Average Error: 0.1 → 0.1
Time: 2.5s
Precision: binary64
\[\left(0 < m \land 0 < v\right) \land v < 0.25\]
\[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
↓
\[m + \left(\frac{m}{\frac{v}{\mathsf{fma}\left(m, m + -2, 1\right)}} + -1\right) \]
(FPCore (m v) :precision binary64 (* (- (/ (* m (- 1.0 m)) v) 1.0) (- 1.0 m)))
↓
(FPCore (m v)
 :precision binary64
 (+ m (+ (/ m (/ v (fma m (+ m -2.0) 1.0))) -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 m + ((m / (v / fma(m, (m + -2.0), 1.0))) + -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 Float64(m + Float64(Float64(m / Float64(v / fma(m, Float64(m + -2.0), 1.0))) + -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[(m + N[(N[(m / N[(v / N[(m * N[(m + -2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]

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

Error

Derivation

  1. Initial program 0.1

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

  2. Simplified0.1

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

  3. Taylor expanded in v around 0 0.1

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

  4. Simplified0.1

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

  5. Taylor expanded in m around 0 0.1

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

  6. Taylor expanded in v around 0 0.1

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

  7. Simplified0.1

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

  8. Final simplification0.1

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

Reproduce

herbie shell --seed 2022204 
(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)))