Average Error: 28.2 → 0.5
Time: 18.6s
Precision: 64
\[1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt a \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt b \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt c \lt 94906265.62425155937671661376953125\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{\left(3 \cdot a\right) \cdot c + \left(b \cdot b - b \cdot b\right)}{\left(b + \sqrt{\frac{{b}^{6} - {\left(\left(3 \cdot a\right) \cdot c\right)}^{3}}{{b}^{4} + \left(\left(3 \cdot a\right) \cdot c + b \cdot b\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)}}\right) \cdot \left(a \cdot \left(-3\right)\right)}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\frac{\left(3 \cdot a\right) \cdot c + \left(b \cdot b - b \cdot b\right)}{\left(b + \sqrt{\frac{{b}^{6} - {\left(\left(3 \cdot a\right) \cdot c\right)}^{3}}{{b}^{4} + \left(\left(3 \cdot a\right) \cdot c + b \cdot b\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)}}\right) \cdot \left(a \cdot \left(-3\right)\right)}
double f(double a, double b, double c) {
        double r91156 = b;
        double r91157 = -r91156;
        double r91158 = r91156 * r91156;
        double r91159 = 3.0;
        double r91160 = a;
        double r91161 = r91159 * r91160;
        double r91162 = c;
        double r91163 = r91161 * r91162;
        double r91164 = r91158 - r91163;
        double r91165 = sqrt(r91164);
        double r91166 = r91157 + r91165;
        double r91167 = r91166 / r91161;
        return r91167;
}
↓
double f(double a, double b, double c) {
        double r91168 = 3.0;
        double r91169 = a;
        double r91170 = r91168 * r91169;
        double r91171 = c;
        double r91172 = r91170 * r91171;
        double r91173 = b;
        double r91174 = r91173 * r91173;
        double r91175 = r91174 - r91174;
        double r91176 = r91172 + r91175;
        double r91177 = 6.0;
        double r91178 = pow(r91173, r91177);
        double r91179 = 3.0;
        double r91180 = pow(r91172, r91179);
        double r91181 = r91178 - r91180;
        double r91182 = 4.0;
        double r91183 = pow(r91173, r91182);
        double r91184 = r91172 + r91174;
        double r91185 = r91184 * r91172;
        double r91186 = r91183 + r91185;
        double r91187 = r91181 / r91186;
        double r91188 = sqrt(r91187);
        double r91189 = r91173 + r91188;
        double r91190 = -r91168;
        double r91191 = r91169 * r91190;
        double r91192 = r91189 * r91191;
        double r91193 = r91176 / r91192;
        return r91193;
}

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

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

  3. Applied flip-+28.2

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

  4. Simplified0.6

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

  5. Simplified0.6

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

  7. Applied associate-*r*0.5

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

  9. Applied div-inv0.5

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

  10. Applied associate-/l*0.5

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

  11. Simplified0.5

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

  13. Applied flip3--0.5

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

  14. Simplified0.5

    \[\leadsto \frac{\left(b \cdot b - b \cdot b\right) + \left(3 \cdot a\right) \cdot c}{\left(-\left(b + \sqrt{\frac{\color{blue}{{b}^{6} - {\left(c \cdot \left(3 \cdot a\right)\right)}^{3}}}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(a \cdot \left(c \cdot 3\right)\right) \cdot \left(a \cdot \left(c \cdot 3\right)\right) + \left(b \cdot b\right) \cdot \left(a \cdot \left(c \cdot 3\right)\right)\right)}}\right)\right) \cdot \left(a \cdot 3\right)}\]

  15. Simplified0.5

    \[\leadsto \frac{\left(b \cdot b - b \cdot b\right) + \left(3 \cdot a\right) \cdot c}{\left(-\left(b + \sqrt{\frac{{b}^{6} - {\left(c \cdot \left(3 \cdot a\right)\right)}^{3}}{\color{blue}{\left(c \cdot \left(3 \cdot a\right)\right) \cdot \left(b \cdot b + c \cdot \left(3 \cdot a\right)\right) + {b}^{4}}}}\right)\right) \cdot \left(a \cdot 3\right)}\]

  16. Final simplification0.5

    \[\leadsto \frac{\left(3 \cdot a\right) \cdot c + \left(b \cdot b - b \cdot b\right)}{\left(b + \sqrt{\frac{{b}^{6} - {\left(\left(3 \cdot a\right) \cdot c\right)}^{3}}{{b}^{4} + \left(\left(3 \cdot a\right) \cdot c + b \cdot b\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)}}\right) \cdot \left(a \cdot \left(-3\right)\right)}\]

Reproduce

herbie shell --seed 2019179 
(FPCore (a b c)
  :name "Cubic critical, narrow range"
  :pre (and (< 1.0536712127723509e-08 a 94906265.62425156) (< 1.0536712127723509e-08 b 94906265.62425156) (< 1.0536712127723509e-08 c 94906265.62425156))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))