Average Error: 28.4 → 0.4
Time: 5.5s
Precision: 64
\[1.05367121277235087 \cdot 10^{-8} \lt a \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt b \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt c \lt 94906265.6242515594\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{\frac{\frac{4 \cdot \left(a \cdot c\right)}{2}}{a}}{\left(-b\right) - \sqrt{\sqrt[3]{{\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)}^{3}}}}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\frac{\frac{\frac{4 \cdot \left(a \cdot c\right)}{2}}{a}}{\left(-b\right) - \sqrt{\sqrt[3]{{\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)}^{3}}}}
double f(double a, double b, double c) {
        double r35697 = b;
        double r35698 = -r35697;
        double r35699 = r35697 * r35697;
        double r35700 = 4.0;
        double r35701 = a;
        double r35702 = r35700 * r35701;
        double r35703 = c;
        double r35704 = r35702 * r35703;
        double r35705 = r35699 - r35704;
        double r35706 = sqrt(r35705);
        double r35707 = r35698 + r35706;
        double r35708 = 2.0;
        double r35709 = r35708 * r35701;
        double r35710 = r35707 / r35709;
        return r35710;
}
↓
double f(double a, double b, double c) {
        double r35711 = 4.0;
        double r35712 = a;
        double r35713 = c;
        double r35714 = r35712 * r35713;
        double r35715 = r35711 * r35714;
        double r35716 = 2.0;
        double r35717 = r35715 / r35716;
        double r35718 = r35717 / r35712;
        double r35719 = b;
        double r35720 = -r35719;
        double r35721 = r35719 * r35719;
        double r35722 = r35711 * r35712;
        double r35723 = r35722 * r35713;
        double r35724 = r35721 - r35723;
        double r35725 = 3.0;
        double r35726 = pow(r35724, r35725);
        double r35727 = cbrt(r35726);
        double r35728 = sqrt(r35727);
        double r35729 = r35720 - r35728;
        double r35730 = r35718 / r35729;
        return r35730;
}

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-+28.5

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

  4. Simplified0.5

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

  6. Applied div-inv0.5

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

  7. Applied associate-/l*0.5

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

  8. Simplified0.4

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

  10. Applied associate-/r*0.3

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

  11. Simplified0.3

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

  13. Applied add-cbrt-cube0.4

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

  14. Simplified0.4

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

  15. Final simplification0.4

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

Reproduce

herbie shell --seed 2020056 
(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)))