a parameter of renormalized beta distribution

Percentage Accurate: 99.8% → 99.7%

Time: 8.8s

Precision: binary64

Cost: 708

• 40.4% of points produce a very large (infinite) output. You may want to add a precondition.

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;m \leq 5.4 \cdot 10^{-14}:\\ \;\;\;\;m \cdot \left(-1 + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - m\right) \cdot \left(m \cdot m\right)}{v}\\ \end{array} \]

FPCore

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

↓

(FPCore (m v)
 :precision binary64
 (if (<= m 5.4e-14) (* m (+ -1.0 (/ m v))) (/ (* (- 1.0 m) (* m 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 <= 5.4e-14) {
		tmp = m * (-1.0 + (m / v));
	} else {
		tmp = ((1.0 - m) * (m * 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 <= 5.4d-14) then
        tmp = m * ((-1.0d0) + (m / v))
    else
        tmp = ((1.0d0 - m) * (m * 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 <= 5.4e-14) {
		tmp = m * (-1.0 + (m / v));
	} else {
		tmp = ((1.0 - m) * (m * 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 <= 5.4e-14:
		tmp = m * (-1.0 + (m / v))
	else:
		tmp = ((1.0 - m) * (m * 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 <= 5.4e-14)
		tmp = Float64(m * Float64(-1.0 + Float64(m / v)));
	else
		tmp = Float64(Float64(Float64(1.0 - m) * Float64(m * 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 <= 5.4e-14)
		tmp = m * (-1.0 + (m / v));
	else
		tmp = ((1.0 - m) * (m * 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, 5.4e-14], N[(m * N[(-1.0 + N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - m), $MachinePrecision] * N[(m * m), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 5.3999999999999997e-14

   1. Initial program 99.8%

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

   2. Simplified 99.8%

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

      [Start] 99.8	\[ \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m \]
      *-commutative [=>] 99.8	\[ \color{blue}{m \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right)} \]
      sub-neg [=>] 99.8	\[ m \cdot \color{blue}{\left(\frac{m \cdot \left(1 - m\right)}{v} + \left(-1\right)\right)} \]
      distribute-lft-in [=>] 99.7	\[ \color{blue}{m \cdot \frac{m \cdot \left(1 - m\right)}{v} + m \cdot \left(-1\right)} \]
      *-commutative [=>] 99.7	\[ \color{blue}{\frac{m \cdot \left(1 - m\right)}{v} \cdot m} + m \cdot \left(-1\right) \]
      associate-*l/ [=>] 80.6	\[ \color{blue}{\frac{\left(m \cdot \left(1 - m\right)\right) \cdot m}{v}} + m \cdot \left(-1\right) \]
      associate-*r/ [<=] 99.7	\[ \color{blue}{\left(m \cdot \left(1 - m\right)\right) \cdot \frac{m}{v}} + m \cdot \left(-1\right) \]
      *-lft-identity [<=] 99.7	\[ \left(m \cdot \left(1 - m\right)\right) \cdot \frac{\color{blue}{1 \cdot m}}{v} + m \cdot \left(-1\right) \]
      associate-*l/ [<=] 99.7	\[ \left(m \cdot \left(1 - m\right)\right) \cdot \color{blue}{\left(\frac{1}{v} \cdot m\right)} + m \cdot \left(-1\right) \]
      associate-*r* [=>] 99.7	\[ \color{blue}{\left(\left(m \cdot \left(1 - m\right)\right) \cdot \frac{1}{v}\right) \cdot m} + m \cdot \left(-1\right) \]
      *-commutative [<=] 99.7	\[ \left(\left(m \cdot \left(1 - m\right)\right) \cdot \frac{1}{v}\right) \cdot m + \color{blue}{\left(-1\right) \cdot m} \]
      distribute-rgt-out [=>] 99.7	\[ \color{blue}{m \cdot \left(\left(m \cdot \left(1 - m\right)\right) \cdot \frac{1}{v} + \left(-1\right)\right)} \]
      associate-*r/ [=>] 99.8	\[ m \cdot \left(\color{blue}{\frac{\left(m \cdot \left(1 - m\right)\right) \cdot 1}{v}} + \left(-1\right)\right) \]
      associate-/l* [=>] 99.8	\[ m \cdot \left(\color{blue}{\frac{m \cdot \left(1 - m\right)}{\frac{v}{1}}} + \left(-1\right)\right) \]
      /-rgt-identity [=>] 99.8	\[ m \cdot \left(\frac{m \cdot \left(1 - m\right)}{\color{blue}{v}} + \left(-1\right)\right) \]
      associate-*l/ [<=] 99.8	\[ m \cdot \left(\color{blue}{\frac{m}{v} \cdot \left(1 - m\right)} + \left(-1\right)\right) \]
      metadata-eval [=>] 99.8	\[ m \cdot \left(\frac{m}{v} \cdot \left(1 - m\right) + \color{blue}{-1}\right) \]

