Quotient of products

Average Error: 11.3 → 3.1

Time: 12.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} = -\infty:\\ \;\;\;\;\frac{a1 \cdot \frac{a2}{b2}}{b1}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -8.843324609247471005376211875053257441102 \cdot 10^{-307}:\\ \;\;\;\;\frac{a1 \cdot a2}{b2 \cdot b1}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -0.0:\\ \;\;\;\;\frac{a1 \cdot \frac{a2}{b2}}{b1}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 1.464542161086205916851616380334111321007 \cdot 10^{288}:\\ \;\;\;\;\frac{a1 \cdot a2}{b2 \cdot b1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a1}{b1} \cdot a2}{b2}\\ \end{array}\]

C

double f(double a1, double a2, double b1, double b2) {
        double r80223 = a1;
        double r80224 = a2;
        double r80225 = r80223 * r80224;
        double r80226 = b1;
        double r80227 = b2;
        double r80228 = r80226 * r80227;
        double r80229 = r80225 / r80228;
        return r80229;
}

↓

double f(double a1, double a2, double b1, double b2) {
        double r80230 = a1;
        double r80231 = a2;
        double r80232 = r80230 * r80231;
        double r80233 = b1;
        double r80234 = b2;
        double r80235 = r80233 * r80234;
        double r80236 = r80232 / r80235;
        double r80237 = -inf.0;
        bool r80238 = r80236 <= r80237;
        double r80239 = r80231 / r80234;
        double r80240 = r80230 * r80239;
        double r80241 = r80240 / r80233;
        double r80242 = -8.843324609247471e-307;
        bool r80243 = r80236 <= r80242;
        double r80244 = r80234 * r80233;
        double r80245 = r80232 / r80244;
        double r80246 = -0.0;
        bool r80247 = r80236 <= r80246;
        double r80248 = 1.464542161086206e+288;
        bool r80249 = r80236 <= r80248;
        double r80250 = r80230 / r80233;
        double r80251 = r80250 * r80231;
        double r80252 = r80251 / r80234;
        double r80253 = r80249 ? r80245 : r80252;
        double r80254 = r80247 ? r80241 : r80253;
        double r80255 = r80243 ? r80245 : r80254;
        double r80256 = r80238 ? r80241 : r80255;
        return r80256;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	11.3
Target	10.9
Herbie	3.1

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

Derivation

1. Split input into 3 regimes

2. if (/ (* a1 a2) (* b1 b2)) < -inf.0 or -8.843324609247471e-307 < (/ (* a1 a2) (* b1 b2)) < -0.0

   1. Initial program 22.4

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

   3. Applied times-frac 3.6

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

   5. Applied associate-*l/ 5.6

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

   if -inf.0 < (/ (* a1 a2) (* b1 b2)) < -8.843324609247471e-307 or -0.0 < (/ (* a1 a2) (* b1 b2)) < 1.464542161086206e+288

   1. Initial program 3.7

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

   3. Applied times-frac 13.1

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

   5. Applied div-inv 13.1

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

   6. Applied associate-*r* 12.1

      \[\leadsto \color{blue}{\left(\frac{a1}{b1} \cdot a2\right) \cdot \frac{1}{b2}}\]
   7. Using strategy rm

   8. Applied associate-*l/ 7.8

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

   9. Applied frac-times 3.7

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

   10. Simplified 3.7

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

   11. Simplified 3.7

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

   if 1.464542161086206e+288 < (/ (* a1 a2) (* b1 b2))

   1. Initial program 59.6

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

   3. Applied times-frac 7.2

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

   5. Applied div-inv 7.3

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

   6. Applied associate-*r* 13.1

      \[\leadsto \color{blue}{\left(\frac{a1}{b1} \cdot a2\right) \cdot \frac{1}{b2}}\]
   7. Using strategy rm

   8. Applied un-div-inv 13.1

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

4. Final simplification 3.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} = -\infty:\\ \;\;\;\;\frac{a1 \cdot \frac{a2}{b2}}{b1}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -8.843324609247471005376211875053257441102 \cdot 10^{-307}:\\ \;\;\;\;\frac{a1 \cdot a2}{b2 \cdot b1}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -0.0:\\ \;\;\;\;\frac{a1 \cdot \frac{a2}{b2}}{b1}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 1.464542161086205916851616380334111321007 \cdot 10^{288}:\\ \;\;\;\;\frac{a1 \cdot a2}{b2 \cdot b1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a1}{b1} \cdot a2}{b2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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