Average Error: 34.3 → 8.9
Time: 15.0s
Precision: binary64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
↓
\[\begin{array}{l} \mathbf{if}\;b \leq -1.4427673897213551 \cdot 10^{+153}:\\ \;\;\;\;\frac{\left(-b\right) - b}{3 \cdot a}\\ \mathbf{elif}\;b \leq 1.2177854566716303 \cdot 10^{-166}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{elif}\;b \leq 124046118059855380:\\ \;\;\;\;\frac{\left(a \cdot \left(c \cdot -3\right)\right) \cdot \frac{0.3333333333333333}{a}}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]
\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 -1.4427673897213551 \cdot 10^{+153}:\\
\;\;\;\;\frac{\left(-b\right) - b}{3 \cdot a}\\

\mathbf{elif}\;b \leq 1.2177854566716303 \cdot 10^{-166}:\\
\;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\

\mathbf{elif}\;b \leq 124046118059855380:\\
\;\;\;\;\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 -1.4427673897213551e+153)
   (/ (- (- b) b) (* 3.0 a))
   (if (<= b 1.2177854566716303e-166)
     (/ (- (sqrt (- (* b b) (* (* 3.0 a) c))) b) (* 3.0 a))
     (if (<= b 124046118059855380.0)
       (/
        (* (* 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 <= -1.4427673897213551e+153) {
		tmp = (-b - b) / (3.0 * a);
	} else if (b <= 1.2177854566716303e-166) {
		tmp = (sqrt((b * b) - ((3.0 * a) * c)) - b) / (3.0 * a);
	} else if (b <= 124046118059855380.0) {
		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 < -1.44276738972135514e153

    1. Initial program 63.1

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

    2. Taylor expanded in b around -inf 3.4

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

    if -1.44276738972135514e153 < b < 1.21778545667163e-166

    1. Initial program 10.3

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

    2. Applied *-commutative_binary6410.3

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

    if 1.21778545667163e-166 < b < 124046118059855376

    1. Initial program 35.4

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

    2. Simplified35.5

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

    3. Applied flip--_binary6435.5

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

    4. Applied associate-*l/_binary6435.5

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

    5. Simplified17.4

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

    if 124046118059855376 < 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 5.1

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

  4. Final simplification8.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.4427673897213551 \cdot 10^{+153}:\\ \;\;\;\;\frac{\left(-b\right) - b}{3 \cdot a}\\ \mathbf{elif}\;b \leq 1.2177854566716303 \cdot 10^{-166}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}\\ \mathbf{elif}\;b \leq 124046118059855380:\\ \;\;\;\;\frac{\left(a \cdot \left(c \cdot -3\right)\right) \cdot \frac{0.3333333333333333}{a}}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Reproduce

herbie shell --seed 2021344 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))