a parameter of renormalized beta distribution

Average Error: 0.2 → 0.2

Time: 5.6s

Precision: binary64

Cost: 704

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

Math

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

↓

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

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(Float64(m / 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[(N[(m / v), $MachinePrecision] * N[(1.0 - m), $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

   \[\leadsto \color{blue}{m \cdot \left(\frac{m}{v} \cdot \left(1 - m\right) + -1\right)} \]
   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)): 10 points increase in error, 4 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)): 26 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))): 3 points increase in error, 4 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)): 9 points increase in error, 26 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)))): 3 points increase in error, 3 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. Final simplification 0.2

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

Alternatives

Alternative 1
Error	24.9
Cost	1244

\[\begin{array}{l} t_0 := m \cdot \frac{m}{v}\\ \mathbf{if}\;v \leq 2.5 \cdot 10^{-185}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;v \leq 4.4 \cdot 10^{-169}:\\ \;\;\;\;-m\\ \mathbf{elif}\;v \leq 2.25 \cdot 10^{-164}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;v \leq 3 \cdot 10^{-156}:\\ \;\;\;\;-m\\ \mathbf{elif}\;v \leq 2.25 \cdot 10^{-141}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;v \leq 6.5 \cdot 10^{-99}:\\ \;\;\;\;-m\\ \mathbf{elif}\;v \leq 8.6 \cdot 10^{-91}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \]

Alternative 2
Error	24.9
Cost	1244

\[\begin{array}{l} t_0 := m \cdot \frac{m}{v}\\ t_1 := \frac{m}{\frac{v}{m}}\\ \mathbf{if}\;v \leq 2.5 \cdot 10^{-185}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;v \leq 5.6 \cdot 10^{-169}:\\ \;\;\;\;-m\\ \mathbf{elif}\;v \leq 7 \cdot 10^{-164}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;v \leq 3.1 \cdot 10^{-156}:\\ \;\;\;\;-m\\ \mathbf{elif}\;v \leq 2.4 \cdot 10^{-141}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;v \leq 6.4 \cdot 10^{-99}:\\ \;\;\;\;-m\\ \mathbf{elif}\;v \leq 8.5 \cdot 10^{-91}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \]

Alternative 3
Error	0.3
Cost	708

\[\begin{array}{l} \mathbf{if}\;m \leq 1.7 \cdot 10^{-14}:\\ \;\;\;\;\frac{m}{\frac{v}{m}} - m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(\frac{m}{v} \cdot \left(1 - m\right)\right)\\ \end{array} \]

Alternative 4
Error	2.3
Cost	644

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

Alternative 5
Error	10.0
Cost	448

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

Alternative 6
Error	10.0
Cost	448

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

Alternative 7
Error	36.6
Cost	128

\[-m \]

Reproduce

herbie shell --seed 2022329 
(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))