a parameter of renormalized beta distribution

Average Error: 0.2 → 0.4

Time: 8.3s

Precision: binary64

Cost: 708

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;m \leq 5.5 \cdot 10^{-15}:\\ \;\;\;\;m \cdot \left(-1 + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - m}{\frac{\frac{v}{m}}{m}}\\ \end{array} \]

FPCore

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

↓

(FPCore (m v)
 :precision binary64
 (if (<= m 5.5e-15) (* m (+ -1.0 (/ m v))) (/ (- 1.0 m) (/ (/ v m) 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 <= 5.5e-15) {
		tmp = m * (-1.0 + (m / v));
	} else {
		tmp = (1.0 - m) / ((v / m) / 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 <= 5.5d-15) then
        tmp = m * ((-1.0d0) + (m / v))
    else
        tmp = (1.0d0 - m) / ((v / m) / 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 <= 5.5e-15) {
		tmp = m * (-1.0 + (m / v));
	} else {
		tmp = (1.0 - m) / ((v / m) / 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 <= 5.5e-15:
		tmp = m * (-1.0 + (m / v))
	else:
		tmp = (1.0 - m) / ((v / m) / 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 <= 5.5e-15)
		tmp = Float64(m * Float64(-1.0 + Float64(m / v)));
	else
		tmp = Float64(Float64(1.0 - m) / Float64(Float64(v / m) / 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 <= 5.5e-15)
		tmp = m * (-1.0 + (m / v));
	else
		tmp = (1.0 - m) / ((v / m) / 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, 5.5e-15], N[(m * N[(-1.0 + N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - m), $MachinePrecision] / N[(N[(v / m), $MachinePrecision] / m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 5.5000000000000002e-15

   1. Initial program 0.1

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

   2. Simplified 0.2

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

      [Start] 0.1	\[ \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m \]
      *-commutative [=>] 0.1	\[ \color{blue}{m \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right)} \]
      associate-*r/ [<=] 0.2	\[ m \cdot \left(\color{blue}{m \cdot \frac{1 - m}{v}} - 1\right) \]
      *-commutative [<=] 0.2	\[ m \cdot \left(\color{blue}{\frac{1 - m}{v} \cdot m} - 1\right) \]
      fma-neg [=>] 0.2	\[ m \cdot \color{blue}{\mathsf{fma}\left(\frac{1 - m}{v}, m, -1\right)} \]
      metadata-eval [=>] 0.2	\[ m \cdot \mathsf{fma}\left(\frac{1 - m}{v}, m, \color{blue}{-1}\right) \]

   3. Taylor expanded in m around 0 0.2

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

   if 5.5000000000000002e-15 < m

   1. Initial program 0.3

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

   2. Simplified 0.3

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

      [Start] 0.3	\[ \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m \]
      *-commutative [=>] 0.3	\[ \color{blue}{m \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right)} \]
      sub-neg [=>] 0.3	\[ m \cdot \color{blue}{\left(\frac{m \cdot \left(1 - m\right)}{v} + \left(-1\right)\right)} \]
      associate-*l/ [<=] 0.3	\[ m \cdot \left(\color{blue}{\frac{m}{v} \cdot \left(1 - m\right)} + \left(-1\right)\right) \]
      metadata-eval [=>] 0.3	\[ m \cdot \left(\frac{m}{v} \cdot \left(1 - m\right) + \color{blue}{-1}\right) \]

   3. Applied egg-rr 0.3

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

   4. Taylor expanded in m around inf 1.3

      \[\leadsto \color{blue}{-1 \cdot \frac{{m}^{3}}{v} + \frac{{m}^{2}}{v}} \]

