b parameter of renormalized beta distribution

Average Error: 0.1 → 0.1

Time: 9.4s

Precision: binary64

Cost: 832

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

Math

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

↓

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

FPCore

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

↓

(FPCore (m v) :precision binary64 (* (- 1.0 m) (+ (/ m (/ v (- 1.0 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 (1.0 - 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) * (1.0d0 - m)
end function

↓

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

↓

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

Python

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

↓

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

↓

function code(m, v)
	return Float64(Float64(1.0 - 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) * (1.0 - m);
end

↓

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

↓

code[m_, v_] := N[(N[(1.0 - m), $MachinePrecision] * N[(N[(m / N[(v / N[(1.0 - m), $MachinePrecision]), $MachinePrecision]), $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

   \[\leadsto \color{blue}{\left(1 - m\right) \cdot \left(\frac{m}{\frac{v}{1 - m}} + -1\right)} \]
   Proof
   (*.f64 (-.f64 1 m) (+.f64 (/.f64 m (/.f64 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)): 8 points increase in error, 4 points decrease in error
   (*.f64 (-.f64 1 m) (+.f64 (/.f64 (*.f64 m (-.f64 1 m)) v) (Rewrite<= metadata-eval (neg.f64 1)))): 0 points increase in error, 0 points decrease in error
   (*.f64 (-.f64 1 m) (Rewrite<= sub-neg_binary64 (-.f64 (/.f64 (*.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) (-.f64 1 m))): 0 points increase in error, 0 points decrease in error

3. Final simplification 0.1

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

Alternatives

Alternative 1
Error	1.8
Cost	964

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

Alternative 2
Error	0.2
Cost	832

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

Alternative 3
Error	0.1
Cost	832

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

Alternative 4
Error	2.2
Cost	708

\[\begin{array}{l} \mathbf{if}\;m \leq 2.35:\\ \;\;\;\;-1 + \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m \cdot \left(m + -1\right)}{v}\\ \end{array} \]

Alternative 5
Error	2.2
Cost	708

\[\begin{array}{l} \mathbf{if}\;m \leq 2.35:\\ \;\;\;\;-1 + \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{\frac{v}{m \cdot m - m}}\\ \end{array} \]

Alternative 6
Error	2.2
Cost	708

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

Alternative 7
Error	2.3
Cost	580

\[\begin{array}{l} \mathbf{if}\;m \leq 0.38:\\ \;\;\;\;-1 + \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{\frac{v}{m \cdot m}}\\ \end{array} \]

Alternative 8
Error	9.6
Cost	320

\[-1 + \frac{m}{v} \]

Alternative 9
Error	37.2
Cost	192

\[m + -1 \]

Alternative 10
Error	37.6
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2022318 
(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)))