Average Error: 6.0 → 1.0
Time: 2.8s
Precision: 64
\[x - \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -4.05710526454107349 \cdot 10^{60}:\\ \;\;\;\;x - y \cdot \frac{z - t}{a}\\ \mathbf{elif}\;y \le 8.84054664567154418 \cdot 10^{-179}:\\ \;\;\;\;x - \frac{y \cdot z + y \cdot \left(-t\right)}{a}\\ \mathbf{elif}\;y \le 3.698579982247782 \cdot 10^{-65}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{z - t}{a}\\ \end{array}\]
x - \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;y \le -4.05710526454107349 \cdot 10^{60}:\\
\;\;\;\;x - y \cdot \frac{z - t}{a}\\

\mathbf{elif}\;y \le 8.84054664567154418 \cdot 10^{-179}:\\
\;\;\;\;x - \frac{y \cdot z + y \cdot \left(-t\right)}{a}\\

\mathbf{elif}\;y \le 3.698579982247782 \cdot 10^{-65}:\\
\;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r325599 = x;
        double r325600 = y;
        double r325601 = z;
        double r325602 = t;
        double r325603 = r325601 - r325602;
        double r325604 = r325600 * r325603;
        double r325605 = a;
        double r325606 = r325604 / r325605;
        double r325607 = r325599 - r325606;
        return r325607;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r325608 = y;
        double r325609 = -4.0571052645410735e+60;
        bool r325610 = r325608 <= r325609;
        double r325611 = x;
        double r325612 = z;
        double r325613 = t;
        double r325614 = r325612 - r325613;
        double r325615 = a;
        double r325616 = r325614 / r325615;
        double r325617 = r325608 * r325616;
        double r325618 = r325611 - r325617;
        double r325619 = 8.840546645671544e-179;
        bool r325620 = r325608 <= r325619;
        double r325621 = r325608 * r325612;
        double r325622 = -r325613;
        double r325623 = r325608 * r325622;
        double r325624 = r325621 + r325623;
        double r325625 = r325624 / r325615;
        double r325626 = r325611 - r325625;
        double r325627 = 3.698579982247782e-65;
        bool r325628 = r325608 <= r325627;
        double r325629 = r325608 / r325615;
        double r325630 = r325613 - r325612;
        double r325631 = fma(r325629, r325630, r325611);
        double r325632 = r325628 ? r325631 : r325618;
        double r325633 = r325620 ? r325626 : r325632;
        double r325634 = r325610 ? r325618 : r325633;
        return r325634;
}

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
Herbie1.0
\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753 \cdot 10^{-10}:\\ \;\;\;\;x - \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.8944268627920891 \cdot 10^{-49}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if y < -4.0571052645410735e+60 or 3.698579982247782e-65 < y

    1. Initial program 14.0

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

    3. Applied *-un-lft-identity14.0

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

    4. Applied times-frac1.3

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

    5. Simplified1.3

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

    if -4.0571052645410735e+60 < y < 8.840546645671544e-179

    1. Initial program 0.8

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

    3. Applied sub-neg0.8

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

    4. Applied distribute-lft-in0.8

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

    if 8.840546645671544e-179 < y < 3.698579982247782e-65

    1. Initial program 0.4

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

    2. Simplified1.4

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -4.05710526454107349 \cdot 10^{60}:\\ \;\;\;\;x - y \cdot \frac{z - t}{a}\\ \mathbf{elif}\;y \le 8.84054664567154418 \cdot 10^{-179}:\\ \;\;\;\;x - \frac{y \cdot z + y \cdot \left(-t\right)}{a}\\ \mathbf{elif}\;y \le 3.698579982247782 \cdot 10^{-65}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{z - t}{a}\\ \end{array}\]

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, 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)))