a parameter of renormalized beta distribution

Average Error: 0.2 → 0.2

Time: 5.1s

Precision: binary64

Cost: 704

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

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

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 * ((-1.0d0) + (m / (v / (1.0d0 - m))))
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 * (-1.0 + (m / (v / (1.0 - m))));
}

Python

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

↓

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

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

MATLAB

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

↓

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

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

3. Applied egg-rr 0.2

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

4. Final simplification 0.2

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

Alternatives

Alternative 1
Error	0.3
Cost	708

\[\begin{array}{l} \mathbf{if}\;m \leq 5.383870556343972 \cdot 10^{-18}:\\ \;\;\;\;m \cdot \left(-1 + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot \left(1 - m\right)}{\frac{v}{m}}\\ \end{array} \]

Alternative 2
Error	0.2
Cost	704

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

Alternative 3
Error	3.3
Cost	644

\[\begin{array}{l} \mathbf{if}\;m \leq 4.667080186403062 \cdot 10^{-13}:\\ \;\;\;\;m \cdot \left(-1 + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot \left(-m \cdot m\right)}{v}\\ \end{array} \]

Alternative 4
Error	3.3
Cost	644

\[\begin{array}{l} \mathbf{if}\;m \leq 4.667080186403062 \cdot 10^{-13}:\\ \;\;\;\;m \cdot \left(-1 + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(m \cdot \frac{-m}{v}\right)\\ \end{array} \]

Alternative 5
Error	25.0
Cost	452

\[\begin{array}{l} \mathbf{if}\;m \leq 1.0830285895521001 \cdot 10^{-128}:\\ \;\;\;\;-m\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot m}{v}\\ \end{array} \]

Alternative 6
Error	23.5
Cost	452

\[\begin{array}{l} \mathbf{if}\;v \leq 1.8468607931819644 \cdot 10^{-162}:\\ \;\;\;\;\frac{m}{\frac{v}{m}}\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \]

Alternative 7
Error	23.5
Cost	452

\[\begin{array}{l} \mathbf{if}\;v \leq 1.8468607931819644 \cdot 10^{-162}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \]

Alternative 8
Error	10.1
Cost	448

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

Alternative 9
Error	10.1
Cost	448

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

Alternative 10
Error	36.6
Cost	128

\[-m \]

Reproduce

herbie shell --seed 2022221 
(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))