b parameter of renormalized beta distribution

Average Error: 99.9% → 99.9%

Time: 8.0s

Precision: binary64

Cost: 832.00

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

Math

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

↓

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

FPCore

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

↓

(FPCore (m v) :precision binary64 (* (- 1.0 m) (+ (* (- 1.0 m) (/ m v)) -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) * (((1.0 - m) * (m / v)) + -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) * (((1.0d0 - m) * (m / v)) + (-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) * (((1.0 - m) * (m / v)) + -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) * (((1.0 - m) * (m / v)) + -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(Float64(1.0 - m) * Float64(m / v)) + -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) * (((1.0 - m) * (m / v)) + -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[(N[(1.0 - m), $MachinePrecision] * N[(m / v), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9

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

2. Simplified 99.9

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

   [Start] 99.9	\[ \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
   *-commutative [=>] 99.9	\[ \color{blue}{\left(1 - m\right) \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right)} \]
   sub-neg [=>] 99.9	\[ \left(1 - m\right) \cdot \color{blue}{\left(\frac{m \cdot \left(1 - m\right)}{v} + \left(-1\right)\right)} \]
   associate-*l/ [<=] 99.9	\[ \left(1 - m\right) \cdot \left(\color{blue}{\frac{m}{v} \cdot \left(1 - m\right)} + \left(-1\right)\right) \]
   metadata-eval [=>] 99.9	\[ \left(1 - m\right) \cdot \left(\frac{m}{v} \cdot \left(1 - m\right) + \color{blue}{-1}\right) \]

3. Final simplification 99.9

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

Alternatives

Alternative 1
Error	99.4%
Cost	836.00

\[\begin{array}{l} \mathbf{if}\;m \leq 4.5 \cdot 10^{-9}:\\ \;\;\;\;\left(1 - m\right) \cdot \left(\frac{m}{v} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - m\right) \cdot \left(\left(1 - m\right) \cdot \frac{m}{v}\right)\\ \end{array} \]

Alternative 2
Error	99.4%
Cost	836.00

\[\begin{array}{l} t_0 := \left(1 - m\right) \cdot \frac{m}{v}\\ \mathbf{if}\;m \leq 4.5 \cdot 10^{-9}:\\ \;\;\;\;t_0 + \left(m + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - m\right) \cdot t_0\\ \end{array} \]

Alternative 3
Error	96.3%
Cost	708.00

\[\begin{array}{l} \mathbf{if}\;m \leq 2.3:\\ \;\;\;\;\frac{m}{v} + \left(m + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(m + -1\right) \cdot \left(m \cdot \frac{m}{v}\right)\\ \end{array} \]

Alternative 4
Error	96.4%
Cost	708.00

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

Alternative 5
Error	96.4%
Cost	708.00

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

Alternative 6
Error	96.4%
Cost	708.00

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

Alternative 7
Error	61.9%
Cost	589.00

\[\begin{array}{l} \mathbf{if}\;v \leq 1.18 \cdot 10^{-129} \lor \neg \left(v \leq 7.8 \cdot 10^{-121}\right) \land v \leq 6.2 \cdot 10^{-114}:\\ \;\;\;\;\frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;m + -1\\ \end{array} \]

Alternative 8
Error	85.3%
Cost	448.00

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

Alternative 9
Error	85.3%
Cost	448.00

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

Alternative 10
Error	85.3%
Cost	320.00

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

Alternative 11
Error	42.5%
Cost	192.00

\[m + -1 \]

Alternative 12
Error	42.0%
Cost	64.00

\[-1 \]

Reproduce

herbie shell --seed 2023093 
(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)))