Average Error: 34.4 → 7.1
Time: 4.8s
Precision: binary64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \leq -6.205007535037304 \cdot 10^{+77}:\\ \;\;\;\;\frac{\left(0.5 \cdot \left(c \cdot \frac{a}{b_2}\right) - b_2\right) - b_2}{a}\\ \mathbf{elif}\;b_2 \leq -1.243685732710164 \cdot 10^{-305}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}{a}\\ \mathbf{elif}\;b_2 \leq 6.9512572766954365 \cdot 10^{+19}:\\ \;\;\;\;\frac{-c}{b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b_2 \cdot 2 + a \cdot \frac{-0.5}{\frac{b_2}{c}}}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \leq -6.205007535037304 \cdot 10^{+77}:\\
\;\;\;\;\frac{\left(0.5 \cdot \left(c \cdot \frac{a}{b_2}\right) - b_2\right) - b_2}{a}\\

\mathbf{elif}\;b_2 \leq -1.243685732710164 \cdot 10^{-305}:\\
\;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}{a}\\

\mathbf{elif}\;b_2 \leq 6.9512572766954365 \cdot 10^{+19}:\\
\;\;\;\;\frac{-c}{b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}}\\

\mathbf{else}:\\
\;\;\;\;\frac{-c}{b_2 \cdot 2 + a \cdot \frac{-0.5}{\frac{b_2}{c}}}\\

\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 -6.205007535037304e+77)
   (/ (- (- (* 0.5 (* c (/ a b_2))) b_2) b_2) a)
   (if (<= b_2 -1.243685732710164e-305)
     (/ (- (sqrt (- (* b_2 b_2) (* c a))) b_2) a)
     (if (<= b_2 6.9512572766954365e+19)
       (/ (- c) (+ b_2 (sqrt (- (* b_2 b_2) (* c a)))))
       (/ (- c) (+ (* b_2 2.0) (* a (/ -0.5 (/ b_2 c)))))))))
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 <= -6.205007535037304e+77)) {
		tmp = (((double) (((double) (((double) (0.5 * ((double) (c * (a / b_2))))) - b_2)) - b_2)) / a);
	} else {
		double tmp_1;
		if ((b_2 <= -1.243685732710164e-305)) {
			tmp_1 = (((double) (((double) sqrt(((double) (((double) (b_2 * b_2)) - ((double) (c * a)))))) - b_2)) / a);
		} else {
			double tmp_2;
			if ((b_2 <= 6.9512572766954365e+19)) {
				tmp_2 = (((double) -(c)) / ((double) (b_2 + ((double) sqrt(((double) (((double) (b_2 * b_2)) - ((double) (c * a)))))))));
			} else {
				tmp_2 = (((double) -(c)) / ((double) (((double) (b_2 * 2.0)) + ((double) (a * (-0.5 / (b_2 / c)))))));
			}
			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 < -6.205007535037304e77

    1. Initial program Error: 41.7 bits

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

    2. SimplifiedError: 41.7 bits

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

    3. Taylor expanded around -inf Error: 8.6 bits

      \[\leadsto \frac{\color{blue}{\left(0.5 \cdot \frac{a \cdot c}{b_2} - b_2\right)} - b_2}{a}\]

    4. SimplifiedError: 3.6 bits

      \[\leadsto \frac{\color{blue}{\left(0.5 \cdot \left(c \cdot \frac{a}{b_2}\right) - b_2\right)} - b_2}{a}\]

    if -6.205007535037304e77 < b_2 < -1.24368573271016397e-305

    1. Initial program Error: 9.9 bits

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

    2. SimplifiedError: 9.9 bits

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

    if -1.24368573271016397e-305 < b_2 < 69512572766954364900

    1. Initial program Error: 28.1 bits

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

    2. SimplifiedError: 28.1 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: 28.1 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: 18.1 bits

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

    6. SimplifiedError: 18.1 bits

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

    8. Applied distribute-rgt-neg-outError: 18.1 bits

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

    9. Applied distribute-frac-negError: 18.1 bits

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

    10. Applied distribute-frac-negError: 18.1 bits

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

    11. SimplifiedError: 10.1 bits

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

    if 69512572766954364900 < b_2

    1. Initial program Error: 56.5 bits

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

    2. SimplifiedError: 56.5 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: 56.5 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: 27.3 bits

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

    6. SimplifiedError: 27.3 bits

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

    8. Applied distribute-rgt-neg-outError: 27.3 bits

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

    9. Applied distribute-frac-negError: 27.3 bits

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

    10. Applied distribute-frac-negError: 27.3 bits

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

    11. SimplifiedError: 23.5 bits

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

    12. Taylor expanded around inf Error: 7.3 bits

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

    13. SimplifiedError: 4.5 bits

      \[\leadsto -\frac{c}{\color{blue}{b_2 \cdot 2 + a \cdot \frac{-0.5}{\frac{b_2}{c}}}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplificationError: 7.1 bits

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -6.205007535037304 \cdot 10^{+77}:\\ \;\;\;\;\frac{\left(0.5 \cdot \left(c \cdot \frac{a}{b_2}\right) - b_2\right) - b_2}{a}\\ \mathbf{elif}\;b_2 \leq -1.243685732710164 \cdot 10^{-305}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}{a}\\ \mathbf{elif}\;b_2 \leq 6.9512572766954365 \cdot 10^{+19}:\\ \;\;\;\;\frac{-c}{b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b_2 \cdot 2 + a \cdot \frac{-0.5}{\frac{b_2}{c}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020204 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))