Average Error: 6.5 → 1.9
Time: 47.1s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le -1.7970758753613627 \cdot 10^{+105}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a}\\ \mathbf{elif}\;a \le 7.171838065420707 \cdot 10^{-249}:\\ \;\;\;\;\frac{\left(z - t\right) \cdot y}{a} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z - t, \frac{y}{a}, x\right)\\ \end{array}\]
x + \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;a \le -1.7970758753613627 \cdot 10^{+105}:\\
\;\;\;\;x + y \cdot \frac{z - t}{a}\\

\mathbf{elif}\;a \le 7.171838065420707 \cdot 10^{-249}:\\
\;\;\;\;\frac{\left(z - t\right) \cdot y}{a} + x\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r19546518 = x;
        double r19546519 = y;
        double r19546520 = z;
        double r19546521 = t;
        double r19546522 = r19546520 - r19546521;
        double r19546523 = r19546519 * r19546522;
        double r19546524 = a;
        double r19546525 = r19546523 / r19546524;
        double r19546526 = r19546518 + r19546525;
        return r19546526;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r19546527 = a;
        double r19546528 = -1.7970758753613627e+105;
        bool r19546529 = r19546527 <= r19546528;
        double r19546530 = x;
        double r19546531 = y;
        double r19546532 = z;
        double r19546533 = t;
        double r19546534 = r19546532 - r19546533;
        double r19546535 = r19546534 / r19546527;
        double r19546536 = r19546531 * r19546535;
        double r19546537 = r19546530 + r19546536;
        double r19546538 = 7.171838065420707e-249;
        bool r19546539 = r19546527 <= r19546538;
        double r19546540 = r19546534 * r19546531;
        double r19546541 = r19546540 / r19546527;
        double r19546542 = r19546541 + r19546530;
        double r19546543 = r19546531 / r19546527;
        double r19546544 = fma(r19546534, r19546543, r19546530);
        double r19546545 = r19546539 ? r19546542 : r19546544;
        double r19546546 = r19546529 ? r19546537 : r19546545;
        return r19546546;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original6.5
Target0.6
Herbie1.9
\[\begin{array}{l} \mathbf{if}\;y \lt -1.0761266216389975 \cdot 10^{-10}:\\ \;\;\;\;x + \frac{1.0}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.894426862792089 \cdot 10^{-49}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if a < -1.7970758753613627e+105

    1. Initial program 11.8

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

    3. Applied *-un-lft-identity11.8

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

    4. Applied times-frac0.6

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

    5. Simplified0.6

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

    if -1.7970758753613627e+105 < a < 7.171838065420707e-249

    1. Initial program 2.4

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

    3. Applied +-commutative2.4

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

    if 7.171838065420707e-249 < a

    1. Initial program 6.8

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

    2. Simplified2.2

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

  4. Final simplification1.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -1.7970758753613627 \cdot 10^{+105}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a}\\ \mathbf{elif}\;a \le 7.171838065420707 \cdot 10^{-249}:\\ \;\;\;\;\frac{\left(z - t\right) \cdot y}{a} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z - t, \frac{y}{a}, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 +o rules:numerics
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, E"

  :herbie-target
  (if (< y -1.0761266216389975e-10) (+ x (/ 1.0 (/ (/ a (- z t)) y))) (if (< y 2.894426862792089e-49) (+ x (/ (* y (- z t)) a)) (+ x (/ y (/ a (- z t))))))

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