Average Error: 28.9 → 0.3
Time: 1.2min
Precision: binary64
Cost: 1856
\[1.0536712127723509 \cdot 10^{-08} < a \land a < 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} < b \land b < 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} < c \land c < 94906265.62425156\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{c}{\left(-b\right) - \sqrt{\frac{{b}^{4} - a \cdot \left(\left(c \cdot c\right) \cdot \left(a \cdot 9\right)\right)}{c \cdot \left(a \cdot 3\right) + b \cdot b}}}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\frac{c}{\left(-b\right) - \sqrt{\frac{{b}^{4} - a \cdot \left(\left(c \cdot c\right) \cdot \left(a \cdot 9\right)\right)}{c \cdot \left(a \cdot 3\right) + b \cdot b}}}
(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))
↓
(FPCore (a b c)
 :precision binary64
 (/
  c
  (-
   (- b)
   (sqrt
    (/
     (- (pow b 4.0) (* a (* (* c c) (* a 9.0))))
     (+ (* c (* a 3.0)) (* b b)))))))
double code(double a, double b, double c) {
	return (-b + sqrt((b * b) - ((3.0 * a) * c))) / (3.0 * a);
}
↓
double code(double a, double b, double c) {
	return c / (-b - sqrt((pow(b, 4.0) - (a * ((c * c) * (a * 9.0)))) / ((c * (a * 3.0)) + (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
Alternative 1
Accuracy0.3
Cost960
\[\frac{c}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)}}\]
Alternative 2
Accuracy0.3
Cost1216
\[a \cdot \frac{\frac{c}{-\left(b + \sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)}\right)}}{a}\]

Derivation

  1. Initial program 28.9

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

  3. Applied flip-+_binary64_175728.9

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

  4. Simplified0.5

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

  6. Applied *-un-lft-identity_binary64_17830.5

    \[\leadsto \frac{\frac{\left(3 \cdot a\right) \cdot c}{\color{blue}{1 \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}}}{3 \cdot a}\]

  7. Applied times-frac_binary64_17890.3

    \[\leadsto \frac{\color{blue}{\frac{3 \cdot a}{1} \cdot \frac{c}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}}{3 \cdot a}\]

  8. Applied times-frac_binary64_17890.4

    \[\leadsto \color{blue}{\frac{\frac{3 \cdot a}{1}}{3} \cdot \frac{\frac{c}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{a}}\]

  9. Simplified0.3

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

  10. Simplified0.3

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

  12. Applied *-un-lft-identity_binary64_17830.3

    \[\leadsto \color{blue}{\left(1 \cdot a\right)} \cdot \frac{\frac{c}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)}}}{a}\]

  13. Applied associate-*l*_binary64_17240.3

    \[\leadsto \color{blue}{1 \cdot \left(a \cdot \frac{\frac{c}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)}}}{a}\right)}\]

  14. Simplified0.3

    \[\leadsto 1 \cdot \color{blue}{\frac{c}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)}}}\]
  15. Using strategy rm

  16. Applied flip--_binary64_17580.3

    \[\leadsto 1 \cdot \frac{c}{\left(-b\right) - \sqrt{\color{blue}{\frac{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(c \cdot \left(3 \cdot a\right)\right) \cdot \left(c \cdot \left(3 \cdot a\right)\right)}{b \cdot b + c \cdot \left(3 \cdot a\right)}}}}\]

  17. Simplified0.3

    \[\leadsto 1 \cdot \frac{c}{\left(-b\right) - \sqrt{\frac{\color{blue}{{b}^{4} - a \cdot \left(\left(c \cdot c\right) \cdot \left(a \cdot 9\right)\right)}}{b \cdot b + c \cdot \left(3 \cdot a\right)}}}\]

  18. Simplified0.3

    \[\leadsto 1 \cdot \frac{c}{\left(-b\right) - \sqrt{\frac{{b}^{4} - a \cdot \left(\left(c \cdot c\right) \cdot \left(a \cdot 9\right)\right)}{\color{blue}{c \cdot \left(3 \cdot a\right) + b \cdot b}}}}\]

  19. Final simplification0.3

    \[\leadsto \frac{c}{\left(-b\right) - \sqrt{\frac{{b}^{4} - a \cdot \left(\left(c \cdot c\right) \cdot \left(a \cdot 9\right)\right)}{c \cdot \left(a \cdot 3\right) + b \cdot b}}}\]

Reproduce

herbie shell --seed 2020322 
(FPCore (a b c)
  :name "Cubic critical, 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) (* (* 3.0 a) c)))) (* 3.0 a)))