Average Error: 2.8 → 0.1
Time: 6.3s
Precision: binary64
\[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
↓
\[\begin{array}{l} t_0 := 1.1283791670955126 \cdot e^{z}\\ \mathbf{if}\;t_0 \leq 6.94725497403986 \cdot 10^{-310}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;t_0 \leq 1.1299496288236466:\\ \;\;\;\;x + \frac{y}{t_0 - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{t_0}\\ \end{array} \]
x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y}
↓
\begin{array}{l}
t_0 := 1.1283791670955126 \cdot e^{z}\\
\mathbf{if}\;t_0 \leq 6.94725497403986 \cdot 10^{-310}:\\
\;\;\;\;x + \frac{-1}{x}\\

\mathbf{elif}\;t_0 \leq 1.1299496288236466:\\
\;\;\;\;x + \frac{y}{t_0 - x \cdot y}\\

\mathbf{else}:\\
\;\;\;\;x + \frac{y}{t_0}\\


\end{array}

(FPCore (x y z)
 :precision binary64
 (+ x (/ y (- (* 1.1283791670955126 (exp z)) (* x y)))))
↓
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* 1.1283791670955126 (exp z))))
   (if (<= t_0 6.94725497403986e-310)
     (+ x (/ -1.0 x))
     (if (<= t_0 1.1299496288236466)
       (+ x (/ y (- t_0 (* x y))))
       (+ x (/ y t_0))))))
double code(double x, double y, double z) {
	return x + (y / ((1.1283791670955126 * exp(z)) - (x * y)));
}
↓
double code(double x, double y, double z) {
	double t_0 = 1.1283791670955126 * exp(z);
	double tmp;
	if (t_0 <= 6.94725497403986e-310) {
		tmp = x + (-1.0 / x);
	} else if (t_0 <= 1.1299496288236466) {
		tmp = x + (y / (t_0 - (x * y)));
	} else {
		tmp = x + (y / t_0);
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original2.8
Target0.0
Herbie0.1
\[x + \frac{1}{\frac{1.1283791670955126}{y} \cdot e^{z} - x} \]

Derivation

  1. Split input into 3 regimes

  2. if (*.f64 5081767996463981/4503599627370496 (exp.f64 z)) < 6.947254974039863e-310

    1. Initial program 8.2

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

    2. Simplified0.0

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

    3. Taylor expanded in y around inf 0.0

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

    if 6.947254974039863e-310 < (*.f64 5081767996463981/4503599627370496 (exp.f64 z)) < 1.12994962882364658

    1. Initial program 0.0

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

    if 1.12994962882364658 < (*.f64 5081767996463981/4503599627370496 (exp.f64 z))

    1. Initial program 3.2

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

    2. Taylor expanded in x around 0 0.3

      \[\leadsto x + \frac{y}{\color{blue}{1.1283791670955126 \cdot e^{z}}} \]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;1.1283791670955126 \cdot e^{z} \leq 6.94725497403986 \cdot 10^{-310}:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;1.1283791670955126 \cdot e^{z} \leq 1.1299496288236466:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 \cdot e^{z}}\\ \end{array} \]

Reproduce

herbie shell --seed 2021215 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:invErfc from math-functions-0.1.5.2, A"
  :precision binary64

  :herbie-target
  (+ x (/ 1.0 (- (* (/ 1.1283791670955126 y) (exp z)) x)))

  (+ x (/ y (- (* 1.1283791670955126 (exp z)) (* x y)))))