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

Average Error: 7.0 → 0.2

Time: 22.1s

Precision: binary64

Cost: 27456

Math

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

↓

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

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)
    (+
     (* -0.5 (pow y 2.0))
     (+ (* -0.3333333333333333 (pow y 3.0)) (+ (- y) (* -0.25 (pow y 4.0)))))))
  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) * ((-0.5 * pow(y, 2.0)) + ((-0.3333333333333333 * pow(y, 3.0)) + (-y + (-0.25 * pow(y, 4.0))))))) - t;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (((x - 1.0d0) * log(y)) + ((z - 1.0d0) * log((1.0d0 - y)))) - t
end function

↓

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (((x - 1.0d0) * log(y)) + ((z - 1.0d0) * (((-0.5d0) * (y ** 2.0d0)) + (((-0.3333333333333333d0) * (y ** 3.0d0)) + (-y + ((-0.25d0) * (y ** 4.0d0))))))) - t
end function

Java

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

↓

public static double code(double x, double y, double z, double t) {
	return (((x - 1.0) * Math.log(y)) + ((z - 1.0) * ((-0.5 * Math.pow(y, 2.0)) + ((-0.3333333333333333 * Math.pow(y, 3.0)) + (-y + (-0.25 * Math.pow(y, 4.0))))))) - t;
}

Python

def code(x, y, z, t):
	return (((x - 1.0) * math.log(y)) + ((z - 1.0) * math.log((1.0 - y)))) - t

↓

def code(x, y, z, t):
	return (((x - 1.0) * math.log(y)) + ((z - 1.0) * ((-0.5 * math.pow(y, 2.0)) + ((-0.3333333333333333 * math.pow(y, 3.0)) + (-y + (-0.25 * math.pow(y, 4.0))))))) - t

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(x - 1.0) * log(y)) + Float64(Float64(z - 1.0) * log(Float64(1.0 - y)))) - t)
end

↓

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(x - 1.0) * log(y)) + Float64(Float64(z - 1.0) * Float64(Float64(-0.5 * (y ^ 2.0)) + Float64(Float64(-0.3333333333333333 * (y ^ 3.0)) + Float64(Float64(-y) + Float64(-0.25 * (y ^ 4.0))))))) - t)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((x - 1.0) * log(y)) + ((z - 1.0) * log((1.0 - y)))) - t;
end

↓

function tmp = code(x, y, z, t)
	tmp = (((x - 1.0) * log(y)) + ((z - 1.0) * ((-0.5 * (y ^ 2.0)) + ((-0.3333333333333333 * (y ^ 3.0)) + (-y + (-0.25 * (y ^ 4.0))))))) - t;
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(x - 1.0), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision] + N[(N[(z - 1.0), $MachinePrecision] * N[Log[N[(1.0 - y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]

↓

code[x_, y_, z_, t_] := N[(N[(N[(N[(x - 1.0), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision] + N[(N[(z - 1.0), $MachinePrecision] * N[(N[(-0.5 * N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-0.3333333333333333 * N[Power[y, 3.0], $MachinePrecision]), $MachinePrecision] + N[((-y) + N[(-0.25 * N[Power[y, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 7.0

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

2. Taylor expanded in y around 0 0.2

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

3. Simplified 0.2

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

   [Start] 0.2	\[ \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(-0.5 \cdot {y}^{2} + \left(-0.25 \cdot {y}^{4} + \left(-0.3333333333333333 \cdot {y}^{3} + -1 \cdot y\right)\right)\right)\right) - t \]
   rational.json-simplify-41 [=>] 0.2	\[ \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(-0.5 \cdot {y}^{2} + \color{blue}{\left(-0.3333333333333333 \cdot {y}^{3} + \left(-1 \cdot y + -0.25 \cdot {y}^{4}\right)\right)}\right)\right) - t \]
   rational.json-simplify-2 [=>] 0.2	\[ \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(-0.5 \cdot {y}^{2} + \left(-0.3333333333333333 \cdot {y}^{3} + \left(\color{blue}{y \cdot -1} + -0.25 \cdot {y}^{4}\right)\right)\right)\right) - t \]
   rational.json-simplify-9 [=>] 0.2	\[ \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(-0.5 \cdot {y}^{2} + \left(-0.3333333333333333 \cdot {y}^{3} + \left(\color{blue}{\left(-y\right)} + -0.25 \cdot {y}^{4}\right)\right)\right)\right) - t \]

