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

Average Accuracy: 88.5% → 99.8%

Time: 18.9s

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 88.5%

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

2. Simplified 99.8%

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

   [Start] 88.5	\[ \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
   +-commutative [=>] 88.5	\[ \color{blue}{\left(\left(z - 1\right) \cdot \log \left(1 - y\right) + \left(x - 1\right) \cdot \log y\right)} - t \]
   fma-def [=>] 88.5	\[ \color{blue}{\mathsf{fma}\left(z - 1, \log \left(1 - y\right), \left(x - 1\right) \cdot \log y\right)} - t \]
   sub-neg [=>] 88.5	\[ \mathsf{fma}\left(z - 1, \log \color{blue}{\left(1 + \left(-y\right)\right)}, \left(x - 1\right) \cdot \log y\right) - t \]
   log1p-def [=>] 99.8	\[ \mathsf{fma}\left(z - 1, \color{blue}{\mathsf{log1p}\left(-y\right)}, \left(x - 1\right) \cdot \log y\right) - t \]
   remove-double-neg [<=] 99.8	\[ \mathsf{fma}\left(z - 1, \mathsf{log1p}\left(-y\right), \color{blue}{\left(-\left(-\left(x - 1\right)\right)\right)} \cdot \log y\right) - t \]
   remove-double-neg [=>] 99.8	\[ \mathsf{fma}\left(z - 1, \mathsf{log1p}\left(-y\right), \color{blue}{\left(x - 1\right)} \cdot \log y\right) - t \]
   sub-neg [=>] 99.8	\[ \mathsf{fma}\left(z - 1, \mathsf{log1p}\left(-y\right), \color{blue}{\left(x + \left(-1\right)\right)} \cdot \log y\right) - t \]
   metadata-eval [=>] 99.8	\[ \mathsf{fma}\left(z - 1, \mathsf{log1p}\left(-y\right), \left(x + \color{blue}{-1}\right) \cdot \log y\right) - t \]

3. Final simplification 99.8%

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

Alternatives

Alternative 1
Accuracy	99.8%
Cost	13696

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

Alternative 2
Accuracy	99.2%
Cost	13504

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

Alternative 3
Accuracy	94.8%
Cost	7496

\[\begin{array}{l} \mathbf{if}\;x + -1 \leq -5 \cdot 10^{+33}:\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{elif}\;x + -1 \leq -0.9999999995:\\ \;\;\;\;\left(\left(y - z \cdot y\right) - \log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot \left(x + -1\right) - t\\ \end{array} \]

Alternative 4
Accuracy	99.2%
Cost	7232

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

Alternative 5
Accuracy	84.5%
Cost	7112

\[\begin{array}{l} \mathbf{if}\;t \leq -1.4 \cdot 10^{+21}:\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{elif}\;t \leq -5.2 \cdot 10^{-81}:\\ \;\;\;\;y - \left(\log y + z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot \left(x + -1\right) - t\\ \end{array} \]

Alternative 6
Accuracy	99.1%
Cost	7104

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

Alternative 7
Accuracy	86.3%
Cost	6985

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 8.8 \cdot 10^{-13}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\left(-\log y\right) - t\\ \end{array} \]

Alternative 8
Accuracy	86.4%
Cost	6985

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 8.8 \cdot 10^{-13}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\left(y - \log y\right) - t\\ \end{array} \]

Alternative 9
Accuracy	60.0%
Cost	6921

\[\begin{array}{l} \mathbf{if}\;z \leq -1.55 \cdot 10^{+40} \lor \neg \left(z \leq 2.4 \cdot 10^{+85}\right):\\ \;\;\;\;z \cdot \left(-y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(-\log y\right) - t\\ \end{array} \]

Alternative 10
Accuracy	43.1%
Cost	520

\[\begin{array}{l} \mathbf{if}\;t \leq -3.3 \cdot 10^{+22}:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 1700000:\\ \;\;\;\;z \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]

Alternative 11
Accuracy	43.0%
Cost	520

\[\begin{array}{l} \mathbf{if}\;t \leq -3.3 \cdot 10^{+22}:\\ \;\;\;\;z \cdot y - t\\ \mathbf{elif}\;t \leq 50000:\\ \;\;\;\;z \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]

Alternative 12
Accuracy	46.6%
Cost	448

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

Alternative 13
Accuracy	46.4%
Cost	384

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

Alternative 14
Accuracy	35.7%
Cost	192

\[y - t \]

Alternative 15
Accuracy	35.4%
Cost	128

\[-t \]

Reproduce

herbie shell --seed 2023141 
(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))