a parameter of renormalized beta distribution

Average Error: 0.2 → 0.4

Time: 7.8s

Precision: binary64

Cost: 708

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;m \leq 3.1 \cdot 10^{-19}:\\ \;\;\;\;m \cdot \left(\frac{m}{v} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - m\right) \cdot \left(m \cdot m\right)}{v}\\ \end{array} \]

FPCore

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

↓

(FPCore (m v)
 :precision binary64
 (if (<= m 3.1e-19) (* m (+ (/ m v) -1.0)) (/ (* (- 1.0 m) (* m m)) v)))

C

double code(double m, double v) {
	return (((m * (1.0 - m)) / v) - 1.0) * m;
}

↓

double code(double m, double v) {
	double tmp;
	if (m <= 3.1e-19) {
		tmp = m * ((m / v) + -1.0);
	} else {
		tmp = ((1.0 - m) * (m * m)) / v;
	}
	return tmp;
}

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
    real(8) :: tmp
    if (m <= 3.1d-19) then
        tmp = m * ((m / v) + (-1.0d0))
    else
        tmp = ((1.0d0 - m) * (m * m)) / v
    end if
    code = tmp
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) {
	double tmp;
	if (m <= 3.1e-19) {
		tmp = m * ((m / v) + -1.0);
	} else {
		tmp = ((1.0 - m) * (m * m)) / v;
	}
	return tmp;
}

Python

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

↓

def code(m, v):
	tmp = 0
	if m <= 3.1e-19:
		tmp = m * ((m / v) + -1.0)
	else:
		tmp = ((1.0 - m) * (m * m)) / v
	return tmp

Julia

function code(m, v)
	return Float64(Float64(Float64(Float64(m * Float64(1.0 - m)) / v) - 1.0) * m)
end

↓

function code(m, v)
	tmp = 0.0
	if (m <= 3.1e-19)
		tmp = Float64(m * Float64(Float64(m / v) + -1.0));
	else
		tmp = Float64(Float64(Float64(1.0 - m) * Float64(m * m)) / v);
	end
	return tmp
end

MATLAB

function tmp = code(m, v)
	tmp = (((m * (1.0 - m)) / v) - 1.0) * m;
end

↓

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 3.1e-19)
		tmp = m * ((m / v) + -1.0);
	else
		tmp = ((1.0 - m) * (m * m)) / v;
	end
	tmp_2 = tmp;
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_] := If[LessEqual[m, 3.1e-19], N[(m * N[(N[(m / v), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - m), $MachinePrecision] * N[(m * m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 3.0999999999999999e-19

   1. Initial program 0.1

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

   2. Taylor expanded in m around 0 0.1

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

   if 3.0999999999999999e-19 < m

   1. Initial program 0.4

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

   2. Simplified 0.4

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

      [Start] 0.4	\[ \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m \]
      *-commutative [=>] 0.4	\[ \color{blue}{m \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right)} \]
      sub-neg [=>] 0.4	\[ m \cdot \color{blue}{\left(\frac{m \cdot \left(1 - m\right)}{v} + \left(-1\right)\right)} \]
      associate-*l/ [<=] 0.4	\[ m \cdot \left(\color{blue}{\frac{m}{v} \cdot \left(1 - m\right)} + \left(-1\right)\right) \]
      metadata-eval [=>] 0.4	\[ m \cdot \left(\frac{m}{v} \cdot \left(1 - m\right) + \color{blue}{-1}\right) \]

   3. Taylor expanded in v around 0 1.4

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

   4. Simplified 1.4

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

      [Start] 1.4	\[ \frac{{m}^{2} \cdot \left(1 - m\right)}{v} \]
      *-commutative [=>] 1.4	\[ \frac{\color{blue}{\left(1 - m\right) \cdot {m}^{2}}}{v} \]
      unpow2 [=>] 1.4	\[ \frac{\left(1 - m\right) \cdot \color{blue}{\left(m \cdot m\right)}}{v} \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 3.1 \cdot 10^{-19}:\\ \;\;\;\;m \cdot \left(\frac{m}{v} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - m\right) \cdot \left(m \cdot m\right)}{v}\\ \end{array} \]

Alternatives

Alternative 1
Error	26.9
Cost	1108

\[\begin{array}{l} \mathbf{if}\;v \leq 7 \cdot 10^{-224}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \mathbf{elif}\;v \leq 6 \cdot 10^{-157}:\\ \;\;\;\;-m\\ \mathbf{elif}\;v \leq 2.3 \cdot 10^{-140}:\\ \;\;\;\;\frac{m \cdot m}{v}\\ \mathbf{elif}\;v \leq 1.3 \cdot 10^{-132}:\\ \;\;\;\;-m\\ \mathbf{elif}\;v \leq 2.35 \cdot 10^{-123}:\\ \;\;\;\;m \cdot \left(m \cdot \frac{1}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \]

Alternative 2
Error	26.9
Cost	982

\[\begin{array}{l} \mathbf{if}\;v \leq 7 \cdot 10^{-224} \lor \neg \left(v \leq 1.6 \cdot 10^{-156}\right) \land \left(v \leq 9.6 \cdot 10^{-141} \lor \neg \left(v \leq 4.3 \cdot 10^{-132}\right) \land v \leq 2.3 \cdot 10^{-123}\right):\\ \;\;\;\;m \cdot \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \]

Alternative 3
Error	26.8
Cost	980

\[\begin{array}{l} t_0 := m \cdot \frac{m}{v}\\ \mathbf{if}\;v \leq 7 \cdot 10^{-224}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;v \leq 2.85 \cdot 10^{-157}:\\ \;\;\;\;-m\\ \mathbf{elif}\;v \leq 2.8 \cdot 10^{-140}:\\ \;\;\;\;\frac{m \cdot m}{v}\\ \mathbf{elif}\;v \leq 2.45 \cdot 10^{-132}:\\ \;\;\;\;-m\\ \mathbf{elif}\;v \leq 2.5 \cdot 10^{-123}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \]

Alternative 4
Error	0.3
Cost	708

\[\begin{array}{l} \mathbf{if}\;m \leq 1.8 \cdot 10^{-17}:\\ \;\;\;\;m \cdot \left(\frac{m}{v} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(m \cdot \frac{1 - m}{v}\right)\\ \end{array} \]

Alternative 5
Error	0.4
Cost	708

\[\begin{array}{l} \mathbf{if}\;m \leq 1.7 \cdot 10^{-19}:\\ \;\;\;\;m \cdot \left(\frac{m}{v} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - m\right) \cdot \frac{m \cdot m}{v}\\ \end{array} \]

Alternative 6
Error	0.4
Cost	708

\[\begin{array}{l} \mathbf{if}\;m \leq 3.1 \cdot 10^{-19}:\\ \;\;\;\;m \cdot \left(\frac{m}{v} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m \cdot \left(1 - m\right)}{v}\\ \end{array} \]

Alternative 7
Error	0.2
Cost	704

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

Alternative 8
Error	0.2
Cost	704

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

Alternative 9
Error	2.3
Cost	644

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

Alternative 10
Error	2.3
Cost	644

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

Alternative 11
Error	2.3
Cost	644

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

Alternative 12
Error	10.4
Cost	448

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

Alternative 13
Error	37.2
Cost	128

\[-m \]

Reproduce

herbie shell --seed 2023083 
(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))