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

?

Percentage Accurate: 95.2% → 97.7%
Time: 18.1s
Precision: binary64
Cost: 20160

?

\[\left(\left(x - \left(y - 1\right) \cdot z\right) - \left(t - 1\right) \cdot a\right) + \left(\left(y + t\right) - 2\right) \cdot b \]
↓
\[\mathsf{fma}\left(y + \left(t + -2\right), b, \mathsf{fma}\left(a, 1 - t, \mathsf{fma}\left(z, 1 - y, x\right)\right)\right) \]
(FPCore (x y z t a b)
 :precision binary64
 (+ (- (- x (* (- y 1.0) z)) (* (- t 1.0) a)) (* (- (+ y t) 2.0) b)))
↓
(FPCore (x y z t a b)
 :precision binary64
 (fma (+ y (+ t -2.0)) b (fma a (- 1.0 t) (fma z (- 1.0 y) x))))
double code(double x, double y, double z, double t, double a, double b) {
	return ((x - ((y - 1.0) * z)) - ((t - 1.0) * a)) + (((y + t) - 2.0) * b);
}
↓
double code(double x, double y, double z, double t, double a, double b) {
	return fma((y + (t + -2.0)), b, fma(a, (1.0 - t), fma(z, (1.0 - y), x)));
}

function code(x, y, z, t, a, b)
	return Float64(Float64(Float64(x - Float64(Float64(y - 1.0) * z)) - Float64(Float64(t - 1.0) * a)) + Float64(Float64(Float64(y + t) - 2.0) * b))
end
↓
function code(x, y, z, t, a, b)
	return fma(Float64(y + Float64(t + -2.0)), b, fma(a, Float64(1.0 - t), fma(z, Float64(1.0 - y), x)))
end

code[x_, y_, z_, t_, a_, b_] := N[(N[(N[(x - N[(N[(y - 1.0), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] - N[(N[(t - 1.0), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(y + t), $MachinePrecision] - 2.0), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]
↓
code[x_, y_, z_, t_, a_, b_] := N[(N[(y + N[(t + -2.0), $MachinePrecision]), $MachinePrecision] * b + N[(a * N[(1.0 - t), $MachinePrecision] + N[(z * N[(1.0 - y), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\left(\left(x - \left(y - 1\right) \cdot z\right) - \left(t - 1\right) \cdot a\right) + \left(\left(y + t\right) - 2\right) \cdot b
↓
\mathsf{fma}\left(y + \left(t + -2\right), b, \mathsf{fma}\left(a, 1 - t, \mathsf{fma}\left(z, 1 - y, x\right)\right)\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 21 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 97.3%

    \[\left(\left(x - \left(y - 1\right) \cdot z\right) - \left(t - 1\right) \cdot a\right) + \left(\left(y + t\right) - 2\right) \cdot b \]

  2. Simplified98.8%

    \[\leadsto \color{blue}{\mathsf{fma}\left(y + \left(t + -2\right), b, \mathsf{fma}\left(a, 1 - t, \mathsf{fma}\left(z, 1 - y, x\right)\right)\right)} \]
    Step-by-step derivation

    [Start]97.3%

    \[ \left(\left(x - \left(y - 1\right) \cdot z\right) - \left(t - 1\right) \cdot a\right) + \left(\left(y + t\right) - 2\right) \cdot b \]

    +-commutative [=>]97.3%

    \[ \color{blue}{\left(\left(y + t\right) - 2\right) \cdot b + \left(\left(x - \left(y - 1\right) \cdot z\right) - \left(t - 1\right) \cdot a\right)} \]

    fma-def [=>]98.0%

    \[ \color{blue}{\mathsf{fma}\left(\left(y + t\right) - 2, b, \left(x - \left(y - 1\right) \cdot z\right) - \left(t - 1\right) \cdot a\right)} \]

    sub-neg [=>]98.0%

    \[ \mathsf{fma}\left(\color{blue}{\left(y + t\right) + \left(-2\right)}, b, \left(x - \left(y - 1\right) \cdot z\right) - \left(t - 1\right) \cdot a\right) \]

    associate-+l+ [=>]98.0%

    \[ \mathsf{fma}\left(\color{blue}{y + \left(t + \left(-2\right)\right)}, b, \left(x - \left(y - 1\right) \cdot z\right) - \left(t - 1\right) \cdot a\right) \]

    metadata-eval [=>]98.0%

    \[ \mathsf{fma}\left(y + \left(t + \color{blue}{-2}\right), b, \left(x - \left(y - 1\right) \cdot z\right) - \left(t - 1\right) \cdot a\right) \]

    sub-neg [=>]98.0%

    \[ \mathsf{fma}\left(y + \left(t + -2\right), b, \color{blue}{\left(x - \left(y - 1\right) \cdot z\right) + \left(-\left(t - 1\right) \cdot a\right)}\right) \]

