Average Error: 28.4 → 0.5
Time: 6.5s
Precision: binary64
\[1.0536712127723509 \cdot 10^{-08} < a \land a < 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} < b \land b < 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} < c \land c < 94906265.62425156\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{\frac{a \cdot \left(c \cdot -4\right)}{b + \sqrt{\frac{-64 \cdot {\left(a \cdot c\right)}^{3} + {b}^{6}}{{b}^{4} + c \cdot \left(\left(a \cdot 4\right) \cdot \left(b \cdot b + c \cdot \left(a \cdot 4\right)\right)\right)}}}}{a \cdot 2}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\frac{\frac{a \cdot \left(c \cdot -4\right)}{b + \sqrt{\frac{-64 \cdot {\left(a \cdot c\right)}^{3} + {b}^{6}}{{b}^{4} + c \cdot \left(\left(a \cdot 4\right) \cdot \left(b \cdot b + c \cdot \left(a \cdot 4\right)\right)\right)}}}}{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
 (/
  (/
   (* a (* c -4.0))
   (+
    b
    (sqrt
     (/
      (+ (* -64.0 (pow (* a c) 3.0)) (pow b 6.0))
      (+ (pow b 4.0) (* c (* (* a 4.0) (+ (* 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 ((a * (c * -4.0)) / (b + sqrt(((-64.0 * pow((a * c), 3.0)) + pow(b, 6.0)) / (pow(b, 4.0) + (c * ((a * 4.0) * ((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. 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 flip3--_binary640.5

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

  9. Simplified0.5

    \[\leadsto \frac{\frac{a \cdot \left(c \cdot -4\right)}{b + \sqrt{\frac{\color{blue}{-64 \cdot {\left(a \cdot c\right)}^{3} + {b}^{6}}}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(4 \cdot a\right) \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c\right)\right)}}}}{a \cdot 2}\]

  10. Simplified0.5

    \[\leadsto \frac{\frac{a \cdot \left(c \cdot -4\right)}{b + \sqrt{\frac{-64 \cdot {\left(a \cdot c\right)}^{3} + {b}^{6}}{\color{blue}{{b}^{4} + c \cdot \left(\left(a \cdot 4\right) \cdot \left(b \cdot b + c \cdot \left(a \cdot 4\right)\right)\right)}}}}}{a \cdot 2}\]

  11. Final simplification0.5

    \[\leadsto \frac{\frac{a \cdot \left(c \cdot -4\right)}{b + \sqrt{\frac{-64 \cdot {\left(a \cdot c\right)}^{3} + {b}^{6}}{{b}^{4} + c \cdot \left(\left(a \cdot 4\right) \cdot \left(b \cdot b + c \cdot \left(a \cdot 4\right)\right)\right)}}}}{a \cdot 2}\]

Reproduce

herbie shell --seed 2020275 
(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)))