a parameter of renormalized beta distribution

Average Accuracy: 99.7% → 99.5%

Time: 8.4s

Precision: binary64

Cost: 708

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;m \leq 2.2 \cdot 10^{-20}:\\ \;\;\;\;m \cdot \frac{m}{v} - m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m \cdot \left(1 - m\right)}{v}\\ \end{array} \]

FPCore

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

↓

(FPCore (m v)
 :precision binary64
 (if (<= m 2.2e-20) (- (* m (/ m v)) m) (* m (/ (* m (- 1.0 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 <= 2.2e-20) {
		tmp = (m * (m / v)) - m;
	} else {
		tmp = m * ((m * (1.0 - 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 <= 2.2d-20) then
        tmp = (m * (m / v)) - m
    else
        tmp = m * ((m * (1.0d0 - 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 <= 2.2e-20) {
		tmp = (m * (m / v)) - m;
	} else {
		tmp = m * ((m * (1.0 - 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 <= 2.2e-20:
		tmp = (m * (m / v)) - m
	else:
		tmp = m * ((m * (1.0 - 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 <= 2.2e-20)
		tmp = Float64(Float64(m * Float64(m / v)) - m);
	else
		tmp = Float64(m * Float64(Float64(m * Float64(1.0 - 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 <= 2.2e-20)
		tmp = (m * (m / v)) - m;
	else
		tmp = m * ((m * (1.0 - 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, 2.2e-20], N[(N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision] - m), $MachinePrecision], N[(m * N[(N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.19999999999999991e-20

   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 87.4%

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

   3. Simplified 99.8%

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

      [Start] 87.4	\[ -1 \cdot m + \frac{{m}^{2}}{v} \]
      neg-mul-1 [<=] 87.4	\[ \color{blue}{\left(-m\right)} + \frac{{m}^{2}}{v} \]
      +-commutative [=>] 87.4	\[ \color{blue}{\frac{{m}^{2}}{v} + \left(-m\right)} \]
      unsub-neg [=>] 87.4	\[ \color{blue}{\frac{{m}^{2}}{v} - m} \]
      unpow2 [=>] 87.4	\[ \frac{\color{blue}{m \cdot m}}{v} - m \]
      associate-*r/ [<=] 99.8	\[ \color{blue}{m \cdot \frac{m}{v}} - m \]

   if 2.19999999999999991e-20 < m

   1. Initial program 99.5%

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

   2. Simplified 99.4%

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

      [Start] 99.5	\[ \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m \]
      *-commutative [=>] 99.5	\[ \color{blue}{m \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right)} \]
      sub-neg [=>] 99.5	\[ m \cdot \color{blue}{\left(\frac{m \cdot \left(1 - m\right)}{v} + \left(-1\right)\right)} \]
      associate-*r/ [<=] 99.4	\[ m \cdot \left(\color{blue}{m \cdot \frac{1 - m}{v}} + \left(-1\right)\right) \]
      fma-def [=>] 99.4	\[ m \cdot \color{blue}{\mathsf{fma}\left(m, \frac{1 - m}{v}, -1\right)} \]
      metadata-eval [=>] 99.4	\[ m \cdot \mathsf{fma}\left(m, \frac{1 - m}{v}, \color{blue}{-1}\right) \]

   3. Taylor expanded in v around 0 98.4%

      \[\leadsto m \cdot \color{blue}{\frac{m \cdot \left(1 - m\right)}{v}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.2 \cdot 10^{-20}:\\ \;\;\;\;m \cdot \frac{m}{v} - m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m \cdot \left(1 - m\right)}{v}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	62.0%
Cost	717

\[\begin{array}{l} \mathbf{if}\;v \leq 1.25 \cdot 10^{-167} \lor \neg \left(v \leq 1.1 \cdot 10^{-156}\right) \land v \leq 1.12 \cdot 10^{-125}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \]

Alternative 2
Accuracy	61.9%
Cost	716

\[\begin{array}{l} \mathbf{if}\;v \leq 5.6 \cdot 10^{-168}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \mathbf{elif}\;v \leq 1.7 \cdot 10^{-156}:\\ \;\;\;\;-m\\ \mathbf{elif}\;v \leq 4.2 \cdot 10^{-125}:\\ \;\;\;\;\frac{m \cdot m}{v}\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \]

Alternative 3
Accuracy	99.5%
Cost	708

\[\begin{array}{l} \mathbf{if}\;m \leq 2.2 \cdot 10^{-20}:\\ \;\;\;\;m \cdot \frac{m}{v} - m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(\left(1 - m\right) \cdot \frac{m}{v}\right)\\ \end{array} \]

Alternative 4
Accuracy	99.7%
Cost	704

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

Alternative 5
Accuracy	99.7%
Cost	704

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

Alternative 6
Accuracy	96.1%
Cost	644

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

Alternative 7
Accuracy	96.1%
Cost	644

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

Alternative 8
Accuracy	83.3%
Cost	448

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

Alternative 9
Accuracy	42.7%
Cost	128

\[-m \]

Reproduce

herbie shell --seed 2023146 
(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))