Average Error: 28.2 → 0.4
Time: 15.6s
Precision: 64
\[1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt a \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt b \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt c \lt 94906265.62425155937671661376953125\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{\frac{1}{\frac{2}{4} \cdot \frac{1}{c}}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\frac{\frac{1}{\frac{2}{4} \cdot \frac{1}{c}}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}
double f(double a, double b, double c) {
        double r42267 = b;
        double r42268 = -r42267;
        double r42269 = r42267 * r42267;
        double r42270 = 4.0;
        double r42271 = a;
        double r42272 = r42270 * r42271;
        double r42273 = c;
        double r42274 = r42272 * r42273;
        double r42275 = r42269 - r42274;
        double r42276 = sqrt(r42275);
        double r42277 = r42268 + r42276;
        double r42278 = 2.0;
        double r42279 = r42278 * r42271;
        double r42280 = r42277 / r42279;
        return r42280;
}
↓
double f(double a, double b, double c) {
        double r42281 = 1.0;
        double r42282 = 2.0;
        double r42283 = 4.0;
        double r42284 = r42282 / r42283;
        double r42285 = c;
        double r42286 = r42281 / r42285;
        double r42287 = r42284 * r42286;
        double r42288 = r42281 / r42287;
        double r42289 = b;
        double r42290 = -r42289;
        double r42291 = r42289 * r42289;
        double r42292 = a;
        double r42293 = r42283 * r42292;
        double r42294 = r42293 * r42285;
        double r42295 = r42291 - r42294;
        double r42296 = sqrt(r42295);
        double r42297 = r42290 - r42296;
        double r42298 = r42288 / r42297;
        return r42298;
}

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 28.2

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

  3. Applied flip-+28.2

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

  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 clear-num0.5

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

  7. Simplified0.5

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

  9. Applied associate-/r/0.5

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

  10. Applied associate-/r*0.4

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

  12. Applied times-frac0.4

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

  13. Simplified0.4

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

  14. Final simplification0.4

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

Reproduce

herbie shell --seed 2019209 
(FPCore (a b c)
  :name "Quadratic roots, narrow range"
  :precision binary64
  :pre (and (< 1.05367121277235087e-8 a 94906265.6242515594) (< 1.05367121277235087e-8 b 94906265.6242515594) (< 1.05367121277235087e-8 c 94906265.6242515594))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))