Average Error: 34.1 → 6.4
Time: 5.0s
Precision: binary64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \leq -9.444009031655424 \cdot 10^{+133}:\\ \;\;\;\;0.5 \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \leq -9.281927902223943 \cdot 10^{-213}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - c \cdot a}}{a} - \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \leq 8.828154833930211 \cdot 10^{+151}:\\ \;\;\;\;-\frac{c}{b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \leq -9.444009031655424 \cdot 10^{+133}:\\
\;\;\;\;0.5 \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\

\mathbf{elif}\;b_2 \leq -9.281927902223943 \cdot 10^{-213}:\\
\;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - c \cdot a}}{a} - \frac{b_2}{a}\\

\mathbf{elif}\;b_2 \leq 8.828154833930211 \cdot 10^{+151}:\\
\;\;\;\;-\frac{c}{b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}}\\

\mathbf{else}:\\
\;\;\;\;\frac{c}{b_2} \cdot -0.5\\

\end{array}
(FPCore (a b_2 c)
 :precision binary64
 (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))
↓
(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -9.444009031655424e+133)
   (- (* 0.5 (/ c b_2)) (* 2.0 (/ b_2 a)))
   (if (<= b_2 -9.281927902223943e-213)
     (- (/ (sqrt (- (* b_2 b_2) (* c a))) a) (/ b_2 a))
     (if (<= b_2 8.828154833930211e+151)
       (- (/ c (+ b_2 (sqrt (- (* b_2 b_2) (* c a))))))
       (* (/ c b_2) -0.5)))))
double code(double a, double b_2, double c) {
	return (((double) (((double) -(b_2)) + ((double) sqrt(((double) (((double) (b_2 * b_2)) - ((double) (a * c)))))))) / a);
}
↓
double code(double a, double b_2, double c) {
	double tmp;
	if ((b_2 <= -9.444009031655424e+133)) {
		tmp = ((double) (((double) (0.5 * (c / b_2))) - ((double) (2.0 * (b_2 / a)))));
	} else {
		double tmp_1;
		if ((b_2 <= -9.281927902223943e-213)) {
			tmp_1 = ((double) ((((double) sqrt(((double) (((double) (b_2 * b_2)) - ((double) (c * a)))))) / a) - (b_2 / a)));
		} else {
			double tmp_2;
			if ((b_2 <= 8.828154833930211e+151)) {
				tmp_2 = ((double) -((c / ((double) (b_2 + ((double) sqrt(((double) (((double) (b_2 * b_2)) - ((double) (c * a)))))))))));
			} else {
				tmp_2 = ((double) ((c / b_2) * -0.5));
			}
			tmp_1 = tmp_2;
		}
		tmp = tmp_1;
	}
	return tmp;
}

Error

Bits error versus a

Bits error versus b_2

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 < -9.444009031655424e133

    1. Initial program Error: 55.1 bits

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

    2. SimplifiedError: 55.1 bits

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}}\]

    3. Taylor expanded around -inf Error: 2.9 bits

      \[\leadsto \color{blue}{0.5 \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}}\]

    if -9.444009031655424e133 < b_2 < -9.2819279022239427e-213

    1. Initial program Error: 7.4 bits

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

    2. SimplifiedError: 7.4 bits

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}}\]
    3. Using strategy rm

    4. Applied div-subError: 7.4 bits

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} - \frac{b_2}{a}}\]

    if -9.2819279022239427e-213 < b_2 < 8.82815483393021091e151

    1. Initial program Error: 31.8 bits

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

    2. SimplifiedError: 31.8 bits

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}}\]
    3. Using strategy rm

    4. Applied flip--Error: 31.9 bits

      \[\leadsto \frac{\color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c} - b_2 \cdot b_2}{\sqrt{b_2 \cdot b_2 - a \cdot c} + b_2}}}{a}\]

    5. SimplifiedError: 15.6 bits

      \[\leadsto \frac{\frac{\color{blue}{-a \cdot c}}{\sqrt{b_2 \cdot b_2 - a \cdot c} + b_2}}{a}\]

    6. SimplifiedError: 15.6 bits

      \[\leadsto \frac{\frac{-a \cdot c}{\color{blue}{b_2 + \sqrt{b_2 \cdot b_2 - a \cdot c}}}}{a}\]
    7. Using strategy rm

    8. Applied distribute-frac-negError: 15.6 bits

      \[\leadsto \frac{\color{blue}{-\frac{a \cdot c}{b_2 + \sqrt{b_2 \cdot b_2 - a \cdot c}}}}{a}\]

    9. Applied distribute-frac-negError: 15.6 bits

      \[\leadsto \color{blue}{-\frac{\frac{a \cdot c}{b_2 + \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}}\]

    10. SimplifiedError: 8.9 bits

      \[\leadsto -\color{blue}{\frac{c}{b_2 + \sqrt{b_2 \cdot b_2 - a \cdot c}}}\]

    if 8.82815483393021091e151 < b_2

    1. Initial program Error: 63.5 bits

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

    2. SimplifiedError: 63.5 bits

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}}\]

    3. Taylor expanded around inf Error: 1.5 bits

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplificationError: 6.4 bits

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -9.444009031655424 \cdot 10^{+133}:\\ \;\;\;\;0.5 \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \leq -9.281927902223943 \cdot 10^{-213}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - c \cdot a}}{a} - \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \leq 8.828154833930211 \cdot 10^{+151}:\\ \;\;\;\;-\frac{c}{b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \end{array}\]

Reproduce

herbie shell --seed 2020205 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))