Average Error: 43.8 → 0.2
Time: 4.3s
Precision: 64
\[1.11022 \cdot 10^{-16} \lt a \lt 9.0072 \cdot 10^{15} \land 1.11022 \cdot 10^{-16} \lt b \lt 9.0072 \cdot 10^{15} \land 1.11022 \cdot 10^{-16} \lt c \lt 9.0072 \cdot 10^{15}\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{-c}{b + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\frac{-c}{b + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}
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(((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 43.8

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

  3. Applied flip-+43.8

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

  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. Simplified0.2

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

  15. Final simplification0.2

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

Reproduce

herbie shell --seed 2020078 +o rules:numerics
(FPCore (a b c)
  :name "Cubic critical, 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) (* (* 3 a) c)))) (* 3 a)))