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

Average Error: 5.9 → 1.2

Time: 21.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \le -2.717101417616620114043785351731635839082 \cdot 10^{48}:\\ \;\;\;\;\left(\left(z - t\right) \cdot \frac{1}{\frac{a}{\sqrt[3]{y}}}\right) \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) + x\\ \mathbf{elif}\;a \le 72135089252230804599899465539374389657600:\\ \;\;\;\;x + \frac{1}{a} \cdot \left(y \cdot \left(z - t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y, -\frac{t}{\frac{a}{y}}\right) + x\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r232673 = x;
        double r232674 = y;
        double r232675 = z;
        double r232676 = t;
        double r232677 = r232675 - r232676;
        double r232678 = r232674 * r232677;
        double r232679 = a;
        double r232680 = r232678 / r232679;
        double r232681 = r232673 + r232680;
        return r232681;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r232682 = a;
        double r232683 = -2.71710141761662e+48;
        bool r232684 = r232682 <= r232683;
        double r232685 = z;
        double r232686 = t;
        double r232687 = r232685 - r232686;
        double r232688 = 1.0;
        double r232689 = y;
        double r232690 = cbrt(r232689);
        double r232691 = r232682 / r232690;
        double r232692 = r232688 / r232691;
        double r232693 = r232687 * r232692;
        double r232694 = r232690 * r232690;
        double r232695 = r232693 * r232694;
        double r232696 = x;
        double r232697 = r232695 + r232696;
        double r232698 = 7.21350892522308e+40;
        bool r232699 = r232682 <= r232698;
        double r232700 = r232688 / r232682;
        double r232701 = r232689 * r232687;
        double r232702 = r232700 * r232701;
        double r232703 = r232696 + r232702;
        double r232704 = r232685 / r232682;
        double r232705 = r232682 / r232689;
        double r232706 = r232686 / r232705;
        double r232707 = -r232706;
        double r232708 = fma(r232704, r232689, r232707);
        double r232709 = r232708 + r232696;
        double r232710 = r232699 ? r232703 : r232709;
        double r232711 = r232684 ? r232697 : r232710;
        return r232711;
}

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	0.7
Herbie	1.2

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

Derivation

1. Split input into 3 regimes

2. if a < -2.71710141761662e+48

   1. Initial program 10.6

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

   2. Simplified 0.5

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

   4. Applied fma-udef 0.5

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

   5. Simplified 0.5

      \[\leadsto \color{blue}{\frac{y}{\frac{a}{z - t}}} + x\]
   6. Using strategy rm

   7. Applied clear-num 0.6

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

   8. Simplified 1.9

      \[\leadsto \frac{1}{\color{blue}{\frac{\frac{a}{y}}{z - t}}} + x\]
   9. Using strategy rm

   10. Applied *-un-lft-identity 1.9

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

   11. Applied add-cube-cbrt 2.2

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

   12. Applied *-un-lft-identity 2.2

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

   13. Applied times-frac 2.2

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

   14. Applied times-frac 0.9

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

   15. Applied add-cube-cbrt 0.9

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

   16. Applied times-frac 0.9

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

   17. Simplified 0.9

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

   18. Simplified 0.9

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

   if -2.71710141761662e+48 < a < 7.21350892522308e+40

   1. Initial program 1.2

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

   2. Simplified 11.8

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

   4. Applied fma-udef 11.8

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

   5. Simplified 10.8

      \[\leadsto \color{blue}{\frac{y}{\frac{a}{z - t}}} + x\]
   6. Using strategy rm

   7. Applied div-inv 10.8

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

   8. Applied *-un-lft-identity 10.8

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

   9. Applied times-frac 1.4

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

   10. Simplified 1.3

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

   if 7.21350892522308e+40 < a

   1. Initial program 9.5

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

   2. Simplified 0.4

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

   4. Applied fma-udef 0.4

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

   5. Simplified 0.5

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

   6. Taylor expanded around 0 9.5

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

   7. Simplified 1.3

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

4. Final simplification 1.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -2.717101417616620114043785351731635839082 \cdot 10^{48}:\\ \;\;\;\;\left(\left(z - t\right) \cdot \frac{1}{\frac{a}{\sqrt[3]{y}}}\right) \cdot \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) + x\\ \mathbf{elif}\;a \le 72135089252230804599899465539374389657600:\\ \;\;\;\;x + \frac{1}{a} \cdot \left(y \cdot \left(z - t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y, -\frac{t}{\frac{a}{y}}\right) + x\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 +o rules:numerics
(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)))