Average Error: 7.4 → 2.7
Time: 18.3s
Precision: 64
\[\frac{x + y}{1 - \frac{y}{z}}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -3.238072608215110164149777448017943406571 \cdot 10^{-21} \lor \neg \left(z \le 2.529936809907233979036830056799149601253 \cdot 10^{-78}\right):\\ \;\;\;\;\frac{y + x}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{y + x} - \frac{y}{\left(y + x\right) \cdot z}}\\ \end{array}\]
\frac{x + y}{1 - \frac{y}{z}}
↓
\begin{array}{l}
\mathbf{if}\;z \le -3.238072608215110164149777448017943406571 \cdot 10^{-21} \lor \neg \left(z \le 2.529936809907233979036830056799149601253 \cdot 10^{-78}\right):\\
\;\;\;\;\frac{y + x}{1 - \frac{y}{z}}\\

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

\end{array}
double f(double x, double y, double z) {
        double r502528 = x;
        double r502529 = y;
        double r502530 = r502528 + r502529;
        double r502531 = 1.0;
        double r502532 = z;
        double r502533 = r502529 / r502532;
        double r502534 = r502531 - r502533;
        double r502535 = r502530 / r502534;
        return r502535;
}
↓
double f(double x, double y, double z) {
        double r502536 = z;
        double r502537 = -3.23807260821511e-21;
        bool r502538 = r502536 <= r502537;
        double r502539 = 2.529936809907234e-78;
        bool r502540 = r502536 <= r502539;
        double r502541 = !r502540;
        bool r502542 = r502538 || r502541;
        double r502543 = y;
        double r502544 = x;
        double r502545 = r502543 + r502544;
        double r502546 = 1.0;
        double r502547 = r502543 / r502536;
        double r502548 = r502546 - r502547;
        double r502549 = r502545 / r502548;
        double r502550 = 1.0;
        double r502551 = r502546 / r502545;
        double r502552 = r502545 * r502536;
        double r502553 = r502543 / r502552;
        double r502554 = r502551 - r502553;
        double r502555 = r502550 / r502554;
        double r502556 = r502542 ? r502549 : r502555;
        return r502556;
}

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
Target4.2
Herbie2.7
\[\begin{array}{l} \mathbf{if}\;y \lt -3.742931076268985646434612946949172132145 \cdot 10^{171}:\\ \;\;\;\;\frac{y + x}{-y} \cdot z\\ \mathbf{elif}\;y \lt 3.553466245608673435460441960303815115662 \cdot 10^{168}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y + x}{-y} \cdot z\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -3.23807260821511e-21 or 2.529936809907234e-78 < z

    1. Initial program 0.6

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

    2. Simplified0.6

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

    if -3.23807260821511e-21 < z < 2.529936809907234e-78

    1. Initial program 17.0

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

    2. Simplified17.0

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

    4. Applied clear-num17.1

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

    5. Simplified17.1

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

    7. Applied div-sub17.1

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

    8. Simplified17.1

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

    9. Simplified5.7

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

  4. Final simplification2.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -3.238072608215110164149777448017943406571 \cdot 10^{-21} \lor \neg \left(z \le 2.529936809907233979036830056799149601253 \cdot 10^{-78}\right):\\ \;\;\;\;\frac{y + x}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{y + x} - \frac{y}{\left(y + x\right) \cdot z}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019196 +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))))