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

Average Error: 5.9 → 5.9

Time: 12.2s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t, double a) {
        double r210168 = x;
        double r210169 = y;
        double r210170 = z;
        double r210171 = t;
        double r210172 = r210170 - r210171;
        double r210173 = r210169 * r210172;
        double r210174 = a;
        double r210175 = r210173 / r210174;
        double r210176 = r210168 - r210175;
        return r210176;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r210177 = t;
        double r210178 = y;
        double r210179 = r210177 * r210178;
        double r210180 = a;
        double r210181 = r210179 / r210180;
        double r210182 = x;
        double r210183 = r210181 + r210182;
        double r210184 = z;
        double r210185 = r210184 * r210178;
        double r210186 = r210185 / r210180;
        double r210187 = r210183 - r210186;
        return r210187;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	5.9
Target	0.8
Herbie	5.9

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

Derivation

1. Split input into 2 regimes

2. if y < -4.220030324223059e+17 or 9.369856241541125e-20 < y

   1. Initial program 14.6

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

   2. Taylor expanded around 0 14.6

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

   3. Simplified 3.7

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

   5. Applied add-cube-cbrt 4.3

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

   6. Applied associate-*r* 4.3

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

   7. Taylor expanded around 0 14.6

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

   9. Applied *-un-lft-identity 14.6

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

   10. Applied *-un-lft-identity 14.6

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

   11. Applied distribute-lft-out-- 14.6

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

   12. Simplified 1.0

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

   if -4.220030324223059e+17 < y < 9.369856241541125e-20

   1. Initial program 0.6

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

   2. Taylor expanded around 0 0.6

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

   3. Simplified 1.7

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

   5. Applied add-cube-cbrt 2.1

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

   6. Applied associate-*r* 2.1

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

   7. Taylor expanded around 0 0.6

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

   9. Applied add-cube-cbrt 0.8

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

4. Final simplification 5.9

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

Reproduce

herbie shell --seed 2019291 
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, F"
  :precision binary64

  :herbie-target
  (if (< y -1.07612662163899753e-10) (- x (/ 1 (/ (/ a (- z t)) y))) (if (< y 2.8944268627920891e-49) (- x (/ (* y (- z t)) a)) (- x (/ y (/ a (- z t))))))

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