speedup: 3.0×

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: 95.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.7% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 91.7%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 0.0 < (exp.f64 z)

   1. Initial program 97.3%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. distribute-neg-frac N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. distribute-neg-in N/A

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

      11. lift-*.f64 N/A

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

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

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

      13. remove-double-neg N/A

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

      14. lower-fma.f64 N/A

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

      15. metadata-eval 97.3

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 97.3

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

   4. Applied rewrites 97.3%

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

   5. Taylor expanded in z around 0

      \[\leadsto x + \frac{-1}{\frac{\color{blue}{\left(x \cdot y + z \cdot \left(z \cdot \left(\frac{-5641895835477563}{30000000000000000} \cdot z - \frac{5641895835477563}{10000000000000000}\right) - \frac{5641895835477563}{5000000000000000}\right)\right) - \frac{5641895835477563}{5000000000000000}}}{y}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. sub-neg N/A

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

      13. *-commutative N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 95.8

          \[\leadsto x + \frac{-1}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.18806319451591877, z, -0.5641895835477563\right), z, -1.1283791670955126\right), z, \color{blue}{\mathsf{fma}\left(y, x, -1.1283791670955126\right)}\right)}{y}} \]

   7. Applied rewrites 95.8%

      \[\leadsto x + \frac{-1}{\frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.18806319451591877, z, -0.5641895835477563\right), z, -1.1283791670955126\right), z, \mathsf{fma}\left(y, x, -1.1283791670955126\right)\right)}}{y}} \]

   8. Taylor expanded in x around inf

      \[\leadsto x + \frac{-1}{\frac{x \cdot \color{blue}{\left(\left(y + \frac{z \cdot \left(z \cdot \left(\frac{-5641895835477563}{30000000000000000} \cdot z - \frac{5641895835477563}{10000000000000000}\right) - \frac{5641895835477563}{5000000000000000}\right)}{x}\right) - \frac{5641895835477563}{5000000000000000} \cdot \frac{1}{x}\right)}}{y}} \]
   9. Step-by-step derivation

   10. Applied rewrites 98.8%

       \[\leadsto x + \frac{-1}{\frac{\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.18806319451591877, z, -0.5641895835477563\right), z, -1.1283791670955126\right), z, -1.1283791670955126\right)}{x} + y\right) \cdot \color{blue}{x}}{y}} \]

   12. Applied rewrites 98.8%

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

4. Final simplification 99.1%

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

Alternative 2: 83.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{x} + x\\ t_1 := \frac{y}{1.1283791670955126 \cdot e^{z} - y \cdot x} + x\\ \mathbf{if}\;t\_1 \leq -40:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 0.04:\\ \;\;\;\;\mathsf{fma}\left(0.8862269254527579, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (/ -1.0 x) x))
        (t_1 (+ (/ y (- (* 1.1283791670955126 (exp z)) (* y x))) x)))
   (if (<= t_1 -40.0)
     t_0
     (if (<= t_1 0.04) (fma 0.8862269254527579 y x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (-1.0 / x) + x;
	double t_1 = (y / ((1.1283791670955126 * exp(z)) - (y * x))) + x;
	double tmp;
	if (t_1 <= -40.0) {
		tmp = t_0;
	} else if (t_1 <= 0.04) {
		tmp = fma(0.8862269254527579, y, x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(-1.0 / x) + x)
	t_1 = Float64(Float64(y / Float64(Float64(1.1283791670955126 * exp(z)) - Float64(y * x))) + x)
	tmp = 0.0
	if (t_1 <= -40.0)
		tmp = t_0;
	elseif (t_1 <= 0.04)
		tmp = fma(0.8862269254527579, y, x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(-1.0 / x), $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y / N[(N[(1.1283791670955126 * N[Exp[z], $MachinePrecision]), $MachinePrecision] - N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t$95$1, -40.0], t$95$0, If[LessEqual[t$95$1, 0.04], N[(0.8862269254527579 * y + 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)))) < -40 or 0.0400000000000000008 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y))))

   1. Initial program 94.7%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 91.8

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

   5. Applied rewrites 91.8%

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

   if -40 < (+.f64 x (/.f64 y (-.f64 (*.f64 #s(literal 5641895835477563/5000000000000000 binary64) (exp.f64 z)) (*.f64 x y)))) < 0.0400000000000000008

