b parameter of renormalized beta distribution

Average Error: 0.1 → 0.1

Time: 6.6s

Precision: binary64

Cost: 832

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

Math

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

↓

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

FPCore

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

↓

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

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 (((m - (m * m)) / v) + -1.0) * (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) * (1.0d0 - m)
end function

↓

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

Python

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

↓

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

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(Float64(Float64(m - Float64(m * m)) / v) + -1.0) * Float64(1.0 - m))
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 = (((m - (m * m)) / v) + -1.0) * (1.0 - m);
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[(N[(N[(m - N[(m * m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision] + -1.0), $MachinePrecision] * N[(1.0 - m), $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. Taylor expanded in m around 0 0.1

   \[\leadsto \left(\frac{\color{blue}{-1 \cdot {m}^{2} + m}}{v} - 1\right) \cdot \left(1 - m\right) \]

3. Simplified 0.1

   \[\leadsto \left(\frac{\color{blue}{m - m \cdot m}}{v} - 1\right) \cdot \left(1 - m\right) \]
   Proof
   (-.f64 m (*.f64 m m)): 0 points increase in error, 0 points decrease in error
   (Rewrite=> cancel-sign-sub-inv_binary64 (+.f64 m (*.f64 (neg.f64 m) m))): 0 points increase in error, 0 points decrease in error
   (Rewrite=> distribute-rgt1-in_binary64 (*.f64 (+.f64 (neg.f64 m) 1) m)): 5 points increase in error, 0 points decrease in error
   (Rewrite<= distribute-lft1-in_binary64 (+.f64 (*.f64 (neg.f64 m) m) m)): 0 points increase in error, 5 points decrease in error
   (+.f64 (Rewrite=> distribute-lft-neg-out_binary64 (neg.f64 (*.f64 m m))) m): 0 points increase in error, 0 points decrease in error
   (+.f64 (neg.f64 (Rewrite<= unpow2_binary64 (pow.f64 m 2))) m): 0 points increase in error, 0 points decrease in error
   (+.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (pow.f64 m 2))) m): 0 points increase in error, 0 points decrease in error

4. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.2
Cost	836

\[\begin{array}{l} \mathbf{if}\;m \leq 2.9 \cdot 10^{-16}:\\ \;\;\;\;\left(1 - m\right) \cdot \left(-1 + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - m}{\frac{v}{m \cdot \left(1 - m\right)}}\\ \end{array} \]

Alternative 2
Error	0.2
Cost	836

\[\begin{array}{l} \mathbf{if}\;m \leq 4.5 \cdot 10^{-16}:\\ \;\;\;\;\left(1 - m\right) \cdot \left(-1 + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - m}{\frac{v}{m - m \cdot m}}\\ \end{array} \]

Alternative 3
Error	0.2
Cost	836

\[\begin{array}{l} \mathbf{if}\;m \leq 8 \cdot 10^{-18}:\\ \;\;\;\;-1 + \left(m + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m - m \cdot m}{\frac{v}{1 - m}}\\ \end{array} \]

Alternative 4
Error	0.1
Cost	832

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

Alternative 5
Error	0.1
Cost	832

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

Alternative 6
Error	0.1
Cost	832

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

Alternative 7
Error	17.0
Cost	712

\[\begin{array}{l} \mathbf{if}\;m \leq 9 \cdot 10^{-146}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 2.6:\\ \;\;\;\;\frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(m \cdot \frac{m}{v}\right)\\ \end{array} \]

Alternative 8
Error	17.0
Cost	712

\[\begin{array}{l} \mathbf{if}\;m \leq 9 \cdot 10^{-146}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 2.6:\\ \;\;\;\;\frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m \cdot m}{v}\\ \end{array} \]

Alternative 9
Error	17.0
Cost	712

\[\begin{array}{l} \mathbf{if}\;m \leq 9.5 \cdot 10^{-146}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 1:\\ \;\;\;\;\left(1 - m\right) \cdot \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m \cdot m}{v}\\ \end{array} \]

Alternative 10
Error	2.4
Cost	708

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

Alternative 11
Error	2.4
Cost	708

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

Alternative 12
Error	2.4
Cost	708

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

Alternative 13
Error	2.4
Cost	708

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

Alternative 14
Error	2.4
Cost	708

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

Alternative 15
Error	2.4
Cost	708

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

Alternative 16
Error	2.4
Cost	580

\[\begin{array}{l} \mathbf{if}\;m \leq 2.6:\\ \;\;\;\;-1 + \left(m + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m \cdot m}{v}\\ \end{array} \]

Alternative 17
Error	24.4
Cost	324

\[\begin{array}{l} \mathbf{if}\;m \leq 8.5 \cdot 10^{-146}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v}\\ \end{array} \]

Alternative 18
Error	37.2
Cost	192

\[m + -1 \]

Alternative 19
Error	37.5
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2022321 
(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)))