Quotient of products

Average Error: 11.3 → 6.0

Time: 5.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b1 \cdot b2 \le -3.07688363857671596 \cdot 10^{290}:\\ \;\;\;\;\frac{\sqrt[3]{a2} \cdot \sqrt[3]{a2}}{b1} \cdot \left(\left(\sqrt[3]{\sqrt[3]{a2} \cdot \sqrt[3]{a2}} \cdot \frac{a1}{b2}\right) \cdot \sqrt[3]{\sqrt[3]{a2}}\right)\\ \mathbf{elif}\;b1 \cdot b2 \le -7.33224774175289027 \cdot 10^{-132}:\\ \;\;\;\;\frac{1}{b1 \cdot b2} \cdot \left(a1 \cdot a2\right)\\ \mathbf{elif}\;b1 \cdot b2 \le 1.1938694317181371 \cdot 10^{-187}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;b1 \cdot b2 \le 6.0946661841822745 \cdot 10^{116}:\\ \;\;\;\;\frac{a1}{\frac{b1 \cdot b2}{a2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b1} \cdot \frac{a1}{\frac{b2}{a2}}\\ \end{array}\]

C

double f(double a1, double a2, double b1, double b2) {
        double r211661 = a1;
        double r211662 = a2;
        double r211663 = r211661 * r211662;
        double r211664 = b1;
        double r211665 = b2;
        double r211666 = r211664 * r211665;
        double r211667 = r211663 / r211666;
        return r211667;
}

↓

double f(double a1, double a2, double b1, double b2) {
        double r211668 = b1;
        double r211669 = b2;
        double r211670 = r211668 * r211669;
        double r211671 = -3.076883638576716e+290;
        bool r211672 = r211670 <= r211671;
        double r211673 = a2;
        double r211674 = cbrt(r211673);
        double r211675 = r211674 * r211674;
        double r211676 = r211675 / r211668;
        double r211677 = cbrt(r211675);
        double r211678 = a1;
        double r211679 = r211678 / r211669;
        double r211680 = r211677 * r211679;
        double r211681 = cbrt(r211674);
        double r211682 = r211680 * r211681;
        double r211683 = r211676 * r211682;
        double r211684 = -7.33224774175289e-132;
        bool r211685 = r211670 <= r211684;
        double r211686 = 1.0;
        double r211687 = r211686 / r211670;
        double r211688 = r211678 * r211673;
        double r211689 = r211687 * r211688;
        double r211690 = 1.193869431718137e-187;
        bool r211691 = r211670 <= r211690;
        double r211692 = r211678 / r211668;
        double r211693 = r211673 / r211669;
        double r211694 = r211692 * r211693;
        double r211695 = 6.0946661841822745e+116;
        bool r211696 = r211670 <= r211695;
        double r211697 = r211670 / r211673;
        double r211698 = r211678 / r211697;
        double r211699 = r211686 / r211668;
        double r211700 = r211669 / r211673;
        double r211701 = r211678 / r211700;
        double r211702 = r211699 * r211701;
        double r211703 = r211696 ? r211698 : r211702;
        double r211704 = r211691 ? r211694 : r211703;
        double r211705 = r211685 ? r211689 : r211704;
        double r211706 = r211672 ? r211683 : r211705;
        return r211706;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	11.3
Target	11.5
Herbie	6.0

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

Derivation

1. Split input into 5 regimes

2. if (* b1 b2) < -3.076883638576716e+290

   1. Initial program 20.2

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

   3. Applied associate-/l* 19.9

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

   5. Applied add-cube-cbrt 20.0

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

   6. Applied times-frac 9.2

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

   7. Applied *-un-lft-identity 9.2

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

   8. Applied times-frac 2.9

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

   9. Simplified 2.8

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

   11. Applied add-cube-cbrt 2.8

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

   12. Applied cbrt-prod 2.8

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

   13. Applied *-un-lft-identity 2.8

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

   14. Applied times-frac 2.8

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

   15. Applied *-un-lft-identity 2.8

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

   16. Applied times-frac 2.7

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

   17. Simplified 2.7

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

   19. Applied associate-/r/ 2.9

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

   20. Applied associate-*r* 2.9

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

   if -3.076883638576716e+290 < (* b1 b2) < -7.33224774175289e-132

   1. Initial program 4.4

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

   3. Applied associate-/l* 3.9

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

   5. Applied div-inv 4.0

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

   6. Applied *-un-lft-identity 4.0

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

   7. Applied times-frac 4.5

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

   8. Simplified 4.5

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

   if -7.33224774175289e-132 < (* b1 b2) < 1.193869431718137e-187

   1. Initial program 26.6

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

   3. Applied times-frac 12.7

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

   if 1.193869431718137e-187 < (* b1 b2) < 6.0946661841822745e+116

   1. Initial program 4.1

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

   3. Applied associate-/l* 3.4

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

   if 6.0946661841822745e+116 < (* b1 b2)

   1. Initial program 12.5

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

   3. Applied associate-/l* 13.0

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

   5. Applied *-un-lft-identity 13.0

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

   6. Applied times-frac 9.3

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

   7. Applied *-un-lft-identity 9.3

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

   8. Applied times-frac 7.0

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

   9. Simplified 7.0

      \[\leadsto \color{blue}{\frac{1}{b1}} \cdot \frac{a1}{\frac{b2}{a2}}\]
3. Recombined 5 regimes into one program.

4. Final simplification 6.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;b1 \cdot b2 \le -3.07688363857671596 \cdot 10^{290}:\\ \;\;\;\;\frac{\sqrt[3]{a2} \cdot \sqrt[3]{a2}}{b1} \cdot \left(\left(\sqrt[3]{\sqrt[3]{a2} \cdot \sqrt[3]{a2}} \cdot \frac{a1}{b2}\right) \cdot \sqrt[3]{\sqrt[3]{a2}}\right)\\ \mathbf{elif}\;b1 \cdot b2 \le -7.33224774175289027 \cdot 10^{-132}:\\ \;\;\;\;\frac{1}{b1 \cdot b2} \cdot \left(a1 \cdot a2\right)\\ \mathbf{elif}\;b1 \cdot b2 \le 1.1938694317181371 \cdot 10^{-187}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;b1 \cdot b2 \le 6.0946661841822745 \cdot 10^{116}:\\ \;\;\;\;\frac{a1}{\frac{b1 \cdot b2}{a2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b1} \cdot \frac{a1}{\frac{b2}{a2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020039 +o rules:numerics
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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