Average Error: 28.6 → 0.6
Time: 6.0s
Precision: 64
\[1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt a \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt b \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt c \lt 94906265.62425155937671661376953125\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{3 \cdot \left(a \cdot c\right)}{3} \cdot \frac{\frac{1}{\left(-b\right) - \sqrt{b \cdot b - \left(\sqrt[3]{\left(3 \cdot a\right) \cdot c} \cdot \sqrt[3]{\left(3 \cdot a\right) \cdot c}\right) \cdot \sqrt[3]{\left(3 \cdot a\right) \cdot c}}}}{a}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\frac{3 \cdot \left(a \cdot c\right)}{3} \cdot \frac{\frac{1}{\left(-b\right) - \sqrt{b \cdot b - \left(\sqrt[3]{\left(3 \cdot a\right) \cdot c} \cdot \sqrt[3]{\left(3 \cdot a\right) \cdot c}\right) \cdot \sqrt[3]{\left(3 \cdot a\right) \cdot c}}}}{a}
double f(double a, double b, double c) {
        double r84209 = b;
        double r84210 = -r84209;
        double r84211 = r84209 * r84209;
        double r84212 = 3.0;
        double r84213 = a;
        double r84214 = r84212 * r84213;
        double r84215 = c;
        double r84216 = r84214 * r84215;
        double r84217 = r84211 - r84216;
        double r84218 = sqrt(r84217);
        double r84219 = r84210 + r84218;
        double r84220 = r84219 / r84214;
        return r84220;
}
↓
double f(double a, double b, double c) {
        double r84221 = 3.0;
        double r84222 = a;
        double r84223 = c;
        double r84224 = r84222 * r84223;
        double r84225 = r84221 * r84224;
        double r84226 = r84225 / r84221;
        double r84227 = 1.0;
        double r84228 = b;
        double r84229 = -r84228;
        double r84230 = r84228 * r84228;
        double r84231 = r84221 * r84222;
        double r84232 = r84231 * r84223;
        double r84233 = cbrt(r84232);
        double r84234 = r84233 * r84233;
        double r84235 = r84234 * r84233;
        double r84236 = r84230 - r84235;
        double r84237 = sqrt(r84236);
        double r84238 = r84229 - r84237;
        double r84239 = r84227 / r84238;
        double r84240 = r84239 / r84222;
        double r84241 = r84226 * r84240;
        return r84241;
}

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

    \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
  2. Using strategy rm

  3. Applied flip-+28.6

    \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}}{3 \cdot a}\]

  4. Simplified0.6

    \[\leadsto \frac{\frac{\color{blue}{\left({b}^{2} - {b}^{2}\right) + 3 \cdot \left(a \cdot c\right)}}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\]
  5. Using strategy rm

  6. Applied div-inv0.6

    \[\leadsto \frac{\color{blue}{\left(\left({b}^{2} - {b}^{2}\right) + 3 \cdot \left(a \cdot c\right)\right) \cdot \frac{1}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}}{3 \cdot a}\]

  7. Applied times-frac0.6

    \[\leadsto \color{blue}{\frac{\left({b}^{2} - {b}^{2}\right) + 3 \cdot \left(a \cdot c\right)}{3} \cdot \frac{\frac{1}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{a}}\]

  8. Simplified0.6

    \[\leadsto \color{blue}{\frac{3 \cdot \left(a \cdot c\right)}{3}} \cdot \frac{\frac{1}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{a}\]
  9. Using strategy rm

  10. Applied add-cube-cbrt0.6

    \[\leadsto \frac{3 \cdot \left(a \cdot c\right)}{3} \cdot \frac{\frac{1}{\left(-b\right) - \sqrt{b \cdot b - \color{blue}{\left(\sqrt[3]{\left(3 \cdot a\right) \cdot c} \cdot \sqrt[3]{\left(3 \cdot a\right) \cdot c}\right) \cdot \sqrt[3]{\left(3 \cdot a\right) \cdot c}}}}}{a}\]

  11. Final simplification0.6

    \[\leadsto \frac{3 \cdot \left(a \cdot c\right)}{3} \cdot \frac{\frac{1}{\left(-b\right) - \sqrt{b \cdot b - \left(\sqrt[3]{\left(3 \cdot a\right) \cdot c} \cdot \sqrt[3]{\left(3 \cdot a\right) \cdot c}\right) \cdot \sqrt[3]{\left(3 \cdot a\right) \cdot c}}}}{a}\]

Reproduce

herbie shell --seed 2020002 
(FPCore (a b c)
  :name "Cubic critical, 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) (* (* 3 a) c)))) (* 3 a)))