b parameter of renormalized beta distribution

Average Error: 0.1 → 0.1

Time: 6.7s

Precision: binary64

Cost: 960

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

Math

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

↓

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

FPCore

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

↓

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

C

double code(double m, double v) {
	return (((m * (1.0 - m)) / v) - 1.0) * (1.0 - m);
}

↓

double code(double m, double v) {
	return (((m / v) - ((m * m) / v)) + -1.0) * (1.0 - m);
}

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
    code = (((m / v) - ((m * m) / v)) + (-1.0d0)) * (1.0d0 - m)
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) {
	return (((m / v) - ((m * m) / v)) + -1.0) * (1.0 - m);
}

Python

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

↓

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

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)
	return Float64(Float64(Float64(Float64(m / v) - Float64(Float64(m * m) / v)) + -1.0) * Float64(1.0 - m))
end

MATLAB

function tmp = code(m, v)
	tmp = (((m * (1.0 - m)) / v) - 1.0) * (1.0 - m);
end

↓

function tmp = code(m, v)
	tmp = (((m / v) - ((m * m) / v)) + -1.0) * (1.0 - m);
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_] := N[(N[(N[(N[(m / v), $MachinePrecision] - N[(N[(m * m), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] * N[(1.0 - m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.1

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

2. Taylor expanded in m around 0 0.1

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

3. Simplified 0.1

   \[\leadsto \left(\color{blue}{\left(\frac{m}{v} - \frac{m \cdot m}{v}\right)} - 1\right) \cdot \left(1 - m\right) \]
   Proof
   (-.f64 (/.f64 m v) (/.f64 (*.f64 m m) v)): 0 points increase in error, 0 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

4. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.2
Cost	836

\[\begin{array}{l} \mathbf{if}\;m \leq 1.5 \cdot 10^{-13}:\\ \;\;\;\;\left(1 - m\right) \cdot \left(\frac{m}{v} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - m\right) \cdot \left(m \cdot \frac{1 - m}{v}\right)\\ \end{array} \]

Alternative 2
Error	0.2
Cost	836

\[\begin{array}{l} \mathbf{if}\;m \leq 2.25 \cdot 10^{-14}:\\ \;\;\;\;\left(1 - m\right) \cdot \left(\frac{m}{v} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - m\right) \cdot \frac{m \cdot \left(1 - m\right)}{v}\\ \end{array} \]

Alternative 3
Error	0.2
Cost	836

\[\begin{array}{l} \mathbf{if}\;m \leq 2.6 \cdot 10^{-14}:\\ \;\;\;\;\left(1 - m\right) \cdot \left(\frac{m}{v} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - m\right) \cdot \left(m - m \cdot m\right)}{v}\\ \end{array} \]

Alternative 4
Error	0.1
Cost	832

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

Alternative 5
Error	0.1
Cost	832

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

Alternative 6
Error	0.1
Cost	832

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

Alternative 7
Error	2.2
Cost	708

\[\begin{array}{l} \mathbf{if}\;m \leq 2.3:\\ \;\;\;\;-1 + \left(m + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(m \cdot \frac{m}{v}\right) \cdot \left(m + -1\right)\\ \end{array} \]

Alternative 8
Error	2.2
Cost	708

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

Alternative 9
Error	2.2
Cost	708

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

Alternative 10
Error	25.3
Cost	588

\[\begin{array}{l} \mathbf{if}\;m \leq 4.8 \cdot 10^{-208}:\\ \;\;\;\;-1\\ \mathbf{elif}\;m \leq 1.06 \cdot 10^{-197}:\\ \;\;\;\;\frac{m}{v}\\ \mathbf{elif}\;m \leq 5.8 \cdot 10^{-140}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v}\\ \end{array} \]

Alternative 11
Error	2.2
Cost	580

\[\begin{array}{l} \mathbf{if}\;m \leq 2.6:\\ \;\;\;\;\frac{m}{v} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v} \cdot \left(m \cdot m\right)\\ \end{array} \]

Alternative 12
Error	2.2
Cost	580

\[\begin{array}{l} \mathbf{if}\;m \leq 2.6:\\ \;\;\;\;-1 + \left(m + \frac{m}{v}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{m}{v} \cdot \left(m \cdot m\right)\\ \end{array} \]

Alternative 13
Error	9.7
Cost	320

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

Alternative 14
Error	37.2
Cost	192

\[m + -1 \]

Alternative 15
Error	37.6
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2022338 
(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)))