Average Error: 6.0 → 0.5
Time: 6.6s
Precision: 64
\[x - \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \le -9.0544991846769346 \cdot 10^{260}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\ \mathbf{elif}\;y \cdot \left(z - t\right) \le 2.8461840642724514 \cdot 10^{164}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \end{array}\]
x - \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;y \cdot \left(z - t\right) \le -9.0544991846769346 \cdot 10^{260}:\\
\;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\

\mathbf{elif}\;y \cdot \left(z - t\right) \le 2.8461840642724514 \cdot 10^{164}:\\
\;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r1656 = x;
        double r1657 = y;
        double r1658 = z;
        double r1659 = t;
        double r1660 = r1658 - r1659;
        double r1661 = r1657 * r1660;
        double r1662 = a;
        double r1663 = r1661 / r1662;
        double r1664 = r1656 - r1663;
        return r1664;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r1665 = y;
        double r1666 = z;
        double r1667 = t;
        double r1668 = r1666 - r1667;
        double r1669 = r1665 * r1668;
        double r1670 = -9.054499184676935e+260;
        bool r1671 = r1669 <= r1670;
        double r1672 = a;
        double r1673 = r1665 / r1672;
        double r1674 = r1667 - r1666;
        double r1675 = x;
        double r1676 = fma(r1673, r1674, r1675);
        double r1677 = 2.8461840642724514e+164;
        bool r1678 = r1669 <= r1677;
        double r1679 = r1669 / r1672;
        double r1680 = r1675 - r1679;
        double r1681 = r1672 / r1668;
        double r1682 = r1665 / r1681;
        double r1683 = r1675 - r1682;
        double r1684 = r1678 ? r1680 : r1683;
        double r1685 = r1671 ? r1676 : r1684;
        return r1685;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original6.0
Target0.7
Herbie0.5
\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753 \cdot 10^{-10}:\\ \;\;\;\;x - \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.8944268627920891 \cdot 10^{-49}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (* y (- z t)) < -9.054499184676935e+260

    1. Initial program 42.1

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

    2. Simplified0.2

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

    if -9.054499184676935e+260 < (* y (- z t)) < 2.8461840642724514e+164

    1. Initial program 0.3

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

    if 2.8461840642724514e+164 < (* y (- z t))

    1. Initial program 23.2

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

    3. Applied associate-/l*1.3

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \le -9.0544991846769346 \cdot 10^{260}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\ \mathbf{elif}\;y \cdot \left(z - t\right) \le 2.8461840642724514 \cdot 10^{164}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020025 +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)))