a parameter of renormalized beta distribution

Average Accuracy: 58.2% → 60.5%

Time: 7.3s

Precision: binary64

Cost: 1476

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

Math

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

↓

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

FPCore

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

↓

(FPCore (m v)
 :precision binary64
 (let* ((t_0 (* m (+ (/ (* m (- 1.0 m)) v) -1.0))))
   (if (<= t_0 (- INFINITY)) (- m) t_0)))

C

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

↓

double code(double m, double v) {
	double t_0 = m * (((m * (1.0 - m)) / v) + -1.0);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = -m;
	} else {
		tmp = t_0;
	}
	return tmp;
}

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 t_0 = m * (((m * (1.0 - m)) / v) + -1.0);
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = -m;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

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

↓

def code(m, v):
	t_0 = m * (((m * (1.0 - m)) / v) + -1.0)
	tmp = 0
	if t_0 <= -math.inf:
		tmp = -m
	else:
		tmp = t_0
	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)
	t_0 = Float64(m * Float64(Float64(Float64(m * Float64(1.0 - m)) / v) + -1.0))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(-m);
	else
		tmp = t_0;
	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)
	t_0 = m * (((m * (1.0 - m)) / v) + -1.0);
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = -m;
	else
		tmp = t_0;
	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_] := Block[{t$95$0 = N[(m * N[(N[(N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], (-m), t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 1 m)) v) 1) m) < -inf.0

   1. Initial program 0.0%

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

   2. Simplified 0.0%

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

      [Start] 0.0	\[ \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m \]
      *-commutative [=>] 0.0	\[ \color{blue}{m \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right)} \]
      sub-neg [=>] 0.0	\[ m \cdot \color{blue}{\left(\frac{m \cdot \left(1 - m\right)}{v} + \left(-1\right)\right)} \]
      distribute-lft-in [=>] 0.0	\[ \color{blue}{m \cdot \frac{m \cdot \left(1 - m\right)}{v} + m \cdot \left(-1\right)} \]
      *-commutative [=>] 0.0	\[ \color{blue}{\frac{m \cdot \left(1 - m\right)}{v} \cdot m} + m \cdot \left(-1\right) \]
      associate-*l/ [=>] 0.0	\[ \color{blue}{\frac{\left(m \cdot \left(1 - m\right)\right) \cdot m}{v}} + m \cdot \left(-1\right) \]
      associate-*r/ [<=] 0.0	\[ \color{blue}{\left(m \cdot \left(1 - m\right)\right) \cdot \frac{m}{v}} + m \cdot \left(-1\right) \]
      *-lft-identity [<=] 0.0	\[ \left(m \cdot \left(1 - m\right)\right) \cdot \frac{\color{blue}{1 \cdot m}}{v} + m \cdot \left(-1\right) \]
      associate-*l/ [<=] 0.0	\[ \left(m \cdot \left(1 - m\right)\right) \cdot \color{blue}{\left(\frac{1}{v} \cdot m\right)} + m \cdot \left(-1\right) \]
      associate-*r* [=>] 0.0	\[ \color{blue}{\left(\left(m \cdot \left(1 - m\right)\right) \cdot \frac{1}{v}\right) \cdot m} + m \cdot \left(-1\right) \]
      *-commutative [<=] 0.0	\[ \left(\left(m \cdot \left(1 - m\right)\right) \cdot \frac{1}{v}\right) \cdot m + \color{blue}{\left(-1\right) \cdot m} \]
      distribute-rgt-out [=>] 0.0	\[ \color{blue}{m \cdot \left(\left(m \cdot \left(1 - m\right)\right) \cdot \frac{1}{v} + \left(-1\right)\right)} \]
      associate-*r/ [=>] 0.0	\[ m \cdot \left(\color{blue}{\frac{\left(m \cdot \left(1 - m\right)\right) \cdot 1}{v}} + \left(-1\right)\right) \]
      associate-/l* [=>] 0.0	\[ m \cdot \left(\color{blue}{\frac{m \cdot \left(1 - m\right)}{\frac{v}{1}}} + \left(-1\right)\right) \]
      /-rgt-identity [=>] 0.0	\[ m \cdot \left(\frac{m \cdot \left(1 - m\right)}{\color{blue}{v}} + \left(-1\right)\right) \]
      associate-*l/ [<=] 0.0	\[ m \cdot \left(\color{blue}{\frac{m}{v} \cdot \left(1 - m\right)} + \left(-1\right)\right) \]
      metadata-eval [=>] 0.0	\[ m \cdot \left(\frac{m}{v} \cdot \left(1 - m\right) + \color{blue}{-1}\right) \]

