Average Error: 6.2 → 1.4
Time: 2.8s
Precision: 64
\[x - \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le -9.123265245159162379441648817674026348617 \cdot 10^{-287}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\ \mathbf{elif}\;a \le 160974762298423667326976:\\ \;\;\;\;x - \frac{1}{\frac{a}{y \cdot \left(z - t\right)}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\sqrt{a}} \cdot \frac{z - t}{\sqrt{a}}\\ \end{array}\]
x - \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;a \le -9.123265245159162379441648817674026348617 \cdot 10^{-287}:\\
\;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\

\mathbf{elif}\;a \le 160974762298423667326976:\\
\;\;\;\;x - \frac{1}{\frac{a}{y \cdot \left(z - t\right)}}\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r278407 = x;
        double r278408 = y;
        double r278409 = z;
        double r278410 = t;
        double r278411 = r278409 - r278410;
        double r278412 = r278408 * r278411;
        double r278413 = a;
        double r278414 = r278412 / r278413;
        double r278415 = r278407 - r278414;
        return r278415;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r278416 = a;
        double r278417 = -9.123265245159162e-287;
        bool r278418 = r278416 <= r278417;
        double r278419 = y;
        double r278420 = r278419 / r278416;
        double r278421 = t;
        double r278422 = z;
        double r278423 = r278421 - r278422;
        double r278424 = x;
        double r278425 = fma(r278420, r278423, r278424);
        double r278426 = 1.6097476229842367e+23;
        bool r278427 = r278416 <= r278426;
        double r278428 = 1.0;
        double r278429 = r278422 - r278421;
        double r278430 = r278419 * r278429;
        double r278431 = r278416 / r278430;
        double r278432 = r278428 / r278431;
        double r278433 = r278424 - r278432;
        double r278434 = sqrt(r278416);
        double r278435 = r278419 / r278434;
        double r278436 = r278429 / r278434;
        double r278437 = r278435 * r278436;
        double r278438 = r278424 - r278437;
        double r278439 = r278427 ? r278433 : r278438;
        double r278440 = r278418 ? r278425 : r278439;
        return r278440;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original6.2
Target0.7
Herbie1.4
\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753216593153715602325729 \cdot 10^{-10}:\\ \;\;\;\;x - \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.894426862792089097262541964056085749132 \cdot 10^{-49}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if a < -9.123265245159162e-287

    1. Initial program 6.4

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

    2. Simplified2.0

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

    if -9.123265245159162e-287 < a < 1.6097476229842367e+23

    1. Initial program 1.0

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

    3. Applied clear-num1.0

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

    if 1.6097476229842367e+23 < a

    1. Initial program 10.0

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

    3. Applied add-sqr-sqrt10.1

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

    4. Applied times-frac0.7

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

  4. Final simplification1.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -9.123265245159162379441648817674026348617 \cdot 10^{-287}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\ \mathbf{elif}\;a \le 160974762298423667326976:\\ \;\;\;\;x - \frac{1}{\frac{a}{y \cdot \left(z - t\right)}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\sqrt{a}} \cdot \frac{z - t}{\sqrt{a}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, F"
  :precision binary64

  :herbie-target
  (if (< y -1.0761266216389975e-10) (- x (/ 1 (/ (/ a (- z t)) y))) (if (< y 2.894426862792089e-49) (- x (/ (* y (- z t)) a)) (- x (/ y (/ a (- z t))))))

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