Quadratic roots, narrow range

Average Error: 28.5 → 0.4

Time: 45.9s

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

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

↓

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

C

double f(double a, double b, double c) {
        double r3507403 = b;
        double r3507404 = -r3507403;
        double r3507405 = r3507403 * r3507403;
        double r3507406 = 4.0;
        double r3507407 = a;
        double r3507408 = r3507406 * r3507407;
        double r3507409 = c;
        double r3507410 = r3507408 * r3507409;
        double r3507411 = r3507405 - r3507410;
        double r3507412 = sqrt(r3507411);
        double r3507413 = r3507404 + r3507412;
        double r3507414 = 2.0;
        double r3507415 = r3507414 * r3507407;
        double r3507416 = r3507413 / r3507415;
        return r3507416;
}

↓

double f(double a, double b, double c) {
        double r3507417 = -2.0;
        double r3507418 = b;
        double r3507419 = a;
        double r3507420 = -4.0;
        double r3507421 = r3507419 * r3507420;
        double r3507422 = c;
        double r3507423 = r3507421 * r3507422;
        double r3507424 = fma(r3507418, r3507418, r3507423);
        double r3507425 = sqrt(r3507424);
        double r3507426 = r3507418 + r3507425;
        double r3507427 = r3507426 / r3507422;
        double r3507428 = r3507417 / r3507427;
        return r3507428;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 28.5

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

2. Simplified 28.5

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

4. Applied *-un-lft-identity 28.5

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

5. Applied div-inv 28.5

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

6. Applied times-frac 28.5

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

7. Simplified 28.5

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

8. Simplified 28.5

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

10. Applied flip-- 28.7

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

11. Applied associate-*l/ 28.7

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

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 -inf 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. Using strategy rm

15. Applied associate-/l* 0.4

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

16. Final simplification 0.4

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

Reproduce

herbie shell --seed 2019121 +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)))