a parameter of renormalized beta distribution

Average Error: 0.2 → 1.5

Time: 7.5s

Precision: binary64

Cost: 708

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;m \leq 3.893290538932841 \cdot 10^{-64}:\\ \;\;\;\;\frac{m}{\frac{v}{m}} - m\\ \mathbf{else}:\\ \;\;\;\;\left(m \cdot \left(1 - m\right)\right) \cdot \frac{m}{v}\\ \end{array} \]

FPCore

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

↓

(FPCore (m v)
 :precision binary64
 (if (<= m 3.893290538932841e-64)
   (- (/ m (/ v m)) m)
   (* (* m (- 1.0 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 <= 3.893290538932841e-64) {
		tmp = (m / (v / m)) - m;
	} else {
		tmp = (m * (1.0 - 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 <= 3.893290538932841d-64) then
        tmp = (m / (v / m)) - m
    else
        tmp = (m * (1.0d0 - 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 <= 3.893290538932841e-64) {
		tmp = (m / (v / m)) - m;
	} else {
		tmp = (m * (1.0 - 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 <= 3.893290538932841e-64:
		tmp = (m / (v / m)) - m
	else:
		tmp = (m * (1.0 - 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 <= 3.893290538932841e-64)
		tmp = Float64(Float64(m / Float64(v / m)) - m);
	else
		tmp = Float64(Float64(m * Float64(1.0 - m)) * Float64(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 <= 3.893290538932841e-64)
		tmp = (m / (v / m)) - m;
	else
		tmp = (m * (1.0 - 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, 3.893290538932841e-64], N[(N[(m / N[(v / m), $MachinePrecision]), $MachinePrecision] - m), $MachinePrecision], N[(N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision] * N[(m / v), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 3.89329053893284102e-64

   1. Initial program 0.1

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

   2. Simplified 0.1

      \[\leadsto \color{blue}{m \cdot \mathsf{fma}\left(1 - m, \frac{m}{v}, -1\right)} \]
      Proof
      (*.f64 m (fma.f64 (-.f64 1 m) (/.f64 m v) -1)): 0 points increase in error, 0 points decrease in error
      (*.f64 m (fma.f64 (-.f64 1 m) (/.f64 m v) (Rewrite<= metadata-eval (neg.f64 1)))): 0 points increase in error, 0 points decrease in error
      (*.f64 m (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 (-.f64 1 m) (/.f64 m v)) 1))): 0 points increase in error, 0 points decrease in error
      (*.f64 m (-.f64 (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 (-.f64 1 m) m) v)) 1)): 5 points increase in error, 8 points decrease in error
      (*.f64 m (-.f64 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 m (-.f64 1 m))) v) 1)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 1 m)) v) 1) m)): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in m around 0 10.4

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

   4. Simplified 0.1

      \[\leadsto \color{blue}{\frac{m}{\frac{v}{m}} - m} \]
      Proof
      (-.f64 (/.f64 m (/.f64 v m)) m): 0 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 m m) v)) m): 64 points increase in error, 17 points decrease in error
      (-.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 m 2)) v) m): 0 points increase in error, 0 points decrease in error
      (Rewrite<= unsub-neg_binary64 (+.f64 (/.f64 (pow.f64 m 2) v) (neg.f64 m))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (pow.f64 m 2) v) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 m))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -1 m) (/.f64 (pow.f64 m 2) v))): 0 points increase in error, 0 points decrease in error

   if 3.89329053893284102e-64 < m

   1. Initial program 0.3

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

   2. Simplified 0.3

      \[\leadsto \color{blue}{m \cdot \mathsf{fma}\left(1 - m, \frac{m}{v}, -1\right)} \]
      Proof
      (*.f64 m (fma.f64 (-.f64 1 m) (/.f64 m v) -1)): 0 points increase in error, 0 points decrease in error
      (*.f64 m (fma.f64 (-.f64 1 m) (/.f64 m v) (Rewrite<= metadata-eval (neg.f64 1)))): 0 points increase in error, 0 points decrease in error
      (*.f64 m (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 (-.f64 1 m) (/.f64 m v)) 1))): 0 points increase in error, 0 points decrease in error
      (*.f64 m (-.f64 (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 (-.f64 1 m) m) v)) 1)): 5 points increase in error, 8 points decrease in error
      (*.f64 m (-.f64 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 m (-.f64 1 m))) v) 1)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 1 m)) v) 1) m)): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in m around inf 4.6

