Average Error: 44.0 → 2.9
Time: 2.3min
Precision: binary64
\[1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992 \land 1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992 \land 1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\left(\frac{c}{b} \cdot \left(-0.5\right) - \left(0.375 \cdot \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}} + 0.5625 \cdot \frac{a \cdot a}{\frac{{b}^{5}}{{c}^{3}}}\right)\right) - 1.0546875 \cdot \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4}}}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\left(\frac{c}{b} \cdot \left(-0.5\right) - \left(0.375 \cdot \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}} + 0.5625 \cdot \frac{a \cdot a}{\frac{{b}^{5}}{{c}^{3}}}\right)\right) - 1.0546875 \cdot \frac{{a}^{3}}{\frac{{b}^{7}}{{c}^{4}}}
(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) (- 0.5))
   (+
    (* 0.375 (/ (* a (* c c)) (pow b 3.0)))
    (* 0.5625 (/ (* a a) (/ (pow b 5.0) (pow c 3.0))))))
  (* 1.0546875 (/ (pow a 3.0) (/ (pow b 7.0) (pow c 4.0))))))
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 (((c / b) * -0.5) - ((0.375 * ((a * (c * c)) / pow(b, 3.0))) + (0.5625 * ((a * a) / (pow(b, 5.0) / pow(c, 3.0)))))) - (1.0546875 * (pow(a, 3.0) / (pow(b, 7.0) / pow(c, 4.0))));
}

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 44.0

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

  2. Simplified44.0

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

  3. Taylor expanded around inf 2.9

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

  4. Simplified2.9

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

  5. Final simplification2.9

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

Reproduce

herbie shell --seed 2021176 
(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)))