Quotient of products

Average Error: 11.1 → 10.8

Time: 16.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a1 \le -2.6262968301438297 \cdot 10^{+154}:\\ \;\;\;\;\frac{a2}{b2 \cdot b1} \cdot a1\\ \mathbf{elif}\;a1 \le -7.197696696298761 \cdot 10^{-23}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{a2}{b2}}}{b1} \cdot \left(\left(\sqrt[3]{\frac{a2}{b2}} \cdot a1\right) \cdot \sqrt[3]{\frac{a2}{b2}}\right)\\ \mathbf{elif}\;a1 \le -1.4115765653207805 \cdot 10^{-101}:\\ \;\;\;\;\frac{a2}{b2 \cdot b1} \cdot a1\\ \mathbf{elif}\;a1 \le -3.75577994435908 \cdot 10^{-192}:\\ \;\;\;\;\frac{\frac{a2 \cdot a1}{b1}}{b2}\\ \mathbf{elif}\;a1 \le -2.7564687177065716 \cdot 10^{-269}:\\ \;\;\;\;\frac{1}{b1} \cdot \left(\frac{a2}{b2} \cdot a1\right)\\ \mathbf{elif}\;a1 \le 8.377469299573798 \cdot 10^{-267}:\\ \;\;\;\;\frac{a2 \cdot a1}{b2 \cdot b1}\\ \mathbf{elif}\;a1 \le 5.6037110240578724 \cdot 10^{-148}:\\ \;\;\;\;\frac{1}{b1} \cdot \left(\frac{a2}{b2} \cdot a1\right)\\ \mathbf{elif}\;a1 \le 1.3929709356464372 \cdot 10^{+88}:\\ \;\;\;\;\frac{a2}{b2 \cdot b1} \cdot a1\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \end{array}\]

C

double f(double a1, double a2, double b1, double b2) {
        double r5231116 = a1;
        double r5231117 = a2;
        double r5231118 = r5231116 * r5231117;
        double r5231119 = b1;
        double r5231120 = b2;
        double r5231121 = r5231119 * r5231120;
        double r5231122 = r5231118 / r5231121;
        return r5231122;
}

↓

double f(double a1, double a2, double b1, double b2) {
        double r5231123 = a1;
        double r5231124 = -2.6262968301438297e+154;
        bool r5231125 = r5231123 <= r5231124;
        double r5231126 = a2;
        double r5231127 = b2;
        double r5231128 = b1;
        double r5231129 = r5231127 * r5231128;
        double r5231130 = r5231126 / r5231129;
        double r5231131 = r5231130 * r5231123;
        double r5231132 = -7.197696696298761e-23;
        bool r5231133 = r5231123 <= r5231132;
        double r5231134 = r5231126 / r5231127;
        double r5231135 = cbrt(r5231134);
        double r5231136 = r5231135 / r5231128;
        double r5231137 = r5231135 * r5231123;
        double r5231138 = r5231137 * r5231135;
        double r5231139 = r5231136 * r5231138;
        double r5231140 = -1.4115765653207805e-101;
        bool r5231141 = r5231123 <= r5231140;
        double r5231142 = -3.75577994435908e-192;
        bool r5231143 = r5231123 <= r5231142;
        double r5231144 = r5231126 * r5231123;
        double r5231145 = r5231144 / r5231128;
        double r5231146 = r5231145 / r5231127;
        double r5231147 = -2.7564687177065716e-269;
        bool r5231148 = r5231123 <= r5231147;
        double r5231149 = 1.0;
        double r5231150 = r5231149 / r5231128;
        double r5231151 = r5231134 * r5231123;
        double r5231152 = r5231150 * r5231151;
        double r5231153 = 8.377469299573798e-267;
        bool r5231154 = r5231123 <= r5231153;
        double r5231155 = r5231144 / r5231129;
        double r5231156 = 5.6037110240578724e-148;
        bool r5231157 = r5231123 <= r5231156;
        double r5231158 = 1.3929709356464372e+88;
        bool r5231159 = r5231123 <= r5231158;
        double r5231160 = r5231123 / r5231128;
        double r5231161 = r5231160 * r5231134;
        double r5231162 = r5231159 ? r5231131 : r5231161;
        double r5231163 = r5231157 ? r5231152 : r5231162;
        double r5231164 = r5231154 ? r5231155 : r5231163;
        double r5231165 = r5231148 ? r5231152 : r5231164;
        double r5231166 = r5231143 ? r5231146 : r5231165;
        double r5231167 = r5231141 ? r5231131 : r5231166;
        double r5231168 = r5231133 ? r5231139 : r5231167;
        double r5231169 = r5231125 ? r5231131 : r5231168;
        return r5231169;
}

