Average Error: 43.7 → 0.4
Time: 19.4s
Precision: 64
\[1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt a \lt 9007199254740992 \land 1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt b \lt 9007199254740992 \land 1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt c \lt 9007199254740992\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{1}{2 \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)} \cdot \frac{c \cdot \left(4 \cdot a\right)}{a}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\frac{1}{2 \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)} \cdot \frac{c \cdot \left(4 \cdot a\right)}{a}
double f(double a, double b, double c) {
        double r39539 = b;
        double r39540 = -r39539;
        double r39541 = r39539 * r39539;
        double r39542 = 4.0;
        double r39543 = a;
        double r39544 = r39542 * r39543;
        double r39545 = c;
        double r39546 = r39544 * r39545;
        double r39547 = r39541 - r39546;
        double r39548 = sqrt(r39547);
        double r39549 = r39540 + r39548;
        double r39550 = 2.0;
        double r39551 = r39550 * r39543;
        double r39552 = r39549 / r39551;
        return r39552;
}
↓
double f(double a, double b, double c) {
        double r39553 = 1.0;
        double r39554 = 2.0;
        double r39555 = b;
        double r39556 = -r39555;
        double r39557 = r39555 * r39555;
        double r39558 = 4.0;
        double r39559 = a;
        double r39560 = r39558 * r39559;
        double r39561 = c;
        double r39562 = r39560 * r39561;
        double r39563 = r39557 - r39562;
        double r39564 = sqrt(r39563);
        double r39565 = r39556 - r39564;
        double r39566 = r39554 * r39565;
        double r39567 = r39553 / r39566;
        double r39568 = r39561 * r39560;
        double r39569 = r39568 / r39559;
        double r39570 = r39567 * r39569;
        return r39570;
}

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 43.7

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

  3. Applied flip-+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

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

  6. Applied div-inv0.5

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

  7. Applied associate-/l*0.5

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

  8. Simplified0.4

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

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

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

  11. Applied times-frac0.4

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

  12. Simplified0.4

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

  13. Final simplification0.4

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

Reproduce

herbie shell --seed 2019325 +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)))