Average Error: 44.1 → 0.5
Time: 16.7s
Precision: 64
\[1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt a \lt 9007199254740992 \land 1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt b \lt 9007199254740992 \land 1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt c \lt 9007199254740992\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{\frac{1}{\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\frac{\frac{1}{\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}
double f(double a, double b, double c) {
        double r44287 = b;
        double r44288 = -r44287;
        double r44289 = r44287 * r44287;
        double r44290 = 4.0;
        double r44291 = a;
        double r44292 = r44290 * r44291;
        double r44293 = c;
        double r44294 = r44292 * r44293;
        double r44295 = r44289 - r44294;
        double r44296 = sqrt(r44295);
        double r44297 = r44288 + r44296;
        double r44298 = 2.0;
        double r44299 = r44298 * r44291;
        double r44300 = r44297 / r44299;
        return r44300;
}
↓
double f(double a, double b, double c) {
        double r44301 = 1.0;
        double r44302 = b;
        double r44303 = -r44302;
        double r44304 = r44302 * r44302;
        double r44305 = 4.0;
        double r44306 = a;
        double r44307 = r44305 * r44306;
        double r44308 = c;
        double r44309 = r44307 * r44308;
        double r44310 = r44304 - r44309;
        double r44311 = sqrt(r44310);
        double r44312 = r44303 - r44311;
        double r44313 = r44306 * r44308;
        double r44314 = r44305 * r44313;
        double r44315 = r44312 / r44314;
        double r44316 = r44301 / r44315;
        double r44317 = 2.0;
        double r44318 = r44317 * r44306;
        double r44319 = r44316 / r44318;
        return r44319;
}

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 44.1

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

  3. Applied flip-+44.1

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

  4. Simplified0.4

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

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

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

  8. Applied clear-num0.5

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

  9. Simplified0.5

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

  10. Final simplification0.5

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

Reproduce

herbie shell --seed 2019326 
(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)))