Average Error: 28.4 → 0.5
Time: 3.9s
Precision: binary64
\[1.0536712127723509 \cdot 10^{-08} < a \land a < 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} < b \land b < 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} < c \land c < 94906265.62425156\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{-4 \cdot \left(a \cdot c\right)}{\left(b + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\right) \cdot \left(a \cdot 2\right)}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\frac{-4 \cdot \left(a \cdot c\right)}{\left(b + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}\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) (* c (* a 4.0))))) (* 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 (-4.0 * (a * c)) / ((b + sqrt((b * b) - (c * (a * 4.0)))) * (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 28.4

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

  2. Simplified28.4

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

  4. Applied flip--_binary6428.4

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

  5. Simplified27.4

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

  6. Simplified27.4

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

  8. Applied *-un-lft-identity_binary6427.4

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

  9. Applied *-un-lft-identity_binary6427.4

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

  10. Applied distribute-lft-out_binary6427.4

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

  11. Applied *-un-lft-identity_binary6427.4

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

  12. Applied times-frac_binary6427.4

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

  13. Simplified27.4

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

  14. Simplified0.5

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

  16. Applied associate-*l/_binary640.5

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

  17. Applied associate-*r/_binary640.5

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

  18. Applied associate-/l/_binary640.5

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

  19. Simplified0.5

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

  20. Final simplification0.5

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

Reproduce

herbie shell --seed 2020288 
(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)))