a parameter of renormalized beta distribution

Average Error: 0.2 → 0.3

Time: 10.4s

Precision: binary64

Cost: 708

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;m \leq 1.25 \cdot 10^{-17}:\\ \;\;\;\;\left(\frac{m}{v} - 1\right) \cdot m\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - m}{v} \cdot \left(m \cdot m\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (m v)
 :precision binary64
 (if (<= m 1.25e-17) (* (- (/ m v) 1.0) m) (* (/ (- 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 <= 1.25e-17) {
		tmp = ((m / v) - 1.0) * m;
	} 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 <= 1.25d-17) then
        tmp = ((m / v) - 1.0d0) * m
    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 <= 1.25e-17) {
		tmp = ((m / v) - 1.0) * m;
	} 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 <= 1.25e-17:
		tmp = ((m / v) - 1.0) * m
	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 <= 1.25e-17)
		tmp = Float64(Float64(Float64(m / v) - 1.0) * m);
	else
		tmp = Float64(Float64(Float64(1.0 - m) / v) * Float64(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 <= 1.25e-17)
		tmp = ((m / v) - 1.0) * m;
	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, 1.25e-17], N[(N[(N[(m / v), $MachinePrecision] - 1.0), $MachinePrecision] * m), $MachinePrecision], N[(N[(N[(1.0 - m), $MachinePrecision] / v), $MachinePrecision] * N[(m * m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.25e-17

   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 0.1

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

   if 1.25e-17 < m

   1. Initial program 0.3

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

   2. Taylor expanded in v around 0 1.2

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

   3. Simplified 1.2

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

      [Start] 1.2	\[ \frac{m \cdot \left(1 - m\right)}{v} \cdot m \]
      rational.json-simplify-2 [=>] 1.2	\[ \frac{\color{blue}{\left(1 - m\right) \cdot m}}{v} \cdot m \]
      rational.json-simplify-49 [=>] 1.2	\[ \color{blue}{\left(m \cdot \frac{1 - m}{v}\right)} \cdot m \]

   4. Applied egg-rr 1.2

      \[\leadsto \color{blue}{m \cdot \left(m \cdot \frac{1 - m}{v}\right) + 0} \]

   5. Simplified 1.2

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

      [Start] 1.2	\[ m \cdot \left(m \cdot \frac{1 - m}{v}\right) + 0 \]
      rational.json-simplify-4 [=>] 1.2	\[ \color{blue}{m \cdot \left(m \cdot \frac{1 - m}{v}\right)} \]
      rational.json-simplify-43 [<=] 1.2	\[ \color{blue}{\frac{1 - m}{v} \cdot \left(m \cdot m\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.25 \cdot 10^{-17}:\\ \;\;\;\;\left(\frac{m}{v} - 1\right) \cdot m\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - m}{v} \cdot \left(m \cdot m\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	24.8
Cost	716

\[\begin{array}{l} t_0 := \frac{m}{v} \cdot m\\ \mathbf{if}\;v \leq 7.5 \cdot 10^{-175}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;v \leq 1.15 \cdot 10^{-170}:\\ \;\;\;\;-m\\ \mathbf{elif}\;v \leq 1.67 \cdot 10^{-136}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \]

Alternative 2
Error	24.8
Cost	716

\[\begin{array}{l} \mathbf{if}\;v \leq 6.5 \cdot 10^{-175}:\\ \;\;\;\;\frac{m}{v} \cdot m\\ \mathbf{elif}\;v \leq 1.4 \cdot 10^{-170}:\\ \;\;\;\;-m\\ \mathbf{elif}\;v \leq 1.67 \cdot 10^{-136}:\\ \;\;\;\;\frac{m}{\frac{v}{m}}\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \]

Alternative 3
Error	24.8
Cost	716

\[\begin{array}{l} \mathbf{if}\;v \leq 5.2 \cdot 10^{-175}:\\ \;\;\;\;\frac{m}{v} \cdot m\\ \mathbf{elif}\;v \leq 1.45 \cdot 10^{-170}:\\ \;\;\;\;-m\\ \mathbf{elif}\;v \leq 1.67 \cdot 10^{-136}:\\ \;\;\;\;\frac{m \cdot m}{v}\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \]

Alternative 4
Error	0.4
Cost	708

\[\begin{array}{l} \mathbf{if}\;m \leq 2 \cdot 10^{-19}:\\ \;\;\;\;\left(\frac{m}{v} - 1\right) \cdot m\\ \mathbf{else}:\\ \;\;\;\;\left(m \cdot \frac{1 - m}{v}\right) \cdot m\\ \end{array} \]

Alternative 5
Error	0.2
Cost	704

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

Alternative 6
Error	0.2
Cost	704

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

Alternative 7
Error	2.5
Cost	644

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

Alternative 8
Error	2.5
Cost	644

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

Alternative 9
Error	10.5
Cost	448

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

Alternative 10
Error	36.7
Cost	128

\[-m \]

Reproduce

herbie shell --seed 2023064 
(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))