Average Error: 7.7 → 6.4
Time: 13.6s
Precision: 64
\[\frac{x + y}{1 - \frac{y}{z}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \le -1.118662450796766229358130903760860563738 \cdot 10^{-273} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \le -0.0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x + y}{\sqrt{1} + \frac{\sqrt{y}}{\sqrt{z}}}}{\sqrt{1} - \frac{\sqrt{y}}{\sqrt{z}}}\\ \end{array}\]
\frac{x + y}{1 - \frac{y}{z}}
↓
\begin{array}{l}
\mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \le -1.118662450796766229358130903760860563738 \cdot 10^{-273} \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{x + y}{\sqrt{1} + \frac{\sqrt{y}}{\sqrt{z}}}}{\sqrt{1} - \frac{\sqrt{y}}{\sqrt{z}}}\\

\end{array}
double f(double x, double y, double z) {
        double r522252 = x;
        double r522253 = y;
        double r522254 = r522252 + r522253;
        double r522255 = 1.0;
        double r522256 = z;
        double r522257 = r522253 / r522256;
        double r522258 = r522255 - r522257;
        double r522259 = r522254 / r522258;
        return r522259;
}
↓
double f(double x, double y, double z) {
        double r522260 = x;
        double r522261 = y;
        double r522262 = r522260 + r522261;
        double r522263 = 1.0;
        double r522264 = z;
        double r522265 = r522261 / r522264;
        double r522266 = r522263 - r522265;
        double r522267 = r522262 / r522266;
        double r522268 = -1.1186624507967662e-273;
        bool r522269 = r522267 <= r522268;
        double r522270 = -0.0;
        bool r522271 = r522267 <= r522270;
        double r522272 = !r522271;
        bool r522273 = r522269 || r522272;
        double r522274 = sqrt(r522263);
        double r522275 = sqrt(r522261);
        double r522276 = sqrt(r522264);
        double r522277 = r522275 / r522276;
        double r522278 = r522274 + r522277;
        double r522279 = r522262 / r522278;
        double r522280 = r522274 - r522277;
        double r522281 = r522279 / r522280;
        double r522282 = r522273 ? r522267 : r522281;
        return r522282;
}

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
Target3.9
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))) < -1.1186624507967662e-273 or -0.0 < (/ (+ x y) (- 1.0 (/ y z)))

    1. Initial program 4.2

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

    if -1.1186624507967662e-273 < (/ (+ x y) (- 1.0 (/ y z))) < -0.0

    1. Initial program 54.8

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

    3. Applied add-sqr-sqrt56.1

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

    4. Applied add-sqr-sqrt60.1

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

    5. Applied times-frac60.1

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

    6. Applied add-sqr-sqrt60.1

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

    7. Applied difference-of-squares60.1

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

    8. Applied associate-/r*34.3

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

  4. Final simplification6.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \le -1.118662450796766229358130903760860563738 \cdot 10^{-273} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \le -0.0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x + y}{\sqrt{1} + \frac{\sqrt{y}}{\sqrt{z}}}}{\sqrt{1} - \frac{\sqrt{y}}{\sqrt{z}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019235 
(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))))