Average Error: 7.4 → 6.3
Time: 13.4s
Precision: 64
\[\frac{x + y}{1 - \frac{y}{z}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{y + x}{1 - \frac{y}{z}} \le -3.397985061883707692793319267642426272456 \cdot 10^{-286} \lor \neg \left(\frac{y + x}{1 - \frac{y}{z}} \le -0.0\right):\\ \;\;\;\;\frac{y + x}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{1} - \frac{\sqrt{y}}{\sqrt{z}}}{\sqrt[3]{y + x}} \cdot \frac{\frac{\sqrt{y}}{\sqrt{z}} + \sqrt{1}}{\sqrt[3]{y + x} \cdot \sqrt[3]{y + x}}}\\ \end{array}\]
\frac{x + y}{1 - \frac{y}{z}}
↓
\begin{array}{l}
\mathbf{if}\;\frac{y + x}{1 - \frac{y}{z}} \le -3.397985061883707692793319267642426272456 \cdot 10^{-286} \lor \neg \left(\frac{y + x}{1 - \frac{y}{z}} \le -0.0\right):\\
\;\;\;\;\frac{y + x}{1 - \frac{y}{z}}\\

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

\end{array}
double f(double x, double y, double z) {
        double r497536 = x;
        double r497537 = y;
        double r497538 = r497536 + r497537;
        double r497539 = 1.0;
        double r497540 = z;
        double r497541 = r497537 / r497540;
        double r497542 = r497539 - r497541;
        double r497543 = r497538 / r497542;
        return r497543;
}
↓
double f(double x, double y, double z) {
        double r497544 = y;
        double r497545 = x;
        double r497546 = r497544 + r497545;
        double r497547 = 1.0;
        double r497548 = z;
        double r497549 = r497544 / r497548;
        double r497550 = r497547 - r497549;
        double r497551 = r497546 / r497550;
        double r497552 = -3.3979850618837077e-286;
        bool r497553 = r497551 <= r497552;
        double r497554 = -0.0;
        bool r497555 = r497551 <= r497554;
        double r497556 = !r497555;
        bool r497557 = r497553 || r497556;
        double r497558 = 1.0;
        double r497559 = sqrt(r497547);
        double r497560 = sqrt(r497544);
        double r497561 = sqrt(r497548);
        double r497562 = r497560 / r497561;
        double r497563 = r497559 - r497562;
        double r497564 = cbrt(r497546);
        double r497565 = r497563 / r497564;
        double r497566 = r497562 + r497559;
        double r497567 = r497564 * r497564;
        double r497568 = r497566 / r497567;
        double r497569 = r497565 * r497568;
        double r497570 = r497558 / r497569;
        double r497571 = r497557 ? r497551 : r497570;
        return r497571;
}

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
Herbie6.3
\[\begin{array}{l} \mathbf{if}\;y \lt -3.742931076268985646434612946949172132145 \cdot 10^{171}:\\ \;\;\;\;\frac{y + x}{-y} \cdot z\\ \mathbf{elif}\;y \lt 3.553466245608673435460441960303815115662 \cdot 10^{168}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y + x}{-y} \cdot z\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (+ x y) (- 1.0 (/ y z))) < -3.3979850618837077e-286 or -0.0 < (/ (+ x y) (- 1.0 (/ y z)))

    1. Initial program 4.1

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

    if -3.3979850618837077e-286 < (/ (+ x y) (- 1.0 (/ y z))) < -0.0

    1. Initial program 56.3

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

    3. Applied clear-num56.3

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

    4. Simplified56.3

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

    6. Applied add-cube-cbrt56.4

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

    7. Applied add-sqr-sqrt57.2

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

    8. Applied add-sqr-sqrt60.7

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

    9. Applied times-frac60.7

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

    10. Applied add-sqr-sqrt60.7

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

    11. Applied difference-of-squares60.7

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

    12. Applied times-frac37.8

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

    13. Simplified37.8

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

  4. Final simplification6.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y + x}{1 - \frac{y}{z}} \le -3.397985061883707692793319267642426272456 \cdot 10^{-286} \lor \neg \left(\frac{y + x}{1 - \frac{y}{z}} \le -0.0\right):\\ \;\;\;\;\frac{y + x}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{1} - \frac{\sqrt{y}}{\sqrt{z}}}{\sqrt[3]{y + x}} \cdot \frac{\frac{\sqrt{y}}{\sqrt{z}} + \sqrt{1}}{\sqrt[3]{y + x} \cdot \sqrt[3]{y + x}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019195 
(FPCore (x y z)
  :name "Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1, A"

  :herbie-target
  (if (< y -3.7429310762689856e+171) (* (/ (+ y x) (- y)) z) (if (< y 3.5534662456086734e+168) (/ (+ x y) (- 1.0 (/ y z))) (* (/ (+ y x) (- y)) z)))

  (/ (+ x y) (- 1.0 (/ y z))))