Average Error: 6.0 → 0.4
Time: 2.9s
Precision: 64
\[x - \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \le -2.58610512582075527 \cdot 10^{286} \lor \neg \left(y \cdot \left(z - t\right) \le 3.55029444822809126 \cdot 10^{190}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \end{array}\]
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}, t - z, 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 r274985 = x;
        double r274986 = y;
        double r274987 = z;
        double r274988 = t;
        double r274989 = r274987 - r274988;
        double r274990 = r274986 * r274989;
        double r274991 = a;
        double r274992 = r274990 / r274991;
        double r274993 = r274985 - r274992;
        return r274993;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r274994 = y;
        double r274995 = z;
        double r274996 = t;
        double r274997 = r274995 - r274996;
        double r274998 = r274994 * r274997;
        double r274999 = -2.5861051258207553e+286;
        bool r275000 = r274998 <= r274999;
        double r275001 = 3.5502944482280913e+190;
        bool r275002 = r274998 <= r275001;
        double r275003 = !r275002;
        bool r275004 = r275000 || r275003;
        double r275005 = a;
        double r275006 = r274994 / r275005;
        double r275007 = r274996 - r274995;
        double r275008 = x;
        double r275009 = fma(r275006, r275007, r275008);
        double r275010 = r274998 / r275005;
        double r275011 = r275008 - r275010;
        double r275012 = r275004 ? r275009 : r275011;
        return r275012;
}

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

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

    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

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

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, F"
  :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)))