Average Error: 52.8 → 0.4
Time: 5.9s
Precision: binary64
\[\left(\left(4.930380657631324 \cdot 10^{-32} < a \land a < 2.028240960365167 \cdot 10^{+31}\right) \land \left(4.930380657631324 \cdot 10^{-32} < b \land b < 2.028240960365167 \cdot 10^{+31}\right)\right) \land \left(4.930380657631324 \cdot 10^{-32} < c \land c < 2.028240960365167 \cdot 10^{+31}\right)\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
↓
\[\begin{array}{l} t_0 := -4 \cdot \left(c \cdot a\right)\\ \frac{\frac{t_0}{b + \sqrt{\mathsf{fma}\left(b, b, t_0\right)}}}{a \cdot 2} \end{array} \]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\begin{array}{l}
t_0 := -4 \cdot \left(c \cdot a\right)\\
\frac{\frac{t_0}{b + \sqrt{\mathsf{fma}\left(b, b, t_0\right)}}}{a \cdot 2}
\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
 (let* ((t_0 (* -4.0 (* c a))))
   (/ (/ t_0 (+ b (sqrt (fma b b t_0)))) (* a 2.0))))
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 t_0 = -4.0 * (c * a);
	return (t_0 / (b + sqrt(fma(b, b, t_0)))) / (a * 2.0);
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Initial program 52.8

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

  2. Applied egg-rr52.6

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

  3. Taylor expanded in b around 0 0.4

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

  4. Final simplification0.4

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

Reproduce

herbie shell --seed 2022129 
(FPCore (a b c)
  :name "Quadratic roots, wide range"
  :precision binary64
  :pre (and (and (and (< 4.930380657631324e-32 a) (< a 2.028240960365167e+31)) (and (< 4.930380657631324e-32 b) (< b 2.028240960365167e+31))) (and (< 4.930380657631324e-32 c) (< c 2.028240960365167e+31)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))