Average Error: 7.4 → 6.2
Time: 5.4s
Precision: 64
\[\frac{x + y}{1 - \frac{y}{z}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \le -3.3763009443507461 \cdot 10^{-298} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \le -0.0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}{\sqrt{1} + \frac{\sqrt{y}}{\sqrt{z}}} \cdot \frac{\sqrt[3]{x + y}}{\sqrt{1} - \frac{\sqrt{y}}{\sqrt{z}}}\\ \end{array}\]
\frac{x + y}{1 - \frac{y}{z}}
↓
\begin{array}{l}
\mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \le -3.3763009443507461 \cdot 10^{-298} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \le -0.0\right):\\
\;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}{\sqrt{1} + \frac{\sqrt{y}}{\sqrt{z}}} \cdot \frac{\sqrt[3]{x + y}}{\sqrt{1} - \frac{\sqrt{y}}{\sqrt{z}}}\\

\end{array}
double f(double x, double y, double z) {
        double r680509 = x;
        double r680510 = y;
        double r680511 = r680509 + r680510;
        double r680512 = 1.0;
        double r680513 = z;
        double r680514 = r680510 / r680513;
        double r680515 = r680512 - r680514;
        double r680516 = r680511 / r680515;
        return r680516;
}
↓
double f(double x, double y, double z) {
        double r680517 = x;
        double r680518 = y;
        double r680519 = r680517 + r680518;
        double r680520 = 1.0;
        double r680521 = z;
        double r680522 = r680518 / r680521;
        double r680523 = r680520 - r680522;
        double r680524 = r680519 / r680523;
        double r680525 = -3.376300944350746e-298;
        bool r680526 = r680524 <= r680525;
        double r680527 = -0.0;
        bool r680528 = r680524 <= r680527;
        double r680529 = !r680528;
        bool r680530 = r680526 || r680529;
        double r680531 = cbrt(r680519);
        double r680532 = r680531 * r680531;
        double r680533 = sqrt(r680520);
        double r680534 = sqrt(r680518);
        double r680535 = sqrt(r680521);
        double r680536 = r680534 / r680535;
        double r680537 = r680533 + r680536;
        double r680538 = r680532 / r680537;
        double r680539 = r680533 - r680536;
        double r680540 = r680531 / r680539;
        double r680541 = r680538 * r680540;
        double r680542 = r680530 ? r680524 : r680541;
        return r680542;
}

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

Original7.4
Target3.9
Herbie6.2
\[\begin{array}{l} \mathbf{if}\;y \lt -3.74293107626898565 \cdot 10^{171}:\\ \;\;\;\;\frac{y + x}{-y} \cdot z\\ \mathbf{elif}\;y \lt 3.55346624560867344 \cdot 10^{168}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y + x}{-y} \cdot z\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (+ x y) (- 1.0 (/ y z))) < -3.376300944350746e-298 or -0.0 < (/ (+ x y) (- 1.0 (/ y z)))

    1. Initial program 0.1

      \[\frac{x + y}{1 - \frac{y}{z}}\]

    if -3.376300944350746e-298 < (/ (+ x y) (- 1.0 (/ y z))) < -0.0

    1. Initial program 59.3

      \[\frac{x + y}{1 - \frac{y}{z}}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt61.3

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

    4. Applied add-sqr-sqrt62.6

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

    5. Applied times-frac62.6

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

    6. Applied add-sqr-sqrt62.6

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

    7. Applied difference-of-squares62.6

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

    8. Applied add-cube-cbrt62.6

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

    9. Applied times-frac49.5

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

  4. Final simplification6.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \le -3.3763009443507461 \cdot 10^{-298} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \le -0.0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}{\sqrt{1} + \frac{\sqrt{y}}{\sqrt{z}}} \cdot \frac{\sqrt[3]{x + y}}{\sqrt{1} - \frac{\sqrt{y}}{\sqrt{z}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 +o rules:numerics
(FPCore (x y z)
  :name "Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1, A"
  :precision binary64

  :herbie-target
  (if (< y -3.7429310762689856e+171) (* (/ (+ y x) (- y)) z) (if (< y 3.5534662456086734e+168) (/ (+ x y) (- 1 (/ y z))) (* (/ (+ y x) (- y)) z)))

  (/ (+ x y) (- 1 (/ y z))))