Average Error: 0.1 → 0.1
Time: 3.3s
Precision: binary64
\[\left(0 < m \land 0 < v\right) \land v < 0.25\]
\[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
↓
\[\mathsf{fma}\left(1, \frac{{m}^{3}}{v}, \left(m + \frac{m}{v}\right) - \mathsf{fma}\left(2, \frac{m \cdot m}{v}, 1\right)\right) \]
(FPCore (m v) :precision binary64 (* (- (/ (* m (- 1.0 m)) v) 1.0) (- 1.0 m)))
↓
(FPCore (m v)
 :precision binary64
 (fma 1.0 (/ (pow m 3.0) v) (- (+ m (/ m v)) (fma 2.0 (/ (* m m) v) 1.0))))
double code(double m, double v) {
	return (((m * (1.0 - m)) / v) - 1.0) * (1.0 - m);
}
↓
double code(double m, double v) {
	return fma(1.0, (pow(m, 3.0) / v), ((m + (m / v)) - fma(2.0, ((m * m) / v), 1.0)));
}

function code(m, v)
	return Float64(Float64(Float64(Float64(m * Float64(1.0 - m)) / v) - 1.0) * Float64(1.0 - m))
end
↓
function code(m, v)
	return fma(1.0, Float64((m ^ 3.0) / v), Float64(Float64(m + Float64(m / v)) - fma(2.0, Float64(Float64(m * m) / v), 1.0)))
end

code[m_, v_] := N[(N[(N[(N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision] - 1.0), $MachinePrecision] * N[(1.0 - m), $MachinePrecision]), $MachinePrecision]
↓
code[m_, v_] := N[(1.0 * N[(N[Power[m, 3.0], $MachinePrecision] / v), $MachinePrecision] + N[(N[(m + N[(m / v), $MachinePrecision]), $MachinePrecision] - N[(2.0 * N[(N[(m * m), $MachinePrecision] / v), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right)
↓
\mathsf{fma}\left(1, \frac{{m}^{3}}{v}, \left(m + \frac{m}{v}\right) - \mathsf{fma}\left(2, \frac{m \cdot m}{v}, 1\right)\right)

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. Taylor expanded in m around 0 0.1

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

  3. Applied egg-rr0.1

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

  4. Final simplification0.1

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

Reproduce

herbie shell --seed 2022148 
(FPCore (m v)
  :name "b parameter of renormalized beta distribution"
  :precision binary64
  :pre (and (and (< 0.0 m) (< 0.0 v)) (< v 0.25))
  (* (- (/ (* m (- 1.0 m)) v) 1.0) (- 1.0 m)))