Average Error: 6.2 → 2.4
Time: 10.3s
Precision: 64
\[x - \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.8095903151115152 \cdot 10^{-138} \lor \neg \left(z \le 6.9390753974297484 \cdot 10^{-149}\right):\\ \;\;\;\;\frac{y}{a} \cdot \left(t - z\right) + x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{a}{t - z}} + x\\ \end{array}\]
x - \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;z \le -1.8095903151115152 \cdot 10^{-138} \lor \neg \left(z \le 6.9390753974297484 \cdot 10^{-149}\right):\\
\;\;\;\;\frac{y}{a} \cdot \left(t - z\right) + x\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r266318 = x;
        double r266319 = y;
        double r266320 = z;
        double r266321 = t;
        double r266322 = r266320 - r266321;
        double r266323 = r266319 * r266322;
        double r266324 = a;
        double r266325 = r266323 / r266324;
        double r266326 = r266318 - r266325;
        return r266326;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r266327 = z;
        double r266328 = -1.8095903151115152e-138;
        bool r266329 = r266327 <= r266328;
        double r266330 = 6.939075397429748e-149;
        bool r266331 = r266327 <= r266330;
        double r266332 = !r266331;
        bool r266333 = r266329 || r266332;
        double r266334 = y;
        double r266335 = a;
        double r266336 = r266334 / r266335;
        double r266337 = t;
        double r266338 = r266337 - r266327;
        double r266339 = r266336 * r266338;
        double r266340 = x;
        double r266341 = r266339 + r266340;
        double r266342 = r266335 / r266338;
        double r266343 = r266334 / r266342;
        double r266344 = r266343 + r266340;
        double r266345 = r266333 ? r266341 : r266344;
        return r266345;
}

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
Herbie2.4
\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753 \cdot 10^{-10}:\\ \;\;\;\;x - \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.8944268627920891 \cdot 10^{-49}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -1.8095903151115152e-138 or 6.939075397429748e-149 < z

    1. Initial program 7.0

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

    2. Simplified6.7

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

    4. Applied clear-num6.7

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{\frac{a}{t - z}}}, y, x\right)\]
    5. Using strategy rm

    6. Applied fma-udef6.7

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

    7. Simplified6.0

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

    9. Applied associate-/r/2.1

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

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

    1. Initial program 4.3

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

    2. Simplified3.6

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

    4. Applied clear-num3.7

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{\frac{a}{t - z}}}, y, x\right)\]
    5. Using strategy rm

    6. Applied fma-udef3.7

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

    7. Simplified3.3

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

  4. Final simplification2.4

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

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(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)))