Average Error: 2.8 → 1.1
Time: 15.9s
Precision: 64
\[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;x + \frac{y}{e^{z} \cdot 1.1283791670955126 - x \cdot y} \le 5.241345650251777 \cdot 10^{+241}:\\ \;\;\;\;x + \frac{y}{e^{z} \cdot 1.1283791670955126 - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{1}{x}\\ \end{array}\]
x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y}
↓
\begin{array}{l}
\mathbf{if}\;x + \frac{y}{e^{z} \cdot 1.1283791670955126 - x \cdot y} \le 5.241345650251777 \cdot 10^{+241}:\\
\;\;\;\;x + \frac{y}{e^{z} \cdot 1.1283791670955126 - x \cdot y}\\

\mathbf{else}:\\
\;\;\;\;x - \frac{1}{x}\\

\end{array}
double f(double x, double y, double z) {
        double r15937317 = x;
        double r15937318 = y;
        double r15937319 = 1.1283791670955126;
        double r15937320 = z;
        double r15937321 = exp(r15937320);
        double r15937322 = r15937319 * r15937321;
        double r15937323 = r15937317 * r15937318;
        double r15937324 = r15937322 - r15937323;
        double r15937325 = r15937318 / r15937324;
        double r15937326 = r15937317 + r15937325;
        return r15937326;
}
↓
double f(double x, double y, double z) {
        double r15937327 = x;
        double r15937328 = y;
        double r15937329 = z;
        double r15937330 = exp(r15937329);
        double r15937331 = 1.1283791670955126;
        double r15937332 = r15937330 * r15937331;
        double r15937333 = r15937327 * r15937328;
        double r15937334 = r15937332 - r15937333;
        double r15937335 = r15937328 / r15937334;
        double r15937336 = r15937327 + r15937335;
        double r15937337 = 5.241345650251777e+241;
        bool r15937338 = r15937336 <= r15937337;
        double r15937339 = 1.0;
        double r15937340 = r15937339 / r15937327;
        double r15937341 = r15937327 - r15937340;
        double r15937342 = r15937338 ? r15937336 : r15937341;
        return r15937342;
}

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
Herbie1.1
\[x + \frac{1}{\frac{1.1283791670955126}{y} \cdot e^{z} - x}\]

Derivation

  1. Split input into 2 regimes

  2. if (+ x (/ y (- (* 1.1283791670955126 (exp z)) (* x y)))) < 5.241345650251777e+241

    1. Initial program 1.1

      \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y}\]
    2. Using strategy rm

    3. Applied div-inv1.3

      \[\leadsto x + \color{blue}{y \cdot \frac{1}{1.1283791670955126 \cdot e^{z} - x \cdot y}}\]
    4. Using strategy rm

    5. Applied un-div-inv1.1

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

    if 5.241345650251777e+241 < (+ x (/ y (- (* 1.1283791670955126 (exp z)) (* x y))))

    1. Initial program 19.3

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

    2. Taylor expanded around inf 0.8

      \[\leadsto \color{blue}{x - \frac{1}{x}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{y}{e^{z} \cdot 1.1283791670955126 - x \cdot y} \le 5.241345650251777 \cdot 10^{+241}:\\ \;\;\;\;x + \frac{y}{e^{z} \cdot 1.1283791670955126 - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{1}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (x y z)
  :name "Numeric.SpecFunctions:invErfc from math-functions-0.1.5.2, A"

  :herbie-target
  (+ x (/ 1 (- (* (/ 1.1283791670955126 y) (exp z)) x)))

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