Average Error: 34.4 → 10.5
Time: 9.1s
Precision: binary64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
↓
\[\begin{array}{l} \mathbf{if}\;b \leq -5.560568612734722 \cdot 10^{+153}:\\ \;\;\;\;\frac{\left(-b\right) - b}{2 \cdot a}\\ \mathbf{elif}\;b \leq 2.0627980826047655 \cdot 10^{-71}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1, b, \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \leq -5.560568612734722 \cdot 10^{+153}:\\
\;\;\;\;\frac{\left(-b\right) - b}{2 \cdot a}\\

\mathbf{elif}\;b \leq 2.0627980826047655 \cdot 10^{-71}:\\
\;\;\;\;\frac{\mathsf{fma}\left(-1, b, \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}\right)}{2 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;-\frac{c}{b}\\


\end{array}

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))
↓
(FPCore (a b c)
 :precision binary64
 (if (<= b -5.560568612734722e+153)
   (/ (- (- b) b) (* 2.0 a))
   (if (<= b 2.0627980826047655e-71)
     (/ (fma -1.0 b (sqrt (- (* b b) (* (* a 4.0) c)))) (* 2.0 a))
     (- (/ c b)))))
double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}
↓
double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.560568612734722e+153) {
		tmp = (-b - b) / (2.0 * a);
	} else if (b <= 2.0627980826047655e-71) {
		tmp = fma(-1.0, b, sqrt(((b * b) - ((a * 4.0) * c)))) / (2.0 * a);
	} else {
		tmp = -(c / b);
	}
	return tmp;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 3 regimes

  2. if b < -5.5605686127347219e153

    1. Initial program 63.9

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

    2. Taylor expanded in b around -inf 2.7

      \[\leadsto \frac{\left(-b\right) + \color{blue}{-1 \cdot b}}{2 \cdot a} \]

    3. Simplified2.7

      \[\leadsto \frac{\left(-b\right) + \color{blue}{\left(-b\right)}}{2 \cdot a} \]

    if -5.5605686127347219e153 < b < 2.0627980826047655e-71

    1. Initial program 12.8

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

    2. Applied neg-mul-1_binary6412.8

      \[\leadsto \frac{\color{blue}{-1 \cdot b} + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

    3. Applied fma-def_binary6412.8

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-1, b, \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a} \]

    if 2.0627980826047655e-71 < b

    1. Initial program 53.1

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]

    2. Taylor expanded in b around inf 9.8

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]

    3. Simplified9.8

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

  4. Final simplification10.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.560568612734722 \cdot 10^{+153}:\\ \;\;\;\;\frac{\left(-b\right) - b}{2 \cdot a}\\ \mathbf{elif}\;b \leq 2.0627980826047655 \cdot 10^{-71}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1, b, \sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]

Reproduce

herbie shell --seed 2022125 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))