   5. Simplified 1.3

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

      [Start] 1.3	\[ -1 \cdot \frac{{m}^{3}}{v} + \frac{{m}^{2}}{v} \]
      mul-1-neg [=>] 1.3	\[ \color{blue}{\left(-\frac{{m}^{3}}{v}\right)} + \frac{{m}^{2}}{v} \]
      distribute-neg-frac [=>] 1.3	\[ \color{blue}{\frac{-{m}^{3}}{v}} + \frac{{m}^{2}}{v} \]
      cube-neg [<=] 1.3	\[ \frac{\color{blue}{{\left(-m\right)}^{3}}}{v} + \frac{{m}^{2}}{v} \]
      unpow3 [=>] 1.4	\[ \frac{\color{blue}{\left(\left(-m\right) \cdot \left(-m\right)\right) \cdot \left(-m\right)}}{v} + \frac{{m}^{2}}{v} \]
      associate-*l/ [<=] 1.3	\[ \color{blue}{\frac{\left(-m\right) \cdot \left(-m\right)}{v} \cdot \left(-m\right)} + \frac{{m}^{2}}{v} \]
      sqr-neg [=>] 1.3	\[ \frac{\color{blue}{m \cdot m}}{v} \cdot \left(-m\right) + \frac{{m}^{2}}{v} \]
      associate-*r/ [<=] 1.4	\[ \color{blue}{\left(m \cdot \frac{m}{v}\right)} \cdot \left(-m\right) + \frac{{m}^{2}}{v} \]
      *-commutative [=>] 1.4	\[ \color{blue}{\left(-m\right) \cdot \left(m \cdot \frac{m}{v}\right)} + \frac{{m}^{2}}{v} \]
      unpow2 [=>] 1.4	\[ \left(-m\right) \cdot \left(m \cdot \frac{m}{v}\right) + \frac{\color{blue}{m \cdot m}}{v} \]
      associate-*r/ [<=] 1.4	\[ \left(-m\right) \cdot \left(m \cdot \frac{m}{v}\right) + \color{blue}{m \cdot \frac{m}{v}} \]
      distribute-lft1-in [=>] 1.3	\[ \color{blue}{\left(\left(-m\right) + 1\right) \cdot \left(m \cdot \frac{m}{v}\right)} \]
      +-commutative [=>] 1.3	\[ \color{blue}{\left(1 + \left(-m\right)\right)} \cdot \left(m \cdot \frac{m}{v}\right) \]
      sub-neg [<=] 1.3	\[ \color{blue}{\left(1 - m\right)} \cdot \left(m \cdot \frac{m}{v}\right) \]
      *-commutative [=>] 1.3	\[ \color{blue}{\left(m \cdot \frac{m}{v}\right) \cdot \left(1 - m\right)} \]
      *-commutative [=>] 1.3	\[ \color{blue}{\left(\frac{m}{v} \cdot m\right)} \cdot \left(1 - m\right) \]
      associate-*l* [=>] 1.3	\[ \color{blue}{\frac{m}{v} \cdot \left(m \cdot \left(1 - m\right)\right)} \]

   6. Applied egg-rr 1.3

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 5.5 \cdot 10^{-15}:\\ \;\;\;\;m \cdot \left(-1 + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - m}{\frac{\frac{v}{m}}{m}}\\ \end{array} \]

Alternatives

Alternative 1
Error	25.9
Cost	982

\[\begin{array}{l} \mathbf{if}\;m \leq 2.8 \cdot 10^{-208} \lor \neg \left(m \leq 1.85 \cdot 10^{-197}\right) \land \left(m \leq 1.8 \cdot 10^{-186} \lor \neg \left(m \leq 7.5 \cdot 10^{-172}\right) \land m \leq 5.8 \cdot 10^{-162}\right):\\ \;\;\;\;-m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \]

Alternative 2
Error	0.4
Cost	708

\[\begin{array}{l} \mathbf{if}\;m \leq 3.15 \cdot 10^{-15}:\\ \;\;\;\;m \cdot \left(-1 + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{\frac{v}{1 - m}}\\ \end{array} \]

Alternative 3
Error	0.4
Cost	708

\[\begin{array}{l} \mathbf{if}\;m \leq 4.8 \cdot 10^{-15}:\\ \;\;\;\;m \cdot \left(-1 + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{\frac{\frac{v}{m}}{1 - m}}\\ \end{array} \]

Alternative 4
Error	0.2
Cost	704

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

Alternative 5
Error	0.2
Cost	704

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

Alternative 6
Error	2.4
Cost	644

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

Alternative 7
Error	10.5
Cost	448

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

Alternative 8
Error	36.8
Cost	128

\[-m \]

Reproduce

herbie shell --seed 2023041 
(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))