Average Error: 0.1 → 0.1
Time: 23.1s
Precision: 64
\[0.0 \lt m \land 0.0 \lt v \land v \lt 0.25\]
\[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right)\]
↓
\[\mathsf{fma}\left(-\sqrt{m}, \sqrt{m}, \sqrt{m} \cdot \sqrt{m}\right) \cdot \left(\frac{m}{\frac{v}{1 - m}} - 1\right) + \left(\frac{m}{\frac{v}{1 - m}} - 1\right) \cdot \mathsf{fma}\left(1, 1, -\sqrt{m} \cdot \sqrt{m}\right)\]
\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right)
↓
\mathsf{fma}\left(-\sqrt{m}, \sqrt{m}, \sqrt{m} \cdot \sqrt{m}\right) \cdot \left(\frac{m}{\frac{v}{1 - m}} - 1\right) + \left(\frac{m}{\frac{v}{1 - m}} - 1\right) \cdot \mathsf{fma}\left(1, 1, -\sqrt{m} \cdot \sqrt{m}\right)
double f(double m, double v) {
        double r1130509 = m;
        double r1130510 = 1.0;
        double r1130511 = r1130510 - r1130509;
        double r1130512 = r1130509 * r1130511;
        double r1130513 = v;
        double r1130514 = r1130512 / r1130513;
        double r1130515 = r1130514 - r1130510;
        double r1130516 = r1130515 * r1130511;
        return r1130516;
}
↓
double f(double m, double v) {
        double r1130517 = m;
        double r1130518 = sqrt(r1130517);
        double r1130519 = -r1130518;
        double r1130520 = r1130518 * r1130518;
        double r1130521 = fma(r1130519, r1130518, r1130520);
        double r1130522 = v;
        double r1130523 = 1.0;
        double r1130524 = r1130523 - r1130517;
        double r1130525 = r1130522 / r1130524;
        double r1130526 = r1130517 / r1130525;
        double r1130527 = r1130526 - r1130523;
        double r1130528 = r1130521 * r1130527;
        double r1130529 = 1.0;
        double r1130530 = -r1130520;
        double r1130531 = fma(r1130529, r1130523, r1130530);
        double r1130532 = r1130527 * r1130531;
        double r1130533 = r1130528 + r1130532;
        return r1130533;
}

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 associate-/l*0.1

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

  5. Applied add-sqr-sqrt0.1

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

  6. Applied *-un-lft-identity0.1

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

  7. Applied prod-diff0.1

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

  8. Applied distribute-rgt-in0.1

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

  9. Final simplification0.1

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

Reproduce

herbie shell --seed 2019171 +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)))