Average Error: 52.9 → 0.4
Time: 4.3s
Precision: binary64
\[4.930380657631324 \cdot 10^{-32} < a \land a < 2.028240960365167 \cdot 10^{+31} \land 4.930380657631324 \cdot 10^{-32} < b \land b < 2.028240960365167 \cdot 10^{+31} \land 4.930380657631324 \cdot 10^{-32} < c \land c < 2.028240960365167 \cdot 10^{+31}\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{1}{\left(b + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{-c}}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\frac{1}{\left(b + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{-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) (((double) (b + ((double) sqrt(((double) (((double) (b * b)) - ((double) (3.0 * ((double) (a * c)))))))))) * (1.0 / ((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 52.9

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

  2. Simplified52.9

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

  4. Applied flip--52.9

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

  5. Simplified0.5

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

  6. Simplified0.5

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

  8. Applied clear-num0.6

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

  9. Simplified0.4

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

  10. Final simplification0.4

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

Reproduce

herbie shell --seed 2020196 
(FPCore (a b c)
  :name "Cubic critical, wide range"
  :precision binary64
  :pre (and (< 4.930380657631324e-32 a 2.028240960365167e+31) (< 4.930380657631324e-32 b 2.028240960365167e+31) (< 4.930380657631324e-32 c 2.028240960365167e+31))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))