Average Error: 7.4 → 0.3
Time: 18.2s
Precision: 64
\[\frac{x + y}{1 - \frac{y}{z}}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -109013814756333918548664182814952914944 \lor \neg \left(y \le 537.7474182801523738817195408046245574951\right):\\ \;\;\;\;\frac{1}{\frac{1}{x + y} - \frac{\frac{y}{x + y}}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 - \frac{y}{z}} \cdot \left(x + y\right)\\ \end{array}\]
\frac{x + y}{1 - \frac{y}{z}}
↓
\begin{array}{l}
\mathbf{if}\;y \le -109013814756333918548664182814952914944 \lor \neg \left(y \le 537.7474182801523738817195408046245574951\right):\\
\;\;\;\;\frac{1}{\frac{1}{x + y} - \frac{\frac{y}{x + y}}{z}}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{1 - \frac{y}{z}} \cdot \left(x + y\right)\\

\end{array}
double f(double x, double y, double z) {
        double r401725 = x;
        double r401726 = y;
        double r401727 = r401725 + r401726;
        double r401728 = 1.0;
        double r401729 = z;
        double r401730 = r401726 / r401729;
        double r401731 = r401728 - r401730;
        double r401732 = r401727 / r401731;
        return r401732;
}
↓
double f(double x, double y, double z) {
        double r401733 = y;
        double r401734 = -1.0901381475633392e+38;
        bool r401735 = r401733 <= r401734;
        double r401736 = 537.7474182801524;
        bool r401737 = r401733 <= r401736;
        double r401738 = !r401737;
        bool r401739 = r401735 || r401738;
        double r401740 = 1.0;
        double r401741 = 1.0;
        double r401742 = x;
        double r401743 = r401742 + r401733;
        double r401744 = r401741 / r401743;
        double r401745 = r401733 / r401743;
        double r401746 = z;
        double r401747 = r401745 / r401746;
        double r401748 = r401744 - r401747;
        double r401749 = r401740 / r401748;
        double r401750 = r401733 / r401746;
        double r401751 = r401741 - r401750;
        double r401752 = r401740 / r401751;
        double r401753 = r401752 * r401743;
        double r401754 = r401739 ? r401749 : r401753;
        return r401754;
}

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
Target4.0
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;y \lt -3.742931076268985646434612946949172132145 \cdot 10^{171}:\\ \;\;\;\;\frac{y + x}{-y} \cdot z\\ \mathbf{elif}\;y \lt 3.553466245608673435460441960303815115662 \cdot 10^{168}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y + x}{-y} \cdot z\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -1.0901381475633392e+38 or 537.7474182801524 < y

    1. Initial program 15.4

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

    2. Simplified15.4

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

    4. Applied clear-num15.6

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

    5. Simplified15.6

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

    7. Applied div-sub15.6

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

    8. Simplified15.6

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

    9. Simplified9.5

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

    11. Applied associate-/r*0.3

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

    12. Simplified0.3

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

    if -1.0901381475633392e+38 < y < 537.7474182801524

    1. Initial program 0.3

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

    2. Simplified0.3

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

    4. Applied div-inv0.3

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -109013814756333918548664182814952914944 \lor \neg \left(y \le 537.7474182801523738817195408046245574951\right):\\ \;\;\;\;\frac{1}{\frac{1}{x + y} - \frac{\frac{y}{x + y}}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 - \frac{y}{z}} \cdot \left(x + y\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 +o rules:numerics
(FPCore (x y z)
  :name "Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1, A"

  :herbie-target
  (if (< y -3.7429310762689856e+171) (* (/ (+ y x) (- y)) z) (if (< y 3.5534662456086734e+168) (/ (+ x y) (- 1.0 (/ y z))) (* (/ (+ y x) (- y)) z)))

  (/ (+ x y) (- 1.0 (/ y z))))