Average Error: 6.0 → 0.9
Time: 3.6s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -31789.4821495515171:\\ \;\;\;\;y \cdot \frac{z - t}{a} + x\\ \mathbf{elif}\;y \le 4.0978388594748642 \cdot 10^{-179}:\\ \;\;\;\;\frac{\left(z - t\right) \cdot y}{a} + x\\ \mathbf{elif}\;y \le 2.78403719146078999 \cdot 10^{-66}:\\ \;\;\;\;\frac{y}{a} \cdot \left(z - t\right) + x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z - t}{a} + x\\ \end{array}\]
x + \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;y \le -31789.4821495515171:\\
\;\;\;\;y \cdot \frac{z - t}{a} + x\\

\mathbf{elif}\;y \le 4.0978388594748642 \cdot 10^{-179}:\\
\;\;\;\;\frac{\left(z - t\right) \cdot y}{a} + x\\

\mathbf{elif}\;y \le 2.78403719146078999 \cdot 10^{-66}:\\
\;\;\;\;\frac{y}{a} \cdot \left(z - t\right) + x\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r274342 = x;
        double r274343 = y;
        double r274344 = z;
        double r274345 = t;
        double r274346 = r274344 - r274345;
        double r274347 = r274343 * r274346;
        double r274348 = a;
        double r274349 = r274347 / r274348;
        double r274350 = r274342 + r274349;
        return r274350;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r274351 = y;
        double r274352 = -31789.482149551517;
        bool r274353 = r274351 <= r274352;
        double r274354 = z;
        double r274355 = t;
        double r274356 = r274354 - r274355;
        double r274357 = a;
        double r274358 = r274356 / r274357;
        double r274359 = r274351 * r274358;
        double r274360 = x;
        double r274361 = r274359 + r274360;
        double r274362 = 4.097838859474864e-179;
        bool r274363 = r274351 <= r274362;
        double r274364 = r274356 * r274351;
        double r274365 = r274364 / r274357;
        double r274366 = r274365 + r274360;
        double r274367 = 2.78403719146079e-66;
        bool r274368 = r274351 <= r274367;
        double r274369 = r274351 / r274357;
        double r274370 = r274369 * r274356;
        double r274371 = r274370 + r274360;
        double r274372 = r274368 ? r274371 : r274361;
        double r274373 = r274363 ? r274366 : r274372;
        double r274374 = r274353 ? r274361 : r274373;
        return r274374;
}

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

Original6.0
Target0.7
Herbie0.9
\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753 \cdot 10^{-10}:\\ \;\;\;\;x + \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.8944268627920891 \cdot 10^{-49}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if y < -31789.482149551517 or 2.78403719146079e-66 < y

    1. Initial program 12.9

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

    2. Simplified3.2

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

    4. Applied fma-udef3.2

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

    6. Applied div-inv3.2

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

    7. Applied associate-*l*1.2

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

    8. Simplified1.1

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

    if -31789.482149551517 < y < 4.097838859474864e-179

    1. Initial program 0.4

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

    2. Simplified2.1

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

    4. Applied fma-udef2.1

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

    5. Taylor expanded around 0 0.4

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

    6. Simplified0.4

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

    if 4.097838859474864e-179 < y < 2.78403719146079e-66

    1. Initial program 0.4

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

    2. Simplified1.4

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

    4. Applied fma-udef1.4

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

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -31789.4821495515171:\\ \;\;\;\;y \cdot \frac{z - t}{a} + x\\ \mathbf{elif}\;y \le 4.0978388594748642 \cdot 10^{-179}:\\ \;\;\;\;\frac{\left(z - t\right) \cdot y}{a} + x\\ \mathbf{elif}\;y \le 2.78403719146078999 \cdot 10^{-66}:\\ \;\;\;\;\frac{y}{a} \cdot \left(z - t\right) + x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z - t}{a} + x\\ \end{array}\]

Reproduce

herbie shell --seed 2020018 +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)))