Average Error: 43.9 → 0.2
Time: 58.8s
Precision: binary64
\[1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992 \land 1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992 \land 1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{-4 \cdot a}{\frac{a}{\frac{\frac{c}{b + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2}}}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\frac{-4 \cdot a}{\frac{a}{\frac{\frac{c}{b + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2}}}
(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))
  (/ a (/ (/ c (+ b (sqrt (- (* b b) (* (* 4.0 a) c))))) 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) (4.0 * a)))) / (a / ((c / ((double) (b + ((double) sqrt(((double) (((double) (b * b)) - ((double) (((double) (4.0 * a)) * c))))))))) / 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 43.9

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

  2. Simplified43.9

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

  4. Applied flip--_binary6443.9

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

  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 *-un-lft-identity_binary640.4

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

  9. Applied distribute-lft-neg-in_binary640.4

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

  10. Applied times-frac_binary640.2

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

  11. Applied associate-/l*_binary640.2

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

  12. Simplified0.2

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

  13. Final simplification0.2

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

Reproduce

herbie shell --seed 2020205 
(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)))