Quotient of products

Average Error: 11.7 → 4.9

Time: 6.1s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b1 \cdot b2 \le -1.0840043817926052 \cdot 10^{194}:\\ \;\;\;\;\left(\frac{\sqrt[3]{a1} \cdot \sqrt[3]{a1}}{b2} \cdot \frac{\sqrt[3]{\sqrt[3]{a1}} \cdot \sqrt[3]{\sqrt[3]{a1}}}{\frac{\sqrt[3]{b1} \cdot \sqrt[3]{b1}}{\sqrt[3]{a2} \cdot \sqrt[3]{a2}}}\right) \cdot \frac{\sqrt[3]{\sqrt[3]{a1}}}{\frac{\sqrt[3]{b1}}{\sqrt[3]{a2}}}\\ \mathbf{elif}\;b1 \cdot b2 \le -3.08257491941769578 \cdot 10^{-195}:\\ \;\;\;\;\frac{1}{\frac{b1 \cdot b2}{a1 \cdot a2}}\\ \mathbf{elif}\;b1 \cdot b2 \le 1.4142553735687107 \cdot 10^{-148}:\\ \;\;\;\;\frac{\frac{a1}{b2}}{\frac{b1}{a2}}\\ \mathbf{elif}\;b1 \cdot b2 \le 3.0707001026583607 \cdot 10^{167}:\\ \;\;\;\;\frac{1}{\frac{b1 \cdot b2}{a1 \cdot a2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt[3]{a1} \cdot \sqrt[3]{a1}}{b2} \cdot \frac{\sqrt[3]{\sqrt[3]{a1}} \cdot \sqrt[3]{\sqrt[3]{a1}}}{\frac{\sqrt[3]{b1} \cdot \sqrt[3]{b1}}{\sqrt[3]{a2} \cdot \sqrt[3]{a2}}}\right) \cdot \frac{\sqrt[3]{\sqrt[3]{a1}}}{\frac{\sqrt[3]{b1}}{\sqrt[3]{a2}}}\\ \end{array}\]

C

double f(double a1, double a2, double b1, double b2) {
        double r137472 = a1;
        double r137473 = a2;
        double r137474 = r137472 * r137473;
        double r137475 = b1;
        double r137476 = b2;
        double r137477 = r137475 * r137476;
        double r137478 = r137474 / r137477;
        return r137478;
}

↓

double f(double a1, double a2, double b1, double b2) {
        double r137479 = b1;
        double r137480 = b2;
        double r137481 = r137479 * r137480;
        double r137482 = -1.0840043817926052e+194;
        bool r137483 = r137481 <= r137482;
        double r137484 = a1;
        double r137485 = cbrt(r137484);
        double r137486 = r137485 * r137485;
        double r137487 = r137486 / r137480;
        double r137488 = cbrt(r137485);
        double r137489 = r137488 * r137488;
        double r137490 = cbrt(r137479);
        double r137491 = r137490 * r137490;
        double r137492 = a2;
        double r137493 = cbrt(r137492);
        double r137494 = r137493 * r137493;
        double r137495 = r137491 / r137494;
        double r137496 = r137489 / r137495;
        double r137497 = r137487 * r137496;
        double r137498 = r137490 / r137493;
        double r137499 = r137488 / r137498;
        double r137500 = r137497 * r137499;
        double r137501 = -3.0825749194176958e-195;
        bool r137502 = r137481 <= r137501;
        double r137503 = 1.0;
        double r137504 = r137484 * r137492;
        double r137505 = r137481 / r137504;
        double r137506 = r137503 / r137505;
        double r137507 = 1.4142553735687107e-148;
        bool r137508 = r137481 <= r137507;
        double r137509 = r137484 / r137480;
        double r137510 = r137479 / r137492;
        double r137511 = r137509 / r137510;
        double r137512 = 3.0707001026583607e+167;
        bool r137513 = r137481 <= r137512;
        double r137514 = r137513 ? r137506 : r137500;
        double r137515 = r137508 ? r137511 : r137514;
        double r137516 = r137502 ? r137506 : r137515;
        double r137517 = r137483 ? r137500 : r137516;
        return r137517;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	11.7
Target	10.7
Herbie	4.9

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

Derivation

1. Split input into 3 regimes

2. if (* b1 b2) < -1.0840043817926052e+194 or 3.0707001026583607e+167 < (* b1 b2)

   1. Initial program 15.2

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

   3. Applied associate-/l* 14.6

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

   4. Simplified 7.9

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

   6. Applied add-cube-cbrt 8.3

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

   7. Applied times-frac 4.7

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

   9. Applied add-cube-cbrt 4.9

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

   10. Applied add-cube-cbrt 4.9

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

   11. Applied times-frac 4.9

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

   12. Applied add-cube-cbrt 5.0

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

   13. Applied times-frac 3.8

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

   14. Applied associate-*r* 2.7

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

   if -1.0840043817926052e+194 < (* b1 b2) < -3.0825749194176958e-195 or 1.4142553735687107e-148 < (* b1 b2) < 3.0707001026583607e+167

   1. Initial program 3.8

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

   3. Applied clear-num 4.2

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

   if -3.0825749194176958e-195 < (* b1 b2) < 1.4142553735687107e-148

   1. Initial program 29.1

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

   3. Applied associate-/l* 28.6

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

   4. Simplified 16.0

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

   6. Applied associate-/r* 11.0

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

4. Final simplification 4.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;b1 \cdot b2 \le -1.0840043817926052 \cdot 10^{194}:\\ \;\;\;\;\left(\frac{\sqrt[3]{a1} \cdot \sqrt[3]{a1}}{b2} \cdot \frac{\sqrt[3]{\sqrt[3]{a1}} \cdot \sqrt[3]{\sqrt[3]{a1}}}{\frac{\sqrt[3]{b1} \cdot \sqrt[3]{b1}}{\sqrt[3]{a2} \cdot \sqrt[3]{a2}}}\right) \cdot \frac{\sqrt[3]{\sqrt[3]{a1}}}{\frac{\sqrt[3]{b1}}{\sqrt[3]{a2}}}\\ \mathbf{elif}\;b1 \cdot b2 \le -3.08257491941769578 \cdot 10^{-195}:\\ \;\;\;\;\frac{1}{\frac{b1 \cdot b2}{a1 \cdot a2}}\\ \mathbf{elif}\;b1 \cdot b2 \le 1.4142553735687107 \cdot 10^{-148}:\\ \;\;\;\;\frac{\frac{a1}{b2}}{\frac{b1}{a2}}\\ \mathbf{elif}\;b1 \cdot b2 \le 3.0707001026583607 \cdot 10^{167}:\\ \;\;\;\;\frac{1}{\frac{b1 \cdot b2}{a1 \cdot a2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt[3]{a1} \cdot \sqrt[3]{a1}}{b2} \cdot \frac{\sqrt[3]{\sqrt[3]{a1}} \cdot \sqrt[3]{\sqrt[3]{a1}}}{\frac{\sqrt[3]{b1} \cdot \sqrt[3]{b1}}{\sqrt[3]{a2} \cdot \sqrt[3]{a2}}}\right) \cdot \frac{\sqrt[3]{\sqrt[3]{a1}}}{\frac{\sqrt[3]{b1}}{\sqrt[3]{a2}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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