b parameter of renormalized beta distribution

Average Error: 0.1 → 0.3

Time: 8.0s

Precision: binary64

Cost: 836

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;m \leq 3.15 \cdot 10^{-21}:\\ \;\;\;\;-1 + \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{\left(1 - m\right) \cdot \left(1 - m\right)}{v}\\ \end{array} \]

FPCore

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

↓

(FPCore (m v)
 :precision binary64
 (if (<= m 3.15e-21) (+ -1.0 (/ m v)) (* m (/ (* (- 1.0 m) (- 1.0 m)) v))))

C

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

↓

double code(double m, double v) {
	double tmp;
	if (m <= 3.15e-21) {
		tmp = -1.0 + (m / v);
	} else {
		tmp = m * (((1.0 - 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) * (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 <= 3.15d-21) then
        tmp = (-1.0d0) + (m / v)
    else
        tmp = m * (((1.0d0 - 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) * (1.0 - m);
}

↓

public static double code(double m, double v) {
	double tmp;
	if (m <= 3.15e-21) {
		tmp = -1.0 + (m / v);
	} else {
		tmp = m * (((1.0 - m) * (1.0 - m)) / v);
	}
	return tmp;
}

Python

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

↓

def code(m, v):
	tmp = 0
	if m <= 3.15e-21:
		tmp = -1.0 + (m / v)
	else:
		tmp = m * (((1.0 - 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) * Float64(1.0 - m))
end

↓

function code(m, v)
	tmp = 0.0
	if (m <= 3.15e-21)
		tmp = Float64(-1.0 + Float64(m / v));
	else
		tmp = Float64(m * Float64(Float64(Float64(1.0 - m) * Float64(1.0 - m)) / v));
	end
	return tmp
end

MATLAB

function tmp = code(m, v)
	tmp = (((m * (1.0 - m)) / v) - 1.0) * (1.0 - m);
end

↓

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 3.15e-21)
		tmp = -1.0 + (m / v);
	else
		tmp = m * (((1.0 - 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] * N[(1.0 - m), $MachinePrecision]), $MachinePrecision]

↓

code[m_, v_] := If[LessEqual[m, 3.15e-21], N[(-1.0 + N[(m / v), $MachinePrecision]), $MachinePrecision], N[(m * N[(N[(N[(1.0 - m), $MachinePrecision] * N[(1.0 - m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 3.15e-21

   1. Initial program 0.0

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

   2. Simplified 0.0

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

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

   3. Taylor expanded in m around 0 0.2

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

   4. Taylor expanded in v around 0 0.0

      \[\leadsto \color{blue}{\frac{m}{v}} - 1 \]

   if 3.15e-21 < m

   1. Initial program 0.3

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

   2. Taylor expanded in v around 0 1.5

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

   3. Applied egg-rr 13.2

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

   4. Applied egg-rr 7.3

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(1 - m\right) \cdot \left(m \cdot \frac{1 - m}{v}\right)\right)} - 1} \]

   5. Simplified 1.5

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

      [Start] 7.3	\[ e^{\mathsf{log1p}\left(\left(1 - m\right) \cdot \left(m \cdot \frac{1 - m}{v}\right)\right)} - 1 \]
      expm1-def [=>] 7.3	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(1 - m\right) \cdot \left(m \cdot \frac{1 - m}{v}\right)\right)\right)} \]
      expm1-log1p [=>] 1.5	\[ \color{blue}{\left(1 - m\right) \cdot \left(m \cdot \frac{1 - m}{v}\right)} \]
      *-commutative [<=] 1.5	\[ \color{blue}{\left(m \cdot \frac{1 - m}{v}\right) \cdot \left(1 - m\right)} \]
      associate-*l* [=>] 1.5	\[ \color{blue}{m \cdot \left(\frac{1 - m}{v} \cdot \left(1 - m\right)\right)} \]
      associate-*l/ [=>] 1.5	\[ m \cdot \color{blue}{\frac{\left(1 - m\right) \cdot \left(1 - m\right)}{v}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 3.15 \cdot 10^{-21}:\\ \;\;\;\;-1 + \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{\left(1 - m\right) \cdot \left(1 - m\right)}{v}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.1
Cost	832

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

Alternative 2
Error	0.1
Cost	832

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

Alternative 3
Error	0.1
Cost	832

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

Alternative 4
Error	2.4
Cost	708

\[\begin{array}{l} \mathbf{if}\;m \leq 2.4:\\ \;\;\;\;-1 + \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;\left(m \cdot \frac{m}{v}\right) \cdot \left(m + -1\right)\\ \end{array} \]

Alternative 5
Error	2.3
Cost	708

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

Alternative 6
Error	2.3
Cost	708

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

Alternative 7
Error	2.3
Cost	708

\[\begin{array}{l} \mathbf{if}\;m \leq 1.9:\\ \;\;\;\;\frac{-1 + \frac{m}{v}}{m + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{\frac{v}{m}} \cdot \left(m + -1\right)\\ \end{array} \]

Alternative 8
Error	2.4
Cost	580

\[\begin{array}{l} \mathbf{if}\;m \leq 0.38:\\ \;\;\;\;-1 + \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{\frac{v}{m \cdot m}}\\ \end{array} \]

Alternative 9
Error	24.4
Cost	324

\[\begin{array}{l} \mathbf{if}\;v \leq 1.9 \cdot 10^{-178}:\\ \;\;\;\;\frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;m + -1\\ \end{array} \]

Alternative 10
Error	9.7
Cost	320

\[-1 + \frac{m}{v} \]

Alternative 11
Error	37.6
Cost	192

\[m + -1 \]

Alternative 12
Error	37.9
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023053 
(FPCore (m v)
  :name "b 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) (- 1.0 m)))