Average Error: 44.0 → 0.4
Time: 19.8s
Precision: 64
\[1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt a \lt 9007199254740992 \land 1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt b \lt 9007199254740992 \land 1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt c \lt 9007199254740992\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{4 \cdot \left(a \cdot c\right)}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{\frac{{b}^{4} - \left(4 \cdot \left(a \cdot c\right)\right) \cdot \left(4 \cdot \left(a \cdot c\right)\right)}{{\left(b \cdot b\right)}^{3} + {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}} \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) - \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)\right)}\right)}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\frac{4 \cdot \left(a \cdot c\right)}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{\frac{{b}^{4} - \left(4 \cdot \left(a \cdot c\right)\right) \cdot \left(4 \cdot \left(a \cdot c\right)\right)}{{\left(b \cdot b\right)}^{3} + {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}} \cdot \left(\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)\right)}\right)}
double f(double a, double b, double c) {
        double r42749 = b;
        double r42750 = -r42749;
        double r42751 = r42749 * r42749;
        double r42752 = 4.0;
        double r42753 = a;
        double r42754 = r42752 * r42753;
        double r42755 = c;
        double r42756 = r42754 * r42755;
        double r42757 = r42751 - r42756;
        double r42758 = sqrt(r42757);
        double r42759 = r42750 + r42758;
        double r42760 = 2.0;
        double r42761 = r42760 * r42753;
        double r42762 = r42759 / r42761;
        return r42762;
}
↓
double f(double a, double b, double c) {
        double r42763 = 4.0;
        double r42764 = a;
        double r42765 = c;
        double r42766 = r42764 * r42765;
        double r42767 = r42763 * r42766;
        double r42768 = 2.0;
        double r42769 = r42768 * r42764;
        double r42770 = b;
        double r42771 = -r42770;
        double r42772 = 4.0;
        double r42773 = pow(r42770, r42772);
        double r42774 = r42767 * r42767;
        double r42775 = r42773 - r42774;
        double r42776 = r42770 * r42770;
        double r42777 = 3.0;
        double r42778 = pow(r42776, r42777);
        double r42779 = r42763 * r42764;
        double r42780 = r42779 * r42765;
        double r42781 = pow(r42780, r42777);
        double r42782 = r42778 + r42781;
        double r42783 = r42775 / r42782;
        double r42784 = r42776 * r42776;
        double r42785 = r42780 * r42780;
        double r42786 = r42776 * r42780;
        double r42787 = r42785 - r42786;
        double r42788 = r42784 + r42787;
        double r42789 = r42783 * r42788;
        double r42790 = sqrt(r42789);
        double r42791 = r42771 - r42790;
        double r42792 = r42769 * r42791;
        double r42793 = r42767 / r42792;
        return r42793;
}

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 44.0

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

  3. Applied flip-+44.0

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

  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 flip--0.4

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

  7. Simplified0.4

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

  9. Applied flip3-+0.4

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

  10. Applied associate-/r/0.4

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

  11. Final simplification0.4

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

Reproduce

herbie shell --seed 2019291 
(FPCore (a b c)
  :name "Quadratic roots, medium range"
  :precision binary64
  :pre (and (< 1.11022e-16 a 9.0072e15) (< 1.11022e-16 b 9.0072e15) (< 1.11022e-16 c 9.0072e15))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))