Average Error: 28.6 → 0.3
Time: 4.1s
Precision: binary64
\[1.0536712127723509 \cdot 10^{-08} < a \land a < 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} < b \land b < 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} < c \land c < 94906265.62425156\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[-2 \cdot \frac{c}{b + \sqrt{\frac{{b}^{4} - a \cdot \left(c \cdot \left(\left(c \cdot a\right) \cdot 16\right)\right)}{b \cdot b + c \cdot \left(4 \cdot a\right)}}}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
-2 \cdot \frac{c}{b + \sqrt{\frac{{b}^{4} - a \cdot \left(c \cdot \left(\left(c \cdot a\right) \cdot 16\right)\right)}{b \cdot b + c \cdot \left(4 \cdot a\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
   (+
    b
    (sqrt
     (/
      (- (pow b 4.0) (* a (* c (* (* c a) 16.0))))
      (+ (* b b) (* c (* 4.0 a)))))))))
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) (-2.0 * (c / ((double) (b + ((double) sqrt((((double) (((double) pow(b, 4.0)) - ((double) (a * ((double) (c * ((double) (((double) (c * a)) * 16.0)))))))) / ((double) (((double) (b * b)) + ((double) (c * ((double) (4.0 * a))))))))))))));
}

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

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

  6. Simplified0.4

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

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

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

  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 times-frac_binary640.3

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

  11. Simplified0.3

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

  12. Simplified0.3

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

  14. Applied flip--_binary640.3

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

  15. Simplified0.3

    \[\leadsto 1 \cdot \left(-2 \cdot \frac{c}{b + \sqrt{\frac{\color{blue}{{b}^{4} - a \cdot \left(c \cdot \left(\left(c \cdot a\right) \cdot 16\right)\right)}}{b \cdot b + c \cdot \left(a \cdot 4\right)}}}\right)\]

  16. Final simplification0.3

    \[\leadsto -2 \cdot \frac{c}{b + \sqrt{\frac{{b}^{4} - a \cdot \left(c \cdot \left(\left(c \cdot a\right) \cdot 16\right)\right)}{b \cdot b + c \cdot \left(4 \cdot a\right)}}}\]

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)))