Average Error: 28.9 → 0.4
Time: 6.5s
Precision: binary64
\[1.0536712127723509 \cdot 10^{-08} < a \land a < 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} < b \land b < 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} < c \land c < 94906265.62425156\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{-4 \cdot \left(a \cdot c\right)}{a} \cdot \frac{0.5}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + b \cdot b}}\]
\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)}{a} \cdot \frac{0.5}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + b \cdot b}}
(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)) a) (/ 0.5 (+ b (sqrt (+ (* -4.0 (* a c)) (* b b)))))))
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)) / a) * (0.5 / (b + sqrt((-4.0 * (a * c)) + (b * b))));
}

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 28.9

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

  2. Simplified28.9

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

  4. Applied flip--_binary6428.9

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

  5. Simplified27.9

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

  6. Simplified27.9

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

  8. Applied div-inv_binary6427.9

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

  9. Applied times-frac_binary6427.9

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

  10. Simplified0.4

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

  11. Simplified0.4

    \[\leadsto \frac{-4 \cdot \left(a \cdot c\right)}{a} \cdot \color{blue}{\frac{0.5}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + b \cdot b}}}\]

  12. Final simplification0.4

    \[\leadsto \frac{-4 \cdot \left(a \cdot c\right)}{a} \cdot \frac{0.5}{b + \sqrt{-4 \cdot \left(a \cdot c\right) + b \cdot b}}\]

Reproduce

herbie shell --seed 2021175 
(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)))