Quadratic roots, wide range

Average Error: 52.5 → 0.4

Time: 9.5s

Precision: 64

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

\[4.93038 \cdot 10^{-32} \lt a \lt 2.02824 \cdot 10^{31} \land 4.93038 \cdot 10^{-32} \lt b \lt 2.02824 \cdot 10^{31} \land 4.93038 \cdot 10^{-32} \lt c \lt 2.02824 \cdot 10^{31}\]

Math

\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]

↓

\[\frac{1}{2} \cdot \frac{4 \cdot \left(a \cdot c\right)}{-\left(\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} \cdot a + b \cdot a\right)}\]

C

double f(double a, double b, double c) {
        double r32749 = b;
        double r32750 = -r32749;
        double r32751 = r32749 * r32749;
        double r32752 = 4.0;
        double r32753 = a;
        double r32754 = r32752 * r32753;
        double r32755 = c;
        double r32756 = r32754 * r32755;
        double r32757 = r32751 - r32756;
        double r32758 = sqrt(r32757);
        double r32759 = r32750 + r32758;
        double r32760 = 2.0;
        double r32761 = r32760 * r32753;
        double r32762 = r32759 / r32761;
        return r32762;
}

↓

double f(double a, double b, double c) {
        double r32763 = 1.0;
        double r32764 = 2.0;
        double r32765 = r32763 / r32764;
        double r32766 = 4.0;
        double r32767 = a;
        double r32768 = c;
        double r32769 = r32767 * r32768;
        double r32770 = r32766 * r32769;
        double r32771 = b;
        double r32772 = 2.0;
        double r32773 = pow(r32771, r32772);
        double r32774 = r32773 - r32770;
        double r32775 = sqrt(r32774);
        double r32776 = r32775 * r32767;
        double r32777 = r32771 * r32767;
        double r32778 = r32776 + r32777;
        double r32779 = -r32778;
        double r32780 = r32770 / r32779;
        double r32781 = r32765 * r32780;
        return r32781;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 52.5

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
2. Using strategy rm

3. Applied flip-+ 52.5

   \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}{2 \cdot a}\]

4. Simplified 0.4

   \[\leadsto \frac{\frac{\color{blue}{b \cdot \left(b - b\right) + 4 \cdot \left(a \cdot c\right)}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\]
5. Using strategy rm

6. Applied add-cbrt-cube 0.4

   \[\leadsto \frac{\frac{b \cdot \left(b - b\right) + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{\color{blue}{\sqrt[3]{\left(\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right) \cdot \left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)\right) \cdot \left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)}}}}}{2 \cdot a}\]

7. Simplified 0.4

   \[\leadsto \frac{\frac{b \cdot \left(b - b\right) + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{\sqrt[3]{\color{blue}{{\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)}^{3}}}}}}{2 \cdot a}\]
8. Using strategy rm

9. Applied *-un-lft-identity 0.4

   \[\leadsto \frac{\frac{b \cdot \left(b - b\right) + 4 \cdot \left(a \cdot c\right)}{\color{blue}{1 \cdot \left(\left(-b\right) - \sqrt{\sqrt[3]{{\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)}^{3}}}\right)}}}{2 \cdot a}\]

10. Applied *-un-lft-identity 0.4

    \[\leadsto \frac{\frac{\color{blue}{1 \cdot \left(b \cdot \left(b - b\right) + 4 \cdot \left(a \cdot c\right)\right)}}{1 \cdot \left(\left(-b\right) - \sqrt{\sqrt[3]{{\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)}^{3}}}\right)}}{2 \cdot a}\]

11. Applied times-frac 0.4

    \[\leadsto \frac{\color{blue}{\frac{1}{1} \cdot \frac{b \cdot \left(b - b\right) + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{\sqrt[3]{{\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)}^{3}}}}}}{2 \cdot a}\]

12. Applied times-frac 0.4

    \[\leadsto \color{blue}{\frac{\frac{1}{1}}{2} \cdot \frac{\frac{b \cdot \left(b - b\right) + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{\sqrt[3]{{\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)}^{3}}}}}{a}}\]

13. Simplified 0.4

    \[\leadsto \color{blue}{\frac{1}{2}} \cdot \frac{\frac{b \cdot \left(b - b\right) + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{\sqrt[3]{{\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)}^{3}}}}}{a}\]

14. Simplified 0.4

    \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{0 + 4 \cdot \left(a \cdot c\right)}{a \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)}}\]
15. Using strategy rm

16. Applied sub-neg 0.4

    \[\leadsto \frac{1}{2} \cdot \frac{0 + 4 \cdot \left(a \cdot c\right)}{a \cdot \color{blue}{\left(\left(-b\right) + \left(-\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)\right)}}\]

17. Applied distribute-lft-in 0.4

    \[\leadsto \frac{1}{2} \cdot \frac{0 + 4 \cdot \left(a \cdot c\right)}{\color{blue}{a \cdot \left(-b\right) + a \cdot \left(-\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)}}\]

18. Simplified 0.4

    \[\leadsto \frac{1}{2} \cdot \frac{0 + 4 \cdot \left(a \cdot c\right)}{\color{blue}{\left(-b\right) \cdot a} + a \cdot \left(-\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)}\]

19. Simplified 0.4

    \[\leadsto \frac{1}{2} \cdot \frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) \cdot a + \color{blue}{\left(-\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right) \cdot a}}\]

20. Final simplification 0.4

    \[\leadsto \frac{1}{2} \cdot \frac{4 \cdot \left(a \cdot c\right)}{-\left(\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} \cdot a + b \cdot a\right)}\]

Reproduce

herbie shell --seed 2020042 
(FPCore (a b c)
  :name "Quadratic roots, 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) (* (* 4 a) c)))) (* 2 a)))