    +-commutative [=>]98.0%

    \[ \mathsf{fma}\left(y + \left(t + -2\right), b, \color{blue}{\left(-\left(t - 1\right) \cdot a\right) + \left(x - \left(y - 1\right) \cdot z\right)}\right) \]

    *-commutative [=>]98.0%

    \[ \mathsf{fma}\left(y + \left(t + -2\right), b, \left(-\color{blue}{a \cdot \left(t - 1\right)}\right) + \left(x - \left(y - 1\right) \cdot z\right)\right) \]

    distribute-rgt-neg-in [=>]98.0%

    \[ \mathsf{fma}\left(y + \left(t + -2\right), b, \color{blue}{a \cdot \left(-\left(t - 1\right)\right)} + \left(x - \left(y - 1\right) \cdot z\right)\right) \]

    fma-def [=>]98.8%

    \[ \mathsf{fma}\left(y + \left(t + -2\right), b, \color{blue}{\mathsf{fma}\left(a, -\left(t - 1\right), x - \left(y - 1\right) \cdot z\right)}\right) \]

    neg-sub0 [=>]98.8%

    \[ \mathsf{fma}\left(y + \left(t + -2\right), b, \mathsf{fma}\left(a, \color{blue}{0 - \left(t - 1\right)}, x - \left(y - 1\right) \cdot z\right)\right) \]

    associate--r- [=>]98.8%

    \[ \mathsf{fma}\left(y + \left(t + -2\right), b, \mathsf{fma}\left(a, \color{blue}{\left(0 - t\right) + 1}, x - \left(y - 1\right) \cdot z\right)\right) \]

    neg-sub0 [<=]98.8%

    \[ \mathsf{fma}\left(y + \left(t + -2\right), b, \mathsf{fma}\left(a, \color{blue}{\left(-t\right)} + 1, x - \left(y - 1\right) \cdot z\right)\right) \]

    +-commutative [=>]98.8%

    \[ \mathsf{fma}\left(y + \left(t + -2\right), b, \mathsf{fma}\left(a, \color{blue}{1 + \left(-t\right)}, x - \left(y - 1\right) \cdot z\right)\right) \]

    sub-neg [<=]98.8%

    \[ \mathsf{fma}\left(y + \left(t + -2\right), b, \mathsf{fma}\left(a, \color{blue}{1 - t}, x - \left(y - 1\right) \cdot z\right)\right) \]

    sub-neg [=>]98.8%

    \[ \mathsf{fma}\left(y + \left(t + -2\right), b, \mathsf{fma}\left(a, 1 - t, \color{blue}{x + \left(-\left(y - 1\right) \cdot z\right)}\right)\right) \]

    +-commutative [=>]98.8%

    \[ \mathsf{fma}\left(y + \left(t + -2\right), b, \mathsf{fma}\left(a, 1 - t, \color{blue}{\left(-\left(y - 1\right) \cdot z\right) + x}\right)\right) \]

    *-commutative [=>]98.8%

    \[ \mathsf{fma}\left(y + \left(t + -2\right), b, \mathsf{fma}\left(a, 1 - t, \left(-\color{blue}{z \cdot \left(y - 1\right)}\right) + x\right)\right) \]

    distribute-rgt-neg-in [=>]98.8%

    \[ \mathsf{fma}\left(y + \left(t + -2\right), b, \mathsf{fma}\left(a, 1 - t, \color{blue}{z \cdot \left(-\left(y - 1\right)\right)} + x\right)\right) \]

    fma-def [=>]98.8%

    \[ \mathsf{fma}\left(y + \left(t + -2\right), b, \mathsf{fma}\left(a, 1 - t, \color{blue}{\mathsf{fma}\left(z, -\left(y - 1\right), x\right)}\right)\right) \]

  3. Final simplification98.8%

    \[\leadsto \mathsf{fma}\left(y + \left(t + -2\right), b, \mathsf{fma}\left(a, 1 - t, \mathsf{fma}\left(z, 1 - y, x\right)\right)\right) \]

