Average Error: 6.0 → 1.2
Time: 15.1s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le -31292472571328848723968 \lor \neg \left(a \le 1.084810071014022110500267120313456275877 \cdot 10^{116}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a} \cdot \left(\left(z - t\right) \cdot y\right) + x\\ \end{array}\]
x + \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;a \le -31292472571328848723968 \lor \neg \left(a \le 1.084810071014022110500267120313456275877 \cdot 10^{116}\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{z - t}{a}, y, x\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{a} \cdot \left(\left(z - t\right) \cdot y\right) + x\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r208336 = x;
        double r208337 = y;
        double r208338 = z;
        double r208339 = t;
        double r208340 = r208338 - r208339;
        double r208341 = r208337 * r208340;
        double r208342 = a;
        double r208343 = r208341 / r208342;
        double r208344 = r208336 + r208343;
        return r208344;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r208345 = a;
        double r208346 = -3.129247257132885e+22;
        bool r208347 = r208345 <= r208346;
        double r208348 = 1.0848100710140221e+116;
        bool r208349 = r208345 <= r208348;
        double r208350 = !r208349;
        bool r208351 = r208347 || r208350;
        double r208352 = z;
        double r208353 = t;
        double r208354 = r208352 - r208353;
        double r208355 = r208354 / r208345;
        double r208356 = y;
        double r208357 = x;
        double r208358 = fma(r208355, r208356, r208357);
        double r208359 = 1.0;
        double r208360 = r208359 / r208345;
        double r208361 = r208354 * r208356;
        double r208362 = r208360 * r208361;
        double r208363 = r208362 + r208357;
        double r208364 = r208351 ? r208358 : r208363;
        return r208364;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original6.0
Target0.8
Herbie1.2
\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753216593153715602325729 \cdot 10^{-10}:\\ \;\;\;\;x + \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.894426862792089097262541964056085749132 \cdot 10^{-49}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if a < -3.129247257132885e+22 or 1.0848100710140221e+116 < a

    1. Initial program 10.5

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

    2. Simplified2.0

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

    4. Applied fma-udef2.0

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

    5. Simplified2.3

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

    7. Applied *-un-lft-identity2.3

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

    8. Applied *-un-lft-identity2.3

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

    9. Applied distribute-lft-out2.3

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

    10. Simplified0.7

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

    if -3.129247257132885e+22 < a < 1.0848100710140221e+116

    1. Initial program 1.6

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

    2. Simplified3.1

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

    4. Applied fma-udef3.1

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

    5. Simplified2.8

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

    7. Applied div-inv2.8

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

    8. Applied *-un-lft-identity2.8

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

    9. Applied times-frac1.7

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

    10. Simplified1.7

      \[\leadsto \frac{1}{a} \cdot \color{blue}{\left(\left(z - t\right) \cdot y\right)} + x\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -31292472571328848723968 \lor \neg \left(a \le 1.084810071014022110500267120313456275877 \cdot 10^{116}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a} \cdot \left(\left(z - t\right) \cdot y\right) + x\\ \end{array}\]

Reproduce

herbie shell --seed 2019306 +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 -1.07612662163899753e-10) (+ x (/ 1 (/ (/ a (- z t)) y))) (if (< y 2.8944268627920891e-49) (+ x (/ (* y (- z t)) a)) (+ x (/ y (/ a (- z t))))))

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