b parameter of renormalized beta distribution

Average Error: 0.1 → 0.1

Time: 8.7s

Precision: binary64

Cost: 964

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;m \leq 1.0530813331283385 \cdot 10^{-9}:\\ \;\;\;\;\frac{m}{v} \cdot \left(1 + m \cdot -2\right) - \left(1 - m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(m \cdot \left(1 - m\right)\right) \cdot \frac{1 - m}{v}\\ \end{array} \]

FPCore

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

↓

(FPCore (m v)
 :precision binary64
 (if (<= m 1.0530813331283385e-9)
   (- (* (/ m v) (+ 1.0 (* m -2.0))) (- 1.0 m))
   (* (* m (- 1.0 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 <= 1.0530813331283385e-9) {
		tmp = ((m / v) * (1.0 + (m * -2.0))) - (1.0 - m);
	} else {
		tmp = (m * (1.0 - 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 <= 1.0530813331283385d-9) then
        tmp = ((m / v) * (1.0d0 + (m * (-2.0d0)))) - (1.0d0 - m)
    else
        tmp = (m * (1.0d0 - 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 <= 1.0530813331283385e-9) {
		tmp = ((m / v) * (1.0 + (m * -2.0))) - (1.0 - m);
	} else {
		tmp = (m * (1.0 - 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 <= 1.0530813331283385e-9:
		tmp = ((m / v) * (1.0 + (m * -2.0))) - (1.0 - m)
	else:
		tmp = (m * (1.0 - 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 <= 1.0530813331283385e-9)
		tmp = Float64(Float64(Float64(m / v) * Float64(1.0 + Float64(m * -2.0))) - Float64(1.0 - m));
	else
		tmp = Float64(Float64(m * Float64(1.0 - m)) * Float64(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 <= 1.0530813331283385e-9)
		tmp = ((m / v) * (1.0 + (m * -2.0))) - (1.0 - m);
	else
		tmp = (m * (1.0 - 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, 1.0530813331283385e-9], N[(N[(N[(m / v), $MachinePrecision] * N[(1.0 + N[(m * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(1.0 - m), $MachinePrecision]), $MachinePrecision], N[(N[(m * N[(1.0 - m), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - m), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.0530813331283385e-9

   1. Initial program 0.0

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

   2. Simplified 0.0

      \[\leadsto \color{blue}{\left(1 - m\right) \cdot \mathsf{fma}\left(1 - m, \frac{m}{v}, -1\right)} \]
      Proof
      (*.f64 (-.f64 1 m) (fma.f64 (-.f64 1 m) (/.f64 m v) -1)): 0 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 1 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 (-.f64 1 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 (-.f64 1 m) (-.f64 (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 m v) (-.f64 1 m))) 1)): 0 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 1 m) (-.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 m (-.f64 1 m)) v)) 1)): 6 points increase in error, 6 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 1 m)) v) 1) (-.f64 1 m))): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in m around 0 0.2

