b parameter of renormalized beta distribution

Average Error: 0.1 → 0.1

Time: 6.1s

Precision: binary64

Cost: 13440

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

Math

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

↓

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

FPCore

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

↓

(FPCore (m v) :precision binary64 (+ m (fma (/ m v) (pow (- 1.0 m) 2.0) -1.0)))

C

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((m / v), pow((1.0 - m), 2.0), -1.0);
}

Julia

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(Float64(m / v), (Float64(1.0 - m) ^ 2.0), -1.0))
end

Wolfram

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

Derivation

1. Initial program 0.1

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

2. Simplified 0.1

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

3. Taylor expanded in v around 0 0.1

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

4. Simplified 0.1

   \[\leadsto \color{blue}{m + \mathsf{fma}\left(\frac{m}{v}, {\left(1 - m\right)}^{2}, -1\right)} \]
   Proof
   (+.f64 m (fma.f64 (/.f64 m v) (pow.f64 (-.f64 1 m) 2) -1)): 0 points increase in error, 0 points decrease in error
   (+.f64 m (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (/.f64 m v) (pow.f64 (-.f64 1 m) 2)) -1))): 1 points increase in error, 0 points decrease in error
   (+.f64 m (+.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 m (pow.f64 (-.f64 1 m) 2)) v)) -1)): 11 points increase in error, 12 points decrease in error
   (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 (/.f64 (*.f64 m (pow.f64 (-.f64 1 m) 2)) v) -1) m)): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-+r+_binary64 (+.f64 (/.f64 (*.f64 m (pow.f64 (-.f64 1 m) 2)) v) (+.f64 -1 m))): 0 points increase in error, 1 points decrease in error
   (+.f64 (/.f64 (*.f64 m (pow.f64 (-.f64 1 m) 2)) v) (+.f64 (Rewrite<= metadata-eval (-.f64 0 1)) m)): 0 points increase in error, 0 points decrease in error
   (+.f64 (/.f64 (*.f64 m (pow.f64 (-.f64 1 m) 2)) v) (Rewrite<= associate--r-_binary64 (-.f64 0 (-.f64 1 m)))): 0 points increase in error, 0 points decrease in error
   (+.f64 (/.f64 (*.f64 m (pow.f64 (-.f64 1 m) 2)) v) (Rewrite<= neg-sub0_binary64 (neg.f64 (-.f64 1 m)))): 0 points increase in error, 0 points decrease in error
   (+.f64 (/.f64 (*.f64 m (pow.f64 (-.f64 1 m) 2)) v) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (-.f64 1 m)))): 0 points increase in error, 0 points decrease in error

5. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.1
Cost	7104

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

Alternative 2
Error	0.2
Cost	964

\[\begin{array}{l} \mathbf{if}\;m \leq 2.526318808437896 \cdot 10^{-15}:\\ \;\;\;\;\left(m + -1\right) + \frac{m}{v} \cdot \left(1 + m \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - m\right) \cdot \frac{m \cdot \left(1 - m\right)}{v}\\ \end{array} \]

Alternative 3
Error	0.2
Cost	836

\[\begin{array}{l} \mathbf{if}\;m \leq 2.526318808437896 \cdot 10^{-15}:\\ \;\;\;\;\left(1 - m\right) \cdot \left(\frac{m}{v} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - m\right) \cdot \frac{m \cdot \left(1 - m\right)}{v}\\ \end{array} \]

Alternative 4
Error	0.2
Cost	832

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

Alternative 5
Error	0.1
Cost	832

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

Alternative 6
Error	1.9
Cost	708

\[\begin{array}{l} \mathbf{if}\;m \leq 0.9765528112922299:\\ \;\;\;\;m + \left(\frac{m}{v} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(m + -2\right) \cdot \left(m \cdot m\right)}{v}\\ \end{array} \]

Alternative 7
Error	1.9
Cost	708

\[\begin{array}{l} \mathbf{if}\;m \leq 0.9765528112922299:\\ \;\;\;\;\left(1 - m\right) \cdot \left(\frac{m}{v} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(m + -2\right) \cdot \left(m \cdot m\right)}{v}\\ \end{array} \]

Alternative 8
Error	25.7
Cost	588

\[\begin{array}{l} \mathbf{if}\;m \leq 6.548066602025169 \cdot 10^{-195}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 1.0114256887997727 \cdot 10^{-159}:\\ \;\;\;\;\frac{m}{v}\\ \mathbf{elif}\;m \leq 1.2477677971843265 \cdot 10^{-117}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v}\\ \end{array} \]

Alternative 9
Error	9.6
Cost	448

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

Alternative 10
Error	61.7
Cost	64

\[m \]

Alternative 11
Error	37.0
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2022311 
(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)))