Average Error: 28.4 → 0.5
Time: 4.1s
Precision: binary64
\[1.0536712127723509 \cdot 10^{-08} < a \land a < 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} < b \land b < 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} < c \land c < 94906265.62425156\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{-2 \cdot c}{a} \cdot \frac{a}{b + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\frac{-2 \cdot c}{a} \cdot \frac{a}{b + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}
(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))
↓
(FPCore (a b c)
 :precision binary64
 (* (/ (* -2.0 c) a) (/ a (+ b (sqrt (- (* b b) (* c (* a 4.0))))))))
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 ((-2.0 * c) / a) * (a / (b + sqrt((b * b) - (c * (a * 4.0)))));
}

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. Simplified0.5

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

  6. Simplified0.5

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

  8. Applied *-un-lft-identity_binary640.5

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

  9. Applied times-frac_binary640.3

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

  10. Applied associate-/l*_binary640.3

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

  11. Simplified0.3

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

  13. Applied associate-*r/_binary640.3

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

  14. Applied associate-/r/_binary640.5

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

  15. Applied add-sqr-sqrt_binary640.5

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

  16. Applied *-un-lft-identity_binary640.5

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

  17. Applied times-frac_binary640.5

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

  18. Applied times-frac_binary640.5

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

  19. Simplified0.5

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

  20. Simplified0.5

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

  21. Final simplification0.5

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

Reproduce

herbie shell --seed 2020220 
(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)))