Average Error: 34.2 → 12.2
Time: 5.0s
Precision: binary64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \leq -2.846912681280906 \cdot 10^{+123}:\\ \;\;\;\;\frac{1.5 \cdot \left(c \cdot \frac{a}{b}\right) + b \cdot -2}{a \cdot 3}\\ \mathbf{elif}\;b \leq 2.0737225518150114 \cdot 10^{-113}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 1.2940411791043692 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{3 \cdot \left(c \cdot a\right)}{\left(-b\right) - \sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)}}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c \cdot \frac{a}{b}\right) \cdot -1.5}{a \cdot 3}\\ \end{array}\]
\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 -2.846912681280906 \cdot 10^{+123}:\\
\;\;\;\;\frac{1.5 \cdot \left(c \cdot \frac{a}{b}\right) + b \cdot -2}{a \cdot 3}\\

\mathbf{elif}\;b \leq 2.0737225518150114 \cdot 10^{-113}:\\
\;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\

\mathbf{elif}\;b \leq 1.2940411791043692 \cdot 10^{+154}:\\
\;\;\;\;\frac{\frac{3 \cdot \left(c \cdot a\right)}{\left(-b\right) - \sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)}}}{a \cdot 3}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(c \cdot \frac{a}{b}\right) \cdot -1.5}{a \cdot 3}\\

\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 -2.846912681280906e+123)
   (/ (+ (* 1.5 (* c (/ a b))) (* b -2.0)) (* a 3.0))
   (if (<= b 2.0737225518150114e-113)
     (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0))
     (if (<= b 1.2940411791043692e+154)
       (/
        (/ (* 3.0 (* c a)) (- (- b) (sqrt (- (* b b) (* 3.0 (* c a))))))
        (* a 3.0))
       (/ (* (* c (/ a b)) -1.5) (* a 3.0))))))
double code(double a, double b, double c) {
	return (((double) (((double) -(b)) + ((double) sqrt(((double) (((double) (b * b)) - ((double) (((double) (3.0 * a)) * c)))))))) / ((double) (3.0 * a)));
}
↓
double code(double a, double b, double c) {
	double tmp;
	if ((b <= -2.846912681280906e+123)) {
		tmp = (((double) (((double) (1.5 * ((double) (c * (a / b))))) + ((double) (b * -2.0)))) / ((double) (a * 3.0)));
	} else {
		double tmp_1;
		if ((b <= 2.0737225518150114e-113)) {
			tmp_1 = (((double) (((double) sqrt(((double) (((double) (b * b)) - ((double) (c * ((double) (a * 3.0)))))))) - b)) / ((double) (a * 3.0)));
		} else {
			double tmp_2;
			if ((b <= 1.2940411791043692e+154)) {
				tmp_2 = ((((double) (3.0 * ((double) (c * a)))) / ((double) (((double) -(b)) - ((double) sqrt(((double) (((double) (b * b)) - ((double) (3.0 * ((double) (c * a))))))))))) / ((double) (a * 3.0)));
			} else {
				tmp_2 = (((double) (((double) (c * (a / b))) * -1.5)) / ((double) (a * 3.0)));
			}
			tmp_1 = tmp_2;
		}
		tmp = tmp_1;
	}
	return tmp;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 4 regimes

  2. if b < -2.84691268128090614e123

    1. Initial program Error: 53.0 bits

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

    2. Taylor expanded around -inf Error: 10.0 bits

      \[\leadsto \frac{\color{blue}{1.5 \cdot \frac{a \cdot c}{b} - 2 \cdot b}}{3 \cdot a}\]

    3. SimplifiedError: 3.1 bits

      \[\leadsto \frac{\color{blue}{1.5 \cdot \left(c \cdot \frac{a}{b}\right) + b \cdot -2}}{3 \cdot a}\]

    if -2.84691268128090614e123 < b < 2.0737225518150114e-113

    1. Initial program Error: 11.8 bits

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

    if 2.0737225518150114e-113 < b < 1.2940411791043692e154

    1. Initial program Error: 41.9 bits

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
    2. Using strategy rm

    3. Applied flip-+Error: 42.0 bits

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

    4. SimplifiedError: 15.7 bits

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

    5. SimplifiedError: 15.7 bits

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

    if 1.2940411791043692e154 < b

    1. Initial program Error: 64.0 bits

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

    2. Taylor expanded around inf Error: 14.7 bits

      \[\leadsto \frac{\color{blue}{-1.5 \cdot \frac{a \cdot c}{b}}}{3 \cdot a}\]

    3. SimplifiedError: 16.0 bits

      \[\leadsto \frac{\color{blue}{\left(c \cdot \frac{a}{b}\right) \cdot -1.5}}{3 \cdot a}\]
  3. Recombined 4 regimes into one program.

  4. Final simplificationError: 12.2 bits

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.846912681280906 \cdot 10^{+123}:\\ \;\;\;\;\frac{1.5 \cdot \left(c \cdot \frac{a}{b}\right) + b \cdot -2}{a \cdot 3}\\ \mathbf{elif}\;b \leq 2.0737225518150114 \cdot 10^{-113}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 1.2940411791043692 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{3 \cdot \left(c \cdot a\right)}{\left(-b\right) - \sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)}}}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(c \cdot \frac{a}{b}\right) \cdot -1.5}{a \cdot 3}\\ \end{array}\]

Reproduce

herbie shell --seed 2020204 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))