Average Error: 43.9 → 3.0
Time: 6.1s
Precision: binary64
\[1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992 \land 1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992 \land 1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\left(-0.5 \cdot \frac{c}{b} - \left(0.375 \cdot \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}} + 0.5625 \cdot \frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{5}}\right)\right) - 1.0546875 \cdot \frac{{a}^{3} \cdot {c}^{4}}{{b}^{7}}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\left(-0.5 \cdot \frac{c}{b} - \left(0.375 \cdot \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}} + 0.5625 \cdot \frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{5}}\right)\right) - 1.0546875 \cdot \frac{{a}^{3} \cdot {c}^{4}}{{b}^{7}}
(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))
↓
(FPCore (a b c)
 :precision binary64
 (-
  (-
   (* -0.5 (/ c b))
   (+
    (* 0.375 (/ (* a (* c c)) (pow b 3.0)))
    (* 0.5625 (/ (* (* a a) (pow c 3.0)) (pow b 5.0)))))
  (* 1.0546875 (/ (* (pow a 3.0) (pow c 4.0)) (pow b 7.0)))))
double code(double a, double b, double c) {
	return (-b + sqrt((b * b) - ((3.0 * a) * c))) / (3.0 * a);
}
↓
double code(double a, double b, double c) {
	return ((-0.5 * (c / b)) - ((0.375 * ((a * (c * c)) / pow(b, 3.0))) + (0.5625 * (((a * a) * pow(c, 3.0)) / pow(b, 5.0))))) - (1.0546875 * ((pow(a, 3.0) * pow(c, 4.0)) / pow(b, 7.0)));
}

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 43.9

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

  2. Simplified43.9

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

  3. Taylor expanded around inf 3.0

    \[\leadsto \color{blue}{-\left(1.0546875 \cdot \frac{{a}^{3} \cdot {c}^{4}}{{b}^{7}} + \left(0.5 \cdot \frac{c}{b} + \left(0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + 0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}}\right)\right)\right)}\]

  4. Simplified3.0

    \[\leadsto \color{blue}{\left(-0.5 \cdot \frac{c}{b} - \left(0.375 \cdot \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}} + 0.5625 \cdot \frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{5}}\right)\right) - 1.0546875 \cdot \frac{{a}^{3} \cdot {c}^{4}}{{b}^{7}}}\]

  5. Final simplification3.0

    \[\leadsto \left(-0.5 \cdot \frac{c}{b} - \left(0.375 \cdot \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}} + 0.5625 \cdot \frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{5}}\right)\right) - 1.0546875 \cdot \frac{{a}^{3} \cdot {c}^{4}}{{b}^{7}}\]

Reproduce

herbie shell --seed 2021173 
(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)))