Average Error: 7.8 → 6.3
Time: 4.3s
Precision: 64
\[\frac{x + y}{1 - \frac{y}{z}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \le -1.0556130076567567683059314597745994606 \cdot 10^{-290} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \le 0.0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{1} + \frac{\sqrt{y}}{\sqrt{z}}} \cdot \frac{x + y}{\sqrt{1} - \frac{\sqrt{y}}{\sqrt{z}}}\\ \end{array}\]
\frac{x + y}{1 - \frac{y}{z}}
↓
\begin{array}{l}
\mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \le -1.0556130076567567683059314597745994606 \cdot 10^{-290} \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}{\sqrt{1} + \frac{\sqrt{y}}{\sqrt{z}}} \cdot \frac{x + y}{\sqrt{1} - \frac{\sqrt{y}}{\sqrt{z}}}\\

\end{array}
double f(double x, double y, double z) {
        double r622661 = x;
        double r622662 = y;
        double r622663 = r622661 + r622662;
        double r622664 = 1.0;
        double r622665 = z;
        double r622666 = r622662 / r622665;
        double r622667 = r622664 - r622666;
        double r622668 = r622663 / r622667;
        return r622668;
}
↓
double f(double x, double y, double z) {
        double r622669 = x;
        double r622670 = y;
        double r622671 = r622669 + r622670;
        double r622672 = 1.0;
        double r622673 = z;
        double r622674 = r622670 / r622673;
        double r622675 = r622672 - r622674;
        double r622676 = r622671 / r622675;
        double r622677 = -1.0556130076567568e-290;
        bool r622678 = r622676 <= r622677;
        double r622679 = 0.0;
        bool r622680 = r622676 <= r622679;
        double r622681 = !r622680;
        bool r622682 = r622678 || r622681;
        double r622683 = 1.0;
        double r622684 = sqrt(r622672);
        double r622685 = sqrt(r622670);
        double r622686 = sqrt(r622673);
        double r622687 = r622685 / r622686;
        double r622688 = r622684 + r622687;
        double r622689 = r622683 / r622688;
        double r622690 = r622684 - r622687;
        double r622691 = r622671 / r622690;
        double r622692 = r622689 * r622691;
        double r622693 = r622682 ? r622676 : r622692;
        return r622693;
}

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

Original7.8
Target4.3
Herbie6.3
\[\begin{array}{l} \mathbf{if}\;y \lt -3.742931076268985646434612946949172132145 \cdot 10^{171}:\\ \;\;\;\;\frac{y + x}{-y} \cdot z\\ \mathbf{elif}\;y \lt 3.553466245608673435460441960303815115662 \cdot 10^{168}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y + x}{-y} \cdot z\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (+ x y) (- 1.0 (/ y z))) < -1.0556130076567568e-290 or 0.0 < (/ (+ x y) (- 1.0 (/ y z)))

    1. Initial program 4.1

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

    if -1.0556130076567568e-290 < (/ (+ x y) (- 1.0 (/ y z))) < 0.0

    1. Initial program 58.4

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

    3. Applied add-sqr-sqrt62.3

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

    4. Applied add-sqr-sqrt63.1

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

    5. Applied times-frac63.1

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

    6. Applied add-sqr-sqrt63.1

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

    7. Applied difference-of-squares63.1

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

    8. Applied *-un-lft-identity63.1

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

    9. Applied times-frac60.3

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

  4. Final simplification6.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \le -1.0556130076567567683059314597745994606 \cdot 10^{-290} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \le 0.0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{1} + \frac{\sqrt{y}}{\sqrt{z}}} \cdot \frac{x + y}{\sqrt{1} - \frac{\sqrt{y}}{\sqrt{z}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 +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 -3.7429310762689856e+171) (* (/ (+ y x) (- y)) z) (if (< y 3.5534662456086734e+168) (/ (+ x y) (- 1 (/ y z))) (* (/ (+ y x) (- y)) z)))

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