Average Error: 7.4 → 6.2
Time: 4.8s
Precision: 64
\[\frac{x + y}{1 - \frac{y}{z}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \le -3.3763009443507461 \cdot 10^{-298} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \le -0.0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}{\sqrt{1} + \frac{\sqrt{y}}{\sqrt{z}}} \cdot \frac{\sqrt[3]{x + y}}{\sqrt{1} - \frac{\sqrt{y}}{\sqrt{z}}}\\ \end{array}\]
\frac{x + y}{1 - \frac{y}{z}}
↓
\begin{array}{l}
\mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \le -3.3763009443507461 \cdot 10^{-298} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \le -0.0\right):\\
\;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}{\sqrt{1} + \frac{\sqrt{y}}{\sqrt{z}}} \cdot \frac{\sqrt[3]{x + y}}{\sqrt{1} - \frac{\sqrt{y}}{\sqrt{z}}}\\

\end{array}
double f(double x, double y, double z) {
        double r673168 = x;
        double r673169 = y;
        double r673170 = r673168 + r673169;
        double r673171 = 1.0;
        double r673172 = z;
        double r673173 = r673169 / r673172;
        double r673174 = r673171 - r673173;
        double r673175 = r673170 / r673174;
        return r673175;
}
↓
double f(double x, double y, double z) {
        double r673176 = x;
        double r673177 = y;
        double r673178 = r673176 + r673177;
        double r673179 = 1.0;
        double r673180 = z;
        double r673181 = r673177 / r673180;
        double r673182 = r673179 - r673181;
        double r673183 = r673178 / r673182;
        double r673184 = -3.376300944350746e-298;
        bool r673185 = r673183 <= r673184;
        double r673186 = -0.0;
        bool r673187 = r673183 <= r673186;
        double r673188 = !r673187;
        bool r673189 = r673185 || r673188;
        double r673190 = cbrt(r673178);
        double r673191 = r673190 * r673190;
        double r673192 = sqrt(r673179);
        double r673193 = sqrt(r673177);
        double r673194 = sqrt(r673180);
        double r673195 = r673193 / r673194;
        double r673196 = r673192 + r673195;
        double r673197 = r673191 / r673196;
        double r673198 = r673192 - r673195;
        double r673199 = r673190 / r673198;
        double r673200 = r673197 * r673199;
        double r673201 = r673189 ? r673183 : r673200;
        return r673201;
}

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

Original7.4
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))) < -3.376300944350746e-298 or -0.0 < (/ (+ x y) (- 1.0 (/ y z)))

    1. Initial program 0.1

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

    if -3.376300944350746e-298 < (/ (+ x y) (- 1.0 (/ y z))) < -0.0

    1. Initial program 59.3

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

    3. Applied add-sqr-sqrt61.3

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

    4. Applied add-sqr-sqrt62.6

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

    5. Applied times-frac62.6

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

    6. Applied add-sqr-sqrt62.6

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

    7. Applied difference-of-squares62.6

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

    8. Applied add-cube-cbrt62.6

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}\right) \cdot \sqrt[3]{x + y}}}{\left(\sqrt{1} + \frac{\sqrt{y}}{\sqrt{z}}\right) \cdot \left(\sqrt{1} - \frac{\sqrt{y}}{\sqrt{z}}\right)}\]

    9. Applied times-frac49.5

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

  4. Final simplification6.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \le -3.3763009443507461 \cdot 10^{-298} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \le -0.0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}{\sqrt{1} + \frac{\sqrt{y}}{\sqrt{z}}} \cdot \frac{\sqrt[3]{x + y}}{\sqrt{1} - \frac{\sqrt{y}}{\sqrt{z}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 
(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))))