Average Error: 33.4 → 6.6
Time: 12.0s
Precision: binary64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \leq -2.0987659544316741 \cdot 10^{+148}:\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \mathbf{elif}\;b_2 \leq 1.2616642087724312 \cdot 10^{-272}:\\ \;\;\;\;\frac{c}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}\\ \mathbf{elif}\;b_2 \leq 2.2347560056365885 \cdot 10^{+106}:\\ \;\;\;\;\frac{b_2}{-a} - \frac{\sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + \frac{c}{b_2} \cdot 0.5\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \leq -2.0987659544316741 \cdot 10^{+148}:\\
\;\;\;\;\frac{c}{b_2} \cdot -0.5\\

\mathbf{elif}\;b_2 \leq 1.2616642087724312 \cdot 10^{-272}:\\
\;\;\;\;\frac{c}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}\\

\mathbf{elif}\;b_2 \leq 2.2347560056365885 \cdot 10^{+106}:\\
\;\;\;\;\frac{b_2}{-a} - \frac{\sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\

\mathbf{else}:\\
\;\;\;\;-2 \cdot \frac{b_2}{a} + \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 -2.0987659544316741e+148)
   (* (/ c b_2) -0.5)
   (if (<= b_2 1.2616642087724312e-272)
     (/ c (- (sqrt (- (* b_2 b_2) (* c a))) b_2))
     (if (<= b_2 2.2347560056365885e+106)
       (- (/ b_2 (- a)) (/ (sqrt (- (* b_2 b_2) (* c a))) a))
       (+ (* -2.0 (/ b_2 a)) (* (/ c b_2) 0.5))))))
double code(double a, double b_2, double c) {
	return (-b_2 - sqrt((b_2 * b_2) - (a * c))) / a;
}
↓
double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.0987659544316741e+148) {
		tmp = (c / b_2) * -0.5;
	} else if (b_2 <= 1.2616642087724312e-272) {
		tmp = c / (sqrt((b_2 * b_2) - (c * a)) - b_2);
	} else if (b_2 <= 2.2347560056365885e+106) {
		tmp = (b_2 / -a) - (sqrt((b_2 * b_2) - (c * a)) / a);
	} else {
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5);
	}
	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 < -2.09876595443167415e148

    1. Initial program 63.8

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
    2. Using strategy rm

    3. Applied div-sub_binary6463.8

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

    4. Simplified63.8

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

    5. Taylor expanded around -inf 1.9

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

    6. Simplified1.9

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

    if -2.09876595443167415e148 < b_2 < 1.26166420877243118e-272

    1. Initial program 32.8

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
    2. Using strategy rm

    3. Applied flip--_binary6432.9

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

    4. Simplified16.2

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

    5. Simplified16.2

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

    7. Applied *-un-lft-identity_binary6416.2

      \[\leadsto \frac{\frac{a \cdot c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{\color{blue}{1 \cdot a}}\]

    8. Applied *-un-lft-identity_binary6416.2

      \[\leadsto \frac{\color{blue}{1 \cdot \frac{a \cdot c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}{1 \cdot a}\]

    9. Applied times-frac_binary6416.2

      \[\leadsto \color{blue}{\frac{1}{1} \cdot \frac{\frac{a \cdot c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a}}\]

    10. Simplified16.2

      \[\leadsto \color{blue}{1} \cdot \frac{\frac{a \cdot c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a}\]

    11. Simplified8.8

      \[\leadsto 1 \cdot \color{blue}{\frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}\]

    if 1.26166420877243118e-272 < b_2 < 2.23475600563658851e106

    1. Initial program 8.1

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
    2. Using strategy rm

    3. Applied div-sub_binary648.1

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

    4. Simplified8.1

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

    if 2.23475600563658851e106 < b_2

    1. Initial program 48.7

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

    2. Taylor expanded around inf 3.8

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

    3. Simplified3.8

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

  4. Final simplification6.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -2.0987659544316741 \cdot 10^{+148}:\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \mathbf{elif}\;b_2 \leq 1.2616642087724312 \cdot 10^{-272}:\\ \;\;\;\;\frac{c}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}\\ \mathbf{elif}\;b_2 \leq 2.2347560056365885 \cdot 10^{+106}:\\ \;\;\;\;\frac{b_2}{-a} - \frac{\sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + \frac{c}{b_2} \cdot 0.5\\ \end{array}\]

Reproduce

herbie shell --seed 2021096 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))