Average Error: 43.9 → 2.9
Time: 5.9s
Precision: binary64
\[\left(\left(1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992\right) \land \left(1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992\right)\right) \land \left(1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\right)\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
↓
\[-2 \cdot \frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{5}} - \mathsf{fma}\left(5, \frac{{a}^{3} \cdot {c}^{4}}{{b}^{7}}, \mathsf{fma}\left(\frac{c \cdot c}{{b}^{3}}, a, \frac{c}{b}\right)\right) \]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
-2 \cdot \frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{5}} - \mathsf{fma}\left(5, \frac{{a}^{3} \cdot {c}^{4}}{{b}^{7}}, \mathsf{fma}\left(\frac{c \cdot c}{{b}^{3}}, a, \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
 (-
  (* -2.0 (/ (* (* a a) (pow c 3.0)) (pow b 5.0)))
  (fma
   5.0
   (/ (* (pow a 3.0) (pow c 4.0)) (pow b 7.0))
   (fma (/ (* c c) (pow b 3.0)) a (/ 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 (-2.0 * (((a * a) * pow(c, 3.0)) / pow(b, 5.0))) - fma(5.0, ((pow(a, 3.0) * pow(c, 4.0)) / pow(b, 7.0)), fma(((c * c) / pow(b, 3.0)), a, (c / b)));
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Initial program 43.9

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

  2. Taylor expanded in b around inf 2.9

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

  3. Simplified2.9

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

  4. Final simplification2.9

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

Reproduce

herbie shell --seed 2022125 
(FPCore (a b c)
  :name "Quadratic roots, medium range"
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a) (< a 9007199254740992.0)) (and (< 1.1102230246251565e-16 b) (< b 9007199254740992.0))) (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))