Average Error: 28.5 → 0.4
Time: 4.7s
Precision: binary64
\[1.0536712127723509 \cdot 10^{-08} < a \land a < 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} < b \land b < 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} < c \land c < 94906265.62425156\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[c \cdot \frac{-2}{b + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
c \cdot \frac{-2}{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
 (* c (/ -2.0 (+ 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 c * (-2.0 / (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 28.5

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

  2. Simplified28.5

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

  4. Applied flip--_binary6428.5

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

  5. Simplified0.5

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

  6. Simplified0.5

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

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

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

  9. Applied times-frac_binary640.5

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

  10. Applied times-frac_binary640.4

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

  11. Simplified0.4

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

  12. Simplified0.4

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

  13. Final simplification0.4

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

Reproduce

herbie shell --seed 2020233 
(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)))