b parameter of renormalized beta distribution

Percentage Accurate: 99.9% → 99.8%

Time: 9.3s

Precision: binary64

Cost: 836

• 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

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

↓

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

FPCore

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

↓

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

↓

function tmp_2 = code(m, v)
	tmp = 0.0;
	if (m <= 5.4e-14)
		tmp = m + ((m / v) + -1.0);
	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, 5.4e-14], N[(m + N[(N[(m / v), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - m), $MachinePrecision] * N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 5.3999999999999997e-14

   1. Initial program 100.0%

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

   2. Simplified 99.8%

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

      [Start] 100.0	\[ \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
      *-commutative [=>] 100.0	\[ \color{blue}{\left(1 - m\right) \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right)} \]
      associate-*r/ [<=] 99.8	\[ \left(1 - m\right) \cdot \left(\color{blue}{m \cdot \frac{1 - m}{v}} - 1\right) \]
      fma-neg [=>] 99.8	\[ \left(1 - m\right) \cdot \color{blue}{\mathsf{fma}\left(m, \frac{1 - m}{v}, -1\right)} \]
      metadata-eval [=>] 99.8	\[ \left(1 - m\right) \cdot \mathsf{fma}\left(m, \frac{1 - m}{v}, \color{blue}{-1}\right) \]

   3. Taylor expanded in m around 0 99.8%

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

   4. Simplified 100.0%

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

      [Start] 99.8	\[ \left(1 + \frac{1}{v}\right) \cdot m - 1 \]
      sub-neg [=>] 99.8	\[ \color{blue}{\left(1 + \frac{1}{v}\right) \cdot m + \left(-1\right)} \]
      metadata-eval [=>] 99.8	\[ \left(1 + \frac{1}{v}\right) \cdot m + \color{blue}{-1} \]
      +-commutative [=>] 99.8	\[ \color{blue}{-1 + \left(1 + \frac{1}{v}\right) \cdot m} \]
      +-commutative [=>] 99.8	\[ -1 + \color{blue}{\left(\frac{1}{v} + 1\right)} \cdot m \]
      distribute-lft1-in [<=] 99.8	\[ -1 + \color{blue}{\left(\frac{1}{v} \cdot m + m\right)} \]
      associate-*l/ [=>] 100.0	\[ -1 + \left(\color{blue}{\frac{1 \cdot m}{v}} + m\right) \]
      *-lft-identity [=>] 100.0	\[ -1 + \left(\frac{\color{blue}{m}}{v} + m\right) \]
      associate-+r+ [=>] 100.0	\[ \color{blue}{\left(-1 + \frac{m}{v}\right) + m} \]

   if 5.3999999999999997e-14 < m

   1. Initial program 99.8%

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

   2. Simplified 99.8%

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

      [Start] 99.8	\[ \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right) \cdot \left(1 - m\right) \]
      *-commutative [=>] 99.8	\[ \color{blue}{\left(1 - m\right) \cdot \left(\frac{m \cdot \left(1 - m\right)}{v} - 1\right)} \]
      associate-*r/ [<=] 99.8	\[ \left(1 - m\right) \cdot \left(\color{blue}{m \cdot \frac{1 - m}{v}} - 1\right) \]
      fma-neg [=>] 99.8	\[ \left(1 - m\right) \cdot \color{blue}{\mathsf{fma}\left(m, \frac{1 - m}{v}, -1\right)} \]
      metadata-eval [=>] 99.8	\[ \left(1 - m\right) \cdot \mathsf{fma}\left(m, \frac{1 - m}{v}, \color{blue}{-1}\right) \]

