b parameter of renormalized beta distribution

Average Error: 0.1 → 0.1

Time: 37.6s

Precision: 64

\[0 \lt m \land 0 \lt v \land v \lt 0.25\]

Math

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

↓

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

C

double f(double m, double v) {
        double r738481 = m;
        double r738482 = 1.0;
        double r738483 = r738482 - r738481;
        double r738484 = r738481 * r738483;
        double r738485 = v;
        double r738486 = r738484 / r738485;
        double r738487 = r738486 - r738482;
        double r738488 = r738487 * r738483;
        return r738488;
}

↓

double f(double m, double v) {
        double r738489 = 1.0;
        double r738490 = m;
        double r738491 = r738489 - r738490;
        double r738492 = v;
        double r738493 = r738490 / r738492;
        double r738494 = -1.0;
        double r738495 = fma(r738491, r738493, r738494);
        double r738496 = r738495 * r738491;
        return r738496;
}

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. Simplified 0.1

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

3. Final simplification 0.1

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

Reproduce

herbie shell --seed 2019144 +o rules:numerics
(FPCore (m v)
  :name "b parameter of renormalized beta distribution"
  :pre (and (< 0 m) (< 0 v) (< v 0.25))
  (* (- (/ (* m (- 1 m)) v) 1) (- 1 m)))