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

Average Error: 6.4 → 0.9

Time: 14.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \le -1272558208.55477428436279296875:\\ \;\;\;\;x - y \cdot \left(\frac{t}{a} - \frac{z}{a}\right)\\ \mathbf{elif}\;a \le 3.071173039328613373912527296282876331673 \cdot 10^{-45}:\\ \;\;\;\;x - \left(\left(t - z\right) \cdot y\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \left(\frac{t}{\frac{a}{y}} - \frac{y}{\frac{a}{z}}\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r124713 = x;
        double r124714 = y;
        double r124715 = z;
        double r124716 = t;
        double r124717 = r124715 - r124716;
        double r124718 = r124714 * r124717;
        double r124719 = a;
        double r124720 = r124718 / r124719;
        double r124721 = r124713 + r124720;
        return r124721;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r124722 = a;
        double r124723 = -1272558208.5547743;
        bool r124724 = r124722 <= r124723;
        double r124725 = x;
        double r124726 = y;
        double r124727 = t;
        double r124728 = r124727 / r124722;
        double r124729 = z;
        double r124730 = r124729 / r124722;
        double r124731 = r124728 - r124730;
        double r124732 = r124726 * r124731;
        double r124733 = r124725 - r124732;
        double r124734 = 3.0711730393286134e-45;
        bool r124735 = r124722 <= r124734;
        double r124736 = r124727 - r124729;
        double r124737 = r124736 * r124726;
        double r124738 = 1.0;
        double r124739 = r124738 / r124722;
        double r124740 = r124737 * r124739;
        double r124741 = r124725 - r124740;
        double r124742 = r124722 / r124726;
        double r124743 = r124727 / r124742;
        double r124744 = r124722 / r124729;
        double r124745 = r124726 / r124744;
        double r124746 = r124743 - r124745;
        double r124747 = r124725 - r124746;
        double r124748 = r124735 ? r124741 : r124747;
        double r124749 = r124724 ? r124733 : r124748;
        return r124749;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	6.4
Target	0.7
Herbie	0.9

\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753216593153715602325729 \cdot 10^{-10}:\\ \;\;\;\;x + \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.894426862792089097262541964056085749132 \cdot 10^{-49}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if a < -1272558208.5547743

   1. Initial program 9.7

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

   2. Simplified 2.1

      \[\leadsto \color{blue}{x - \frac{y}{a} \cdot \left(t - z\right)}\]
   3. Using strategy rm

   4. Applied add-cube-cbrt 2.5

      \[\leadsto x - \color{blue}{\left(\left(\sqrt[3]{\frac{y}{a}} \cdot \sqrt[3]{\frac{y}{a}}\right) \cdot \sqrt[3]{\frac{y}{a}}\right)} \cdot \left(t - z\right)\]

   5. Applied associate-*l* 2.5

      \[\leadsto x - \color{blue}{\left(\sqrt[3]{\frac{y}{a}} \cdot \sqrt[3]{\frac{y}{a}}\right) \cdot \left(\sqrt[3]{\frac{y}{a}} \cdot \left(t - z\right)\right)}\]

   6. Taylor expanded around 0 9.7

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

   7. Simplified 0.5

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

   if -1272558208.5547743 < a < 3.0711730393286134e-45

   1. Initial program 1.0

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

   2. Simplified 4.1

      \[\leadsto \color{blue}{x - \frac{y}{a} \cdot \left(t - z\right)}\]
   3. Using strategy rm

   4. Applied *-commutative 4.1

      \[\leadsto x - \color{blue}{\left(t - z\right) \cdot \frac{y}{a}}\]
   5. Using strategy rm

   6. Applied div-inv 4.1

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

   7. Applied associate-*r* 1.0

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

   if 3.0711730393286134e-45 < a

   1. Initial program 8.9

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

   2. Simplified 1.6

      \[\leadsto \color{blue}{x - \frac{y}{a} \cdot \left(t - z\right)}\]
   3. Using strategy rm

   4. Applied add-cube-cbrt 2.1

      \[\leadsto x - \color{blue}{\left(\left(\sqrt[3]{\frac{y}{a}} \cdot \sqrt[3]{\frac{y}{a}}\right) \cdot \sqrt[3]{\frac{y}{a}}\right)} \cdot \left(t - z\right)\]

   5. Applied associate-*l* 2.1

      \[\leadsto x - \color{blue}{\left(\sqrt[3]{\frac{y}{a}} \cdot \sqrt[3]{\frac{y}{a}}\right) \cdot \left(\sqrt[3]{\frac{y}{a}} \cdot \left(t - z\right)\right)}\]
   6. Using strategy rm

   7. Applied add-cube-cbrt 2.1

      \[\leadsto x - \left(\sqrt[3]{\frac{y}{a}} \cdot \sqrt[3]{\frac{y}{\color{blue}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}}}\right) \cdot \left(\sqrt[3]{\frac{y}{a}} \cdot \left(t - z\right)\right)\]

   8. Applied add-cube-cbrt 2.1

      \[\leadsto x - \left(\sqrt[3]{\frac{y}{a}} \cdot \sqrt[3]{\frac{\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}}\right) \cdot \left(\sqrt[3]{\frac{y}{a}} \cdot \left(t - z\right)\right)\]

   9. Applied times-frac 2.1

      \[\leadsto x - \left(\sqrt[3]{\frac{y}{a}} \cdot \sqrt[3]{\color{blue}{\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{a}}}}\right) \cdot \left(\sqrt[3]{\frac{y}{a}} \cdot \left(t - z\right)\right)\]

   10. Applied cbrt-prod 2.1

       \[\leadsto x - \left(\sqrt[3]{\frac{y}{a}} \cdot \color{blue}{\left(\sqrt[3]{\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{a} \cdot \sqrt[3]{a}}} \cdot \sqrt[3]{\frac{\sqrt[3]{y}}{\sqrt[3]{a}}}\right)}\right) \cdot \left(\sqrt[3]{\frac{y}{a}} \cdot \left(t - z\right)\right)\]

   11. Simplified 2.1

       \[\leadsto x - \left(\sqrt[3]{\frac{y}{a}} \cdot \left(\color{blue}{\sqrt[3]{\frac{\sqrt[3]{y} \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{a}}}{\sqrt[3]{a}}}} \cdot \sqrt[3]{\frac{\sqrt[3]{y}}{\sqrt[3]{a}}}\right)\right) \cdot \left(\sqrt[3]{\frac{y}{a}} \cdot \left(t - z\right)\right)\]

   12. Taylor expanded around 0 8.9

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

   13. Simplified 1.3

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

4. Final simplification 0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -1272558208.55477428436279296875:\\ \;\;\;\;x - y \cdot \left(\frac{t}{a} - \frac{z}{a}\right)\\ \mathbf{elif}\;a \le 3.071173039328613373912527296282876331673 \cdot 10^{-45}:\\ \;\;\;\;x - \left(\left(t - z\right) \cdot y\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \left(\frac{t}{\frac{a}{y}} - \frac{y}{\frac{a}{z}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019196 
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, E"

  :herbie-target
  (if (< y -1.0761266216389975e-10) (+ x (/ 1.0 (/ (/ a (- z t)) y))) (if (< y 2.894426862792089e-49) (+ x (/ (* y (- z t)) a)) (+ x (/ y (/ a (- z t))))))

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