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

Average Error: 6.2 → 1.3

Time: 16.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -4.156241843490642492342216522498114807292 \cdot 10^{-54}:\\ \;\;\;\;x - \left(y \cdot \frac{t}{a} - \frac{z}{a} \cdot y\right)\\ \mathbf{elif}\;y \le 7.465710341714553384297243632530613668098 \cdot 10^{-77}:\\ \;\;\;\;x - \frac{1}{a} \cdot \left(y \cdot \left(t - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{1}{\frac{a}{y}} \cdot \left(t - z\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r277130 = x;
        double r277131 = y;
        double r277132 = z;
        double r277133 = t;
        double r277134 = r277132 - r277133;
        double r277135 = r277131 * r277134;
        double r277136 = a;
        double r277137 = r277135 / r277136;
        double r277138 = r277130 + r277137;
        return r277138;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r277139 = y;
        double r277140 = -4.1562418434906425e-54;
        bool r277141 = r277139 <= r277140;
        double r277142 = x;
        double r277143 = t;
        double r277144 = a;
        double r277145 = r277143 / r277144;
        double r277146 = r277139 * r277145;
        double r277147 = z;
        double r277148 = r277147 / r277144;
        double r277149 = r277148 * r277139;
        double r277150 = r277146 - r277149;
        double r277151 = r277142 - r277150;
        double r277152 = 7.465710341714553e-77;
        bool r277153 = r277139 <= r277152;
        double r277154 = 1.0;
        double r277155 = r277154 / r277144;
        double r277156 = r277143 - r277147;
        double r277157 = r277139 * r277156;
        double r277158 = r277155 * r277157;
        double r277159 = r277142 - r277158;
        double r277160 = r277144 / r277139;
        double r277161 = r277154 / r277160;
        double r277162 = r277161 * r277156;
        double r277163 = r277142 - r277162;
        double r277164 = r277153 ? r277159 : r277163;
        double r277165 = r277141 ? r277151 : r277164;
        return r277165;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	6.2
Target	0.8
Herbie	1.3

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

Derivation

1. Split input into 3 regimes

2. if y < -4.1562418434906425e-54

   1. Initial program 12.7

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

   2. Simplified 3.0

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

   4. Applied sub-neg 3.0

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

   5. Applied distribute-lft-in 3.0

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

   6. Simplified 3.7

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

   7. Taylor expanded around 0 12.7

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

   8. Simplified 2.8

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

   10. Applied div-inv 3.2

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

   11. Applied div-inv 3.2

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

   12. Applied distribute-rgt-out-- 3.2

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

   13. Taylor expanded around 0 12.7

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

   14. Simplified 1.3

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

   if -4.1562418434906425e-54 < y < 7.465710341714553e-77

   1. Initial program 0.5

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

   2. Simplified 2.1

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

   4. Applied sub-neg 2.1

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

   5. Applied distribute-lft-in 2.1

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

   6. Simplified 7.0

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

   7. Taylor expanded around 0 0.5

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

   8. Simplified 2.3

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

   10. Applied div-inv 2.3

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

   11. Applied div-inv 2.3

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

   12. Applied distribute-rgt-out-- 2.3

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

   14. Applied div-inv 2.4

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

   15. Applied add-sqr-sqrt 2.4

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

   16. Applied times-frac 2.2

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

   17. Applied associate-*l* 0.6

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

   18. Simplified 0.6

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

   if 7.465710341714553e-77 < y

   1. Initial program 11.6

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

   2. Simplified 2.7

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

   4. Applied sub-neg 2.7

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

   5. Applied distribute-lft-in 2.7

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

   6. Simplified 3.6

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

   7. Taylor expanded around 0 11.6

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

   8. Simplified 2.4

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

   10. Applied div-inv 2.8

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

   11. Applied div-inv 2.8

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

   12. Applied distribute-rgt-out-- 2.8

       \[\leadsto x - \color{blue}{\frac{1}{\frac{a}{y}} \cdot \left(t - z\right)}\]
3. Recombined 3 regimes into one program.

4. Final simplification 1.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -4.156241843490642492342216522498114807292 \cdot 10^{-54}:\\ \;\;\;\;x - \left(y \cdot \frac{t}{a} - \frac{z}{a} \cdot y\right)\\ \mathbf{elif}\;y \le 7.465710341714553384297243632530613668098 \cdot 10^{-77}:\\ \;\;\;\;x - \frac{1}{a} \cdot \left(y \cdot \left(t - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{1}{\frac{a}{y}} \cdot \left(t - z\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019195 
(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)))