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

   4. Simplified 0.0

      \[\leadsto \color{blue}{\left(-1 + m\right) + \frac{m}{v} \cdot \left(m \cdot -2 + 1\right)} \]
      Proof
      (+.f64 (+.f64 -1 m) (*.f64 (/.f64 m v) (+.f64 (*.f64 m -2) 1))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 -1 m) (*.f64 (/.f64 m v) (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 -2 m)) 1))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 -1 m) (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 (/.f64 m v) (*.f64 -2 m)) (*.f64 (/.f64 m v) 1)))): 1 points increase in error, 3 points decrease in error
      (+.f64 (+.f64 -1 m) (+.f64 (*.f64 (/.f64 m v) (Rewrite=> *-commutative_binary64 (*.f64 m -2))) (*.f64 (/.f64 m v) 1))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 -1 m) (+.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 m v) m) -2)) (*.f64 (/.f64 m v) 1))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 -1 m) (+.f64 (*.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 m m) v)) -2) (*.f64 (/.f64 m v) 1))): 12 points increase in error, 8 points decrease in error
      (+.f64 (+.f64 -1 m) (+.f64 (*.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 m 2)) v) -2) (*.f64 (/.f64 m v) 1))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 -1 m) (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 -2 (/.f64 (pow.f64 m 2) v))) (*.f64 (/.f64 m v) 1))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 -1 m) (+.f64 (*.f64 -2 (/.f64 (pow.f64 m 2) v)) (Rewrite=> *-rgt-identity_binary64 (/.f64 m v)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 -1 m) (Rewrite<= +-commutative_binary64 (+.f64 (/.f64 m v) (*.f64 -2 (/.f64 (pow.f64 m 2) v))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-+r+_binary64 (+.f64 -1 (+.f64 m (+.f64 (/.f64 m v) (*.f64 -2 (/.f64 (pow.f64 m 2) v)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 -1 (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 m (/.f64 m v)) (*.f64 -2 (/.f64 (pow.f64 m 2) v))))): 0 points increase in error, 0 points decrease in error
      (+.f64 -1 (+.f64 (Rewrite=> +-commutative_binary64 (+.f64 (/.f64 m v) m)) (*.f64 -2 (/.f64 (pow.f64 m 2) v)))): 0 points increase in error, 0 points decrease in error
      (+.f64 -1 (+.f64 (+.f64 (/.f64 (Rewrite<= *-lft-identity_binary64 (*.f64 1 m)) v) m) (*.f64 -2 (/.f64 (pow.f64 m 2) v)))): 0 points increase in error, 0 points decrease in error
      (+.f64 -1 (+.f64 (+.f64 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 1 v) m)) m) (*.f64 -2 (/.f64 (pow.f64 m 2) v)))): 29 points increase in error, 0 points decrease in error
      (+.f64 -1 (+.f64 (Rewrite=> distribute-lft1-in_binary64 (*.f64 (+.f64 (/.f64 1 v) 1) m)) (*.f64 -2 (/.f64 (pow.f64 m 2) v)))): 0 points increase in error, 0 points decrease in error
      (+.f64 -1 (+.f64 (*.f64 (Rewrite<= +-commutative_binary64 (+.f64 1 (/.f64 1 v))) m) (*.f64 -2 (/.f64 (pow.f64 m 2) v)))): 0 points increase in error, 0 points decrease in error
      (+.f64 -1 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -2 (/.f64 (pow.f64 m 2) v)) (*.f64 (+.f64 1 (/.f64 1 v)) m)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 (*.f64 -2 (/.f64 (pow.f64 m 2) v)) (*.f64 (+.f64 1 (/.f64 1 v)) m)) -1)): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (*.f64 -2 (/.f64 (pow.f64 m 2) v)) (*.f64 (+.f64 1 (/.f64 1 v)) m)) (Rewrite<= metadata-eval (neg.f64 1))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= sub-neg_binary64 (-.f64 (+.f64 (*.f64 -2 (/.f64 (pow.f64 m 2) v)) (*.f64 (+.f64 1 (/.f64 1 v)) m)) 1)): 0 points increase in error, 0 points decrease in error

   if 1.0530813331283385e-9 < m

   1. Initial program 0.4

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

   2. Simplified 0.4

      \[\leadsto \color{blue}{\left(1 - m\right) \cdot \mathsf{fma}\left(1 - m, \frac{m}{v}, -1\right)} \]
      Proof
      (*.f64 (-.f64 1 m) (fma.f64 (-.f64 1 m) (/.f64 m v) -1)): 0 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 1 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 (-.f64 1 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 (-.f64 1 m) (-.f64 (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 m v) (-.f64 1 m))) 1)): 0 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 1 m) (-.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 m (-.f64 1 m)) v)) 1)): 6 points increase in error, 6 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 (-.f64 (/.f64 (*.f64 m (-.f64 1 m)) v) 1) (-.f64 1 m))): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in m around inf 0.7

