b parameter of renormalized beta distribution

Average Error: 0.1 → 0.1

Time: 6.8s

Precision: binary64

Cost: 13440

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

Math

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

↓

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

FPCore

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

↓

(FPCore (m v) :precision binary64 (fma (pow (- 1.0 m) 2.0) (/ m v) (+ m -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 fma(pow((1.0 - m), 2.0), (m / v), (m + -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 fma((Float64(1.0 - m) ^ 2.0), Float64(m / v), Float64(m + -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[(N[Power[N[(1.0 - m), $MachinePrecision], 2.0], $MachinePrecision] * N[(m / v), $MachinePrecision] + N[(m + -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. Taylor expanded in v around 0 0.1

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

3. Simplified 0.1

   \[\leadsto \color{blue}{\mathsf{fma}\left({\left(1 - m\right)}^{2}, \frac{m}{v}, -1 + m\right)} \]
   Proof
   (fma.f64 (pow.f64 (-.f64 1 m) 2) (/.f64 m v) (+.f64 -1 m)): 0 points increase in error, 0 points decrease in error
   (fma.f64 (pow.f64 (-.f64 1 m) 2) (/.f64 m v) (+.f64 (Rewrite<= metadata-eval (-.f64 0 1)) m)): 0 points increase in error, 0 points decrease in error
   (fma.f64 (pow.f64 (-.f64 1 m) 2) (/.f64 m v) (Rewrite<= associate--r-_binary64 (-.f64 0 (-.f64 1 m)))): 0 points increase in error, 0 points decrease in error
   (fma.f64 (pow.f64 (-.f64 1 m) 2) (/.f64 m v) (Rewrite<= neg-sub0_binary64 (neg.f64 (-.f64 1 m)))): 0 points increase in error, 0 points decrease in error
   (fma.f64 (pow.f64 (-.f64 1 m) 2) (/.f64 m v) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (-.f64 1 m)))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (pow.f64 (-.f64 1 m) 2) (/.f64 m v)) (*.f64 -1 (-.f64 1 m)))): 0 points increase in error, 0 points decrease in error
   (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 m v) (pow.f64 (-.f64 1 m) 2))) (*.f64 -1 (-.f64 1 m))): 0 points increase in error, 0 points decrease in error
   (+.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 m (pow.f64 (-.f64 1 m) 2)) v)) (*.f64 -1 (-.f64 1 m))): 13 points increase in error, 5 points decrease in error

4. Final simplification 0.1

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

Alternatives

Alternative 1
Error	17.1
Cost	976

\[\begin{array}{l} \mathbf{if}\;m \leq 1.0534508925246646 \cdot 10^{-160}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 8.576845461150992 \cdot 10^{-138}:\\ \;\;\;\;\frac{m}{v}\\ \mathbf{elif}\;m \leq 4.649357006449268 \cdot 10^{-128}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 1.8274290652065521:\\ \;\;\;\;\frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot m}{\frac{v}{m}}\\ \end{array} \]

Alternative 2
Error	17.1
Cost	976

\[\begin{array}{l} \mathbf{if}\;m \leq 1.0534508925246646 \cdot 10^{-160}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 8.576845461150992 \cdot 10^{-138}:\\ \;\;\;\;\frac{m}{v}\\ \mathbf{elif}\;m \leq 4.649357006449268 \cdot 10^{-128}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 0.11484052024384418:\\ \;\;\;\;\frac{m \cdot \left(1 - m\right)}{v}\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot m}{\frac{v}{m}}\\ \end{array} \]

Alternative 3
Error	0.3
Cost	836

\[\begin{array}{l} \mathbf{if}\;m \leq 3.8489667410869433 \cdot 10^{-23}:\\ \;\;\;\;\frac{m}{v} + \left(m + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - m}{\frac{v}{m \cdot \left(1 - m\right)}}\\ \end{array} \]

Alternative 4
Error	0.1
Cost	832

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

Alternative 5
Error	0.1
Cost	832

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

Alternative 6
Error	1.7
Cost	708

\[\begin{array}{l} \mathbf{if}\;m \leq 1.8274290652065521:\\ \;\;\;\;\frac{m}{v} + \left(m + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(m + -2\right) \cdot \frac{m \cdot m}{v}\\ \end{array} \]

Alternative 7
Error	1.7
Cost	708

\[\begin{array}{l} \mathbf{if}\;m \leq 1.8274290652065521:\\ \;\;\;\;\frac{m}{v} + \left(m + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(m + -2\right) \cdot \left(m \cdot m\right)}{v}\\ \end{array} \]

Alternative 8
Error	2.3
Cost	580

\[\begin{array}{l} \mathbf{if}\;m \leq 1.8274290652065521:\\ \;\;\;\;\frac{m}{v} + \left(m + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot m}{\frac{v}{m}}\\ \end{array} \]

Alternative 9
Error	24.7
Cost	324

\[\begin{array}{l} \mathbf{if}\;v \leq 2.4 \cdot 10^{-186}:\\ \;\;\;\;\frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 10
Error	24.4
Cost	324

\[\begin{array}{l} \mathbf{if}\;v \leq 2.4 \cdot 10^{-186}:\\ \;\;\;\;\frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;m + -1\\ \end{array} \]

Alternative 11
Error	37.2
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2022293 
(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)))