a parameter of renormalized beta distribution

Average Error: 0.2 → 0.2

Time: 7.7s

Precision: binary64

Cost: 704

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

Math

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

↓

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

FPCore

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

↓

(FPCore (m v) :precision binary64 (- (* (/ m v) (* m (- 1.0 m))) 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 / v) * (m * (1.0 - m))) - 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 / v) * (m * (1.0d0 - m))) - 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 / v) * (m * (1.0 - m))) - m;
}

Python

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

↓

def code(m, v):
	return ((m / v) * (m * (1.0 - m))) - 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(Float64(Float64(m / v) * Float64(m * Float64(1.0 - m))) - 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 / v) * (m * (1.0 - m))) - 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[(N[(N[(m / v), $MachinePrecision] * N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - m), $MachinePrecision]

Derivation

1. Initial program 0.2

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

2. Simplified 0.2

   \[\leadsto \color{blue}{m \cdot \left(\frac{m}{v} \cdot \left(1 - m\right) + -1\right)} \]
   Proof
   (*.f64 m (+.f64 (*.f64 (/.f64 m v) (-.f64 1 m)) -1)): 0 points increase in error, 0 points decrease in error
   (*.f64 m (+.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 m (-.f64 1 m)) v)) -1)): 8 points increase in error, 10 points decrease in error
   (*.f64 m (+.f64 (Rewrite<= *-rgt-identity_binary64 (*.f64 (/.f64 (*.f64 m (-.f64 1 m)) v) 1)) -1)): 0 points increase in error, 0 points decrease in error
   (*.f64 m (+.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 (*.f64 m (-.f64 1 m)) 1) v)) -1)): 0 points increase in error, 0 points decrease in error
   (*.f64 m (+.f64 (Rewrite<= associate-*r/_binary64 (*.f64 (*.f64 m (-.f64 1 m)) (/.f64 1 v))) -1)): 19 points increase in error, 8 points decrease in error
   (*.f64 m (+.f64 (*.f64 (*.f64 m (-.f64 1 m)) (/.f64 1 v)) (Rewrite<= metadata-eval (neg.f64 1)))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 (*.f64 (*.f64 m (-.f64 1 m)) (/.f64 1 v)) m) (*.f64 (neg.f64 1) m))): 2 points increase in error, 7 points decrease in error
   (+.f64 (*.f64 (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 (*.f64 m (-.f64 1 m)) 1) v)) m) (*.f64 (neg.f64 1) m)): 8 points increase in error, 18 points decrease in error
   (+.f64 (*.f64 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 (*.f64 m (-.f64 1 m)) v) 1)) m) (*.f64 (neg.f64 1) m)): 0 points increase in error, 0 points decrease in error
   (+.f64 (*.f64 (Rewrite=> *-rgt-identity_binary64 (/.f64 (*.f64 m (-.f64 1 m)) v)) m) (*.f64 (neg.f64 1) m)): 0 points increase in error, 0 points decrease in error
   (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 m (/.f64 (*.f64 m (-.f64 1 m)) v))) (*.f64 (neg.f64 1) m)): 0 points increase in error, 0 points decrease in error
   (+.f64 (*.f64 m (/.f64 (*.f64 m (-.f64 1 m)) v)) (Rewrite=> *-commutative_binary64 (*.f64 m (neg.f64 1)))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= distribute-lft-in_binary64 (*.f64 m (+.f64 (/.f64 (*.f64 m (-.f64 1 m)) v) (neg.f64 1)))): 6 points increase in error, 2 points decrease in error
   (*.f64 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) m)): 0 points increase in error, 0 points decrease in error

3. Applied egg-rr 0.2

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

4. Final simplification 0.2

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

Alternatives

Alternative 1
Error	0.4
Cost	708

\[\begin{array}{l} t_0 := \frac{m}{\frac{v}{m}}\\ \mathbf{if}\;m \leq 4.1 \cdot 10^{-10}:\\ \;\;\;\;t_0 - m\\ \mathbf{else}:\\ \;\;\;\;\left(1 - m\right) \cdot t_0\\ \end{array} \]

Alternative 2
Error	0.2
Cost	704

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

Alternative 3
Error	0.2
Cost	704

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

Alternative 4
Error	0.2
Cost	704

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

Alternative 5
Error	2.6
Cost	644

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

Alternative 6
Error	24.0
Cost	580

\[\begin{array}{l} \mathbf{if}\;v \leq 6.5 \cdot 10^{-141}:\\ \;\;\;\;\frac{m}{v + \frac{v}{m}}\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \]

Alternative 7
Error	14.6
Cost	580

\[\begin{array}{l} \mathbf{if}\;v \leq 6 \cdot 10^{-201}:\\ \;\;\;\;\frac{m}{v + \frac{v}{m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot m}{v} - m\\ \end{array} \]

Alternative 8
Error	24.7
Cost	452

\[\begin{array}{l} \mathbf{if}\;m \leq 2.4 \cdot 10^{-139}:\\ \;\;\;\;-m\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot m}{v}\\ \end{array} \]

Alternative 9
Error	24.3
Cost	452

\[\begin{array}{l} \mathbf{if}\;v \leq 6 \cdot 10^{-141}:\\ \;\;\;\;\frac{m}{\frac{v}{m}}\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \]

Alternative 10
Error	10.5
Cost	448

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

Alternative 11
Error	10.5
Cost	448

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

Alternative 12
Error	36.5
Cost	128

\[-m \]

Reproduce

herbie shell --seed 2022296 
(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))