Average Error: 43.3 → 0.4
Time: 6.9s
Precision: 64
\[1.11022 \cdot 10^{-16} \lt a \lt 9.0072 \cdot 10^{15} \land 1.11022 \cdot 10^{-16} \lt b \lt 9.0072 \cdot 10^{15} \land 1.11022 \cdot 10^{-16} \lt c \lt 9.0072 \cdot 10^{15}\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{\left({b}^{2} + 4 \cdot \left(a \cdot c\right)\right) \cdot \frac{{b}^{2} - 4 \cdot \left(a \cdot c\right)}{b \cdot b + \left(4 \cdot a\right) \cdot c}}}}{2 \cdot a}\]
\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{\left({b}^{2} + 4 \cdot \left(a \cdot c\right)\right) \cdot \frac{{b}^{2} - 4 \cdot \left(a \cdot c\right)}{b \cdot b + \left(4 \cdot a\right) \cdot c}}}}{2 \cdot a}
double code(double a, double b, double c) {
	return ((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a));
}
↓
double code(double a, double b, double c) {
	return (((0.0 + (4.0 * (a * c))) / (-b - sqrt(((pow(b, 2.0) + (4.0 * (a * c))) * ((pow(b, 2.0) - (4.0 * (a * c))) / ((b * b) + ((4.0 * a) * c))))))) / (2.0 * a));
}

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 43.3

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

  3. Applied flip-+43.3

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

  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 *-un-lft-identity0.4

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

  10. Applied sqr-pow0.4

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

  11. Applied difference-of-squares0.4

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

  12. Applied times-frac0.4

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

  13. Simplified0.4

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

  14. Simplified0.4

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

  15. Final simplification0.4

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

Reproduce

herbie shell --seed 2020058 
(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)))