Average Error: 7.6 → 6.4
Time: 17.3s
Precision: 64
\[\frac{x + y}{1 - \frac{y}{z}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \le -2.869692918571675329109593574597536633393 \cdot 10^{-291} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \le 0.0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{y + x}}{\sqrt{1} + \frac{\sqrt{y}}{\sqrt{z}}} \cdot \frac{\sqrt{y + x}}{\sqrt{1} - \frac{\sqrt{y}}{\sqrt{z}}}\\ \end{array}\]
\frac{x + y}{1 - \frac{y}{z}}
↓
\begin{array}{l}
\mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \le -2.869692918571675329109593574597536633393 \cdot 10^{-291} \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{y + x}}{\sqrt{1} + \frac{\sqrt{y}}{\sqrt{z}}} \cdot \frac{\sqrt{y + x}}{\sqrt{1} - \frac{\sqrt{y}}{\sqrt{z}}}\\

\end{array}
double f(double x, double y, double z) {
        double r409999 = x;
        double r410000 = y;
        double r410001 = r409999 + r410000;
        double r410002 = 1.0;
        double r410003 = z;
        double r410004 = r410000 / r410003;
        double r410005 = r410002 - r410004;
        double r410006 = r410001 / r410005;
        return r410006;
}
↓
double f(double x, double y, double z) {
        double r410007 = x;
        double r410008 = y;
        double r410009 = r410007 + r410008;
        double r410010 = 1.0;
        double r410011 = z;
        double r410012 = r410008 / r410011;
        double r410013 = r410010 - r410012;
        double r410014 = r410009 / r410013;
        double r410015 = -2.8696929185716753e-291;
        bool r410016 = r410014 <= r410015;
        double r410017 = 0.0;
        bool r410018 = r410014 <= r410017;
        double r410019 = !r410018;
        bool r410020 = r410016 || r410019;
        double r410021 = r410008 + r410007;
        double r410022 = sqrt(r410021);
        double r410023 = sqrt(r410010);
        double r410024 = sqrt(r410008);
        double r410025 = sqrt(r410011);
        double r410026 = r410024 / r410025;
        double r410027 = r410023 + r410026;
        double r410028 = r410022 / r410027;
        double r410029 = r410023 - r410026;
        double r410030 = r410022 / r410029;
        double r410031 = r410028 * r410030;
        double r410032 = r410020 ? r410014 : r410031;
        return r410032;
}

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

Original7.6
Target4.2
Herbie6.4
\[\begin{array}{l} \mathbf{if}\;y \lt -3.742931076268985646434612946949172132145 \cdot 10^{171}:\\ \;\;\;\;\frac{y + x}{-y} \cdot z\\ \mathbf{elif}\;y \lt 3.553466245608673435460441960303815115662 \cdot 10^{168}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y + x}{-y} \cdot z\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (+ x y) (- 1.0 (/ y z))) < -2.8696929185716753e-291 or 0.0 < (/ (+ x y) (- 1.0 (/ y z)))

    1. Initial program 4.1

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

    if -2.8696929185716753e-291 < (/ (+ x y) (- 1.0 (/ y z))) < 0.0

    1. Initial program 58.6

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

    3. Applied clear-num58.6

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

    5. Applied add-sqr-sqrt61.5

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

    6. Applied add-sqr-sqrt63.3

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

    7. Applied add-sqr-sqrt63.4

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

    8. Applied times-frac63.4

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

    9. Applied add-sqr-sqrt63.4

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

    10. Applied difference-of-squares63.4

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

    11. Applied times-frac63.4

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

    12. Applied add-cube-cbrt63.4

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

    13. Applied times-frac63.4

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

    14. Simplified63.4

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

    15. Simplified63.4

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

  4. Final simplification6.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \le -2.869692918571675329109593574597536633393 \cdot 10^{-291} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \le 0.0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{y + x}}{\sqrt{1} + \frac{\sqrt{y}}{\sqrt{z}}} \cdot \frac{\sqrt{y + x}}{\sqrt{1} - \frac{\sqrt{y}}{\sqrt{z}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 +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))))