Average Error: 6.2 → 1.7
Time: 3.1s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le -1.8744657462895403389224078436647417432 \cdot 10^{-286} \lor \neg \left(a \le 3146.272963083657941751880571246147155762\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \end{array}\]
x + \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;a \le -1.8744657462895403389224078436647417432 \cdot 10^{-286} \lor \neg \left(a \le 3146.272963083657941751880571246147155762\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, 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 r264004 = x;
        double r264005 = y;
        double r264006 = z;
        double r264007 = t;
        double r264008 = r264006 - r264007;
        double r264009 = r264005 * r264008;
        double r264010 = a;
        double r264011 = r264009 / r264010;
        double r264012 = r264004 + r264011;
        return r264012;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r264013 = a;
        double r264014 = -1.8744657462895403e-286;
        bool r264015 = r264013 <= r264014;
        double r264016 = 3146.272963083658;
        bool r264017 = r264013 <= r264016;
        double r264018 = !r264017;
        bool r264019 = r264015 || r264018;
        double r264020 = y;
        double r264021 = r264020 / r264013;
        double r264022 = z;
        double r264023 = t;
        double r264024 = r264022 - r264023;
        double r264025 = x;
        double r264026 = fma(r264021, r264024, r264025);
        double r264027 = r264020 * r264024;
        double r264028 = r264027 / r264013;
        double r264029 = r264025 + r264028;
        double r264030 = r264019 ? r264026 : r264029;
        return r264030;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original6.2
Target0.7
Herbie1.7
\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753216593153715602325729 \cdot 10^{-10}:\\ \;\;\;\;x + \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.894426862792089097262541964056085749132 \cdot 10^{-49}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if a < -1.8744657462895403e-286 or 3146.272963083658 < a

    1. Initial program 7.6

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

    2. Simplified1.9

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

    if -1.8744657462895403e-286 < a < 3146.272963083658

    1. Initial program 1.0

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

  4. Final simplification1.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -1.8744657462895403389224078436647417432 \cdot 10^{-286} \lor \neg \left(a \le 3146.272963083657941751880571246147155762\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 +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)))