Average Error: 2.9 → 0.5
Time: 9.8s
Precision: 64
\[x + \frac{y}{1.12837916709551256 \cdot e^{z} - x \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -744.98628896478272 \lor \neg \left(z \le 1.00942494343838114 \cdot 10^{-88}\right):\\ \;\;\;\;x + \frac{1}{\frac{1.12837916709551256}{\sqrt[3]{{\left(\frac{y}{e^{z}}\right)}^{3}}} - x}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{1.12837916709551256 \cdot e^{z} - x \cdot y}\\ \end{array}\]
x + \frac{y}{1.12837916709551256 \cdot e^{z} - x \cdot y}
↓
\begin{array}{l}
\mathbf{if}\;z \le -744.98628896478272 \lor \neg \left(z \le 1.00942494343838114 \cdot 10^{-88}\right):\\
\;\;\;\;x + \frac{1}{\frac{1.12837916709551256}{\sqrt[3]{{\left(\frac{y}{e^{z}}\right)}^{3}}} - x}\\

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

\end{array}
double f(double x, double y, double z) {
        double r487260 = x;
        double r487261 = y;
        double r487262 = 1.1283791670955126;
        double r487263 = z;
        double r487264 = exp(r487263);
        double r487265 = r487262 * r487264;
        double r487266 = r487260 * r487261;
        double r487267 = r487265 - r487266;
        double r487268 = r487261 / r487267;
        double r487269 = r487260 + r487268;
        return r487269;
}
↓
double f(double x, double y, double z) {
        double r487270 = z;
        double r487271 = -744.9862889647827;
        bool r487272 = r487270 <= r487271;
        double r487273 = 1.0094249434383811e-88;
        bool r487274 = r487270 <= r487273;
        double r487275 = !r487274;
        bool r487276 = r487272 || r487275;
        double r487277 = x;
        double r487278 = 1.0;
        double r487279 = 1.1283791670955126;
        double r487280 = y;
        double r487281 = exp(r487270);
        double r487282 = r487280 / r487281;
        double r487283 = 3.0;
        double r487284 = pow(r487282, r487283);
        double r487285 = cbrt(r487284);
        double r487286 = r487279 / r487285;
        double r487287 = r487286 - r487277;
        double r487288 = r487278 / r487287;
        double r487289 = r487277 + r487288;
        double r487290 = r487279 * r487281;
        double r487291 = r487277 * r487280;
        double r487292 = r487290 - r487291;
        double r487293 = r487280 / r487292;
        double r487294 = r487277 + r487293;
        double r487295 = r487276 ? r487289 : r487294;
        return r487295;
}

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

Original2.9
Target0.0
Herbie0.5
\[x + \frac{1}{\frac{1.12837916709551256}{y} \cdot e^{z} - x}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -744.9862889647827 or 1.0094249434383811e-88 < z

    1. Initial program 5.0

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

    3. Applied clear-num5.0

      \[\leadsto x + \color{blue}{\frac{1}{\frac{1.12837916709551256 \cdot e^{z} - x \cdot y}{y}}}\]
    4. Using strategy rm

    5. Applied div-sub5.0

      \[\leadsto x + \frac{1}{\color{blue}{\frac{1.12837916709551256 \cdot e^{z}}{y} - \frac{x \cdot y}{y}}}\]

    6. Simplified5.0

      \[\leadsto x + \frac{1}{\color{blue}{\frac{1.12837916709551256}{\frac{y}{e^{z}}}} - \frac{x \cdot y}{y}}\]

    7. Simplified0.0

      \[\leadsto x + \frac{1}{\frac{1.12837916709551256}{\frac{y}{e^{z}}} - \color{blue}{x}}\]
    8. Using strategy rm

    9. Applied add-cbrt-cube0.0

      \[\leadsto x + \frac{1}{\frac{1.12837916709551256}{\frac{y}{\color{blue}{\sqrt[3]{\left(e^{z} \cdot e^{z}\right) \cdot e^{z}}}}} - x}\]

    10. Applied add-cbrt-cube18.7

      \[\leadsto x + \frac{1}{\frac{1.12837916709551256}{\frac{\color{blue}{\sqrt[3]{\left(y \cdot y\right) \cdot y}}}{\sqrt[3]{\left(e^{z} \cdot e^{z}\right) \cdot e^{z}}}} - x}\]

    11. Applied cbrt-undiv18.7

      \[\leadsto x + \frac{1}{\frac{1.12837916709551256}{\color{blue}{\sqrt[3]{\frac{\left(y \cdot y\right) \cdot y}{\left(e^{z} \cdot e^{z}\right) \cdot e^{z}}}}} - x}\]

    12. Simplified0.9

      \[\leadsto x + \frac{1}{\frac{1.12837916709551256}{\sqrt[3]{\color{blue}{{\left(\frac{y}{e^{z}}\right)}^{3}}}} - x}\]

    if -744.9862889647827 < z < 1.0094249434383811e-88

    1. Initial program 0.0

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -744.98628896478272 \lor \neg \left(z \le 1.00942494343838114 \cdot 10^{-88}\right):\\ \;\;\;\;x + \frac{1}{\frac{1.12837916709551256}{\sqrt[3]{{\left(\frac{y}{e^{z}}\right)}^{3}}} - x}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{1.12837916709551256 \cdot e^{z} - x \cdot y}\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:invErfc from math-functions-0.1.5.2, A"
  :precision binary64

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

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