a parameter of renormalized beta distribution

Average Error: 0.2 → 0.2

Time: 7.4s

Precision: binary64

Cost: 6976

\[\left(0 < m \land 0 < v\right) \land v < 0.25\]

Math

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

↓

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

FPCore

(FPCore (m v) :precision binary64 (* (- (/ (* m (- 1.0 m)) v) 1.0) m))

↓

(FPCore (m v) :precision binary64 (* m (fma (- 1.0 m) (/ m v) -1.0)))

C

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 - m), (m / v), -1.0);
}

Julia

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 - m), Float64(m / v), -1.0))
end

Wolfram

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

Derivation

1. Initial program 0.2

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

2. Simplified 0.2

   \[\leadsto \color{blue}{m \cdot \mathsf{fma}\left(1 - m, \frac{m}{v}, -1\right)} \]
   Proof
   (*.f64 m (fma.f64 (-.f64 1 m) (/.f64 m v) -1)): 0 points increase in error, 0 points decrease in error
   (*.f64 m (fma.f64 (-.f64 1 m) (/.f64 m v) (Rewrite<= metadata-eval (neg.f64 1)))): 0 points increase in error, 0 points decrease in error
   (*.f64 m (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 (-.f64 1 m) (/.f64 m v)) 1))): 1 points increase in error, 0 points decrease in error
   (*.f64 m (-.f64 (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 (-.f64 1 m) m) v)) 1)): 11 points increase in error, 10 points decrease in error
   (*.f64 m (-.f64 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 m (-.f64 1 m))) v) 1)): 0 points increase in error, 0 points decrease in error
   (Rewrite<= *-commutative_binary64 (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 1 m)) v) 1) m)): 0 points increase in error, 0 points decrease in error

3. Final simplification 0.2

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

Alternatives

Alternative 1
Error	0.3
Cost	708

\[\begin{array}{l} \mathbf{if}\;m \leq 5.3793573023771413 \cdot 10^{-20}:\\ \;\;\;\;\frac{m}{\frac{v}{m}} - m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(\left(1 - m\right) \cdot \frac{m}{v}\right)\\ \end{array} \]

Alternative 2
Error	0.3
Cost	708

\[\begin{array}{l} \mathbf{if}\;m \leq 5.3793573023771413 \cdot 10^{-20}:\\ \;\;\;\;\frac{m}{\frac{v}{m}} - m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(m \cdot \frac{1 - m}{v}\right)\\ \end{array} \]

Alternative 3
Error	0.2
Cost	704

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

Alternative 4
Error	2.5
Cost	644

\[\begin{array}{l} \mathbf{if}\;m \leq 0.09856972644641661:\\ \;\;\;\;\frac{m}{\frac{v}{m}} - m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{-m}{\frac{v}{m}}\\ \end{array} \]

Alternative 5
Error	2.5
Cost	644

\[\begin{array}{l} \mathbf{if}\;m \leq 0.09856972644641661:\\ \;\;\;\;\frac{m}{\frac{v}{m}} - m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(m \cdot \frac{-m}{v}\right)\\ \end{array} \]

Alternative 6
Error	23.8
Cost	452

\[\begin{array}{l} \mathbf{if}\;v \leq 4.5 \cdot 10^{-161}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \]

Alternative 7
Error	23.8
Cost	452

\[\begin{array}{l} \mathbf{if}\;v \leq 4.5 \cdot 10^{-161}:\\ \;\;\;\;\frac{m}{\frac{v}{m}}\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \]

Alternative 8
Error	10.4
Cost	448

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

Alternative 9
Error	10.4
Cost	448

\[\frac{m}{\frac{v}{m}} - m \]

Alternative 10
Error	36.1
Cost	128

\[-m \]

Reproduce

herbie shell --seed 2022308 
(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))