Average Error: 7.6 → 6.4
Time: 2.5s
Precision: 64
\[\frac{x + y}{1 - \frac{y}{z}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \le -9.340220995992204435937211078763363155616 \cdot 10^{-288} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \le -0.0\right):\\ \;\;\;\;1 \cdot \frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{x + y}}{\sqrt{1} + \frac{\sqrt{y}}{\sqrt{z}}}}{\frac{\sqrt{1} - \frac{\sqrt{y}}{\sqrt{z}}}{\sqrt{x + y}}}\\ \end{array}\]
\frac{x + y}{1 - \frac{y}{z}}
↓
\begin{array}{l}
\mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \le -9.340220995992204435937211078763363155616 \cdot 10^{-288} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \le -0.0\right):\\
\;\;\;\;1 \cdot \frac{x + y}{1 - \frac{y}{z}}\\

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

\end{array}
double f(double x, double y, double z) {
        double r580198 = x;
        double r580199 = y;
        double r580200 = r580198 + r580199;
        double r580201 = 1.0;
        double r580202 = z;
        double r580203 = r580199 / r580202;
        double r580204 = r580201 - r580203;
        double r580205 = r580200 / r580204;
        return r580205;
}
↓
double f(double x, double y, double z) {
        double r580206 = x;
        double r580207 = y;
        double r580208 = r580206 + r580207;
        double r580209 = 1.0;
        double r580210 = z;
        double r580211 = r580207 / r580210;
        double r580212 = r580209 - r580211;
        double r580213 = r580208 / r580212;
        double r580214 = -9.340220995992204e-288;
        bool r580215 = r580213 <= r580214;
        double r580216 = -0.0;
        bool r580217 = r580213 <= r580216;
        double r580218 = !r580217;
        bool r580219 = r580215 || r580218;
        double r580220 = 1.0;
        double r580221 = r580220 * r580213;
        double r580222 = sqrt(r580208);
        double r580223 = sqrt(r580209);
        double r580224 = sqrt(r580207);
        double r580225 = sqrt(r580210);
        double r580226 = r580224 / r580225;
        double r580227 = r580223 + r580226;
        double r580228 = r580222 / r580227;
        double r580229 = r580223 - r580226;
        double r580230 = r580229 / r580222;
        double r580231 = r580228 / r580230;
        double r580232 = r580219 ? r580221 : r580231;
        return r580232;
}

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

Derivation

  1. Split input into 2 regimes

  2. if (/ (+ x y) (- 1.0 (/ y z))) < -9.340220995992204e-288 or -0.0 < (/ (+ x y) (- 1.0 (/ y z)))

    1. Initial program 4.1

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

    3. Applied div-inv4.2

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

    5. Applied *-un-lft-identity4.2

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

    6. Applied associate-*l*4.2

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

    7. Simplified4.1

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

    if -9.340220995992204e-288 < (/ (+ x y) (- 1.0 (/ y z))) < -0.0

    1. Initial program 57.3

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

    3. Applied clear-num57.3

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

    5. Applied add-sqr-sqrt60.9

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

    6. Applied add-sqr-sqrt61.0

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

    7. Applied add-sqr-sqrt61.0

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

    8. Applied times-frac61.0

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

    9. Applied add-sqr-sqrt61.0

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

    10. Applied difference-of-squares61.0

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

    11. Applied times-frac35.5

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

    12. Applied associate-/r*35.5

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

    13. Simplified35.4

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

  4. Final simplification6.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \le -9.340220995992204435937211078763363155616 \cdot 10^{-288} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \le -0.0\right):\\ \;\;\;\;1 \cdot \frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{x + y}}{\sqrt{1} + \frac{\sqrt{y}}{\sqrt{z}}}}{\frac{\sqrt{1} - \frac{\sqrt{y}}{\sqrt{z}}}{\sqrt{x + y}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 
(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))))