Average Error: 28.6 → 0.4
Time: 27.6s
Precision: 64
\[1.05367121277235087 \cdot 10^{-8} \lt a \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt b \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt c \lt 94906265.6242515594\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{\frac{4 \cdot \left(a \cdot c\right)}{2 \cdot a}}{\left(-b\right) - \sqrt{\sqrt[3]{{\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)}^{3}}}}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\frac{\frac{4 \cdot \left(a \cdot c\right)}{2 \cdot 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 r43651 = b;
        double r43652 = -r43651;
        double r43653 = r43651 * r43651;
        double r43654 = 4.0;
        double r43655 = a;
        double r43656 = r43654 * r43655;
        double r43657 = c;
        double r43658 = r43656 * r43657;
        double r43659 = r43653 - r43658;
        double r43660 = sqrt(r43659);
        double r43661 = r43652 + r43660;
        double r43662 = 2.0;
        double r43663 = r43662 * r43655;
        double r43664 = r43661 / r43663;
        return r43664;
}
↓
double f(double a, double b, double c) {
        double r43665 = 4.0;
        double r43666 = a;
        double r43667 = c;
        double r43668 = r43666 * r43667;
        double r43669 = r43665 * r43668;
        double r43670 = 2.0;
        double r43671 = r43670 * r43666;
        double r43672 = r43669 / r43671;
        double r43673 = b;
        double r43674 = -r43673;
        double r43675 = r43673 * r43673;
        double r43676 = r43665 * r43666;
        double r43677 = r43676 * r43667;
        double r43678 = r43675 - r43677;
        double r43679 = 3.0;
        double r43680 = pow(r43678, r43679);
        double r43681 = cbrt(r43680);
        double r43682 = sqrt(r43681);
        double r43683 = r43674 - r43682;
        double r43684 = r43672 / r43683;
        return r43684;
}

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 28.6

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

  3. Applied flip-+28.6

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

  4. Simplified0.5

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

  6. Applied div-inv0.5

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

  8. Applied add-cbrt-cube0.5

    \[\leadsto \frac{0 + \left(4 \cdot a\right) \cdot c}{\left(-b\right) - \sqrt{\color{blue}{\sqrt[3]{\left(\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right) \cdot \left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)\right) \cdot \left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)}}}} \cdot \frac{1}{2 \cdot a}\]

  9. Simplified0.5

    \[\leadsto \frac{0 + \left(4 \cdot a\right) \cdot c}{\left(-b\right) - \sqrt{\sqrt[3]{\color{blue}{{\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)}^{3}}}}} \cdot \frac{1}{2 \cdot a}\]
  10. Using strategy rm

  11. Applied associate-*l/0.5

    \[\leadsto \color{blue}{\frac{\left(0 + \left(4 \cdot a\right) \cdot c\right) \cdot \frac{1}{2 \cdot a}}{\left(-b\right) - \sqrt{\sqrt[3]{{\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)}^{3}}}}}\]

  12. Simplified0.4

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

  13. Final simplification0.4

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

Reproduce

herbie shell --seed 2019199 
(FPCore (a b c)
  :name "Quadratic roots, 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) (* (* 4.0 a) c)))) (* 2.0 a)))