Average Error: 6.3 → 0.7
Time: 17.7s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\left(z - t\right) \cdot y}{a} = -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\ \mathbf{elif}\;\frac{\left(z - t\right) \cdot y}{a} \le 7.322226024676649204991107570780988530293 \cdot 10^{282}:\\ \;\;\;\;\frac{\left(z - t\right) \cdot y}{a} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{a}, x\right)\\ \end{array}\]
x + \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;\frac{\left(z - t\right) \cdot y}{a} = -\infty:\\
\;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\

\mathbf{elif}\;\frac{\left(z - t\right) \cdot y}{a} \le 7.322226024676649204991107570780988530293 \cdot 10^{282}:\\
\;\;\;\;\frac{\left(z - t\right) \cdot y}{a} + x\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r198287 = x;
        double r198288 = y;
        double r198289 = z;
        double r198290 = t;
        double r198291 = r198289 - r198290;
        double r198292 = r198288 * r198291;
        double r198293 = a;
        double r198294 = r198292 / r198293;
        double r198295 = r198287 + r198294;
        return r198295;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r198296 = z;
        double r198297 = t;
        double r198298 = r198296 - r198297;
        double r198299 = y;
        double r198300 = r198298 * r198299;
        double r198301 = a;
        double r198302 = r198300 / r198301;
        double r198303 = -inf.0;
        bool r198304 = r198302 <= r198303;
        double r198305 = r198299 / r198301;
        double r198306 = x;
        double r198307 = fma(r198305, r198298, r198306);
        double r198308 = 7.322226024676649e+282;
        bool r198309 = r198302 <= r198308;
        double r198310 = r198302 + r198306;
        double r198311 = r198298 / r198301;
        double r198312 = fma(r198299, r198311, r198306);
        double r198313 = r198309 ? r198310 : r198312;
        double r198314 = r198304 ? r198307 : r198313;
        return r198314;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original6.3
Target0.6
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753216593153715602325729 \cdot 10^{-10}:\\ \;\;\;\;x + \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.894426862792089097262541964056085749132 \cdot 10^{-49}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

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

    1. Initial program 64.0

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

    2. Simplified0.2

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

    4. Applied fma-udef0.2

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

    5. Simplified0.2

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

    7. Applied *-un-lft-identity0.2

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

    8. Applied *-un-lft-identity0.2

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

    9. Applied distribute-lft-out0.2

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

    10. Simplified0.2

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

    if -inf.0 < (/ (* y (- z t)) a) < 7.322226024676649e+282

    1. Initial program 0.3

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

    if 7.322226024676649e+282 < (/ (* y (- z t)) a)

    1. Initial program 50.5

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

    2. Simplified7.1

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(z - t\right) \cdot y}{a} = -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\ \mathbf{elif}\;\frac{\left(z - t\right) \cdot y}{a} \le 7.322226024676649204991107570780988530293 \cdot 10^{282}:\\ \;\;\;\;\frac{\left(z - t\right) \cdot y}{a} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{a}, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 +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)))