Quadratic roots, narrow range

Average Error: 28.3 → 25.4

Time: 40.1s

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

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

↓

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

C

double f(double a, double b, double c) {
        double r1764616 = b;
        double r1764617 = -r1764616;
        double r1764618 = r1764616 * r1764616;
        double r1764619 = 4.0;
        double r1764620 = a;
        double r1764621 = r1764619 * r1764620;
        double r1764622 = c;
        double r1764623 = r1764621 * r1764622;
        double r1764624 = r1764618 - r1764623;
        double r1764625 = sqrt(r1764624);
        double r1764626 = r1764617 + r1764625;
        double r1764627 = 2.0;
        double r1764628 = r1764627 * r1764620;
        double r1764629 = r1764626 / r1764628;
        return r1764629;
}

↓

double f(double a, double b, double c) {
        double r1764630 = b;
        double r1764631 = r1764630 * r1764630;
        double r1764632 = -r1764631;
        double r1764633 = r1764630 * r1764631;
        double r1764634 = fma(r1764632, r1764630, r1764633);
        double r1764635 = a;
        double r1764636 = -4.0;
        double r1764637 = r1764635 * r1764636;
        double r1764638 = c;
        double r1764639 = fma(r1764637, r1764638, r1764631);
        double r1764640 = sqrt(r1764639);
        double r1764641 = r1764632 * r1764630;
        double r1764642 = fma(r1764631, r1764640, r1764641);
        double r1764643 = r1764637 * r1764638;
        double r1764644 = r1764643 * r1764640;
        double r1764645 = r1764642 + r1764644;
        double r1764646 = r1764634 + r1764645;
        double r1764647 = r1764630 + r1764640;
        double r1764648 = fma(r1764630, r1764647, r1764639);
        double r1764649 = r1764646 / r1764648;
        double r1764650 = 2.0;
        double r1764651 = r1764649 / r1764650;
        double r1764652 = r1764651 / r1764635;
        return r1764652;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 28.3

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

2. Simplified 28.3

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

4. Applied flip3-- 28.4

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

5. Simplified 27.7

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

6. Simplified 27.7

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

8. Applied prod-diff 27.5

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

9. Simplified 27.2

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

11. Applied fma-udef 27.2

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

12. Applied distribute-rgt-in 27.0

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

13. Applied associate--l+ 26.8

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

15. Applied fma-neg 25.4

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

16. Final simplification 25.4

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

Reproduce

herbie shell --seed 2019133 +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)))