b parameter of renormalized beta distribution

Average Error: 0.1 → 0.5

Time: 5.7s

Precision: binary64

Cost: 836

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;m \leq 1.648114941356343 \cdot 10^{-32}:\\ \;\;\;\;\frac{m}{v} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot m - m}{\frac{v}{m + -1}}\\ \end{array} \]

FPCore

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

↓

(FPCore (m v)
 :precision binary64
 (if (<= m 1.648114941356343e-32)
   (+ (/ m v) -1.0)
   (/ (- (* m m) m) (/ v (+ m -1.0)))))

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 <= 1.648114941356343e-32) {
		tmp = (m / v) + -1.0;
	} else {
		tmp = ((m * m) - m) / (v / (m + -1.0));
	}
	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 <= 1.648114941356343d-32) then
        tmp = (m / v) + (-1.0d0)
    else
        tmp = ((m * m) - m) / (v / (m + (-1.0d0)))
    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 <= 1.648114941356343e-32) {
		tmp = (m / v) + -1.0;
	} else {
		tmp = ((m * m) - m) / (v / (m + -1.0));
	}
	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 <= 1.648114941356343e-32:
		tmp = (m / v) + -1.0
	else:
		tmp = ((m * m) - m) / (v / (m + -1.0))
	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 <= 1.648114941356343e-32)
		tmp = Float64(Float64(m / v) + -1.0);
	else
		tmp = Float64(Float64(Float64(m * m) - m) / Float64(v / Float64(m + -1.0)));
	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 <= 1.648114941356343e-32)
		tmp = (m / v) + -1.0;
	else
		tmp = ((m * m) - m) / (v / (m + -1.0));
	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, 1.648114941356343e-32], N[(N[(m / v), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(N[(m * m), $MachinePrecision] - m), $MachinePrecision] / N[(v / N[(m + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.64811494135634304e-32

   1. Initial program 0.0

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

   2. Taylor expanded in m around 0 0.1

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

   3. Taylor expanded in v around 0 0.0

      \[\leadsto \color{blue}{\frac{m}{v}} - 1 \]

