Average Error: 28.5 → 0.5
Time: 6.2s
Precision: 64
\[1.05367121277235087 \cdot 10^{-8} \lt a \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt b \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt c \lt 94906265.6242515594\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{\frac{{b}^{6} - 64 \cdot \left({a}^{3} \cdot {c}^{3}\right)}{\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c + {b}^{2}\right) + \left(b \cdot b\right) \cdot \left(b \cdot b\right)}}}}{2 \cdot a}\]
\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{{b}^{6} - 64 \cdot \left({a}^{3} \cdot {c}^{3}\right)}{\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c + {b}^{2}\right) + \left(b \cdot b\right) \cdot \left(b \cdot b\right)}}}}{2 \cdot a}
double f(double a, double b, double c) {
        double r33155 = b;
        double r33156 = -r33155;
        double r33157 = r33155 * r33155;
        double r33158 = 4.0;
        double r33159 = a;
        double r33160 = r33158 * r33159;
        double r33161 = c;
        double r33162 = r33160 * r33161;
        double r33163 = r33157 - r33162;
        double r33164 = sqrt(r33163);
        double r33165 = r33156 + r33164;
        double r33166 = 2.0;
        double r33167 = r33166 * r33159;
        double r33168 = r33165 / r33167;
        return r33168;
}
↓
double f(double a, double b, double c) {
        double r33169 = 0.0;
        double r33170 = 4.0;
        double r33171 = a;
        double r33172 = c;
        double r33173 = r33171 * r33172;
        double r33174 = r33170 * r33173;
        double r33175 = r33169 + r33174;
        double r33176 = b;
        double r33177 = -r33176;
        double r33178 = 6.0;
        double r33179 = pow(r33176, r33178);
        double r33180 = 64.0;
        double r33181 = 3.0;
        double r33182 = pow(r33171, r33181);
        double r33183 = pow(r33172, r33181);
        double r33184 = r33182 * r33183;
        double r33185 = r33180 * r33184;
        double r33186 = r33179 - r33185;
        double r33187 = r33170 * r33171;
        double r33188 = r33187 * r33172;
        double r33189 = 2.0;
        double r33190 = pow(r33176, r33189);
        double r33191 = r33188 + r33190;
        double r33192 = r33188 * r33191;
        double r33193 = r33176 * r33176;
        double r33194 = r33193 * r33193;
        double r33195 = r33192 + r33194;
        double r33196 = r33186 / r33195;
        double r33197 = sqrt(r33196);
        double r33198 = r33177 - r33197;
        double r33199 = r33175 / r33198;
        double r33200 = 2.0;
        double r33201 = r33200 * r33171;
        double r33202 = r33199 / r33201;
        return r33202;
}

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 28.5

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

  3. Applied flip-+28.5

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

  4. Simplified0.4

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

  6. Applied flip3--0.5

    \[\leadsto \frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{\color{blue}{\frac{{\left(b \cdot b\right)}^{3} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}}}{2 \cdot a}\]

  7. Simplified0.5

    \[\leadsto \frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{\frac{\color{blue}{{b}^{6} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}}{2 \cdot a}\]

  8. Simplified0.5

    \[\leadsto \frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{\frac{{b}^{6} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}{\color{blue}{\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c + {b}^{2}\right) + \left(b \cdot b\right) \cdot \left(b \cdot b\right)}}}}}{2 \cdot a}\]

  9. Taylor expanded around 0 0.5

    \[\leadsto \frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{\frac{{b}^{6} - \color{blue}{64 \cdot \left({a}^{3} \cdot {c}^{3}\right)}}{\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c + {b}^{2}\right) + \left(b \cdot b\right) \cdot \left(b \cdot b\right)}}}}{2 \cdot a}\]

  10. Final simplification0.5

    \[\leadsto \frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{\frac{{b}^{6} - 64 \cdot \left({a}^{3} \cdot {c}^{3}\right)}{\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c + {b}^{2}\right) + \left(b \cdot b\right) \cdot \left(b \cdot b\right)}}}}{2 \cdot a}\]

Reproduce

herbie shell --seed 2020059 
(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)))