Average Error: 7.6 → 6.6
Time: 7.4s
Precision: 64
\[\frac{x + y}{1 - \frac{y}{z}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \le -2.833440705943035823332213321994716089663 \cdot 10^{-251} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \le 0.0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x + y}{\sqrt{1} + \frac{\sqrt{y}}{\sqrt{z}}}}{\sqrt{1} - \frac{\sqrt{y}}{\sqrt{z}}}\\ \end{array}\]
\frac{x + y}{1 - \frac{y}{z}}
↓
\begin{array}{l}
\mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \le -2.833440705943035823332213321994716089663 \cdot 10^{-251} \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 r531128 = x;
        double r531129 = y;
        double r531130 = r531128 + r531129;
        double r531131 = 1.0;
        double r531132 = z;
        double r531133 = r531129 / r531132;
        double r531134 = r531131 - r531133;
        double r531135 = r531130 / r531134;
        return r531135;
}
↓
double f(double x, double y, double z) {
        double r531136 = x;
        double r531137 = y;
        double r531138 = r531136 + r531137;
        double r531139 = 1.0;
        double r531140 = z;
        double r531141 = r531137 / r531140;
        double r531142 = r531139 - r531141;
        double r531143 = r531138 / r531142;
        double r531144 = -2.833440705943036e-251;
        bool r531145 = r531143 <= r531144;
        double r531146 = 0.0;
        bool r531147 = r531143 <= r531146;
        double r531148 = !r531147;
        bool r531149 = r531145 || r531148;
        double r531150 = sqrt(r531139);
        double r531151 = sqrt(r531137);
        double r531152 = sqrt(r531140);
        double r531153 = r531151 / r531152;
        double r531154 = r531150 + r531153;
        double r531155 = r531138 / r531154;
        double r531156 = r531150 - r531153;
        double r531157 = r531155 / r531156;
        double r531158 = r531149 ? r531143 : r531157;
        return r531158;
}

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
Target4.0
Herbie6.6
\[\begin{array}{l} \mathbf{if}\;y \lt -3.742931076268985646434612946949172132145 \cdot 10^{171}:\\ \;\;\;\;\frac{y + x}{-y} \cdot z\\ \mathbf{elif}\;y \lt 3.553466245608673435460441960303815115662 \cdot 10^{168}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y + x}{-y} \cdot z\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (+ x y) (- 1.0 (/ y z))) < -2.833440705943036e-251 or 0.0 < (/ (+ x y) (- 1.0 (/ y z)))

    1. Initial program 4.1

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

    if -2.833440705943036e-251 < (/ (+ x y) (- 1.0 (/ y z))) < 0.0

    1. Initial program 49.3

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

    3. Applied add-sqr-sqrt54.7

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

    4. Applied add-sqr-sqrt60.3

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

    5. Applied times-frac60.3

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

    6. Applied add-sqr-sqrt60.3

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

    7. Applied difference-of-squares60.3

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

    8. Applied associate-/r*57.8

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

  4. Final simplification6.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \le -2.833440705943035823332213321994716089663 \cdot 10^{-251} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \le 0.0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x + y}{\sqrt{1} + \frac{\sqrt{y}}{\sqrt{z}}}}{\sqrt{1} - \frac{\sqrt{y}}{\sqrt{z}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019354 +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))))