Average Error: 34.3 → 9.0
Time: 13.6s
Precision: binary64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
↓
\[\begin{array}{l} \mathbf{if}\;b \leq -3.9352038855422006 \cdot 10^{+98}:\\ \;\;\;\;\frac{\left(-b\right) - b}{3 \cdot a}\\ \mathbf{elif}\;b \leq 2.543157513544152 \cdot 10^{-126}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{elif}\;b \leq 1645583.6515607869:\\ \;\;\;\;\frac{\left(a \cdot \left(c \cdot -3\right)\right) \cdot \frac{0.3333333333333333}{a}}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]
\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 -3.9352038855422006 \cdot 10^{+98}:\\
\;\;\;\;\frac{\left(-b\right) - b}{3 \cdot a}\\

\mathbf{elif}\;b \leq 2.543157513544152 \cdot 10^{-126}:\\
\;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\

\mathbf{elif}\;b \leq 1645583.6515607869:\\
\;\;\;\;\frac{\left(a \cdot \left(c \cdot -3\right)\right) \cdot \frac{0.3333333333333333}{a}}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}\\

\mathbf{else}:\\
\;\;\;\;-0.5 \cdot \frac{c}{b}\\


\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 -3.9352038855422006e+98)
   (/ (- (- b) b) (* 3.0 a))
   (if (<= b 2.543157513544152e-126)
     (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a))
     (if (<= b 1645583.6515607869)
       (/
        (* (* a (* c -3.0)) (/ 0.3333333333333333 a))
        (+ b (sqrt (fma a (* c -3.0) (* b b)))))
       (* -0.5 (/ c b))))))
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 <= -3.9352038855422006e+98) {
		tmp = (-b - b) / (3.0 * a);
	} else if (b <= 2.543157513544152e-126) {
		tmp = (sqrt((b * b) - ((3.0 * a) * c)) - b) / (3.0 * a);
	} else if (b <= 1645583.6515607869) {
		tmp = ((a * (c * -3.0)) * (0.3333333333333333 / a)) / (b + sqrt(fma(a, (c * -3.0), (b * b))));
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 4 regimes

  2. if b < -3.9352038855422006e98

    1. Initial program 46.6

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

    2. Taylor expanded in b around -inf 3.6

      \[\leadsto \frac{\left(-b\right) + \color{blue}{-1 \cdot b}}{3 \cdot a} \]

    3. Simplified3.6

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

    if -3.9352038855422006e98 < b < 2.543157513544152e-126

    1. Initial program 11.9

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

    2. Applied *-un-lft-identity_binary6411.9

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{1 \cdot \left(\left(3 \cdot a\right) \cdot c\right)}}}{3 \cdot a} \]

    if 2.543157513544152e-126 < b < 1645583.65156078688

    1. Initial program 37.2

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

    2. Simplified37.2

      \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)} - b\right) \cdot \frac{0.3333333333333333}{a}} \]

    3. Applied flip--_binary6437.2

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)} - b \cdot b}{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)} + b}} \cdot \frac{0.3333333333333333}{a} \]

    4. Applied associate-*l/_binary6437.2

      \[\leadsto \color{blue}{\frac{\left(\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)} - b \cdot b\right) \cdot \frac{0.3333333333333333}{a}}{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)} + b}} \]

    5. Simplified16.5

      \[\leadsto \frac{\color{blue}{\left(a \cdot \left(c \cdot -3\right) + 0\right) \cdot \frac{0.3333333333333333}{a}}}{\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)} + b} \]

    if 1645583.65156078688 < b

    1. Initial program 56.1

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

    2. Simplified56.1

      \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)} - b\right) \cdot \frac{0.3333333333333333}{a}} \]

    3. Taylor expanded in a around 0 5.5

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
  3. Recombined 4 regimes into one program.

  4. Final simplification9.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.9352038855422006 \cdot 10^{+98}:\\ \;\;\;\;\frac{\left(-b\right) - b}{3 \cdot a}\\ \mathbf{elif}\;b \leq 2.543157513544152 \cdot 10^{-126}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{elif}\;b \leq 1645583.6515607869:\\ \;\;\;\;\frac{\left(a \cdot \left(c \cdot -3\right)\right) \cdot \frac{0.3333333333333333}{a}}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Reproduce

herbie shell --seed 2022125 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))