Average Error: 52.7 → 0.5
Time: 8.4s
Precision: 64
\[4.93038 \cdot 10^{-32} \lt a \lt 2.02824 \cdot 10^{31} \land 4.93038 \cdot 10^{-32} \lt b \lt 2.02824 \cdot 10^{31} \land 4.93038 \cdot 10^{-32} \lt c \lt 2.02824 \cdot 10^{31}\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{\frac{0 - \left(4 \cdot \left(a \cdot c\right)\right) \cdot \left(4 \cdot \left(a \cdot c\right)\right)}{\left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \left(0 - 4 \cdot \left(a \cdot c\right)\right)}}{2 \cdot a}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\frac{\frac{0 - \left(4 \cdot \left(a \cdot c\right)\right) \cdot \left(4 \cdot \left(a \cdot c\right)\right)}{\left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \left(0 - 4 \cdot \left(a \cdot c\right)\right)}}{2 \cdot a}
double f(double a, double b, double c) {
        double r38543 = b;
        double r38544 = -r38543;
        double r38545 = r38543 * r38543;
        double r38546 = 4.0;
        double r38547 = a;
        double r38548 = r38546 * r38547;
        double r38549 = c;
        double r38550 = r38548 * r38549;
        double r38551 = r38545 - r38550;
        double r38552 = sqrt(r38551);
        double r38553 = r38544 + r38552;
        double r38554 = 2.0;
        double r38555 = r38554 * r38547;
        double r38556 = r38553 / r38555;
        return r38556;
}
↓
double f(double a, double b, double c) {
        double r38557 = 0.0;
        double r38558 = 4.0;
        double r38559 = a;
        double r38560 = c;
        double r38561 = r38559 * r38560;
        double r38562 = r38558 * r38561;
        double r38563 = r38562 * r38562;
        double r38564 = r38557 - r38563;
        double r38565 = b;
        double r38566 = -r38565;
        double r38567 = r38565 * r38565;
        double r38568 = r38558 * r38559;
        double r38569 = r38568 * r38560;
        double r38570 = r38567 - r38569;
        double r38571 = sqrt(r38570);
        double r38572 = r38566 - r38571;
        double r38573 = r38557 - r38562;
        double r38574 = r38572 * r38573;
        double r38575 = r38564 / r38574;
        double r38576 = 2.0;
        double r38577 = r38576 * r38559;
        double r38578 = r38575 / r38577;
        return r38578;
}

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 52.7

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

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

  7. Applied associate-/l/0.5

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

  8. Final simplification0.5

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

Reproduce

herbie shell --seed 2020035 +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)))