Average Error: 6.0 → 1.5
Time: 14.5s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le -7.397051135373071984248927387992949000155 \cdot 10^{-100}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{a}, y, x\right)\\ \mathbf{elif}\;a \le 2.618974838534455632423168448609278208763 \cdot 10^{-91}:\\ \;\;\;\;\frac{1}{a} \cdot \left(\left(z - t\right) \cdot y\right) + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\ \end{array}\]
x + \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;a \le -7.397051135373071984248927387992949000155 \cdot 10^{-100}:\\
\;\;\;\;\mathsf{fma}\left(\frac{z - t}{a}, y, x\right)\\

\mathbf{elif}\;a \le 2.618974838534455632423168448609278208763 \cdot 10^{-91}:\\
\;\;\;\;\frac{1}{a} \cdot \left(\left(z - t\right) \cdot y\right) + x\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r182644 = x;
        double r182645 = y;
        double r182646 = z;
        double r182647 = t;
        double r182648 = r182646 - r182647;
        double r182649 = r182645 * r182648;
        double r182650 = a;
        double r182651 = r182649 / r182650;
        double r182652 = r182644 + r182651;
        return r182652;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r182653 = a;
        double r182654 = -7.397051135373072e-100;
        bool r182655 = r182653 <= r182654;
        double r182656 = z;
        double r182657 = t;
        double r182658 = r182656 - r182657;
        double r182659 = r182658 / r182653;
        double r182660 = y;
        double r182661 = x;
        double r182662 = fma(r182659, r182660, r182661);
        double r182663 = 2.6189748385344556e-91;
        bool r182664 = r182653 <= r182663;
        double r182665 = 1.0;
        double r182666 = r182665 / r182653;
        double r182667 = r182658 * r182660;
        double r182668 = r182666 * r182667;
        double r182669 = r182668 + r182661;
        double r182670 = r182660 / r182653;
        double r182671 = fma(r182670, r182658, r182661);
        double r182672 = r182664 ? r182669 : r182671;
        double r182673 = r182655 ? r182662 : r182672;
        return r182673;
}

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
Herbie1.5
\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753216593153715602325729 \cdot 10^{-10}:\\ \;\;\;\;x + \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.894426862792089097262541964056085749132 \cdot 10^{-49}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if a < -7.397051135373072e-100

    1. Initial program 7.8

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

    2. Simplified1.8

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

    4. Applied fma-udef1.8

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

    5. Simplified1.9

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

    7. Applied associate-/r/1.5

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

    8. Applied fma-def1.5

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

    if -7.397051135373072e-100 < a < 2.6189748385344556e-91

    1. Initial program 1.1

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

    2. Simplified5.3

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

    4. Applied fma-udef5.3

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

    5. Simplified4.5

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

    7. Applied div-inv4.6

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

    8. Applied *-un-lft-identity4.6

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

    9. Applied times-frac1.2

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

    10. Simplified1.2

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

    if 2.6189748385344556e-91 < a

    1. Initial program 7.4

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

    2. Simplified1.7

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

  4. Final simplification1.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -7.397051135373071984248927387992949000155 \cdot 10^{-100}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{a}, y, x\right)\\ \mathbf{elif}\;a \le 2.618974838534455632423168448609278208763 \cdot 10^{-91}:\\ \;\;\;\;\frac{1}{a} \cdot \left(\left(z - t\right) \cdot y\right) + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 +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)))