   if 1.64811494135634304e-32 < m

   1. Initial program 0.3

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

   2. Taylor expanded in m around 0 0.3

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

   3. Simplified 0.3

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

   4. Taylor expanded in v around 0 2.3

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

   5. Simplified 2.4

      \[\leadsto \color{blue}{\frac{m \cdot m - m}{\frac{v}{m + -1}}} \]
      Proof
      (/.f64 (-.f64 (*.f64 m m) m) (/.f64 v (+.f64 m -1))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 (*.f64 m m) (Rewrite<= *-lft-identity_binary64 (*.f64 1 m))) (/.f64 v (+.f64 m -1))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite=> distribute-rgt-out--_binary64 (*.f64 m (-.f64 m 1))) (/.f64 v (+.f64 m -1))): 4 points increase in error, 1 points decrease in error
      (/.f64 (*.f64 m (Rewrite=> sub-neg_binary64 (+.f64 m (neg.f64 1)))) (/.f64 v (+.f64 m -1))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 m (+.f64 (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 m))) (neg.f64 1))) (/.f64 v (+.f64 m -1))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 m (Rewrite<= distribute-neg-in_binary64 (neg.f64 (+.f64 (neg.f64 m) 1)))) (/.f64 v (+.f64 m -1))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 m (neg.f64 (Rewrite<= +-commutative_binary64 (+.f64 1 (neg.f64 m))))) (/.f64 v (+.f64 m -1))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 m (neg.f64 (Rewrite<= sub-neg_binary64 (-.f64 1 m)))) (/.f64 v (+.f64 m -1))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite=> distribute-rgt-neg-out_binary64 (neg.f64 (*.f64 m (-.f64 1 m)))) (/.f64 v (+.f64 m -1))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= distribute-lft-neg-out_binary64 (*.f64 (neg.f64 m) (-.f64 1 m))) (/.f64 v (+.f64 m -1))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (neg.f64 m) (-.f64 1 m)) (/.f64 v (Rewrite<= +-commutative_binary64 (+.f64 -1 m)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (neg.f64 m) (-.f64 1 m)) (/.f64 v (+.f64 (Rewrite<= metadata-eval (-.f64 0 1)) m))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (neg.f64 m) (-.f64 1 m)) (/.f64 v (Rewrite<= associate--r-_binary64 (-.f64 0 (-.f64 1 m))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (neg.f64 m) (-.f64 1 m)) (/.f64 v (Rewrite<= neg-sub0_binary64 (neg.f64 (-.f64 1 m))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (neg.f64 m) (-.f64 1 m)) (/.f64 v (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (-.f64 1 m))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (neg.f64 m) (-.f64 1 m)) (Rewrite<= associate-/l/_binary64 (/.f64 (/.f64 v (-.f64 1 m)) -1))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (*.f64 (neg.f64 m) (-.f64 1 m)) -1) (/.f64 v (-.f64 1 m)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 (-.f64 1 m) (neg.f64 m))) -1) (/.f64 v (-.f64 1 m))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= associate-*r*_binary64 (*.f64 (-.f64 1 m) (*.f64 (neg.f64 m) -1))) (/.f64 v (-.f64 1 m))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (-.f64 1 m) (Rewrite<= *-commutative_binary64 (*.f64 -1 (neg.f64 m)))) (/.f64 v (-.f64 1 m))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (-.f64 1 m) (Rewrite<= neg-mul-1_binary64 (neg.f64 (neg.f64 m)))) (/.f64 v (-.f64 1 m))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (-.f64 1 m) (Rewrite=> remove-double-neg_binary64 m)) (/.f64 v (-.f64 1 m))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite=> *-commutative_binary64 (*.f64 m (-.f64 1 m))) (/.f64 v (-.f64 1 m))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= distribute-rgt-out--_binary64 (-.f64 (*.f64 1 m) (*.f64 m m))) (/.f64 v (-.f64 1 m))): 1 points increase in error, 4 points decrease in error
      (/.f64 (-.f64 (Rewrite=> *-lft-identity_binary64 m) (*.f64 m m)) (/.f64 v (-.f64 1 m))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 m (Rewrite<= unpow2_binary64 (pow.f64 m 2))) (/.f64 v (-.f64 1 m))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (-.f64 m (pow.f64 m 2)) (-.f64 1 m)) v)): 8 points increase in error, 6 points decrease in error
3. Recombined 2 regimes into one program.

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.648114941356343 \cdot 10^{-32}:\\ \;\;\;\;\frac{m}{v} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{m \cdot m - m}{\frac{v}{m + -1}}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.1
Cost	832

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

Alternative 2
Error	0.2
Cost	832

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

Alternative 3
Error	0.1
Cost	832

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

Alternative 4
Error	1.8
Cost	708

\[\begin{array}{l} \mathbf{if}\;m \leq 0.2592220715079121:\\ \;\;\;\;\frac{m}{v} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(m \cdot m\right) \cdot \left(m + -2\right)}{v}\\ \end{array} \]

Alternative 5
Error	1.7
Cost	708

\[\begin{array}{l} \mathbf{if}\;m \leq 0.2592220715079121:\\ \;\;\;\;\left(1 - m\right) \cdot \left(\frac{m}{v} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(m \cdot m\right) \cdot \left(m + -2\right)}{v}\\ \end{array} \]

Alternative 6
Error	24.8
Cost	588

\[\begin{array}{l} \mathbf{if}\;m \leq 3.7799908927755998 \cdot 10^{-162}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 2.884230712415029 \cdot 10^{-149}:\\ \;\;\;\;\frac{m}{v}\\ \mathbf{elif}\;m \leq 4.581973094259714 \cdot 10^{-127}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v}\\ \end{array} \]

Alternative 7
Error	9.6
Cost	448

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

Alternative 8
Error	9.6
Cost	320

\[\frac{m}{v} + -1 \]

Alternative 9
Error	36.7
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2022297 
(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)))