Average Error: 2.6 → 0.9
Time: 3.3s
Precision: binary64
\[x + \frac{y}{1.12837916709551256 \cdot e^{z} - x \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{z} \le 4.8589319155281618 \cdot 10^{-309}:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{1}{1.12837916709551256 \cdot e^{z} - x \cdot y}\\ \end{array}\]
x + \frac{y}{1.12837916709551256 \cdot e^{z} - x \cdot y}
↓
\begin{array}{l}
\mathbf{if}\;e^{z} \le 4.8589319155281618 \cdot 10^{-309}:\\
\;\;\;\;x - \frac{1}{x}\\

\mathbf{else}:\\
\;\;\;\;x + y \cdot \frac{1}{1.12837916709551256 \cdot e^{z} - x \cdot y}\\

\end{array}
double code(double x, double y, double z) {
	return ((double) (x + ((double) (y / ((double) (((double) (1.1283791670955126 * ((double) exp(z)))) - ((double) (x * y))))))));
}
↓
double code(double x, double y, double z) {
	double VAR;
	if ((((double) exp(z)) <= 4.85893191552816e-309)) {
		VAR = ((double) (x - ((double) (1.0 / x))));
	} else {
		VAR = ((double) (x + ((double) (y * ((double) (1.0 / ((double) (((double) (1.1283791670955126 * ((double) exp(z)))) - ((double) (x * y))))))))));
	}
	return VAR;
}

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

Original2.6
Target0.0
Herbie0.9
\[x + \frac{1}{\frac{1.12837916709551256}{y} \cdot e^{z} - x}\]

Derivation

  1. Split input into 2 regimes

  2. if (exp z) < 4.8589319155281618e-309

    1. Initial program 7.0

      \[x + \frac{y}{1.12837916709551256 \cdot e^{z} - x \cdot y}\]

    2. Taylor expanded around inf 0.0

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

    if 4.8589319155281618e-309 < (exp z)

    1. Initial program 1.1

      \[x + \frac{y}{1.12837916709551256 \cdot e^{z} - x \cdot y}\]
    2. Using strategy rm

    3. Applied div-inv1.1

      \[\leadsto x + \color{blue}{y \cdot \frac{1}{1.12837916709551256 \cdot e^{z} - x \cdot y}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \le 4.8589319155281618 \cdot 10^{-309}:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{1}{1.12837916709551256 \cdot e^{z} - x \cdot y}\\ \end{array}\]

Reproduce

herbie shell --seed 2020147 
(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)))))