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-lft-identity N/A

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

      3. associate-*l/ N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lower-exp.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 61.0%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{1.1283791670955126 \cdot e^{z} - y \cdot x} + x \leq -40:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;\frac{y}{1.1283791670955126 \cdot e^{z} - y \cdot x} + x \leq 0.04:\\ \;\;\;\;\mathsf{fma}\left(0.8862269254527579, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x} + x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;e^{z} \leq 2:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(0.5641895835477563, z, 1.1283791670955126\right), z, 1.1283791670955126\right) - y \cdot x} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.8862269254527579}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, z, 0.5\right), z, 1\right), z, 1\right)}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (+ (/ -1.0 x) x)
   (if (<= (exp z) 2.0)
     (+
      (/
       y
       (-
        (fma
         (fma 0.5641895835477563 z 1.1283791670955126)
         z
         1.1283791670955126)
        (* y x)))
      x)
     (fma
      (/
       0.8862269254527579
       (fma (fma (fma 0.16666666666666666 z 0.5) z 1.0) z 1.0))
      y
      x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (exp(z) <= 0.0) {
		tmp = (-1.0 / x) + x;
	} else if (exp(z) <= 2.0) {
		tmp = (y / (fma(fma(0.5641895835477563, z, 1.1283791670955126), z, 1.1283791670955126) - (y * x))) + x;
	} else {
		tmp = fma((0.8862269254527579 / fma(fma(fma(0.16666666666666666, z, 0.5), z, 1.0), z, 1.0)), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (exp(z) <= 0.0)
		tmp = Float64(Float64(-1.0 / x) + x);
	elseif (exp(z) <= 2.0)
		tmp = Float64(Float64(y / Float64(fma(fma(0.5641895835477563, z, 1.1283791670955126), z, 1.1283791670955126) - Float64(y * x))) + x);
	else
		tmp = fma(Float64(0.8862269254527579 / fma(fma(fma(0.16666666666666666, z, 0.5), z, 1.0), z, 1.0)), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[Exp[z], $MachinePrecision], 0.0], N[(N[(-1.0 / x), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[N[Exp[z], $MachinePrecision], 2.0], N[(N[(y / N[(N[(N[(0.5641895835477563 * z + 1.1283791670955126), $MachinePrecision] * z + 1.1283791670955126), $MachinePrecision] - N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(N[(0.8862269254527579 / N[(N[(N[(0.16666666666666666 * z + 0.5), $MachinePrecision] * z + 1.0), $MachinePrecision] * z + 1.0), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 91.7%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 0.0 < (exp.f64 z) < 2

   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. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 99.6

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

   5. Applied rewrites 99.6%

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

   if 2 < (exp.f64 z)

   1. Initial program 92.4%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-lft-identity N/A

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

      3. associate-*l/ N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(\frac{\frac{5000000000000000}{5641895835477563}}{1 + z \cdot \left(1 + z \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot z\right)\right)}, y, x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 92.7%

      \[\leadsto \mathsf{fma}\left(\frac{0.8862269254527579}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, z, 0.5\right), z, 1\right), z, 1\right)}, y, x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;e^{z} \leq 2:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(0.5641895835477563, z, 1.1283791670955126\right), z, 1.1283791670955126\right) - y \cdot x} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.8862269254527579}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, z, 0.5\right), z, 1\right), z, 1\right)}, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, z, 0.5\right), z, 1\right), z, 1\right) \cdot 1.1283791670955126 - y \cdot x} + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (exp z) 0.0)
   (+ (/ -1.0 x) x)
   (+
    (/
     y
     (-
      (*
       (fma (fma (fma 0.16666666666666666 z 0.5) z 1.0) z 1.0)
       1.1283791670955126)
      (* y x)))
    x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (exp(z) <= 0.0) {
		tmp = (-1.0 / x) + x;
	} else {
		tmp = (y / ((fma(fma(fma(0.16666666666666666, z, 0.5), z, 1.0), z, 1.0) * 1.1283791670955126) - (y * x))) + x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (exp(z) <= 0.0)
		tmp = Float64(Float64(-1.0 / x) + x);
	else
		tmp = Float64(Float64(y / Float64(Float64(fma(fma(fma(0.16666666666666666, z, 0.5), z, 1.0), z, 1.0) * 1.1283791670955126) - Float64(y * x))) + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[Exp[z], $MachinePrecision], 0.0], N[(N[(-1.0 / x), $MachinePrecision] + x), $MachinePrecision], N[(N[(y / N[(N[(N[(N[(N[(0.16666666666666666 * z + 0.5), $MachinePrecision] * z + 1.0), $MachinePrecision] * z + 1.0), $MachinePrecision] * 1.1283791670955126), $MachinePrecision] - N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 91.7%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 0.0 < (exp.f64 z)

