Quotient of products

Average Error: 10.8 → 5.3

Time: 4.7m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a1 \cdot a2 \le -6.897241169256815 \cdot 10^{+159}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{a1}}{\sqrt[3]{b1}} \cdot \frac{\sqrt[3]{a1}}{\sqrt[3]{b1}}}{\frac{b2}{\frac{1}{\frac{\frac{\sqrt[3]{b1}}{a2}}{\sqrt[3]{a1}}}}}\\ \mathbf{elif}\;a1 \cdot a2 \le -4.213300454402738 \cdot 10^{-210}:\\ \;\;\;\;\left(a1 \cdot a2\right) \cdot \frac{\frac{1}{b1}}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \le 2.8391901767890553 \cdot 10^{-163}:\\ \;\;\;\;\frac{\frac{a2}{b1}}{\frac{b2}{a1}}\\ \mathbf{elif}\;a1 \cdot a2 \le 1.3171664498037926 \cdot 10^{+299}:\\ \;\;\;\;\left(a1 \cdot a2\right) \cdot \frac{\frac{1}{b1}}{b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{b2} \cdot \frac{a1}{b1}\\ \end{array}\]

C

double f(double a1, double a2, double b1, double b2) {
        double r22638947 = a1;
        double r22638948 = a2;
        double r22638949 = r22638947 * r22638948;
        double r22638950 = b1;
        double r22638951 = b2;
        double r22638952 = r22638950 * r22638951;
        double r22638953 = r22638949 / r22638952;
        return r22638953;
}

↓

double f(double a1, double a2, double b1, double b2) {
        double r22638954 = a1;
        double r22638955 = a2;
        double r22638956 = r22638954 * r22638955;
        double r22638957 = -6.897241169256815e+159;
        bool r22638958 = r22638956 <= r22638957;
        double r22638959 = cbrt(r22638954);
        double r22638960 = b1;
        double r22638961 = cbrt(r22638960);
        double r22638962 = r22638959 / r22638961;
        double r22638963 = r22638962 * r22638962;
        double r22638964 = b2;
        double r22638965 = 1.0;
        double r22638966 = r22638961 / r22638955;
        double r22638967 = r22638966 / r22638959;
        double r22638968 = r22638965 / r22638967;
        double r22638969 = r22638964 / r22638968;
        double r22638970 = r22638963 / r22638969;
        double r22638971 = -4.213300454402738e-210;
        bool r22638972 = r22638956 <= r22638971;
        double r22638973 = r22638965 / r22638960;
        double r22638974 = r22638973 / r22638964;
        double r22638975 = r22638956 * r22638974;
        double r22638976 = 2.8391901767890553e-163;
        bool r22638977 = r22638956 <= r22638976;
        double r22638978 = r22638955 / r22638960;
        double r22638979 = r22638964 / r22638954;
        double r22638980 = r22638978 / r22638979;
        double r22638981 = 1.3171664498037926e+299;
        bool r22638982 = r22638956 <= r22638981;
        double r22638983 = r22638955 / r22638964;
        double r22638984 = r22638954 / r22638960;
        double r22638985 = r22638983 * r22638984;
        double r22638986 = r22638982 ? r22638975 : r22638985;
        double r22638987 = r22638977 ? r22638980 : r22638986;
        double r22638988 = r22638972 ? r22638975 : r22638987;
        double r22638989 = r22638958 ? r22638970 : r22638988;
        return r22638989;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	10.8
Target	10.9
Herbie	5.3

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

Derivation

1. Split input into 4 regimes

2. if (* a1 a2) < -6.897241169256815e+159

   1. Initial program 26.1

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

   3. Applied associate-/r* 26.4

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

   5. Applied associate-/l* 15.6

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

   7. Applied *-un-lft-identity 15.6

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

   8. Applied associate-/l* 15.7

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

   10. Applied add-cube-cbrt 16.3

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

   11. Applied *-un-lft-identity 16.3

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

   12. Applied add-cube-cbrt 16.5

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

   13. Applied times-frac 16.5

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

   14. Applied times-frac 15.3

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

   15. Applied *-un-lft-identity 15.3

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

   16. Applied times-frac 15.4

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

   17. Applied associate-/l* 8.9

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

   18. Simplified 8.9

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

   if -6.897241169256815e+159 < (* a1 a2) < -4.213300454402738e-210 or 2.8391901767890553e-163 < (* a1 a2) < 1.3171664498037926e+299

   1. Initial program 4.7

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

   3. Applied associate-/r* 4.3

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

   5. Applied *-un-lft-identity 4.3

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

   6. Applied div-inv 4.4

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

   7. Applied times-frac 4.8

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

   8. Simplified 4.8

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

   if -4.213300454402738e-210 < (* a1 a2) < 2.8391901767890553e-163

   1. Initial program 13.5

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

   3. Applied associate-/r* 13.3

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

   5. Applied associate-/l* 7.6

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

   7. Applied *-un-lft-identity 7.6

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

   8. Applied associate-/l* 8.0

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

   10. Applied associate-/r/ 7.7

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

   11. Applied associate-/l* 5.3

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

   12. Simplified 5.0

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

   if 1.3171664498037926e+299 < (* a1 a2)

   1. Initial program 57.6

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

   3. Applied associate-/r* 57.9

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

   5. Applied associate-/l* 24.9

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

   7. Applied *-un-lft-identity 24.9

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

   8. Applied associate-/r/ 25.0

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

   9. Applied times-frac 7.4

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

   10. Simplified 7.4

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

4. Final simplification 5.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;a1 \cdot a2 \le -6.897241169256815 \cdot 10^{+159}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{a1}}{\sqrt[3]{b1}} \cdot \frac{\sqrt[3]{a1}}{\sqrt[3]{b1}}}{\frac{b2}{\frac{1}{\frac{\frac{\sqrt[3]{b1}}{a2}}{\sqrt[3]{a1}}}}}\\ \mathbf{elif}\;a1 \cdot a2 \le -4.213300454402738 \cdot 10^{-210}:\\ \;\;\;\;\left(a1 \cdot a2\right) \cdot \frac{\frac{1}{b1}}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \le 2.8391901767890553 \cdot 10^{-163}:\\ \;\;\;\;\frac{\frac{a2}{b1}}{\frac{b2}{a1}}\\ \mathbf{elif}\;a1 \cdot a2 \le 1.3171664498037926 \cdot 10^{+299}:\\ \;\;\;\;\left(a1 \cdot a2\right) \cdot \frac{\frac{1}{b1}}{b2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a2}{b2} \cdot \frac{a1}{b1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019124 +o rules:numerics
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"

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

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