Average Error: 43.7 → 3.0
Time: 7.7s
Precision: binary64
\[1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992 \land 1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992 \land 1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
↓
\[-\mathsf{fma}\left(0.5625, \frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{5}}, \mathsf{fma}\left(1.0546875, \frac{{a}^{3} \cdot {c}^{4}}{{b}^{7}}, \mathsf{fma}\left(\frac{c}{b}, 0.5, 0.375 \cdot \frac{c \cdot \left(a \cdot c\right)}{{b}^{3}}\right)\right)\right) \]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
-\mathsf{fma}\left(0.5625, \frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{5}}, \mathsf{fma}\left(1.0546875, \frac{{a}^{3} \cdot {c}^{4}}{{b}^{7}}, \mathsf{fma}\left(\frac{c}{b}, 0.5, 0.375 \cdot \frac{c \cdot \left(a \cdot c\right)}{{b}^{3}}\right)\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
 (-
  (fma
   0.5625
   (/ (* (* a a) (pow c 3.0)) (pow b 5.0))
   (fma
    1.0546875
    (/ (* (pow a 3.0) (pow c 4.0)) (pow b 7.0))
    (fma (/ c b) 0.5 (* 0.375 (/ (* c (* a c)) (pow b 3.0))))))))
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 -fma(0.5625, (((a * a) * pow(c, 3.0)) / pow(b, 5.0)), fma(1.0546875, ((pow(a, 3.0) * pow(c, 4.0)) / pow(b, 7.0)), fma((c / b), 0.5, (0.375 * ((c * (a * c)) / pow(b, 3.0))))));
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Initial program 43.7

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

  2. Taylor expanded around inf 3.0

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

  3. Simplified3.0

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

  4. Final simplification3.0

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

Reproduce

herbie shell --seed 2021211 
(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)))