Logistic distribution

?

Percentage Accurate: 99.5% → 99.6%
Time: 23.0s
Precision: binary32
Cost: 16448

?

\[0 \leq s \land s \leq 1.0651631\]
\[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
↓
\[\frac{1}{\left(1 + e^{\frac{\left|x\right|}{-s}}\right) \cdot \mathsf{fma}\left(s, e^{\frac{\left|x\right|}{s}}, s\right)} \]
(FPCore (x s)
 :precision binary32
 (/
  (exp (/ (- (fabs x)) s))
  (* (* s (+ 1.0 (exp (/ (- (fabs x)) s)))) (+ 1.0 (exp (/ (- (fabs x)) s))))))
↓
(FPCore (x s)
 :precision binary32
 (/ 1.0 (* (+ 1.0 (exp (/ (fabs x) (- s)))) (fma s (exp (/ (fabs x) s)) s))))
float code(float x, float s) {
	return expf((-fabsf(x) / s)) / ((s * (1.0f + expf((-fabsf(x) / s)))) * (1.0f + expf((-fabsf(x) / s))));
}
↓
float code(float x, float s) {
	return 1.0f / ((1.0f + expf((fabsf(x) / -s))) * fmaf(s, expf((fabsf(x) / s)), s));
}

function code(x, s)
	return Float32(exp(Float32(Float32(-abs(x)) / s)) / Float32(Float32(s * Float32(Float32(1.0) + exp(Float32(Float32(-abs(x)) / s)))) * Float32(Float32(1.0) + exp(Float32(Float32(-abs(x)) / s)))))
end
↓
function code(x, s)
	return Float32(Float32(1.0) / Float32(Float32(Float32(1.0) + exp(Float32(abs(x) / Float32(-s)))) * fma(s, exp(Float32(abs(x) / s)), s)))
end

\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}
↓
\frac{1}{\left(1 + e^{\frac{\left|x\right|}{-s}}\right) \cdot \mathsf{fma}\left(s, e^{\frac{\left|x\right|}{s}}, s\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 16 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 99.7%

    \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]

  2. Simplified99.7%

    \[\leadsto \color{blue}{\frac{1}{\left(e^{\frac{\left|x\right|}{-s}} + 1\right) \cdot \mathsf{fma}\left(s, e^{\frac{\left|x\right|}{s}}, s\right)}} \]
    Step-by-step derivation

    [Start]99.7%

    \[ \frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]

    *-lft-identity [<=]99.7%

    \[ \color{blue}{1 \cdot \frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

    associate-*r/ [=>]99.7%

    \[ \color{blue}{\frac{1 \cdot e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]

    associate-/l* [=>]99.7%

    \[ \color{blue}{\frac{1}{\frac{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{e^{\frac{-\left|x\right|}{s}}}}} \]

    distribute-frac-neg [=>]99.7%

    \[ \frac{1}{\frac{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{e^{\color{blue}{-\frac{\left|x\right|}{s}}}}} \]

    exp-neg [=>]99.7%

    \[ \frac{1}{\frac{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{\color{blue}{\frac{1}{e^{\frac{\left|x\right|}{s}}}}}} \]

    associate-/r/ [=>]99.7%

    \[ \frac{1}{\color{blue}{\frac{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{1} \cdot e^{\frac{\left|x\right|}{s}}}} \]

    /-rgt-identity [=>]99.7%

    \[ \frac{1}{\color{blue}{\left(\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right)} \cdot e^{\frac{\left|x\right|}{s}}} \]

    associate-*l* [=>]99.7%

    \[ \frac{1}{\color{blue}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot e^{\frac{\left|x\right|}{s}}\right)}} \]

  3. Final simplification99.7%

    \[\leadsto \frac{1}{\left(1 + e^{\frac{\left|x\right|}{-s}}\right) \cdot \mathsf{fma}\left(s, e^{\frac{\left|x\right|}{s}}, s\right)} \]

