Average Error: 0.2 → 0.2
Time: 2.7s
Precision: binary64
\[\left(0 < m \land 0 < v\right) \land v < 0.25\]
\[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m \]
↓
\[-\mathsf{fma}\left(\frac{m}{v}, m \cdot m - m, m\right) \]
(FPCore (m v) :precision binary64 (* (- (/ (* m (- 1.0 m)) v) 1.0) m))
↓
(FPCore (m v) :precision binary64 (- (fma (/ m v) (- (* m m) m) 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 / v), ((m * m) - m), 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 Float64(-fma(Float64(m / v), Float64(Float64(m * m) - m), 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[(N[(m / v), $MachinePrecision] * N[(N[(m * m), $MachinePrecision] - m), $MachinePrecision] + m), $MachinePrecision])

\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m
↓
-\mathsf{fma}\left(\frac{m}{v}, m \cdot m - m, 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-rr0.4

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

  3. Taylor expanded in m around 0 7.1

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

  4. Simplified0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2022169 
(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))