Quotient of products

Average Error: 11.4 → 2.0

Time: 15.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} = -\infty:\\ \;\;\;\;\frac{\frac{a1}{\sqrt[3]{b2}}}{\sqrt[3]{b2}} \cdot \frac{\frac{a2}{b1}}{\sqrt[3]{b2}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -1.5091135102985783 \cdot 10^{-293}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 0.0:\\ \;\;\;\;\left(\frac{\frac{a1}{\sqrt[3]{b2}}}{\sqrt[3]{b2}} \cdot \frac{\sqrt[3]{\frac{\sqrt[3]{a2} \cdot \sqrt[3]{a2}}{\sqrt[3]{b1} \cdot \sqrt[3]{b1}}} \cdot \sqrt[3]{\frac{\sqrt[3]{a2} \cdot \sqrt[3]{a2}}{\sqrt[3]{b1} \cdot \sqrt[3]{b1}}}}{\frac{1}{\frac{\sqrt[3]{\sqrt[3]{a2} \cdot \sqrt[3]{a2}}}{\sqrt[3]{\sqrt[3]{b1} \cdot \sqrt[3]{b1}}}}}\right) \cdot \frac{\sqrt[3]{\frac{\sqrt[3]{a2} \cdot \sqrt[3]{a2}}{\sqrt[3]{b1} \cdot \sqrt[3]{b1}}}}{\frac{\sqrt[3]{b2}}{\frac{\sqrt[3]{\sqrt[3]{a2}}}{\sqrt[3]{\sqrt[3]{b1}}}}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 2.7924622985567088 \cdot 10^{265}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \end{array}\]

C

double f(double a1, double a2, double b1, double b2) {
        double r232571 = a1;
        double r232572 = a2;
        double r232573 = r232571 * r232572;
        double r232574 = b1;
        double r232575 = b2;
        double r232576 = r232574 * r232575;
        double r232577 = r232573 / r232576;
        return r232577;
}

↓

double f(double a1, double a2, double b1, double b2) {
        double r232578 = a1;
        double r232579 = a2;
        double r232580 = r232578 * r232579;
        double r232581 = b1;
        double r232582 = b2;
        double r232583 = r232581 * r232582;
        double r232584 = r232580 / r232583;
        double r232585 = -inf.0;
        bool r232586 = r232584 <= r232585;
        double r232587 = cbrt(r232582);
        double r232588 = r232578 / r232587;
        double r232589 = r232588 / r232587;
        double r232590 = r232579 / r232581;
        double r232591 = r232590 / r232587;
        double r232592 = r232589 * r232591;
        double r232593 = -1.5091135102985783e-293;
        bool r232594 = r232584 <= r232593;
        double r232595 = 0.0;
        bool r232596 = r232584 <= r232595;
        double r232597 = cbrt(r232579);
        double r232598 = r232597 * r232597;
        double r232599 = cbrt(r232581);
        double r232600 = r232599 * r232599;
        double r232601 = r232598 / r232600;
        double r232602 = cbrt(r232601);
        double r232603 = r232602 * r232602;
        double r232604 = 1.0;
        double r232605 = cbrt(r232598);
        double r232606 = cbrt(r232600);
        double r232607 = r232605 / r232606;
        double r232608 = r232604 / r232607;
        double r232609 = r232603 / r232608;
        double r232610 = r232589 * r232609;
        double r232611 = cbrt(r232597);
        double r232612 = cbrt(r232599);
        double r232613 = r232611 / r232612;
        double r232614 = r232587 / r232613;
        double r232615 = r232602 / r232614;
        double r232616 = r232610 * r232615;
        double r232617 = 2.7924622985567088e+265;
        bool r232618 = r232584 <= r232617;
        double r232619 = r232578 / r232581;
        double r232620 = r232579 / r232582;
        double r232621 = r232619 * r232620;
        double r232622 = r232618 ? r232584 : r232621;
        double r232623 = r232596 ? r232616 : r232622;
        double r232624 = r232594 ? r232584 : r232623;
        double r232625 = r232586 ? r232592 : r232624;
        return r232625;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	11.4
Target	11.4
Herbie	2.0

