Average Error: 33.7 → 9.7
Time: 8.2s
Precision: binary64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \leq -1.5343846642293365 \cdot 10^{-36}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 5.039495121971627 \cdot 10^{+140}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b_2}, -2 \cdot \frac{b_2}{a}\right)\\ \end{array} \]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \leq -1.5343846642293365 \cdot 10^{-36}:\\
\;\;\;\;-0.5 \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \leq 5.039495121971627 \cdot 10^{+140}:\\
\;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b_2}, -2 \cdot \frac{b_2}{a}\right)\\


\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 -1.5343846642293365e-36)
   (* -0.5 (/ c b_2))
   (if (<= b_2 5.039495121971627e+140)
     (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* c a)))) a)
     (fma 0.5 (/ c b_2) (* -2.0 (/ b_2 a))))))
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 <= -1.5343846642293365e-36) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 5.039495121971627e+140) {
		tmp = (-b_2 - sqrt((b_2 * b_2) - (c * a))) / a;
	} else {
		tmp = fma(0.5, (c / b_2), (-2.0 * (b_2 / a)));
	}
	return tmp;
}

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Derivation

  1. Split input into 3 regimes

  2. if b_2 < -1.53438466422933652e-36

    1. Initial program 54.4

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

    2. Taylor expanded in b_2 around -inf 7.6

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

    if -1.53438466422933652e-36 < b_2 < 5.0394951219716266e140

    1. Initial program 12.8

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

    2. Applied *-un-lft-identity_binary6412.8

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

    3. Applied sqrt-prod_binary6412.8

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

    if 5.0394951219716266e140 < b_2

    1. Initial program 59.4

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

    2. Taylor expanded in b_2 around inf 2.5

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

    3. Simplified2.5

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \frac{c}{b_2}, -2 \cdot \frac{b_2}{a}\right)} \]
  3. Recombined 3 regimes into one program.

  4. Final simplification9.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -1.5343846642293365 \cdot 10^{-36}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 5.039495121971627 \cdot 10^{+140}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b_2}, -2 \cdot \frac{b_2}{a}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022067 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))