Quadratic roots, medium range

Average Error: 43.7 → 0.2

Time: 6.0s

Precision: 64

\[1.11022 \cdot 10^{-16} \lt a \lt 9.0072 \cdot 10^{15} \land 1.11022 \cdot 10^{-16} \lt b \lt 9.0072 \cdot 10^{15} \land 1.11022 \cdot 10^{-16} \lt c \lt 9.0072 \cdot 10^{15}\]

Math

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

↓

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

C

double f(double a, double b, double c) {
        double r34255 = b;
        double r34256 = -r34255;
        double r34257 = r34255 * r34255;
        double r34258 = 4.0;
        double r34259 = a;
        double r34260 = r34258 * r34259;
        double r34261 = c;
        double r34262 = r34260 * r34261;
        double r34263 = r34257 - r34262;
        double r34264 = sqrt(r34263);
        double r34265 = r34256 + r34264;
        double r34266 = 2.0;
        double r34267 = r34266 * r34259;
        double r34268 = r34265 / r34267;
        return r34268;
}

↓

double f(double a, double b, double c) {
        double r34269 = 1.0;
        double r34270 = 2.0;
        double r34271 = r34269 / r34270;
        double r34272 = 4.0;
        double r34273 = c;
        double r34274 = r34272 * r34273;
        double r34275 = -r34274;
        double r34276 = b;
        double r34277 = r34276 * r34276;
        double r34278 = a;
        double r34279 = r34272 * r34278;
        double r34280 = r34279 * r34273;
        double r34281 = r34277 - r34280;
        double r34282 = sqrt(r34281);
        double r34283 = r34276 + r34282;
        double r34284 = r34275 / r34283;
        double r34285 = r34271 * r34284;
        return r34285;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 43.7

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

3. Applied flip-+ 43.7

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

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 add-sqr-sqrt 0.5

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

7. Applied distribute-rgt-neg-in 0.5

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

8. Applied fma-neg 0.4

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

10. Applied *-un-lft-identity 0.4

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

11. Applied *-un-lft-identity 0.4

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

12. Applied times-frac 0.4

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

13. Applied times-frac 0.4

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

14. Simplified 0.4

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

15. Simplified 0.2

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

17. Applied frac-2neg 0.2

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

18. Simplified 0.2

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

19. Simplified 0.2

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

20. Final simplification 0.2

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

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, medium range"
  :precision binary64
  :pre (and (< 1.11022e-16 a 9.0072e+15) (< 1.11022e-16 b 9.0072e+15) (< 1.11022e-16 c 9.0072e+15))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))