a parameter of renormalized beta distribution

Average Error: 0.2 → 0.3

Time: 6.8s

Precision: binary64

Cost: 708

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;m \leq 4.6 \cdot 10^{-16}:\\ \;\;\;\;\frac{m}{\frac{v}{m}} - m\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot \left(m \cdot \left(1 - m\right)\right)}{v}\\ \end{array} \]

FPCore

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

↓

(FPCore (m v)
 :precision binary64
 (if (<= m 4.6e-16) (- (/ m (/ v m)) 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 <= 4.6e-16) {
		tmp = (m / (v / m)) - 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 <= 4.6d-16) then
        tmp = (m / (v / m)) - 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 <= 4.6e-16) {
		tmp = (m / (v / m)) - 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 <= 4.6e-16:
		tmp = (m / (v / m)) - 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 <= 4.6e-16)
		tmp = Float64(Float64(m / Float64(v / m)) - m);
	else
		tmp = Float64(Float64(m * 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 <= 4.6e-16)
		tmp = (m / (v / m)) - 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, 4.6e-16], N[(N[(m / N[(v / m), $MachinePrecision]), $MachinePrecision] - m), $MachinePrecision], N[(N[(m * N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 4.5999999999999998e-16

   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 8.4

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

   3. Simplified 0.1

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

      [Start] 8.4	\[ -1 \cdot m + \frac{{m}^{2}}{v} \]
      +-commutative [=>] 8.4	\[ \color{blue}{\frac{{m}^{2}}{v} + -1 \cdot m} \]
      mul-1-neg [=>] 8.4	\[ \frac{{m}^{2}}{v} + \color{blue}{\left(-m\right)} \]
      unsub-neg [=>] 8.4	\[ \color{blue}{\frac{{m}^{2}}{v} - m} \]
      unpow2 [=>] 8.4	\[ \frac{\color{blue}{m \cdot m}}{v} - m \]
      associate-/l* [=>] 0.1	\[ \color{blue}{\frac{m}{\frac{v}{m}}} - m \]

   if 4.5999999999999998e-16 < m

   1. Initial program 0.4

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

   2. Simplified 0.4

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

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

   3. Taylor expanded in v around 0 1.1

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

   4. Simplified 1.1

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

      [Start] 1.1	\[ \frac{{m}^{2} \cdot \left(1 - m\right)}{v} \]
      unpow2 [=>] 1.1	\[ \frac{\color{blue}{\left(m \cdot m\right)} \cdot \left(1 - m\right)}{v} \]
      associate-*r* [<=] 1.1	\[ \frac{\color{blue}{m \cdot \left(m \cdot \left(1 - m\right)\right)}}{v} \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 4.6 \cdot 10^{-16}:\\ \;\;\;\;\frac{m}{\frac{v}{m}} - m\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot \left(m \cdot \left(1 - m\right)\right)}{v}\\ \end{array} \]

Alternatives

Alternative 1
Error	23.9
Cost	717

\[\begin{array}{l} \mathbf{if}\;v \leq 6.8 \cdot 10^{-167} \lor \neg \left(v \leq 4.2 \cdot 10^{-134}\right) \land v \leq 4.5 \cdot 10^{-126}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \]

Alternative 2
Error	23.9
Cost	716

\[\begin{array}{l} \mathbf{if}\;v \leq 3.6 \cdot 10^{-166}:\\ \;\;\;\;\frac{m}{\frac{v}{m}}\\ \mathbf{elif}\;v \leq 9 \cdot 10^{-134}:\\ \;\;\;\;-m\\ \mathbf{elif}\;v \leq 2.5 \cdot 10^{-126}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \]

Alternative 3
Error	23.9
Cost	716

\[\begin{array}{l} \mathbf{if}\;v \leq 1.26 \cdot 10^{-166}:\\ \;\;\;\;\frac{m}{\frac{v}{m}}\\ \mathbf{elif}\;v \leq 7.5 \cdot 10^{-133}:\\ \;\;\;\;-m\\ \mathbf{elif}\;v \leq 2 \cdot 10^{-126}:\\ \;\;\;\;\frac{m \cdot m}{v}\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \]

Alternative 4
Error	0.4
Cost	708

\[\begin{array}{l} \mathbf{if}\;m \leq 7.5 \cdot 10^{-21}:\\ \;\;\;\;\frac{m}{\frac{v}{m}} - m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(m \cdot \frac{1 - m}{v}\right)\\ \end{array} \]

Alternative 5
Error	0.4
Cost	708

\[\begin{array}{l} \mathbf{if}\;m \leq 7.5 \cdot 10^{-21}:\\ \;\;\;\;\frac{m}{\frac{v}{m}} - m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{\frac{v}{1 - m}}\\ \end{array} \]

Alternative 6
Error	0.4
Cost	708

\[\begin{array}{l} \mathbf{if}\;m \leq 1.5 \cdot 10^{-24}:\\ \;\;\;\;\frac{m}{\frac{v}{m}} - m\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{\frac{v}{m \cdot \left(1 - m\right)}}\\ \end{array} \]

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

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

Alternative 9
Error	0.2
Cost	704

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

Alternative 10
Error	2.4
Cost	644

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

Alternative 11
Error	2.4
Cost	644

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

Alternative 12
Error	2.4
Cost	644

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

Alternative 13
Error	10.4
Cost	448

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

Alternative 14
Error	10.4
Cost	448

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

Alternative 15
Error	37.0
Cost	128

\[-m \]

Reproduce

herbie shell --seed 2023045 
(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))