b parameter of renormalized beta distribution

Average Error: 0.1 → 0.2

Time: 8.0s

Precision: binary64

Cost: 836

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;m \leq 2.8 \cdot 10^{-20}:\\ \;\;\;\;-1 + \left(m + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - m\right) \cdot \left(m \cdot \frac{1 - m}{v}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (m v)
 :precision binary64
 (if (<= m 2.8e-20)
   (+ -1.0 (+ m (/ m v)))
   (* (- 1.0 m) (* 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 <= 2.8e-20) {
		tmp = -1.0 + (m + (m / v));
	} else {
		tmp = (1.0 - 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) * (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 <= 2.8d-20) then
        tmp = (-1.0d0) + (m + (m / v))
    else
        tmp = (1.0d0 - 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) * (1.0 - m);
}

↓

public static double code(double m, double v) {
	double tmp;
	if (m <= 2.8e-20) {
		tmp = -1.0 + (m + (m / v));
	} else {
		tmp = (1.0 - m) * (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 <= 2.8e-20:
		tmp = -1.0 + (m + (m / v))
	else:
		tmp = (1.0 - 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) * Float64(1.0 - m))
end

↓

function code(m, v)
	tmp = 0.0
	if (m <= 2.8e-20)
		tmp = Float64(-1.0 + Float64(m + Float64(m / v)));
	else
		tmp = Float64(Float64(1.0 - m) * Float64(m * Float64(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 <= 2.8e-20)
		tmp = -1.0 + (m + (m / v));
	else
		tmp = (1.0 - 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] * N[(1.0 - m), $MachinePrecision]), $MachinePrecision]

↓

code[m_, v_] := If[LessEqual[m, 2.8e-20], N[(-1.0 + N[(m + N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - m), $MachinePrecision] * N[(m * N[(N[(1.0 - m), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.8000000000000003e-20

   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

      \[\leadsto \color{blue}{\left(1 - m\right) \cdot \left(\frac{m}{\frac{v}{1 - m}} + -1\right)} \]
      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	\[ \left(1 - m\right) \cdot \left(\color{blue}{\frac{m}{\frac{v}{1 - m}}} + \left(-1\right)\right) \]
      metadata-eval [=>] 0.0	\[ \left(1 - m\right) \cdot \left(\frac{m}{\frac{v}{1 - m}} + \color{blue}{-1}\right) \]

   3. Taylor expanded in m around 0 0.1

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

   4. Simplified 0.0

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

      [Start] 0.1	\[ \left(1 + \frac{1}{v}\right) \cdot m - 1 \]
      sub-neg [=>] 0.1	\[ \color{blue}{\left(1 + \frac{1}{v}\right) \cdot m + \left(-1\right)} \]
      metadata-eval [=>] 0.1	\[ \left(1 + \frac{1}{v}\right) \cdot m + \color{blue}{-1} \]
      +-commutative [=>] 0.1	\[ \color{blue}{-1 + \left(1 + \frac{1}{v}\right) \cdot m} \]
      *-commutative [=>] 0.1	\[ -1 + \color{blue}{m \cdot \left(1 + \frac{1}{v}\right)} \]
      distribute-rgt-in [=>] 0.1	\[ -1 + \color{blue}{\left(1 \cdot m + \frac{1}{v} \cdot m\right)} \]
      *-lft-identity [=>] 0.1	\[ -1 + \left(\color{blue}{m} + \frac{1}{v} \cdot m\right) \]
      associate-*l/ [=>] 0.0	\[ -1 + \left(m + \color{blue}{\frac{1 \cdot m}{v}}\right) \]
      associate-*r/ [<=] 0.0	\[ -1 + \left(m + \color{blue}{1 \cdot \frac{m}{v}}\right) \]
      *-lft-identity [=>] 0.0	\[ -1 + \left(m + \color{blue}{\frac{m}{v}}\right) \]

   if 2.8000000000000003e-20 < m

   1. Initial program 0.4

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

   2. Taylor expanded in v around 0 1.2

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

   3. Applied egg-rr 1.1

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

   4. Applied egg-rr 1.2

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.8 \cdot 10^{-20}:\\ \;\;\;\;-1 + \left(m + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - m\right) \cdot \left(m \cdot \frac{1 - m}{v}\right)\\ \end{array} \]

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	2.2
Cost	708

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

Alternative 5
Error	2.1
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	2.1
Cost	708

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

Alternative 7
Error	24.0
Cost	588

\[\begin{array}{l} \mathbf{if}\;m \leq 5.3 \cdot 10^{-163}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 1.15 \cdot 10^{-151}:\\ \;\;\;\;\frac{m}{v}\\ \mathbf{elif}\;m \leq 1.05 \cdot 10^{-134}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v}\\ \end{array} \]

Alternative 8
Error	2.2
Cost	580

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

Alternative 9
Error	9.4
Cost	448

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

Alternative 10
Error	36.5
Cost	192

\[m + -1 \]

Alternative 11
Error	36.8
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023016 
(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)))