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

Average Error: 6.4 → 1.9

Time: 17.4s

Precision: 64

Math

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

↓

\[\mathsf{fma}\left(\frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\frac{1}{\sqrt[3]{y} \cdot \sqrt[3]{y}}}, \frac{\sqrt[3]{z - t}}{\frac{a}{\sqrt[3]{y}}}, x\right)\]

C

double f(double x, double y, double z, double t, double a) {
        double r189394 = x;
        double r189395 = y;
        double r189396 = z;
        double r189397 = t;
        double r189398 = r189396 - r189397;
        double r189399 = r189395 * r189398;
        double r189400 = a;
        double r189401 = r189399 / r189400;
        double r189402 = r189394 + r189401;
        return r189402;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r189403 = z;
        double r189404 = t;
        double r189405 = r189403 - r189404;
        double r189406 = cbrt(r189405);
        double r189407 = r189406 * r189406;
        double r189408 = 1.0;
        double r189409 = y;
        double r189410 = cbrt(r189409);
        double r189411 = r189410 * r189410;
        double r189412 = r189408 / r189411;
        double r189413 = r189407 / r189412;
        double r189414 = a;
        double r189415 = r189414 / r189410;
        double r189416 = r189406 / r189415;
        double r189417 = x;
        double r189418 = fma(r189413, r189416, r189417);
        return r189418;
}

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	1.9

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

Derivation

1. Initial program 6.4

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

2. Simplified 2.5

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

4. Applied fma-udef 2.5

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

5. Simplified 2.4

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

7. Applied add-cube-cbrt 2.9

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

8. Applied *-un-lft-identity 2.9

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

9. Applied times-frac 2.9

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

10. Applied add-cube-cbrt 3.0

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

11. Applied times-frac 1.9

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

12. Applied fma-def 1.9

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

13. Final simplification 1.9

    \[\leadsto \mathsf{fma}\left(\frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\frac{1}{\sqrt[3]{y} \cdot \sqrt[3]{y}}}, \frac{\sqrt[3]{z - t}}{\frac{a}{\sqrt[3]{y}}}, x\right)\]

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, E"
  :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)))