Average Error: 7.2 → 6.6
Time: 18.3s
Precision: 64
\[\frac{x + y}{1 - \frac{y}{z}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \le -1.194944994013871437284920410941093036324 \cdot 10^{-247} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \le -0.0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{x + y}}{\sqrt{1} + \frac{\sqrt{y}}{\sqrt{z}}} \cdot \frac{\sqrt{x + y}}{\sqrt{1} - \frac{\sqrt{y}}{\sqrt{z}}}\\ \end{array}\]
\frac{x + y}{1 - \frac{y}{z}}
↓
\begin{array}{l}
\mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \le -1.194944994013871437284920410941093036324 \cdot 10^{-247} \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{x + y}}{\sqrt{1} + \frac{\sqrt{y}}{\sqrt{z}}} \cdot \frac{\sqrt{x + y}}{\sqrt{1} - \frac{\sqrt{y}}{\sqrt{z}}}\\

\end{array}
double f(double x, double y, double z) {
        double r442287 = x;
        double r442288 = y;
        double r442289 = r442287 + r442288;
        double r442290 = 1.0;
        double r442291 = z;
        double r442292 = r442288 / r442291;
        double r442293 = r442290 - r442292;
        double r442294 = r442289 / r442293;
        return r442294;
}
↓
double f(double x, double y, double z) {
        double r442295 = x;
        double r442296 = y;
        double r442297 = r442295 + r442296;
        double r442298 = 1.0;
        double r442299 = z;
        double r442300 = r442296 / r442299;
        double r442301 = r442298 - r442300;
        double r442302 = r442297 / r442301;
        double r442303 = -1.1949449940138714e-247;
        bool r442304 = r442302 <= r442303;
        double r442305 = -0.0;
        bool r442306 = r442302 <= r442305;
        double r442307 = !r442306;
        bool r442308 = r442304 || r442307;
        double r442309 = sqrt(r442297);
        double r442310 = sqrt(r442298);
        double r442311 = sqrt(r442296);
        double r442312 = sqrt(r442299);
        double r442313 = r442311 / r442312;
        double r442314 = r442310 + r442313;
        double r442315 = r442309 / r442314;
        double r442316 = r442310 - r442313;
        double r442317 = r442309 / r442316;
        double r442318 = r442315 * r442317;
        double r442319 = r442308 ? r442302 : r442318;
        return r442319;
}

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

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

Derivation

  1. Split input into 2 regimes

  2. if (/ (+ x y) (- 1.0 (/ y z))) < -1.1949449940138714e-247 or -0.0 < (/ (+ x y) (- 1.0 (/ y z)))

    1. Initial program 3.8

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

    if -1.1949449940138714e-247 < (/ (+ x y) (- 1.0 (/ y z))) < -0.0

    1. Initial program 48.9

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

    3. Applied add-sqr-sqrt52.4

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

    4. Applied add-sqr-sqrt59.0

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

    5. Applied times-frac59.0

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

    6. Applied add-sqr-sqrt59.0

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

    7. Applied difference-of-squares59.0

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

    8. Applied add-sqr-sqrt59.4

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

    9. Applied times-frac38.0

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

  4. Final simplification6.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \le -1.194944994013871437284920410941093036324 \cdot 10^{-247} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \le -0.0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{x + y}}{\sqrt{1} + \frac{\sqrt{y}}{\sqrt{z}}} \cdot \frac{\sqrt{x + y}}{\sqrt{1} - \frac{\sqrt{y}}{\sqrt{z}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 +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))))