Cubic critical, medium range

Average Error: 43.7 → 0.2

Time: 11.2s

Precision: 64

\[1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt a \lt 9007199254740992 \land 1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt b \lt 9007199254740992 \land 1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt c \lt 9007199254740992\]

Math

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

↓

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

C

double f(double a, double b, double c) {
        double r69317 = b;
        double r69318 = -r69317;
        double r69319 = r69317 * r69317;
        double r69320 = 3.0;
        double r69321 = a;
        double r69322 = r69320 * r69321;
        double r69323 = c;
        double r69324 = r69322 * r69323;
        double r69325 = r69319 - r69324;
        double r69326 = sqrt(r69325);
        double r69327 = r69318 + r69326;
        double r69328 = r69327 / r69322;
        return r69328;
}

↓

double f(double a, double b, double c) {
        double r69329 = c;
        double r69330 = b;
        double r69331 = -r69330;
        double r69332 = r69330 * r69330;
        double r69333 = 3.0;
        double r69334 = a;
        double r69335 = r69333 * r69334;
        double r69336 = r69335 * r69329;
        double r69337 = r69332 - r69336;
        double r69338 = sqrt(r69337);
        double r69339 = r69331 - r69338;
        double r69340 = r69329 / r69339;
        return r69340;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 43.7

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

3. Applied flip-+ 43.7

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

4. Simplified 0.5

   \[\leadsto \frac{\frac{\color{blue}{0 + 3 \cdot \left(a \cdot c\right)}}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\]
5. Using strategy rm

6. Applied div-inv 0.6

   \[\leadsto \frac{\color{blue}{\left(0 + 3 \cdot \left(a \cdot c\right)\right) \cdot \frac{1}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}}{3 \cdot a}\]

7. Applied times-frac 0.5

   \[\leadsto \color{blue}{\frac{0 + 3 \cdot \left(a \cdot c\right)}{3} \cdot \frac{\frac{1}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{a}}\]

8. Simplified 0.5

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

10. Applied pow1 0.5

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

11. Applied pow1 0.5

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

12. Applied pow-prod-down 0.5

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

13. Simplified 0.4

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

15. Applied times-frac 0.2

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

16. Simplified 0.2

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

17. Final simplification 0.2

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

Reproduce

herbie shell --seed 2019308 
(FPCore (a b c)
  :name "Cubic critical, medium range"
  :precision binary64
  :pre (and (< 1.11022e-16 a 9.0072e15) (< 1.11022e-16 b 9.0072e15) (< 1.11022e-16 c 9.0072e15))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))