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

Average Accuracy: 89.3% → 99.8%

Time: 21.1s

Precision: binary64

Cost: 19968

Math

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

↓

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

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

Derivation

1. Initial program 91.4%

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

2. Simplified 99.8%

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

   [Start] 91.4	\[ \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
   cancel-sign-sub [<=] 91.4	\[ \color{blue}{\left(\left(x - 1\right) \cdot \log y - \left(-\left(z - 1\right)\right) \cdot \log \left(1 - y\right)\right)} - t \]
   distribute-lft-neg-in [<=] 91.4	\[ \left(\left(x - 1\right) \cdot \log y - \color{blue}{\left(-\left(z - 1\right) \cdot \log \left(1 - y\right)\right)}\right) - t \]
   fma-neg [=>] 91.4	\[ \color{blue}{\mathsf{fma}\left(x - 1, \log y, -\left(-\left(z - 1\right) \cdot \log \left(1 - y\right)\right)\right)} - t \]
   remove-double-neg [=>] 91.4	\[ \mathsf{fma}\left(x - 1, \log y, \color{blue}{\left(z - 1\right) \cdot \log \left(1 - y\right)}\right) - t \]
   sub-neg [=>] 91.4	\[ \mathsf{fma}\left(x - 1, \log y, \left(z - 1\right) \cdot \log \color{blue}{\left(1 + \left(-y\right)\right)}\right) - t \]
   log1p-def [=>] 99.8	\[ \mathsf{fma}\left(x - 1, \log y, \left(z - 1\right) \cdot \color{blue}{\mathsf{log1p}\left(-y\right)}\right) - t \]

3. Final simplification 99.8%

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

Alternatives

Alternative 1
Accuracy	99.2%
Cost	13632

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

Alternative 2
Accuracy	88.3%
Cost	7497

\[\begin{array}{l} \mathbf{if}\;x + -1 \leq 1.2 \lor \neg \left(x + -1 \leq 1\right):\\ \;\;\;\;\left(x + -1\right) \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \left(1 - z\right) - \log y\right) - t\\ \end{array} \]

Alternative 3
Accuracy	80.6%
Cost	7497

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

Alternative 4
Accuracy	99.1%
Cost	7104

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

Alternative 5
Accuracy	70.1%
Cost	6985

\[\begin{array}{l} \mathbf{if}\;x \leq 1 \lor \neg \left(x \leq 2.3 \cdot 10^{-13}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\left(-\log y\right) - t\\ \end{array} \]

Alternative 6
Accuracy	44.9%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;z \leq 4.8 \cdot 10^{+218}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(x + -1\right) \cdot \log y - t\\ \end{array} \]

Alternative 7
Accuracy	36.9%
Cost	6921

\[\begin{array}{l} \mathbf{if}\;x \leq 1.2 \cdot 10^{+47} \lor \neg \left(x \leq 2.7 \cdot 10^{+45}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\left(-\log y\right) - t\\ \end{array} \]

Alternative 8
Accuracy	36.9%
Cost	6857

\[\begin{array}{l} \mathbf{if}\;x \leq 2.9 \cdot 10^{+46} \lor \neg \left(x \leq 5.3 \cdot 10^{+40}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot \left(-1 + y \cdot -0.5\right)\right) - t\\ \end{array} \]

Alternative 9
Accuracy	45.9%
Cost	704

\[z \cdot \left(y \cdot \left(-1 + y \cdot -0.5\right)\right) - t \]

Alternative 10
Accuracy	35.8%
Cost	520

\[\begin{array}{l} \mathbf{if}\;t \leq 3.4 \cdot 10^{-28}:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 235000:\\ \;\;\;\;-y \cdot z\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]

Alternative 11
Accuracy	45.9%
Cost	448

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

Alternative 12
Accuracy	45.7%
Cost	384

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

Alternative 13
Accuracy	35.4%
Cost	128

\[-t \]

Reproduce

herbie shell --seed 2023159 
(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))