Average Error: 6.1 → 0.4
Time: 29.1s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) = -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\ \mathbf{elif}\;y \cdot \left(z - t\right) \le 9.67279530102474374 \cdot 10^{204}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{a}, y, x\right)\\ \end{array}\]
x + \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;y \cdot \left(z - t\right) = -\infty:\\
\;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\

\mathbf{elif}\;y \cdot \left(z - t\right) \le 9.67279530102474374 \cdot 10^{204}:\\
\;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r274335 = x;
        double r274336 = y;
        double r274337 = z;
        double r274338 = t;
        double r274339 = r274337 - r274338;
        double r274340 = r274336 * r274339;
        double r274341 = a;
        double r274342 = r274340 / r274341;
        double r274343 = r274335 + r274342;
        return r274343;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r274344 = y;
        double r274345 = z;
        double r274346 = t;
        double r274347 = r274345 - r274346;
        double r274348 = r274344 * r274347;
        double r274349 = -inf.0;
        bool r274350 = r274348 <= r274349;
        double r274351 = a;
        double r274352 = r274344 / r274351;
        double r274353 = x;
        double r274354 = fma(r274352, r274347, r274353);
        double r274355 = 9.672795301024744e+204;
        bool r274356 = r274348 <= r274355;
        double r274357 = r274348 / r274351;
        double r274358 = r274353 + r274357;
        double r274359 = r274347 / r274351;
        double r274360 = fma(r274359, r274344, r274353);
        double r274361 = r274356 ? r274358 : r274360;
        double r274362 = r274350 ? r274354 : r274361;
        return r274362;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original6.1
Target0.6
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753 \cdot 10^{-10}:\\ \;\;\;\;x + \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.8944268627920891 \cdot 10^{-49}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (* y (- z t)) < -inf.0

    1. Initial program 64.0

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

    2. Simplified0.2

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

    if -inf.0 < (* y (- z t)) < 9.672795301024744e+204

    1. Initial program 0.3

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

    if 9.672795301024744e+204 < (* y (- z t))

    1. Initial program 29.7

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

    2. Simplified0.8

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

    4. Applied fma-udef0.8

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

    5. Simplified0.8

      \[\leadsto \color{blue}{\frac{z - t}{\frac{a}{y}}} + x\]
    6. Using strategy rm

    7. Applied *-un-lft-identity0.8

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

    8. Applied *-un-lft-identity0.8

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

    9. Applied distribute-lft-out0.8

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

    10. Simplified0.9

      \[\leadsto 1 \cdot \color{blue}{\mathsf{fma}\left(\frac{z - t}{a}, y, x\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) = -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\ \mathbf{elif}\;y \cdot \left(z - t\right) \le 9.67279530102474374 \cdot 10^{204}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{a}, y, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019199 +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)))