Average Error: 44.0 → 0.4
Time: 40.7s
Precision: 64
\[1.11022 \cdot 10^{-16} \lt a \lt 9.0072 \cdot 10^{15} \land 1.11022 \cdot 10^{-16} \lt b \lt 9.0072 \cdot 10^{15} \land 1.11022 \cdot 10^{-16} \lt c \lt 9.0072 \cdot 10^{15}\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{\frac{\left({b}^{2} - {b}^{2}\right) + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{\left(\left(\sqrt{4} \cdot \sqrt{a}\right) \cdot \sqrt{c} + b\right) \cdot \left(b - \left(\sqrt{4} \cdot \sqrt{a}\right) \cdot \sqrt{c}\right)}}}{2 \cdot a}\]
\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)}{\left(-b\right) - \sqrt{\left(\left(\sqrt{4} \cdot \sqrt{a}\right) \cdot \sqrt{c} + b\right) \cdot \left(b - \left(\sqrt{4} \cdot \sqrt{a}\right) \cdot \sqrt{c}\right)}}}{2 \cdot a}
double code(double a, double b, double c) {
	return ((double) (((double) (((double) -(b)) + ((double) sqrt(((double) (((double) (b * b)) - ((double) (((double) (4.0 * a)) * c)))))))) / ((double) (2.0 * a))));
}
↓
double code(double a, double b, double c) {
	return ((double) (((double) (((double) (((double) (((double) pow(b, 2.0)) - ((double) pow(b, 2.0)))) + ((double) (4.0 * ((double) (a * c)))))) / ((double) (((double) -(b)) - ((double) sqrt(((double) (((double) (((double) (((double) (((double) sqrt(4.0)) * ((double) sqrt(a)))) * ((double) sqrt(c)))) + b)) * ((double) (b - ((double) (((double) (((double) sqrt(4.0)) * ((double) sqrt(a)))) * ((double) sqrt(c)))))))))))))) / ((double) (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 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

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

  7. Applied add-sqr-sqrt0.4

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

  8. Applied add-sqr-sqrt0.4

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

  9. Applied unswap-sqr0.4

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

  10. Applied unswap-sqr0.4

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

  11. Applied difference-of-squares0.4

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

  12. Simplified0.4

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

  13. Final simplification0.4

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

Reproduce

herbie shell --seed 2020114 
(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)))