a parameter of renormalized beta distribution

Average Error: 0.2 → 0.5

Time: 6.0s

Precision: binary64

Cost: 708

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;m \leq 1.45 \cdot 10^{-27}:\\ \;\;\;\;\frac{m}{\frac{v}{m}} - m\\ \mathbf{else}:\\ \;\;\;\;\left(1 - m\right) \cdot \left(m \cdot \frac{m}{v}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (m v)
 :precision binary64
 (if (<= m 1.45e-27) (- (/ m (/ v m)) m) (* (- 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 <= 1.45e-27) {
		tmp = (m / (v / m)) - m;
	} 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 <= 1.45d-27) then
        tmp = (m / (v / m)) - m
    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 <= 1.45e-27) {
		tmp = (m / (v / m)) - m;
	} 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 <= 1.45e-27:
		tmp = (m / (v / m)) - m
	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 <= 1.45e-27)
		tmp = Float64(Float64(m / Float64(v / m)) - m);
	else
		tmp = Float64(Float64(1.0 - m) * Float64(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 <= 1.45e-27)
		tmp = (m / (v / m)) - m;
	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, 1.45e-27], N[(N[(m / N[(v / m), $MachinePrecision]), $MachinePrecision] - m), $MachinePrecision], N[(N[(1.0 - m), $MachinePrecision] * N[(m * N[(m / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.45000000000000002e-27

   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, 5 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)): 26 points increase in error, 9 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))): 1 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)): 10 points increase in error, 24 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, 2 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 8.5

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

   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-/r/_binary64 (*.f64 (/.f64 m v) m)) m): 31 points increase in error, 27 points decrease in error
      (Rewrite<= unsub-neg_binary64 (+.f64 (*.f64 (/.f64 m v) m) (neg.f64 m))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 (neg.f64 m) (*.f64 (/.f64 m v) m))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite=> neg-mul-1_binary64 (*.f64 -1 m)) (*.f64 (/.f64 m v) m)): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 -1 m) (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 m m) v))): 44 points increase in error, 21 points decrease in error
      (+.f64 (*.f64 -1 m) (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 m 2)) v)): 0 points increase in error, 0 points decrease in error

   if 1.45000000000000002e-27 < m

   1. Initial program 0.3

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

   2. Simplified 0.3

      \[\leadsto \color{blue}{m \cdot \left(\frac{m}{\frac{v}{1 - m}} + -1\right)} \]
      Proof
      (*.f64 m (+.f64 (/.f64 m (/.f64 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)): 7 points increase in error, 7 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)): 26 points increase in error, 9 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))): 1 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)): 10 points increase in error, 24 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, 2 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 0.3

      \[\leadsto \color{blue}{-1 \cdot \frac{{m}^{3}}{v} + \left(-1 \cdot m + \frac{{m}^{2}}{v}\right)} \]

   4. Simplified 0.3

      \[\leadsto \color{blue}{\left(\frac{m \cdot m}{v} - m\right) - \frac{{m}^{3}}{v}} \]
      Proof
      (-.f64 (-.f64 (/.f64 (*.f64 m m) v) m) (/.f64 (pow.f64 m 3) v)): 0 points increase in error, 0 points decrease in error
      (-.f64 (-.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 m 2)) v) m) (/.f64 (pow.f64 m 3) v)): 0 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite<= unsub-neg_binary64 (+.f64 (/.f64 (pow.f64 m 2) v) (neg.f64 m))) (/.f64 (pow.f64 m 3) v)): 0 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite<= +-commutative_binary64 (+.f64 (neg.f64 m) (/.f64 (pow.f64 m 2) v))) (/.f64 (pow.f64 m 3) v)): 0 points increase in error, 0 points decrease in error
      (-.f64 (+.f64 (Rewrite=> neg-mul-1_binary64 (*.f64 -1 m)) (/.f64 (pow.f64 m 2) v)) (/.f64 (pow.f64 m 3) v)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= unsub-neg_binary64 (+.f64 (+.f64 (*.f64 -1 m) (/.f64 (pow.f64 m 2) v)) (neg.f64 (/.f64 (pow.f64 m 3) v)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (*.f64 -1 m) (/.f64 (pow.f64 m 2) v)) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 (pow.f64 m 3) v)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -1 (/.f64 (pow.f64 m 3) v)) (+.f64 (*.f64 -1 m) (/.f64 (pow.f64 m 2) v)))): 0 points increase in error, 0 points decrease in error

