Average Error: 6.3 → 1.3
Time: 29.2s
Precision: 64
\[x + \frac{y \cdot \left(z - x\right)}{t}\]
↓
\[\begin{array}{l} \mathbf{if}\;x + \frac{y \cdot \left(z - x\right)}{t} \le -1.12539201893726255 \cdot 10^{306}:\\ \;\;\;\;x + \frac{z - x}{\frac{t}{y}}\\ \mathbf{elif}\;x + \frac{y \cdot \left(z - x\right)}{t} \le -5.52428703018855934 \cdot 10^{-210}:\\ \;\;\;\;x + \frac{y \cdot \left(z - x\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - x, x\right)\\ \end{array}\]
x + \frac{y \cdot \left(z - x\right)}{t}
↓
\begin{array}{l}
\mathbf{if}\;x + \frac{y \cdot \left(z - x\right)}{t} \le -1.12539201893726255 \cdot 10^{306}:\\
\;\;\;\;x + \frac{z - x}{\frac{t}{y}}\\

\mathbf{elif}\;x + \frac{y \cdot \left(z - x\right)}{t} \le -5.52428703018855934 \cdot 10^{-210}:\\
\;\;\;\;x + \frac{y \cdot \left(z - x\right)}{t}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - x, x\right)\\

\end{array}
double f(double x, double y, double z, double t) {
        double r272407 = x;
        double r272408 = y;
        double r272409 = z;
        double r272410 = r272409 - r272407;
        double r272411 = r272408 * r272410;
        double r272412 = t;
        double r272413 = r272411 / r272412;
        double r272414 = r272407 + r272413;
        return r272414;
}
↓
double f(double x, double y, double z, double t) {
        double r272415 = x;
        double r272416 = y;
        double r272417 = z;
        double r272418 = r272417 - r272415;
        double r272419 = r272416 * r272418;
        double r272420 = t;
        double r272421 = r272419 / r272420;
        double r272422 = r272415 + r272421;
        double r272423 = -1.1253920189372625e+306;
        bool r272424 = r272422 <= r272423;
        double r272425 = r272420 / r272416;
        double r272426 = r272418 / r272425;
        double r272427 = r272415 + r272426;
        double r272428 = -5.5242870301885593e-210;
        bool r272429 = r272422 <= r272428;
        double r272430 = r272416 / r272420;
        double r272431 = fma(r272430, r272418, r272415);
        double r272432 = r272429 ? r272422 : r272431;
        double r272433 = r272424 ? r272427 : r272432;
        return r272433;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original6.3
Target2.0
Herbie1.3
\[x - \left(x \cdot \frac{y}{t} + \left(-z\right) \cdot \frac{y}{t}\right)\]

Derivation

  1. Split input into 3 regimes

  2. if (+ x (/ (* y (- z x)) t)) < -1.1253920189372625e+306

    1. Initial program 61.5

      \[x + \frac{y \cdot \left(z - x\right)}{t}\]

    2. Taylor expanded around 0 61.5

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

    3. Simplified0.3

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

    if -1.1253920189372625e+306 < (+ x (/ (* y (- z x)) t)) < -5.5242870301885593e-210

    1. Initial program 0.3

      \[x + \frac{y \cdot \left(z - x\right)}{t}\]

    if -5.5242870301885593e-210 < (+ x (/ (* y (- z x)) t))

    1. Initial program 6.4

      \[x + \frac{y \cdot \left(z - x\right)}{t}\]

    2. Simplified2.2

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{t}, z - x, x\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification1.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{y \cdot \left(z - x\right)}{t} \le -1.12539201893726255 \cdot 10^{306}:\\ \;\;\;\;x + \frac{z - x}{\frac{t}{y}}\\ \mathbf{elif}\;x + \frac{y \cdot \left(z - x\right)}{t} \le -5.52428703018855934 \cdot 10^{-210}:\\ \;\;\;\;x + \frac{y \cdot \left(z - x\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - x, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 +o rules:numerics
(FPCore (x y z t)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, D"
  :precision binary64

  :herbie-target
  (- x (+ (* x (/ y t)) (* (- z) (/ y t))))

  (+ x (/ (* y (- z x)) t)))