Average Error: 7.5 → 6.2
Time: 16.1s
Precision: 64
\[\frac{x + y}{1 - \frac{y}{z}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \le -1.801559318634141379499295463502958423038 \cdot 10^{-285} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \le -0.0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{1} + \frac{\sqrt{y}}{\sqrt{z}}}{\frac{x + y}{\sqrt{1} - \frac{\sqrt{y}}{\sqrt{z}}}}}\\ \end{array}\]
\frac{x + y}{1 - \frac{y}{z}}
↓
\begin{array}{l}
\mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \le -1.801559318634141379499295463502958423038 \cdot 10^{-285} \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 r859947 = x;
        double r859948 = y;
        double r859949 = r859947 + r859948;
        double r859950 = 1.0;
        double r859951 = z;
        double r859952 = r859948 / r859951;
        double r859953 = r859950 - r859952;
        double r859954 = r859949 / r859953;
        return r859954;
}
↓
double f(double x, double y, double z) {
        double r859955 = x;
        double r859956 = y;
        double r859957 = r859955 + r859956;
        double r859958 = 1.0;
        double r859959 = z;
        double r859960 = r859956 / r859959;
        double r859961 = r859958 - r859960;
        double r859962 = r859957 / r859961;
        double r859963 = -1.8015593186341414e-285;
        bool r859964 = r859962 <= r859963;
        double r859965 = -0.0;
        bool r859966 = r859962 <= r859965;
        double r859967 = !r859966;
        bool r859968 = r859964 || r859967;
        double r859969 = 1.0;
        double r859970 = sqrt(r859958);
        double r859971 = sqrt(r859956);
        double r859972 = sqrt(r859959);
        double r859973 = r859971 / r859972;
        double r859974 = r859970 + r859973;
        double r859975 = r859970 - r859973;
        double r859976 = r859957 / r859975;
        double r859977 = r859974 / r859976;
        double r859978 = r859969 / r859977;
        double r859979 = r859968 ? r859962 : r859978;
        return r859979;
}

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

Original7.5
Target4.1
Herbie6.2
\[\begin{array}{l} \mathbf{if}\;y \lt -3.742931076268985646434612946949172132145 \cdot 10^{171}:\\ \;\;\;\;\frac{y + x}{-y} \cdot z\\ \mathbf{elif}\;y \lt 3.553466245608673435460441960303815115662 \cdot 10^{168}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y + x}{-y} \cdot z\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (+ x y) (- 1.0 (/ y z))) < -1.8015593186341414e-285 or -0.0 < (/ (+ x y) (- 1.0 (/ y z)))

    1. Initial program 4.3

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

    if -1.8015593186341414e-285 < (/ (+ x y) (- 1.0 (/ y z))) < -0.0

    1. Initial program 56.0

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

    3. Applied clear-num56.0

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

    5. Applied add-sqr-sqrt56.7

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

    6. Applied add-sqr-sqrt60.8

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

    7. Applied times-frac60.8

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

    8. Applied add-sqr-sqrt60.8

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

    9. Applied difference-of-squares60.8

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

    10. Applied associate-/l*34.5

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

  4. Final simplification6.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \le -1.801559318634141379499295463502958423038 \cdot 10^{-285} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \le -0.0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{1} + \frac{\sqrt{y}}{\sqrt{z}}}{\frac{x + y}{\sqrt{1} - \frac{\sqrt{y}}{\sqrt{z}}}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019305 +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 -3.74293107626898565e171) (* (/ (+ y x) (- y)) z) (if (< y 3.55346624560867344e168) (/ (+ x y) (- 1 (/ y z))) (* (/ (+ y x) (- y)) z)))

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