Average Error: 6.4 → 1.6
Time: 2.4s
Precision: 64
\[x + \frac{y \cdot \left(z - x\right)}{t}\]
↓
\[\begin{array}{l} \mathbf{if}\;x + \frac{y \cdot \left(z - x\right)}{t} \le 8.98940625938930919 \cdot 10^{-113} \lor \neg \left(x + \frac{y \cdot \left(z - x\right)}{t} \le 1.0276911230637797 \cdot 10^{260}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(z - x\right)}{t}\\ \end{array}\]
x + \frac{y \cdot \left(z - x\right)}{t}
↓
\begin{array}{l}
\mathbf{if}\;x + \frac{y \cdot \left(z - x\right)}{t} \le 8.98940625938930919 \cdot 10^{-113} \lor \neg \left(x + \frac{y \cdot \left(z - x\right)}{t} \le 1.0276911230637797 \cdot 10^{260}\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 r397255 = x;
        double r397256 = y;
        double r397257 = z;
        double r397258 = r397257 - r397255;
        double r397259 = r397256 * r397258;
        double r397260 = t;
        double r397261 = r397259 / r397260;
        double r397262 = r397255 + r397261;
        return r397262;
}
↓
double f(double x, double y, double z, double t) {
        double r397263 = x;
        double r397264 = y;
        double r397265 = z;
        double r397266 = r397265 - r397263;
        double r397267 = r397264 * r397266;
        double r397268 = t;
        double r397269 = r397267 / r397268;
        double r397270 = r397263 + r397269;
        double r397271 = 8.989406259389309e-113;
        bool r397272 = r397270 <= r397271;
        double r397273 = 1.0276911230637797e+260;
        bool r397274 = r397270 <= r397273;
        double r397275 = !r397274;
        bool r397276 = r397272 || r397275;
        double r397277 = r397264 / r397268;
        double r397278 = fma(r397277, r397266, r397263);
        double r397279 = r397276 ? r397278 : r397270;
        return r397279;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original6.4
Target2.2
Herbie1.6
\[x - \left(x \cdot \frac{y}{t} + \left(-z\right) \cdot \frac{y}{t}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if (+ x (/ (* y (- z x)) t)) < 8.989406259389309e-113 or 1.0276911230637797e+260 < (+ x (/ (* y (- z x)) t))

    1. Initial program 9.6

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

    2. Simplified2.4

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

    if 8.989406259389309e-113 < (+ x (/ (* y (- z x)) t)) < 1.0276911230637797e+260

    1. Initial program 0.2

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

  4. Final simplification1.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{y \cdot \left(z - x\right)}{t} \le 8.98940625938930919 \cdot 10^{-113} \lor \neg \left(x + \frac{y \cdot \left(z - x\right)}{t} \le 1.0276911230637797 \cdot 10^{260}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(z - x\right)}{t}\\ \end{array}\]

Reproduce

herbie shell --seed 2020020 +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)))