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

Average Error: 6.3 → 0.9

Time: 15.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -4.597123127540302396685313985725525489425 \cdot 10^{-46}:\\ \;\;\;\;y \cdot \frac{t - z}{a} + x\\ \mathbf{elif}\;y \le 5.972044538660694606197555323074244957271 \cdot 10^{-31}:\\ \;\;\;\;\frac{y \cdot \left(t - z\right)}{a} + x\\ \mathbf{else}:\\ \;\;\;\;x + \sqrt{y} \cdot \left(\frac{\sqrt{y}}{a} \cdot \left(t - z\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r207266 = x;
        double r207267 = y;
        double r207268 = z;
        double r207269 = t;
        double r207270 = r207268 - r207269;
        double r207271 = r207267 * r207270;
        double r207272 = a;
        double r207273 = r207271 / r207272;
        double r207274 = r207266 - r207273;
        return r207274;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r207275 = y;
        double r207276 = -4.5971231275403024e-46;
        bool r207277 = r207275 <= r207276;
        double r207278 = t;
        double r207279 = z;
        double r207280 = r207278 - r207279;
        double r207281 = a;
        double r207282 = r207280 / r207281;
        double r207283 = r207275 * r207282;
        double r207284 = x;
        double r207285 = r207283 + r207284;
        double r207286 = 5.972044538660695e-31;
        bool r207287 = r207275 <= r207286;
        double r207288 = r207275 * r207280;
        double r207289 = r207288 / r207281;
        double r207290 = r207289 + r207284;
        double r207291 = sqrt(r207275);
        double r207292 = r207291 / r207281;
        double r207293 = r207292 * r207280;
        double r207294 = r207291 * r207293;
        double r207295 = r207284 + r207294;
        double r207296 = r207287 ? r207290 : r207295;
        double r207297 = r207277 ? r207285 : r207296;
        return r207297;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	6.3
Target	0.6
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 y < -4.5971231275403024e-46

   1. Initial program 13.2

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

   2. Simplified 3.2

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

   4. Applied fma-udef 3.2

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

   6. Applied div-inv 3.3

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

   7. Applied associate-*l* 1.4

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

   8. Simplified 1.3

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

   if -4.5971231275403024e-46 < y < 5.972044538660695e-31

   1. Initial program 0.3

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

   2. Simplified 2.0

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

   4. Applied fma-udef 2.0

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

   6. Applied add-cube-cbrt 2.4

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

   7. Applied *-un-lft-identity 2.4

      \[\leadsto \frac{\color{blue}{1 \cdot y}}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}} \cdot \left(t - z\right) + x\]

   8. Applied times-frac 2.4

      \[\leadsto \color{blue}{\left(\frac{1}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{y}{\sqrt[3]{a}}\right)} \cdot \left(t - z\right) + x\]

   9. Applied associate-*l* 0.8

      \[\leadsto \color{blue}{\frac{1}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \left(\frac{y}{\sqrt[3]{a}} \cdot \left(t - z\right)\right)} + x\]
   10. Using strategy rm

   11. Applied pow1 0.8

       \[\leadsto \frac{1}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \left(\frac{y}{\sqrt[3]{a}} \cdot \color{blue}{{\left(t - z\right)}^{1}}\right) + x\]

   12. Applied pow1 0.8

       \[\leadsto \frac{1}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \left(\color{blue}{{\left(\frac{y}{\sqrt[3]{a}}\right)}^{1}} \cdot {\left(t - z\right)}^{1}\right) + x\]

   13. Applied pow-prod-down 0.8

       \[\leadsto \frac{1}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \color{blue}{{\left(\frac{y}{\sqrt[3]{a}} \cdot \left(t - z\right)\right)}^{1}} + x\]

   14. Applied pow1 0.8

       \[\leadsto \color{blue}{{\left(\frac{1}{\sqrt[3]{a} \cdot \sqrt[3]{a}}\right)}^{1}} \cdot {\left(\frac{y}{\sqrt[3]{a}} \cdot \left(t - z\right)\right)}^{1} + x\]

   15. Applied pow-prod-down 0.8

       \[\leadsto \color{blue}{{\left(\frac{1}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \left(\frac{y}{\sqrt[3]{a}} \cdot \left(t - z\right)\right)\right)}^{1}} + x\]

   16. Simplified 0.3

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

   if 5.972044538660695e-31 < y

   1. Initial program 13.5

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

   2. Simplified 3.5

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

   4. Applied fma-udef 3.5

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

   6. Applied *-un-lft-identity 3.5

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

   7. Applied add-sqr-sqrt 3.7

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

   8. Applied times-frac 3.7

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

   9. Applied associate-*l* 1.8

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

4. Final simplification 0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -4.597123127540302396685313985725525489425 \cdot 10^{-46}:\\ \;\;\;\;y \cdot \frac{t - z}{a} + x\\ \mathbf{elif}\;y \le 5.972044538660694606197555323074244957271 \cdot 10^{-31}:\\ \;\;\;\;\frac{y \cdot \left(t - z\right)}{a} + x\\ \mathbf{else}:\\ \;\;\;\;x + \sqrt{y} \cdot \left(\frac{\sqrt{y}}{a} \cdot \left(t - z\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019322 +o rules:numerics
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, F"
  :precision binary64

  :herbie-target
  (if (< y -1.0761266216389975e-10) (- x (/ 1 (/ (/ a (- z t)) y))) (if (< y 2.894426862792089e-49) (- x (/ (* y (- z t)) a)) (- x (/ y (/ a (- z t))))))

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