Average Error: 6.2 → 2.5
Time: 7.2s
Precision: 64
\[x - \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.8095903151115152 \cdot 10^{-138}:\\ \;\;\;\;x - \frac{y}{a} \cdot \left(z - t\right)\\ \mathbf{elif}\;z \le 6.9390753974297484 \cdot 10^{-149}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z - t}{\frac{a}{y}}\\ \end{array}\]
x - \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;z \le -1.8095903151115152 \cdot 10^{-138}:\\
\;\;\;\;x - \frac{y}{a} \cdot \left(z - t\right)\\

\mathbf{elif}\;z \le 6.9390753974297484 \cdot 10^{-149}:\\
\;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r369360 = x;
        double r369361 = y;
        double r369362 = z;
        double r369363 = t;
        double r369364 = r369362 - r369363;
        double r369365 = r369361 * r369364;
        double r369366 = a;
        double r369367 = r369365 / r369366;
        double r369368 = r369360 - r369367;
        return r369368;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r369369 = z;
        double r369370 = -1.8095903151115152e-138;
        bool r369371 = r369369 <= r369370;
        double r369372 = x;
        double r369373 = y;
        double r369374 = a;
        double r369375 = r369373 / r369374;
        double r369376 = t;
        double r369377 = r369369 - r369376;
        double r369378 = r369375 * r369377;
        double r369379 = r369372 - r369378;
        double r369380 = 6.939075397429748e-149;
        bool r369381 = r369369 <= r369380;
        double r369382 = r369374 / r369377;
        double r369383 = r369373 / r369382;
        double r369384 = r369372 - r369383;
        double r369385 = r369374 / r369373;
        double r369386 = r369377 / r369385;
        double r369387 = r369372 - r369386;
        double r369388 = r369381 ? r369384 : r369387;
        double r369389 = r369371 ? r369379 : r369388;
        return r369389;
}

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

Original6.2
Target0.7
Herbie2.5
\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753 \cdot 10^{-10}:\\ \;\;\;\;x - \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.8944268627920891 \cdot 10^{-49}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -1.8095903151115152e-138

    1. Initial program 7.4

      \[x - \frac{y \cdot \left(z - t\right)}{a}\]
    2. Using strategy rm

    3. Applied associate-/l*6.1

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

    5. Applied *-un-lft-identity6.1

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

    6. Applied *-un-lft-identity6.1

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

    7. Applied times-frac6.1

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

    8. Applied *-un-lft-identity6.1

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

    9. Applied times-frac6.1

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

    10. Simplified6.1

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

    11. Simplified2.1

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

    if -1.8095903151115152e-138 < z < 6.939075397429748e-149

    1. Initial program 4.3

      \[x - \frac{y \cdot \left(z - t\right)}{a}\]
    2. Using strategy rm

    3. Applied associate-/l*3.3

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

    if 6.939075397429748e-149 < z

    1. Initial program 6.5

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

    2. Taylor expanded around 0 6.5

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

    3. Simplified2.2

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

  4. Final simplification2.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.8095903151115152 \cdot 10^{-138}:\\ \;\;\;\;x - \frac{y}{a} \cdot \left(z - t\right)\\ \mathbf{elif}\;z \le 6.9390753974297484 \cdot 10^{-149}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z - t}{\frac{a}{y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, F"
  :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)))