Average Error: 7.6 → 6.2
Time: 4.6s
Precision: 64
\[\frac{x + y}{1 - \frac{y}{z}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \le -4.1818847110488908 \cdot 10^{-283} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \le -0.0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{1} + \frac{\sqrt{y}}{\sqrt{z}}}}{\frac{\sqrt{1} - \frac{\sqrt{y}}{\sqrt{z}}}{x + y}}\\ \end{array}\]
\frac{x + y}{1 - \frac{y}{z}}
↓
\begin{array}{l}
\mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \le -4.1818847110488908 \cdot 10^{-283} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \le -0.0\right):\\
\;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\

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

\end{array}
double f(double x, double y, double z) {
        double r577895 = x;
        double r577896 = y;
        double r577897 = r577895 + r577896;
        double r577898 = 1.0;
        double r577899 = z;
        double r577900 = r577896 / r577899;
        double r577901 = r577898 - r577900;
        double r577902 = r577897 / r577901;
        return r577902;
}
↓
double f(double x, double y, double z) {
        double r577903 = x;
        double r577904 = y;
        double r577905 = r577903 + r577904;
        double r577906 = 1.0;
        double r577907 = z;
        double r577908 = r577904 / r577907;
        double r577909 = r577906 - r577908;
        double r577910 = r577905 / r577909;
        double r577911 = -4.181884711048891e-283;
        bool r577912 = r577910 <= r577911;
        double r577913 = -0.0;
        bool r577914 = r577910 <= r577913;
        double r577915 = !r577914;
        bool r577916 = r577912 || r577915;
        double r577917 = 1.0;
        double r577918 = sqrt(r577906);
        double r577919 = sqrt(r577904);
        double r577920 = sqrt(r577907);
        double r577921 = r577919 / r577920;
        double r577922 = r577918 + r577921;
        double r577923 = r577917 / r577922;
        double r577924 = r577918 - r577921;
        double r577925 = r577924 / r577905;
        double r577926 = r577923 / r577925;
        double r577927 = r577916 ? r577910 : r577926;
        return r577927;
}

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

Original7.6
Target3.9
Herbie6.2
\[\begin{array}{l} \mathbf{if}\;y \lt -3.74293107626898565 \cdot 10^{171}:\\ \;\;\;\;\frac{y + x}{-y} \cdot z\\ \mathbf{elif}\;y \lt 3.55346624560867344 \cdot 10^{168}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y + x}{-y} \cdot z\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (+ x y) (- 1.0 (/ y z))) < -4.181884711048891e-283 or -0.0 < (/ (+ x y) (- 1.0 (/ y z)))

    1. Initial program 0.1

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

    if -4.181884711048891e-283 < (/ (+ 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 clear-num58.4

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

    5. Applied *-un-lft-identity58.4

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

    6. Applied add-sqr-sqrt60.5

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

    7. Applied add-sqr-sqrt62.5

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

    8. Applied times-frac62.5

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

    9. Applied add-sqr-sqrt62.5

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

    10. Applied difference-of-squares62.5

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

    11. Applied times-frac47.8

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

    12. Applied associate-/r*47.8

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

    13. Simplified47.8

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

  4. Final simplification6.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \le -4.1818847110488908 \cdot 10^{-283} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \le -0.0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{1} + \frac{\sqrt{y}}{\sqrt{z}}}}{\frac{\sqrt{1} - \frac{\sqrt{y}}{\sqrt{z}}}{x + y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020065 +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))))