Average Error: 5.9 → 1.0
Time: 1.1m
Precision: 64
\[x - \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(z - t\right) \cdot y \le -4.463702555680383129591610974344041882665 \cdot 10^{202}:\\ \;\;\;\;x + \left(\frac{t}{a} - \frac{z}{a}\right) \cdot y\\ \mathbf{elif}\;\left(z - t\right) \cdot y \le 9.417385554814111584786435793986354092013 \cdot 10^{65}:\\ \;\;\;\;x - \frac{1}{\frac{a}{\left(z - t\right) \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t - z, \frac{y}{a}, x\right)\\ \end{array}\]
x - \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;\left(z - t\right) \cdot y \le -4.463702555680383129591610974344041882665 \cdot 10^{202}:\\
\;\;\;\;x + \left(\frac{t}{a} - \frac{z}{a}\right) \cdot y\\

\mathbf{elif}\;\left(z - t\right) \cdot y \le 9.417385554814111584786435793986354092013 \cdot 10^{65}:\\
\;\;\;\;x - \frac{1}{\frac{a}{\left(z - t\right) \cdot y}}\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r18057470 = x;
        double r18057471 = y;
        double r18057472 = z;
        double r18057473 = t;
        double r18057474 = r18057472 - r18057473;
        double r18057475 = r18057471 * r18057474;
        double r18057476 = a;
        double r18057477 = r18057475 / r18057476;
        double r18057478 = r18057470 - r18057477;
        return r18057478;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r18057479 = z;
        double r18057480 = t;
        double r18057481 = r18057479 - r18057480;
        double r18057482 = y;
        double r18057483 = r18057481 * r18057482;
        double r18057484 = -4.463702555680383e+202;
        bool r18057485 = r18057483 <= r18057484;
        double r18057486 = x;
        double r18057487 = a;
        double r18057488 = r18057480 / r18057487;
        double r18057489 = r18057479 / r18057487;
        double r18057490 = r18057488 - r18057489;
        double r18057491 = r18057490 * r18057482;
        double r18057492 = r18057486 + r18057491;
        double r18057493 = 9.417385554814112e+65;
        bool r18057494 = r18057483 <= r18057493;
        double r18057495 = 1.0;
        double r18057496 = r18057487 / r18057483;
        double r18057497 = r18057495 / r18057496;
        double r18057498 = r18057486 - r18057497;
        double r18057499 = r18057480 - r18057479;
        double r18057500 = r18057482 / r18057487;
        double r18057501 = fma(r18057499, r18057500, r18057486);
        double r18057502 = r18057494 ? r18057498 : r18057501;
        double r18057503 = r18057485 ? r18057492 : r18057502;
        return r18057503;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original5.9
Target0.8
Herbie1.0
\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753216593153715602325729 \cdot 10^{-10}:\\ \;\;\;\;x - \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.894426862792089097262541964056085749132 \cdot 10^{-49}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (* y (- z t)) < -4.463702555680383e+202

    1. Initial program 27.3

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

    2. Taylor expanded around 0 27.3

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

    3. Simplified1.6

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

    if -4.463702555680383e+202 < (* y (- z t)) < 9.417385554814112e+65

    1. Initial program 0.5

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

    3. Applied clear-num0.6

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

    if 9.417385554814112e+65 < (* y (- z t))

    1. Initial program 13.6

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

    2. Simplified2.0

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z - t\right) \cdot y \le -4.463702555680383129591610974344041882665 \cdot 10^{202}:\\ \;\;\;\;x + \left(\frac{t}{a} - \frac{z}{a}\right) \cdot y\\ \mathbf{elif}\;\left(z - t\right) \cdot y \le 9.417385554814111584786435793986354092013 \cdot 10^{65}:\\ \;\;\;\;x - \frac{1}{\frac{a}{\left(z - t\right) \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t - z, \frac{y}{a}, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, F"

  :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)))