   1. Initial program 97.3%

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

   3. Taylor expanded in z around 0

      \[\leadsto x + \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot \color{blue}{\left(1 + z \cdot \left(1 + z \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot z\right)\right)\right)} - x \cdot y} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x + \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot \color{blue}{\left(z \cdot \left(1 + z \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot z\right)\right) + 1\right)} - x \cdot y} \]

      2. *-commutative N/A

         \[\leadsto x + \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot \left(\color{blue}{\left(1 + z \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot z\right)\right) \cdot z} + 1\right) - x \cdot y} \]

      3. lower-fma.f64 N/A

         \[\leadsto x + \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot \color{blue}{\mathsf{fma}\left(1 + z \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot z\right), z, 1\right)} - x \cdot y} \]

      4. +-commutative N/A

         \[\leadsto x + \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot \mathsf{fma}\left(\color{blue}{z \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot z\right) + 1}, z, 1\right) - x \cdot y} \]

      5. *-commutative N/A

         \[\leadsto x + \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot \mathsf{fma}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot z\right) \cdot z} + 1, z, 1\right) - x \cdot y} \]

      6. lower-fma.f64 N/A

         \[\leadsto x + \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{2} + \frac{1}{6} \cdot z, z, 1\right)}, z, 1\right) - x \cdot y} \]

      7. +-commutative N/A

         \[\leadsto x + \frac{y}{\frac{5641895835477563}{5000000000000000} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{6} \cdot z + \frac{1}{2}}, z, 1\right), z, 1\right) - x \cdot y} \]

      8. lower-fma.f64 95.9

         \[\leadsto x + \frac{y}{1.1283791670955126 \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.16666666666666666, z, 0.5\right)}, z, 1\right), z, 1\right) - x \cdot y} \]

   5. Applied rewrites 95.9%

      \[\leadsto x + \frac{y}{1.1283791670955126 \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, z, 0.5\right), z, 1\right), z, 1\right)} - x \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, z, 0.5\right), z, 1\right), z, 1\right) \cdot 1.1283791670955126 - y \cdot x} + x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 91.7%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 0.0 < (exp.f64 z)

   1. Initial program 97.3%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 95.9

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

   5. Applied rewrites 95.9%

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

4. Final simplification 96.8%

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

Alternative 6: 96.1% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 91.7%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 0.0 < (exp.f64 z)

   1. Initial program 97.3%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 95.1

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

   5. Applied rewrites 95.1%

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

4. Final simplification 96.2%

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

Alternative 7: 96.2% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -29000000:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;z \leq 6.4 \cdot 10^{+65}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(z, 1.1283791670955126, 1.1283791670955126\right) - y \cdot x} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.8862269254527579}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, z, 1\right), z, 1\right)}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -29000000.0)
   (+ (/ -1.0 x) x)
   (if (<= z 6.4e+65)
     (+ (/ y (- (fma z 1.1283791670955126 1.1283791670955126) (* y x))) x)
     (fma (/ 0.8862269254527579 (fma (fma 0.5 z 1.0) z 1.0)) y x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -29000000.0) {
		tmp = (-1.0 / x) + x;
	} else if (z <= 6.4e+65) {
		tmp = (y / (fma(z, 1.1283791670955126, 1.1283791670955126) - (y * x))) + x;
	} else {
		tmp = fma((0.8862269254527579 / fma(fma(0.5, z, 1.0), z, 1.0)), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -29000000.0)
		tmp = Float64(Float64(-1.0 / x) + x);
	elseif (z <= 6.4e+65)
		tmp = Float64(Float64(y / Float64(fma(z, 1.1283791670955126, 1.1283791670955126) - Float64(y * x))) + x);
	else
		tmp = fma(Float64(0.8862269254527579 / fma(fma(0.5, z, 1.0), z, 1.0)), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -29000000.0], N[(N[(-1.0 / x), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 6.4e+65], N[(N[(y / N[(N[(z * 1.1283791670955126 + 1.1283791670955126), $MachinePrecision] - N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(N[(0.8862269254527579 / N[(N[(0.5 * z + 1.0), $MachinePrecision] * z + 1.0), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.9e7

