Average Error: 8.0 → 6.4
Time: 11.8s
Precision: 64
\[\frac{x + y}{1.0 - \frac{y}{z}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{y + x}{1.0 - \frac{y}{z}} \le -1.6574953719104506 \cdot 10^{-297}:\\ \;\;\;\;\frac{y + x}{1.0 - \frac{y}{z}}\\ \mathbf{elif}\;\frac{y + x}{1.0 - \frac{y}{z}} \le -0.0:\\ \;\;\;\;\frac{\sqrt{y + x}}{\sqrt{1.0} - \frac{\sqrt{y}}{\sqrt{z}}} \cdot \frac{\sqrt{y + x}}{\sqrt{1.0} + \frac{\sqrt{y}}{\sqrt{z}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y + x}{1.0 - \frac{y}{z}}\\ \end{array}\]
\frac{x + y}{1.0 - \frac{y}{z}}
↓
\begin{array}{l}
\mathbf{if}\;\frac{y + x}{1.0 - \frac{y}{z}} \le -1.6574953719104506 \cdot 10^{-297}:\\
\;\;\;\;\frac{y + x}{1.0 - \frac{y}{z}}\\

\mathbf{elif}\;\frac{y + x}{1.0 - \frac{y}{z}} \le -0.0:\\
\;\;\;\;\frac{\sqrt{y + x}}{\sqrt{1.0} - \frac{\sqrt{y}}{\sqrt{z}}} \cdot \frac{\sqrt{y + x}}{\sqrt{1.0} + \frac{\sqrt{y}}{\sqrt{z}}}\\

\mathbf{else}:\\
\;\;\;\;\frac{y + x}{1.0 - \frac{y}{z}}\\

\end{array}
double f(double x, double y, double z) {
        double r11077749 = x;
        double r11077750 = y;
        double r11077751 = r11077749 + r11077750;
        double r11077752 = 1.0;
        double r11077753 = z;
        double r11077754 = r11077750 / r11077753;
        double r11077755 = r11077752 - r11077754;
        double r11077756 = r11077751 / r11077755;
        return r11077756;
}
↓
double f(double x, double y, double z) {
        double r11077757 = y;
        double r11077758 = x;
        double r11077759 = r11077757 + r11077758;
        double r11077760 = 1.0;
        double r11077761 = z;
        double r11077762 = r11077757 / r11077761;
        double r11077763 = r11077760 - r11077762;
        double r11077764 = r11077759 / r11077763;
        double r11077765 = -1.6574953719104506e-297;
        bool r11077766 = r11077764 <= r11077765;
        double r11077767 = -0.0;
        bool r11077768 = r11077764 <= r11077767;
        double r11077769 = sqrt(r11077759);
        double r11077770 = sqrt(r11077760);
        double r11077771 = sqrt(r11077757);
        double r11077772 = sqrt(r11077761);
        double r11077773 = r11077771 / r11077772;
        double r11077774 = r11077770 - r11077773;
        double r11077775 = r11077769 / r11077774;
        double r11077776 = r11077770 + r11077773;
        double r11077777 = r11077769 / r11077776;
        double r11077778 = r11077775 * r11077777;
        double r11077779 = r11077768 ? r11077778 : r11077764;
        double r11077780 = r11077766 ? r11077764 : r11077779;
        return r11077780;
}

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

Original8.0
Target3.9
Herbie6.4
\[\begin{array}{l} \mathbf{if}\;y \lt -3.7429310762689856 \cdot 10^{+171}:\\ \;\;\;\;\frac{y + x}{-y} \cdot z\\ \mathbf{elif}\;y \lt 3.5534662456086734 \cdot 10^{+168}:\\ \;\;\;\;\frac{x + y}{1.0 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y + x}{-y} \cdot z\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (+ x y) (- 1.0 (/ y z))) < -1.6574953719104506e-297 or -0.0 < (/ (+ x y) (- 1.0 (/ y z)))

    1. Initial program 0.1

      \[\frac{x + y}{1.0 - \frac{y}{z}}\]

    if -1.6574953719104506e-297 < (/ (+ x y) (- 1.0 (/ y z))) < -0.0

    1. Initial program 59.4

      \[\frac{x + y}{1.0 - \frac{y}{z}}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt60.7

      \[\leadsto \frac{x + y}{1.0 - \frac{y}{\color{blue}{\sqrt{z} \cdot \sqrt{z}}}}\]

    4. Applied add-sqr-sqrt61.9

      \[\leadsto \frac{x + y}{1.0 - \frac{\color{blue}{\sqrt{y} \cdot \sqrt{y}}}{\sqrt{z} \cdot \sqrt{z}}}\]

    5. Applied times-frac61.9

      \[\leadsto \frac{x + y}{1.0 - \color{blue}{\frac{\sqrt{y}}{\sqrt{z}} \cdot \frac{\sqrt{y}}{\sqrt{z}}}}\]

    6. Applied add-sqr-sqrt61.9

      \[\leadsto \frac{x + y}{\color{blue}{\sqrt{1.0} \cdot \sqrt{1.0}} - \frac{\sqrt{y}}{\sqrt{z}} \cdot \frac{\sqrt{y}}{\sqrt{z}}}\]

    7. Applied difference-of-squares61.9

      \[\leadsto \frac{x + y}{\color{blue}{\left(\sqrt{1.0} + \frac{\sqrt{y}}{\sqrt{z}}\right) \cdot \left(\sqrt{1.0} - \frac{\sqrt{y}}{\sqrt{z}}\right)}}\]

    8. Applied add-sqr-sqrt61.9

      \[\leadsto \frac{\color{blue}{\sqrt{x + y} \cdot \sqrt{x + y}}}{\left(\sqrt{1.0} + \frac{\sqrt{y}}{\sqrt{z}}\right) \cdot \left(\sqrt{1.0} - \frac{\sqrt{y}}{\sqrt{z}}\right)}\]

    9. Applied times-frac47.2

      \[\leadsto \color{blue}{\frac{\sqrt{x + y}}{\sqrt{1.0} + \frac{\sqrt{y}}{\sqrt{z}}} \cdot \frac{\sqrt{x + y}}{\sqrt{1.0} - \frac{\sqrt{y}}{\sqrt{z}}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification6.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y + x}{1.0 - \frac{y}{z}} \le -1.6574953719104506 \cdot 10^{-297}:\\ \;\;\;\;\frac{y + x}{1.0 - \frac{y}{z}}\\ \mathbf{elif}\;\frac{y + x}{1.0 - \frac{y}{z}} \le -0.0:\\ \;\;\;\;\frac{\sqrt{y + x}}{\sqrt{1.0} - \frac{\sqrt{y}}{\sqrt{z}}} \cdot \frac{\sqrt{y + x}}{\sqrt{1.0} + \frac{\sqrt{y}}{\sqrt{z}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y + x}{1.0 - \frac{y}{z}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019156 +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))))