quad2p (problem 3.2.1, positive)

Average Error: 34.2 → 11.0

Time: 37.2s

Precision: 64

Math

\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

↓

\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.961160193587751793384768751503345323313 \cdot 10^{167}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-c, a, 0\right)}{\frac{\frac{1}{2} \cdot a}{\frac{b_2}{c}}}}{a}\\ \mathbf{elif}\;b_2 \le -3.224291431715844638574408383287037485194 \cdot 10^{-253}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-c, a, b_2 \cdot b_2\right)}}{a} - \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le 7.866133526806334455497723933726682386756 \cdot 10^{146}:\\ \;\;\;\;-\frac{c}{b_2 + \sqrt{\mathsf{fma}\left(a, -c, b_2 \cdot b_2\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \frac{c}{b_2 + b_2}\right) \cdot \frac{-1}{\sqrt[3]{a} \cdot \sqrt[3]{a}}\\ \end{array}\]

C

double f(double a, double b_2, double c) {
        double r39790 = b_2;
        double r39791 = -r39790;
        double r39792 = r39790 * r39790;
        double r39793 = a;
        double r39794 = c;
        double r39795 = r39793 * r39794;
        double r39796 = r39792 - r39795;
        double r39797 = sqrt(r39796);
        double r39798 = r39791 + r39797;
        double r39799 = r39798 / r39793;
        return r39799;
}

↓

double f(double a, double b_2, double c) {
        double r39800 = b_2;
        double r39801 = -1.9611601935877518e+167;
        bool r39802 = r39800 <= r39801;
        double r39803 = c;
        double r39804 = -r39803;
        double r39805 = a;
        double r39806 = 0.0;
        double r39807 = fma(r39804, r39805, r39806);
        double r39808 = 0.5;
        double r39809 = r39808 * r39805;
        double r39810 = r39800 / r39803;
        double r39811 = r39809 / r39810;
        double r39812 = r39807 / r39811;
        double r39813 = r39812 / r39805;
        double r39814 = -3.2242914317158446e-253;
        bool r39815 = r39800 <= r39814;
        double r39816 = r39800 * r39800;
        double r39817 = fma(r39804, r39805, r39816);
        double r39818 = sqrt(r39817);
        double r39819 = r39818 / r39805;
        double r39820 = r39800 / r39805;
        double r39821 = r39819 - r39820;
        double r39822 = 7.8661335268063345e+146;
        bool r39823 = r39800 <= r39822;
        double r39824 = fma(r39805, r39804, r39816);
        double r39825 = sqrt(r39824);
        double r39826 = r39800 + r39825;
        double r39827 = r39803 / r39826;
        double r39828 = -r39827;
        double r39829 = cbrt(r39805);
        double r39830 = r39829 * r39829;
        double r39831 = r39800 + r39800;
        double r39832 = r39803 / r39831;
        double r39833 = r39830 * r39832;
        double r39834 = -1.0;
        double r39835 = r39834 / r39830;
        double r39836 = r39833 * r39835;
        double r39837 = r39823 ? r39828 : r39836;
        double r39838 = r39815 ? r39821 : r39837;
        double r39839 = r39802 ? r39813 : r39838;
        return r39839;
}

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Derivation

1. Split input into 4 regimes

2. if b_2 < -1.9611601935877518e+167

   1. Initial program 64.0

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

   2. Simplified 64.0

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(c, -a, b_2 \cdot b_2\right)} - b_2}{a}}\]
   3. Using strategy rm

   4. Applied flip-- 64.0

      \[\leadsto \frac{\color{blue}{\frac{\sqrt{\mathsf{fma}\left(c, -a, b_2 \cdot b_2\right)} \cdot \sqrt{\mathsf{fma}\left(c, -a, b_2 \cdot b_2\right)} - b_2 \cdot b_2}{\sqrt{\mathsf{fma}\left(c, -a, b_2 \cdot b_2\right)} + b_2}}}{a}\]