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

   4. Simplified 0.7

      \[\leadsto \left(1 - m\right) \cdot \color{blue}{\left(m \cdot \frac{1 - m}{v}\right)} \]
      Proof
      (*.f64 m (/.f64 (-.f64 1 m) v)): 0 points increase in error, 0 points decrease in error
      (*.f64 m (Rewrite=> div-sub_binary64 (-.f64 (/.f64 1 v) (/.f64 m v)))): 2 points increase in error, 2 points decrease in error
      (Rewrite<= distribute-rgt-out--_binary64 (-.f64 (*.f64 (/.f64 1 v) m) (*.f64 (/.f64 m v) m))): 6 points increase in error, 2 points decrease in error
      (-.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 1 m) v)) (*.f64 (/.f64 m v) m)): 18 points increase in error, 44 points decrease in error
      (-.f64 (/.f64 (Rewrite=> *-lft-identity_binary64 m) v) (*.f64 (/.f64 m v) m)): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 m v) (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 m m) v))): 12 points increase in error, 8 points decrease in error
      (-.f64 (/.f64 m v) (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 m 2)) v)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= unsub-neg_binary64 (+.f64 (/.f64 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 0.7

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

   6. Applied egg-rr 0.7

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.0530813331283385 \cdot 10^{-9}:\\ \;\;\;\;\frac{m}{v} \cdot \left(1 + m \cdot -2\right) - \left(1 - m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(m \cdot \left(1 - m\right)\right) \cdot \frac{1 - m}{v}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.1
Cost	7104

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

Alternative 2
Error	0.3
Cost	836

\[\begin{array}{l} \mathbf{if}\;m \leq 1.3018175972005351 \cdot 10^{-19}:\\ \;\;\;\;\frac{m}{v} - \left(1 - m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(m \cdot \left(1 - m\right)\right) \cdot \frac{1 - m}{v}\\ \end{array} \]

Alternative 3
Error	0.2
Cost	832

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

Alternative 4
Error	0.1
Cost	832

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

Alternative 5
Error	17.1
Cost	712

\[\begin{array}{l} \mathbf{if}\;m \leq 1.632848001758723 \cdot 10^{-152}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 0.03584537202507449:\\ \;\;\;\;\frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{\frac{\frac{v}{m}}{m}}\\ \end{array} \]

Alternative 6
Error	17.1
Cost	712

\[\begin{array}{l} \mathbf{if}\;m \leq 1.632848001758723 \cdot 10^{-152}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 0.03584537202507449:\\ \;\;\;\;\frac{1 - m}{\frac{v}{m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{\frac{\frac{v}{m}}{m}}\\ \end{array} \]

Alternative 7
Error	17.0
Cost	712

\[\begin{array}{l} \mathbf{if}\;m \leq 1.632848001758723 \cdot 10^{-152}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 0.03584537202507449:\\ \;\;\;\;\left(1 - m\right) \cdot \frac{m}{v}\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{\frac{\frac{v}{m}}{m}}\\ \end{array} \]

Alternative 8
Error	1.8
Cost	708

\[\begin{array}{l} \mathbf{if}\;m \leq 0.03584537202507449:\\ \;\;\;\;\frac{m}{v} - \left(1 - m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m + -2}{\frac{v}{m \cdot m}}\\ \end{array} \]

Alternative 9
Error	1.8
Cost	708

\[\begin{array}{l} \mathbf{if}\;m \leq 0.03584537202507449:\\ \;\;\;\;\left(1 - m\right) \cdot \left(\frac{m}{v} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m + -2}{\frac{v}{m \cdot m}}\\ \end{array} \]

Alternative 10
Error	2.3
Cost	580

\[\begin{array}{l} \mathbf{if}\;m \leq 0.03584537202507449:\\ \;\;\;\;\frac{m}{v} - \left(1 - m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{\frac{\frac{v}{m}}{m}}\\ \end{array} \]

Alternative 11
Error	24.3
Cost	324

\[\begin{array}{l} \mathbf{if}\;m \leq 1.632848001758723 \cdot 10^{-152}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v}\\ \end{array} \]

Alternative 12
Error	61.7
Cost	64

\[m \]

Alternative 13
Error	37.6
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2022318 
(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)))