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

Average Error: 6.7 → 0.3

Time: 19.4s

Precision: binary64

Cost: 20736

Math

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

↓

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

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)
    (+ (- y) (+ (* -0.5 (pow y 2.0)) (* -0.3333333333333333 (pow y 3.0))))))
  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) * (-y + ((-0.5 * pow(y, 2.0)) + (-0.3333333333333333 * pow(y, 3.0)))))) - 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) * (-y + (((-0.5d0) * (y ** 2.0d0)) + ((-0.3333333333333333d0) * (y ** 3.0d0)))))) - 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) * (-y + ((-0.5 * Math.pow(y, 2.0)) + (-0.3333333333333333 * Math.pow(y, 3.0)))))) - 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) * (-y + ((-0.5 * math.pow(y, 2.0)) + (-0.3333333333333333 * math.pow(y, 3.0)))))) - 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(-y) + Float64(Float64(-0.5 * (y ^ 2.0)) + Float64(-0.3333333333333333 * (y ^ 3.0)))))) - 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) * (-y + ((-0.5 * (y ^ 2.0)) + (-0.3333333333333333 * (y ^ 3.0)))))) - 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[((-y) + N[(N[(-0.5 * N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision] + N[(-0.3333333333333333 * N[Power[y, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 6.7

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

2. Taylor expanded in y around 0 0.3

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

3. Simplified 0.3

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

   [Start] 0.3	\[ \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(-0.5 \cdot {y}^{2} + \left(-0.3333333333333333 \cdot {y}^{3} + -1 \cdot y\right)\right)\right) - t \]
   rational.json-simplify-41 [<=] 0.3	\[ \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\left(-1 \cdot y + \left(-0.5 \cdot {y}^{2} + -0.3333333333333333 \cdot {y}^{3}\right)\right)}\right) - t \]
   rational.json-simplify-2 [=>] 0.3	\[ \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\color{blue}{y \cdot -1} + \left(-0.5 \cdot {y}^{2} + -0.3333333333333333 \cdot {y}^{3}\right)\right)\right) - t \]
   rational.json-simplify-9 [=>] 0.3	\[ \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\color{blue}{\left(-y\right)} + \left(-0.5 \cdot {y}^{2} + -0.3333333333333333 \cdot {y}^{3}\right)\right)\right) - t \]

4. Final simplification 0.3

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

Alternatives

Alternative 1
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 2
Error	2.6
Cost	7560

\[\begin{array}{l} t_1 := \left(x - 1\right) \cdot \log y - t\\ \mathbf{if}\;x - 1 \leq -20:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x - 1 \leq -1:\\ \;\;\;\;\left(y \cdot \left(1 - z\right) + \left(-\log y\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	27.8
Cost	7384

\[\begin{array}{l} t_1 := -\log y\\ t_2 := \log y \cdot x\\ \mathbf{if}\;t \leq -1.75 \cdot 10^{+56}:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq -2.85 \cdot 10^{-77}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -9 \cdot 10^{-203}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{-282}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{-180}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 5.9 \cdot 10^{+91}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]

Alternative 4
Error	0.5
Cost	7232

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

Alternative 5
Error	7.9
Cost	6984

\[\begin{array}{l} t_1 := \log y \cdot x - t\\ \mathbf{if}\;t \leq -29000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 330:\\ \;\;\;\;\left(x - 1\right) \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	7.3
Cost	6976

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

Alternative 7
Error	15.3
Cost	6920

\[\begin{array}{l} t_1 := \log y \cdot x\\ \mathbf{if}\;x \leq -1.95 \cdot 10^{+29}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{+64}:\\ \;\;\;\;\left(-\log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	7.4
Cost	6848

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

Alternative 9
Error	30.1
Cost	6792

\[\begin{array}{l} \mathbf{if}\;t \leq -0.26:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{+15}:\\ \;\;\;\;-\log y\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]

Alternative 10
Error	36.2
Cost	584

\[\begin{array}{l} \mathbf{if}\;t \leq -28500:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 8.6 \cdot 10^{+15}:\\ \;\;\;\;y \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]

Alternative 11
Error	36.4
Cost	520

\[\begin{array}{l} \mathbf{if}\;t \leq -28500:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 1.35 \cdot 10^{+16}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]

Alternative 12
Error	40.6
Cost	128

\[-t \]

Reproduce

herbie shell --seed 2023077 
(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))