Average Error: 8.1 → 6.6
Time: 31.2s
Precision: 64
\[\frac{x + y}{1 - \frac{y}{z}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{y + x}{1 - \frac{y}{z}} \le -2.713922652192975851304047207930983359652 \cdot 10^{-295}:\\ \;\;\;\;\frac{y + x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;\frac{y + x}{1 - \frac{y}{z}} \le -0.0:\\ \;\;\;\;\frac{1}{\frac{\sqrt{1} - \frac{\sqrt{y}}{\sqrt{z}}}{\sqrt{y + x}} \cdot \frac{\sqrt{1} + \frac{\sqrt{y}}{\sqrt{z}}}{\sqrt{y + x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y + x}{1 - \frac{y}{z}}\\ \end{array}\]
\frac{x + y}{1 - \frac{y}{z}}
↓
\begin{array}{l}
\mathbf{if}\;\frac{y + x}{1 - \frac{y}{z}} \le -2.713922652192975851304047207930983359652 \cdot 10^{-295}:\\
\;\;\;\;\frac{y + x}{1 - \frac{y}{z}}\\

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

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

\end{array}
double f(double x, double y, double z) {
        double r26466347 = x;
        double r26466348 = y;
        double r26466349 = r26466347 + r26466348;
        double r26466350 = 1.0;
        double r26466351 = z;
        double r26466352 = r26466348 / r26466351;
        double r26466353 = r26466350 - r26466352;
        double r26466354 = r26466349 / r26466353;
        return r26466354;
}
↓
double f(double x, double y, double z) {
        double r26466355 = y;
        double r26466356 = x;
        double r26466357 = r26466355 + r26466356;
        double r26466358 = 1.0;
        double r26466359 = z;
        double r26466360 = r26466355 / r26466359;
        double r26466361 = r26466358 - r26466360;
        double r26466362 = r26466357 / r26466361;
        double r26466363 = -2.713922652192976e-295;
        bool r26466364 = r26466362 <= r26466363;
        double r26466365 = -0.0;
        bool r26466366 = r26466362 <= r26466365;
        double r26466367 = 1.0;
        double r26466368 = sqrt(r26466358);
        double r26466369 = sqrt(r26466355);
        double r26466370 = sqrt(r26466359);
        double r26466371 = r26466369 / r26466370;
        double r26466372 = r26466368 - r26466371;
        double r26466373 = sqrt(r26466357);
        double r26466374 = r26466372 / r26466373;
        double r26466375 = r26466368 + r26466371;
        double r26466376 = r26466375 / r26466373;
        double r26466377 = r26466374 * r26466376;
        double r26466378 = r26466367 / r26466377;
        double r26466379 = r26466366 ? r26466378 : r26466362;
        double r26466380 = r26466364 ? r26466362 : r26466379;
        return r26466380;
}

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

Original8.1
Target4.5
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))) < -2.713922652192976e-295 or -0.0 < (/ (+ x y) (- 1.0 (/ y z)))

    1. Initial program 0.1

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

    if -2.713922652192976e-295 < (/ (+ x y) (- 1.0 (/ y z))) < -0.0

    1. Initial program 59.6

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

    3. Applied clear-num59.7

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

    5. Applied add-sqr-sqrt61.9

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

    6. Applied add-sqr-sqrt62.8

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

    7. Applied add-sqr-sqrt62.9

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

    8. Applied times-frac62.9

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

    9. Applied add-sqr-sqrt62.9

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

    10. Applied difference-of-squares62.9

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

    11. Applied times-frac48.9

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

  4. Final simplification6.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y + x}{1 - \frac{y}{z}} \le -2.713922652192975851304047207930983359652 \cdot 10^{-295}:\\ \;\;\;\;\frac{y + x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;\frac{y + x}{1 - \frac{y}{z}} \le -0.0:\\ \;\;\;\;\frac{1}{\frac{\sqrt{1} - \frac{\sqrt{y}}{\sqrt{z}}}{\sqrt{y + x}} \cdot \frac{\sqrt{1} + \frac{\sqrt{y}}{\sqrt{z}}}{\sqrt{y + x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y + x}{1 - \frac{y}{z}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 +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))))