Quadratic roots, narrow range

Average Error: 28.7 → 0.4

Time: 11.4s

Precision: 64

\[1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt a \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt b \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt c \lt 94906265.62425155937671661376953125\]

Math

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

↓

\[\frac{\frac{\frac{0 - \left(4 \cdot \left(a \cdot c\right)\right) \cdot \left(4 \cdot \left(a \cdot c\right)\right)}{0 - 4 \cdot \left(a \cdot c\right)}}{a}}{2 \cdot \left(\left(-b\right) - \sqrt{b \cdot b + \left(0 - 4 \cdot \left(a \cdot c\right)\right)}\right)}\]

C

double f(double a, double b, double c) {
        double r42573 = b;
        double r42574 = -r42573;
        double r42575 = r42573 * r42573;
        double r42576 = 4.0;
        double r42577 = a;
        double r42578 = r42576 * r42577;
        double r42579 = c;
        double r42580 = r42578 * r42579;
        double r42581 = r42575 - r42580;
        double r42582 = sqrt(r42581);
        double r42583 = r42574 + r42582;
        double r42584 = 2.0;
        double r42585 = r42584 * r42577;
        double r42586 = r42583 / r42585;
        return r42586;
}

↓

double f(double a, double b, double c) {
        double r42587 = 0.0;
        double r42588 = 4.0;
        double r42589 = a;
        double r42590 = c;
        double r42591 = r42589 * r42590;
        double r42592 = r42588 * r42591;
        double r42593 = r42592 * r42592;
        double r42594 = r42587 - r42593;
        double r42595 = r42587 - r42592;
        double r42596 = r42594 / r42595;
        double r42597 = r42596 / r42589;
        double r42598 = 2.0;
        double r42599 = b;
        double r42600 = -r42599;
        double r42601 = r42599 * r42599;
        double r42602 = r42601 + r42595;
        double r42603 = sqrt(r42602);
        double r42604 = r42600 - r42603;
        double r42605 = r42598 * r42604;
        double r42606 = r42597 / r42605;
        return r42606;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 28.7

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

3. Applied flip-+ 28.7

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

4. Simplified 0.4

   \[\leadsto \frac{\frac{\color{blue}{0 + 4 \cdot \left(a \cdot c\right)}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\]
5. Using strategy rm

6. Applied sub-neg 0.4

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

7. Simplified 0.4

   \[\leadsto \frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b + \color{blue}{\left(0 - 4 \cdot \left(a \cdot c\right)\right)}}}}{2 \cdot a}\]
8. Using strategy rm

9. Applied flip-+ 0.5

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

10. Applied associate-/l/ 0.5

    \[\leadsto \frac{\color{blue}{\frac{0 \cdot 0 - \left(4 \cdot \left(a \cdot c\right)\right) \cdot \left(4 \cdot \left(a \cdot c\right)\right)}{\left(\left(-b\right) - \sqrt{b \cdot b + \left(0 - 4 \cdot \left(a \cdot c\right)\right)}\right) \cdot \left(0 - 4 \cdot \left(a \cdot c\right)\right)}}}{2 \cdot a}\]
11. Using strategy rm

12. Applied *-un-lft-identity 0.5

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

13. Applied times-frac 0.5

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

14. Applied times-frac 0.5

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

15. Simplified 0.5

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

16. Final simplification 0.4

    \[\leadsto \frac{\frac{\frac{0 - \left(4 \cdot \left(a \cdot c\right)\right) \cdot \left(4 \cdot \left(a \cdot c\right)\right)}{0 - 4 \cdot \left(a \cdot c\right)}}{a}}{2 \cdot \left(\left(-b\right) - \sqrt{b \cdot b + \left(0 - 4 \cdot \left(a \cdot c\right)\right)}\right)}\]

Reproduce

herbie shell --seed 2019304 
(FPCore (a b c)
  :name "Quadratic roots, narrow range"
  :precision binary64
  :pre (and (< 1.05367121277235087e-8 a 94906265.6242515594) (< 1.05367121277235087e-8 b 94906265.6242515594) (< 1.05367121277235087e-8 c 94906265.6242515594))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))