   3. Taylor expanded in m around inf 54.9%

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

   4. Simplified 99.8%

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

      [Start] 54.9	\[ \left(1 - m\right) \cdot \left(\frac{m}{v} + -1 \cdot \frac{{m}^{2}}{v}\right) \]
      mul-1-neg [=>] 54.9	\[ \left(1 - m\right) \cdot \left(\frac{m}{v} + \color{blue}{\left(-\frac{{m}^{2}}{v}\right)}\right) \]
      unsub-neg [=>] 54.9	\[ \left(1 - m\right) \cdot \color{blue}{\left(\frac{m}{v} - \frac{{m}^{2}}{v}\right)} \]
      *-lft-identity [<=] 54.9	\[ \left(1 - m\right) \cdot \left(\frac{\color{blue}{1 \cdot m}}{v} - \frac{{m}^{2}}{v}\right) \]
      associate-/l* [=>] 54.9	\[ \left(1 - m\right) \cdot \left(\color{blue}{\frac{1}{\frac{v}{m}}} - \frac{{m}^{2}}{v}\right) \]
      unpow2 [=>] 54.9	\[ \left(1 - m\right) \cdot \left(\frac{1}{\frac{v}{m}} - \frac{\color{blue}{m \cdot m}}{v}\right) \]
      associate-/l* [=>] 54.9	\[ \left(1 - m\right) \cdot \left(\frac{1}{\frac{v}{m}} - \color{blue}{\frac{m}{\frac{v}{m}}}\right) \]
      div-sub [<=] 99.8	\[ \left(1 - m\right) \cdot \color{blue}{\frac{1 - m}{\frac{v}{m}}} \]
      associate-/l* [<=] 99.8	\[ \left(1 - m\right) \cdot \color{blue}{\frac{\left(1 - m\right) \cdot m}{v}} \]
      *-commutative [<=] 99.8	\[ \left(1 - m\right) \cdot \frac{\color{blue}{m \cdot \left(1 - m\right)}}{v} \]
      associate-*r/ [<=] 99.8	\[ \left(1 - m\right) \cdot \color{blue}{\left(m \cdot \frac{1 - m}{v}\right)} \]

   5. Applied egg-rr 99.9%

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

      [Start] 99.8	\[ \left(1 - m\right) \cdot \left(m \cdot \frac{1 - m}{v}\right) \]
      associate-*r/ [=>] 99.8	\[ \left(1 - m\right) \cdot \color{blue}{\frac{m \cdot \left(1 - m\right)}{v}} \]
      associate-*r/ [=>] 99.9	\[ \color{blue}{\frac{\left(1 - m\right) \cdot \left(m \cdot \left(1 - m\right)\right)}{v}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

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

Alternatives

Alternative 1
Accuracy	72.4%
Cost	848

\[\begin{array}{l} \mathbf{if}\;m \leq 1.05 \cdot 10^{-197}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 4.4 \cdot 10^{-152}:\\ \;\;\;\;\frac{m}{v}\\ \mathbf{elif}\;m \leq 1.66 \cdot 10^{-143}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 0.28:\\ \;\;\;\;\frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \]

Alternative 2
Accuracy	99.8%
Cost	836

\[\begin{array}{l} \mathbf{if}\;m \leq 1.35 \cdot 10^{-13}:\\ \;\;\;\;m + \left(\frac{m}{v} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - m\right) \cdot \left(m \cdot \frac{1 - m}{v}\right)\\ \end{array} \]

Alternative 3
Accuracy	99.8%
Cost	836

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

Alternative 4
Accuracy	99.8%
Cost	836

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

Alternative 5
Accuracy	99.9%
Cost	832

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

Alternative 6
Accuracy	97.5%
Cost	708

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

Alternative 7
Accuracy	97.6%
Cost	708

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

Alternative 8
Accuracy	97.6%
Cost	708

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

Alternative 9
Accuracy	98.1%
Cost	708

\[\begin{array}{l} \mathbf{if}\;m \leq 2.4:\\ \;\;\;\;m + \left(\frac{m}{v} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(m \cdot m\right) \cdot \left(m + -2\right)}{v}\\ \end{array} \]

Alternative 10
Accuracy	60.8%
Cost	588

\[\begin{array}{l} \mathbf{if}\;m \leq 1.02 \cdot 10^{-198}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 4.35 \cdot 10^{-156}:\\ \;\;\;\;\frac{m}{v}\\ \mathbf{elif}\;m \leq 1.8 \cdot 10^{-143}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v}\\ \end{array} \]

Alternative 11
Accuracy	86.7%
Cost	580

\[\begin{array}{l} \mathbf{if}\;m \leq 0.28:\\ \;\;\;\;m + \left(\frac{m}{v} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \]

Alternative 12
Accuracy	86.7%
Cost	452

\[\begin{array}{l} \mathbf{if}\;m \leq 0.28:\\ \;\;\;\;\frac{m}{v} + -1\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \]

Alternative 13
Accuracy	27.1%
Cost	192

\[m + -1 \]

Alternative 14
Accuracy	24.7%
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023161 
(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)))