Quotient of products

Average Error: 11.5 → 2.7

Time: 2.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -1.002329447227761808147469830815412761507 \cdot 10^{264}:\\ \;\;\;\;\frac{\frac{a1}{b1}}{\frac{b2}{a2}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -2.941470907051121937881790391096071314592 \cdot 10^{-294}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 2.304552242205900595612811993092939692084 \cdot 10^{-302}:\\ \;\;\;\;\frac{\frac{a1}{b1}}{\frac{b2}{a2}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 1.501180578243183430701187487175576593932 \cdot 10^{300}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b2} \cdot \left(\frac{1}{b1} \cdot a2\right)\\ \end{array}\]

C

double f(double a1, double a2, double b1, double b2) {
        double r138839 = a1;
        double r138840 = a2;
        double r138841 = r138839 * r138840;
        double r138842 = b1;
        double r138843 = b2;
        double r138844 = r138842 * r138843;
        double r138845 = r138841 / r138844;
        return r138845;
}

↓

double f(double a1, double a2, double b1, double b2) {
        double r138846 = a1;
        double r138847 = a2;
        double r138848 = r138846 * r138847;
        double r138849 = b1;
        double r138850 = b2;
        double r138851 = r138849 * r138850;
        double r138852 = r138848 / r138851;
        double r138853 = -1.0023294472277618e+264;
        bool r138854 = r138852 <= r138853;
        double r138855 = r138846 / r138849;
        double r138856 = r138850 / r138847;
        double r138857 = r138855 / r138856;
        double r138858 = -2.941470907051122e-294;
        bool r138859 = r138852 <= r138858;
        double r138860 = 2.3045522422059006e-302;
        bool r138861 = r138852 <= r138860;
        double r138862 = 1.5011805782431834e+300;
        bool r138863 = r138852 <= r138862;
        double r138864 = r138846 / r138850;
        double r138865 = 1.0;
        double r138866 = r138865 / r138849;
        double r138867 = r138866 * r138847;
        double r138868 = r138864 * r138867;
        double r138869 = r138863 ? r138852 : r138868;
        double r138870 = r138861 ? r138857 : r138869;
        double r138871 = r138859 ? r138852 : r138870;
        double r138872 = r138854 ? r138857 : r138871;
        return r138872;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	11.5
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)) < -1.0023294472277618e+264 or -2.941470907051122e-294 < (/ (* a1 a2) (* b1 b2)) < 2.3045522422059006e-302

   1. Initial program 17.9

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

   3. Applied times-frac 4.8

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

   5. Applied associate-*r/ 6.4

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

   7. Applied associate-/l* 4.8

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

   if -1.0023294472277618e+264 < (/ (* a1 a2) (* b1 b2)) < -2.941470907051122e-294 or 2.3045522422059006e-302 < (/ (* a1 a2) (* b1 b2)) < 1.5011805782431834e+300

   1. Initial program 0.8

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

   3. Applied times-frac 16.5

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

   5. Applied associate-*r/ 14.4

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

   7. Applied associate-/l* 16.4

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

   8. Taylor expanded around 0 0.8

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

   if 1.5011805782431834e+300 < (/ (* a1 a2) (* b1 b2))

   1. Initial program 61.7

      \[\frac{a1 \cdot a2}{b1 \cdot b2}\]
   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 associate-*r/ 15.4

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

   7. Applied associate-/l* 7.0

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

   9. Applied div-inv 7.1

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

   10. Applied div-inv 7.1

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

   11. Applied times-frac 5.8

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

   12. Simplified 5.8

       \[\leadsto \frac{a1}{b2} \cdot \color{blue}{\left(\frac{1}{b1} \cdot a2\right)}\]
3. Recombined 3 regimes into one program.

4. Final simplification 2.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -1.002329447227761808147469830815412761507 \cdot 10^{264}:\\ \;\;\;\;\frac{\frac{a1}{b1}}{\frac{b2}{a2}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le -2.941470907051121937881790391096071314592 \cdot 10^{-294}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 2.304552242205900595612811993092939692084 \cdot 10^{-302}:\\ \;\;\;\;\frac{\frac{a1}{b1}}{\frac{b2}{a2}}\\ \mathbf{elif}\;\frac{a1 \cdot a2}{b1 \cdot b2} \le 1.501180578243183430701187487175576593932 \cdot 10^{300}:\\ \;\;\;\;\frac{a1 \cdot a2}{b1 \cdot b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b2} \cdot \left(\frac{1}{b1} \cdot a2\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019354 +o rules:numerics
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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