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

Average Error: 6.7 → 0.1

Time: 22.1s

Precision: binary64

Cost: 26304

Math

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

↓

\[\mathsf{fma}\left(z + -1, \mathsf{log1p}\left(-y\right), \mathsf{fma}\left(\log y, x + -1, -t\right)\right) \]

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
 (fma (+ z -1.0) (log1p (- y)) (fma (log y) (+ x -1.0) (- 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 fma((z + -1.0), log1p(-y), fma(log(y), (x + -1.0), -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 fma(Float64(z + -1.0), log1p(Float64(-y)), fma(log(y), Float64(x + -1.0), Float64(-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[(z + -1.0), $MachinePrecision] * N[Log[1 + (-y)], $MachinePrecision] + N[(N[Log[y], $MachinePrecision] * N[(x + -1.0), $MachinePrecision] + (-t)), $MachinePrecision]), $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. Simplified 0.1

   \[\leadsto \color{blue}{\mathsf{fma}\left(z + -1, \mathsf{log1p}\left(-y\right), \mathsf{fma}\left(\log y, x + -1, -t\right)\right)} \]
   Proof

Alternatives

Alternative 1
Error	0.1
Cost	13696

\[\left(\left(z + -1\right) \cdot \mathsf{log1p}\left(-y\right) - \left(1 - x\right) \cdot \log y\right) - t \]

Alternative 2
Error	2.9
Cost	7368

\[\begin{array}{l} t_1 := \left(x - 1\right) \cdot \log y\\ \mathbf{if}\;t \leq -28:\\ \;\;\;\;t_1 - t\\ \mathbf{elif}\;t \leq 35000:\\ \;\;\;\;t_1 - \left(z - 1\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot x - t\\ \end{array} \]

Alternative 3
Error	0.6
Cost	7232

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

Alternative 4
Error	6.8
Cost	7108

\[\begin{array}{l} \mathbf{if}\;z \leq -2.2 \cdot 10^{+225}:\\ \;\;\;\;\left(-y \cdot z\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(y + \left(x - 1\right) \cdot \log y\right) - t\\ \end{array} \]

Alternative 5
Error	14.8
Cost	6984

\[\begin{array}{l} t_1 := \log y \cdot x - t\\ \mathbf{if}\;x \leq -4.7 \cdot 10^{+14}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 175000:\\ \;\;\;\;y \cdot \left(1 - z\right) - t\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	7.9
Cost	6984

\[\begin{array}{l} t_1 := \log y \cdot x - t\\ \mathbf{if}\;x \leq -1550:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-6}:\\ \;\;\;\;\left(y - \log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	6.9
Cost	6980

\[\begin{array}{l} \mathbf{if}\;z \leq -2.7 \cdot 10^{+233}:\\ \;\;\;\;\left(-y \cdot z\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(x - 1\right) \cdot \log y - t\\ \end{array} \]

Alternative 8
Error	22.0
Cost	6856

\[\begin{array}{l} t_1 := \log y \cdot x\\ \mathbf{if}\;x \leq -5.5 \cdot 10^{+80}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 17000000000:\\ \;\;\;\;y \cdot \left(1 - z\right) - t\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	36.7
Cost	520

\[\begin{array}{l} \mathbf{if}\;t \leq -0.009:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 1800000000000:\\ \;\;\;\;-y \cdot z\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]

Alternative 10
Error	34.6
Cost	448

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

Alternative 11
Error	34.8
Cost	384

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

Alternative 12
Error	41.0
Cost	192

\[y - t \]

Alternative 13
Error	41.1
Cost	128

\[-t \]

Reproduce

herbie shell --seed 2023010 
(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))