Average Error: 44.0 → 0.4
Time: 4.2s
Precision: binary64
\[1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992 \land 1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992 \land 1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{-\left(4 \cdot a\right) \cdot c}{\left(b + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \left(a \cdot 2\right)}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\frac{-\left(4 \cdot a\right) \cdot c}{\left(b + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \left(a \cdot 2\right)}
(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))
↓
(FPCore (a b c)
 :precision binary64
 (/
  (- (* (* 4.0 a) c))
  (* (+ b (sqrt (- (* b b) (* (* 4.0 a) c)))) (* a 2.0))))
double code(double a, double b, double c) {
	return (((double) (((double) -(b)) + ((double) sqrt(((double) (((double) (b * b)) - ((double) (((double) (4.0 * a)) * c)))))))) / ((double) (2.0 * a)));
}
↓
double code(double a, double b, double c) {
	return (((double) -(((double) (((double) (4.0 * a)) * c)))) / ((double) (((double) (b + ((double) sqrt(((double) (((double) (b * b)) - ((double) (((double) (4.0 * a)) * c)))))))) * ((double) (a * 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 44.0

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

  2. Simplified44.0

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

  4. Applied flip--_binary6444.0

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

  5. Simplified0.4

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

  6. Simplified0.4

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

  8. Applied div-inv_binary640.5

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

  9. Applied associate-/l*_binary640.5

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

  10. Simplified0.4

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

  11. Final simplification0.4

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

Reproduce

herbie shell --seed 2020210 
(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)))