Average Error: 28.6 → 0.5
Time: 7.6s
Precision: 64
\[1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt a \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt b \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt c \lt 94906265.62425155937671661376953125\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{\frac{\sqrt{{b}^{4} - \left(4 \cdot \left(a \cdot c\right)\right) \cdot \left(4 \cdot \left(a \cdot c\right)\right)}}{\sqrt{b \cdot b + \left(4 \cdot a\right) \cdot c}} \cdot \frac{\sqrt{{b}^{4} - \left(4 \cdot \left(a \cdot c\right)\right) \cdot \left(4 \cdot \left(a \cdot c\right)\right)}}{\sqrt{b \cdot b + \left(4 \cdot a\right) \cdot c}}}}}{2 \cdot a}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{\frac{\sqrt{{b}^{4} - \left(4 \cdot \left(a \cdot c\right)\right) \cdot \left(4 \cdot \left(a \cdot c\right)\right)}}{\sqrt{b \cdot b + \left(4 \cdot a\right) \cdot c}} \cdot \frac{\sqrt{{b}^{4} - \left(4 \cdot \left(a \cdot c\right)\right) \cdot \left(4 \cdot \left(a \cdot c\right)\right)}}{\sqrt{b \cdot b + \left(4 \cdot a\right) \cdot c}}}}}{2 \cdot a}
double f(double a, double b, double c) {
        double r43252 = b;
        double r43253 = -r43252;
        double r43254 = r43252 * r43252;
        double r43255 = 4.0;
        double r43256 = a;
        double r43257 = r43255 * r43256;
        double r43258 = c;
        double r43259 = r43257 * r43258;
        double r43260 = r43254 - r43259;
        double r43261 = sqrt(r43260);
        double r43262 = r43253 + r43261;
        double r43263 = 2.0;
        double r43264 = r43263 * r43256;
        double r43265 = r43262 / r43264;
        return r43265;
}
↓
double f(double a, double b, double c) {
        double r43266 = 0.0;
        double r43267 = 4.0;
        double r43268 = a;
        double r43269 = c;
        double r43270 = r43268 * r43269;
        double r43271 = r43267 * r43270;
        double r43272 = r43266 + r43271;
        double r43273 = b;
        double r43274 = -r43273;
        double r43275 = 4.0;
        double r43276 = pow(r43273, r43275);
        double r43277 = r43271 * r43271;
        double r43278 = r43276 - r43277;
        double r43279 = sqrt(r43278);
        double r43280 = r43273 * r43273;
        double r43281 = r43267 * r43268;
        double r43282 = r43281 * r43269;
        double r43283 = r43280 + r43282;
        double r43284 = sqrt(r43283);
        double r43285 = r43279 / r43284;
        double r43286 = r43285 * r43285;
        double r43287 = sqrt(r43286);
        double r43288 = r43274 - r43287;
        double r43289 = r43272 / r43288;
        double r43290 = 2.0;
        double r43291 = r43290 * r43268;
        double r43292 = r43289 / r43291;
        return r43292;
}

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 28.6

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

  3. Applied flip-+28.6

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

  4. Simplified0.5

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

  6. Applied flip--0.5

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

  7. Simplified0.5

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

  9. Applied add-sqr-sqrt0.5

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

  10. Applied add-sqr-sqrt0.5

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

  11. Applied times-frac0.5

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

  12. Final simplification0.5

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

Reproduce

herbie shell --seed 2019344 
(FPCore (a b c)
  :name "Quadratic roots, narrow range"
  :precision binary64
  :pre (and (< 1.0536712127723509e-08 a 94906265.62425156) (< 1.0536712127723509e-08 b 94906265.62425156) (< 1.0536712127723509e-08 c 94906265.62425156))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))