a parameter of renormalized beta distribution

Average Accuracy: 99.7% → 99.5%

Time: 7.8s

Precision: binary64

Cost: 708

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;m \leq 1.85 \cdot 10^{-16}:\\ \;\;\;\;m \cdot \frac{m}{v} - m\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot \left(m \cdot \left(1 - m\right)\right)}{v}\\ \end{array} \]

FPCore

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

↓

(FPCore (m v)
 :precision binary64
 (if (<= m 1.85e-16) (- (* m (/ m v)) 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 <= 1.85e-16) {
		tmp = (m * (m / v)) - 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 <= 1.85d-16) then
        tmp = (m * (m / v)) - 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 <= 1.85e-16) {
		tmp = (m * (m / v)) - 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 <= 1.85e-16:
		tmp = (m * (m / v)) - 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 <= 1.85e-16)
		tmp = Float64(Float64(m * Float64(m / v)) - 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 <= 1.85e-16)
		tmp = (m * (m / v)) - 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, 1.85e-16], N[(N[(m * N[(m / v), $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 < 1.85e-16

   1. Initial program 99.8%

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

   2. Applied egg-rr 99.6%

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

      [Start] 99.8	\[ \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m \]
      add-sqr-sqrt [=>] 99.6	\[ \left(\frac{\color{blue}{\sqrt{m \cdot \left(1 - m\right)} \cdot \sqrt{m \cdot \left(1 - m\right)}}}{v} - 1\right) \cdot m \]
      pow2 [=>] 99.6	\[ \left(\frac{\color{blue}{{\left(\sqrt{m \cdot \left(1 - m\right)}\right)}^{2}}}{v} - 1\right) \cdot m \]

   3. Taylor expanded in m around 0 86.8%

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

   4. Simplified 99.8%

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

      [Start] 86.8	\[ -1 \cdot m + \frac{{m}^{2}}{v} \]
      neg-mul-1 [<=] 86.8	\[ \color{blue}{\left(-m\right)} + \frac{{m}^{2}}{v} \]
      +-commutative [=>] 86.8	\[ \color{blue}{\frac{{m}^{2}}{v} + \left(-m\right)} \]
      unsub-neg [=>] 86.8	\[ \color{blue}{\frac{{m}^{2}}{v} - m} \]
      unpow2 [=>] 86.8	\[ \frac{\color{blue}{m \cdot m}}{v} - m \]
      associate-/l* [=>] 99.8	\[ \color{blue}{\frac{m}{\frac{v}{m}}} - m \]
      *-lft-identity [<=] 99.8	\[ \frac{\color{blue}{1 \cdot m}}{\frac{v}{m}} - m \]
      *-rgt-identity [<=] 99.8	\[ \frac{1 \cdot m}{\color{blue}{\frac{v}{m} \cdot 1}} - m \]
      times-frac [=>] 99.7	\[ \color{blue}{\frac{1}{\frac{v}{m}} \cdot \frac{m}{1}} - m \]
      metadata-eval [<=] 99.7	\[ \frac{\color{blue}{-1 \cdot -1}}{\frac{v}{m}} \cdot \frac{m}{1} - m \]
      rem-square-sqrt [<=] 0.0	\[ \frac{-1 \cdot \color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)}}{\frac{v}{m}} \cdot \frac{m}{1} - m \]
      unpow2 [<=] 0.0	\[ \frac{-1 \cdot \color{blue}{{\left(\sqrt{-1}\right)}^{2}}}{\frac{v}{m}} \cdot \frac{m}{1} - m \]
      associate-/l* [<=] 0.0	\[ \color{blue}{\frac{\left(-1 \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot m}{v}} \cdot \frac{m}{1} - m \]
      unpow2 [=>] 0.0	\[ \frac{\left(-1 \cdot \color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)}\right) \cdot m}{v} \cdot \frac{m}{1} - m \]
      rem-square-sqrt [=>] 99.8	\[ \frac{\left(-1 \cdot \color{blue}{-1}\right) \cdot m}{v} \cdot \frac{m}{1} - m \]
      metadata-eval [=>] 99.8	\[ \frac{\color{blue}{1} \cdot m}{v} \cdot \frac{m}{1} - m \]
      *-lft-identity [=>] 99.8	\[ \frac{\color{blue}{m}}{v} \cdot \frac{m}{1} - m \]
      /-rgt-identity [=>] 99.8	\[ \frac{m}{v} \cdot \color{blue}{m} - m \]

