Average Error: 28.4 → 0.5
Time: 6.5s
Precision: 64
\[1.05367121277235087 \cdot 10^{-8} \lt a \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt b \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt c \lt 94906265.6242515594\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{1 \cdot \frac{4}{\frac{\frac{-b}{a}}{c} - \frac{\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a}}{c}}}{2 \cdot a}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\frac{1 \cdot \frac{4}{\frac{\frac{-b}{a}}{c} - \frac{\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a}}{c}}}{2 \cdot a}
double f(double a, double b, double c) {
        double r42146 = b;
        double r42147 = -r42146;
        double r42148 = r42146 * r42146;
        double r42149 = 4.0;
        double r42150 = a;
        double r42151 = r42149 * r42150;
        double r42152 = c;
        double r42153 = r42151 * r42152;
        double r42154 = r42148 - r42153;
        double r42155 = sqrt(r42154);
        double r42156 = r42147 + r42155;
        double r42157 = 2.0;
        double r42158 = r42157 * r42150;
        double r42159 = r42156 / r42158;
        return r42159;
}
↓
double f(double a, double b, double c) {
        double r42160 = 1.0;
        double r42161 = 4.0;
        double r42162 = b;
        double r42163 = -r42162;
        double r42164 = a;
        double r42165 = r42163 / r42164;
        double r42166 = c;
        double r42167 = r42165 / r42166;
        double r42168 = r42162 * r42162;
        double r42169 = r42161 * r42164;
        double r42170 = r42169 * r42166;
        double r42171 = r42168 - r42170;
        double r42172 = sqrt(r42171);
        double r42173 = r42172 / r42164;
        double r42174 = r42173 / r42166;
        double r42175 = r42167 - r42174;
        double r42176 = r42161 / r42175;
        double r42177 = r42160 * r42176;
        double r42178 = 2.0;
        double r42179 = r42178 * r42164;
        double r42180 = r42177 / r42179;
        return r42180;
}

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 28.4

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

  3. Applied flip-+28.4

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

  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 *-un-lft-identity0.5

    \[\leadsto \frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\color{blue}{1 \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}}{2 \cdot a}\]

  7. Applied *-un-lft-identity0.5

    \[\leadsto \frac{\frac{\color{blue}{1 \cdot \left(0 + 4 \cdot \left(a \cdot c\right)\right)}}{1 \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a}\]

  8. Applied times-frac0.5

    \[\leadsto \frac{\color{blue}{\frac{1}{1} \cdot \frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}{2 \cdot a}\]

  9. Simplified0.5

    \[\leadsto \frac{\color{blue}{1} \cdot \frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\]

  10. Simplified0.5

    \[\leadsto \frac{1 \cdot \color{blue}{\frac{4}{\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot c}}}}{2 \cdot a}\]
  11. Using strategy rm

  12. Applied associate-/r*0.5

    \[\leadsto \frac{1 \cdot \frac{4}{\color{blue}{\frac{\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a}}{c}}}}{2 \cdot a}\]
  13. Using strategy rm

  14. Applied div-sub0.5

    \[\leadsto \frac{1 \cdot \frac{4}{\frac{\color{blue}{\frac{-b}{a} - \frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a}}}{c}}}{2 \cdot a}\]

  15. Applied div-sub0.5

    \[\leadsto \frac{1 \cdot \frac{4}{\color{blue}{\frac{\frac{-b}{a}}{c} - \frac{\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a}}{c}}}}{2 \cdot a}\]

  16. Final simplification0.5

    \[\leadsto \frac{1 \cdot \frac{4}{\frac{\frac{-b}{a}}{c} - \frac{\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a}}{c}}}{2 \cdot a}\]

Reproduce

herbie shell --seed 2020062 
(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)))