a parameter of renormalized beta distribution

Average Error: 0.2 → 0.3

Time: 7.8s

Precision: binary64

Cost: 708

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;m \leq 1.1 \cdot 10^{-17}:\\ \;\;\;\;m \cdot \left(-1 + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - m}{\frac{v}{m \cdot m}}\\ \end{array} \]

FPCore

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

↓

(FPCore (m v)
 :precision binary64
 (if (<= m 1.1e-17) (* m (+ -1.0 (/ m v))) (/ (- 1.0 m) (/ v (* m m)))))

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 <= 1.1e-17) {
		tmp = m * (-1.0 + (m / v));
	} else {
		tmp = (1.0 - m) / (v / (m * m));
	}
	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 <= 1.1d-17) then
        tmp = m * ((-1.0d0) + (m / v))
    else
        tmp = (1.0d0 - m) / (v / (m * m))
    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 <= 1.1e-17) {
		tmp = m * (-1.0 + (m / v));
	} else {
		tmp = (1.0 - m) / (v / (m * m));
	}
	return tmp;
}

Python

def code(m, v):
	return (((m * (1.0 - m)) / v) - 1.0) * m

↓

def code(m, v):
	tmp = 0
	if m <= 1.1e-17:
		tmp = m * (-1.0 + (m / v))
	else:
		tmp = (1.0 - m) / (v / (m * m))
	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 <= 1.1e-17)
		tmp = Float64(m * Float64(-1.0 + Float64(m / v)));
	else
		tmp = Float64(Float64(1.0 - m) / Float64(v / Float64(m * m)));
	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 <= 1.1e-17)
		tmp = m * (-1.0 + (m / v));
	else
		tmp = (1.0 - m) / (v / (m * m));
	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, 1.1e-17], N[(m * N[(-1.0 + N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - m), $MachinePrecision] / N[(v / N[(m * m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.1e-17

   1. Initial program 0.1

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

   2. Simplified 0.1

      \[\leadsto \color{blue}{m \cdot \left(\frac{m}{v} \cdot \left(1 - m\right) + -1\right)} \]
      Proof
      (*.f64 m (+.f64 (*.f64 (/.f64 m v) (-.f64 1 m)) -1)): 0 points increase in error, 0 points decrease in error
      (*.f64 m (+.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 m (-.f64 1 m)) v)) -1)): 6 points increase in error, 6 points decrease in error
      (*.f64 m (+.f64 (Rewrite<= *-rgt-identity_binary64 (*.f64 (/.f64 (*.f64 m (-.f64 1 m)) v) 1)) -1)): 0 points increase in error, 0 points decrease in error
      (*.f64 m (+.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 (*.f64 m (-.f64 1 m)) 1) v)) -1)): 0 points increase in error, 0 points decrease in error
      (*.f64 m (+.f64 (Rewrite<= associate-*r/_binary64 (*.f64 (*.f64 m (-.f64 1 m)) (/.f64 1 v))) -1)): 23 points increase in error, 7 points decrease in error
      (*.f64 m (+.f64 (*.f64 (*.f64 m (-.f64 1 m)) (/.f64 1 v)) (Rewrite<= metadata-eval (neg.f64 1)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 (*.f64 (*.f64 m (-.f64 1 m)) (/.f64 1 v)) m) (*.f64 (neg.f64 1) m))): 2 points increase in error, 5 points decrease in error
      (+.f64 (*.f64 (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 (*.f64 m (-.f64 1 m)) 1) v)) m) (*.f64 (neg.f64 1) m)): 9 points increase in error, 21 points decrease in error
      (+.f64 (*.f64 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 (*.f64 m (-.f64 1 m)) v) 1)) m) (*.f64 (neg.f64 1) m)): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 (Rewrite=> *-rgt-identity_binary64 (/.f64 (*.f64 m (-.f64 1 m)) v)) m) (*.f64 (neg.f64 1) m)): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 m (/.f64 (*.f64 m (-.f64 1 m)) v))) (*.f64 (neg.f64 1) m)): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 m (/.f64 (*.f64 m (-.f64 1 m)) v)) (Rewrite=> *-commutative_binary64 (*.f64 m (neg.f64 1)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= distribute-lft-in_binary64 (*.f64 m (+.f64 (/.f64 (*.f64 m (-.f64 1 m)) v) (neg.f64 1)))): 3 points increase in error, 4 points decrease in error
      (*.f64 m (Rewrite<= sub-neg_binary64 (-.f64 (/.f64 (*.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 0.1

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

   if 1.1e-17 < m

   1. Initial program 0.4

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

   2. Taylor expanded in v around 0 1.2

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

   3. Simplified 1.2

      \[\leadsto \color{blue}{\frac{1 - m}{\frac{v}{m \cdot m}}} \]
      Proof
      (/.f64 (-.f64 1 m) (/.f64 v (*.f64 m m))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 1 m) (/.f64 v (Rewrite<= unpow2_binary64 (pow.f64 m 2)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (-.f64 1 m) (pow.f64 m 2)) v)): 17 points increase in error, 26 points decrease in error
      (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (pow.f64 m 2) (-.f64 1 m))) v): 0 points increase in error, 0 points decrease in error
3. Recombined 2 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.1 \cdot 10^{-17}:\\ \;\;\;\;m \cdot \left(-1 + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - m}{\frac{v}{m \cdot m}}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.4
Cost	708

\[\begin{array}{l} \mathbf{if}\;m \leq 4 \cdot 10^{-19}:\\ \;\;\;\;m \cdot \left(-1 + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - m}{\frac{\frac{v}{m}}{m}}\\ \end{array} \]

Alternative 2
Error	0.3
Cost	708

\[\begin{array}{l} \mathbf{if}\;m \leq 1.1 \cdot 10^{-16}:\\ \;\;\;\;m \cdot \left(-1 + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(m \cdot m\right) \cdot \frac{1 - m}{v}\\ \end{array} \]

Alternative 3
Error	0.2
Cost	704

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

Alternative 4
Error	0.2
Cost	704

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

Alternative 5
Error	2.3
Cost	644

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

Alternative 6
Error	24.7
Cost	452

\[\begin{array}{l} \mathbf{if}\;m \leq 2.6 \cdot 10^{-152}:\\ \;\;\;\;-m\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot m}{v}\\ \end{array} \]

Alternative 7
Error	24.5
Cost	452

\[\begin{array}{l} \mathbf{if}\;v \leq 9 \cdot 10^{-180}:\\ \;\;\;\;\frac{m}{\frac{v}{m}}\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \]

Alternative 8
Error	24.5
Cost	452

\[\begin{array}{l} \mathbf{if}\;v \leq 9.5 \cdot 10^{-180}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;-m\\ \end{array} \]

Alternative 9
Error	10.3
Cost	448

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

Alternative 10
Error	10.3
Cost	448

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

Alternative 11
Error	36.9
Cost	128

\[-m \]

Reproduce

herbie shell --seed 2022318 
(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))