Quadratic roots, wide range

Average Error: 52.6 → 51.8

Time: 31.7s

Precision: 64

\[4.930380657631324 \cdot 10^{-32} \lt a \lt 2.028240960365167 \cdot 10^{+31} \land 4.930380657631324 \cdot 10^{-32} \lt b \lt 2.028240960365167 \cdot 10^{+31} \land 4.930380657631324 \cdot 10^{-32} \lt c \lt 2.028240960365167 \cdot 10^{+31}\]

Math

\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]

↓

\[\left(\sqrt[3]{\frac{\mathsf{fma}\left(\left(\sqrt{\sqrt{\mathsf{fma}\left(c, \left(a \cdot -4\right), \left(b \cdot b\right)\right)}}\right), \left(\sqrt{\sqrt{\mathsf{fma}\left(c, \left(a \cdot -4\right), \left(b \cdot b\right)\right)}}\right), \left(-b\right)\right)}{2} \cdot \frac{1}{a}} \cdot \sqrt[3]{\frac{\mathsf{fma}\left(\left(\sqrt{\sqrt{\mathsf{fma}\left(c, \left(a \cdot -4\right), \left(b \cdot b\right)\right)}}\right), \left(\sqrt{\sqrt{\mathsf{fma}\left(c, \left(a \cdot -4\right), \left(b \cdot b\right)\right)}}\right), \left(-b\right)\right)}{2} \cdot \frac{1}{a}}\right) \cdot \frac{\sqrt[3]{\frac{1}{a} \cdot \mathsf{fma}\left(\left(\sqrt{\sqrt{\mathsf{fma}\left(c, \left(a \cdot -4\right), \left(b \cdot b\right)\right)}}\right), \left(\sqrt{\sqrt{\mathsf{fma}\left(c, \left(a \cdot -4\right), \left(b \cdot b\right)\right)}}\right), \left(-b\right)\right)}}{\sqrt[3]{2}}\]

C

double f(double a, double b, double c) {
        double r1266823 = b;
        double r1266824 = -r1266823;
        double r1266825 = r1266823 * r1266823;
        double r1266826 = 4.0;
        double r1266827 = a;
        double r1266828 = r1266826 * r1266827;
        double r1266829 = c;
        double r1266830 = r1266828 * r1266829;
        double r1266831 = r1266825 - r1266830;
        double r1266832 = sqrt(r1266831);
        double r1266833 = r1266824 + r1266832;
        double r1266834 = 2.0;
        double r1266835 = r1266834 * r1266827;
        double r1266836 = r1266833 / r1266835;
        return r1266836;
}

↓

double f(double a, double b, double c) {
        double r1266837 = c;
        double r1266838 = a;
        double r1266839 = -4.0;
        double r1266840 = r1266838 * r1266839;
        double r1266841 = b;
        double r1266842 = r1266841 * r1266841;
        double r1266843 = fma(r1266837, r1266840, r1266842);
        double r1266844 = sqrt(r1266843);
        double r1266845 = sqrt(r1266844);
        double r1266846 = -r1266841;
        double r1266847 = fma(r1266845, r1266845, r1266846);
        double r1266848 = 2.0;
        double r1266849 = r1266847 / r1266848;
        double r1266850 = 1.0;
        double r1266851 = r1266850 / r1266838;
        double r1266852 = r1266849 * r1266851;
        double r1266853 = cbrt(r1266852);
        double r1266854 = r1266853 * r1266853;
        double r1266855 = r1266851 * r1266847;
        double r1266856 = cbrt(r1266855);
        double r1266857 = cbrt(r1266848);
        double r1266858 = r1266856 / r1266857;
        double r1266859 = r1266854 * r1266858;
        return r1266859;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 52.6

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]

2. Simplified 52.6

   \[\leadsto \color{blue}{\frac{\frac{\sqrt{\mathsf{fma}\left(c, \left(-4 \cdot a\right), \left(b \cdot b\right)\right)} - b}{2}}{a}}\]
3. Using strategy rm