Alternatives

Alternative 1
Accuracy97.7%
Cost20160
\[\mathsf{fma}\left(y + \left(t + -2\right), b, \mathsf{fma}\left(a, 1 - t, \mathsf{fma}\left(z, 1 - y, x\right)\right)\right) \]
Alternative 2
Accuracy97.6%
Cost13888
\[\mathsf{fma}\left(t + \left(y - 2\right), b, x - \mathsf{fma}\left(y + -1, z, a \cdot \left(t + -1\right)\right)\right) \]
Alternative 3
Accuracy97.9%
Cost2756
\[\begin{array}{l} t_1 := \left(\left(x + z \cdot \left(1 - y\right)\right) + a \cdot \left(1 - t\right)\right) + b \cdot \left(\left(y + t\right) - 2\right)\\ \mathbf{if}\;t_1 \leq \infty:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\left(z + x\right) + t \cdot \left(b - a\right)\\ \end{array} \]
Alternative 4
Accuracy78.6%
Cost1492
\[\begin{array}{l} t_1 := z \cdot \left(1 - y\right)\\ t_2 := x + t_1\\ t_3 := t_2 + y \cdot b\\ t_4 := t_2 + t \cdot \left(b - a\right)\\ \mathbf{if}\;t \leq -3800000000:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t \leq -3.3 \cdot 10^{-257}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq -2.2 \cdot 10^{-294}:\\ \;\;\;\;\left(a + x\right) + b \cdot \left(y - 2\right)\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{-162}:\\ \;\;\;\;\left(a + x\right) + t_1\\ \mathbf{elif}\;t \leq 2.7 \cdot 10^{+17}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_4\\ \end{array} \]
Alternative 5
Accuracy81.9%
Cost1360
\[\begin{array}{l} t_1 := x + z \cdot \left(1 - y\right)\\ t_2 := t_1 + \left(a - t \cdot a\right)\\ t_3 := b \cdot \left(\left(y + t\right) - 2\right)\\ \mathbf{if}\;b \leq -2.9 \cdot 10^{+93}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;b \leq 2.55 \cdot 10^{-133}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq 2.95 \cdot 10^{-15}:\\ \;\;\;\;t_1 + t \cdot \left(b - a\right)\\ \mathbf{elif}\;b \leq 2.4 \cdot 10^{+85}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]
Alternative 6
Accuracy59.5%
Cost1240
\[\begin{array}{l} t_1 := x + t \cdot \left(b - a\right)\\ t_2 := y \cdot \left(b - z\right)\\ t_3 := z + \left(a + x\right)\\ \mathbf{if}\;y \leq -2.6 \cdot 10^{+17}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -6.5 \cdot 10^{-56}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -7.5 \cdot 10^{-223}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{-308}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{-194}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y \leq 2.35 \cdot 10^{+101}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
Alternative 7
Accuracy93.5%
Cost1225
\[\begin{array}{l} t_1 := x + z \cdot \left(1 - y\right)\\ \mathbf{if}\;t \leq -190000 \lor \neg \left(t \leq 9.5 \cdot 10^{+17}\right):\\ \;\;\;\;t_1 + t \cdot \left(b - a\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 + \left(a + b \cdot \left(y - 2\right)\right)\\ \end{array} \]
Alternative 8
Accuracy32.1%
Cost1180
\[\begin{array}{l} t_1 := t \cdot \left(-a\right)\\ \mathbf{if}\;y \leq -8.8 \cdot 10^{+16}:\\ \;\;\;\;y \cdot b\\ \mathbf{elif}\;y \leq -4.4 \cdot 10^{-30}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{-193}:\\ \;\;\;\;z + x\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{-118}:\\ \;\;\;\;t \cdot b\\ \mathbf{elif}\;y \leq 9 \cdot 10^{-67}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 11000000:\\ \;\;\;\;a\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{+103}:\\ \;\;\;\;t \cdot b\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \end{array} \]
Alternative 9
Accuracy37.4%
Cost1112
\[\begin{array}{l} t_1 := t \cdot \left(b - a\right)\\ \mathbf{if}\;y \leq -3.9 \cdot 10^{+68}:\\ \;\;\;\;y \cdot b\\ \mathbf{elif}\;y \leq -4.4 \cdot 10^{-59}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -9 \cdot 10^{-259}:\\ \;\;\;\;z + x\\ \mathbf{elif}\;y \leq -9 \cdot 10^{-298}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.02 \cdot 10^{-194}:\\ \;\;\;\;z + x\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{+109}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \end{array} \]
Alternative 10
Accuracy50.4%
Cost1112
