Average Error: 52.4 → 0.1
Time: 4.7s
Precision: binary64
\[4.930380657631324 \cdot 10^{-32} < a \land a < 2.028240960365167 \cdot 10^{+31} \land 4.930380657631324 \cdot 10^{-32} < b \land b < 2.028240960365167 \cdot 10^{+31} \land 4.930380657631324 \cdot 10^{-32} < c \land c < 2.028240960365167 \cdot 10^{+31}\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{c}{\left(-b\right) - \sqrt{\frac{{b}^{4} - 3 \cdot \left(3 \cdot \left(a \cdot \left(a \cdot \left(c \cdot c\right)\right)\right)\right)}{3 \cdot \left(c \cdot a\right) + b \cdot b}}}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\frac{c}{\left(-b\right) - \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 (c / ((double) (((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: 52.4 bits

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

  3. Applied flip-+Error: 52.4 bits

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

  4. SimplifiedError: 0.5 bits

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

  5. SimplifiedError: 0.5 bits

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

  7. Applied *-un-lft-identityError: 0.5 bits

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

  8. Applied times-fracError: 0.5 bits

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

  9. Applied times-fracError: 0.4 bits

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

  10. SimplifiedError: 0.4 bits

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

  11. SimplifiedError: 0.1 bits

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

  13. Applied flip--Error: 0.1 bits

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

  14. SimplifiedError: 0.1 bits

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

  15. SimplifiedError: 0.1 bits

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

  16. Final simplificationError: 0.1 bits

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

Reproduce

herbie shell --seed 2020203 
(FPCore (a b c)
  :name "Cubic critical, 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)))