Average Error: 28.3 → 27.3
Time: 12.9s
Precision: binary64
\[1.0536712127723509 \cdot 10^{-08} < a \land a < 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} < b \land b < 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} < c \land c < 94906265.62425156\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{\sqrt[3]{{\left(\frac{\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right) - b \cdot b}{b + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right)}^{3}}}{a \cdot 2}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\frac{\sqrt[3]{{\left(\frac{\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right) - b \cdot b}{b + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right)}^{3}}}{a \cdot 2}
(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))
↓
(FPCore (a b c)
 :precision binary64
 (/
  (cbrt
   (pow
    (/
     (- (- (* b b) (* (* 4.0 a) c)) (* b b))
     (+ b (sqrt (- (* b b) (* (* 4.0 a) c)))))
    3.0))
  (* a 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) {
	return cbrt(pow(((((b * b) - ((4.0 * a) * c)) - (b * b)) / (b + sqrt((b * b) - ((4.0 * a) * c)))), 3.0)) / (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 28.3

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

  2. Simplified28.3

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

  4. Applied flip--_binary6428.3

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

  5. Simplified27.3

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

  6. Simplified27.3

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

  8. Applied add-cbrt-cube_binary6427.3

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

  9. Simplified27.3

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

  10. Final simplification27.3

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

Reproduce

herbie shell --seed 2020277 
(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)))