Cubic critical, narrow range

Average Error: 28.8 → 0.5

Time: 4.3s

Precision: 64

\[1.05367121277235087 \cdot 10^{-8} \lt a \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt b \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt c \lt 94906265.6242515594\]

Math

\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]

↓

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

C

double code(double a, double b, double c) {
	return ((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a));
}

↓

double code(double a, double b, double c) {
	return -(a / (((b + sqrt(fma(b, b, -((a * c) * 3.0)))) / c) * a));
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 28.8

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
2. Using strategy rm

3. Applied flip-+ 28.9

   \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}}{3 \cdot a}\]

4. Simplified 0.6

   \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(3 \cdot c, a, 0\right)}}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\]
5. Using strategy rm

6. Applied frac-2neg 0.6

   \[\leadsto \frac{\color{blue}{\frac{-\mathsf{fma}\left(3 \cdot c, a, 0\right)}{-\left(\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}}}{3 \cdot a}\]

7. Simplified 0.4

   \[\leadsto \frac{\frac{\color{blue}{-\left(3 \cdot a\right) \cdot c}}{-\left(\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}}{3 \cdot a}\]

8. Simplified 0.4

   \[\leadsto \frac{\frac{-\left(3 \cdot a\right) \cdot c}{\color{blue}{b + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}}{3 \cdot a}\]
9. Using strategy rm

10. Applied *-commutative 0.4

    \[\leadsto \frac{\frac{-\color{blue}{\left(a \cdot 3\right)} \cdot c}{b + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\]

11. Applied associate-*l* 0.6

    \[\leadsto \frac{\frac{-\color{blue}{a \cdot \left(3 \cdot c\right)}}{b + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\]

12. Applied distribute-lft-neg-in 0.6

    \[\leadsto \frac{\frac{\color{blue}{\left(-a\right) \cdot \left(3 \cdot c\right)}}{b + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\]

13. Applied associate-/l* 0.6

    \[\leadsto \frac{\color{blue}{\frac{-a}{\frac{b + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot c}}}}{3 \cdot a}\]

14. Applied associate-/l/ 0.6

    \[\leadsto \color{blue}{\frac{-a}{\left(3 \cdot a\right) \cdot \frac{b + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot c}}}\]

15. Simplified 0.5

    \[\leadsto \frac{-a}{\color{blue}{\left(1 \cdot a\right) \cdot \frac{b + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{c}}}\]
16. Using strategy rm

17. Applied fma-neg 0.5

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

18. Simplified 0.5

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

19. Final simplification 0.5

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

Reproduce

herbie shell --seed 2020071 +o rules:numerics
(FPCore (a b c)
  :name "Cubic critical, narrow range"
  :precision binary64
  :pre (and (< 1.0536712127723509e-08 a 94906265.62425156) (< 1.0536712127723509e-08 b 94906265.62425156) (< 1.0536712127723509e-08 c 94906265.62425156))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))