b parameter of renormalized beta distribution

Average Error: 0.1 → 0.1

Time: 8.6s

Precision: 64

\[0.0 \lt m \land 0.0 \lt v \land v \lt 0.25\]

Math

\[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right)\]

↓

\[\left(\frac{1 \cdot m + \left(-m\right) \cdot m}{v} - 1\right) \cdot 1 + \left(\left(-1\right) + \frac{\sqrt{m}}{\frac{\frac{v}{1 - m}}{\sqrt{m}}}\right) \cdot \left(-m\right)\]

C

double f(double m, double v) {
        double r25766 = m;
        double r25767 = 1.0;
        double r25768 = r25767 - r25766;
        double r25769 = r25766 * r25768;
        double r25770 = v;
        double r25771 = r25769 / r25770;
        double r25772 = r25771 - r25767;
        double r25773 = r25772 * r25768;
        return r25773;
}

↓

double f(double m, double v) {
        double r25774 = 1.0;
        double r25775 = m;
        double r25776 = r25774 * r25775;
        double r25777 = -r25775;
        double r25778 = r25777 * r25775;
        double r25779 = r25776 + r25778;
        double r25780 = v;
        double r25781 = r25779 / r25780;
        double r25782 = r25781 - r25774;
        double r25783 = r25782 * r25774;
        double r25784 = -r25774;
        double r25785 = sqrt(r25775);
        double r25786 = r25774 - r25775;
        double r25787 = r25780 / r25786;
        double r25788 = r25787 / r25785;
        double r25789 = r25785 / r25788;
        double r25790 = r25784 + r25789;
        double r25791 = r25790 * r25777;
        double r25792 = r25783 + r25791;
        return r25792;
}

Error

Bits error versus m

Bits error versus v

Derivation

1. Initial program 0.1

   \[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right)\]
2. Using strategy rm

3. Applied sub-neg 0.1

   \[\leadsto \left(\frac{m \cdot \color{blue}{\left(1 + \left(-m\right)\right)}}{v} - 1\right) \cdot \left(1 - m\right)\]

4. Applied distribute-lft-in 0.1

   \[\leadsto \left(\frac{\color{blue}{m \cdot 1 + m \cdot \left(-m\right)}}{v} - 1\right) \cdot \left(1 - m\right)\]

5. Simplified 0.1

   \[\leadsto \left(\frac{\color{blue}{1 \cdot m} + m \cdot \left(-m\right)}{v} - 1\right) \cdot \left(1 - m\right)\]

6. Simplified 0.1

   \[\leadsto \left(\frac{1 \cdot m + \color{blue}{\left(-m\right) \cdot m}}{v} - 1\right) \cdot \left(1 - m\right)\]
7. Using strategy rm

8. Applied sub-neg 0.1

   \[\leadsto \left(\frac{1 \cdot m + \left(-m\right) \cdot m}{v} - 1\right) \cdot \color{blue}{\left(1 + \left(-m\right)\right)}\]

9. Applied distribute-lft-in 0.1

   \[\leadsto \color{blue}{\left(\frac{1 \cdot m + \left(-m\right) \cdot m}{v} - 1\right) \cdot 1 + \left(\frac{1 \cdot m + \left(-m\right) \cdot m}{v} - 1\right) \cdot \left(-m\right)}\]

10. Simplified 0.1

    \[\leadsto \left(\frac{1 \cdot m + \left(-m\right) \cdot m}{v} - 1\right) \cdot 1 + \color{blue}{\left(\left(-1\right) + \frac{m}{\frac{v}{1 - m}}\right) \cdot \left(-m\right)}\]
11. Using strategy rm

12. Applied add-sqr-sqrt 0.1

    \[\leadsto \left(\frac{1 \cdot m + \left(-m\right) \cdot m}{v} - 1\right) \cdot 1 + \left(\left(-1\right) + \frac{\color{blue}{\sqrt{m} \cdot \sqrt{m}}}{\frac{v}{1 - m}}\right) \cdot \left(-m\right)\]

13. Applied associate-/l* 0.1

    \[\leadsto \left(\frac{1 \cdot m + \left(-m\right) \cdot m}{v} - 1\right) \cdot 1 + \left(\left(-1\right) + \color{blue}{\frac{\sqrt{m}}{\frac{\frac{v}{1 - m}}{\sqrt{m}}}}\right) \cdot \left(-m\right)\]

14. Final simplification 0.1

    \[\leadsto \left(\frac{1 \cdot m + \left(-m\right) \cdot m}{v} - 1\right) \cdot 1 + \left(\left(-1\right) + \frac{\sqrt{m}}{\frac{\frac{v}{1 - m}}{\sqrt{m}}}\right) \cdot \left(-m\right)\]

Reproduce

herbie shell --seed 2019346 
(FPCore (m v)
  :name "b parameter of renormalized beta distribution"
  :precision binary64
  :pre (and (< 0.0 m) (< 0.0 v) (< v 0.25))
  (* (- (/ (* m (- 1 m)) v) 1) (- 1 m)))