Average Error: 43.4 → 3.1
Time: 9.7s
Precision: binary64
\[1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992 \land 1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992 \land 1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[-\left(5 \cdot \frac{{a}^{3} \cdot {c}^{4}}{{b}^{7}} + \left(\frac{c}{b} + \left(\frac{a \cdot {c}^{2}}{{b}^{3}} + 2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}}\right)\right)\right)\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
-\left(5 \cdot \frac{{a}^{3} \cdot {c}^{4}}{{b}^{7}} + \left(\frac{c}{b} + \left(\frac{a \cdot {c}^{2}}{{b}^{3}} + 2 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}}\right)\right)\right)
(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))
↓
(FPCore (a b c)
 :precision binary64
 (-
  (+
   (* 5.0 (/ (* (pow a 3.0) (pow c 4.0)) (pow b 7.0)))
   (+
    (/ c b)
    (+
     (/ (* a (pow c 2.0)) (pow b 3.0))
     (* 2.0 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 5.0))))))))
double code(double a, double b, double c) {
	return (-b + sqrt((b * b) - ((4.0 * a) * c))) / (2.0 * a);
}
↓
double code(double a, double b, double c) {
	return -((5.0 * ((pow(a, 3.0) * pow(c, 4.0)) / pow(b, 7.0))) + ((c / b) + (((a * pow(c, 2.0)) / pow(b, 3.0)) + (2.0 * ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 5.0))))));
}

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 43.4

    \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]

  2. Simplified43.4

    \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2}}\]

  3. Taylor expanded around inf 3.1

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

  4. Final simplification3.1

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

Reproduce

herbie shell --seed 2021019 
(FPCore (a b c)
  :name "Quadratic roots, 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) (* (* 4.0 a) c)))) (* 2.0 a)))