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

Average Error: 6.3 → 2.1

Time: 2.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -735739220375.78149:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - x, x\right)\\ \mathbf{elif}\;y \le 8.34934423511015436 \cdot 10^{-219}:\\ \;\;\;\;x + \frac{y \cdot z + y \cdot \left(-x\right)}{t}\\ \mathbf{elif}\;y \le 7.97693725199553989 \cdot 10^{-106}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - x, x\right)\\ \mathbf{elif}\;y \le 1.96514479102399276 \cdot 10^{280}:\\ \;\;\;\;x + y \cdot \frac{z - x}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - x, x\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r234711 = x;
        double r234712 = y;
        double r234713 = z;
        double r234714 = r234713 - r234711;
        double r234715 = r234712 * r234714;
        double r234716 = t;
        double r234717 = r234715 / r234716;
        double r234718 = r234711 + r234717;
        return r234718;
}

↓

double f(double x, double y, double z, double t) {
        double r234719 = y;
        double r234720 = -735739220375.7815;
        bool r234721 = r234719 <= r234720;
        double r234722 = t;
        double r234723 = r234719 / r234722;
        double r234724 = z;
        double r234725 = x;
        double r234726 = r234724 - r234725;
        double r234727 = fma(r234723, r234726, r234725);
        double r234728 = 8.349344235110154e-219;
        bool r234729 = r234719 <= r234728;
        double r234730 = r234719 * r234724;
        double r234731 = -r234725;
        double r234732 = r234719 * r234731;
        double r234733 = r234730 + r234732;
        double r234734 = r234733 / r234722;
        double r234735 = r234725 + r234734;
        double r234736 = 7.97693725199554e-106;
        bool r234737 = r234719 <= r234736;
        double r234738 = 1.9651447910239928e+280;
        bool r234739 = r234719 <= r234738;
        double r234740 = r234726 / r234722;
        double r234741 = r234719 * r234740;
        double r234742 = r234725 + r234741;
        double r234743 = r234739 ? r234742 : r234727;
        double r234744 = r234737 ? r234727 : r234743;
        double r234745 = r234729 ? r234735 : r234744;
        double r234746 = r234721 ? r234727 : r234745;
        return r234746;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	6.3
Target	1.9
Herbie	2.1

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

Derivation

1. Split input into 3 regimes

2. if y < -735739220375.7815 or 8.349344235110154e-219 < y < 7.97693725199554e-106 or 1.9651447910239928e+280 < y

   1. Initial program 11.1

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

   2. Simplified 2.9

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

   if -735739220375.7815 < y < 8.349344235110154e-219

   1. Initial program 0.9

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

   3. Applied sub-neg 0.9

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

   4. Applied distribute-lft-in 0.9

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

   if 7.97693725199554e-106 < y < 1.9651447910239928e+280

   1. Initial program 9.2

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

   3. Applied *-un-lft-identity 9.2

      \[\leadsto x + \frac{y \cdot \left(z - x\right)}{\color{blue}{1 \cdot t}}\]

   4. Applied times-frac 3.1

      \[\leadsto x + \color{blue}{\frac{y}{1} \cdot \frac{z - x}{t}}\]

   5. Simplified 3.1

      \[\leadsto x + \color{blue}{y} \cdot \frac{z - x}{t}\]
3. Recombined 3 regimes into one program.

4. Final simplification 2.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -735739220375.78149:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - x, x\right)\\ \mathbf{elif}\;y \le 8.34934423511015436 \cdot 10^{-219}:\\ \;\;\;\;x + \frac{y \cdot z + y \cdot \left(-x\right)}{t}\\ \mathbf{elif}\;y \le 7.97693725199553989 \cdot 10^{-106}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - x, x\right)\\ \mathbf{elif}\;y \le 1.96514479102399276 \cdot 10^{280}:\\ \;\;\;\;x + y \cdot \frac{z - x}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - x, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020027 +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)))