Average Error: 7.7 → 6.4
Time: 10.8s
Precision: 64
\[\frac{x + y}{1 - \frac{y}{z}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \le -7.66629167620117969354071927698147699138 \cdot 10^{-261} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \le -0.0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{1} + \frac{\sqrt{y}}{\sqrt{z}}}{\frac{x + y}{\sqrt{1} - \frac{\sqrt{y}}{\sqrt{z}}}}}\\ \end{array}\]
\frac{x + y}{1 - \frac{y}{z}}
↓
\begin{array}{l}
\mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \le -7.66629167620117969354071927698147699138 \cdot 10^{-261} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \le -0.0\right):\\
\;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\

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

\end{array}
double f(double x, double y, double z) {
        double r490868 = x;
        double r490869 = y;
        double r490870 = r490868 + r490869;
        double r490871 = 1.0;
        double r490872 = z;
        double r490873 = r490869 / r490872;
        double r490874 = r490871 - r490873;
        double r490875 = r490870 / r490874;
        return r490875;
}
↓
double f(double x, double y, double z) {
        double r490876 = x;
        double r490877 = y;
        double r490878 = r490876 + r490877;
        double r490879 = 1.0;
        double r490880 = z;
        double r490881 = r490877 / r490880;
        double r490882 = r490879 - r490881;
        double r490883 = r490878 / r490882;
        double r490884 = -7.66629167620118e-261;
        bool r490885 = r490883 <= r490884;
        double r490886 = -0.0;
        bool r490887 = r490883 <= r490886;
        double r490888 = !r490887;
        bool r490889 = r490885 || r490888;
        double r490890 = 1.0;
        double r490891 = sqrt(r490879);
        double r490892 = sqrt(r490877);
        double r490893 = sqrt(r490880);
        double r490894 = r490892 / r490893;
        double r490895 = r490891 + r490894;
        double r490896 = r490891 - r490894;
        double r490897 = r490878 / r490896;
        double r490898 = r490895 / r490897;
        double r490899 = r490890 / r490898;
        double r490900 = r490889 ? r490883 : r490899;
        return r490900;
}

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
Target4.1
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))) < -7.66629167620118e-261 or -0.0 < (/ (+ x y) (- 1.0 (/ y z)))

    1. Initial program 3.7

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

    if -7.66629167620118e-261 < (/ (+ x y) (- 1.0 (/ y z))) < -0.0

    1. Initial program 53.7

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

    3. Applied clear-num53.7

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

    5. Applied add-sqr-sqrt55.2

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

    6. Applied add-sqr-sqrt59.9

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

    7. Applied times-frac59.9

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

    8. Applied add-sqr-sqrt59.9

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

    9. Applied difference-of-squares59.9

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

    10. Applied associate-/l*36.7

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

  4. Final simplification6.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \le -7.66629167620117969354071927698147699138 \cdot 10^{-261} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \le -0.0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{1} + \frac{\sqrt{y}}{\sqrt{z}}}{\frac{x + y}{\sqrt{1} - \frac{\sqrt{y}}{\sqrt{z}}}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019212 
(FPCore (x y z)
  :name "Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1, A"
  :precision binary64

  :herbie-target
  (if (< y -3.74293107626898565e171) (* (/ (+ y x) (- y)) z) (if (< y 3.55346624560867344e168) (/ (+ x y) (- 1 (/ y z))) (* (/ (+ y x) (- y)) z)))

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