b parameter of renormalized beta distribution

Average Error: 0.1 → 0.1

Time: 5.3s

Precision: 64

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

Math

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

↓

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

C

double f(double m, double v) {
        double r17170 = m;
        double r17171 = 1.0;
        double r17172 = r17171 - r17170;
        double r17173 = r17170 * r17172;
        double r17174 = v;
        double r17175 = r17173 / r17174;
        double r17176 = r17175 - r17171;
        double r17177 = r17176 * r17172;
        return r17177;
}

↓

double f(double m, double v) {
        double r17178 = 1.0;
        double r17179 = m;
        double r17180 = v;
        double r17181 = 1.0;
        double r17182 = r17181 * r17181;
        double r17183 = r17179 * r17179;
        double r17184 = r17182 - r17183;
        double r17185 = r17180 / r17184;
        double r17186 = r17179 / r17185;
        double r17187 = r17181 + r17179;
        double r17188 = r17186 / r17187;
        double r17189 = r17178 * r17188;
        double r17190 = r17189 - r17181;
        double r17191 = r17181 - r17179;
        double r17192 = r17190 * r17191;
        return r17192;
}

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 *-un-lft-identity 0.1

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

9. Applied *-un-lft-identity 0.1

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

10. Applied times-frac 0.1

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

11. Simplified 0.1

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

12. Simplified 0.1

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

14. Applied flip-- 0.1

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

15. Applied associate-/r/ 0.1

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

16. Applied associate-/r* 0.1

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

17. Final simplification 0.1

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

Reproduce

herbie shell --seed 2019354 
(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)))