   1. Initial program 91.5%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -2.9e7 < z < 6.40000000000000014e65

   1. Initial program 99.1%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 95.9

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

   5. Applied rewrites 95.9%

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

   if 6.40000000000000014e65 < z

   1. Initial program 92.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-lft-identity N/A

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

      3. associate-*l/ N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(\frac{\frac{5000000000000000}{5641895835477563}}{1 + z \cdot \left(1 + \frac{1}{2} \cdot z\right)}, y, x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 96.0%

      \[\leadsto \mathsf{fma}\left(\frac{0.8862269254527579}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, z, 1\right), z, 1\right)}, y, x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -29000000:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;z \leq 6.4 \cdot 10^{+65}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(z, 1.1283791670955126, 1.1283791670955126\right) - y \cdot x} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.8862269254527579}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, z, 1\right), z, 1\right)}, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 92.8% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -29000000:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{+99}:\\ \;\;\;\;\frac{-1}{x - \frac{1.1283791670955126}{y}} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.8862269254527579}{1 + z}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -29000000.0)
   (+ (/ -1.0 x) x)
   (if (<= z 1.4e+99)
     (+ (/ -1.0 (- x (/ 1.1283791670955126 y))) x)
     (fma (/ 0.8862269254527579 (+ 1.0 z)) y x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -29000000.0) {
		tmp = (-1.0 / x) + x;
	} else if (z <= 1.4e+99) {
		tmp = (-1.0 / (x - (1.1283791670955126 / y))) + x;
	} else {
		tmp = fma((0.8862269254527579 / (1.0 + z)), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -29000000.0)
		tmp = Float64(Float64(-1.0 / x) + x);
	elseif (z <= 1.4e+99)
		tmp = Float64(Float64(-1.0 / Float64(x - Float64(1.1283791670955126 / y))) + x);
	else
		tmp = fma(Float64(0.8862269254527579 / Float64(1.0 + z)), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -29000000.0], N[(N[(-1.0 / x), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 1.4e+99], N[(N[(-1.0 / N[(x - N[(1.1283791670955126 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(N[(0.8862269254527579 / N[(1.0 + z), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.9e7

   1. Initial program 91.5%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -2.9e7 < z < 1.4e99

   1. Initial program 98.5%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. distribute-neg-frac N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. distribute-neg-in N/A

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

      11. lift-*.f64 N/A

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

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

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

      13. remove-double-neg N/A

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

      14. lower-fma.f64 N/A

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

      15. metadata-eval 98.5

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 98.5

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

   4. Applied rewrites 98.5%

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

   5. Taylor expanded in z around 0

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

      1. div-sub N/A

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

      2. associate-/l* N/A

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

      3. *-inverses N/A

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

      4. *-rgt-identity N/A

         \[\leadsto x + \frac{-1}{\color{blue}{x} - \frac{\frac{5641895835477563}{5000000000000000}}{y}} \]

      5. metadata-eval N/A

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

      6. associate-*r/ N/A

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

      7. lower--.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

         \[\leadsto x + \frac{-1}{x - \frac{\color{blue}{\frac{5641895835477563}{5000000000000000}}}{y}} \]

      10. lower-/.f64 94.2

          \[\leadsto x + \frac{-1}{x - \color{blue}{\frac{1.1283791670955126}{y}}} \]

   7. Applied rewrites 94.2%

      \[\leadsto x + \frac{-1}{\color{blue}{x - \frac{1.1283791670955126}{y}}} \]

   if 1.4e99 < z

   1. Initial program 93.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-lft-identity N/A

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

      3. associate-*l/ N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 77.6%

      \[\leadsto \mathsf{fma}\left(\frac{0.8862269254527579}{1 + z}, y, x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -29000000:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{+99}:\\ \;\;\;\;\frac{-1}{x - \frac{1.1283791670955126}{y}} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.8862269254527579}{1 + z}, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 93.6% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -29000000:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(z, 1.1283791670955126, 1.1283791670955126\right) - y \cdot x} + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -29000000.0)
   (+ (/ -1.0 x) x)
   (+ (/ y (- (fma z 1.1283791670955126 1.1283791670955126) (* y x))) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -29000000.0) {
		tmp = (-1.0 / x) + x;
	} else {
		tmp = (y / (fma(z, 1.1283791670955126, 1.1283791670955126) - (y * x))) + x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -29000000.0)
		tmp = Float64(Float64(-1.0 / x) + x);
	else
		tmp = Float64(Float64(y / Float64(fma(z, 1.1283791670955126, 1.1283791670955126) - Float64(y * x))) + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -29000000.0], N[(N[(-1.0 / x), $MachinePrecision] + x), $MachinePrecision], N[(N[(y / N[(N[(z * 1.1283791670955126 + 1.1283791670955126), $MachinePrecision] - N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.9e7

