Quotient of products

Average Error: 11.3 → 2.4

Time: 2.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} = -\infty:\\ \;\;\;\;\frac{a1 \cdot \frac{a2}{b1}}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -3.0991765939 \cdot 10^{-315}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -0.0:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 8.3923045265864167 \cdot 10^{294}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \end{array}\]

C

double f(double a1, double a2, double b1, double b2) {
        double r102472 = a1;
        double r102473 = a2;
        double r102474 = r102472 * r102473;
        double r102475 = b1;
        double r102476 = b2;
        double r102477 = r102475 * r102476;
        double r102478 = r102474 / r102477;
        return r102478;
}

↓

double f(double a1, double a2, double b1, double b2) {
        double r102479 = a1;
        double r102480 = a2;
        double r102481 = r102479 * r102480;
        double r102482 = b1;
        double r102483 = b2;
        double r102484 = r102482 * r102483;
        double r102485 = r102481 / r102484;
        double r102486 = -inf.0;
        bool r102487 = r102485 <= r102486;
        double r102488 = r102480 / r102482;
        double r102489 = r102479 * r102488;
        double r102490 = r102489 / r102483;
        double r102491 = -3.099176593887e-315;
        bool r102492 = r102485 <= r102491;
        double r102493 = -0.0;
        bool r102494 = r102485 <= r102493;
        double r102495 = r102479 / r102482;
        double r102496 = r102480 / r102483;
        double r102497 = r102495 * r102496;
        double r102498 = 8.392304526586417e+294;
        bool r102499 = r102485 <= r102498;
        double r102500 = r102499 ? r102485 : r102497;
        double r102501 = r102494 ? r102497 : r102500;
        double r102502 = r102492 ? r102485 : r102501;
        double r102503 = r102487 ? r102490 : r102502;
        return r102503;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	11.3
Target	11.3
Herbie	2.4

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

Derivation

1. Split input into 3 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 times-frac 11.1

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

   5. Applied associate-*r/ 15.8

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

   7. Applied div-inv 15.8

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

   8. Applied associate-*l* 15.8

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

   9. Simplified 15.8

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

   if -inf.0 < (/ (* a1 a2) (* b1 b2)) < -3.099176593887e-315 or -0.0 < (/ (* a1 a2) (* b1 b2)) < 8.392304526586417e+294

   1. Initial program 3.8

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

   if -3.099176593887e-315 < (/ (* a1 a2) (* b1 b2)) < -0.0 or 8.392304526586417e+294 < (/ (* a1 a2) (* b1 b2))

   1. Initial program 27.9

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

   3. Applied times-frac 4.4

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

4. Final simplification 2.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} = -\infty:\\ \;\;\;\;\frac{a1 \cdot \frac{a2}{b1}}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -3.0991765939 \cdot 10^{-315}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -0.0:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 8.3923045265864167 \cdot 10^{294}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020021 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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