Average Error: 52.6 → 0.4
Time: 5.8s
Precision: binary64
\[4.930380657631324 \cdot 10^{-32} < a \land a < 2.028240960365167 \cdot 10^{+31} \land 4.930380657631324 \cdot 10^{-32} < b \land b < 2.028240960365167 \cdot 10^{+31} \land 4.930380657631324 \cdot 10^{-32} < c \land c < 2.028240960365167 \cdot 10^{+31}\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
↓
\[\begin{array}{l} t_0 := -4 \cdot \left(c \cdot a\right)\\ \frac{t_0}{a} \cdot \frac{0.5}{b + \sqrt{t_0 + b \cdot b}} \end{array} \]
\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{t_0}{a} \cdot \frac{0.5}{b + \sqrt{t_0 + b \cdot 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
 (let* ((t_0 (* -4.0 (* c a))))
   (* (/ t_0 a) (/ 0.5 (+ b (sqrt (+ t_0 (* b 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 t_0 = -4.0 * (c * a);
	return (t_0 / a) * (0.5 / (b + sqrt(t_0 + (b * b))));
}

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 52.6

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

  2. Simplified52.6

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

  4. Applied flip--_binary6452.6

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

  5. Simplified52.3

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

  6. Simplified52.3

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

  8. Applied div-inv_binary6452.3

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

  9. Applied times-frac_binary6452.3

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

  10. Simplified0.4

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

  11. Simplified0.4

    \[\leadsto \frac{-4 \cdot \left(c \cdot a\right)}{a} \cdot \color{blue}{\frac{0.5}{b + \sqrt{-4 \cdot \left(c \cdot a\right) + b \cdot b}}} \]

  12. Final simplification0.4

    \[\leadsto \frac{-4 \cdot \left(c \cdot a\right)}{a} \cdot \frac{0.5}{b + \sqrt{-4 \cdot \left(c \cdot a\right) + b \cdot b}} \]

Reproduce

herbie shell --seed 2021202 
(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)))