Quotient of products

Average Error: 11.5 → 3.0

Time: 26.5s

Precision: 64

Math

\[\frac{a1 \cdot a2}{b1 \cdot b2}\]

↓

\[\frac{\frac{\sqrt[3]{a2}}{\sqrt[3]{b2}} \cdot \frac{\sqrt[3]{a2}}{\sqrt[3]{b2}}}{\sqrt[3]{b1} \cdot \sqrt[3]{b1}} \cdot \frac{a1}{\frac{\sqrt[3]{b1}}{\frac{\sqrt[3]{a2}}{\sqrt[3]{b2}}}}\]

C

double f(double a1, double a2, double b1, double b2) {
        double r6884490 = a1;
        double r6884491 = a2;
        double r6884492 = r6884490 * r6884491;
        double r6884493 = b1;
        double r6884494 = b2;
        double r6884495 = r6884493 * r6884494;
        double r6884496 = r6884492 / r6884495;
        return r6884496;
}

↓

double f(double a1, double a2, double b1, double b2) {
        double r6884497 = a2;
        double r6884498 = cbrt(r6884497);
        double r6884499 = b2;
        double r6884500 = cbrt(r6884499);
        double r6884501 = r6884498 / r6884500;
        double r6884502 = r6884501 * r6884501;
        double r6884503 = b1;
        double r6884504 = cbrt(r6884503);
        double r6884505 = r6884504 * r6884504;
        double r6884506 = r6884502 / r6884505;
        double r6884507 = a1;
        double r6884508 = r6884504 / r6884501;
        double r6884509 = r6884507 / r6884508;
        double r6884510 = r6884506 * r6884509;
        return r6884510;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	11.5
Target	11.3
Herbie	3.0

\[\frac{a1}{b1} \cdot \frac{a2}{b2}\]

Derivation

1. Initial program 11.5

   \[\frac{a1 \cdot a2}{b1 \cdot b2}\]
2. Using strategy rm

3. Applied associate-/l* 11.1

   \[\leadsto \color{blue}{\frac{a1}{\frac{b1 \cdot b2}{a2}}}\]
4. Using strategy rm

5. Applied associate-/l* 11.0

   \[\leadsto \frac{a1}{\color{blue}{\frac{b1}{\frac{a2}{b2}}}}\]
6. Using strategy rm

7. Applied add-cube-cbrt 11.6

   \[\leadsto \frac{a1}{\frac{b1}{\frac{a2}{\color{blue}{\left(\sqrt[3]{b2} \cdot \sqrt[3]{b2}\right) \cdot \sqrt[3]{b2}}}}}\]

8. Applied add-cube-cbrt 11.7

   \[\leadsto \frac{a1}{\frac{b1}{\frac{\color{blue}{\left(\sqrt[3]{a2} \cdot \sqrt[3]{a2}\right) \cdot \sqrt[3]{a2}}}{\left(\sqrt[3]{b2} \cdot \sqrt[3]{b2}\right) \cdot \sqrt[3]{b2}}}}\]

9. Applied times-frac 11.7

   \[\leadsto \frac{a1}{\frac{b1}{\color{blue}{\frac{\sqrt[3]{a2} \cdot \sqrt[3]{a2}}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}} \cdot \frac{\sqrt[3]{a2}}{\sqrt[3]{b2}}}}}\]

10. Applied add-cube-cbrt 11.8

    \[\leadsto \frac{a1}{\frac{\color{blue}{\left(\sqrt[3]{b1} \cdot \sqrt[3]{b1}\right) \cdot \sqrt[3]{b1}}}{\frac{\sqrt[3]{a2} \cdot \sqrt[3]{a2}}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}} \cdot \frac{\sqrt[3]{a2}}{\sqrt[3]{b2}}}}\]

11. Applied times-frac 8.0

    \[\leadsto \frac{a1}{\color{blue}{\frac{\sqrt[3]{b1} \cdot \sqrt[3]{b1}}{\frac{\sqrt[3]{a2} \cdot \sqrt[3]{a2}}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}}} \cdot \frac{\sqrt[3]{b1}}{\frac{\sqrt[3]{a2}}{\sqrt[3]{b2}}}}}\]

12. Applied *-un-lft-identity 8.0

    \[\leadsto \frac{\color{blue}{1 \cdot a1}}{\frac{\sqrt[3]{b1} \cdot \sqrt[3]{b1}}{\frac{\sqrt[3]{a2} \cdot \sqrt[3]{a2}}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}}} \cdot \frac{\sqrt[3]{b1}}{\frac{\sqrt[3]{a2}}{\sqrt[3]{b2}}}}\]

13. Applied times-frac 3.1

    \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt[3]{b1} \cdot \sqrt[3]{b1}}{\frac{\sqrt[3]{a2} \cdot \sqrt[3]{a2}}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}}}} \cdot \frac{a1}{\frac{\sqrt[3]{b1}}{\frac{\sqrt[3]{a2}}{\sqrt[3]{b2}}}}}\]

14. Simplified 3.0

    \[\leadsto \color{blue}{\frac{\frac{\sqrt[3]{a2}}{\sqrt[3]{b2}} \cdot \frac{\sqrt[3]{a2}}{\sqrt[3]{b2}}}{\sqrt[3]{b1} \cdot \sqrt[3]{b1}}} \cdot \frac{a1}{\frac{\sqrt[3]{b1}}{\frac{\sqrt[3]{a2}}{\sqrt[3]{b2}}}}\]

15. Final simplification 3.0

    \[\leadsto \frac{\frac{\sqrt[3]{a2}}{\sqrt[3]{b2}} \cdot \frac{\sqrt[3]{a2}}{\sqrt[3]{b2}}}{\sqrt[3]{b1} \cdot \sqrt[3]{b1}} \cdot \frac{a1}{\frac{\sqrt[3]{b1}}{\frac{\sqrt[3]{a2}}{\sqrt[3]{b2}}}}\]

Reproduce

herbie shell --seed 2019169 +o rules:numerics
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"

  :herbie-target
  (* (/ a1 b1) (/ a2 b2))

  (/ (* a1 a2) (* b1 b2)))