Quadratic roots, narrow range

Average Error: 28.2 → 0.5

Time: 7.3s

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

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

↓

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

C

double f(double a, double b, double c) {
        double r41458 = b;
        double r41459 = -r41458;
        double r41460 = r41458 * r41458;
        double r41461 = 4.0;
        double r41462 = a;
        double r41463 = r41461 * r41462;
        double r41464 = c;
        double r41465 = r41463 * r41464;
        double r41466 = r41460 - r41465;
        double r41467 = sqrt(r41466);
        double r41468 = r41459 + r41467;
        double r41469 = 2.0;
        double r41470 = r41469 * r41462;
        double r41471 = r41468 / r41470;
        return r41471;
}

↓

double f(double a, double b, double c) {
        double r41472 = 0.0;
        double r41473 = 4.0;
        double r41474 = a;
        double r41475 = c;
        double r41476 = r41474 * r41475;
        double r41477 = r41473 * r41476;
        double r41478 = r41472 + r41477;
        double r41479 = b;
        double r41480 = -r41479;
        double r41481 = 1.0;
        double r41482 = 4.0;
        double r41483 = pow(r41479, r41482);
        double r41484 = r41473 * r41474;
        double r41485 = r41484 * r41475;
        double r41486 = 2.0;
        double r41487 = pow(r41479, r41486);
        double r41488 = r41485 + r41487;
        double r41489 = r41485 * r41488;
        double r41490 = r41483 + r41489;
        double r41491 = 6.0;
        double r41492 = pow(r41479, r41491);
        double r41493 = 3.0;
        double r41494 = pow(r41485, r41493);
        double r41495 = r41492 - r41494;
        double r41496 = r41490 / r41495;
        double r41497 = r41481 / r41496;
        double r41498 = sqrt(r41497);
        double r41499 = r41480 - r41498;
        double r41500 = r41478 / r41499;
        double r41501 = 2.0;
        double r41502 = r41501 * r41474;
        double r41503 = r41500 / r41502;
        return r41503;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 28.2

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

3. Applied flip-+ 28.2

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

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-- 0.5

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

7. Simplified 0.5

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

8. Simplified 0.5

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

10. Applied clear-num 0.5

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

11. Simplified 0.5

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

12. Final simplification 0.5

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

Reproduce

herbie shell --seed 2019362 
(FPCore (a b c)
  :name "Quadratic roots, 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) (* (* 4 a) c)))) (* 2 a)))