Average Error: 7.7 → 6.4
Time: 15.5s
Precision: 64
\[\frac{x + y}{1 - \frac{y}{z}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \le -7.66629167620117969354071927698147699138 \cdot 10^{-261} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \le -0.0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{1} + \frac{\sqrt{y}}{\sqrt{z}}}{\frac{x + y}{\sqrt{1} - \frac{\sqrt{y}}{\sqrt{z}}}}}\\ \end{array}\]
\frac{x + y}{1 - \frac{y}{z}}
↓
\begin{array}{l}
\mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \le -7.66629167620117969354071927698147699138 \cdot 10^{-261} \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}{\frac{\sqrt{1} + \frac{\sqrt{y}}{\sqrt{z}}}{\frac{x + y}{\sqrt{1} - \frac{\sqrt{y}}{\sqrt{z}}}}}\\

\end{array}
double f(double x, double y, double z) {
        double r465379 = x;
        double r465380 = y;
        double r465381 = r465379 + r465380;
        double r465382 = 1.0;
        double r465383 = z;
        double r465384 = r465380 / r465383;
        double r465385 = r465382 - r465384;
        double r465386 = r465381 / r465385;
        return r465386;
}
↓
double f(double x, double y, double z) {
        double r465387 = x;
        double r465388 = y;
        double r465389 = r465387 + r465388;
        double r465390 = 1.0;
        double r465391 = z;
        double r465392 = r465388 / r465391;
        double r465393 = r465390 - r465392;
        double r465394 = r465389 / r465393;
        double r465395 = -7.66629167620118e-261;
        bool r465396 = r465394 <= r465395;
        double r465397 = -0.0;
        bool r465398 = r465394 <= r465397;
        double r465399 = !r465398;
        bool r465400 = r465396 || r465399;
        double r465401 = 1.0;
        double r465402 = sqrt(r465390);
        double r465403 = sqrt(r465388);
        double r465404 = sqrt(r465391);
        double r465405 = r465403 / r465404;
        double r465406 = r465402 + r465405;
        double r465407 = r465402 - r465405;
        double r465408 = r465389 / r465407;
        double r465409 = r465406 / r465408;
        double r465410 = r465401 / r465409;
        double r465411 = r465400 ? r465394 : r465410;
        return r465411;
}

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

Original7.7
Target4.1
Herbie6.4
\[\begin{array}{l} \mathbf{if}\;y \lt -3.742931076268985646434612946949172132145 \cdot 10^{171}:\\ \;\;\;\;\frac{y + x}{-y} \cdot z\\ \mathbf{elif}\;y \lt 3.553466245608673435460441960303815115662 \cdot 10^{168}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y + x}{-y} \cdot z\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (+ x y) (- 1.0 (/ y z))) < -7.66629167620118e-261 or -0.0 < (/ (+ x y) (- 1.0 (/ y z)))

    1. Initial program 3.7

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

    if -7.66629167620118e-261 < (/ (+ x y) (- 1.0 (/ y z))) < -0.0

    1. Initial program 53.7

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

    3. Applied clear-num53.7

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

    5. Applied add-sqr-sqrt55.2

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

    6. Applied add-sqr-sqrt59.9

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

    7. Applied times-frac59.9

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

    8. Applied add-sqr-sqrt59.9

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

    9. Applied difference-of-squares59.9

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

    10. Applied associate-/l*36.7

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

  4. Final simplification6.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \le -7.66629167620117969354071927698147699138 \cdot 10^{-261} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \le -0.0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{1} + \frac{\sqrt{y}}{\sqrt{z}}}{\frac{x + y}{\sqrt{1} - \frac{\sqrt{y}}{\sqrt{z}}}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019212 +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 -3.74293107626898565e171) (* (/ (+ y x) (- y)) z) (if (< y 3.55346624560867344e168) (/ (+ x y) (- 1 (/ y z))) (* (/ (+ y x) (- y)) z)))

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