\[\begin{array}{l} t_1 := t \cdot \left(b - a\right)\\ t_2 := y \cdot \left(b - z\right)\\ \mathbf{if}\;y \leq -1.3 \cdot 10^{+17}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -2.8 \cdot 10^{-57}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -3.6 \cdot 10^{-257}:\\ \;\;\;\;z + x\\ \mathbf{elif}\;y \leq -3.5 \cdot 10^{-290}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{-194}:\\ \;\;\;\;z + x\\ \mathbf{elif}\;y \leq 3.05 \cdot 10^{+93}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
Alternative 11
Accuracy68.4%
Cost1104
\[\begin{array}{l} t_1 := \left(a + x\right) + z \cdot \left(1 - y\right)\\ t_2 := b \cdot \left(\left(y + t\right) - 2\right)\\ \mathbf{if}\;b \leq -2.2 \cdot 10^{+77}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq 2.7 \cdot 10^{-64}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 12:\\ \;\;\;\;z + t \cdot \left(b - a\right)\\ \mathbf{elif}\;b \leq 6.5 \cdot 10^{+68}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
Alternative 12
Accuracy71.5%
Cost969
\[\begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{+21} \lor \neg \left(y \leq 3.6 \cdot 10^{+92}\right):\\ \;\;\;\;\left(x + z \cdot \left(1 - y\right)\right) + y \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(z + x\right) + t \cdot \left(b - a\right)\\ \end{array} \]
Alternative 13
Accuracy54.2%
Cost848
\[\begin{array}{l} t_1 := t \cdot \left(b - a\right)\\ t_2 := y \cdot \left(b - z\right)\\ \mathbf{if}\;y \leq -1.25 \cdot 10^{+19}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -1.2 \cdot 10^{-51}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 8.8 \cdot 10^{-194}:\\ \;\;\;\;z + \left(a + x\right)\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{+97}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
Alternative 14
Accuracy70.3%
Cost841
\[\begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{+17} \lor \neg \left(y \leq 8.7 \cdot 10^{+92}\right):\\ \;\;\;\;y \cdot \left(b - z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z + x\right) + t \cdot \left(b - a\right)\\ \end{array} \]
Alternative 15
Accuracy23.9%
Cost720
\[\begin{array}{l} \mathbf{if}\;t \leq -3.3 \cdot 10^{+127}:\\ \;\;\;\;t \cdot b\\ \mathbf{elif}\;t \leq 8.2 \cdot 10^{-202}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{-163}:\\ \;\;\;\;a\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{+88}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;t \cdot b\\ \end{array} \]
Alternative 16
Accuracy32.7%
Cost652
\[\begin{array}{l} \mathbf{if}\;y \leq -4.6 \cdot 10^{+33}:\\ \;\;\;\;y \cdot b\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{-193}:\\ \;\;\;\;z + x\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{+103}:\\ \;\;\;\;t \cdot b\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \end{array} \]
Alternative 17
Accuracy25.8%
Cost588
\[\begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-61}:\\ \;\;\;\;y \cdot b\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-194}:\\ \;\;\;\;z\\ \mathbf{elif}\;y \leq 2.15 \cdot 10^{+94}:\\ \;\;\;\;t \cdot b\\ \mathbf{else}:\\ \;\;\;\;y \cdot b\\ \end{array} \]
Alternative 18
Accuracy32.4%
Cost588
\[\begin{array}{l} \mathbf{if}\;y \leq -2.15 \cdot 10^{+33}:\\ \;\;\;\;y \cdot b\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{-193}:\\ \;\;\;\;z + x\\ \mathbf{elif}\;y \leq 2.25 \cdot 10^{+95}:\\ \;\;\;\;t \cdot b\\ \mathbf{else}:\\ \;\;\;\;y \cdot b\\ \end{array} \]
Alternative 19
Accuracy20.2%
Cost460
\[\begin{array}{l} \mathbf{if}\;z \leq -3.6 \cdot 10^{+16}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq -2.8 \cdot 10^{-172}:\\ \;\;\;\;a\\ \mathbf{elif}\;z \leq 5 \cdot 10^{+90}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]
Alternative 20
Accuracy19.4%
Cost328
\[\begin{array}{l} \mathbf{if}\;x \leq -3.75 \cdot 10^{+149}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-130}:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
Alternative 21
Accuracy11.0%
Cost64
\[a \]

Reproduce?

herbie shell --seed 2023263 
(FPCore (x y z t a b)
  :name "Statistics.Distribution.Beta:$centropy from math-functions-0.1.5.2"
  :precision binary64
  (+ (- (- x (* (- y 1.0) z)) (* (- t 1.0) a)) (* (- (+ y t) 2.0) b)))