Average Error: 52.4 → 6.5
Time: 7.0s
Precision: 64
\[4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt a \lt 20282409603651670423947251286016 \land 4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt b \lt 20282409603651670423947251286016 \land 4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt c \lt 20282409603651670423947251286016\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{-0.5}{b} \cdot \frac{a \cdot c}{a}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\frac{-0.5}{b} \cdot \frac{a \cdot c}{a}
double f(double a, double b, double c) {
        double r100429 = b;
        double r100430 = -r100429;
        double r100431 = r100429 * r100429;
        double r100432 = 3.0;
        double r100433 = a;
        double r100434 = r100432 * r100433;
        double r100435 = c;
        double r100436 = r100434 * r100435;
        double r100437 = r100431 - r100436;
        double r100438 = sqrt(r100437);
        double r100439 = r100430 + r100438;
        double r100440 = r100439 / r100434;
        return r100440;
}
↓
double f(double a, double b, double c) {
        double r100441 = -0.5;
        double r100442 = b;
        double r100443 = r100441 / r100442;
        double r100444 = a;
        double r100445 = c;
        double r100446 = r100444 * r100445;
        double r100447 = r100446 / r100444;
        double r100448 = r100443 * r100447;
        return r100448;
}

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 52.4

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

  2. Simplified52.4

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

  3. Taylor expanded around inf 6.6

    \[\leadsto \frac{\color{blue}{-0.5 \cdot \frac{a \cdot c}{b}}}{a}\]
  4. Using strategy rm

  5. Applied associate-*r/6.6

    \[\leadsto \frac{\color{blue}{\frac{-0.5 \cdot \left(a \cdot c\right)}{b}}}{a}\]

  6. Applied associate-/l/6.5

    \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(a \cdot c\right)}{a \cdot b}}\]

  7. Simplified6.5

    \[\leadsto \frac{-0.5 \cdot \left(a \cdot c\right)}{\color{blue}{b \cdot a}}\]
  8. Using strategy rm

  9. Applied times-frac6.5

    \[\leadsto \color{blue}{\frac{-0.5}{b} \cdot \frac{a \cdot c}{a}}\]

  10. Final simplification6.5

    \[\leadsto \frac{-0.5}{b} \cdot \frac{a \cdot c}{a}\]

Reproduce

herbie shell --seed 2019351 
(FPCore (a b c)
  :name "Cubic critical, wide range"
  :precision binary64
  :pre (and (< 4.9303800000000003e-32 a 2.02824e+31) (< 4.9303800000000003e-32 b 2.02824e+31) (< 4.9303800000000003e-32 c 2.02824e+31))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))