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

?

Percentage Accurate: 89.1% → 99.8%
Time: 18.4s
Precision: binary64
Cost: 19968

?

\[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
↓
\[\mathsf{fma}\left(z + -1, \mathsf{log1p}\left(-y\right), \left(x + -1\right) \cdot \log y - t\right) \]
(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)) (- (* (+ x -1.0) (log y)) t)))
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), (((x + -1.0) * log(y)) - t));
}

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)), Float64(Float64(Float64(x + -1.0) * log(y)) - t))
end

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[(N[(x + -1.0), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]

\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), \left(x + -1\right) \cdot \log y - t\right)

Local Percentage Accuracy vs ?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Herbie found 15 alternatives:

AlternativeAccuracySpeedup

Accuracy vs Speed

The accuracy (vertical axis) and speed (horizontal axis) of each alternatives. Up and to the right is better. The red square shows the initial program, and each blue circle shows an alternative.The line shows the best available speed-accuracy tradeoffs.

Bogosity?

Bogosity

Derivation?

  1. Initial program 88.4%

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

  2. Simplified99.8%

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

    [Start]88.4%

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

    +-commutative [=>]88.4%

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

    associate--l+ [=>]88.4%

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

    fma-def [=>]88.4%

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

    sub-neg [=>]88.4%

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

    log1p-def [=>]99.8%

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

  3. Final simplification99.8%

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

Alternatives

Alternative 1
Accuracy99.8%
Cost19968
\[\mathsf{fma}\left(z + -1, \mathsf{log1p}\left(-y\right), \left(x + -1\right) \cdot \log y - t\right) \]
Alternative 2
Accuracy99.8%
Cost19968
\[\mathsf{fma}\left(x + -1, \log y, \left(z + -1\right) \cdot \mathsf{log1p}\left(-y\right)\right) - t \]
Alternative 3
Accuracy99.5%
Cost7616
\[\left(\left(x + -1\right) \cdot \log y + \left(z + -1\right) \cdot \left(y \cdot \left(y \cdot -0.5\right) - y\right)\right) - t \]
Alternative 4
Accuracy88.1%
Cost7496
\[\begin{array}{l} t_1 := \left(x + -1\right) \cdot \log y\\ \mathbf{if}\;x + -1 \leq -1:\\ \;\;\;\;t_1 - t\\ \mathbf{elif}\;x + -1 \leq -0.999999999999:\\ \;\;\;\;\left(y \cdot \left(1 - z\right) - \log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(y + t_1\right) - t\\ \end{array} \]
Alternative 5
Accuracy95.7%
Cost7369
\[\begin{array}{l} t_1 := \left(x + -1\right) \cdot \log y\\ \mathbf{if}\;t \leq -8 \cdot 10^{-6} \lor \neg \left(t \leq 620\right):\\ \;\;\;\;t_1 - t\\ \mathbf{else}:\\ \;\;\;\;t_1 + y \cdot \left(1 - z\right)\\ \end{array} \]
Alternative 6
Accuracy99.2%
Cost7232
\[\left(\left(x + -1\right) \cdot \log y + y \cdot \left(1 - z\right)\right) - t \]
Alternative 7
Accuracy87.3%
Cost6985
\[\begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.2\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\left(-\log y\right) - t\\ \end{array} \]
Alternative 8
Accuracy88.1%
Cost6976
\[\left(y + \left(x + -1\right) \cdot \log y\right) - t \]
Alternative 9
Accuracy60.6%
Cost6921
\[\begin{array}{l} \mathbf{if}\;z \leq -1.7 \cdot 10^{+149} \lor \neg \left(z \leq 1.25 \cdot 10^{+79}\right):\\ \;\;\;\;z \cdot \left(-y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(-\log y\right) - t\\ \end{array} \]
Alternative 10
Accuracy88.0%
Cost6848
\[\left(x + -1\right) \cdot \log y - t \]
Alternative 11
Accuracy42.6%
Cost584
\[\begin{array}{l} \mathbf{if}\;t \leq -3.4 \cdot 10^{-24}:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 65000000000:\\ \;\;\;\;y \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]
Alternative 12
Accuracy42.3%
Cost520
\[\begin{array}{l} \mathbf{if}\;t \leq -3.1 \cdot 10^{-24}:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 255000000000:\\ \;\;\;\;z \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]
Alternative 13
Accuracy45.9%
Cost448
\[\left(y - z \cdot y\right) - t \]
Alternative 14
Accuracy45.7%
Cost384
\[z \cdot \left(-y\right) - t \]
Alternative 15
Accuracy35.2%
Cost128
\[-t \]

Reproduce?

herbie shell --seed 2023165 
(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))