Average Error: 6.1 → 0.5
Time: 11.0s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \le -8.00872266911859742407687626933145763664 \cdot 10^{184} \lor \neg \left(y \cdot \left(z - t\right) \le 1.251434998882354340368841462679706875691 \cdot 10^{193}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \end{array}\]
x + \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;y \cdot \left(z - t\right) \le -8.00872266911859742407687626933145763664 \cdot 10^{184} \lor \neg \left(y \cdot \left(z - t\right) \le 1.251434998882354340368841462679706875691 \cdot 10^{193}\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{z - t}{a}, y, x\right)\\

\mathbf{else}:\\
\;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r216801 = x;
        double r216802 = y;
        double r216803 = z;
        double r216804 = t;
        double r216805 = r216803 - r216804;
        double r216806 = r216802 * r216805;
        double r216807 = a;
        double r216808 = r216806 / r216807;
        double r216809 = r216801 + r216808;
        return r216809;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r216810 = y;
        double r216811 = z;
        double r216812 = t;
        double r216813 = r216811 - r216812;
        double r216814 = r216810 * r216813;
        double r216815 = -8.008722669118597e+184;
        bool r216816 = r216814 <= r216815;
        double r216817 = 1.2514349988823543e+193;
        bool r216818 = r216814 <= r216817;
        double r216819 = !r216818;
        bool r216820 = r216816 || r216819;
        double r216821 = a;
        double r216822 = r216813 / r216821;
        double r216823 = x;
        double r216824 = fma(r216822, r216810, r216823);
        double r216825 = r216814 / r216821;
        double r216826 = r216823 + r216825;
        double r216827 = r216820 ? r216824 : r216826;
        return r216827;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original6.1
Target0.6
Herbie0.5
\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753216593153715602325729 \cdot 10^{-10}:\\ \;\;\;\;x + \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.894426862792089097262541964056085749132 \cdot 10^{-49}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* y (- z t)) < -8.008722669118597e+184 or 1.2514349988823543e+193 < (* y (- z t))

    1. Initial program 26.3

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

    2. Simplified0.9

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

    4. Applied fma-udef0.9

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

    5. Simplified0.9

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

    7. Applied *-un-lft-identity0.9

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

    8. Applied *-un-lft-identity0.9

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

    9. Applied distribute-lft-out0.9

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

    10. Simplified1.0

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

    if -8.008722669118597e+184 < (* y (- z t)) < 1.2514349988823543e+193

    1. Initial program 0.4

      \[x + \frac{y \cdot \left(z - t\right)}{a}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \le -8.00872266911859742407687626933145763664 \cdot 10^{184} \lor \neg \left(y \cdot \left(z - t\right) \le 1.251434998882354340368841462679706875691 \cdot 10^{193}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019347 +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)))