Average Error: 0.1 → 0.1
Time: 3.4s
Precision: binary64
\[\left(0 < m \land 0 < v\right) \land v < 0.25\]
\[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
↓
\[-1 + \left(m - \frac{m \cdot \mathsf{fma}\left(m, 2 - m, -1\right)}{v}\right) \]
(FPCore (m v) :precision binary64 (* (- (/ (* m (- 1.0 m)) v) 1.0) (- 1.0 m)))
↓
(FPCore (m v)
 :precision binary64
 (+ -1.0 (- m (/ (* m (fma m (- 2.0 m) -1.0)) v))))
double code(double m, double v) {
	return (((m * (1.0 - m)) / v) - 1.0) * (1.0 - m);
}
↓
double code(double m, double v) {
	return -1.0 + (m - ((m * fma(m, (2.0 - m), -1.0)) / v));
}

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(-1.0 + Float64(m - Float64(Float64(m * fma(m, Float64(2.0 - m), -1.0)) / v)))
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[(m - N[(N[(m * N[(m * N[(2.0 - m), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right)
↓
-1 + \left(m - \frac{m \cdot \mathsf{fma}\left(m, 2 - m, -1\right)}{v}\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. Taylor expanded in m around 0 0.2

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

  3. Taylor expanded in m around 0 0.2

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

  4. Simplified0.1

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

  5. Final simplification0.1

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

Reproduce

herbie shell --seed 2022186 
(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)))