Average Error: 52.6 → 0.1
Time: 4.1s
Precision: binary64
\[4.930380657631324 \cdot 10^{-32} < a \land a < 2.028240960365167 \cdot 10^{+31} \land 4.930380657631324 \cdot 10^{-32} < b \land b < 2.028240960365167 \cdot 10^{+31} \land 4.930380657631324 \cdot 10^{-32} < c \land c < 2.028240960365167 \cdot 10^{+31}\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{-c}{b + \sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)}}\]
\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 - 3 \cdot \left(c \cdot a\right)}}
(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))
↓
(FPCore (a b c)
 :precision binary64
 (/ (- c) (+ b (sqrt (- (* b b) (* 3.0 (* c a)))))))
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 (((double) -(c)) / ((double) (b + ((double) sqrt(((double) (((double) (b * b)) - ((double) (3.0 * ((double) (c * 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 Error: 52.6 bits

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

  2. SimplifiedError: 52.6 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: 52.6 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 distribute-rgt-neg-outError: 0.5 bits

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

  9. Applied distribute-rgt-neg-outError: 0.5 bits

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

  10. Applied distribute-frac-negError: 0.5 bits

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

  11. Applied distribute-frac-negError: 0.5 bits

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

  12. SimplifiedError: 0.1 bits

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

  13. Final simplificationError: 0.1 bits

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

Reproduce

herbie shell --seed 2020204 
(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)))