Optimisation.CirclePacking:place from circle-packing-0.1.0.4, D

Average Error: 6.5 → 1.2

Time: 2.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x + \frac{y \cdot \left(z - x\right)}{t} = -\infty:\\ \;\;\;\;x + \frac{y}{\frac{t}{z - x}}\\ \mathbf{elif}\;x + \frac{y \cdot \left(z - x\right)}{t} \le 3.909385011385941616558339325394184542054 \cdot 10^{208}:\\ \;\;\;\;x + \frac{y \cdot \left(z - x\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - x, x\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r307510 = x;
        double r307511 = y;
        double r307512 = z;
        double r307513 = r307512 - r307510;
        double r307514 = r307511 * r307513;
        double r307515 = t;
        double r307516 = r307514 / r307515;
        double r307517 = r307510 + r307516;
        return r307517;
}

↓

double f(double x, double y, double z, double t) {
        double r307518 = x;
        double r307519 = y;
        double r307520 = z;
        double r307521 = r307520 - r307518;
        double r307522 = r307519 * r307521;
        double r307523 = t;
        double r307524 = r307522 / r307523;
        double r307525 = r307518 + r307524;
        double r307526 = -inf.0;
        bool r307527 = r307525 <= r307526;
        double r307528 = r307523 / r307521;
        double r307529 = r307519 / r307528;
        double r307530 = r307518 + r307529;
        double r307531 = 3.9093850113859416e+208;
        bool r307532 = r307525 <= r307531;
        double r307533 = r307519 / r307523;
        double r307534 = fma(r307533, r307521, r307518);
        double r307535 = r307532 ? r307525 : r307534;
        double r307536 = r307527 ? r307530 : r307535;
        return r307536;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	6.5
Target	2.1
Herbie	1.2

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

Derivation

1. Split input into 3 regimes

2. if (+ x (/ (* y (- z x)) t)) < -inf.0

   1. Initial program 64.0

      \[x + \frac{y \cdot \left(z - x\right)}{t}\]
   2. Using strategy rm

   3. Applied associate-/l* 0.2

      \[\leadsto x + \color{blue}{\frac{y}{\frac{t}{z - x}}}\]

   if -inf.0 < (+ x (/ (* y (- z x)) t)) < 3.9093850113859416e+208

   1. Initial program 0.8

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

   if 3.9093850113859416e+208 < (+ x (/ (* y (- z x)) t))

   1. Initial program 20.1

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

   2. Simplified 3.7

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{t}, z - x, x\right)}\]
3. Recombined 3 regimes into one program.

4. Final simplification 1.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{y \cdot \left(z - x\right)}{t} = -\infty:\\ \;\;\;\;x + \frac{y}{\frac{t}{z - x}}\\ \mathbf{elif}\;x + \frac{y \cdot \left(z - x\right)}{t} \le 3.909385011385941616558339325394184542054 \cdot 10^{208}:\\ \;\;\;\;x + \frac{y \cdot \left(z - x\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - x, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 +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)))