Alternatives

Alternative 1
Accuracy99.6%
Cost16448
\[\frac{1}{\left(1 + e^{\frac{\left|x\right|}{-s}}\right) \cdot \mathsf{fma}\left(s, e^{\frac{\left|x\right|}{s}}, s\right)} \]
Alternative 2
Accuracy99.1%
Cost13248
\[\frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(e^{\frac{\left|x\right|}{-s}} + 2\right)} \]
Alternative 3
Accuracy97.1%
Cost10080
\[\frac{1}{\left(1 + e^{\frac{x}{s}}\right) \cdot \left(s + s \cdot e^{\frac{\left|x\right|}{s}}\right)} \]
Alternative 4
Accuracy96.9%
Cost10016
\[\frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(2 + e^{\frac{x}{s}}\right)} \]
Alternative 5
Accuracy96.6%
Cost6688
\[\frac{1}{s \cdot \left(e^{\frac{\left|x\right|}{s}} + 3\right)} \]
Alternative 6
Accuracy95.1%
Cost3620
\[\begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \mathbf{if}\;x \leq -1.000000046701102 \cdot 10^{-34}:\\ \;\;\;\;\frac{t_0}{s + s \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(1 + t_0\right) \cdot \left(s \cdot 2\right)}\\ \end{array} \]
Alternative 7
Accuracy84.6%
Cost3556
\[\begin{array}{l} \mathbf{if}\;x \leq -1.000000046701102 \cdot 10^{-34}:\\ \;\;\;\;\frac{e^{\frac{x}{s}}}{s + s \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{s}{\frac{-1}{\left(-4\right) - \frac{x}{s} \cdot \left(2 + \frac{x}{s}\right)}}}\\ \end{array} \]
Alternative 8
Accuracy73.9%
Cost553
\[\begin{array}{l} \mathbf{if}\;x \leq -4.9999998413276127 \cdot 10^{-20} \lor \neg \left(x \leq 4.999999943633011 \cdot 10^{-27}\right):\\ \;\;\;\;\frac{1}{s \cdot \left(\frac{x}{s} \cdot \left(2 + \frac{x}{s}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]
Alternative 9
Accuracy73.5%
Cost480
\[\frac{\frac{1}{s}}{4 + \frac{x}{s} \cdot \left(2 + \frac{x}{s}\right)} \]
Alternative 10
Accuracy58.0%
Cost424
\[\begin{array}{l} \mathbf{if}\;x \leq -4.9999998413276127 \cdot 10^{-20}:\\ \;\;\;\;\frac{1}{\frac{x \cdot x}{s}}\\ \mathbf{elif}\;x \leq 2.50000003337858 \cdot 10^{-11}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(2 + \frac{x}{s}\right)}\\ \end{array} \]
Alternative 11
Accuracy53.6%
Cost420
\[\begin{array}{l} \mathbf{if}\;x \leq -4.999999999099794 \cdot 10^{-24}:\\ \;\;\;\;\frac{1}{x \cdot \frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(4 + 2 \cdot \frac{x}{s}\right)}\\ \end{array} \]
Alternative 12
Accuracy58.0%
Cost361
\[\begin{array}{l} \mathbf{if}\;x \leq -4.9999998413276127 \cdot 10^{-20} \lor \neg \left(x \leq 2.50000003337858 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{1}{x \cdot \frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]
Alternative 13
Accuracy58.0%
Cost360
\[\begin{array}{l} \mathbf{if}\;x \leq -4.9999998413276127 \cdot 10^{-20}:\\ \;\;\;\;\frac{1}{\frac{x \cdot x}{s}}\\ \mathbf{elif}\;x \leq 2.50000003337858 \cdot 10^{-11}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \frac{x}{s}}\\ \end{array} \]
Alternative 14
Accuracy56.9%
Cost297
\[\begin{array}{l} \mathbf{if}\;x \leq -4.9999998413276127 \cdot 10^{-20} \lor \neg \left(x \leq 2.50000003337858 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{s}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]
Alternative 15
Accuracy56.8%
Cost297
\[\begin{array}{l} \mathbf{if}\;x \leq -4.9999998413276127 \cdot 10^{-20} \lor \neg \left(x \leq 2.50000003337858 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{\frac{s}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]
Alternative 16
Accuracy15.2%
Cost96
\[\frac{0.25}{s} \]

Reproduce?

herbie shell --seed 2023256 
(FPCore (x s)
  :name "Logistic distribution"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ (exp (/ (- (fabs x)) s)) (* (* s (+ 1.0 (exp (/ (- (fabs x)) s)))) (+ 1.0 (exp (/ (- (fabs x)) s))))))