Average Error: 52.5 → 1.5
Time: 11.5s
Precision: binary64
\[4.930380657631324 \cdot 10^{-32} < a \land a < 2.028240960365167 \cdot 10^{+31} \land 4.930380657631324 \cdot 10^{-32} < b \land b < 2.028240960365167 \cdot 10^{+31} \land 4.930380657631324 \cdot 10^{-32} < c \land c < 2.028240960365167 \cdot 10^{+31}\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
↓
\[-\mathsf{fma}\left(2, \frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{5}}, \mathsf{fma}\left(5, \frac{{a}^{3} \cdot {c}^{4}}{{b}^{7}}, \frac{c}{b} + \frac{c \cdot \left(a \cdot c\right)}{{b}^{3}}\right)\right) \]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
-\mathsf{fma}\left(2, \frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{5}}, \mathsf{fma}\left(5, \frac{{a}^{3} \cdot {c}^{4}}{{b}^{7}}, \frac{c}{b} + \frac{c \cdot \left(a \cdot c\right)}{{b}^{3}}\right)\right)

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))
↓
(FPCore (a b c)
 :precision binary64
 (-
  (fma
   2.0
   (/ (* (* a a) (pow c 3.0)) (pow b 5.0))
   (fma
    5.0
    (/ (* (pow a 3.0) (pow c 4.0)) (pow b 7.0))
    (+ (/ c b) (/ (* c (* a c)) (pow b 3.0)))))))
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) {
	return -fma(2.0, (((a * a) * pow(c, 3.0)) / pow(b, 5.0)), fma(5.0, ((pow(a, 3.0) * pow(c, 4.0)) / pow(b, 7.0)), ((c / b) + ((c * (a * c)) / pow(b, 3.0)))));
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Initial program 52.5

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

  2. Taylor expanded in b around inf 1.5

    \[\leadsto \color{blue}{-\left(2 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(5 \cdot \frac{{c}^{4} \cdot {a}^{3}}{{b}^{7}} + \left(\frac{{c}^{2} \cdot a}{{b}^{3}} + \frac{c}{b}\right)\right)\right)} \]

  3. Simplified1.5

    \[\leadsto \color{blue}{-\mathsf{fma}\left(2, \frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{5}}, \mathsf{fma}\left(5, \frac{{a}^{3} \cdot {c}^{4}}{{b}^{7}}, \frac{c}{b} + \frac{c \cdot \left(c \cdot a\right)}{{b}^{3}}\right)\right)} \]

  4. Final simplification1.5

    \[\leadsto -\mathsf{fma}\left(2, \frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{5}}, \mathsf{fma}\left(5, \frac{{a}^{3} \cdot {c}^{4}}{{b}^{7}}, \frac{c}{b} + \frac{c \cdot \left(a \cdot c\right)}{{b}^{3}}\right)\right) \]

Reproduce

herbie shell --seed 2021224 
(FPCore (a b c)
  :name "Quadratic roots, wide range"
  :precision binary64
  :pre (and (< 4.930380657631324e-32 a 2.028240960365167e+31) (< 4.930380657631324e-32 b 2.028240960365167e+31) (< 4.930380657631324e-32 c 2.028240960365167e+31))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))