Quotient of products

Average Error: 11.4 → 4.9

Time: 11.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b1 \cdot b2 \le -6.898490520999014 \cdot 10^{+277}:\\ \;\;\;\;\frac{\frac{1}{b1}}{\frac{\frac{1}{a2}}{\frac{a1}{b2}}}\\ \mathbf{elif}\;b1 \cdot b2 \le -9.006559789728174 \cdot 10^{-250}:\\ \;\;\;\;a1 \cdot \frac{a2}{b1 \cdot b2}\\ \mathbf{elif}\;b1 \cdot b2 \le 1.30715975193335 \cdot 10^{-196}:\\ \;\;\;\;\frac{\frac{a1}{b1}}{\frac{b2}{a2}}\\ \mathbf{elif}\;b1 \cdot b2 \le 7.676958632971629 \cdot 10^{+257}:\\ \;\;\;\;\frac{\frac{a1}{b1 \cdot b2}}{\frac{1}{a2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{\frac{b2}{\sqrt[3]{a2}}} \cdot \frac{1}{\frac{b1}{\sqrt[3]{a2} \cdot \sqrt[3]{a2}}}\\ \end{array}\]

C

double f(double a1, double a2, double b1, double b2) {
        double r2861061 = a1;
        double r2861062 = a2;
        double r2861063 = r2861061 * r2861062;
        double r2861064 = b1;
        double r2861065 = b2;
        double r2861066 = r2861064 * r2861065;
        double r2861067 = r2861063 / r2861066;
        return r2861067;
}

↓

double f(double a1, double a2, double b1, double b2) {
        double r2861068 = b1;
        double r2861069 = b2;
        double r2861070 = r2861068 * r2861069;
        double r2861071 = -6.898490520999014e+277;
        bool r2861072 = r2861070 <= r2861071;
        double r2861073 = 1.0;
        double r2861074 = r2861073 / r2861068;
        double r2861075 = a2;
        double r2861076 = r2861073 / r2861075;
        double r2861077 = a1;
        double r2861078 = r2861077 / r2861069;
        double r2861079 = r2861076 / r2861078;
        double r2861080 = r2861074 / r2861079;
        double r2861081 = -9.006559789728174e-250;
        bool r2861082 = r2861070 <= r2861081;
        double r2861083 = r2861075 / r2861070;
        double r2861084 = r2861077 * r2861083;
        double r2861085 = 1.30715975193335e-196;
        bool r2861086 = r2861070 <= r2861085;
        double r2861087 = r2861077 / r2861068;
        double r2861088 = r2861069 / r2861075;
        double r2861089 = r2861087 / r2861088;
        double r2861090 = 7.676958632971629e+257;
        bool r2861091 = r2861070 <= r2861090;
        double r2861092 = r2861077 / r2861070;
        double r2861093 = r2861092 / r2861076;
        double r2861094 = cbrt(r2861075);
        double r2861095 = r2861069 / r2861094;
        double r2861096 = r2861077 / r2861095;
        double r2861097 = r2861094 * r2861094;
        double r2861098 = r2861068 / r2861097;
        double r2861099 = r2861073 / r2861098;
        double r2861100 = r2861096 * r2861099;
        double r2861101 = r2861091 ? r2861093 : r2861100;
        double r2861102 = r2861086 ? r2861089 : r2861101;
        double r2861103 = r2861082 ? r2861084 : r2861102;
        double r2861104 = r2861072 ? r2861080 : r2861103;
        return r2861104;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	11.4
Target	11.0
Herbie	4.9

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

Derivation

1. Split input into 5 regimes

2. if (* b1 b2) < -6.898490520999014e+277

   1. Initial program 20.9

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

   3. Applied associate-/l* 20.8

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

   5. Applied div-inv 20.8

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

   6. Applied associate-/r* 20.9

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

   8. Applied *-un-lft-identity 20.9

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

   9. Applied times-frac 7.2

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

   10. Applied associate-/l* 2.4

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

   if -6.898490520999014e+277 < (* b1 b2) < -9.006559789728174e-250

   1. Initial program 5.0

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

   3. Applied associate-/l* 5.2

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

   5. Applied div-inv 5.2

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

   6. Applied associate-/r* 4.9

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

   8. Applied *-un-lft-identity 4.9

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

   9. Applied add-cube-cbrt 4.9

      \[\leadsto \frac{\frac{a1}{b1 \cdot b2}}{\frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{1 \cdot a2}}\]

