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

Average Error: 6.6 → 0.2

Time: 17.6s

Precision: binary64

Cost: 28224

Math

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

↓

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

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))
   (+
    (* -0.3333333333333333 (* (pow y 3.0) (+ -1.0 z)))
    (+
     (* (- y) (+ -1.0 z))
     (+
      (* -0.5 (* (pow y 2.0) (+ -1.0 z)))
      (* -0.25 (* (pow y 4.0) (+ -1.0 z)))))))
  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)) + ((-0.3333333333333333 * (pow(y, 3.0) * (-1.0 + z))) + ((-y * (-1.0 + z)) + ((-0.5 * (pow(y, 2.0) * (-1.0 + z))) + (-0.25 * (pow(y, 4.0) * (-1.0 + z))))))) - 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)) + (((-0.3333333333333333d0) * ((y ** 3.0d0) * ((-1.0d0) + z))) + ((-y * ((-1.0d0) + z)) + (((-0.5d0) * ((y ** 2.0d0) * ((-1.0d0) + z))) + ((-0.25d0) * ((y ** 4.0d0) * ((-1.0d0) + z))))))) - 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)) + ((-0.3333333333333333 * (Math.pow(y, 3.0) * (-1.0 + z))) + ((-y * (-1.0 + z)) + ((-0.5 * (Math.pow(y, 2.0) * (-1.0 + z))) + (-0.25 * (Math.pow(y, 4.0) * (-1.0 + z))))))) - 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)) + ((-0.3333333333333333 * (math.pow(y, 3.0) * (-1.0 + z))) + ((-y * (-1.0 + z)) + ((-0.5 * (math.pow(y, 2.0) * (-1.0 + z))) + (-0.25 * (math.pow(y, 4.0) * (-1.0 + z))))))) - 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(-0.3333333333333333 * Float64((y ^ 3.0) * Float64(-1.0 + z))) + Float64(Float64(Float64(-y) * Float64(-1.0 + z)) + Float64(Float64(-0.5 * Float64((y ^ 2.0) * Float64(-1.0 + z))) + Float64(-0.25 * Float64((y ^ 4.0) * Float64(-1.0 + z))))))) - 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)) + ((-0.3333333333333333 * ((y ^ 3.0) * (-1.0 + z))) + ((-y * (-1.0 + z)) + ((-0.5 * ((y ^ 2.0) * (-1.0 + z))) + (-0.25 * ((y ^ 4.0) * (-1.0 + z))))))) - 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[(-0.3333333333333333 * N[(N[Power[y, 3.0], $MachinePrecision] * N[(-1.0 + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[((-y) * N[(-1.0 + z), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.5 * N[(N[Power[y, 2.0], $MachinePrecision] * N[(-1.0 + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.25 * N[(N[Power[y, 4.0], $MachinePrecision] * N[(-1.0 + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 6.6

