Quotient of products

Average Error: 11.4 → 3.4

Time: 2.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -1.68280036335944034 \cdot 10^{301}:\\ \;\;\;\;\left(a1 \cdot \frac{a2}{b1}\right) \cdot \frac{1}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -4.5204447990899316 \cdot 10^{-307}:\\ \;\;\;\;{\left(\frac{a1 \cdot a2}{b1 \cdot b2}\right)}^{1}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 0.0:\\ \;\;\;\;\frac{a1}{\frac{b1}{\frac{a2}{b2}}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 4.51733405101081216 \cdot 10^{288}:\\ \;\;\;\;{\left(\frac{a1 \cdot a2}{b1 \cdot b2}\right)}^{1}\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot \frac{a2}{b1}\right) \cdot \frac{1}{b2}\\ \end{array}\]

C

double f(double a1, double a2, double b1, double b2) {
        double r173620 = a1;
        double r173621 = a2;
        double r173622 = r173620 * r173621;
        double r173623 = b1;
        double r173624 = b2;
        double r173625 = r173623 * r173624;
        double r173626 = r173622 / r173625;
        return r173626;
}

↓

double f(double a1, double a2, double b1, double b2) {
        double r173627 = a1;
        double r173628 = a2;
        double r173629 = r173627 * r173628;
        double r173630 = b1;
        double r173631 = b2;
        double r173632 = r173630 * r173631;
        double r173633 = r173629 / r173632;
        double r173634 = -1.6828003633594403e+301;
        bool r173635 = r173633 <= r173634;
        double r173636 = r173628 / r173630;
        double r173637 = r173627 * r173636;
        double r173638 = 1.0;
        double r173639 = r173638 / r173631;
        double r173640 = r173637 * r173639;
        double r173641 = -4.520444799089932e-307;
        bool r173642 = r173633 <= r173641;
        double r173643 = pow(r173633, r173638);
        double r173644 = 0.0;
        bool r173645 = r173633 <= r173644;
        double r173646 = r173628 / r173631;
        double r173647 = r173630 / r173646;
        double r173648 = r173627 / r173647;
        double r173649 = 4.517334051010812e+288;
        bool r173650 = r173633 <= r173649;
        double r173651 = r173650 ? r173643 : r173640;
        double r173652 = r173645 ? r173648 : r173651;
        double r173653 = r173642 ? r173643 : r173652;
        double r173654 = r173635 ? r173640 : r173653;
        return r173654;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	11.4
Target	11.5
Herbie	3.4

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

Derivation

1. Split input into 3 regimes

2. if (/ (* a1 a2) (* b1 b2)) < -1.6828003633594403e+301 or 4.517334051010812e+288 < (/ (* a1 a2) (* b1 b2))

   1. Initial program 59.3

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

   3. Applied associate-/l* 39.8

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

   5. Applied div-inv 39.9

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

   6. Simplified 39.8

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

   8. Applied associate-/r* 15.2

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

   10. Applied div-inv 15.3

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

   11. Applied associate-*r* 14.9

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

   if -1.6828003633594403e+301 < (/ (* a1 a2) (* b1 b2)) < -4.520444799089932e-307 or 0.0 < (/ (* a1 a2) (* b1 b2)) < 4.517334051010812e+288

   1. Initial program 0.8

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

   3. Applied associate-/l* 7.3

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

   5. Applied div-inv 7.6

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

   6. Simplified 7.3

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

   8. Applied associate-/r* 13.7

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

   10. Applied pow1 13.7

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

   11. Applied pow1 13.7

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

   12. Applied pow-prod-down 13.7

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

   13. Simplified 0.8

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

   if -4.520444799089932e-307 < (/ (* a1 a2) (* b1 b2)) < 0.0

   1. Initial program 13.1

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

   3. Applied associate-/l* 7.1

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

   5. Applied associate-/l* 3.8

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

4. Final simplification 3.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -1.68280036335944034 \cdot 10^{301}:\\ \;\;\;\;\left(a1 \cdot \frac{a2}{b1}\right) \cdot \frac{1}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -4.5204447990899316 \cdot 10^{-307}:\\ \;\;\;\;{\left(\frac{a1 \cdot a2}{b1 \cdot b2}\right)}^{1}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 0.0:\\ \;\;\;\;\frac{a1}{\frac{b1}{\frac{a2}{b2}}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 4.51733405101081216 \cdot 10^{288}:\\ \;\;\;\;{\left(\frac{a1 \cdot a2}{b1 \cdot b2}\right)}^{1}\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot \frac{a2}{b1}\right) \cdot \frac{1}{b2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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