Cubic critical, medium range

Average Error: 43.8 → 0.6

Time: 7.1s

Precision: 64

\[1.11022 \cdot 10^{-16} \lt a \lt 9.0072 \cdot 10^{15} \land 1.11022 \cdot 10^{-16} \lt b \lt 9.0072 \cdot 10^{15} \land 1.11022 \cdot 10^{-16} \lt c \lt 9.0072 \cdot 10^{15}\]

Math

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

↓

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

C

double f(double a, double b, double c) {
        double r106752 = b;
        double r106753 = -r106752;
        double r106754 = r106752 * r106752;
        double r106755 = 3.0;
        double r106756 = a;
        double r106757 = r106755 * r106756;
        double r106758 = c;
        double r106759 = r106757 * r106758;
        double r106760 = r106754 - r106759;
        double r106761 = sqrt(r106760);
        double r106762 = r106753 + r106761;
        double r106763 = r106762 / r106757;
        return r106763;
}

↓

double f(double a, double b, double c) {
        double r106764 = 1.0;
        double r106765 = cbrt(r106764);
        double r106766 = r106765 * r106765;
        double r106767 = r106764 / r106766;
        double r106768 = 3.0;
        double r106769 = a;
        double r106770 = r106768 * r106769;
        double r106771 = r106770 / r106768;
        double r106772 = r106767 / r106771;
        double r106773 = r106764 / r106765;
        double r106774 = b;
        double r106775 = -r106774;
        double r106776 = r106774 * r106774;
        double r106777 = c;
        double r106778 = r106770 * r106777;
        double r106779 = r106776 - r106778;
        double r106780 = sqrt(r106779);
        double r106781 = r106775 - r106780;
        double r106782 = r106769 * r106777;
        double r106783 = r106781 / r106782;
        double r106784 = r106773 / r106783;
        double r106785 = r106772 * r106784;
        return r106785;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 43.8

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

3. Applied flip-+ 43.8

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

4. Simplified 0.5

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

6. Applied *-un-lft-identity 0.5

   \[\leadsto \frac{\frac{\left({b}^{2} - {b}^{2}\right) + 3 \cdot \left(a \cdot c\right)}{\color{blue}{1 \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}}}{3 \cdot a}\]

7. Applied *-un-lft-identity 0.5

   \[\leadsto \frac{\frac{\color{blue}{1 \cdot \left(\left({b}^{2} - {b}^{2}\right) + 3 \cdot \left(a \cdot c\right)\right)}}{1 \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}}{3 \cdot a}\]

8. Applied times-frac 0.5

   \[\leadsto \frac{\color{blue}{\frac{1}{1} \cdot \frac{\left({b}^{2} - {b}^{2}\right) + 3 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}}{3 \cdot a}\]

9. Applied associate-/l* 0.6

   \[\leadsto \color{blue}{\frac{\frac{1}{1}}{\frac{3 \cdot a}{\frac{\left({b}^{2} - {b}^{2}\right) + 3 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}}}\]

10. Simplified 0.6

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

12. Applied times-frac 0.5

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

13. Applied add-cube-cbrt 0.5

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

14. Applied add-sqr-sqrt 0.5

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

15. Applied times-frac 0.5

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

16. Applied times-frac 0.6

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

17. Simplified 0.6

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

18. Simplified 0.6

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

19. Final simplification 0.6

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

Reproduce

herbie shell --seed 2020100 
(FPCore (a b c)
  :name "Cubic critical, medium range"
  :precision binary64
  :pre (and (< 1.11022e-16 a 9.0072e+15) (< 1.11022e-16 b 9.0072e+15) (< 1.11022e-16 c 9.0072e+15))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))