Average Error: 43.6 → 0.4
Time: 8.6s
Precision: 64
\[1.11022 \cdot 10^{-16} \lt a \lt 9.0072 \cdot 10^{15} \land 1.11022 \cdot 10^{-16} \lt b \lt 9.0072 \cdot 10^{15} \land 1.11022 \cdot 10^{-16} \lt c \lt 9.0072 \cdot 10^{15}\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{1}{2} \cdot \frac{4 \cdot \left(a \cdot c\right)}{a \cdot \left(\left(-b\right) - \sqrt{\frac{{b}^{4} - \left(4 \cdot \left(a \cdot c\right)\right) \cdot \left(4 \cdot \left(a \cdot c\right)\right)}{\mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot c\right)}}\right)}\]
\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 \left(a \cdot c\right)}{a \cdot \left(\left(-b\right) - \sqrt{\frac{{b}^{4} - \left(4 \cdot \left(a \cdot c\right)\right) \cdot \left(4 \cdot \left(a \cdot c\right)\right)}{\mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot c\right)}}\right)}
double f(double a, double b, double c) {
        double r35440 = b;
        double r35441 = -r35440;
        double r35442 = r35440 * r35440;
        double r35443 = 4.0;
        double r35444 = a;
        double r35445 = r35443 * r35444;
        double r35446 = c;
        double r35447 = r35445 * r35446;
        double r35448 = r35442 - r35447;
        double r35449 = sqrt(r35448);
        double r35450 = r35441 + r35449;
        double r35451 = 2.0;
        double r35452 = r35451 * r35444;
        double r35453 = r35450 / r35452;
        return r35453;
}
↓
double f(double a, double b, double c) {
        double r35454 = 1.0;
        double r35455 = 2.0;
        double r35456 = r35454 / r35455;
        double r35457 = 4.0;
        double r35458 = a;
        double r35459 = c;
        double r35460 = r35458 * r35459;
        double r35461 = r35457 * r35460;
        double r35462 = b;
        double r35463 = -r35462;
        double r35464 = 4.0;
        double r35465 = pow(r35462, r35464);
        double r35466 = r35461 * r35461;
        double r35467 = r35465 - r35466;
        double r35468 = r35457 * r35458;
        double r35469 = r35468 * r35459;
        double r35470 = fma(r35462, r35462, r35469);
        double r35471 = r35467 / r35470;
        double r35472 = sqrt(r35471);
        double r35473 = r35463 - r35472;
        double r35474 = r35458 * r35473;
        double r35475 = r35461 / r35474;
        double r35476 = r35456 * r35475;
        return r35476;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Initial program 43.6

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

  3. Applied flip-+43.6

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

  4. Simplified0.4

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

  6. Applied flip--0.4

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

  7. Simplified0.4

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

  8. Simplified0.4

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

  10. Applied *-un-lft-identity0.4

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

  11. Applied *-un-lft-identity0.4

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

  12. Applied times-frac0.4

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

  13. Applied times-frac0.4

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

  14. Simplified0.4

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

  15. Simplified0.4

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

  16. Final simplification0.4

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

Reproduce

herbie shell --seed 2020035 +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)))