Average Error: 28.8 → 0.5
Time: 4.2s
Precision: binary64
\[1.0536712127723509 \cdot 10^{-08} < a \land a < 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} < b \land b < 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} < c \land c < 94906265.62425156\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{c \cdot \frac{-a}{b + \sqrt{b \cdot b - \sqrt[3]{3 \cdot \left(a \cdot c\right)} \cdot \left(\sqrt[3]{3 \cdot \left(a \cdot c\right)} \cdot \sqrt[3]{3 \cdot \left(a \cdot c\right)}\right)}}}{a}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\frac{c \cdot \frac{-a}{b + \sqrt{b \cdot b - \sqrt[3]{3 \cdot \left(a \cdot c\right)} \cdot \left(\sqrt[3]{3 \cdot \left(a \cdot c\right)} \cdot \sqrt[3]{3 \cdot \left(a \cdot c\right)}\right)}}}{a}
(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))
↓
(FPCore (a b c)
 :precision binary64
 (/
  (*
   c
   (/
    (- a)
    (+
     b
     (sqrt
      (-
       (* b b)
       (*
        (cbrt (* 3.0 (* a c)))
        (* (cbrt (* 3.0 (* a c))) (cbrt (* 3.0 (* a c))))))))))
  a))
double code(double a, double b, double c) {
	return (((double) (((double) -(b)) + ((double) sqrt(((double) (((double) (b * b)) - ((double) (((double) (3.0 * a)) * c)))))))) / ((double) (3.0 * a)));
}
↓
double code(double a, double b, double c) {
	return (((double) (c * (((double) -(a)) / ((double) (b + ((double) sqrt(((double) (((double) (b * b)) - ((double) (((double) cbrt(((double) (3.0 * ((double) (a * c)))))) * ((double) (((double) cbrt(((double) (3.0 * ((double) (a * c)))))) * ((double) cbrt(((double) (3.0 * ((double) (a * c))))))))))))))))))) / 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 28.8

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

  2. Simplified28.8

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

  4. Applied flip--28.8

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

  5. Simplified0.6

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

  6. Simplified0.6

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

  8. Applied associate-/r*0.6

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

  9. Simplified0.4

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

  11. Applied add-cube-cbrt0.5

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

  12. Final simplification0.5

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

Reproduce

herbie shell --seed 2020198 
(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)))