Average Error: 43.5 → 0.3
Time: 6.5s
Precision: 64
\[1.11022 \cdot 10^{-16} \lt a \lt 9.0072 \cdot 10^{15} \land 1.11022 \cdot 10^{-16} \lt b \lt 9.0072 \cdot 10^{15} \land 1.11022 \cdot 10^{-16} \lt c \lt 9.0072 \cdot 10^{15}\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{\frac{1}{1}}{\frac{\frac{2}{4} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}{c}}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\frac{\frac{1}{1}}{\frac{\frac{2}{4} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}{c}}
double f(double a, double b, double c) {
        double r31806 = b;
        double r31807 = -r31806;
        double r31808 = r31806 * r31806;
        double r31809 = 4.0;
        double r31810 = a;
        double r31811 = r31809 * r31810;
        double r31812 = c;
        double r31813 = r31811 * r31812;
        double r31814 = r31808 - r31813;
        double r31815 = sqrt(r31814);
        double r31816 = r31807 + r31815;
        double r31817 = 2.0;
        double r31818 = r31817 * r31810;
        double r31819 = r31816 / r31818;
        return r31819;
}
↓
double f(double a, double b, double c) {
        double r31820 = 1.0;
        double r31821 = r31820 / r31820;
        double r31822 = 2.0;
        double r31823 = 4.0;
        double r31824 = r31822 / r31823;
        double r31825 = b;
        double r31826 = -r31825;
        double r31827 = r31825 * r31825;
        double r31828 = a;
        double r31829 = r31823 * r31828;
        double r31830 = c;
        double r31831 = r31829 * r31830;
        double r31832 = r31827 - r31831;
        double r31833 = sqrt(r31832);
        double r31834 = r31826 - r31833;
        double r31835 = r31824 * r31834;
        double r31836 = r31835 / r31830;
        double r31837 = r31821 / r31836;
        return r31837;
}

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 43.5

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

  3. Applied flip-+43.6

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

  4. Simplified0.4

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

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

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

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

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

  8. Applied times-frac0.4

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

  9. Applied associate-/l*0.5

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

  10. Simplified0.5

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

  12. Applied div-inv0.5

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

  13. Simplified0.4

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

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

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

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

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

  17. Applied times-frac0.4

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

  18. Applied associate-/l*0.5

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

  19. Simplified0.3

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

  20. Final simplification0.3

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

Reproduce

herbie shell --seed 2020083 
(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)))