   if 1.85e-16 < m

   1. Initial program 99.4%

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

   2. Simplified 99.4%

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

      [Start] 99.4	\[ \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m \]
      *-commutative [=>] 99.4	\[ \color{blue}{m \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right)} \]
      sub-neg [=>] 99.4	\[ m \cdot \color{blue}{\left(\frac{m \cdot \left(1 - m\right)}{v} + \left(-1\right)\right)} \]
      distribute-lft-in [=>] 99.4	\[ \color{blue}{m \cdot \frac{m \cdot \left(1 - m\right)}{v} + m \cdot \left(-1\right)} \]
      *-commutative [=>] 99.4	\[ \color{blue}{\frac{m \cdot \left(1 - m\right)}{v} \cdot m} + m \cdot \left(-1\right) \]
      associate-*l/ [=>] 99.5	\[ \color{blue}{\frac{\left(m \cdot \left(1 - m\right)\right) \cdot m}{v}} + m \cdot \left(-1\right) \]
      associate-*r/ [<=] 99.4	\[ \color{blue}{\left(m \cdot \left(1 - m\right)\right) \cdot \frac{m}{v}} + m \cdot \left(-1\right) \]
      *-lft-identity [<=] 99.4	\[ \left(m \cdot \left(1 - m\right)\right) \cdot \frac{\color{blue}{1 \cdot m}}{v} + m \cdot \left(-1\right) \]
      associate-*l/ [<=] 99.3	\[ \left(m \cdot \left(1 - m\right)\right) \cdot \color{blue}{\left(\frac{1}{v} \cdot m\right)} + m \cdot \left(-1\right) \]
      associate-*r* [=>] 99.3	\[ \color{blue}{\left(\left(m \cdot \left(1 - m\right)\right) \cdot \frac{1}{v}\right) \cdot m} + m \cdot \left(-1\right) \]
      *-commutative [<=] 99.3	\[ \left(\left(m \cdot \left(1 - m\right)\right) \cdot \frac{1}{v}\right) \cdot m + \color{blue}{\left(-1\right) \cdot m} \]
      distribute-rgt-out [=>] 99.3	\[ \color{blue}{m \cdot \left(\left(m \cdot \left(1 - m\right)\right) \cdot \frac{1}{v} + \left(-1\right)\right)} \]
      associate-*r/ [=>] 99.4	\[ m \cdot \left(\color{blue}{\frac{\left(m \cdot \left(1 - m\right)\right) \cdot 1}{v}} + \left(-1\right)\right) \]
      associate-/l* [=>] 99.4	\[ m \cdot \left(\color{blue}{\frac{m \cdot \left(1 - m\right)}{\frac{v}{1}}} + \left(-1\right)\right) \]
      /-rgt-identity [=>] 99.4	\[ m \cdot \left(\frac{m \cdot \left(1 - m\right)}{\color{blue}{v}} + \left(-1\right)\right) \]
      associate-/l* [=>] 99.4	\[ m \cdot \left(\color{blue}{\frac{m}{\frac{v}{1 - m}}} + \left(-1\right)\right) \]
      metadata-eval [=>] 99.4	\[ m \cdot \left(\frac{m}{\frac{v}{1 - m}} + \color{blue}{-1}\right) \]

   3. Taylor expanded in v around 0 98.3%

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

   4. Simplified 98.2%

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

      [Start] 98.3	\[ \frac{{m}^{2} \cdot \left(1 - m\right)}{v} \]
      associate-*r/ [<=] 98.2	\[ \color{blue}{{m}^{2} \cdot \frac{1 - m}{v}} \]
      unpow2 [=>] 98.2	\[ \color{blue}{\left(m \cdot m\right)} \cdot \frac{1 - m}{v} \]

   5. Applied egg-rr 98.3%

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

      [Start] 98.2	\[ \left(m \cdot m\right) \cdot \frac{1 - m}{v} \]
      associate-*r/ [=>] 98.3	\[ \color{blue}{\frac{\left(m \cdot m\right) \cdot \left(1 - m\right)}{v}} \]
      associate-*l* [=>] 98.3	\[ \frac{\color{blue}{m \cdot \left(m \cdot \left(1 - m\right)\right)}}{v} \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.85 \cdot 10^{-16}:\\ \;\;\;\;m \cdot \frac{m}{v} - m\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot \left(m \cdot \left(1 - m\right)\right)}{v}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	60.2%
Cost	717

\[\begin{array}{l} \mathbf{if}\;m \leq 1.2 \cdot 10^{-181} \lor \neg \left(m \leq 3.6 \cdot 10^{-162}\right) \land m \leq 3.5 \cdot 10^{-135}:\\ \;\;\;\;-m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \]

Alternative 2
Accuracy	60.3%
Cost	716

\[\begin{array}{l} \mathbf{if}\;m \leq 7 \cdot 10^{-182}:\\ \;\;\;\;-m\\ \mathbf{elif}\;m \leq 4.6 \cdot 10^{-162}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \mathbf{elif}\;m \leq 1.25 \cdot 10^{-138}:\\ \;\;\;\;-m\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{\frac{v}{m}}\\ \end{array} \]

Alternative 3
Accuracy	99.5%
Cost	708

\[\begin{array}{l} \mathbf{if}\;m \leq 1.95 \cdot 10^{-16}:\\ \;\;\;\;m \cdot \frac{m}{v} - m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(m \cdot \frac{1 - m}{v}\right)\\ \end{array} \]

Alternative 4
Accuracy	99.5%
Cost	708

\[\begin{array}{l} \mathbf{if}\;m \leq 2.5 \cdot 10^{-16}:\\ \;\;\;\;m \cdot \frac{m}{v} - m\\ \mathbf{else}:\\ \;\;\;\;\left(m \cdot m\right) \cdot \frac{1 - m}{v}\\ \end{array} \]

Alternative 5
Accuracy	99.7%
Cost	704

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

Alternative 6
Accuracy	99.7%
Cost	704

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

Alternative 7
Accuracy	96.4%
Cost	644

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

Alternative 8
Accuracy	96.4%
Cost	644

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

Alternative 9
Accuracy	83.6%
Cost	448

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

Alternative 10
Accuracy	83.6%
Cost	448

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

Alternative 11
Accuracy	83.6%
Cost	448

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

Alternative 12
Accuracy	42.6%
Cost	128

\[-m \]

Reproduce

herbie shell --seed 2023151 
(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))