Average Error: 43.7 → 0.4
Time: 8.7s
Precision: 64
\[1.11022 \cdot 10^{-16} \lt a \lt 9.0072 \cdot 10^{15} \land 1.11022 \cdot 10^{-16} \lt b \lt 9.0072 \cdot 10^{15} \land 1.11022 \cdot 10^{-16} \lt c \lt 9.0072 \cdot 10^{15}\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{\frac{\sqrt{{b}^{4} - \left(4 \cdot \left(a \cdot c\right)\right) \cdot \left(4 \cdot \left(a \cdot c\right)\right)} \cdot \frac{\sqrt{{b}^{4} - \left(4 \cdot \left(a \cdot c\right)\right) \cdot \left(4 \cdot \left(a \cdot c\right)\right)}}{\sqrt{b \cdot b + \left(4 \cdot a\right) \cdot c}}}{\sqrt{b \cdot b + \left(4 \cdot a\right) \cdot c}}}}}{2 \cdot a}\]
\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{\sqrt{{b}^{4} - \left(4 \cdot \left(a \cdot c\right)\right) \cdot \left(4 \cdot \left(a \cdot c\right)\right)} \cdot \frac{\sqrt{{b}^{4} - \left(4 \cdot \left(a \cdot c\right)\right) \cdot \left(4 \cdot \left(a \cdot c\right)\right)}}{\sqrt{b \cdot b + \left(4 \cdot a\right) \cdot c}}}{\sqrt{b \cdot b + \left(4 \cdot a\right) \cdot c}}}}}{2 \cdot a}
double f(double a, double b, double c) {
        double r42773 = b;
        double r42774 = -r42773;
        double r42775 = r42773 * r42773;
        double r42776 = 4.0;
        double r42777 = a;
        double r42778 = r42776 * r42777;
        double r42779 = c;
        double r42780 = r42778 * r42779;
        double r42781 = r42775 - r42780;
        double r42782 = sqrt(r42781);
        double r42783 = r42774 + r42782;
        double r42784 = 2.0;
        double r42785 = r42784 * r42777;
        double r42786 = r42783 / r42785;
        return r42786;
}
↓
double f(double a, double b, double c) {
        double r42787 = 0.0;
        double r42788 = 4.0;
        double r42789 = a;
        double r42790 = c;
        double r42791 = r42789 * r42790;
        double r42792 = r42788 * r42791;
        double r42793 = r42787 + r42792;
        double r42794 = b;
        double r42795 = -r42794;
        double r42796 = 4.0;
        double r42797 = pow(r42794, r42796);
        double r42798 = r42792 * r42792;
        double r42799 = r42797 - r42798;
        double r42800 = sqrt(r42799);
        double r42801 = r42794 * r42794;
        double r42802 = r42788 * r42789;
        double r42803 = r42802 * r42790;
        double r42804 = r42801 + r42803;
        double r42805 = sqrt(r42804);
        double r42806 = r42800 / r42805;
        double r42807 = r42800 * r42806;
        double r42808 = r42807 / r42805;
        double r42809 = sqrt(r42808);
        double r42810 = r42795 - r42809;
        double r42811 = r42793 / r42810;
        double r42812 = 2.0;
        double r42813 = r42812 * r42789;
        double r42814 = r42811 / r42813;
        return r42814;
}

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 43.7

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

  3. Applied flip-+43.7

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

  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 add-sqr-sqrt0.4

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

  10. Applied associate-/r*0.4

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

  12. Applied *-un-lft-identity0.4

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

  13. Applied sqrt-prod0.4

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

  14. Applied add-sqr-sqrt0.5

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

  15. Applied times-frac0.4

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

  16. Simplified0.4

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

  17. Final simplification0.4

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

Reproduce

herbie shell --seed 2020047 
(FPCore (a b c)
  :name "Quadratic roots, medium range"
  :precision binary64
  :pre (and (< 1.11022e-16 a 9.0072e+15) (< 1.11022e-16 b 9.0072e+15) (< 1.11022e-16 c 9.0072e+15))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))