Average Error: 34.1 → 10.2
Time: 15.3s
Precision: binary64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \leq -5.676559387027347 \cdot 10^{+135}:\\ \;\;\;\;\frac{\left(-b_2\right) - b_2}{a}\\ \mathbf{elif}\;b_2 \leq 2.466272322022482 \cdot 10^{-84}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} - \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \leq -5.676559387027347 \cdot 10^{+135}:\\
\;\;\;\;\frac{\left(-b_2\right) - b_2}{a}\\

\mathbf{elif}\;b_2 \leq 2.466272322022482 \cdot 10^{-84}:\\
\;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} - \frac{b_2}{a}\\

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

\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 -5.676559387027347e+135)
   (/ (- (- b_2) b_2) a)
   (if (<= b_2 2.466272322022482e-84)
     (- (/ (sqrt (- (* b_2 b_2) (* a c))) a) (/ b_2 a))
     (* -0.5 (/ c b_2)))))
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 <= -5.676559387027347e+135) {
		tmp = (-b_2 - b_2) / a;
	} else if (b_2 <= 2.466272322022482e-84) {
		tmp = (sqrt((b_2 * b_2) - (a * c)) / a) - (b_2 / a);
	} else {
		tmp = -0.5 * (c / b_2);
	}
	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 3 regimes

  2. if b_2 < -5.6765593870273467e135

    1. Initial program 55.4

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

    2. Simplified55.4

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

    3. Taylor expanded around -inf 2.8

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

    4. Simplified2.8

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

    if -5.6765593870273467e135 < b_2 < 2.4662723220224822e-84

    1. Initial program 12.4

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

    2. Simplified12.4

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

    4. Applied div-sub_binary6412.4

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

    6. Applied *-un-lft-identity_binary6412.4

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

    7. Applied *-un-lft-identity_binary6412.4

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

    8. Applied sqrt-prod_binary6412.4

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

    9. Applied times-frac_binary6412.4

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

    if 2.4662723220224822e-84 < b_2

    1. Initial program 52.2

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

    2. Simplified52.2

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

    3. Taylor expanded around inf 10.0

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

  4. Final simplification10.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -5.676559387027347 \cdot 10^{+135}:\\ \;\;\;\;\frac{\left(-b_2\right) - b_2}{a}\\ \mathbf{elif}\;b_2 \leq 2.466272322022482 \cdot 10^{-84}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} - \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2021075 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))