Average Error: 0.2 → 0.2
Time: 3.2s
Precision: binary64
\[\left(0 < m \land 0 < v\right) \land v < 0.25\]
\[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m \]
↓
\[\mathsf{fma}\left(m, \frac{m - {m}^{3}}{\mathsf{fma}\left(m, v, v\right)}, -m\right) \]
(FPCore (m v) :precision binary64 (* (- (/ (* m (- 1.0 m)) v) 1.0) m))
↓
(FPCore (m v)
 :precision binary64
 (fma m (/ (- m (pow m 3.0)) (fma m v v)) (- m)))
double code(double m, double v) {
	return (((m * (1.0 - m)) / v) - 1.0) * m;
}
↓
double code(double m, double v) {
	return fma(m, ((m - pow(m, 3.0)) / fma(m, v, v)), -m);
}

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

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

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

Error

Bits error versus m

Bits error versus v

Derivation

  1. Initial program 0.2

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

  2. Applied egg-rr0.2

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

  3. Taylor expanded in v around 0 6.9

    \[\leadsto \color{blue}{\frac{{m}^{2}}{v \cdot \left(1 + m\right)} - \left(\frac{{m}^{4}}{v \cdot \left(1 + m\right)} + m\right)} \]

  4. Simplified0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2022153 
(FPCore (m v)
  :name "a 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) m))