   10. Applied times-frac 4.9

       \[\leadsto \frac{\frac{a1}{b1 \cdot b2}}{\color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1} \cdot \frac{\sqrt[3]{1}}{a2}}}\]

   11. Applied div-inv 4.9

       \[\leadsto \frac{\color{blue}{a1 \cdot \frac{1}{b1 \cdot b2}}}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1} \cdot \frac{\sqrt[3]{1}}{a2}}\]

   12. Applied times-frac 5.3

       \[\leadsto \color{blue}{\frac{a1}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1}} \cdot \frac{\frac{1}{b1 \cdot b2}}{\frac{\sqrt[3]{1}}{a2}}}\]

   13. Simplified 5.3

       \[\leadsto \color{blue}{a1} \cdot \frac{\frac{1}{b1 \cdot b2}}{\frac{\sqrt[3]{1}}{a2}}\]

   14. Simplified 5.2

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

   if -9.006559789728174e-250 < (* b1 b2) < 1.30715975193335e-196

   1. Initial program 35.6

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

   3. Applied associate-/l* 35.7

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

   5. Applied *-un-lft-identity 35.7

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

   6. Applied times-frac 17.2

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

   7. Applied associate-/r* 10.0

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

   8. Simplified 10.0

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

   if 1.30715975193335e-196 < (* b1 b2) < 7.676958632971629e+257

   1. Initial program 5.0

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

   3. Applied associate-/l* 4.7

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

   5. Applied div-inv 4.7

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

   6. Applied associate-/r* 4.3

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

   if 7.676958632971629e+257 < (* b1 b2)

   1. Initial program 18.3

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

   3. Applied associate-/l* 18.4

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

   5. Applied add-cube-cbrt 18.5

      \[\leadsto \frac{a1}{\frac{b1 \cdot b2}{\color{blue}{\left(\sqrt[3]{a2} \cdot \sqrt[3]{a2}\right) \cdot \sqrt[3]{a2}}}}\]

   6. Applied times-frac 7.9

      \[\leadsto \frac{a1}{\color{blue}{\frac{b1}{\sqrt[3]{a2} \cdot \sqrt[3]{a2}} \cdot \frac{b2}{\sqrt[3]{a2}}}}\]

   7. Applied *-un-lft-identity 7.9

      \[\leadsto \frac{\color{blue}{1 \cdot a1}}{\frac{b1}{\sqrt[3]{a2} \cdot \sqrt[3]{a2}} \cdot \frac{b2}{\sqrt[3]{a2}}}\]

   8. Applied times-frac 2.4

      \[\leadsto \color{blue}{\frac{1}{\frac{b1}{\sqrt[3]{a2} \cdot \sqrt[3]{a2}}} \cdot \frac{a1}{\frac{b2}{\sqrt[3]{a2}}}}\]
3. Recombined 5 regimes into one program.

4. Final simplification 4.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;b1 \cdot b2 \le -6.898490520999014 \cdot 10^{+277}:\\ \;\;\;\;\frac{\frac{1}{b1}}{\frac{\frac{1}{a2}}{\frac{a1}{b2}}}\\ \mathbf{elif}\;b1 \cdot b2 \le -9.006559789728174 \cdot 10^{-250}:\\ \;\;\;\;a1 \cdot \frac{a2}{b1 \cdot b2}\\ \mathbf{elif}\;b1 \cdot b2 \le 1.30715975193335 \cdot 10^{-196}:\\ \;\;\;\;\frac{\frac{a1}{b1}}{\frac{b2}{a2}}\\ \mathbf{elif}\;b1 \cdot b2 \le 7.676958632971629 \cdot 10^{+257}:\\ \;\;\;\;\frac{\frac{a1}{b1 \cdot b2}}{\frac{1}{a2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{\frac{b2}{\sqrt[3]{a2}}} \cdot \frac{1}{\frac{b1}{\sqrt[3]{a2} \cdot \sqrt[3]{a2}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019153 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"

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

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