Cubic critical, wide range

Average Error: 52.1 → 0.5

Time: 12.1s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

\[4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt a \lt 20282409603651670423947251286016 \land 4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt b \lt 20282409603651670423947251286016 \land 4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt c \lt 20282409603651670423947251286016\]

Math

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

↓

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

C

double f(double a, double b, double c) {
        double r80328 = b;
        double r80329 = -r80328;
        double r80330 = r80328 * r80328;
        double r80331 = 3.0;
        double r80332 = a;
        double r80333 = r80331 * r80332;
        double r80334 = c;
        double r80335 = r80333 * r80334;
        double r80336 = r80330 - r80335;
        double r80337 = sqrt(r80336);
        double r80338 = r80329 + r80337;
        double r80339 = r80338 / r80333;
        return r80339;
}

↓

double f(double a, double b, double c) {
        double r80340 = c;
        double r80341 = a;
        double r80342 = r80340 * r80341;
        double r80343 = 1.0;
        double r80344 = cbrt(r80343);
        double r80345 = b;
        double r80346 = -r80345;
        double r80347 = r80345 * r80345;
        double r80348 = 3.0;
        double r80349 = r80348 * r80341;
        double r80350 = r80349 * r80340;
        double r80351 = r80347 - r80350;
        double r80352 = sqrt(r80351);
        double r80353 = r80346 - r80352;
        double r80354 = r80344 / r80353;
        double r80355 = r80354 / r80341;
        double r80356 = r80342 * r80355;
        return r80356;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 52.1

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

3. Applied flip-+ 52.1

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

4. Simplified 0.5

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

6. Applied div-inv 0.6

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

7. Applied times-frac 0.5

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

8. Simplified 0.5

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

10. Applied *-un-lft-identity 0.5

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

11. Applied *-un-lft-identity 0.5

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

12. Applied add-cube-cbrt 0.5

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

13. Applied times-frac 0.5

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

14. Applied times-frac 0.5

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

15. Applied associate-*r* 0.5

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

16. Simplified 0.5

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

17. Final simplification 0.5

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

Reproduce

herbie shell --seed 2019350 +o rules:numerics
(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)))