   5. Simplified 62.7

      \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(-c, a, 0\right)}}{\sqrt{\mathsf{fma}\left(c, -a, b_2 \cdot b_2\right)} + b_2}}{a}\]

   6. Simplified 62.7

      \[\leadsto \frac{\frac{\mathsf{fma}\left(-c, a, 0\right)}{\color{blue}{b_2 + \sqrt{\mathsf{fma}\left(-c, a, b_2 \cdot b_2\right)}}}}{a}\]

   7. Taylor expanded around -inf 20.8

      \[\leadsto \frac{\frac{\mathsf{fma}\left(-c, a, 0\right)}{\color{blue}{\frac{1}{2} \cdot \frac{a \cdot c}{b_2}}}}{a}\]

   8. Simplified 35.0

      \[\leadsto \frac{\frac{\mathsf{fma}\left(-c, a, 0\right)}{\color{blue}{\frac{a \cdot \frac{1}{2}}{\frac{b_2}{c}}}}}{a}\]

   if -1.9611601935877518e+167 < b_2 < -3.2242914317158446e-253

   1. Initial program 9.5

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

   2. Simplified 9.5

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(c, -a, b_2 \cdot b_2\right)} - b_2}{a}}\]
   3. Using strategy rm

   4. Applied div-sub 9.5

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(c, -a, b_2 \cdot b_2\right)}}{a} - \frac{b_2}{a}}\]

   5. Simplified 9.5

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(-c, a, b_2 \cdot b_2\right)}}{a}} - \frac{b_2}{a}\]

   if -3.2242914317158446e-253 < b_2 < 7.8661335268063345e+146

   1. Initial program 32.5

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

   2. Simplified 32.5

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(c, -a, b_2 \cdot b_2\right)} - b_2}{a}}\]
   3. Using strategy rm

   4. Applied flip-- 32.5

      \[\leadsto \frac{\color{blue}{\frac{\sqrt{\mathsf{fma}\left(c, -a, b_2 \cdot b_2\right)} \cdot \sqrt{\mathsf{fma}\left(c, -a, b_2 \cdot b_2\right)} - b_2 \cdot b_2}{\sqrt{\mathsf{fma}\left(c, -a, b_2 \cdot b_2\right)} + b_2}}}{a}\]

   5. Simplified 15.2

      \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(-c, a, 0\right)}}{\sqrt{\mathsf{fma}\left(c, -a, b_2 \cdot b_2\right)} + b_2}}{a}\]

   6. Simplified 15.2

      \[\leadsto \frac{\frac{\mathsf{fma}\left(-c, a, 0\right)}{\color{blue}{b_2 + \sqrt{\mathsf{fma}\left(-c, a, b_2 \cdot b_2\right)}}}}{a}\]
   7. Using strategy rm

   8. Applied add-cube-cbrt 15.9

      \[\leadsto \frac{\frac{\mathsf{fma}\left(-c, a, 0\right)}{b_2 + \sqrt{\mathsf{fma}\left(-c, a, b_2 \cdot b_2\right)}}}{\color{blue}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}}\]

   9. Applied *-un-lft-identity 15.9

      \[\leadsto \frac{\frac{\mathsf{fma}\left(-c, a, 0\right)}{\color{blue}{1 \cdot \left(b_2 + \sqrt{\mathsf{fma}\left(-c, a, b_2 \cdot b_2\right)}\right)}}}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}\]

   10. Applied *-un-lft-identity 15.9

       \[\leadsto \frac{\frac{\color{blue}{1 \cdot \mathsf{fma}\left(-c, a, 0\right)}}{1 \cdot \left(b_2 + \sqrt{\mathsf{fma}\left(-c, a, b_2 \cdot b_2\right)}\right)}}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}\]

   11. Applied times-frac 15.9

       \[\leadsto \frac{\color{blue}{\frac{1}{1} \cdot \frac{\mathsf{fma}\left(-c, a, 0\right)}{b_2 + \sqrt{\mathsf{fma}\left(-c, a, b_2 \cdot b_2\right)}}}}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}\]