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

2. Taylor expanded in y around 0 0.2

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

3. Simplified 0.2

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

   [Start] 0.2	\[ \left(\left(x - 1\right) \cdot \log y + \left(-0.5 \cdot \left(\left(z - 1\right) \cdot {y}^{2}\right) + \left(-0.3333333333333333 \cdot \left(\left(z - 1\right) \cdot {y}^{3}\right) + \left(-1 \cdot \left(\left(z - 1\right) \cdot y\right) + -0.25 \cdot \left(\left(z - 1\right) \cdot {y}^{4}\right)\right)\right)\right)\right) - t \]
   rational_best_oopsla_all_46_json_45_simplify-82 [=>] 0.2	\[ \left(\left(x - 1\right) \cdot \log y + \color{blue}{\left(-0.3333333333333333 \cdot \left(\left(z - 1\right) \cdot {y}^{3}\right) + \left(-0.5 \cdot \left(\left(z - 1\right) \cdot {y}^{2}\right) + \left(-1 \cdot \left(\left(z - 1\right) \cdot y\right) + -0.25 \cdot \left(\left(z - 1\right) \cdot {y}^{4}\right)\right)\right)\right)}\right) - t \]
   rational_best_oopsla_all_46_json_45_simplify-45 [=>] 0.2	\[ \left(\left(x - 1\right) \cdot \log y + \left(-0.3333333333333333 \cdot \left(\color{blue}{\left(z + -1\right)} \cdot {y}^{3}\right) + \left(-0.5 \cdot \left(\left(z - 1\right) \cdot {y}^{2}\right) + \left(-1 \cdot \left(\left(z - 1\right) \cdot y\right) + -0.25 \cdot \left(\left(z - 1\right) \cdot {y}^{4}\right)\right)\right)\right)\right) - t \]
   rational_best_oopsla_all_46_json_45_simplify-74 [=>] 0.2	\[ \left(\left(x - 1\right) \cdot \log y + \left(-0.3333333333333333 \cdot \color{blue}{\left({y}^{3} \cdot \left(z + -1\right)\right)} + \left(-0.5 \cdot \left(\left(z - 1\right) \cdot {y}^{2}\right) + \left(-1 \cdot \left(\left(z - 1\right) \cdot y\right) + -0.25 \cdot \left(\left(z - 1\right) \cdot {y}^{4}\right)\right)\right)\right)\right) - t \]
   rational_best_oopsla_all_46_json_45_simplify-35 [=>] 0.2	\[ \left(\left(x - 1\right) \cdot \log y + \left(-0.3333333333333333 \cdot \left({y}^{3} \cdot \color{blue}{\left(-1 + z\right)}\right) + \left(-0.5 \cdot \left(\left(z - 1\right) \cdot {y}^{2}\right) + \left(-1 \cdot \left(\left(z - 1\right) \cdot y\right) + -0.25 \cdot \left(\left(z - 1\right) \cdot {y}^{4}\right)\right)\right)\right)\right) - t \]
   rational_best_oopsla_all_46_json_45_simplify-82 [=>] 0.2	\[ \left(\left(x - 1\right) \cdot \log y + \left(-0.3333333333333333 \cdot \left({y}^{3} \cdot \left(-1 + z\right)\right) + \color{blue}{\left(-1 \cdot \left(\left(z - 1\right) \cdot y\right) + \left(-0.5 \cdot \left(\left(z - 1\right) \cdot {y}^{2}\right) + -0.25 \cdot \left(\left(z - 1\right) \cdot {y}^{4}\right)\right)\right)}\right)\right) - t \]
   rational_best_oopsla_all_46_json_45_simplify-7 [=>] 0.2	\[ \left(\left(x - 1\right) \cdot \log y + \left(-0.3333333333333333 \cdot \left({y}^{3} \cdot \left(-1 + z\right)\right) + \left(\color{blue}{\left(z - 1\right) \cdot \left(-1 \cdot y\right)} + \left(-0.5 \cdot \left(\left(z - 1\right) \cdot {y}^{2}\right) + -0.25 \cdot \left(\left(z - 1\right) \cdot {y}^{4}\right)\right)\right)\right)\right) - t \]
   rational_best_oopsla_all_46_json_45_simplify-45 [=>] 0.2	\[ \left(\left(x - 1\right) \cdot \log y + \left(-0.3333333333333333 \cdot \left({y}^{3} \cdot \left(-1 + z\right)\right) + \left(\color{blue}{\left(z + -1\right)} \cdot \left(-1 \cdot y\right) + \left(-0.5 \cdot \left(\left(z - 1\right) \cdot {y}^{2}\right) + -0.25 \cdot \left(\left(z - 1\right) \cdot {y}^{4}\right)\right)\right)\right)\right) - t \]
   rational_best_oopsla_all_46_json_45_simplify-74 [=>] 0.2	\[ \left(\left(x - 1\right) \cdot \log y + \left(-0.3333333333333333 \cdot \left({y}^{3} \cdot \left(-1 + z\right)\right) + \left(\color{blue}{\left(-1 \cdot y\right) \cdot \left(z + -1\right)} + \left(-0.5 \cdot \left(\left(z - 1\right) \cdot {y}^{2}\right) + -0.25 \cdot \left(\left(z - 1\right) \cdot {y}^{4}\right)\right)\right)\right)\right) - t \]
   rational_best_oopsla_all_46_json_45_simplify-74 [=>] 0.2	\[ \left(\left(x - 1\right) \cdot \log y + \left(-0.3333333333333333 \cdot \left({y}^{3} \cdot \left(-1 + z\right)\right) + \left(\color{blue}{\left(y \cdot -1\right)} \cdot \left(z + -1\right) + \left(-0.5 \cdot \left(\left(z - 1\right) \cdot {y}^{2}\right) + -0.25 \cdot \left(\left(z - 1\right) \cdot {y}^{4}\right)\right)\right)\right)\right) - t \]
   rational_best_oopsla_all_46_json_45_simplify-92 [=>] 0.2	\[ \left(\left(x - 1\right) \cdot \log y + \left(-0.3333333333333333 \cdot \left({y}^{3} \cdot \left(-1 + z\right)\right) + \left(\color{blue}{\left(-y\right)} \cdot \left(z + -1\right) + \left(-0.5 \cdot \left(\left(z - 1\right) \cdot {y}^{2}\right) + -0.25 \cdot \left(\left(z - 1\right) \cdot {y}^{4}\right)\right)\right)\right)\right) - t \]
   rational_best_oopsla_all_46_json_45_simplify-35 [=>] 0.2	\[ \left(\left(x - 1\right) \cdot \log y + \left(-0.3333333333333333 \cdot \left({y}^{3} \cdot \left(-1 + z\right)\right) + \left(\left(-y\right) \cdot \color{blue}{\left(-1 + z\right)} + \left(-0.5 \cdot \left(\left(z - 1\right) \cdot {y}^{2}\right) + -0.25 \cdot \left(\left(z - 1\right) \cdot {y}^{4}\right)\right)\right)\right)\right) - t \]
   rational_best_oopsla_all_46_json_45_simplify-45 [=>] 0.2	\[ \left(\left(x - 1\right) \cdot \log y + \left(-0.3333333333333333 \cdot \left({y}^{3} \cdot \left(-1 + z\right)\right) + \left(\left(-y\right) \cdot \left(-1 + z\right) + \left(-0.5 \cdot \left(\color{blue}{\left(z + -1\right)} \cdot {y}^{2}\right) + -0.25 \cdot \left(\left(z - 1\right) \cdot {y}^{4}\right)\right)\right)\right)\right) - t \]
   rational_best_oopsla_all_46_json_45_simplify-74 [=>] 0.2	\[ \left(\left(x - 1\right) \cdot \log y + \left(-0.3333333333333333 \cdot \left({y}^{3} \cdot \left(-1 + z\right)\right) + \left(\left(-y\right) \cdot \left(-1 + z\right) + \left(-0.5 \cdot \color{blue}{\left({y}^{2} \cdot \left(z + -1\right)\right)} + -0.25 \cdot \left(\left(z - 1\right) \cdot {y}^{4}\right)\right)\right)\right)\right) - t \]
   rational_best_oopsla_all_46_json_45_simplify-35 [=>] 0.2	\[ \left(\left(x - 1\right) \cdot \log y + \left(-0.3333333333333333 \cdot \left({y}^{3} \cdot \left(-1 + z\right)\right) + \left(\left(-y\right) \cdot \left(-1 + z\right) + \left(-0.5 \cdot \left({y}^{2} \cdot \color{blue}{\left(-1 + z\right)}\right) + -0.25 \cdot \left(\left(z - 1\right) \cdot {y}^{4}\right)\right)\right)\right)\right) - t \]

