Average Error: 44.1 → 0.5
Time: 5.7s
Precision: binary64
\[1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992 \land 1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992 \land 1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
↓
\[\begin{array}{l} t_0 := -4 \cdot \left(a \cdot c\right)\\ \frac{1}{\frac{a}{\frac{\frac{t_0}{b + \sqrt{t_0 + b \cdot b}}}{2}}} \end{array} \]
\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(a \cdot c\right)\\
\frac{1}{\frac{a}{\frac{\frac{t_0}{b + \sqrt{t_0 + b \cdot b}}}{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 (* a c))))
   (/ 1.0 (/ a (/ (/ t_0 (+ b (sqrt (+ t_0 (* b b))))) 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 * (a * c);
	return 1.0 / (a / ((t_0 / (b + sqrt(t_0 + (b * b)))) / 2.0));
}

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 44.1

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

  2. Simplified44.1

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

  4. Applied flip--_binary6444.1

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

  5. Simplified43.5

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

  6. Simplified43.5

    \[\leadsto \frac{\frac{\left(b \cdot b - 4 \cdot \left(c \cdot a\right)\right) - b \cdot b}{\color{blue}{b + \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}}}{a \cdot 2} \]
  7. Using strategy rm

  8. Applied clear-num_binary6443.5

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

  9. Simplified0.5

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

  10. Final simplification0.5

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

Reproduce

herbie shell --seed 2021205 
(FPCore (a b c)
  :name "Quadratic roots, medium range"
  :precision binary64
  :pre (and (< 1.1102230246251565e-16 a 9007199254740992.0) (< 1.1102230246251565e-16 b 9007199254740992.0) (< 1.1102230246251565e-16 c 9007199254740992.0))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))