Error

Bits error versus a1

Bits error versus a2

Bits error versus b1

Bits error versus b2

Target

Original	11.1
Target	11.2
Herbie	10.8

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

Derivation

1. Split input into 6 regimes

2. if a1 < -2.6262968301438297e+154 or -7.197696696298761e-23 < a1 < -1.4115765653207805e-101 or 5.6037110240578724e-148 < a1 < 1.3929709356464372e+88

   1. Initial program 10.6

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

   3. Applied times-frac 10.6

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

   5. Applied div-inv 10.6

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

   6. Applied associate-*l* 10.0

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

   7. Simplified 10.0

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

   8. Taylor expanded around 0 10.7

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

   if -2.6262968301438297e+154 < a1 < -7.197696696298761e-23

   1. Initial program 9.7

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

   3. Applied times-frac 10.2

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

   5. Applied div-inv 10.3

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

   6. Applied associate-*l* 10.2

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

   7. Simplified 10.2

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

   9. Applied *-un-lft-identity 10.2

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

   10. Applied add-cube-cbrt 10.9

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

   11. Applied times-frac 10.9

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

   12. Applied associate-*r* 8.5

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

   13. Simplified 8.5

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

   if -1.4115765653207805e-101 < a1 < -3.75577994435908e-192

   1. Initial program 8.3

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

   3. Applied associate-/r* 8.5

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

   if -3.75577994435908e-192 < a1 < -2.7564687177065716e-269 or 8.377469299573798e-267 < a1 < 5.6037110240578724e-148

   1. Initial program 11.0

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

   3. Applied times-frac 10.9

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

   5. Applied div-inv 10.9

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

   6. Applied associate-*l* 9.6

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

   7. Simplified 9.6

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

   9. Applied div-inv 9.6

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

   10. Applied associate-*r* 10.0

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

   if -2.7564687177065716e-269 < a1 < 8.377469299573798e-267

   1. Initial program 11.8

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

   3. Applied times-frac 10.1

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

   5. Applied div-inv 10.1

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

   6. Applied associate-*l* 10.3

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

   7. Simplified 10.3

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

   9. Applied *-un-lft-identity 10.3

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

   10. Applied add-cube-cbrt 10.6

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

   11. Applied times-frac 10.6

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

   12. Applied associate-*r* 14.1

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

   13. Simplified 14.1

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

   14. Taylor expanded around -inf 11.8

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

   if 1.3929709356464372e+88 < a1

   1. Initial program 15.6

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

   3. Applied times-frac 15.4

      \[\leadsto \color{blue}{\frac{a1}{b1} \cdot \frac{a2}{b2}}\]
3. Recombined 6 regimes into one program.

4. Final simplification 10.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;a1 \le -2.6262968301438297 \cdot 10^{+154}:\\ \;\;\;\;\frac{a2}{b2 \cdot b1} \cdot a1\\ \mathbf{elif}\;a1 \le -7.197696696298761 \cdot 10^{-23}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{a2}{b2}}}{b1} \cdot \left(\left(\sqrt[3]{\frac{a2}{b2}} \cdot a1\right) \cdot \sqrt[3]{\frac{a2}{b2}}\right)\\ \mathbf{elif}\;a1 \le -1.4115765653207805 \cdot 10^{-101}:\\ \;\;\;\;\frac{a2}{b2 \cdot b1} \cdot a1\\ \mathbf{elif}\;a1 \le -3.75577994435908 \cdot 10^{-192}:\\ \;\;\;\;\frac{\frac{a2 \cdot a1}{b1}}{b2}\\ \mathbf{elif}\;a1 \le -2.7564687177065716 \cdot 10^{-269}:\\ \;\;\;\;\frac{1}{b1} \cdot \left(\frac{a2}{b2} \cdot a1\right)\\ \mathbf{elif}\;a1 \le 8.377469299573798 \cdot 10^{-267}:\\ \;\;\;\;\frac{a2 \cdot a1}{b2 \cdot b1}\\ \mathbf{elif}\;a1 \le 5.6037110240578724 \cdot 10^{-148}:\\ \;\;\;\;\frac{1}{b1} \cdot \left(\frac{a2}{b2} \cdot a1\right)\\ \mathbf{elif}\;a1 \le 1.3929709356464372 \cdot 10^{+88}:\\ \;\;\;\;\frac{a2}{b2 \cdot b1} \cdot a1\\ \mathbf{else}:\\ \;\;\;\;\frac{a1}{b1} \cdot \frac{a2}{b2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019151 
(FPCore (a1 a2 b1 b2)
  :name "Quotient of products"

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

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