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

Derivation

1. Split input into 4 regimes

2. if (/ (* a1 a2) (* b1 b2)) < -inf.0

   1. Initial program 64.0

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

   3. Applied associate-/r* 34.4

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

   5. Applied add-cube-cbrt 34.9

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

   6. Applied *-un-lft-identity 34.9

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

   7. Applied times-frac 16.1

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

   8. Applied times-frac 9.7

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

   9. Simplified 9.7

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

   if -inf.0 < (/ (* a1 a2) (* b1 b2)) < -1.5091135102985783e-293 or 0.0 < (/ (* a1 a2) (* b1 b2)) < 2.7924622985567088e+265

   1. Initial program 0.9

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

   if -1.5091135102985783e-293 < (/ (* a1 a2) (* b1 b2)) < 0.0

   1. Initial program 13.6

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

   3. Applied associate-/r* 6.7

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

   5. Applied add-cube-cbrt 6.8

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

   6. Applied *-un-lft-identity 6.8

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

   7. Applied times-frac 4.0

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

   8. Applied times-frac 2.7

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

   9. Simplified 2.7

      \[\leadsto \color{blue}{\frac{\frac{a1}{\sqrt[3]{b2}}}{\sqrt[3]{b2}}} \cdot \frac{\frac{a2}{b1}}{\sqrt[3]{b2}}\]
   10. Using strategy rm

   11. Applied add-cube-cbrt 2.7

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

   12. Applied add-cube-cbrt 2.8

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

   13. Applied times-frac 2.8

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

   14. Applied associate-/l* 2.5

       \[\leadsto \frac{\frac{a1}{\sqrt[3]{b2}}}{\sqrt[3]{b2}} \cdot \color{blue}{\frac{\frac{\sqrt[3]{a2} \cdot \sqrt[3]{a2}}{\sqrt[3]{b1} \cdot \sqrt[3]{b1}}}{\frac{\sqrt[3]{b2}}{\frac{\sqrt[3]{a2}}{\sqrt[3]{b1}}}}}\]
   15. Using strategy rm

   16. Applied add-cube-cbrt 2.5

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

   17. Applied cbrt-prod 2.5

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

   18. Applied add-cube-cbrt 2.5

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

   19. Applied cbrt-prod 2.5

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

   20. Applied times-frac 2.5

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

   21. Applied *-un-lft-identity 2.5

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

   22. Applied times-frac 2.5

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

   23. Applied add-cube-cbrt 2.5

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

   24. Applied times-frac 2.5

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

   25. Applied associate-*r* 1.1

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

   if 2.7924622985567088e+265 < (/ (* a1 a2) (* b1 b2))

   1. Initial program 54.3

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

   3. Applied times-frac 10.1

      \[\leadsto \color{blue}{\frac{a1}{b1} \cdot \frac{a2}{b2}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 2.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} = -\infty:\\ \;\;\;\;\frac{\frac{a1}{\sqrt[3]{b2}}}{\sqrt[3]{b2}} \cdot \frac{\frac{a2}{b1}}{\sqrt[3]{b2}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -1.5091135102985783 \cdot 10^{-293}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 0.0:\\ \;\;\;\;\left(\frac{\frac{a1}{\sqrt[3]{b2}}}{\sqrt[3]{b2}} \cdot \frac{\sqrt[3]{\frac{\sqrt[3]{a2} \cdot \sqrt[3]{a2}}{\sqrt[3]{b1} \cdot \sqrt[3]{b1}}} \cdot \sqrt[3]{\frac{\sqrt[3]{a2} \cdot \sqrt[3]{a2}}{\sqrt[3]{b1} \cdot \sqrt[3]{b1}}}}{\frac{1}{\frac{\sqrt[3]{\sqrt[3]{a2} \cdot \sqrt[3]{a2}}}{\sqrt[3]{\sqrt[3]{b1} \cdot \sqrt[3]{b1}}}}}\right) \cdot \frac{\sqrt[3]{\frac{\sqrt[3]{a2} \cdot \sqrt[3]{a2}}{\sqrt[3]{b1} \cdot \sqrt[3]{b1}}}}{\frac{\sqrt[3]{b2}}{\frac{\sqrt[3]{\sqrt[3]{a2}}}{\sqrt[3]{\sqrt[3]{b1}}}}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 2.7924622985567088 \cdot 10^{265}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020033 +o rules:numerics
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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