Average Error: 0.2 → 0.3
Time: 4.7s
Precision: binary64
\[\left(0 < m \land 0 < v\right) \land v < 0.25\]
\[\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m \]
↓
\[m \cdot \mathsf{fma}\left(\frac{1}{v}, m \cdot \left(1 - m\right), -1\right) \]
(FPCore (m v) :precision binary64 (* (- (/ (* m (- 1.0 m)) v) 1.0) m))
↓
(FPCore (m v) :precision binary64 (* m (fma (/ 1.0 v) (* m (- 1.0 m)) -1.0)))
double code(double m, double v) {
	return (((m * (1.0 - m)) / v) - 1.0) * m;
}
↓
double code(double m, double v) {
	return m * fma((1.0 / v), (m * (1.0 - m)), -1.0);
}

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

\left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m
↓
m \cdot \mathsf{fma}\left(\frac{1}{v}, m \cdot \left(1 - m\right), -1\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 clear-num_binary640.2

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

  3. Applied associate-/r/_binary640.3

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

  4. Applied fma-neg_binary640.3

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

  5. Simplified0.3

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

  6. Final simplification0.3

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

Reproduce

herbie shell --seed 2022131 
(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))