Statistics.Distribution.Beta:$cdensity from math-functions-0.1.5.2

Average Error: 6.8 → 0.2

Time: 10.5s

Precision: binary64

Cost: 14592

Math

\[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t\]

↓

\[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left({y}^{4} \cdot -0.25 - \left(y + \left(y \cdot y\right) \cdot \left(y \cdot 0.3333333333333333 + 0.5\right)\right)\right)\right) - t\]

FPCore

(FPCore (x y z t)
 :precision binary64
 (- (+ (* (- x 1.0) (log y)) (* (- z 1.0) (log (- 1.0 y)))) t))

↓

(FPCore (x y z t)
 :precision binary64
 (-
  (+
   (* (- x 1.0) (log y))
   (*
    (- z 1.0)
    (-
     (* (pow y 4.0) -0.25)
     (+ y (* (* y y) (+ (* y 0.3333333333333333) 0.5))))))
  t))

C

double code(double x, double y, double z, double t) {
	return (((x - 1.0) * log(y)) + ((z - 1.0) * log(1.0 - y))) - t;
}

↓

double code(double x, double y, double z, double t) {
	return (((x - 1.0) * log(y)) + ((z - 1.0) * ((pow(y, 4.0) * -0.25) - (y + ((y * y) * ((y * 0.3333333333333333) + 0.5)))))) - t;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Alternatives

Alternative 1
Error	0.3
Cost	8640

\[\left(\left(x - 1\right) \cdot \log y + \left(y + \left(y \cdot y\right) \cdot \left(y \cdot 0.3333333333333333 + 0.5\right)\right)\right) - \left(t + \left(y \cdot z + z \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot 0.3333333333333333 + 0.5\right)\right)\right)\right)\]

Alternative 2
Error	0.3
Cost	7872

\[\left(\left(x - 1\right) \cdot \log y + \left(y + \left(y \cdot y\right) \cdot \left(y \cdot 0.3333333333333333 + 0.5\right)\right) \cdot \left(1 - z\right)\right) - t\]

Alternative 3
Error	0.4
Cost	7616

\[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(y \cdot \left(-1 - y \cdot 0.5\right)\right)\right) - t\]

Alternative 4
Error	0.6
Cost	7232

\[\left(\left(x - 1\right) \cdot \log y + y \cdot \left(1 - z\right)\right) - t\]

Alternative 5
Error	7.5
Cost	6848

\[\left(x - 1\right) \cdot \log y - t\]

Alternative 6
Error	26.6
Cost	6920

\[\begin{array}{l} \mathbf{if}\;x \leq -4935272682.862067 \lor \neg \left(x \leq 1.2110646529640444 \cdot 10^{+73}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array}\]

Alternative 7
Error	39.8
Cost	456

\[\begin{array}{l} \mathbf{if}\;t \leq -1.02292342243375 \lor \neg \left(t \leq 0.03464102882019671\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

Alternative 8
Error	60.8
Cost	64

\[1\]

Derivation

1. Initial program 6.8

   \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t\]

2. Taylor expanded around 0 0.2

   \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\left(-\left(0.25 \cdot {y}^{4} + \left(y + \left(0.3333333333333333 \cdot {y}^{3} + 0.5 \cdot {y}^{2}\right)\right)\right)\right)}\right) - t\]

3. Simplified 0.2

   \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\left({y}^{4} \cdot -0.25 - \left(y + \left(y \cdot y\right) \cdot \left(0.5 + 0.3333333333333333 \cdot y\right)\right)\right)}\right) - t\]

4. Simplified 0.2

   \[\leadsto \color{blue}{\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left({y}^{4} \cdot -0.25 - \left(y + \left(y \cdot y\right) \cdot \left(y \cdot 0.3333333333333333 + 0.5\right)\right)\right)\right) - t}\]

5. Final simplification 0.2

   \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left({y}^{4} \cdot -0.25 - \left(y + \left(y \cdot y\right) \cdot \left(y \cdot 0.3333333333333333 + 0.5\right)\right)\right)\right) - t\]

Reproduce

herbie shell --seed 2021044 
(FPCore (x y z t)
  :name "Statistics.Distribution.Beta:$cdensity from math-functions-0.1.5.2"
  :precision binary64
  (- (+ (* (- x 1.0) (log y)) (* (- z 1.0) (log (- 1.0 y)))) t))