Average Error: 28.9 → 0.3
Time: 4.1s
Precision: binary64
\[1.0536712127723509 \cdot 10^{-08} < a \land a < 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} < b \land b < 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} < c \land c < 94906265.62425156\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{-c}{b + \sqrt{\frac{{b}^{4} - 3 \cdot \left(3 \cdot \left(a \cdot \left(a \cdot \left(c \cdot c\right)\right)\right)\right)}{3 \cdot \left(c \cdot a\right) + b \cdot b}}}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\frac{-c}{b + \sqrt{\frac{{b}^{4} - 3 \cdot \left(3 \cdot \left(a \cdot \left(a \cdot \left(c \cdot c\right)\right)\right)\right)}{3 \cdot \left(c \cdot a\right) + b \cdot b}}}
(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
    (/
     (- (pow b 4.0) (* 3.0 (* 3.0 (* a (* a (* c c))))))
     (+ (* 3.0 (* c a)) (* b b)))))))
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) pow(b, 4.0)) - ((double) (3.0 * ((double) (3.0 * ((double) (a * ((double) (a * ((double) (c * c)))))))))))) / ((double) (((double) (3.0 * ((double) (c * a)))) + ((double) (b * b))))))))));
}

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: 28.9 bits

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

  2. SimplifiedError: 28.9 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: 28.9 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: 0.6 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: 0.6 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: 0.6 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: 0.6 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: 0.6 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: 0.6 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: 0.3 bits

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

  14. Applied flip--Error: 0.3 bits

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

  15. SimplifiedError: 0.3 bits

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

  16. SimplifiedError: 0.3 bits

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

  17. Final simplificationError: 0.3 bits

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

Reproduce

herbie shell --seed 2020203 
(FPCore (a b c)
  :name "Cubic critical, narrow range"
  :precision binary64
  :pre (and (< 1.0536712127723509e-08 a 94906265.62425156) (< 1.0536712127723509e-08 b 94906265.62425156) (< 1.0536712127723509e-08 c 94906265.62425156))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))