Average Error: 52.5 → 0.3
Time: 21.2s
Precision: 64
\[4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt a \lt 20282409603651670423947251286016 \land 4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt b \lt 20282409603651670423947251286016 \land 4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt c \lt 20282409603651670423947251286016\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{1}{-\left(\frac{2}{4} \cdot \frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{c} + \frac{2}{4} \cdot \frac{b}{c}\right)}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\frac{1}{-\left(\frac{2}{4} \cdot \frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{c} + \frac{2}{4} \cdot \frac{b}{c}\right)}
double f(double a, double b, double c) {
        double r30284 = b;
        double r30285 = -r30284;
        double r30286 = r30284 * r30284;
        double r30287 = 4.0;
        double r30288 = a;
        double r30289 = r30287 * r30288;
        double r30290 = c;
        double r30291 = r30289 * r30290;
        double r30292 = r30286 - r30291;
        double r30293 = sqrt(r30292);
        double r30294 = r30285 + r30293;
        double r30295 = 2.0;
        double r30296 = r30295 * r30288;
        double r30297 = r30294 / r30296;
        return r30297;
}
↓
double f(double a, double b, double c) {
        double r30298 = 1.0;
        double r30299 = 2.0;
        double r30300 = 4.0;
        double r30301 = r30299 / r30300;
        double r30302 = b;
        double r30303 = r30302 * r30302;
        double r30304 = a;
        double r30305 = r30300 * r30304;
        double r30306 = c;
        double r30307 = r30305 * r30306;
        double r30308 = r30303 - r30307;
        double r30309 = sqrt(r30308);
        double r30310 = r30309 / r30306;
        double r30311 = r30301 * r30310;
        double r30312 = r30302 / r30306;
        double r30313 = r30301 * r30312;
        double r30314 = r30311 + r30313;
        double r30315 = -r30314;
        double r30316 = r30298 / r30315;
        return r30316;
}

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 52.5

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

  3. Applied flip-+52.4

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

  4. Simplified0.4

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

  6. Applied clear-num0.4

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

  7. Simplified0.4

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

  9. Applied sub-neg0.4

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

  10. Applied distribute-lft-in0.4

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

  11. Simplified0.4

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

  12. Simplified0.3

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

  13. Final simplification0.3

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

Reproduce

herbie shell --seed 2019235 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, wide range"
  :precision binary64
  :pre (and (< 4.93038e-32 a 2.02824e31) (< 4.93038e-32 b 2.02824e31) (< 4.93038e-32 c 2.02824e31))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))