Quadratic roots, wide range

Average Error: 52.7 → 0.4

Time: 8.4s

Precision: 64

\[4.93038 \cdot 10^{-32} \lt a \lt 2.02824 \cdot 10^{31} \land 4.93038 \cdot 10^{-32} \lt b \lt 2.02824 \cdot 10^{31} \land 4.93038 \cdot 10^{-32} \lt c \lt 2.02824 \cdot 10^{31}\]

Math

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

↓

\[\frac{1 \cdot \frac{4 \cdot a}{\frac{\left(-b\right) - \sqrt{\frac{{b}^{6} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}{\mathsf{fma}\left(4, \left(a \cdot c\right) \cdot \mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot c\right), {b}^{4}\right)}}}{c}}}{2 \cdot a}\]

C

double f(double a, double b, double c) {
        double r38398 = b;
        double r38399 = -r38398;
        double r38400 = r38398 * r38398;
        double r38401 = 4.0;
        double r38402 = a;
        double r38403 = r38401 * r38402;
        double r38404 = c;
        double r38405 = r38403 * r38404;
        double r38406 = r38400 - r38405;
        double r38407 = sqrt(r38406);
        double r38408 = r38399 + r38407;
        double r38409 = 2.0;
        double r38410 = r38409 * r38402;
        double r38411 = r38408 / r38410;
        return r38411;
}

↓

double f(double a, double b, double c) {
        double r38412 = 1.0;
        double r38413 = 4.0;
        double r38414 = a;
        double r38415 = r38413 * r38414;
        double r38416 = b;
        double r38417 = -r38416;
        double r38418 = 6.0;
        double r38419 = pow(r38416, r38418);
        double r38420 = c;
        double r38421 = r38415 * r38420;
        double r38422 = 3.0;
        double r38423 = pow(r38421, r38422);
        double r38424 = r38419 - r38423;
        double r38425 = r38414 * r38420;
        double r38426 = fma(r38416, r38416, r38421);
        double r38427 = r38425 * r38426;
        double r38428 = 4.0;
        double r38429 = pow(r38416, r38428);
        double r38430 = fma(r38413, r38427, r38429);
        double r38431 = r38424 / r38430;
        double r38432 = sqrt(r38431);
        double r38433 = r38417 - r38432;
        double r38434 = r38433 / r38420;
        double r38435 = r38415 / r38434;
        double r38436 = r38412 * r38435;
        double r38437 = 2.0;
        double r38438 = r38437 * r38414;
        double r38439 = r38436 / r38438;
        return r38439;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 52.7

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

3. Applied flip-+ 52.7

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

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 flip3-- 0.4

   \[\leadsto \frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{\color{blue}{\frac{{\left(b \cdot b\right)}^{3} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}}}{2 \cdot a}\]

7. Simplified 0.4

   \[\leadsto \frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{\frac{\color{blue}{{b}^{6} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}}{2 \cdot a}\]

8. Simplified 0.4

   \[\leadsto \frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{\frac{{b}^{6} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}{\color{blue}{\mathsf{fma}\left(4, \left(a \cdot c\right) \cdot \mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot c\right), {b}^{4}\right)}}}}}{2 \cdot a}\]
9. Using strategy rm

10. Applied *-un-lft-identity 0.4

    \[\leadsto \frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\color{blue}{1 \cdot \left(\left(-b\right) - \sqrt{\frac{{b}^{6} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}{\mathsf{fma}\left(4, \left(a \cdot c\right) \cdot \mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot c\right), {b}^{4}\right)}}\right)}}}{2 \cdot a}\]

11. Applied *-un-lft-identity 0.4

    \[\leadsto \frac{\frac{\color{blue}{1 \cdot \left(0 + 4 \cdot \left(a \cdot c\right)\right)}}{1 \cdot \left(\left(-b\right) - \sqrt{\frac{{b}^{6} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}{\mathsf{fma}\left(4, \left(a \cdot c\right) \cdot \mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot c\right), {b}^{4}\right)}}\right)}}{2 \cdot a}\]

12. Applied times-frac 0.4

    \[\leadsto \frac{\color{blue}{\frac{1}{1} \cdot \frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{\frac{{b}^{6} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}{\mathsf{fma}\left(4, \left(a \cdot c\right) \cdot \mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot c\right), {b}^{4}\right)}}}}}{2 \cdot a}\]

13. Simplified 0.4

    \[\leadsto \frac{\color{blue}{1} \cdot \frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{\frac{{b}^{6} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}{\mathsf{fma}\left(4, \left(a \cdot c\right) \cdot \mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot c\right), {b}^{4}\right)}}}}{2 \cdot a}\]

14. Simplified 0.4

    \[\leadsto \frac{1 \cdot \color{blue}{\frac{4 \cdot a}{\frac{\left(-b\right) - \sqrt{\frac{{b}^{6} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}{\mathsf{fma}\left(4, \left(a \cdot c\right) \cdot \mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot c\right), {b}^{4}\right)}}}{c}}}}{2 \cdot a}\]

15. Final simplification 0.4

    \[\leadsto \frac{1 \cdot \frac{4 \cdot a}{\frac{\left(-b\right) - \sqrt{\frac{{b}^{6} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}{\mathsf{fma}\left(4, \left(a \cdot c\right) \cdot \mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot c\right), {b}^{4}\right)}}}{c}}}{2 \cdot a}\]

Reproduce

herbie shell --seed 2020034 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, wide range"
  :precision binary64
  :pre (and (< 4.9303800000000003e-32 a 2.02824e+31) (< 4.9303800000000003e-32 b 2.02824e+31) (< 4.9303800000000003e-32 c 2.02824e+31))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))