Quadratic roots, narrow range

Average Error: 28.8 → 0.3

Time: 46.1s

Precision: 64

\[1.0536712127723509 \cdot 10^{-08} \lt a \lt 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} \lt b \lt 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} \lt c \lt 94906265.62425156\]

Math

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

↓

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

C

double f(double a, double b, double c) {
        double r3440647 = b;
        double r3440648 = -r3440647;
        double r3440649 = r3440647 * r3440647;
        double r3440650 = 4.0;
        double r3440651 = a;
        double r3440652 = r3440650 * r3440651;
        double r3440653 = c;
        double r3440654 = r3440652 * r3440653;
        double r3440655 = r3440649 - r3440654;
        double r3440656 = sqrt(r3440655);
        double r3440657 = r3440648 + r3440656;
        double r3440658 = 2.0;
        double r3440659 = r3440658 * r3440651;
        double r3440660 = r3440657 / r3440659;
        return r3440660;
}

↓

double f(double a, double b, double c) {
        double r3440661 = -2.0;
        double r3440662 = c;
        double r3440663 = r3440661 * r3440662;
        double r3440664 = b;
        double r3440665 = a;
        double r3440666 = r3440665 * r3440662;
        double r3440667 = -4.0;
        double r3440668 = r3440664 * r3440664;
        double r3440669 = fma(r3440666, r3440667, r3440668);
        double r3440670 = sqrt(r3440669);
        double r3440671 = r3440664 + r3440670;
        double r3440672 = r3440663 / r3440671;
        return r3440672;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 28.8

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

2. Simplified 28.8

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

4. Applied *-un-lft-identity 28.8

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

5. Applied div-inv 28.8

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

6. Applied times-frac 28.8

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

7. Simplified 28.7

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

8. Simplified 28.7

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

10. Applied flip-- 28.9

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

11. Applied associate-*l/ 28.9

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

12. Simplified 0.4

    \[\leadsto \frac{\color{blue}{\frac{\frac{1}{2}}{a} \cdot \mathsf{fma}\left(a, \left(c \cdot -4\right), 0\right)}}{\sqrt{\mathsf{fma}\left(b, b, \left(\left(a \cdot -4\right) \cdot c\right)\right)} + b}\]

13. Taylor expanded around 0 0.3

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

14. Taylor expanded around 0 0.3

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

15. Simplified 0.3

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

16. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019124 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, narrow range"
  :pre (and (< 1.0536712127723509e-08 a 94906265.62425156) (< 1.0536712127723509e-08 b 94906265.62425156) (< 1.0536712127723509e-08 c 94906265.62425156))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))