Average Error: 6.7 → 2.1
Time: 20.7s
Precision: 64
\[x + \frac{y \cdot \left(z - x\right)}{t}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 3.926176551076473834370923735405751547934 \cdot 10^{-240} \lor \neg \left(x \le 8.301025463282562919137701221500573529604 \cdot 10^{-126}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(z - x\right)}{t}\\ \end{array}\]
x + \frac{y \cdot \left(z - x\right)}{t}
↓
\begin{array}{l}
\mathbf{if}\;x \le 3.926176551076473834370923735405751547934 \cdot 10^{-240} \lor \neg \left(x \le 8.301025463282562919137701221500573529604 \cdot 10^{-126}\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - x, x\right)\\

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

\end{array}
double f(double x, double y, double z, double t) {
        double r177798 = x;
        double r177799 = y;
        double r177800 = z;
        double r177801 = r177800 - r177798;
        double r177802 = r177799 * r177801;
        double r177803 = t;
        double r177804 = r177802 / r177803;
        double r177805 = r177798 + r177804;
        return r177805;
}
↓
double f(double x, double y, double z, double t) {
        double r177806 = x;
        double r177807 = 3.926176551076474e-240;
        bool r177808 = r177806 <= r177807;
        double r177809 = 8.301025463282563e-126;
        bool r177810 = r177806 <= r177809;
        double r177811 = !r177810;
        bool r177812 = r177808 || r177811;
        double r177813 = y;
        double r177814 = t;
        double r177815 = r177813 / r177814;
        double r177816 = z;
        double r177817 = r177816 - r177806;
        double r177818 = fma(r177815, r177817, r177806);
        double r177819 = r177813 * r177817;
        double r177820 = r177819 / r177814;
        double r177821 = r177806 + r177820;
        double r177822 = r177812 ? r177818 : r177821;
        return r177822;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original6.7
Target2.0
Herbie2.1
\[x - \left(x \cdot \frac{y}{t} + \left(-z\right) \cdot \frac{y}{t}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if x < 3.926176551076474e-240 or 8.301025463282563e-126 < x

    1. Initial program 7.0

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

    2. Simplified1.8

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

    if 3.926176551076474e-240 < x < 8.301025463282563e-126

    1. Initial program 4.6

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

  4. Final simplification2.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 3.926176551076473834370923735405751547934 \cdot 10^{-240} \lor \neg \left(x \le 8.301025463282562919137701221500573529604 \cdot 10^{-126}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(z - x\right)}{t}\\ \end{array}\]

Reproduce

herbie shell --seed 2019322 +o rules:numerics
(FPCore (x y z t)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, D"
  :precision binary64

  :herbie-target
  (- x (+ (* x (/ y t)) (* (- z) (/ y t))))

  (+ x (/ (* y (- z x)) t)))