Average Error: 0.1 → 0.1
Time: 3.2s
Precision: binary64
\[\left(0 < m \land 0 < v\right) \land v < 0.25\]
\[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
↓
\[m + \mathsf{fma}\left(\mathsf{fma}\left(m, m + -2, 1\right), \frac{m}{v}, -1\right) \]
(FPCore (m v) :precision binary64 (* (- (/ (* m (- 1.0 m)) v) 1.0) (- 1.0 m)))
↓
(FPCore (m v)
 :precision binary64
 (+ m (fma (fma m (+ m -2.0) 1.0) (/ 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 m + fma(fma(m, (m + -2.0), 1.0), (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 Float64(m + fma(fma(m, Float64(m + -2.0), 1.0), Float64(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[(m + N[(N[(m * N[(m + -2.0), $MachinePrecision] + 1.0), $MachinePrecision] * N[(m / v), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]

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

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

  4. Simplified0.1

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

  5. Taylor expanded in m around 0 0.2

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

  6. Simplified0.1

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

  7. Taylor expanded in v around 0 0.1

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

  8. Simplified0.1

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

  9. Applied egg-rr0.1

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

  10. Final simplification0.1

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

Reproduce

herbie shell --seed 2022211 
(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)))