Average Error: 6.5 → 0.4
Time: 9.2s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \le -1.04741031975931943017991088976937445358 \cdot 10^{207}:\\ \;\;\;\;\frac{\frac{z - t}{a}}{\frac{1}{y}} + x\\ \mathbf{elif}\;y \cdot \left(z - t\right) \le 1.091273540682466379682658620694282505865 \cdot 10^{263}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{z - t}{\frac{a}{y}} + x\\ \end{array}\]
x + \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;y \cdot \left(z - t\right) \le -1.04741031975931943017991088976937445358 \cdot 10^{207}:\\
\;\;\;\;\frac{\frac{z - t}{a}}{\frac{1}{y}} + x\\

\mathbf{elif}\;y \cdot \left(z - t\right) \le 1.091273540682466379682658620694282505865 \cdot 10^{263}:\\
\;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r368333 = x;
        double r368334 = y;
        double r368335 = z;
        double r368336 = t;
        double r368337 = r368335 - r368336;
        double r368338 = r368334 * r368337;
        double r368339 = a;
        double r368340 = r368338 / r368339;
        double r368341 = r368333 + r368340;
        return r368341;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r368342 = y;
        double r368343 = z;
        double r368344 = t;
        double r368345 = r368343 - r368344;
        double r368346 = r368342 * r368345;
        double r368347 = -1.0474103197593194e+207;
        bool r368348 = r368346 <= r368347;
        double r368349 = a;
        double r368350 = r368345 / r368349;
        double r368351 = 1.0;
        double r368352 = r368351 / r368342;
        double r368353 = r368350 / r368352;
        double r368354 = x;
        double r368355 = r368353 + r368354;
        double r368356 = 1.0912735406824664e+263;
        bool r368357 = r368346 <= r368356;
        double r368358 = r368346 / r368349;
        double r368359 = r368354 + r368358;
        double r368360 = r368349 / r368342;
        double r368361 = r368345 / r368360;
        double r368362 = r368361 + r368354;
        double r368363 = r368357 ? r368359 : r368362;
        double r368364 = r368348 ? r368355 : r368363;
        return r368364;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original6.5
Target0.7
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753216593153715602325729 \cdot 10^{-10}:\\ \;\;\;\;x + \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.894426862792089097262541964056085749132 \cdot 10^{-49}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (* y (- z t)) < -1.0474103197593194e+207

    1. Initial program 31.0

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

    2. Simplified0.4

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

    4. Applied fma-udef0.4

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

    5. Simplified0.5

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

    7. Applied div-inv0.5

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

    8. Applied associate-/r*0.8

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

    if -1.0474103197593194e+207 < (* y (- z t)) < 1.0912735406824664e+263

    1. Initial program 0.4

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

    if 1.0912735406824664e+263 < (* y (- z t))

    1. Initial program 44.3

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

    2. Simplified0.2

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

    4. Applied fma-udef0.2

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

    5. Simplified0.2

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \le -1.04741031975931943017991088976937445358 \cdot 10^{207}:\\ \;\;\;\;\frac{\frac{z - t}{a}}{\frac{1}{y}} + x\\ \mathbf{elif}\;y \cdot \left(z - t\right) \le 1.091273540682466379682658620694282505865 \cdot 10^{263}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{z - t}{\frac{a}{y}} + x\\ \end{array}\]

Reproduce

herbie shell --seed 2019351 +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)))