Average Error: 34.1 → 10.1
Time: 11.5s
Precision: binary64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \leq -9.527912883871295 \cdot 10^{+86}:\\ \;\;\;\;\frac{\left(-b_2\right) - b_2}{a}\\ \mathbf{elif}\;b_2 \leq 2.500953026669186 \cdot 10^{-66}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1, \sqrt{b_2 \cdot b_2 - a \cdot c}, -b_2\right) + \mathsf{fma}\left(-b_2, 1, b_2\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \end{array} \]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \leq -9.527912883871295 \cdot 10^{+86}:\\
\;\;\;\;\frac{\left(-b_2\right) - b_2}{a}\\

\mathbf{elif}\;b_2 \leq 2.500953026669186 \cdot 10^{-66}:\\
\;\;\;\;\frac{\mathsf{fma}\left(1, \sqrt{b_2 \cdot b_2 - a \cdot c}, -b_2\right) + \mathsf{fma}\left(-b_2, 1, b_2\right)}{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 -9.527912883871295e+86)
   (/ (- (- b_2) b_2) a)
   (if (<= b_2 2.500953026669186e-66)
     (/
      (+
       (fma 1.0 (sqrt (- (* b_2 b_2) (* a c))) (- b_2))
       (fma (- b_2) 1.0 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 <= -9.527912883871295e+86) {
		tmp = (-b_2 - b_2) / a;
	} else if (b_2 <= 2.500953026669186e-66) {
		tmp = (fma(1.0, sqrt((b_2 * b_2) - (a * c)), -b_2) + fma(-b_2, 1.0, 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

Derivation

  1. Split input into 3 regimes

  2. if b_2 < -9.52791288387129465e86

    1. Initial program 45.2

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

    2. Simplified45.2

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

    3. Taylor expanded in b_2 around -inf 4.9

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

    4. Simplified4.9

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

    if -9.52791288387129465e86 < b_2 < 2.5009530266691861e-66

    1. Initial program 13.5

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

    2. Simplified13.5

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

    3. Applied *-un-lft-identity_binary6413.5

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

    4. Applied *-un-lft-identity_binary6413.5

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

    5. Applied prod-diff_binary6413.5

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

    if 2.5009530266691861e-66 < b_2

    1. Initial program 53.4

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

    2. Simplified53.4

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

    3. Taylor expanded in b_2 around inf 8.5

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

  4. Final simplification10.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -9.527912883871295 \cdot 10^{+86}:\\ \;\;\;\;\frac{\left(-b_2\right) - b_2}{a}\\ \mathbf{elif}\;b_2 \leq 2.500953026669186 \cdot 10^{-66}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1, \sqrt{b_2 \cdot b_2 - a \cdot c}, -b_2\right) + \mathsf{fma}\left(-b_2, 1, b_2\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Reproduce

herbie shell --seed 2021329 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))