Average Error: 0.1 → 0.1
Time: 21.2s
Precision: 64
\[0.0 \lt m \land 0.0 \lt v \land v \lt 0.25\]
\[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right)\]
↓
\[\left(\mathsf{fma}\left(m, 1, \frac{m}{\frac{v}{m \cdot m}}\right) - \frac{1 \cdot \left(m \cdot m\right)}{v}\right) + 1 \cdot \left(\frac{\left(1 - m\right) \cdot m}{v} - 1\right)\]
\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right)
↓
\left(\mathsf{fma}\left(m, 1, \frac{m}{\frac{v}{m \cdot m}}\right) - \frac{1 \cdot \left(m \cdot m\right)}{v}\right) + 1 \cdot \left(\frac{\left(1 - m\right) \cdot m}{v} - 1\right)
double f(double m, double v) {
        double r1037083 = m;
        double r1037084 = 1.0;
        double r1037085 = r1037084 - r1037083;
        double r1037086 = r1037083 * r1037085;
        double r1037087 = v;
        double r1037088 = r1037086 / r1037087;
        double r1037089 = r1037088 - r1037084;
        double r1037090 = r1037089 * r1037085;
        return r1037090;
}
↓
double f(double m, double v) {
        double r1037091 = m;
        double r1037092 = 1.0;
        double r1037093 = v;
        double r1037094 = r1037091 * r1037091;
        double r1037095 = r1037093 / r1037094;
        double r1037096 = r1037091 / r1037095;
        double r1037097 = fma(r1037091, r1037092, r1037096);
        double r1037098 = r1037092 * r1037094;
        double r1037099 = r1037098 / r1037093;
        double r1037100 = r1037097 - r1037099;
        double r1037101 = r1037092 - r1037091;
        double r1037102 = r1037101 * r1037091;
        double r1037103 = r1037102 / r1037093;
        double r1037104 = r1037103 - r1037092;
        double r1037105 = r1037092 * r1037104;
        double r1037106 = r1037100 + r1037105;
        return r1037106;
}

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-neg0.1

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

  4. Applied distribute-rgt-in0.1

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

  5. Taylor expanded around 0 0.1

    \[\leadsto 1 \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) + \color{blue}{\left(\left(1 \cdot m + \frac{{m}^{3}}{v}\right) - 1 \cdot \frac{{m}^{2}}{v}\right)}\]

  6. Simplified0.1

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

  8. Applied associate-/l*0.1

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

  9. Final simplification0.1

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

Reproduce

herbie shell --seed 2019169 +o rules:numerics
(FPCore (m v)
  :name "b parameter of renormalized beta distribution"
  :pre (and (< 0.0 m) (< 0.0 v) (< v 0.25))
  (* (- (/ (* m (- 1.0 m)) v) 1.0) (- 1.0 m)))