Quotient of products

Average Error: 11.6 → 5.0

Time: 8.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a1 \cdot a2 \le -1.4931232845681356 \cdot 10^{168}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \le -2.0866604942007535 \cdot 10^{-124}:\\ \;\;\;\;\frac{1}{\frac{b2}{\frac{a1 \cdot a2}{b1}}}\\ \mathbf{elif}\;a1 \cdot a2 \le 1.01600999395086732 \cdot 10^{-257}:\\ \;\;\;\;\frac{\frac{\frac{a1}{\sqrt[3]{b1} \cdot \sqrt[3]{b1}}}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}} \cdot \left(\sqrt[3]{a2} \cdot \sqrt[3]{a2}\right)}{\sqrt[3]{\sqrt[3]{b2} \cdot \sqrt[3]{b2}}} \cdot \frac{\frac{\sqrt[3]{a2}}{\sqrt[3]{b1}}}{\sqrt[3]{\sqrt[3]{b2}}}\\ \mathbf{elif}\;a1 \cdot a2 \le 1.59608712487395094 \cdot 10^{126}:\\ \;\;\;\;\frac{1}{\frac{b2}{\frac{a1 \cdot a2}{b1}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a1}{\sqrt[3]{b1} \cdot \sqrt[3]{b1}}}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}} \cdot \frac{\frac{a2}{\sqrt[3]{b1}}}{\sqrt[3]{b2}}\\ \end{array}\]

C

double f(double a1, double a2, double b1, double b2) {
        double r176036 = a1;
        double r176037 = a2;
        double r176038 = r176036 * r176037;
        double r176039 = b1;
        double r176040 = b2;
        double r176041 = r176039 * r176040;
        double r176042 = r176038 / r176041;
        return r176042;
}

↓

double f(double a1, double a2, double b1, double b2) {
        double r176043 = a1;
        double r176044 = a2;
        double r176045 = r176043 * r176044;
        double r176046 = -1.4931232845681356e+168;
        bool r176047 = r176045 <= r176046;
        double r176048 = b1;
        double r176049 = r176043 / r176048;
        double r176050 = b2;
        double r176051 = r176044 / r176050;
        double r176052 = r176049 * r176051;
        double r176053 = -2.0866604942007535e-124;
        bool r176054 = r176045 <= r176053;
        double r176055 = 1.0;
        double r176056 = r176045 / r176048;
        double r176057 = r176050 / r176056;
        double r176058 = r176055 / r176057;
        double r176059 = 1.0160099939508673e-257;
        bool r176060 = r176045 <= r176059;
        double r176061 = cbrt(r176048);
        double r176062 = r176061 * r176061;
        double r176063 = r176043 / r176062;
        double r176064 = cbrt(r176050);
        double r176065 = r176064 * r176064;
        double r176066 = r176063 / r176065;
        double r176067 = cbrt(r176044);
        double r176068 = r176067 * r176067;
        double r176069 = r176066 * r176068;
        double r176070 = cbrt(r176065);
        double r176071 = r176069 / r176070;
        double r176072 = r176067 / r176061;
        double r176073 = cbrt(r176064);
        double r176074 = r176072 / r176073;
        double r176075 = r176071 * r176074;
        double r176076 = 1.596087124873951e+126;
        bool r176077 = r176045 <= r176076;
        double r176078 = r176044 / r176061;
        double r176079 = r176078 / r176064;
        double r176080 = r176066 * r176079;
        double r176081 = r176077 ? r176058 : r176080;
        double r176082 = r176060 ? r176075 : r176081;
        double r176083 = r176054 ? r176058 : r176082;
        double r176084 = r176047 ? r176052 : r176083;
        return r176084;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	11.6
Target	11.3
Herbie	5.0

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

Derivation

1. Split input into 4 regimes

2. if (* a1 a2) < -1.4931232845681356e+168

   1. Initial program 30.4

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

   3. Applied times-frac 12.5

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

   if -1.4931232845681356e+168 < (* a1 a2) < -2.0866604942007535e-124 or 1.0160099939508673e-257 < (* a1 a2) < 1.596087124873951e+126

   1. Initial program 4.0

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

   3. Applied associate-/r* 3.7

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

   5. Applied clear-num 4.1

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

   if -2.0866604942007535e-124 < (* a1 a2) < 1.0160099939508673e-257

   1. Initial program 14.7

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

   3. Applied associate-/r* 14.7

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

   5. Applied add-cube-cbrt 15.0

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

   6. Applied add-cube-cbrt 15.1

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

   7. Applied times-frac 8.4

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

   8. Applied times-frac 3.5

      \[\leadsto \color{blue}{\frac{\frac{a1}{\sqrt[3]{b1} \cdot \sqrt[3]{b1}}}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}} \cdot \frac{\frac{a2}{\sqrt[3]{b1}}}{\sqrt[3]{b2}}}\]
   9. Using strategy rm

   10. Applied add-cube-cbrt 3.5

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

   11. Applied cbrt-prod 3.6

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

   12. Applied *-un-lft-identity 3.6

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

   13. Applied add-cube-cbrt 3.6

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

   14. Applied times-frac 3.6

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

   15. Applied times-frac 3.0

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

   16. Applied associate-*r* 3.0

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

   17. Simplified 3.0

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

   if 1.596087124873951e+126 < (* a1 a2)

   1. Initial program 25.3

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

   3. Applied associate-/r* 27.1

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

   5. Applied add-cube-cbrt 27.7

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

   6. Applied add-cube-cbrt 27.8

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

   7. Applied times-frac 17.4

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

   8. Applied times-frac 9.4

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

4. Final simplification 5.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;a1 \cdot a2 \le -1.4931232845681356 \cdot 10^{168}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \mathbf{elif}\;a1 \cdot a2 \le -2.0866604942007535 \cdot 10^{-124}:\\ \;\;\;\;\frac{1}{\frac{b2}{\frac{a1 \cdot a2}{b1}}}\\ \mathbf{elif}\;a1 \cdot a2 \le 1.01600999395086732 \cdot 10^{-257}:\\ \;\;\;\;\frac{\frac{\frac{a1}{\sqrt[3]{b1} \cdot \sqrt[3]{b1}}}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}} \cdot \left(\sqrt[3]{a2} \cdot \sqrt[3]{a2}\right)}{\sqrt[3]{\sqrt[3]{b2} \cdot \sqrt[3]{b2}}} \cdot \frac{\frac{\sqrt[3]{a2}}{\sqrt[3]{b1}}}{\sqrt[3]{\sqrt[3]{b2}}}\\ \mathbf{elif}\;a1 \cdot a2 \le 1.59608712487395094 \cdot 10^{126}:\\ \;\;\;\;\frac{1}{\frac{b2}{\frac{a1 \cdot a2}{b1}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a1}{\sqrt[3]{b1} \cdot \sqrt[3]{b1}}}{\sqrt[3]{b2} \cdot \sqrt[3]{b2}} \cdot \frac{\frac{a2}{\sqrt[3]{b1}}}{\sqrt[3]{b2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"
  :precision binary64

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

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