Average Error: 6.0 → 1.1
Time: 48.0s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le -7.263328688146093211874950167035565067353 \cdot 10^{44}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{a} - \frac{t}{a}, x\right)\\ \mathbf{elif}\;a \le 7.661482294517787913285118129263252490351 \cdot 10^{-69}:\\ \;\;\;\;\left(\frac{z \cdot y}{a} - \frac{t \cdot y}{a}\right) + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\ \end{array}\]
x + \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;a \le -7.263328688146093211874950167035565067353 \cdot 10^{44}:\\
\;\;\;\;\mathsf{fma}\left(y, \frac{z}{a} - \frac{t}{a}, x\right)\\

\mathbf{elif}\;a \le 7.661482294517787913285118129263252490351 \cdot 10^{-69}:\\
\;\;\;\;\left(\frac{z \cdot y}{a} - \frac{t \cdot y}{a}\right) + x\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r290352 = x;
        double r290353 = y;
        double r290354 = z;
        double r290355 = t;
        double r290356 = r290354 - r290355;
        double r290357 = r290353 * r290356;
        double r290358 = a;
        double r290359 = r290357 / r290358;
        double r290360 = r290352 + r290359;
        return r290360;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r290361 = a;
        double r290362 = -7.263328688146093e+44;
        bool r290363 = r290361 <= r290362;
        double r290364 = y;
        double r290365 = z;
        double r290366 = r290365 / r290361;
        double r290367 = t;
        double r290368 = r290367 / r290361;
        double r290369 = r290366 - r290368;
        double r290370 = x;
        double r290371 = fma(r290364, r290369, r290370);
        double r290372 = 7.661482294517788e-69;
        bool r290373 = r290361 <= r290372;
        double r290374 = r290365 * r290364;
        double r290375 = r290374 / r290361;
        double r290376 = r290367 * r290364;
        double r290377 = r290376 / r290361;
        double r290378 = r290375 - r290377;
        double r290379 = r290378 + r290370;
        double r290380 = r290364 / r290361;
        double r290381 = r290365 - r290367;
        double r290382 = fma(r290380, r290381, r290370);
        double r290383 = r290373 ? r290379 : r290382;
        double r290384 = r290363 ? r290371 : r290383;
        return r290384;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original6.0
Target0.7
Herbie1.1
\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753216593153715602325729 \cdot 10^{-10}:\\ \;\;\;\;x + \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.894426862792089097262541964056085749132 \cdot 10^{-49}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if a < -7.263328688146093e+44

    1. Initial program 10.6

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

    2. Simplified1.9

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

    4. Applied fma-udef1.9

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

    5. Simplified2.1

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

    7. Applied div-sub2.1

      \[\leadsto \color{blue}{\left(\frac{z}{\frac{a}{y}} - \frac{t}{\frac{a}{y}}\right)} + x\]
    8. Using strategy rm

    9. Applied associate-/r/1.4

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

    10. Applied associate-/r/0.6

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

    11. Applied distribute-rgt-out--0.6

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

    12. Applied fma-def0.6

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

    if -7.263328688146093e+44 < a < 7.661482294517788e-69

    1. Initial program 1.0

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

    2. Simplified4.1

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

    4. Applied fma-udef4.1

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

    5. Simplified3.5

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

    7. Applied div-sub3.5

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

    8. Taylor expanded around 0 1.0

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

    if 7.661482294517788e-69 < a

    1. Initial program 7.7

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

    2. Simplified1.6

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

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -7.263328688146093211874950167035565067353 \cdot 10^{44}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{a} - \frac{t}{a}, x\right)\\ \mathbf{elif}\;a \le 7.661482294517787913285118129263252490351 \cdot 10^{-69}:\\ \;\;\;\;\left(\frac{z \cdot y}{a} - \frac{t \cdot y}{a}\right) + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 +o rules:numerics
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, E"
  :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)))