a parameter of renormalized beta distribution

Average Accuracy: 99.7% → 99.7%

Time: 9.0s

Precision: binary64

Cost: 704

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

Math

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

↓

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

FPCore

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

↓

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

Fortran

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    code = (((m * (1.0d0 - m)) / v) - 1.0d0) * m
end function

↓

real(8) function code(m, v)
    real(8), intent (in) :: m
    real(8), intent (in) :: v
    code = m * ((m / (v / (1.0d0 - m))) + (-1.0d0))
end function

Java

public static double code(double m, double v) {
	return (((m * (1.0 - m)) / v) - 1.0) * m;
}

↓

public static double code(double m, double v) {
	return m * ((m / (v / (1.0 - m))) + -1.0);
}

Python

def code(m, v):
	return (((m * (1.0 - m)) / v) - 1.0) * m

↓

def code(m, v):
	return m * ((m / (v / (1.0 - m))) + -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 * Float64(Float64(m / Float64(v / Float64(1.0 - m))) + -1.0))
end

MATLAB

function tmp = code(m, v)
	tmp = (((m * (1.0 - m)) / v) - 1.0) * m;
end

↓

function tmp = code(m, v)
	tmp = m * ((m / (v / (1.0 - m))) + -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[(m / N[(v / N[(1.0 - m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

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

2. Simplified 99.7%

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

   [Start] 99.7	\[ \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m \]
   *-commutative [=>] 99.7	\[ \color{blue}{m \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right)} \]
   sub-neg [=>] 99.7	\[ m \cdot \color{blue}{\left(\frac{m \cdot \left(1 - m\right)}{v} + \left(-1\right)\right)} \]
   associate-/l* [=>] 99.7	\[ m \cdot \left(\color{blue}{\frac{m}{\frac{v}{1 - m}}} + \left(-1\right)\right) \]
   metadata-eval [=>] 99.7	\[ m \cdot \left(\frac{m}{\frac{v}{1 - m}} + \color{blue}{-1}\right) \]

3. Final simplification 99.7%

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

Alternatives

Alternative 1
Accuracy	99.5%
Cost	708

\[\begin{array}{l} \mathbf{if}\;m \leq 3.6 \cdot 10^{-17}:\\ \;\;\;\;m \cdot \frac{m}{v} - m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(m \cdot \frac{1 - m}{v}\right)\\ \end{array} \]

Alternative 2
Accuracy	99.5%
Cost	708

\[\begin{array}{l} \mathbf{if}\;m \leq 3.6 \cdot 10^{-17}:\\ \;\;\;\;m \cdot \frac{m}{v} - m\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot \left(1 - m\right)}{\frac{v}{m}}\\ \end{array} \]

Alternative 3
Accuracy	99.7%
Cost	704

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

Alternative 4
Accuracy	96.2%
Cost	644

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

Alternative 5
Accuracy	96.2%
Cost	644

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

Alternative 6
Accuracy	96.2%
Cost	644

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

Alternative 7
Accuracy	96.2%
Cost	644

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

Alternative 8
Accuracy	63.1%
Cost	452

\[\begin{array}{l} \mathbf{if}\;v \leq 2.1 \cdot 10^{-164}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \]

Alternative 9
Accuracy	63.1%
Cost	452

\[\begin{array}{l} \mathbf{if}\;v \leq 3.35 \cdot 10^{-164}:\\ \;\;\;\;\frac{m}{\frac{v}{m}}\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \]

Alternative 10
Accuracy	83.7%
Cost	448

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

Alternative 11
Accuracy	83.7%
Cost	448

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

Alternative 12
Accuracy	42.9%
Cost	128

\[-m \]

Reproduce

herbie shell --seed 2023129 
(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))