Average Error: 6.5 → 1.7
Time: 3.2s
Precision: 64
\[x + \frac{y \cdot \left(z - x\right)}{t}\]
↓
\[\begin{array}{l} \mathbf{if}\;x + \frac{y \cdot \left(z - x\right)}{t} \le 3.331195896825248809143966500129953449664 \cdot 10^{-77}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - x, x\right)\\ \mathbf{elif}\;x + \frac{y \cdot \left(z - x\right)}{t} \le 9.924781450718653455992743727606336837787 \cdot 10^{287}:\\ \;\;\;\;x + \frac{y \cdot \left(z - x\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{t}{z - x}}\\ \end{array}\]
x + \frac{y \cdot \left(z - x\right)}{t}
↓
\begin{array}{l}
\mathbf{if}\;x + \frac{y \cdot \left(z - x\right)}{t} \le 3.331195896825248809143966500129953449664 \cdot 10^{-77}:\\
\;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - x, x\right)\\

\mathbf{elif}\;x + \frac{y \cdot \left(z - x\right)}{t} \le 9.924781450718653455992743727606336837787 \cdot 10^{287}:\\
\;\;\;\;x + \frac{y \cdot \left(z - x\right)}{t}\\

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

\end{array}
double f(double x, double y, double z, double t) {
        double r298506 = x;
        double r298507 = y;
        double r298508 = z;
        double r298509 = r298508 - r298506;
        double r298510 = r298507 * r298509;
        double r298511 = t;
        double r298512 = r298510 / r298511;
        double r298513 = r298506 + r298512;
        return r298513;
}
↓
double f(double x, double y, double z, double t) {
        double r298514 = x;
        double r298515 = y;
        double r298516 = z;
        double r298517 = r298516 - r298514;
        double r298518 = r298515 * r298517;
        double r298519 = t;
        double r298520 = r298518 / r298519;
        double r298521 = r298514 + r298520;
        double r298522 = 3.331195896825249e-77;
        bool r298523 = r298521 <= r298522;
        double r298524 = r298515 / r298519;
        double r298525 = fma(r298524, r298517, r298514);
        double r298526 = 9.924781450718653e+287;
        bool r298527 = r298521 <= r298526;
        double r298528 = r298519 / r298517;
        double r298529 = r298515 / r298528;
        double r298530 = r298514 + r298529;
        double r298531 = r298527 ? r298521 : r298530;
        double r298532 = r298523 ? r298525 : r298531;
        return r298532;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original6.5
Target2.2
Herbie1.7
\[x - \left(x \cdot \frac{y}{t} + \left(-z\right) \cdot \frac{y}{t}\right)\]

Derivation

  1. Split input into 3 regimes

  2. if (+ x (/ (* y (- z x)) t)) < 3.331195896825249e-77

    1. Initial program 6.0

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

    2. Simplified2.0

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

    if 3.331195896825249e-77 < (+ x (/ (* y (- z x)) t)) < 9.924781450718653e+287

    1. Initial program 0.2

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

    if 9.924781450718653e+287 < (+ x (/ (* y (- z x)) t))

    1. Initial program 45.0

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

    3. Applied associate-/l*6.9

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

  4. Final simplification1.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{y \cdot \left(z - x\right)}{t} \le 3.331195896825248809143966500129953449664 \cdot 10^{-77}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - x, x\right)\\ \mathbf{elif}\;x + \frac{y \cdot \left(z - x\right)}{t} \le 9.924781450718653455992743727606336837787 \cdot 10^{287}:\\ \;\;\;\;x + \frac{y \cdot \left(z - x\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{t}{z - x}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019356 +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)))