   1. Initial program 91.5%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -2.9e7 < z

   1. Initial program 97.3%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 90.8

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

   5. Applied rewrites 90.8%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -29000000:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(z, 1.1283791670955126, 1.1283791670955126\right) - y \cdot x} + x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 92.8% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -29000000:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{+99}:\\ \;\;\;\;x - \frac{y}{\mathsf{fma}\left(y, x, -1.1283791670955126\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.8862269254527579}{1 + z}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -29000000.0)
   (+ (/ -1.0 x) x)
   (if (<= z 1.4e+99)
     (- x (/ y (fma y x -1.1283791670955126)))
     (fma (/ 0.8862269254527579 (+ 1.0 z)) y x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -29000000.0) {
		tmp = (-1.0 / x) + x;
	} else if (z <= 1.4e+99) {
		tmp = x - (y / fma(y, x, -1.1283791670955126));
	} else {
		tmp = fma((0.8862269254527579 / (1.0 + z)), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -29000000.0)
		tmp = Float64(Float64(-1.0 / x) + x);
	elseif (z <= 1.4e+99)
		tmp = Float64(x - Float64(y / fma(y, x, -1.1283791670955126)));
	else
		tmp = fma(Float64(0.8862269254527579 / Float64(1.0 + z)), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -29000000.0], N[(N[(-1.0 / x), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 1.4e+99], N[(x - N[(y / N[(y * x + -1.1283791670955126), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.8862269254527579 / N[(1.0 + z), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.9e7

   1. Initial program 91.5%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -2.9e7 < z < 1.4e99

   1. Initial program 98.5%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 61.2

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

   5. Applied rewrites 61.2%

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

   6. Taylor expanded in z around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. sub-neg N/A

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

      7. distribute-neg-frac2 N/A

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

      8. unsub-neg N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. sub-neg N/A

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

      12. *-commutative N/A

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 94.2

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

   8. Applied rewrites 94.2%

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

   if 1.4e99 < z

   1. Initial program 93.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-lft-identity N/A

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

      3. associate-*l/ N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 77.6%

      \[\leadsto \mathsf{fma}\left(\frac{0.8862269254527579}{1 + z}, y, x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -29000000:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{+99}:\\ \;\;\;\;x - \frac{y}{\mathsf{fma}\left(y, x, -1.1283791670955126\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.8862269254527579}{1 + z}, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 90.4% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -29000000:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\mathsf{fma}\left(y, x, -1.1283791670955126\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -29000000.0)
   (+ (/ -1.0 x) x)
   (- x (/ y (fma y x -1.1283791670955126)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -29000000.0) {
		tmp = (-1.0 / x) + x;
	} else {
		tmp = x - (y / fma(y, x, -1.1283791670955126));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -29000000.0)
		tmp = Float64(Float64(-1.0 / x) + x);
	else
		tmp = Float64(x - Float64(y / fma(y, x, -1.1283791670955126)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -29000000.0], N[(N[(-1.0 / x), $MachinePrecision] + x), $MachinePrecision], N[(x - N[(y / N[(y * x + -1.1283791670955126), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.9e7

   1. Initial program 91.5%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -2.9e7 < z

   1. Initial program 97.3%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 59.3

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

   5. Applied rewrites 59.3%

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

   6. Taylor expanded in z around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. sub-neg N/A

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

      7. distribute-neg-frac2 N/A

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

      8. unsub-neg N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. sub-neg N/A

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

      12. *-commutative N/A

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 86.3

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

   8. Applied rewrites 86.3%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -29000000:\\ \;\;\;\;\frac{-1}{x} + x\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\mathsf{fma}\left(y, x, -1.1283791670955126\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 60.2% accurate, 18.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 96.0%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. *-lft-identity N/A

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

   3. associate-*l/ N/A

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

   4. associate-*l* N/A

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

   5. lower-fma.f64 N/A

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

   6. associate-*r/ N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f64 N/A

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

   9. lower-exp.f64 62.3

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

5. Applied rewrites 62.3%

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

6. Taylor expanded in z around 0

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

8. Applied rewrites 55.7%

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

Alternative 13: 14.7% accurate, 21.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.0%

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

3. Taylor expanded in y around inf

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

   1. lower-/.f64 68.5

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

5. Applied rewrites 68.5%

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

6. Taylor expanded in z around 0

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. metadata-eval N/A

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

   4. distribute-neg-in N/A

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

   5. metadata-eval N/A

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

   6. sub-neg N/A

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

   7. distribute-neg-frac2 N/A

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

   8. unsub-neg N/A

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

   9. lower--.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. sub-neg N/A

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

   12. *-commutative N/A

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

   13. metadata-eval N/A

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

   14. lower-fma.f64 82.6

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

8. Applied rewrites 82.6%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 14.3%

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

Developer Target 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024243 
(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)))))