b parameter of renormalized beta distribution

Average Error: 0.1 → 0.1

Time: 13.8s

Precision: binary64

Cost: 1344

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

Math

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

↓

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

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) v) (* m m))) (+ 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) / v) * (m * m))) + (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) / v) * (m * m))) + (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) / v) * (m * m))) + (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) / v) * (m * m))) + (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(Float64(Float64(Float64(1.0 - m) * m) / v) + Float64(Float64(Float64(-1.0 + m) / v) * Float64(m * m))) + Float64(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) / v) * (m * m))) + (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[(N[(N[(N[(1.0 - m), $MachinePrecision] * m), $MachinePrecision] / v), $MachinePrecision] + N[(N[(N[(-1.0 + m), $MachinePrecision] / v), $MachinePrecision] * N[(m * m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(m + -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. Applied egg-rr 0.1

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

3. Simplified 0.1

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

4. Applied egg-rr 0.1

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

5. Applied egg-rr 0.1

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

Alternatives

Alternative 1
Error	0.1
Cost	1280

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

Alternative 2
Error	0.4
Cost	900

\[\begin{array}{l} \mathbf{if}\;m \leq 8.2 \cdot 10^{-30}:\\ \;\;\;\;\frac{m}{v} - 1\\ \mathbf{else}:\\ \;\;\;\;-\left(m + -1\right) \cdot \left(m \cdot \frac{1 - m}{v}\right)\\ \end{array} \]

Alternative 3
Error	0.4
Cost	900

\[\begin{array}{l} \mathbf{if}\;m \leq 8.2 \cdot 10^{-30}:\\ \;\;\;\;\frac{m}{v} - 1\\ \mathbf{else}:\\ \;\;\;\;-\frac{1 - m}{v} \cdot \left(\left(m + -1\right) \cdot m\right)\\ \end{array} \]

Alternative 4
Error	0.4
Cost	836

\[\begin{array}{l} \mathbf{if}\;m \leq 8 \cdot 10^{-30}:\\ \;\;\;\;\frac{m}{v} - 1\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot \left(1 - m\right)}{v} \cdot \left(1 - m\right)\\ \end{array} \]

Alternative 5
Error	0.1
Cost	832

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

Alternative 6
Error	24.9
Cost	716

\[\begin{array}{l} t_0 := -\frac{-m}{v}\\ \mathbf{if}\;v \leq 3.2 \cdot 10^{-172}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;v \leq 4 \cdot 10^{-126}:\\ \;\;\;\;m - 1\\ \mathbf{elif}\;v \leq 1.45 \cdot 10^{-112}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;m - 1\\ \end{array} \]

Alternative 7
Error	9.7
Cost	448

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

Alternative 8
Error	9.7
Cost	320

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

Alternative 9
Error	36.8
Cost	192

\[m - 1 \]

Alternative 10
Error	37.1
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023010 
(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)))