4. Final simplification 0.2

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

Alternatives

Alternative 1
Error	0.2
Cost	27456

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

Alternative 2
Error	0.2
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	20544

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

Alternative 4
Error	0.3
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 5
Error	1.2
Cost	7624

\[\begin{array}{l} t_1 := \left(\left(y + \log y \cdot x\right) - y \cdot z\right) - t\\ \mathbf{if}\;x - 1 \leq -50:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x - 1 \leq -0.99999999:\\ \;\;\;\;\left(\left(y - z \cdot y\right) - \log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	3.0
Cost	7496

\[\begin{array}{l} \mathbf{if}\;x - 1 \leq -50:\\ \;\;\;\;\left(x - 1\right) \cdot \log y - t\\ \mathbf{elif}\;x - 1 \leq 2 \cdot 10^{+23}:\\ \;\;\;\;\left(\left(y - z \cdot y\right) - \log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot x - t\\ \end{array} \]

Alternative 7
Error	0.6
Cost	7424

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

Alternative 8
Error	0.5
Cost	7296

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

Alternative 9
Error	7.9
Cost	7048

\[\begin{array}{l} t_1 := \log y \cdot x - t\\ \mathbf{if}\;x \leq -2.9:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-8}:\\ \;\;\;\;\left(\left(-\log y\right) + y\right) - t\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	7.9
Cost	6984

\[\begin{array}{l} t_1 := \log y \cdot x - t\\ \mathbf{if}\;x \leq -2.9:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-8}:\\ \;\;\;\;\left(-\log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	9.0
Cost	6980

\[\begin{array}{l} \mathbf{if}\;y \leq 2.8 \cdot 10^{-43}:\\ \;\;\;\;\left(x - 1\right) \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\left(y - z \cdot y\right) - t\\ \end{array} \]

Alternative 12
Error	15.3
Cost	6920

\[\begin{array}{l} t_1 := \log y \cdot x\\ \mathbf{if}\;x \leq -2.5 \cdot 10^{+25}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{+52}:\\ \;\;\;\;\left(-\log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 13
Error	21.3
Cost	6856

\[\begin{array}{l} t_1 := \log y \cdot x\\ \mathbf{if}\;x \leq -6.6 \cdot 10^{+59}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{+52}:\\ \;\;\;\;\left(y - z \cdot y\right) - t\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 14
Error	37.1
Cost	520

\[\begin{array}{l} \mathbf{if}\;t \leq -6.6 \cdot 10^{+26}:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 4.6 \cdot 10^{-5}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]

Alternative 15
Error	34.7
Cost	448

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

Alternative 16
Error	34.8
Cost	384

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

Alternative 17
Error	41.1
Cost	128

\[-t \]

Alternative 18
Error	62.2
Cost	64

\[y \]

Reproduce

herbie shell --seed 2023090 
(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))