4. Applied add-sqr-sqrt 52.6

   \[\leadsto \frac{\frac{\sqrt{\color{blue}{\sqrt{\mathsf{fma}\left(c, \left(-4 \cdot a\right), \left(b \cdot b\right)\right)} \cdot \sqrt{\mathsf{fma}\left(c, \left(-4 \cdot a\right), \left(b \cdot b\right)\right)}}} - b}{2}}{a}\]

5. Applied sqrt-prod 52.4

   \[\leadsto \frac{\frac{\color{blue}{\sqrt{\sqrt{\mathsf{fma}\left(c, \left(-4 \cdot a\right), \left(b \cdot b\right)\right)}} \cdot \sqrt{\sqrt{\mathsf{fma}\left(c, \left(-4 \cdot a\right), \left(b \cdot b\right)\right)}}} - b}{2}}{a}\]

6. Applied fma-neg 51.8

   \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(\left(\sqrt{\sqrt{\mathsf{fma}\left(c, \left(-4 \cdot a\right), \left(b \cdot b\right)\right)}}\right), \left(\sqrt{\sqrt{\mathsf{fma}\left(c, \left(-4 \cdot a\right), \left(b \cdot b\right)\right)}}\right), \left(-b\right)\right)}}{2}}{a}\]
7. Using strategy rm

8. Applied div-inv 51.8

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\sqrt{\sqrt{\mathsf{fma}\left(c, \left(-4 \cdot a\right), \left(b \cdot b\right)\right)}}\right), \left(\sqrt{\sqrt{\mathsf{fma}\left(c, \left(-4 \cdot a\right), \left(b \cdot b\right)\right)}}\right), \left(-b\right)\right)}{2} \cdot \frac{1}{a}}\]
9. Using strategy rm

10. Applied add-cube-cbrt 51.8

    \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{\mathsf{fma}\left(\left(\sqrt{\sqrt{\mathsf{fma}\left(c, \left(-4 \cdot a\right), \left(b \cdot b\right)\right)}}\right), \left(\sqrt{\sqrt{\mathsf{fma}\left(c, \left(-4 \cdot a\right), \left(b \cdot b\right)\right)}}\right), \left(-b\right)\right)}{2} \cdot \frac{1}{a}} \cdot \sqrt[3]{\frac{\mathsf{fma}\left(\left(\sqrt{\sqrt{\mathsf{fma}\left(c, \left(-4 \cdot a\right), \left(b \cdot b\right)\right)}}\right), \left(\sqrt{\sqrt{\mathsf{fma}\left(c, \left(-4 \cdot a\right), \left(b \cdot b\right)\right)}}\right), \left(-b\right)\right)}{2} \cdot \frac{1}{a}}\right) \cdot \sqrt[3]{\frac{\mathsf{fma}\left(\left(\sqrt{\sqrt{\mathsf{fma}\left(c, \left(-4 \cdot a\right), \left(b \cdot b\right)\right)}}\right), \left(\sqrt{\sqrt{\mathsf{fma}\left(c, \left(-4 \cdot a\right), \left(b \cdot b\right)\right)}}\right), \left(-b\right)\right)}{2} \cdot \frac{1}{a}}}\]
11. Using strategy rm

