Quotient of products

Average Error: 11.6 → 2.7

Time: 13.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} = -\infty:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -9.782499787656681574696062098790783172828 \cdot 10^{-322}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 1.421277284710987870204738960879832034274 \cdot 10^{-315}:\\ \;\;\;\;\frac{\frac{a1}{b2}}{b1} \cdot a2\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 2.286175403506427577716209346328092058444 \cdot 10^{301}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \end{array}\]

C

double f(double a1, double a2, double b1, double b2) {
        double r99237 = a1;
        double r99238 = a2;
        double r99239 = r99237 * r99238;
        double r99240 = b1;
        double r99241 = b2;
        double r99242 = r99240 * r99241;
        double r99243 = r99239 / r99242;
        return r99243;
}

↓

double f(double a1, double a2, double b1, double b2) {
        double r99244 = a1;
        double r99245 = a2;
        double r99246 = r99244 * r99245;
        double r99247 = b1;
        double r99248 = b2;
        double r99249 = r99247 * r99248;
        double r99250 = r99246 / r99249;
        double r99251 = -inf.0;
        bool r99252 = r99250 <= r99251;
        double r99253 = r99244 / r99247;
        double r99254 = r99245 / r99248;
        double r99255 = r99253 * r99254;
        double r99256 = -9.7824997876567e-322;
        bool r99257 = r99250 <= r99256;
        double r99258 = 1.421277284711e-315;
        bool r99259 = r99250 <= r99258;
        double r99260 = r99244 / r99248;
        double r99261 = r99260 / r99247;
        double r99262 = r99261 * r99245;
        double r99263 = 2.2861754035064276e+301;
        bool r99264 = r99250 <= r99263;
        double r99265 = r99264 ? r99250 : r99255;
        double r99266 = r99259 ? r99262 : r99265;
        double r99267 = r99257 ? r99250 : r99266;
        double r99268 = r99252 ? r99255 : r99267;
        return r99268;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	11.6
Target	11.4
Herbie	2.7

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

Derivation

1. Split input into 3 regimes

2. if (/ (* a1 a2) (* b1 b2)) < -inf.0 or 2.2861754035064276e+301 < (/ (* a1 a2) (* b1 b2))

   1. Initial program 62.7

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

   3. Applied times-frac 9.3

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

   if -inf.0 < (/ (* a1 a2) (* b1 b2)) < -9.7824997876567e-322 or 1.421277284711e-315 < (/ (* a1 a2) (* b1 b2)) < 2.2861754035064276e+301

   1. Initial program 0.9

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

   3. Applied associate-/l* 8.3

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

   5. Applied *-un-lft-identity 8.3

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

   6. Applied times-frac 14.7

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

   7. Applied *-un-lft-identity 14.7

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

   8. Applied times-frac 14.5

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

   9. Simplified 14.5

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

   10. Simplified 14.5

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

   11. Taylor expanded around 0 0.9

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

   if -9.7824997876567e-322 < (/ (* a1 a2) (* b1 b2)) < 1.421277284711e-315

   1. Initial program 13.6

      \[\frac{a1 \cdot a2}{b1 \cdot b2}\]
   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 *-un-lft-identity 7.3

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

   6. Applied times-frac 3.9

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

   7. Applied *-un-lft-identity 3.9

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

   8. Applied times-frac 3.4

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

   9. Simplified 3.4

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

   10. Simplified 3.5

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

   12. Applied associate-*r* 3.8

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

   13. Simplified 3.8

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

4. Final simplification 2.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} = -\infty:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -9.782499787656681574696062098790783172828 \cdot 10^{-322}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 1.421277284710987870204738960879832034274 \cdot 10^{-315}:\\ \;\;\;\;\frac{\frac{a1}{b2}}{b1} \cdot a2\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 2.286175403506427577716209346328092058444 \cdot 10^{301}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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