Average Error: 52.7 → 0.5
Time: 18.6s
Precision: 64
\[4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt a \lt 20282409603651670423947251286016 \land 4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt b \lt 20282409603651670423947251286016 \land 4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt c \lt 20282409603651670423947251286016\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{\frac{\left(c \cdot 3\right) \cdot a}{\left(-b\right) - \sqrt{{b}^{2} - 3 \cdot \left(a \cdot c\right)}}}{3 \cdot a}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\frac{\frac{\left(c \cdot 3\right) \cdot a}{\left(-b\right) - \sqrt{{b}^{2} - 3 \cdot \left(a \cdot c\right)}}}{3 \cdot a}
double f(double a, double b, double c) {
        double r58056 = b;
        double r58057 = -r58056;
        double r58058 = r58056 * r58056;
        double r58059 = 3.0;
        double r58060 = a;
        double r58061 = r58059 * r58060;
        double r58062 = c;
        double r58063 = r58061 * r58062;
        double r58064 = r58058 - r58063;
        double r58065 = sqrt(r58064);
        double r58066 = r58057 + r58065;
        double r58067 = r58066 / r58061;
        return r58067;
}
↓
double f(double a, double b, double c) {
        double r58068 = c;
        double r58069 = 3.0;
        double r58070 = r58068 * r58069;
        double r58071 = a;
        double r58072 = r58070 * r58071;
        double r58073 = b;
        double r58074 = -r58073;
        double r58075 = 2.0;
        double r58076 = pow(r58073, r58075);
        double r58077 = r58071 * r58068;
        double r58078 = r58069 * r58077;
        double r58079 = r58076 - r58078;
        double r58080 = sqrt(r58079);
        double r58081 = r58074 - r58080;
        double r58082 = r58072 / r58081;
        double r58083 = r58069 * r58071;
        double r58084 = r58082 / r58083;
        return r58084;
}

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 52.7

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

  3. Applied flip-+52.7

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

  4. Simplified0.5

    \[\leadsto \frac{\frac{\color{blue}{0 + \left(a \cdot c\right) \cdot 3}}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\]

  5. Simplified0.5

    \[\leadsto \frac{\frac{0 + \left(a \cdot c\right) \cdot 3}{\color{blue}{\left(-b\right) - \sqrt{{b}^{2} - 3 \cdot \left(a \cdot c\right)}}}}{3 \cdot a}\]
  6. Using strategy rm

  7. Applied associate-*l*0.5

    \[\leadsto \frac{\frac{0 + \color{blue}{a \cdot \left(c \cdot 3\right)}}{\left(-b\right) - \sqrt{{b}^{2} - 3 \cdot \left(a \cdot c\right)}}}{3 \cdot a}\]

  8. Final simplification0.5

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

Reproduce

herbie shell --seed 2019304 
(FPCore (a b c)
  :name "Cubic critical, 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) (* (* 3 a) c)))) (* 3 a)))