Average Error: 34.0 → 8.7
Time: 6.2s
Precision: binary64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \leq -1.3450496563396583 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \leq 4.525283768938065 \cdot 10^{-198}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}{a}\\ \mathbf{elif}\;b_2 \leq 2.7099763392697605 \cdot 10^{+106}:\\ \;\;\;\;\frac{\frac{-c \cdot a}{a}}{b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \leq -1.3450496563396583 \cdot 10^{+154}:\\
\;\;\;\;0.5 \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\

\mathbf{elif}\;b_2 \leq 4.525283768938065 \cdot 10^{-198}:\\
\;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}{a}\\

\mathbf{elif}\;b_2 \leq 2.7099763392697605 \cdot 10^{+106}:\\
\;\;\;\;\frac{\frac{-c \cdot a}{a}}{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 -1.3450496563396583e+154)
   (- (* 0.5 (/ c b_2)) (* 2.0 (/ b_2 a)))
   (if (<= b_2 4.525283768938065e-198)
     (/ (- (sqrt (- (* b_2 b_2) (* c a))) b_2) a)
     (if (<= b_2 2.7099763392697605e+106)
       (/ (/ (- (* c a)) a) (+ b_2 (sqrt (- (* b_2 b_2) (* c 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 <= -1.3450496563396583e+154) {
		tmp = (0.5 * (c / b_2)) - (2.0 * (b_2 / a));
	} else if (b_2 <= 4.525283768938065e-198) {
		tmp = (sqrt((b_2 * b_2) - (c * a)) - b_2) / a;
	} else if (b_2 <= 2.7099763392697605e+106) {
		tmp = (-(c * a) / a) / (b_2 + sqrt((b_2 * b_2) - (c * a)));
	} else {
		tmp = (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 < -1.34504965633965828e154

    1. Initial program 64.0

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

    2. Simplified64.0

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

    3. Taylor expanded around -inf 2.0

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

    if -1.34504965633965828e154 < b_2 < 4.525283768938065e-198

    1. Initial program 10.4

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

    2. Simplified10.4

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

    if 4.525283768938065e-198 < b_2 < 2.7099763392697605e106

    1. Initial program 36.7

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

    2. Simplified36.7

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

    4. Applied div-inv_binary6436.7

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

    6. Applied flip--_binary6436.7

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

    7. Applied associate-*l/_binary6436.8

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

    8. Simplified14.5

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

    if 2.7099763392697605e106 < b_2

    1. Initial program 60.2

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

    2. Simplified60.2

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

    3. Taylor expanded around inf 2.5

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

  4. Final simplification8.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -1.3450496563396583 \cdot 10^{+154}:\\ \;\;\;\;0.5 \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \leq 4.525283768938065 \cdot 10^{-198}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}{a}\\ \mathbf{elif}\;b_2 \leq 2.7099763392697605 \cdot 10^{+106}:\\ \;\;\;\;\frac{\frac{-c \cdot a}{a}}{b_2 + \sqrt{b_2 \cdot b_2 - c \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \end{array}\]

Reproduce

herbie shell --seed 2020288 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))