Average Error: 0.2 → 0.2
Time: 4.8s
Precision: binary64
\[\left(0 < m \land 0 < v\right) \land v < 0.25\]
\[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m \]
↓
\[\mathsf{fma}\left(\frac{m - m \cdot m}{v}, m, -m\right) \]
(FPCore (m v) :precision binary64 (* (- (/ (* m (- 1.0 m)) v) 1.0) m))
↓
(FPCore (m v) :precision binary64 (fma (/ (- m (* m m)) v) 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 - (m * m)) / v), 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 fma(Float64(Float64(m - Float64(m * m)) / v), m, 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[(N[(N[(m - N[(m * m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision] * m + (-m)), $MachinePrecision]

\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m
↓
\mathsf{fma}\left(\frac{m - m \cdot m}{v}, 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 add-cube-cbrt_binary640.8

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

  3. Applied associate-*l*_binary640.8

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

  4. Taylor expanded in m around 0 6.8

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

  5. Simplified0.2

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

  6. Final simplification0.2

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

Reproduce

herbie shell --seed 2022129 
(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))