Average Error: 6.0 → 0.8
Time: 16.4s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le -1.598281406670949442324391521520754455565 \cdot 10^{-10}:\\ \;\;\;\;\frac{y}{\frac{a}{z - t}} + x\\ \mathbf{elif}\;a \le 6.452314596038023650887609022413351253596 \cdot 10^{-142}:\\ \;\;\;\;\frac{y \cdot \left(z - t\right)}{a} + x\\ \mathbf{elif}\;a \le 332600248696025938523062272:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{a}{z - t}} + x\\ \end{array}\]
x + \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;a \le -1.598281406670949442324391521520754455565 \cdot 10^{-10}:\\
\;\;\;\;\frac{y}{\frac{a}{z - t}} + x\\

\mathbf{elif}\;a \le 6.452314596038023650887609022413351253596 \cdot 10^{-142}:\\
\;\;\;\;\frac{y \cdot \left(z - t\right)}{a} + x\\

\mathbf{elif}\;a \le 332600248696025938523062272:\\
\;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r198605 = x;
        double r198606 = y;
        double r198607 = z;
        double r198608 = t;
        double r198609 = r198607 - r198608;
        double r198610 = r198606 * r198609;
        double r198611 = a;
        double r198612 = r198610 / r198611;
        double r198613 = r198605 + r198612;
        return r198613;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r198614 = a;
        double r198615 = -1.5982814066709494e-10;
        bool r198616 = r198614 <= r198615;
        double r198617 = y;
        double r198618 = z;
        double r198619 = t;
        double r198620 = r198618 - r198619;
        double r198621 = r198614 / r198620;
        double r198622 = r198617 / r198621;
        double r198623 = x;
        double r198624 = r198622 + r198623;
        double r198625 = 6.452314596038024e-142;
        bool r198626 = r198614 <= r198625;
        double r198627 = r198617 * r198620;
        double r198628 = r198627 / r198614;
        double r198629 = r198628 + r198623;
        double r198630 = 3.3260024869602594e+26;
        bool r198631 = r198614 <= r198630;
        double r198632 = r198617 / r198614;
        double r198633 = fma(r198632, r198620, r198623);
        double r198634 = r198631 ? r198633 : r198624;
        double r198635 = r198626 ? r198629 : r198634;
        double r198636 = r198616 ? r198624 : r198635;
        return r198636;
}

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
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753216593153715602325729 \cdot 10^{-10}:\\ \;\;\;\;x + \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.894426862792089097262541964056085749132 \cdot 10^{-49}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if a < -1.5982814066709494e-10 or 3.3260024869602594e+26 < a

    1. Initial program 9.3

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

    2. Simplified0.4

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

    4. Applied fma-udef0.5

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

    5. Simplified0.5

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

    if -1.5982814066709494e-10 < a < 6.452314596038024e-142

    1. Initial program 1.1

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

    if 6.452314596038024e-142 < a < 3.3260024869602594e+26

    1. Initial program 0.7

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

    2. Simplified7.2

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

    4. Applied fma-udef7.2

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

    5. Simplified6.1

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

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

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

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

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

    9. Applied distribute-lft-out6.1

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

    10. Simplified1.3

      \[\leadsto 1 \cdot \color{blue}{\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -1.598281406670949442324391521520754455565 \cdot 10^{-10}:\\ \;\;\;\;\frac{y}{\frac{a}{z - t}} + x\\ \mathbf{elif}\;a \le 6.452314596038023650887609022413351253596 \cdot 10^{-142}:\\ \;\;\;\;\frac{y \cdot \left(z - t\right)}{a} + x\\ \mathbf{elif}\;a \le 332600248696025938523062272:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{a}{z - t}} + x\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 +o rules:numerics
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, E"

  :herbie-target
  (if (< y -1.0761266216389975e-10) (+ x (/ 1.0 (/ (/ a (- z t)) y))) (if (< y 2.894426862792089e-49) (+ x (/ (* y (- z t)) a)) (+ x (/ y (/ a (- z t))))))

  (+ x (/ (* y (- z t)) a)))