   5. Taylor expanded in v around 0 1.9

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

   6. Simplified 1.9

      \[\leadsto \color{blue}{\left(m \cdot \frac{m}{v}\right) \cdot \left(1 - m\right)} \]
      Proof
      (*.f64 (*.f64 m (/.f64 m v)) (-.f64 1 m)): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 m v) m)) (-.f64 1 m)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= distribute-lft-out--_binary64 (-.f64 (*.f64 (*.f64 (/.f64 m v) m) 1) (*.f64 (*.f64 (/.f64 m v) m) m))): 7 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite=> *-rgt-identity_binary64 (*.f64 (/.f64 m v) m)) (*.f64 (*.f64 (/.f64 m v) m) m)): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 (/.f64 m v) m) (Rewrite=> associate-*l*_binary64 (*.f64 (/.f64 m v) (*.f64 m m)))): 6 points increase in error, 4 points decrease in error
      (-.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 m m) v)) (*.f64 (/.f64 m v) (*.f64 m m))): 51 points increase in error, 49 points decrease in error
      (-.f64 (/.f64 (*.f64 m m) v) (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 m (*.f64 m m)) v))): 6 points increase in error, 4 points decrease in error
      (-.f64 (/.f64 (*.f64 m m) v) (/.f64 (Rewrite<= cube-mult_binary64 (pow.f64 m 3)) v)): 1 points increase in error, 8 points decrease in error
      (Rewrite<= div-sub_binary64 (/.f64 (-.f64 (*.f64 m m) (pow.f64 m 3)) v)): 4 points increase in error, 4 points decrease in error
      (/.f64 (-.f64 (Rewrite<= unpow2_binary64 (pow.f64 m 2)) (pow.f64 m 3)) v): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 (pow.f64 m 2) (pow.f64 m (Rewrite<= metadata-eval (+.f64 2 1)))) v): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 (pow.f64 m 2) (Rewrite<= pow-plus_binary64 (*.f64 (pow.f64 m 2) m))) v): 8 points increase in error, 1 points decrease in error
3. Recombined 2 regimes into one program.

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.45 \cdot 10^{-27}:\\ \;\;\;\;\frac{m}{\frac{v}{m}} - m\\ \mathbf{else}:\\ \;\;\;\;\left(1 - m\right) \cdot \left(m \cdot \frac{m}{v}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	24.6
Cost	716

\[\begin{array}{l} t_0 := m \cdot \frac{m}{v}\\ \mathbf{if}\;m \leq 1.45 \cdot 10^{-172}:\\ \;\;\;\;-m\\ \mathbf{elif}\;m \leq 1.4 \cdot 10^{-145}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;m \leq 1.85 \cdot 10^{-122}:\\ \;\;\;\;-m\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	24.6
Cost	716

\[\begin{array}{l} \mathbf{if}\;m \leq 3.3 \cdot 10^{-173}:\\ \;\;\;\;-m\\ \mathbf{elif}\;m \leq 1.4 \cdot 10^{-145}:\\ \;\;\;\;\frac{m}{\frac{v}{m}}\\ \mathbf{elif}\;m \leq 8.5 \cdot 10^{-122}:\\ \;\;\;\;-m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \frac{m}{v}\\ \end{array} \]

Alternative 3
Error	0.4
Cost	708

\[\begin{array}{l} \mathbf{if}\;m \leq 6.8 \cdot 10^{-21}:\\ \;\;\;\;\frac{m}{\frac{v}{m}} - m\\ \mathbf{else}:\\ \;\;\;\;m \cdot \left(m \cdot \frac{1 - m}{v}\right)\\ \end{array} \]

Alternative 4
Error	0.2
Cost	704

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

Alternative 5
Error	2.5
Cost	644

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

Alternative 6
Error	10.2
Cost	448

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

Alternative 7
Error	10.2
Cost	448

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

Alternative 8
Error	37.1
Cost	128

\[-m \]

Reproduce

herbie shell --seed 2022335 
(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))