Average Error: 52.5 → 1.4
Time: 6.2s
Precision: binary64
\[4.930380657631324 \cdot 10^{-32} < a \land a < 2.028240960365167 \cdot 10^{+31} \land 4.930380657631324 \cdot 10^{-32} < b \land b < 2.028240960365167 \cdot 10^{+31} \land 4.930380657631324 \cdot 10^{-32} < c \land c < 2.028240960365167 \cdot 10^{+31}\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
↓
\[\frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} \cdot -0.5625 - \left(1.0546875 \cdot \frac{{c}^{4} \cdot {a}^{3}}{{b}^{7}} + \left(0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + 0.5 \cdot \frac{c}{b}\right)\right) \]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} \cdot -0.5625 - \left(1.0546875 \cdot \frac{{c}^{4} \cdot {a}^{3}}{{b}^{7}} + \left(0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + 0.5 \cdot \frac{c}{b}\right)\right)

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))
↓
(FPCore (a b c)
 :precision binary64
 (-
  (* (/ (* (pow c 3.0) (pow a 2.0)) (pow b 5.0)) -0.5625)
  (+
   (* 1.0546875 (/ (* (pow c 4.0) (pow a 3.0)) (pow b 7.0)))
   (+ (* 0.375 (/ (* a (pow c 2.0)) (pow b 3.0))) (* 0.5 (/ c b))))))
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 (((pow(c, 3.0) * pow(a, 2.0)) / pow(b, 5.0)) * -0.5625) - ((1.0546875 * ((pow(c, 4.0) * pow(a, 3.0)) / pow(b, 7.0))) + ((0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0))) + (0.5 * (c / 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 52.5

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

  2. Taylor expanded in b around inf 1.4

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

  3. Final simplification1.4

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

Reproduce

herbie shell --seed 2021224 
(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)))