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

   4. Simplified 4.7

      \[\leadsto m \cdot \color{blue}{\frac{1 - m}{\frac{v}{m}}} \]
      Proof
      (/.f64 (-.f64 1 m) (/.f64 v m)): 0 points increase in error, 0 points decrease in error
      (Rewrite=> div-sub_binary64 (-.f64 (/.f64 1 (/.f64 v m)) (/.f64 m (/.f64 v m)))): 3 points increase in error, 4 points decrease in error
      (-.f64 (/.f64 1 (/.f64 v m)) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 m m) v))): 7 points increase in error, 10 points decrease in error
      (-.f64 (/.f64 1 (/.f64 v m)) (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 m 2)) v)): 0 points increase in error, 0 points decrease in error
      (Rewrite=> sub-neg_binary64 (+.f64 (/.f64 1 (/.f64 v m)) (neg.f64 (/.f64 (pow.f64 m 2) v)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 1 m) v)) (neg.f64 (/.f64 (pow.f64 m 2) v))): 27 points increase in error, 37 points decrease in error
      (+.f64 (/.f64 (Rewrite=> *-lft-identity_binary64 m) v) (neg.f64 (/.f64 (pow.f64 m 2) v))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 m v) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 (pow.f64 m 2) v)))): 0 points increase in error, 0 points decrease in error

   5. Applied egg-rr 4.7

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

   6. Applied egg-rr 4.6

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

4. Final simplification 1.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 3.893290538932841 \cdot 10^{-64}:\\ \;\;\;\;\frac{m}{\frac{v}{m}} - m\\ \mathbf{else}:\\ \;\;\;\;\left(m \cdot \left(1 - m\right)\right) \cdot \frac{m}{v}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.7
Cost	708

\[\begin{array}{l} \mathbf{if}\;m \leq 4.086538403100834 \cdot 10^{-41}:\\ \;\;\;\;\frac{m}{\frac{v}{m}} - m\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot \left(m \cdot \left(1 - m\right)\right)}{v}\\ \end{array} \]

Alternative 2
Error	0.2
Cost	704

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

Alternative 3
Error	2.3
Cost	644

\[\begin{array}{l} \mathbf{if}\;m \leq 0.44820996333954455:\\ \;\;\;\;\frac{m}{\frac{v}{m}} - m\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{\frac{\frac{-v}{m}}{m}}\\ \end{array} \]

Alternative 4
Error	2.3
Cost	644

\[\begin{array}{l} \mathbf{if}\;m \leq 0.44820996333954455:\\ \;\;\;\;\frac{m}{\frac{v}{m}} - m\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v} \cdot \left(m \cdot \left(-m\right)\right)\\ \end{array} \]

Alternative 5
Error	24.0
Cost	580

\[\begin{array}{l} \mathbf{if}\;v \leq 2.4 \cdot 10^{-186}:\\ \;\;\;\;\frac{m}{v + \frac{v}{m}}\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \]

Alternative 6
Error	24.2
Cost	452

\[\begin{array}{l} \mathbf{if}\;v \leq 2.4 \cdot 10^{-186}:\\ \;\;\;\;\frac{m}{\frac{v}{m}}\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \]

Alternative 7
Error	10.1
Cost	448

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

Alternative 8
Error	36.6
Cost	128

\[-m \]

Reproduce

herbie shell --seed 2022293 
(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))