Average Error: 5.9 → 1.2
Time: 8.0s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le -1.39950285805274381 \cdot 10^{104}:\\ \;\;\;\;\frac{y}{\frac{a}{z - t}} + x\\ \mathbf{elif}\;a \le 3.75740059193433464 \cdot 10^{65}:\\ \;\;\;\;\frac{y \cdot \left(z - t\right)}{a} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{a}, y, x\right)\\ \end{array}\]
x + \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;a \le -1.39950285805274381 \cdot 10^{104}:\\
\;\;\;\;\frac{y}{\frac{a}{z - t}} + x\\

\mathbf{elif}\;a \le 3.75740059193433464 \cdot 10^{65}:\\
\;\;\;\;\frac{y \cdot \left(z - t\right)}{a} + x\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r333022 = x;
        double r333023 = y;
        double r333024 = z;
        double r333025 = t;
        double r333026 = r333024 - r333025;
        double r333027 = r333023 * r333026;
        double r333028 = a;
        double r333029 = r333027 / r333028;
        double r333030 = r333022 + r333029;
        return r333030;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r333031 = a;
        double r333032 = -1.3995028580527438e+104;
        bool r333033 = r333031 <= r333032;
        double r333034 = y;
        double r333035 = z;
        double r333036 = t;
        double r333037 = r333035 - r333036;
        double r333038 = r333031 / r333037;
        double r333039 = r333034 / r333038;
        double r333040 = x;
        double r333041 = r333039 + r333040;
        double r333042 = 3.7574005919343346e+65;
        bool r333043 = r333031 <= r333042;
        double r333044 = r333034 * r333037;
        double r333045 = r333044 / r333031;
        double r333046 = r333045 + r333040;
        double r333047 = r333037 / r333031;
        double r333048 = fma(r333047, r333034, r333040);
        double r333049 = r333043 ? r333046 : r333048;
        double r333050 = r333033 ? r333041 : r333049;
        return r333050;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original5.9
Target0.7
Herbie1.2
\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753 \cdot 10^{-10}:\\ \;\;\;\;x + \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.8944268627920891 \cdot 10^{-49}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if a < -1.3995028580527438e+104

    1. Initial program 12.4

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

    2. Simplified2.3

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

    4. Applied fma-udef2.3

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

    5. Simplified12.4

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

    7. Applied associate-/l*0.7

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

    if -1.3995028580527438e+104 < a < 3.7574005919343346e+65

    1. Initial program 1.6

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

    2. Simplified3.0

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

    4. Applied fma-udef3.0

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

    5. Simplified1.6

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

    if 3.7574005919343346e+65 < a

    1. Initial program 10.0

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

    2. Simplified2.0

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

    4. Applied fma-udef2.0

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

    5. Simplified10.0

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

    6. Taylor expanded around 0 10.0

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

    7. Simplified2.2

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

    9. Applied *-un-lft-identity2.2

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

    10. Applied *-un-lft-identity2.2

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

    11. Applied distribute-lft-out2.2

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

    12. Simplified0.7

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

  4. Final simplification1.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -1.39950285805274381 \cdot 10^{104}:\\ \;\;\;\;\frac{y}{\frac{a}{z - t}} + x\\ \mathbf{elif}\;a \le 3.75740059193433464 \cdot 10^{65}:\\ \;\;\;\;\frac{y \cdot \left(z - t\right)}{a} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{a}, y, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 +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)))