Average Error: 52.6 → 0.1
Time: 4.3s
Precision: 64
\[4.93038 \cdot 10^{-32} \lt a \lt 2.02824 \cdot 10^{31} \land 4.93038 \cdot 10^{-32} \lt b \lt 2.02824 \cdot 10^{31} \land 4.93038 \cdot 10^{-32} \lt c \lt 2.02824 \cdot 10^{31}\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{-c}{b + \sqrt{\frac{\mathsf{fma}\left(-\left(3 \cdot a\right) \cdot \left(3 \cdot a\right), c \cdot c, {b}^{4}\right)}{\mathsf{fma}\left(b, b, \left(3 \cdot a\right) \cdot c\right)}}}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\frac{-c}{b + \sqrt{\frac{\mathsf{fma}\left(-\left(3 \cdot a\right) \cdot \left(3 \cdot a\right), c \cdot c, {b}^{4}\right)}{\mathsf{fma}\left(b, b, \left(3 \cdot a\right) \cdot c\right)}}}
double code(double a, double b, double c) {
	return ((-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a));
}
↓
double code(double a, double b, double c) {
	return (-c / (b + sqrt((fma(-((3.0 * a) * (3.0 * a)), (c * c), pow(b, 4.0)) / fma(b, b, ((3.0 * a) * c))))));
}

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 52.6

    \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
  2. Using strategy rm

  3. Applied flip-+52.6

    \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}}{3 \cdot a}\]

  4. Simplified0.5

    \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(3 \cdot c, a, 0\right)}}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\]
  5. Using strategy rm

  6. Applied frac-2neg0.5

    \[\leadsto \frac{\color{blue}{\frac{-\mathsf{fma}\left(3 \cdot c, a, 0\right)}{-\left(\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}}}{3 \cdot a}\]

  7. Simplified0.4

    \[\leadsto \frac{\frac{\color{blue}{-\left(3 \cdot a\right) \cdot c}}{-\left(\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}}{3 \cdot a}\]

  8. Simplified0.4

    \[\leadsto \frac{\frac{-\left(3 \cdot a\right) \cdot c}{\color{blue}{b + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}}{3 \cdot a}\]
  9. Using strategy rm

  10. Applied neg-sub00.4

    \[\leadsto \frac{\frac{\color{blue}{0 - \left(3 \cdot a\right) \cdot c}}{b + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\]

  11. Applied div-sub0.4

    \[\leadsto \frac{\color{blue}{\frac{0}{b + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}} - \frac{\left(3 \cdot a\right) \cdot c}{b + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}}{3 \cdot a}\]

  12. Applied div-sub0.4

    \[\leadsto \color{blue}{\frac{\frac{0}{b + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a} - \frac{\frac{\left(3 \cdot a\right) \cdot c}{b + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}}\]

  13. Simplified0.4

    \[\leadsto \color{blue}{0} - \frac{\frac{\left(3 \cdot a\right) \cdot c}{b + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\]

  14. Simplified0.1

    \[\leadsto 0 - \color{blue}{\frac{1 \cdot c}{b + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}\]
  15. Using strategy rm

  16. Applied flip--0.1

    \[\leadsto 0 - \frac{1 \cdot c}{b + \sqrt{\color{blue}{\frac{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)}{b \cdot b + \left(3 \cdot a\right) \cdot c}}}}\]

  17. Simplified0.1

    \[\leadsto 0 - \frac{1 \cdot c}{b + \sqrt{\frac{\color{blue}{\mathsf{fma}\left(-\left(3 \cdot a\right) \cdot \left(3 \cdot a\right), c \cdot c, {b}^{4}\right)}}{b \cdot b + \left(3 \cdot a\right) \cdot c}}}\]

  18. Simplified0.1

    \[\leadsto 0 - \frac{1 \cdot c}{b + \sqrt{\frac{\mathsf{fma}\left(-\left(3 \cdot a\right) \cdot \left(3 \cdot a\right), c \cdot c, {b}^{4}\right)}{\color{blue}{\mathsf{fma}\left(b, b, \left(3 \cdot a\right) \cdot c\right)}}}}\]

  19. Final simplification0.1

    \[\leadsto \frac{-c}{b + \sqrt{\frac{\mathsf{fma}\left(-\left(3 \cdot a\right) \cdot \left(3 \cdot a\right), c \cdot c, {b}^{4}\right)}{\mathsf{fma}\left(b, b, \left(3 \cdot a\right) \cdot c\right)}}}\]

Reproduce

herbie shell --seed 2020071 +o rules:numerics
(FPCore (a b c)
  :name "Cubic critical, wide range"
  :precision binary64
  :pre (and (< 4.9303800000000003e-32 a 2.02824e+31) (< 4.9303800000000003e-32 b 2.02824e+31) (< 4.9303800000000003e-32 c 2.02824e+31))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))