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

Average Error: 10.29% → 0.18%

Time: 18.6s

Precision: binary64

Cost: 19968

Math

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

↓

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

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

Derivation

1. Initial program 10.29

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

2. Simplified 0.18

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

   [Start] 10.29	\[ \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
   +-commutative [=>] 10.29	\[ \color{blue}{\left(\left(z - 1\right) \cdot \log \left(1 - y\right) + \left(x - 1\right) \cdot \log y\right)} - t \]
   fma-def [=>] 10.29	\[ \color{blue}{\mathsf{fma}\left(z - 1, \log \left(1 - y\right), \left(x - 1\right) \cdot \log y\right)} - t \]
   sub-neg [=>] 10.29	\[ \mathsf{fma}\left(z - 1, \log \color{blue}{\left(1 + \left(-y\right)\right)}, \left(x - 1\right) \cdot \log y\right) - t \]
   log1p-def [=>] 0.18	\[ \mathsf{fma}\left(z - 1, \color{blue}{\mathsf{log1p}\left(-y\right)}, \left(x - 1\right) \cdot \log y\right) - t \]
   remove-double-neg [<=] 0.18	\[ \mathsf{fma}\left(z - 1, \mathsf{log1p}\left(-y\right), \color{blue}{\left(-\left(-\left(x - 1\right)\right)\right)} \cdot \log y\right) - t \]
   remove-double-neg [=>] 0.18	\[ \mathsf{fma}\left(z - 1, \mathsf{log1p}\left(-y\right), \color{blue}{\left(x - 1\right)} \cdot \log y\right) - t \]
   sub-neg [=>] 0.18	\[ \mathsf{fma}\left(z - 1, \mathsf{log1p}\left(-y\right), \color{blue}{\left(x + \left(-1\right)\right)} \cdot \log y\right) - t \]
   metadata-eval [=>] 0.18	\[ \mathsf{fma}\left(z - 1, \mathsf{log1p}\left(-y\right), \left(x + \color{blue}{-1}\right) \cdot \log y\right) - t \]

3. Final simplification 0.18

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

Alternatives

Alternative 1
Error	4.48%
Cost	7497

\[\begin{array}{l} \mathbf{if}\;x + -1 \leq -1000 \lor \neg \left(x + -1 \leq 10^{+19}\right):\\ \;\;\;\;\log y \cdot \left(x + -1\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y - z \cdot y\right) - \log y\right) - t\\ \end{array} \]

Alternative 2
Error	12.31%
Cost	7241

\[\begin{array}{l} \mathbf{if}\;x + -1 \leq -1.000001 \lor \neg \left(x + -1 \leq -0.96\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;-\left(\log y + t\right)\\ \end{array} \]

Alternative 3
Error	0.91%
Cost	7232

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

Alternative 4
Error	35.88%
Cost	7120

\[\begin{array}{l} t_1 := x \cdot \log y\\ t_2 := y \cdot \left(1 - z\right) - t\\ \mathbf{if}\;x \leq -1250:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -5.5 \cdot 10^{-291}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 2.35 \cdot 10^{-251}:\\ \;\;\;\;-\log y\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{+19}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	9.53%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;z \leq -3.1 \cdot 10^{+228} \lor \neg \left(z \leq 1.3 \cdot 10^{+196}\right):\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot \left(x + -1\right) - t\\ \end{array} \]

Alternative 6
Error	47.41%
Cost	7056

\[\begin{array}{l} t_1 := -\log y\\ t_2 := y \cdot \left(1 - z\right) - t\\ \mathbf{if}\;t \leq -3.6 \cdot 10^{-102}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{-212}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 3.7 \cdot 10^{-153}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 1.8 \cdot 10^{-28}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) - z \cdot y\\ \end{array} \]

Alternative 7
Error	23.89%
Cost	6921

\[\begin{array}{l} \mathbf{if}\;x \leq -6.6 \cdot 10^{+40} \lor \neg \left(x \leq 1800000000000\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;-\left(\log y + t\right)\\ \end{array} \]

Alternative 8
Error	24.05%
Cost	6920

\[\begin{array}{l} \mathbf{if}\;x \leq -210:\\ \;\;\;\;\log y \cdot \left(x + -1\right)\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{+16}:\\ \;\;\;\;-\left(\log y + t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y\\ \end{array} \]

Alternative 9
Error	54.44%
Cost	448

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

Alternative 10
Error	54.64%
Cost	384

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

Alternative 11
Error	64.37%
Cost	128

\[-t \]

Alternative 12
Error	97.76%
Cost	64

\[t \]

Reproduce

herbie shell --seed 2023088 
(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))