Average Error: 28.4 → 5.8
Time: 4.6s
Precision: binary64
\[\left(\left(1.0536712127723509 \cdot 10^{-8} < a \land a < 94906265.62425156\right) \land \left(1.0536712127723509 \cdot 10^{-8} < b \land b < 94906265.62425156\right)\right) \land \left(1.0536712127723509 \cdot 10^{-8} < c \land c < 94906265.62425156\right)\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
↓
\[\frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} \cdot -2 - \left(5 \cdot \frac{{c}^{4} \cdot {a}^{3}}{{b}^{7}} + \left(\frac{a \cdot {c}^{2}}{{b}^{3}} + \frac{c}{b}\right)\right) \]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} \cdot -2 - \left(5 \cdot \frac{{c}^{4} \cdot {a}^{3}}{{b}^{7}} + \left(\frac{a \cdot {c}^{2}}{{b}^{3}} + \frac{c}{b}\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
 (-
  (* (/ (* (pow c 3.0) (pow a 2.0)) (pow b 5.0)) -2.0)
  (+
   (* 5.0 (/ (* (pow c 4.0) (pow a 3.0)) (pow b 7.0)))
   (+ (/ (* a (pow c 2.0)) (pow b 3.0)) (/ c b)))))
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 (((pow(c, 3.0) * pow(a, 2.0)) / pow(b, 5.0)) * -2.0) - ((5.0 * ((pow(c, 4.0) * pow(a, 3.0)) / pow(b, 7.0))) + (((a * pow(c, 2.0)) / pow(b, 3.0)) + (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 28.4

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

  2. Taylor expanded in b around inf 5.8

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

  3. Final simplification5.8

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

Reproduce

herbie shell --seed 2022129 
(FPCore (a b c)
  :name "Quadratic roots, narrow range"
  :precision binary64
  :pre (and (and (and (< 1.0536712127723509e-8 a) (< a 94906265.62425156)) (and (< 1.0536712127723509e-8 b) (< b 94906265.62425156))) (and (< 1.0536712127723509e-8 c) (< c 94906265.62425156)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))