   3. Taylor expanded in m around 0 5.7%

      \[\leadsto \color{blue}{-1 \cdot m} \]

   4. Simplified 5.7%

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

      [Start] 5.7	\[ -1 \cdot m \]
      neg-mul-1 [<=] 5.7	\[ \color{blue}{-m} \]

   if -inf.0 < (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 1 m)) v) 1) m)

   1. Initial program 99.7%

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

4. Final simplification 60.5%

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

Alternatives

Alternative 1
Accuracy	60.0%
Cost	840

\[\begin{array}{l} \mathbf{if}\;m \leq 3.8 \cdot 10^{-15}:\\ \;\;\;\;\frac{m}{\frac{v}{m}} - m\\ \mathbf{elif}\;m \leq 7.4 \cdot 10^{+80}:\\ \;\;\;\;m \cdot \left(m \cdot \frac{1 - m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \]

Alternative 2
Accuracy	60.0%
Cost	840

\[\begin{array}{l} \mathbf{if}\;m \leq 1.35 \cdot 10^{-15}:\\ \;\;\;\;\frac{m}{\frac{v}{m}} - m\\ \mathbf{elif}\;m \leq 3.1 \cdot 10^{+82}:\\ \;\;\;\;\frac{m}{v} \cdot \left(m - m \cdot m\right)\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \]

Alternative 3
Accuracy	60.0%
Cost	840

\[\begin{array}{l} \mathbf{if}\;m \leq 3 \cdot 10^{-15}:\\ \;\;\;\;\frac{m}{\frac{v}{m}} - m\\ \mathbf{elif}\;m \leq 3.1 \cdot 10^{+82}:\\ \;\;\;\;\frac{m}{\frac{\frac{v}{1 - m}}{m}}\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \]

Alternative 4
Accuracy	60.1%
Cost	836

\[\begin{array}{l} \mathbf{if}\;m \leq 3.1 \cdot 10^{+82}:\\ \;\;\;\;m \cdot \left(\left(1 - m\right) \cdot \frac{m}{v} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \]

Alternative 5
Accuracy	58.1%
Cost	776

\[\begin{array}{l} \mathbf{if}\;m \leq 1:\\ \;\;\;\;\frac{m}{\frac{v}{m}} - m\\ \mathbf{elif}\;m \leq 3 \cdot 10^{+82}:\\ \;\;\;\;m \cdot \left(m \cdot \frac{-m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \]

Alternative 6
Accuracy	58.1%
Cost	776

\[\begin{array}{l} \mathbf{if}\;m \leq 1:\\ \;\;\;\;\frac{m}{\frac{v}{m}} - m\\ \mathbf{elif}\;m \leq 3.1 \cdot 10^{+82}:\\ \;\;\;\;\left(-m\right) \cdot \frac{m \cdot m}{v}\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \]

Alternative 7
Accuracy	58.1%
Cost	776

\[\begin{array}{l} \mathbf{if}\;m \leq 1:\\ \;\;\;\;\frac{m}{\frac{v}{m}} - m\\ \mathbf{elif}\;m \leq 3.1 \cdot 10^{+82}:\\ \;\;\;\;\frac{m}{\frac{-v}{m \cdot m}}\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \]

Alternative 8
Accuracy	38.4%
Cost	712

\[\begin{array}{l} \mathbf{if}\;m \leq 1.22 \cdot 10^{-164}:\\ \;\;\;\;-m\\ \mathbf{elif}\;m \leq 1:\\ \;\;\;\;m \cdot \left(m \cdot \frac{1}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \]

Alternative 9
Accuracy	38.5%
Cost	584

\[\begin{array}{l} \mathbf{if}\;m \leq 1.8 \cdot 10^{-163}:\\ \;\;\;\;-m\\ \mathbf{elif}\;m \leq 1:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \]

Alternative 10
Accuracy	38.5%
Cost	584

\[\begin{array}{l} \mathbf{if}\;m \leq 9.5 \cdot 10^{-159}:\\ \;\;\;\;-m\\ \mathbf{elif}\;m \leq 1:\\ \;\;\;\;\frac{m}{\frac{v}{m}}\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \]

Alternative 11
Accuracy	51.5%
Cost	580

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

Alternative 12
Accuracy	51.5%
Cost	580

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

Alternative 13
Accuracy	27.1%
Cost	128

\[-m \]

Reproduce

herbie shell --seed 2023153 
(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))