4. Final simplification 0.2

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

Alternatives

Alternative 1
Error	0.3
Cost	20800

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

Alternative 2
Error	0.3
Cost	20736

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

Alternative 3
Error	0.4
Cost	14016

\[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\left(-y\right) + -0.5 \cdot {y}^{2}\right)\right) - t \]

Alternative 4
Error	7.7
Cost	7560

\[\begin{array}{l} t_1 := x - 1 \leq -1\\ t_2 := \left(x - 1\right) \cdot \log y - t\\ \mathbf{if}\;t_1:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1:\\ \;\;\;\;\left(y \cdot \left(1 - z\right) + \left(-\log y\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Error	6.1
Cost	7372

\[\begin{array}{l} t_1 := y \cdot \left(1 - z\right)\\ t_2 := \log y \cdot x + t_1\\ \mathbf{if}\;z \leq -2.1 \cdot 10^{+228}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 6.6 \cdot 10^{+240}:\\ \;\;\;\;\left(x - 1\right) \cdot \log y - t\\ \mathbf{elif}\;z \leq 2 \cdot 10^{+295}:\\ \;\;\;\;t_1 - t\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 6
Error	3.7
Cost	7368

\[\begin{array}{l} t_1 := \left(\log y \cdot x + y \cdot \left(1 - z\right)\right) - t\\ \mathbf{if}\;z \leq -6.2 \cdot 10^{+93}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 10^{+233}:\\ \;\;\;\;\left(x - 1\right) \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	0.5
Cost	7232

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

Alternative 8
Error	13.6
Cost	6984

\[\begin{array}{l} t_1 := y \cdot \left(1 - z\right) - t\\ \mathbf{if}\;t \leq -165000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 15000000000000:\\ \;\;\;\;\left(x - 1\right) \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	8.2
Cost	6984

\[\begin{array}{l} t_1 := \log y \cdot x - t\\ \mathbf{if}\;t \leq -18000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.08 \cdot 10^{-5}:\\ \;\;\;\;\left(x - 1\right) \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	7.1
Cost	6980

\[\begin{array}{l} \mathbf{if}\;z \leq 3.5 \cdot 10^{+238}:\\ \;\;\;\;\left(x - 1\right) \cdot \log y - t\\ \mathbf{elif}\;z \leq 4.4 \cdot 10^{+296}:\\ \;\;\;\;y \cdot \left(1 - z\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot x\\ \end{array} \]

Alternative 11
Error	15.2
Cost	6920

\[\begin{array}{l} t_1 := \log y \cdot x\\ \mathbf{if}\;x \leq -5.6 \cdot 10^{+50}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2.95 \cdot 10^{+35}:\\ \;\;\;\;\left(-\log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 12
Error	24.8
Cost	6856

\[\begin{array}{l} t_1 := y \cdot \left(1 - z\right) - t\\ \mathbf{if}\;t \leq -100000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 320000000000:\\ \;\;\;\;\log y \cdot x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 13
Error	29.1
Cost	6792

\[\begin{array}{l} t_1 := y \cdot \left(1 - z\right) - t\\ \mathbf{if}\;t \leq -0.32:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 3.85 \cdot 10^{-84}:\\ \;\;\;\;-\log y\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 14
Error	36.0
Cost	584

\[\begin{array}{l} \mathbf{if}\;t \leq -15800000000000:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 3.6:\\ \;\;\;\;y \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]

Alternative 15
Error	36.2
Cost	520

\[\begin{array}{l} \mathbf{if}\;t \leq -370000000000:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 3.6:\\ \;\;\;\;z \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]

Alternative 16
Error	34.2
Cost	448

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

Alternative 17
Error	41.1
Cost	128

\[-t \]

Reproduce

herbie shell --seed 2023074 
(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))