Quotient of products

Average Error: 11.2 → 5.6

Time: 32.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a1 \cdot a2 \le -1.445905988366334040639285224787715187975 \cdot 10^{244}:\\ \;\;\;\;\frac{a1}{\frac{b2}{\frac{a2}{b1}}}\\ \mathbf{elif}\;a1 \cdot a2 \le -9.609520285206617620294914143781194869795 \cdot 10^{-248}:\\ \;\;\;\;\frac{\frac{1}{\frac{b1}{a1 \cdot a2}}}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \le 1.307444751099091752185460323607005997068 \cdot 10^{-272}:\\ \;\;\;\;\frac{a1}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}} \cdot \frac{\frac{a2}{b1}}{\sqrt[3]{b2}}\\ \mathbf{elif}\;a1 \cdot a2 \le 1.182783549232119682193057561507006002421 \cdot 10^{164}:\\ \;\;\;\;\frac{\frac{1}{\frac{b1}{a1 \cdot a2}}}{b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{\frac{b2}{\frac{a2}{b1}}}\\ \end{array}\]

C

double f(double a1, double a2, double b1, double b2) {
        double r5820720 = a1;
        double r5820721 = a2;
        double r5820722 = r5820720 * r5820721;
        double r5820723 = b1;
        double r5820724 = b2;
        double r5820725 = r5820723 * r5820724;
        double r5820726 = r5820722 / r5820725;
        return r5820726;
}

↓

double f(double a1, double a2, double b1, double b2) {
        double r5820727 = a1;
        double r5820728 = a2;
        double r5820729 = r5820727 * r5820728;
        double r5820730 = -1.445905988366334e+244;
        bool r5820731 = r5820729 <= r5820730;
        double r5820732 = b2;
        double r5820733 = b1;
        double r5820734 = r5820728 / r5820733;
        double r5820735 = r5820732 / r5820734;
        double r5820736 = r5820727 / r5820735;
        double r5820737 = -9.609520285206618e-248;
        bool r5820738 = r5820729 <= r5820737;
        double r5820739 = 1.0;
        double r5820740 = r5820733 / r5820729;
        double r5820741 = r5820739 / r5820740;
        double r5820742 = r5820741 / r5820732;
        double r5820743 = 1.3074447510990918e-272;
        bool r5820744 = r5820729 <= r5820743;
        double r5820745 = cbrt(r5820732);
        double r5820746 = r5820745 * r5820745;
        double r5820747 = r5820727 / r5820746;
        double r5820748 = r5820734 / r5820745;
        double r5820749 = r5820747 * r5820748;
        double r5820750 = 1.1827835492321197e+164;
        bool r5820751 = r5820729 <= r5820750;
        double r5820752 = r5820751 ? r5820742 : r5820736;
        double r5820753 = r5820744 ? r5820749 : r5820752;
        double r5820754 = r5820738 ? r5820742 : r5820753;
        double r5820755 = r5820731 ? r5820736 : r5820754;
        return r5820755;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	11.2
Target	11.3
Herbie	5.6

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

Derivation

1. Split input into 3 regimes

2. if (* a1 a2) < -1.445905988366334e+244 or 1.1827835492321197e+164 < (* a1 a2)

   1. Initial program 34.5

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

   3. Applied associate-/r* 34.9

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

   5. Applied *-un-lft-identity 34.9

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

   6. Applied times-frac 19.7

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

   7. Applied associate-/l* 10.7

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

   if -1.445905988366334e+244 < (* a1 a2) < -9.609520285206618e-248 or 1.3074447510990918e-272 < (* a1 a2) < 1.1827835492321197e+164

   1. Initial program 4.8

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

   3. Applied associate-/r* 5.0

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

   5. Applied clear-num 5.4

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

   if -9.609520285206618e-248 < (* a1 a2) < 1.3074447510990918e-272

   1. Initial program 17.2

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

   3. Applied associate-/r* 16.2

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

   5. Applied add-cube-cbrt 16.4

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

   6. Applied *-un-lft-identity 16.4

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

   7. Applied times-frac 7.5

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

   8. Applied times-frac 2.9

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

   9. Simplified 2.9

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

4. Final simplification 5.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;a1 \cdot a2 \le -1.445905988366334040639285224787715187975 \cdot 10^{244}:\\ \;\;\;\;\frac{a1}{\frac{b2}{\frac{a2}{b1}}}\\ \mathbf{elif}\;a1 \cdot a2 \le -9.609520285206617620294914143781194869795 \cdot 10^{-248}:\\ \;\;\;\;\frac{\frac{1}{\frac{b1}{a1 \cdot a2}}}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \le 1.307444751099091752185460323607005997068 \cdot 10^{-272}:\\ \;\;\;\;\frac{a1}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}} \cdot \frac{\frac{a2}{b1}}{\sqrt[3]{b2}}\\ \mathbf{elif}\;a1 \cdot a2 \le 1.182783549232119682193057561507006002421 \cdot 10^{164}:\\ \;\;\;\;\frac{\frac{1}{\frac{b1}{a1 \cdot a2}}}{b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{\frac{b2}{\frac{a2}{b1}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"

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

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