Average Error: 7.7 → 6.4
Time: 20.0s
Precision: 64
\[\frac{x + y}{1 - \frac{y}{z}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{y + x}{1 - \frac{y}{z}} \le -4.61291755031202367093527746970482296077 \cdot 10^{-270}:\\ \;\;\;\;\frac{y + x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;\frac{y + x}{1 - \frac{y}{z}} \le 0.0:\\ \;\;\;\;\frac{1}{\sqrt{1} + \frac{\sqrt{y}}{\sqrt{z}}} \cdot \left(\left(y + x\right) \cdot \frac{1}{\sqrt{1} - \frac{\sqrt{y}}{\sqrt{z}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y + x}{1 - \frac{y}{z}}\\ \end{array}\]
\frac{x + y}{1 - \frac{y}{z}}
↓
\begin{array}{l}
\mathbf{if}\;\frac{y + x}{1 - \frac{y}{z}} \le -4.61291755031202367093527746970482296077 \cdot 10^{-270}:\\
\;\;\;\;\frac{y + x}{1 - \frac{y}{z}}\\

\mathbf{elif}\;\frac{y + x}{1 - \frac{y}{z}} \le 0.0:\\
\;\;\;\;\frac{1}{\sqrt{1} + \frac{\sqrt{y}}{\sqrt{z}}} \cdot \left(\left(y + x\right) \cdot \frac{1}{\sqrt{1} - \frac{\sqrt{y}}{\sqrt{z}}}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{y + x}{1 - \frac{y}{z}}\\

\end{array}
double f(double x, double y, double z) {
        double r20456916 = x;
        double r20456917 = y;
        double r20456918 = r20456916 + r20456917;
        double r20456919 = 1.0;
        double r20456920 = z;
        double r20456921 = r20456917 / r20456920;
        double r20456922 = r20456919 - r20456921;
        double r20456923 = r20456918 / r20456922;
        return r20456923;
}
↓
double f(double x, double y, double z) {
        double r20456924 = y;
        double r20456925 = x;
        double r20456926 = r20456924 + r20456925;
        double r20456927 = 1.0;
        double r20456928 = z;
        double r20456929 = r20456924 / r20456928;
        double r20456930 = r20456927 - r20456929;
        double r20456931 = r20456926 / r20456930;
        double r20456932 = -4.612917550312024e-270;
        bool r20456933 = r20456931 <= r20456932;
        double r20456934 = 0.0;
        bool r20456935 = r20456931 <= r20456934;
        double r20456936 = 1.0;
        double r20456937 = sqrt(r20456927);
        double r20456938 = sqrt(r20456924);
        double r20456939 = sqrt(r20456928);
        double r20456940 = r20456938 / r20456939;
        double r20456941 = r20456937 + r20456940;
        double r20456942 = r20456936 / r20456941;
        double r20456943 = r20456937 - r20456940;
        double r20456944 = r20456936 / r20456943;
        double r20456945 = r20456926 * r20456944;
        double r20456946 = r20456942 * r20456945;
        double r20456947 = r20456935 ? r20456946 : r20456931;
        double r20456948 = r20456933 ? r20456931 : r20456947;
        return r20456948;
}

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

Original7.7
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))) < -4.612917550312024e-270 or 0.0 < (/ (+ x y) (- 1.0 (/ y z)))

    1. Initial program 0.1

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

    if -4.612917550312024e-270 < (/ (+ x y) (- 1.0 (/ y z))) < 0.0

    1. Initial program 57.1

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

    3. Applied clear-num57.1

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

    5. Applied div-inv57.1

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

    6. Applied associate-/r*57.1

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

    8. Applied *-un-lft-identity57.1

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

    9. Applied add-cube-cbrt57.1

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

    10. Applied times-frac57.1

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

    11. Applied add-sqr-sqrt59.5

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

    12. Applied add-sqr-sqrt61.9

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

    13. Applied times-frac61.9

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

    14. Applied add-sqr-sqrt61.9

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

    15. Applied difference-of-squares61.9

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

    16. Applied add-cube-cbrt61.9

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

    17. Applied times-frac61.2

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

    18. Applied times-frac47.7

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

    19. Simplified47.7

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

    20. Simplified47.7

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

  4. Final simplification6.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y + x}{1 - \frac{y}{z}} \le -4.61291755031202367093527746970482296077 \cdot 10^{-270}:\\ \;\;\;\;\frac{y + x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;\frac{y + x}{1 - \frac{y}{z}} \le 0.0:\\ \;\;\;\;\frac{1}{\sqrt{1} + \frac{\sqrt{y}}{\sqrt{z}}} \cdot \left(\left(y + x\right) \cdot \frac{1}{\sqrt{1} - \frac{\sqrt{y}}{\sqrt{z}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y + x}{1 - \frac{y}{z}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(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))))