   3. Taylor expanded in m around 0 99.8%

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

   if 5.3999999999999997e-14 < m

   1. Initial program 99.9%

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

   2. Simplified 99.9%

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

      [Start] 99.9	\[ \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot m \]
      *-commutative [=>] 99.9	\[ \color{blue}{m \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right)} \]
      sub-neg [=>] 99.9	\[ m \cdot \color{blue}{\left(\frac{m \cdot \left(1 - m\right)}{v} + \left(-1\right)\right)} \]
      associate-*r/ [<=] 99.9	\[ m \cdot \left(\color{blue}{m \cdot \frac{1 - m}{v}} + \left(-1\right)\right) \]
      fma-def [=>] 99.9	\[ m \cdot \color{blue}{\mathsf{fma}\left(m, \frac{1 - m}{v}, -1\right)} \]
      metadata-eval [=>] 99.9	\[ m \cdot \mathsf{fma}\left(m, \frac{1 - m}{v}, \color{blue}{-1}\right) \]

   3. Taylor expanded in m around inf 55.0%

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

   4. Simplified 99.9%

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

      [Start] 55.0	\[ m \cdot \left(\frac{m}{v} + -1 \cdot \frac{{m}^{2}}{v}\right) \]
      mul-1-neg [=>] 55.0	\[ m \cdot \left(\frac{m}{v} + \color{blue}{\left(-\frac{{m}^{2}}{v}\right)}\right) \]
      unsub-neg [=>] 55.0	\[ m \cdot \color{blue}{\left(\frac{m}{v} - \frac{{m}^{2}}{v}\right)} \]
      *-lft-identity [<=] 55.0	\[ m \cdot \left(\frac{\color{blue}{1 \cdot m}}{v} - \frac{{m}^{2}}{v}\right) \]
      associate-/l* [=>] 54.9	\[ m \cdot \left(\color{blue}{\frac{1}{\frac{v}{m}}} - \frac{{m}^{2}}{v}\right) \]
      unpow2 [=>] 54.9	\[ m \cdot \left(\frac{1}{\frac{v}{m}} - \frac{\color{blue}{m \cdot m}}{v}\right) \]
      associate-/l* [=>] 54.9	\[ m \cdot \left(\frac{1}{\frac{v}{m}} - \color{blue}{\frac{m}{\frac{v}{m}}}\right) \]
      div-sub [<=] 99.9	\[ m \cdot \color{blue}{\frac{1 - m}{\frac{v}{m}}} \]
      associate-/l* [<=] 99.9	\[ m \cdot \color{blue}{\frac{\left(1 - m\right) \cdot m}{v}} \]
      *-commutative [<=] 99.9	\[ m \cdot \frac{\color{blue}{m \cdot \left(1 - m\right)}}{v} \]
      associate-*r/ [<=] 99.9	\[ m \cdot \color{blue}{\left(m \cdot \frac{1 - m}{v}\right)} \]

   5. Applied egg-rr 100.0%

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

      [Start] 99.9	\[ m \cdot \left(m \cdot \frac{1 - m}{v}\right) \]
      associate-*r* [=>] 99.9	\[ \color{blue}{\left(m \cdot m\right) \cdot \frac{1 - m}{v}} \]
      associate-*r/ [=>] 100.0	\[ \color{blue}{\frac{\left(m \cdot m\right) \cdot \left(1 - m\right)}{v}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 5.4 \cdot 10^{-14}:\\ \;\;\;\;m \cdot \left(-1 + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - m\right) \cdot \left(m \cdot m\right)}{v}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.8%
Cost	832

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

Alternative 2
Accuracy	36.6%
Cost	717

\[\begin{array}{l} \mathbf{if}\;v \leq 4.5 \cdot 10^{-199} \lor \neg \left(v \leq 2.1 \cdot 10^{-165}\right) \land v \leq 1.42 \cdot 10^{-123}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \]

Alternative 3
Accuracy	36.6%
Cost	716

\[\begin{array}{l} \mathbf{if}\;v \leq 5.4 \cdot 10^{-199}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \mathbf{elif}\;v \leq 1.15 \cdot 10^{-165}:\\ \;\;\;\;-m\\ \mathbf{elif}\;v \leq 4 \cdot 10^{-123}:\\ \;\;\;\;\frac{m \cdot m}{v}\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \]

Alternative 4
Accuracy	99.7%
Cost	708

\[\begin{array}{l} \mathbf{if}\;m \leq 6.6 \cdot 10^{-14}:\\ \;\;\;\;m \cdot \left(-1 + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(m \cdot \frac{1 - m}{v}\right)\\ \end{array} \]

Alternative 5
Accuracy	99.7%
Cost	708

\[\begin{array}{l} \mathbf{if}\;m \leq 5.3 \cdot 10^{-14}:\\ \;\;\;\;m \cdot \left(-1 + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot m}{\frac{v}{1 - m}}\\ \end{array} \]

Alternative 6
Accuracy	99.8%
Cost	704

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

Alternative 7
Accuracy	97.5%
Cost	644

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

Alternative 8
Accuracy	97.5%
Cost	644

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

Alternative 9
Accuracy	97.5%
Cost	644

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

Alternative 10
Accuracy	52.2%
Cost	580

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

Alternative 11
Accuracy	27.4%
Cost	128

\[-m \]

Reproduce

herbie shell --seed 2023161 
(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))