Average Error: 7.1 → 2.6
Time: 3.0s
Precision: 64
\[x + \frac{y \cdot \left(z - x\right)}{t}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.15904748216736652 \cdot 10^{-174} \lor \neg \left(z \le 5.14916335796076412 \cdot 10^{-75}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - x}{t}\\ \end{array}\]
x + \frac{y \cdot \left(z - x\right)}{t}
↓
\begin{array}{l}
\mathbf{if}\;z \le -1.15904748216736652 \cdot 10^{-174} \lor \neg \left(z \le 5.14916335796076412 \cdot 10^{-75}\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - x, x\right)\\

\mathbf{else}:\\
\;\;\;\;x + y \cdot \frac{z - x}{t}\\

\end{array}
double f(double x, double y, double z, double t) {
        double r271407 = x;
        double r271408 = y;
        double r271409 = z;
        double r271410 = r271409 - r271407;
        double r271411 = r271408 * r271410;
        double r271412 = t;
        double r271413 = r271411 / r271412;
        double r271414 = r271407 + r271413;
        return r271414;
}
↓
double f(double x, double y, double z, double t) {
        double r271415 = z;
        double r271416 = -1.1590474821673665e-174;
        bool r271417 = r271415 <= r271416;
        double r271418 = 5.149163357960764e-75;
        bool r271419 = r271415 <= r271418;
        double r271420 = !r271419;
        bool r271421 = r271417 || r271420;
        double r271422 = y;
        double r271423 = t;
        double r271424 = r271422 / r271423;
        double r271425 = x;
        double r271426 = r271415 - r271425;
        double r271427 = fma(r271424, r271426, r271425);
        double r271428 = r271426 / r271423;
        double r271429 = r271422 * r271428;
        double r271430 = r271425 + r271429;
        double r271431 = r271421 ? r271427 : r271430;
        return r271431;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original7.1
Target2.0
Herbie2.6
\[x - \left(x \cdot \frac{y}{t} + \left(-z\right) \cdot \frac{y}{t}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if z < -1.1590474821673665e-174 or 5.149163357960764e-75 < z

    1. Initial program 8.2

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

    2. Simplified1.6

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{t}, z - x, x\right)}\]

    if -1.1590474821673665e-174 < z < 5.149163357960764e-75

    1. Initial program 4.8

      \[x + \frac{y \cdot \left(z - x\right)}{t}\]
    2. Using strategy rm

    3. Applied *-un-lft-identity4.8

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

    4. Applied times-frac4.7

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

    5. Simplified4.7

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

  4. Final simplification2.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.15904748216736652 \cdot 10^{-174} \lor \neg \left(z \le 5.14916335796076412 \cdot 10^{-75}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - x}{t}\\ \end{array}\]

Reproduce

herbie shell --seed 2020034 +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)))