Quotient of products

Average Error: 11.6 → 5.0

Time: 7.7s

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 r174058 = a1;
        double r174059 = a2;
        double r174060 = r174058 * r174059;
        double r174061 = b1;
        double r174062 = b2;
        double r174063 = r174061 * r174062;
        double r174064 = r174060 / r174063;
        return r174064;
}

↓

double f(double a1, double a2, double b1, double b2) {
        double r174065 = a1;
        double r174066 = a2;
        double r174067 = r174065 * r174066;
        double r174068 = -1.4931232845681356e+168;
        bool r174069 = r174067 <= r174068;
        double r174070 = b1;
        double r174071 = r174065 / r174070;
        double r174072 = b2;
        double r174073 = r174066 / r174072;
        double r174074 = r174071 * r174073;
        double r174075 = -2.0866604942007535e-124;
        bool r174076 = r174067 <= r174075;
        double r174077 = 1.0;
        double r174078 = r174067 / r174070;
        double r174079 = r174072 / r174078;
        double r174080 = r174077 / r174079;
        double r174081 = 1.0160099939508673e-257;
        bool r174082 = r174067 <= r174081;
        double r174083 = cbrt(r174070);
        double r174084 = r174083 * r174083;
        double r174085 = r174065 / r174084;
        double r174086 = cbrt(r174072);
        double r174087 = r174086 * r174086;
        double r174088 = r174085 / r174087;
        double r174089 = cbrt(r174066);
        double r174090 = r174089 * r174089;
        double r174091 = r174088 * r174090;
        double r174092 = cbrt(r174087);
        double r174093 = r174091 / r174092;
        double r174094 = r174089 / r174083;
        double r174095 = cbrt(r174086);
        double r174096 = r174094 / r174095;
        double r174097 = r174093 * r174096;
        double r174098 = 1.596087124873951e+126;
        bool r174099 = r174067 <= r174098;
        double r174100 = r174066 / r174083;
        double r174101 = r174100 / r174086;
        double r174102 = r174088 * r174101;
        double r174103 = r174099 ? r174080 : r174102;
        double r174104 = r174082 ? r174097 : r174103;
        double r174105 = r174076 ? r174080 : r174104;
        double r174106 = r174069 ? r174074 : r174105;
        return r174106;
}

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)))