   12. Applied times-frac 15.9

       \[\leadsto \color{blue}{\frac{\frac{1}{1}}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{\frac{\mathsf{fma}\left(-c, a, 0\right)}{b_2 + \sqrt{\mathsf{fma}\left(-c, a, b_2 \cdot b_2\right)}}}{\sqrt[3]{a}}}\]

   13. Simplified 15.9

       \[\leadsto \color{blue}{\frac{1}{\sqrt[3]{a} \cdot \sqrt[3]{a}}} \cdot \frac{\frac{\mathsf{fma}\left(-c, a, 0\right)}{b_2 + \sqrt{\mathsf{fma}\left(-c, a, b_2 \cdot b_2\right)}}}{\sqrt[3]{a}}\]

   14. Simplified 12.0

       \[\leadsto \frac{1}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \color{blue}{\left(\frac{a}{\sqrt[3]{a}} \cdot \frac{-c}{b_2 + \sqrt{\mathsf{fma}\left(a, -c, b_2 \cdot b_2\right)}}\right)}\]
   15. Using strategy rm

   16. Applied *-un-lft-identity 12.0

       \[\leadsto \frac{1}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \left(\frac{a}{\sqrt[3]{\color{blue}{1 \cdot a}}} \cdot \frac{-c}{b_2 + \sqrt{\mathsf{fma}\left(a, -c, b_2 \cdot b_2\right)}}\right)\]

   17. Applied cbrt-prod 12.0

       \[\leadsto \frac{1}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \left(\frac{a}{\color{blue}{\sqrt[3]{1} \cdot \sqrt[3]{a}}} \cdot \frac{-c}{b_2 + \sqrt{\mathsf{fma}\left(a, -c, b_2 \cdot b_2\right)}}\right)\]

   18. Applied add-cube-cbrt 11.3

       \[\leadsto \frac{1}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \left(\frac{\color{blue}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}}{\sqrt[3]{1} \cdot \sqrt[3]{a}} \cdot \frac{-c}{b_2 + \sqrt{\mathsf{fma}\left(a, -c, b_2 \cdot b_2\right)}}\right)\]

   19. Applied times-frac 11.2

       \[\leadsto \frac{1}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \left(\color{blue}{\left(\frac{\sqrt[3]{a} \cdot \sqrt[3]{a}}{\sqrt[3]{1}} \cdot \frac{\sqrt[3]{a}}{\sqrt[3]{a}}\right)} \cdot \frac{-c}{b_2 + \sqrt{\mathsf{fma}\left(a, -c, b_2 \cdot b_2\right)}}\right)\]

   20. Applied associate-*l* 11.2

       \[\leadsto \frac{1}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \color{blue}{\left(\frac{\sqrt[3]{a} \cdot \sqrt[3]{a}}{\sqrt[3]{1}} \cdot \left(\frac{\sqrt[3]{a}}{\sqrt[3]{a}} \cdot \frac{-c}{b_2 + \sqrt{\mathsf{fma}\left(a, -c, b_2 \cdot b_2\right)}}\right)\right)}\]

   21. Simplified 11.2

       \[\leadsto \frac{1}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \left(\frac{\sqrt[3]{a} \cdot \sqrt[3]{a}}{\sqrt[3]{1}} \cdot \color{blue}{\frac{-c}{b_2 + \sqrt{\mathsf{fma}\left(a, -c, b_2 \cdot b_2\right)}}}\right)\]
   22. Using strategy rm

   23. Applied distribute-frac-neg 11.2

       \[\leadsto \frac{1}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \left(\frac{\sqrt[3]{a} \cdot \sqrt[3]{a}}{\sqrt[3]{1}} \cdot \color{blue}{\left(-\frac{c}{b_2 + \sqrt{\mathsf{fma}\left(a, -c, b_2 \cdot b_2\right)}}\right)}\right)\]

