Average Error: 6.0 → 0.4
Time: 2.8s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \le -2.58610512582075527 \cdot 10^{286} \lor \neg \left(y \cdot \left(z - t\right) \le 3.55029444822809126 \cdot 10^{190}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \end{array}\]
x + \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;y \cdot \left(z - t\right) \le -2.58610512582075527 \cdot 10^{286} \lor \neg \left(y \cdot \left(z - t\right) \le 3.55029444822809126 \cdot 10^{190}\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 r279082 = x;
        double r279083 = y;
        double r279084 = z;
        double r279085 = t;
        double r279086 = r279084 - r279085;
        double r279087 = r279083 * r279086;
        double r279088 = a;
        double r279089 = r279087 / r279088;
        double r279090 = r279082 + r279089;
        return r279090;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r279091 = y;
        double r279092 = z;
        double r279093 = t;
        double r279094 = r279092 - r279093;
        double r279095 = r279091 * r279094;
        double r279096 = -2.5861051258207553e+286;
        bool r279097 = r279095 <= r279096;
        double r279098 = 3.5502944482280913e+190;
        bool r279099 = r279095 <= r279098;
        double r279100 = !r279099;
        bool r279101 = r279097 || r279100;
        double r279102 = a;
        double r279103 = r279091 / r279102;
        double r279104 = x;
        double r279105 = fma(r279103, r279094, r279104);
        double r279106 = r279095 / r279102;
        double r279107 = r279104 + r279106;
        double r279108 = r279101 ? r279105 : r279107;
        return r279108;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original6.0
Target0.6
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753 \cdot 10^{-10}:\\ \;\;\;\;x + \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.8944268627920891 \cdot 10^{-49}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* y (- z t)) < -2.5861051258207553e+286 or 3.5502944482280913e+190 < (* y (- z t))

    1. Initial program 34.5

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

    2. Simplified0.5

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

    if -2.5861051258207553e+286 < (* y (- z t)) < 3.5502944482280913e+190

    1. Initial program 0.3

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \le -2.58610512582075527 \cdot 10^{286} \lor \neg \left(y \cdot \left(z - t\right) \le 3.55029444822809126 \cdot 10^{190}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020062 +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)))