Average Error: 28.6 → 5.2
Time: 8.8s
Precision: binary64
\[\left(\left(1.0536712127723509 \cdot 10^{-8} < a \land a < 94906265.62425156\right) \land \left(1.0536712127723509 \cdot 10^{-8} < b \land b < 94906265.62425156\right)\right) \land \left(1.0536712127723509 \cdot 10^{-8} < c \land c < 94906265.62425156\right)\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
↓
\[\begin{array}{l} \mathbf{if}\;b \leq 0.4982759683134188:\\ \;\;\;\;\begin{array}{l} t_0 := \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\\ \frac{b \cdot b - t_0 \cdot t_0}{\left(a \cdot -3\right) \cdot \left(b + t_0\right)} \end{array}\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(0.5625, \frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{5}}, \mathsf{fma}\left(1.0546875, \frac{{a}^{3} \cdot {c}^{4}}{{b}^{7}}, \mathsf{fma}\left(0.5, \frac{c}{b}, 0.375 \cdot \frac{c \cdot \left(a \cdot c\right)}{{b}^{3}}\right)\right)\right)\\ \end{array} \]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \leq 0.4982759683134188:\\
\;\;\;\;\begin{array}{l}
t_0 := \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\\
\frac{b \cdot b - t_0 \cdot t_0}{\left(a \cdot -3\right) \cdot \left(b + t_0\right)}
\end{array}\\

\mathbf{else}:\\
\;\;\;\;-\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(0.5, \frac{c}{b}, 0.375 \cdot \frac{c \cdot \left(a \cdot c\right)}{{b}^{3}}\right)\right)\right)\\


\end{array}

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))
↓
(FPCore (a b c)
 :precision binary64
 (if (<= b 0.4982759683134188)
   (let* ((t_0 (sqrt (- (* b b) (* (* 3.0 a) c)))))
     (/ (- (* b b) (* t_0 t_0)) (* (* a -3.0) (+ b t_0))))
   (-
    (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 0.5 (/ c b) (* 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) {
	double tmp;
	if (b <= 0.4982759683134188) {
		double t_0_1 = sqrt((b * b) - ((3.0 * a) * c));
		tmp = ((b * b) - (t_0_1 * t_0_1)) / ((a * -3.0) * (b + t_0_1));
	} else {
		tmp = -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(0.5, (c / b), (0.375 * ((c * (a * c)) / pow(b, 3.0))))));
	}
	return tmp;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 2 regimes

  2. if b < 0.4982759683134188

    1. Initial program 11.1

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

    2. Applied flip-+_binary6411.1

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

    3. Applied associate-/l/_binary6411.1

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

    if 0.4982759683134188 < b

    1. Initial program 31.5

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

    2. Taylor expanded in b around inf 4.3

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

    3. Simplified4.3

      \[\leadsto \color{blue}{-\mathsf{fma}\left(0.5625, \frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{5}}, \mathsf{fma}\left(1.0546875, \frac{{a}^{3} \cdot {c}^{4}}{{b}^{7}}, \mathsf{fma}\left(0.5, \frac{c}{b}, 0.375 \cdot \frac{c \cdot \left(c \cdot a\right)}{{b}^{3}}\right)\right)\right)} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification5.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.4982759683134188:\\ \;\;\;\;\frac{b \cdot b - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\left(a \cdot -3\right) \cdot \left(b + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(0.5625, \frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{5}}, \mathsf{fma}\left(1.0546875, \frac{{a}^{3} \cdot {c}^{4}}{{b}^{7}}, \mathsf{fma}\left(0.5, \frac{c}{b}, 0.375 \cdot \frac{c \cdot \left(a \cdot c\right)}{{b}^{3}}\right)\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2021357 
(FPCore (a b c)
  :name "Cubic critical, 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) (* (* 3.0 a) c)))) (* 3.0 a)))