a parameter of renormalized beta distribution

Average Error: 99.7% → 99.5%

Time: 7.3s

Precision: binary64

Cost: 708.00

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;m \leq 1.35 \cdot 10^{-16}:\\ \;\;\;\;\frac{m}{\frac{v}{m}} - m\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot \left(1 - m\right)}{\frac{v}{m}}\\ \end{array} \]

FPCore

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

↓

(FPCore (m v)
 :precision binary64
 (if (<= m 1.35e-16) (- (/ m (/ v m)) m) (/ (* m (- 1.0 m)) (/ v m))))

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 <= 1.35e-16) {
		tmp = (m / (v / m)) - m;
	} else {
		tmp = (m * (1.0 - m)) / (v / m);
	}
	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 <= 1.35d-16) then
        tmp = (m / (v / m)) - m
    else
        tmp = (m * (1.0d0 - m)) / (v / m)
    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 <= 1.35e-16) {
		tmp = (m / (v / m)) - m;
	} else {
		tmp = (m * (1.0 - m)) / (v / m);
	}
	return tmp;
}

Python

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

↓

def code(m, v):
	tmp = 0
	if m <= 1.35e-16:
		tmp = (m / (v / m)) - m
	else:
		tmp = (m * (1.0 - m)) / (v / m)
	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 <= 1.35e-16)
		tmp = Float64(Float64(m / Float64(v / m)) - m);
	else
		tmp = Float64(Float64(m * Float64(1.0 - m)) / Float64(v / m));
	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 <= 1.35e-16)
		tmp = (m / (v / m)) - m;
	else
		tmp = (m * (1.0 - m)) / (v / m);
	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, 1.35e-16], N[(N[(m / N[(v / m), $MachinePrecision]), $MachinePrecision] - m), $MachinePrecision], N[(N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision] / N[(v / m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.35e-16

   1. Initial program 99.8

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

   2. Taylor expanded in m around 0 86.0

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

   3. Simplified 99.8

      \[\leadsto \color{blue}{\frac{m}{\frac{v}{m}} - m} \]
      Proof

      [Start] 86.0	\[ -1 \cdot m + \frac{{m}^{2}}{v} \]
      +-commutative [=>] 86.0	\[ \color{blue}{\frac{{m}^{2}}{v} + -1 \cdot m} \]
      mul-1-neg [=>] 86.0	\[ \frac{{m}^{2}}{v} + \color{blue}{\left(-m\right)} \]
      unsub-neg [=>] 86.0	\[ \color{blue}{\frac{{m}^{2}}{v} - m} \]
      unpow2 [=>] 86.0	\[ \frac{\color{blue}{m \cdot m}}{v} - m \]
      associate-/l* [=>] 99.8	\[ \color{blue}{\frac{m}{\frac{v}{m}}} - m \]

   if 1.35e-16 < m

   1. Initial program 99.5

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

   2. Taylor expanded in v around 0 98.4

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

   3. Simplified 98.4

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

      [Start] 98.4	\[ \frac{{m}^{2} \cdot \left(1 - m\right)}{v} \]
      associate-/l* [=>] 98.4	\[ \color{blue}{\frac{{m}^{2}}{\frac{v}{1 - m}}} \]
      associate-/r/ [=>] 98.4	\[ \color{blue}{\frac{{m}^{2}}{v} \cdot \left(1 - m\right)} \]
      unpow2 [=>] 98.4	\[ \frac{\color{blue}{m \cdot m}}{v} \cdot \left(1 - m\right) \]
      associate-/l* [=>] 98.4	\[ \color{blue}{\frac{m}{\frac{v}{m}}} \cdot \left(1 - m\right) \]

   4. Applied egg-rr 98.4

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

      [Start] 98.4	\[ \frac{m}{\frac{v}{m}} \cdot \left(1 - m\right) \]
      associate-*l/ [=>] 98.4	\[ \color{blue}{\frac{m \cdot \left(1 - m\right)}{\frac{v}{m}}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.35 \cdot 10^{-16}:\\ \;\;\;\;\frac{m}{\frac{v}{m}} - m\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot \left(1 - m\right)}{\frac{v}{m}}\\ \end{array} \]

Alternatives

Alternative 1
Error	99.5%
Cost	708.00

\[\begin{array}{l} \mathbf{if}\;m \leq 5.2 \cdot 10^{-17}:\\ \;\;\;\;\frac{m}{\frac{v}{m}} - m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(m \cdot \frac{1 - m}{v}\right)\\ \end{array} \]

Alternative 2
Error	99.5%
Cost	708.00

\[\begin{array}{l} \mathbf{if}\;m \leq 1.35 \cdot 10^{-16}:\\ \;\;\;\;\frac{m}{\frac{v}{m}} - m\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot m}{\frac{v}{1 - m}}\\ \end{array} \]

Alternative 3
Error	99.7%
Cost	704.00

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

Alternative 4
Error	99.7%
Cost	704.00

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

Alternative 5
Error	96.3%
Cost	644.00

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

Alternative 6
Error	96.3%
Cost	644.00

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

Alternative 7
Error	63.7%
Cost	452.00

\[\begin{array}{l} \mathbf{if}\;v \leq 6.4 \cdot 10^{-146}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \]

Alternative 8
Error	63.6%
Cost	452.00

\[\begin{array}{l} \mathbf{if}\;v \leq 4.2 \cdot 10^{-146}:\\ \;\;\;\;\frac{m}{\frac{v}{m}}\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \]

Alternative 9
Error	84.3%
Cost	448.00

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

Alternative 10
Error	84.3%
Cost	448.00

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

Alternative 11
Error	41.9%
Cost	128.00

\[-m \]

Reproduce

herbie shell --seed 2023121 
(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))