Average Error: 44.2 → 0.3
Time: 4.0s
Precision: binary64
\[1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992 \land 1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992 \land 1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{1}{\frac{b + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{-c}}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\frac{1}{\frac{b + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{-c}}
double code(double a, double b, double c) {
	return (((double) (((double) -(b)) + ((double) sqrt(((double) (((double) (b * b)) - ((double) (((double) (3.0 * a)) * c)))))))) / ((double) (3.0 * a)));
}
↓
double code(double a, double b, double c) {
	return (1.0 / (((double) (b + ((double) sqrt(((double) (((double) (b * b)) - ((double) (3.0 * ((double) (a * c)))))))))) / ((double) -(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 Error: 44.2 bits

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

  2. SimplifiedError: 44.2 bits

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

  4. Applied flip--Error: 44.3 bits

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

  5. SimplifiedError: 0.5 bits

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

  6. SimplifiedError: 0.5 bits

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

  8. Applied associate-/r*Error: 0.5 bits

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

  9. SimplifiedError: 0.4 bits

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

  11. Applied clear-numError: 0.5 bits

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

  12. SimplifiedError: 0.3 bits

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

  13. Final simplificationError: 0.3 bits

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

Reproduce

herbie shell --seed 2020200 
(FPCore (a b c)
  :name "Cubic critical, medium range"
  :precision binary64
  :pre (and (< 1.1102230246251565e-16 a 9007199254740992.0) (< 1.1102230246251565e-16 b 9007199254740992.0) (< 1.1102230246251565e-16 c 9007199254740992.0))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))