12. Applied associate-*l/ 51.8

    \[\leadsto \left(\sqrt[3]{\frac{\mathsf{fma}\left(\left(\sqrt{\sqrt{\mathsf{fma}\left(c, \left(-4 \cdot a\right), \left(b \cdot b\right)\right)}}\right), \left(\sqrt{\sqrt{\mathsf{fma}\left(c, \left(-4 \cdot a\right), \left(b \cdot b\right)\right)}}\right), \left(-b\right)\right)}{2} \cdot \frac{1}{a}} \cdot \sqrt[3]{\frac{\mathsf{fma}\left(\left(\sqrt{\sqrt{\mathsf{fma}\left(c, \left(-4 \cdot a\right), \left(b \cdot b\right)\right)}}\right), \left(\sqrt{\sqrt{\mathsf{fma}\left(c, \left(-4 \cdot a\right), \left(b \cdot b\right)\right)}}\right), \left(-b\right)\right)}{2} \cdot \frac{1}{a}}\right) \cdot \sqrt[3]{\color{blue}{\frac{\mathsf{fma}\left(\left(\sqrt{\sqrt{\mathsf{fma}\left(c, \left(-4 \cdot a\right), \left(b \cdot b\right)\right)}}\right), \left(\sqrt{\sqrt{\mathsf{fma}\left(c, \left(-4 \cdot a\right), \left(b \cdot b\right)\right)}}\right), \left(-b\right)\right) \cdot \frac{1}{a}}{2}}}\]

13. Applied cbrt-div 51.8

    \[\leadsto \left(\sqrt[3]{\frac{\mathsf{fma}\left(\left(\sqrt{\sqrt{\mathsf{fma}\left(c, \left(-4 \cdot a\right), \left(b \cdot b\right)\right)}}\right), \left(\sqrt{\sqrt{\mathsf{fma}\left(c, \left(-4 \cdot a\right), \left(b \cdot b\right)\right)}}\right), \left(-b\right)\right)}{2} \cdot \frac{1}{a}} \cdot \sqrt[3]{\frac{\mathsf{fma}\left(\left(\sqrt{\sqrt{\mathsf{fma}\left(c, \left(-4 \cdot a\right), \left(b \cdot b\right)\right)}}\right), \left(\sqrt{\sqrt{\mathsf{fma}\left(c, \left(-4 \cdot a\right), \left(b \cdot b\right)\right)}}\right), \left(-b\right)\right)}{2} \cdot \frac{1}{a}}\right) \cdot \color{blue}{\frac{\sqrt[3]{\mathsf{fma}\left(\left(\sqrt{\sqrt{\mathsf{fma}\left(c, \left(-4 \cdot a\right), \left(b \cdot b\right)\right)}}\right), \left(\sqrt{\sqrt{\mathsf{fma}\left(c, \left(-4 \cdot a\right), \left(b \cdot b\right)\right)}}\right), \left(-b\right)\right) \cdot \frac{1}{a}}}{\sqrt[3]{2}}}\]

14. Final simplification 51.8

    \[\leadsto \left(\sqrt[3]{\frac{\mathsf{fma}\left(\left(\sqrt{\sqrt{\mathsf{fma}\left(c, \left(a \cdot -4\right), \left(b \cdot b\right)\right)}}\right), \left(\sqrt{\sqrt{\mathsf{fma}\left(c, \left(a \cdot -4\right), \left(b \cdot b\right)\right)}}\right), \left(-b\right)\right)}{2} \cdot \frac{1}{a}} \cdot \sqrt[3]{\frac{\mathsf{fma}\left(\left(\sqrt{\sqrt{\mathsf{fma}\left(c, \left(a \cdot -4\right), \left(b \cdot b\right)\right)}}\right), \left(\sqrt{\sqrt{\mathsf{fma}\left(c, \left(a \cdot -4\right), \left(b \cdot b\right)\right)}}\right), \left(-b\right)\right)}{2} \cdot \frac{1}{a}}\right) \cdot \frac{\sqrt[3]{\frac{1}{a} \cdot \mathsf{fma}\left(\left(\sqrt{\sqrt{\mathsf{fma}\left(c, \left(a \cdot -4\right), \left(b \cdot b\right)\right)}}\right), \left(\sqrt{\sqrt{\mathsf{fma}\left(c, \left(a \cdot -4\right), \left(b \cdot b\right)\right)}}\right), \left(-b\right)\right)}}{\sqrt[3]{2}}\]

Reproduce

herbie shell --seed 2019132 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, wide range"
  :pre (and (< 4.930380657631324e-32 a 2.028240960365167e+31) (< 4.930380657631324e-32 b 2.028240960365167e+31) (< 4.930380657631324e-32 c 2.028240960365167e+31))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))