Quotient of products

Average Error: 11.5 → 11.5

Time: 7.5s

Precision: 64

Math

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

↓

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

C

double f(double a1, double a2, double b1, double b2) {
        double r79509 = a1;
        double r79510 = a2;
        double r79511 = r79509 * r79510;
        double r79512 = b1;
        double r79513 = b2;
        double r79514 = r79512 * r79513;
        double r79515 = r79511 / r79514;
        return r79515;
}

↓

double f(double a1, double a2, double b1, double b2) {
        double r79516 = a1;
        double r79517 = a2;
        double r79518 = r79516 * r79517;
        double r79519 = b1;
        double r79520 = b2;
        double r79521 = r79519 * r79520;
        double r79522 = r79518 / r79521;
        return r79522;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	11.5
Target	11.2
Herbie	11.5

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

Derivation

1. Split input into 4 regimes

2. if (* b1 b2) < -1.9533036250714963e+210

   1. Initial program 16.9

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

   3. Applied times-frac 4.6

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

   if -1.9533036250714963e+210 < (* b1 b2) < -1.1259899127088251e-109 or 1.2432106094940164e-234 < (* b1 b2) < 2.771822019487987e+122

   1. Initial program 3.8

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

   3. Applied div-inv 3.9

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

   if -1.1259899127088251e-109 < (* b1 b2) < 1.2432106094940164e-234

   1. Initial program 26.9

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

   3. Applied associate-/r* 15.0

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

   5. Applied associate-/l* 11.6

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

   if 2.771822019487987e+122 < (* b1 b2)

   1. Initial program 13.9

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

   3. Applied associate-/r* 8.8

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

   5. Applied associate-/l* 6.9

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

   7. Applied associate-/r/ 6.3

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

   8. Applied associate-/l* 7.2

      \[\leadsto \color{blue}{\frac{\frac{a1}{b1}}{\frac{b2}{a2}}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 11.5

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

Reproduce

herbie shell --seed 2019297 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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