   24. Applied distribute-rgt-neg-out 11.2

       \[\leadsto \frac{1}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \color{blue}{\left(-\frac{\sqrt[3]{a} \cdot \sqrt[3]{a}}{\sqrt[3]{1}} \cdot \frac{c}{b_2 + \sqrt{\mathsf{fma}\left(a, -c, b_2 \cdot b_2\right)}}\right)}\]

   25. Applied distribute-rgt-neg-out 11.2

       \[\leadsto \color{blue}{-\frac{1}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \left(\frac{\sqrt[3]{a} \cdot \sqrt[3]{a}}{\sqrt[3]{1}} \cdot \frac{c}{b_2 + \sqrt{\mathsf{fma}\left(a, -c, b_2 \cdot b_2\right)}}\right)}\]

   26. Simplified 8.3

       \[\leadsto -\color{blue}{1 \cdot \frac{c}{b_2 + \sqrt{\mathsf{fma}\left(a, -c, b_2 \cdot b_2\right)}}}\]

   if 7.8661335268063345e+146 < b_2

   1. Initial program 63.1

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

   2. Simplified 63.1

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(c, -a, b_2 \cdot b_2\right)} - b_2}{a}}\]
   3. Using strategy rm

   4. Applied flip-- 63.1

      \[\leadsto \frac{\color{blue}{\frac{\sqrt{\mathsf{fma}\left(c, -a, b_2 \cdot b_2\right)} \cdot \sqrt{\mathsf{fma}\left(c, -a, b_2 \cdot b_2\right)} - b_2 \cdot b_2}{\sqrt{\mathsf{fma}\left(c, -a, b_2 \cdot b_2\right)} + b_2}}}{a}\]

   5. Simplified 37.5

      \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(-c, a, 0\right)}}{\sqrt{\mathsf{fma}\left(c, -a, b_2 \cdot b_2\right)} + b_2}}{a}\]

   6. Simplified 37.5

      \[\leadsto \frac{\frac{\mathsf{fma}\left(-c, a, 0\right)}{\color{blue}{b_2 + \sqrt{\mathsf{fma}\left(-c, a, b_2 \cdot b_2\right)}}}}{a}\]
   7. Using strategy rm

   8. Applied add-cube-cbrt 37.5

      \[\leadsto \frac{\frac{\mathsf{fma}\left(-c, a, 0\right)}{b_2 + \sqrt{\mathsf{fma}\left(-c, a, b_2 \cdot b_2\right)}}}{\color{blue}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}}\]

   9. Applied *-un-lft-identity 37.5

      \[\leadsto \frac{\frac{\mathsf{fma}\left(-c, a, 0\right)}{\color{blue}{1 \cdot \left(b_2 + \sqrt{\mathsf{fma}\left(-c, a, b_2 \cdot b_2\right)}\right)}}}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}\]

   10. Applied *-un-lft-identity 37.5

       \[\leadsto \frac{\frac{\color{blue}{1 \cdot \mathsf{fma}\left(-c, a, 0\right)}}{1 \cdot \left(b_2 + \sqrt{\mathsf{fma}\left(-c, a, b_2 \cdot b_2\right)}\right)}}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}\]

   11. Applied times-frac 37.5

       \[\leadsto \frac{\color{blue}{\frac{1}{1} \cdot \frac{\mathsf{fma}\left(-c, a, 0\right)}{b_2 + \sqrt{\mathsf{fma}\left(-c, a, b_2 \cdot b_2\right)}}}}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}\]

   12. Applied times-frac 37.5

       \[\leadsto \color{blue}{\frac{\frac{1}{1}}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{\frac{\mathsf{fma}\left(-c, a, 0\right)}{b_2 + \sqrt{\mathsf{fma}\left(-c, a, b_2 \cdot b_2\right)}}}{\sqrt[3]{a}}}\]

   13. Simplified 37.5

       \[\leadsto \color{blue}{\frac{1}{\sqrt[3]{a} \cdot \sqrt[3]{a}}} \cdot \frac{\frac{\mathsf{fma}\left(-c, a, 0\right)}{b_2 + \sqrt{\mathsf{fma}\left(-c, a, b_2 \cdot b_2\right)}}}{\sqrt[3]{a}}\]

