Average Error: 28.5 → 5.9
Time: 6.2s
Precision: binary64
\[1.0536712127723509 \cdot 10^{-08} < a \land a < 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} < b \land b < 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} < c \land c < 94906265.62425156\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\left(\left(-2 \cdot \frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{5}} - \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}}\right) - \frac{c}{b}\right) + \frac{{a}^{3} \cdot {c}^{4}}{{b}^{7}} \cdot -5\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\left(\left(-2 \cdot \frac{\left(a \cdot a\right) \cdot {c}^{3}}{{b}^{5}} - \frac{a \cdot \left(c \cdot c\right)}{{b}^{3}}\right) - \frac{c}{b}\right) + \frac{{a}^{3} \cdot {c}^{4}}{{b}^{7}} \cdot -5
(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))
↓
(FPCore (a b c)
 :precision binary64
 (+
  (-
   (-
    (* -2.0 (/ (* (* a a) (pow c 3.0)) (pow b 5.0)))
    (/ (* a (* c c)) (pow b 3.0)))
   (/ c b))
  (* (/ (* (pow a 3.0) (pow c 4.0)) (pow b 7.0)) -5.0)))
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 (((-2.0 * (((a * a) * pow(c, 3.0)) / pow(b, 5.0))) - ((a * (c * c)) / pow(b, 3.0))) - (c / b)) + (((pow(a, 3.0) * pow(c, 4.0)) / pow(b, 7.0)) * -5.0);
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 28.5

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

  2. Simplified28.5

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

  3. Taylor expanded around inf 5.9

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

  4. Simplified5.9

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

  5. Final simplification5.9

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

Reproduce

herbie shell --seed 2021174 
(FPCore (a b c)
  :name "Quadratic roots, narrow range"
  :precision binary64
  :pre (and (< 1.0536712127723509e-08 a 94906265.62425156) (< 1.0536712127723509e-08 b 94906265.62425156) (< 1.0536712127723509e-08 c 94906265.62425156))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))