Average Error: 28.6 → 0.3
Time: 4.7s
Precision: binary64
\[1.0536712127723509 \cdot 10^{-08} < a \land a < 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} < b \land b < 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} < c \land c < 94906265.62425156\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\left(-4\right) \cdot \frac{\frac{c}{b + \sqrt{\frac{{b}^{4} - \left(c \cdot \left(4 \cdot a\right)\right) \cdot \left(c \cdot \left(4 \cdot a\right)\right)}{c \cdot \left(4 \cdot a\right) + b \cdot b}}}}{2}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\left(-4\right) \cdot \frac{\frac{c}{b + \sqrt{\frac{{b}^{4} - \left(c \cdot \left(4 \cdot a\right)\right) \cdot \left(c \cdot \left(4 \cdot a\right)\right)}{c \cdot \left(4 \cdot a\right) + b \cdot b}}}}{2}
(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)
  (/
   (/
    c
    (+
     b
     (sqrt
      (/
       (- (pow b 4.0) (* (* c (* 4.0 a)) (* c (* 4.0 a))))
       (+ (* c (* 4.0 a)) (* b b))))))
   2.0)))
double code(double a, double b, double c) {
	return (((double) (((double) -(b)) + ((double) sqrt(((double) (((double) (b * b)) - ((double) (((double) (4.0 * a)) * c)))))))) / ((double) (2.0 * a)));
}
↓
double code(double a, double b, double c) {
	return ((double) (((double) -(4.0)) * ((c / ((double) (b + ((double) sqrt((((double) (((double) pow(b, 4.0)) - ((double) (((double) (c * ((double) (4.0 * a)))) * ((double) (c * ((double) (4.0 * a)))))))) / ((double) (((double) (c * ((double) (4.0 * a)))) + ((double) (b * b)))))))))) / 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 28.6

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

  2. Simplified28.6

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

  4. Applied flip--_binary6428.6

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

  5. Simplified0.4

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

  6. Simplified0.4

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

  8. Applied *-un-lft-identity_binary640.4

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

  9. Applied distribute-lft-neg-in_binary640.4

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

  10. Applied times-frac_binary640.3

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

  11. Applied times-frac_binary640.3

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

  12. Simplified0.3

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

  14. Applied flip--_binary640.3

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

  15. Simplified0.3

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

  16. Final simplification0.3

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

Reproduce

herbie shell --seed 2020210 
(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)))