Average Error: 33.6 → 8.0
Time: 18.6s
Precision: binary64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \leq -4.5426505460560854 \cdot 10^{+27}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq -1.1138808105799072 \cdot 10^{-298}:\\ \;\;\;\;\frac{c}{\mathsf{hypot}\left(\sqrt{-c \cdot a}, b_2\right) - b_2}\\ \mathbf{elif}\;b_2 \leq 2.145068448284974 \cdot 10^{+83}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b_2}, -2 \cdot \frac{b_2}{a}\right)\\ \end{array} \]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \leq -4.5426505460560854 \cdot 10^{+27}:\\
\;\;\;\;-0.5 \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \leq -1.1138808105799072 \cdot 10^{-298}:\\
\;\;\;\;\frac{c}{\mathsf{hypot}\left(\sqrt{-c \cdot a}, b_2\right) - b_2}\\

\mathbf{elif}\;b_2 \leq 2.145068448284974 \cdot 10^{+83}:\\
\;\;\;\;\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 -4.5426505460560854e+27)
   (* -0.5 (/ c b_2))
   (if (<= b_2 -1.1138808105799072e-298)
     (/ c (- (hypot (sqrt (- (* c a))) b_2) b_2))
     (if (<= b_2 2.145068448284974e+83)
       (/ (- (- 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 <= -4.5426505460560854e+27) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= -1.1138808105799072e-298) {
		tmp = c / (hypot(sqrt(-(c * a)), b_2) - b_2);
	} else if (b_2 <= 2.145068448284974e+83) {
		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 4 regimes

  2. if b_2 < -4.5426505460560854e27

    1. Initial program 56.3

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

    2. Taylor expanded in b_2 around -inf 4.7

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

    if -4.5426505460560854e27 < b_2 < -1.11388081057990716e-298

    1. Initial program 28.2

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

    2. Applied flip--_binary6428.2

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

    3. Simplified17.4

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

    4. Simplified23.4

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

    5. Applied *-un-lft-identity_binary6423.4

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

    6. Applied *-un-lft-identity_binary6423.4

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

    7. Applied times-frac_binary6423.4

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

    8. Simplified23.4

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

    9. Simplified14.1

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

    if -1.11388081057990716e-298 < b_2 < 2.1450684482849739e83

    1. Initial program 8.8

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

    if 2.1450684482849739e83 < b_2

    1. Initial program 42.5

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

    2. Taylor expanded in b_2 around inf 4.2

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

    3. Simplified4.2

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \frac{c}{b_2}, -2 \cdot \frac{b_2}{a}\right)} \]
  3. Recombined 4 regimes into one program.

  4. Final simplification8.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -4.5426505460560854 \cdot 10^{+27}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq -1.1138808105799072 \cdot 10^{-298}:\\ \;\;\;\;\frac{c}{\mathsf{hypot}\left(\sqrt{-c \cdot a}, b_2\right) - b_2}\\ \mathbf{elif}\;b_2 \leq 2.145068448284974 \cdot 10^{+83}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b_2}, -2 \cdot \frac{b_2}{a}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022068 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))