Average Error: 52.3 → 0.4
Time: 6.5s
Precision: 64
\[4.93038 \cdot 10^{-32} \lt a \lt 2.02824 \cdot 10^{31} \land 4.93038 \cdot 10^{-32} \lt b \lt 2.02824 \cdot 10^{31} \land 4.93038 \cdot 10^{-32} \lt c \lt 2.02824 \cdot 10^{31}\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{\frac{{b}^{4} - \left(4 \cdot \left(a \cdot c\right)\right) \cdot \left(4 \cdot \left(a \cdot c\right)\right)}{b \cdot b + \left(4 \cdot a\right) \cdot c}}}}{2 \cdot a}\]
\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{{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}
double f(double a, double b, double c) {
        double r36495 = b;
        double r36496 = -r36495;
        double r36497 = r36495 * r36495;
        double r36498 = 4.0;
        double r36499 = a;
        double r36500 = r36498 * r36499;
        double r36501 = c;
        double r36502 = r36500 * r36501;
        double r36503 = r36497 - r36502;
        double r36504 = sqrt(r36503);
        double r36505 = r36496 + r36504;
        double r36506 = 2.0;
        double r36507 = r36506 * r36499;
        double r36508 = r36505 / r36507;
        return r36508;
}
↓
double f(double a, double b, double c) {
        double r36509 = 0.0;
        double r36510 = 4.0;
        double r36511 = a;
        double r36512 = c;
        double r36513 = r36511 * r36512;
        double r36514 = r36510 * r36513;
        double r36515 = r36509 + r36514;
        double r36516 = b;
        double r36517 = -r36516;
        double r36518 = 4.0;
        double r36519 = pow(r36516, r36518);
        double r36520 = r36514 * r36514;
        double r36521 = r36519 - r36520;
        double r36522 = r36516 * r36516;
        double r36523 = r36510 * r36511;
        double r36524 = r36523 * r36512;
        double r36525 = r36522 + r36524;
        double r36526 = r36521 / r36525;
        double r36527 = sqrt(r36526);
        double r36528 = r36517 - r36527;
        double r36529 = r36515 / r36528;
        double r36530 = 2.0;
        double r36531 = r36530 * r36511;
        double r36532 = r36529 / r36531;
        return r36532;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 52.3

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

  3. Applied flip-+52.3

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

  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. Final simplification0.4

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

Reproduce

herbie shell --seed 2020060 
(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)))