quad2p (problem 3.2.1, positive)

Average Error: 34.6 → 10.0

Time: 6.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b_2 \le -4.706781135059311758856471716413486308072 \cdot 10^{-92}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le 6.385814412780331293336851171468331234192 \cdot 10^{98}:\\ \;\;\;\;\frac{1}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}} \cdot \frac{1}{\frac{1}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

C

double f(double a, double b_2, double c) {
        double r22735 = b_2;
        double r22736 = -r22735;
        double r22737 = r22735 * r22735;
        double r22738 = a;
        double r22739 = c;
        double r22740 = r22738 * r22739;
        double r22741 = r22737 - r22740;
        double r22742 = sqrt(r22741);
        double r22743 = r22736 + r22742;
        double r22744 = r22743 / r22738;
        return r22744;
}

↓

double f(double a, double b_2, double c) {
        double r22745 = b_2;
        double r22746 = -4.706781135059312e-92;
        bool r22747 = r22745 <= r22746;
        double r22748 = 0.5;
        double r22749 = c;
        double r22750 = r22749 / r22745;
        double r22751 = r22748 * r22750;
        double r22752 = 2.0;
        double r22753 = a;
        double r22754 = r22745 / r22753;
        double r22755 = r22752 * r22754;
        double r22756 = r22751 - r22755;
        double r22757 = 6.385814412780331e+98;
        bool r22758 = r22745 <= r22757;
        double r22759 = 1.0;
        double r22760 = -r22745;
        double r22761 = r22745 * r22745;
        double r22762 = r22753 * r22749;
        double r22763 = r22761 - r22762;
        double r22764 = sqrt(r22763);
        double r22765 = r22760 - r22764;
        double r22766 = r22759 / r22765;
        double r22767 = r22759 / r22749;
        double r22768 = r22759 / r22767;
        double r22769 = r22766 * r22768;
        double r22770 = -0.5;
        double r22771 = r22770 * r22750;
        double r22772 = r22758 ? r22769 : r22771;
        double r22773 = r22747 ? r22756 : r22772;
        return r22773;
}

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Derivation

1. Split input into 3 regimes

2. if b_2 < -4.706781135059312e-92

   1. Initial program 26.6

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

   2. Taylor expanded around -inf 12.8

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}}\]

   if -4.706781135059312e-92 < b_2 < 6.385814412780331e+98

   1. Initial program 26.4

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
   2. Using strategy rm

   3. Applied flip-+ 28.7

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

   4. Simplified 17.8

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

   6. Applied *-un-lft-identity 17.8

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

   7. Applied associate-/r* 17.8

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

   8. Simplified 16.3

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

   10. Applied add-sqr-sqrt 40.3

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

   11. Applied div-inv 40.3

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

   12. Applied add-sqr-sqrt 40.2

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

   13. Applied times-frac 40.4

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

   14. Applied times-frac 39.3

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

   15. Simplified 39.3

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

   16. Simplified 12.4

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

   if 6.385814412780331e+98 < b_2

   1. Initial program 59.2

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

   2. Taylor expanded around inf 2.5

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b_2}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 10.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -4.706781135059311758856471716413486308072 \cdot 10^{-92}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le 6.385814412780331293336851171468331234192 \cdot 10^{98}:\\ \;\;\;\;\frac{1}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}} \cdot \frac{1}{\frac{1}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))