   14. Simplified 37.0

       \[\leadsto \frac{1}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \color{blue}{\left(\frac{a}{\sqrt[3]{a}} \cdot \frac{-c}{b_2 + \sqrt{\mathsf{fma}\left(a, -c, b_2 \cdot b_2\right)}}\right)}\]
   15. Using strategy rm

   16. Applied *-un-lft-identity 37.0

       \[\leadsto \frac{1}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \left(\frac{a}{\sqrt[3]{\color{blue}{1 \cdot a}}} \cdot \frac{-c}{b_2 + \sqrt{\mathsf{fma}\left(a, -c, b_2 \cdot b_2\right)}}\right)\]

   17. Applied cbrt-prod 37.0

       \[\leadsto \frac{1}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \left(\frac{a}{\color{blue}{\sqrt[3]{1} \cdot \sqrt[3]{a}}} \cdot \frac{-c}{b_2 + \sqrt{\mathsf{fma}\left(a, -c, b_2 \cdot b_2\right)}}\right)\]

   18. Applied add-cube-cbrt 37.0

       \[\leadsto \frac{1}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \left(\frac{\color{blue}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}}{\sqrt[3]{1} \cdot \sqrt[3]{a}} \cdot \frac{-c}{b_2 + \sqrt{\mathsf{fma}\left(a, -c, b_2 \cdot b_2\right)}}\right)\]

   19. Applied times-frac 37.0

       \[\leadsto \frac{1}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \left(\color{blue}{\left(\frac{\sqrt[3]{a} \cdot \sqrt[3]{a}}{\sqrt[3]{1}} \cdot \frac{\sqrt[3]{a}}{\sqrt[3]{a}}\right)} \cdot \frac{-c}{b_2 + \sqrt{\mathsf{fma}\left(a, -c, b_2 \cdot b_2\right)}}\right)\]

   20. Applied associate-*l* 37.0

       \[\leadsto \frac{1}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \color{blue}{\left(\frac{\sqrt[3]{a} \cdot \sqrt[3]{a}}{\sqrt[3]{1}} \cdot \left(\frac{\sqrt[3]{a}}{\sqrt[3]{a}} \cdot \frac{-c}{b_2 + \sqrt{\mathsf{fma}\left(a, -c, b_2 \cdot b_2\right)}}\right)\right)}\]

   21. Simplified 37.0

       \[\leadsto \frac{1}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \left(\frac{\sqrt[3]{a} \cdot \sqrt[3]{a}}{\sqrt[3]{1}} \cdot \color{blue}{\frac{-c}{b_2 + \sqrt{\mathsf{fma}\left(a, -c, b_2 \cdot b_2\right)}}}\right)\]

   22. Taylor expanded around 0 6.3

       \[\leadsto \frac{1}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \left(\frac{\sqrt[3]{a} \cdot \sqrt[3]{a}}{\sqrt[3]{1}} \cdot \frac{-c}{b_2 + \color{blue}{b_2}}\right)\]
3. Recombined 4 regimes into one program.

4. Final simplification 11.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.961160193587751793384768751503345323313 \cdot 10^{167}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-c, a, 0\right)}{\frac{\frac{1}{2} \cdot a}{\frac{b_2}{c}}}}{a}\\ \mathbf{elif}\;b_2 \le -3.224291431715844638574408383287037485194 \cdot 10^{-253}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-c, a, b_2 \cdot b_2\right)}}{a} - \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le 7.866133526806334455497723933726682386756 \cdot 10^{146}:\\ \;\;\;\;-\frac{c}{b_2 + \sqrt{\mathsf{fma}\left(a, -c, b_2 \cdot b_2\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \frac{c}{b_2 + b_2}\right) \cdot \frac{-1}{\sqrt[3]{a} \cdot \sqrt[3]{a}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))