Average Error: 7.5 → 6.4
Time: 3.2s
Precision: 64
\[\frac{x + y}{1 - \frac{y}{z}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \le -1.4563325680088959 \cdot 10^{-254} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \le 0.0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\sqrt{1} + \frac{\sqrt{y}}{\sqrt{z}}\right) \cdot \frac{\sqrt{1} - \frac{\sqrt{y}}{\sqrt{z}}}{x + y}}\\ \end{array}\]
\frac{x + y}{1 - \frac{y}{z}}
↓
\begin{array}{l}
\mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \le -1.4563325680088959 \cdot 10^{-254} \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}{\left(\sqrt{1} + \frac{\sqrt{y}}{\sqrt{z}}\right) \cdot \frac{\sqrt{1} - \frac{\sqrt{y}}{\sqrt{z}}}{x + y}}\\

\end{array}
double f(double x, double y, double z) {
        double r747523 = x;
        double r747524 = y;
        double r747525 = r747523 + r747524;
        double r747526 = 1.0;
        double r747527 = z;
        double r747528 = r747524 / r747527;
        double r747529 = r747526 - r747528;
        double r747530 = r747525 / r747529;
        return r747530;
}
↓
double f(double x, double y, double z) {
        double r747531 = x;
        double r747532 = y;
        double r747533 = r747531 + r747532;
        double r747534 = 1.0;
        double r747535 = z;
        double r747536 = r747532 / r747535;
        double r747537 = r747534 - r747536;
        double r747538 = r747533 / r747537;
        double r747539 = -1.456332568008896e-254;
        bool r747540 = r747538 <= r747539;
        double r747541 = 0.0;
        bool r747542 = r747538 <= r747541;
        double r747543 = !r747542;
        bool r747544 = r747540 || r747543;
        double r747545 = 1.0;
        double r747546 = sqrt(r747534);
        double r747547 = sqrt(r747532);
        double r747548 = sqrt(r747535);
        double r747549 = r747547 / r747548;
        double r747550 = r747546 + r747549;
        double r747551 = r747546 - r747549;
        double r747552 = r747551 / r747533;
        double r747553 = r747550 * r747552;
        double r747554 = r747545 / r747553;
        double r747555 = r747544 ? r747538 : r747554;
        return r747555;
}

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
Target3.8
Herbie6.4
\[\begin{array}{l} \mathbf{if}\;y \lt -3.74293107626898565 \cdot 10^{171}:\\ \;\;\;\;\frac{y + x}{-y} \cdot z\\ \mathbf{elif}\;y \lt 3.55346624560867344 \cdot 10^{168}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y + x}{-y} \cdot z\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (+ x y) (- 1.0 (/ y z))) < -1.456332568008896e-254 or 0.0 < (/ (+ x y) (- 1.0 (/ y z)))

    1. Initial program 0.1

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

    if -1.456332568008896e-254 < (/ (+ x y) (- 1.0 (/ y z))) < 0.0

    1. Initial program 55.0

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

    3. Applied clear-num55.1

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

    5. Applied *-un-lft-identity55.1

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

    6. Applied add-sqr-sqrt57.9

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

    7. Applied add-sqr-sqrt61.2

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

    8. Applied times-frac61.2

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

    9. Applied add-sqr-sqrt61.2

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

    10. Applied difference-of-squares61.2

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

    11. Applied times-frac47.0

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

    12. Simplified47.0

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

  4. Final simplification6.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \le -1.4563325680088959 \cdot 10^{-254} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \le 0.0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\sqrt{1} + \frac{\sqrt{y}}{\sqrt{z}}\right) \cdot \frac{\sqrt{1} - \frac{\sqrt{y}}{\sqrt{z}}}{x + y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020036 
(FPCore (x y z)
  :name "Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1, A"
  :precision binary64

  :herbie-target
  (if (< y -3.7429310762689856e+171) (* (/ (+ y x) (- y)) z) (if (< y 3.5534662456086734e+168) (/ (+ x y) (- 1 (/ y z))) (* (/ (+ y x) (- y)) z)))

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