Average Error: 44.0 → 0.2
Time: 14.0s
Precision: 64
\[1.11022 \cdot 10^{-16} \lt a \lt 9.0072 \cdot 10^{15} \land 1.11022 \cdot 10^{-16} \lt b \lt 9.0072 \cdot 10^{15} \land 1.11022 \cdot 10^{-16} \lt c \lt 9.0072 \cdot 10^{15}\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{\frac{\left({b}^{2} - {b}^{2}\right) + 4 \cdot \left(a \cdot c\right)}{a}}{\left(-b\right) - \sqrt{\left(\sqrt{\left(4 \cdot a\right) \cdot c} + b\right) \cdot \left(b - \sqrt{\left(4 \cdot a\right) \cdot c}\right)}} \cdot \frac{1}{2}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\frac{\frac{\left({b}^{2} - {b}^{2}\right) + 4 \cdot \left(a \cdot c\right)}{a}}{\left(-b\right) - \sqrt{\left(\sqrt{\left(4 \cdot a\right) \cdot c} + b\right) \cdot \left(b - \sqrt{\left(4 \cdot a\right) \cdot c}\right)}} \cdot \frac{1}{2}
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 (((((pow(b, 2.0) - pow(b, 2.0)) + (4.0 * (a * c))) / a) / (-b - sqrt(((sqrt(((4.0 * a) * c)) + b) * (b - sqrt(((4.0 * a) * c))))))) * (1.0 / 2.0));
}

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

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

  6. Applied add-sqr-sqrt0.4

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

  7. Applied difference-of-squares0.4

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

  8. Simplified0.4

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

  10. Applied *-commutative0.4

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

  11. Applied div-inv0.5

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

  12. Applied times-frac0.4

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

  14. Applied div-inv0.4

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

  15. Applied associate-*r*0.4

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

  16. Simplified0.2

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

  17. Final simplification0.2

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

Reproduce

herbie shell --seed 2020066 
(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)))