Logistic distribution

?

Percentage Accurate: 99.5% → 99.6%
Time: 18.1s
Precision: binary32
Cost: 13248

?

\[0 \leq s \land s \leq 1.0651631\]
\[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
↓
\[\frac{1}{s \cdot \left(e^{\frac{-\left|x\right|}{s}} + \left(2 + e^{\frac{\left|x\right|}{s}}\right)\right)} \]
(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 (* s (+ (exp (/ (- (fabs x)) s)) (+ 2.0 (exp (/ (fabs x) 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 / (s * (expf((-fabsf(x) / s)) + (2.0f + expf((fabsf(x) / s)))));
}

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = exp((-abs(x) / s)) / ((s * (1.0e0 + exp((-abs(x) / s)))) * (1.0e0 + exp((-abs(x) / s))))
end function
↓
real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 1.0e0 / (s * (exp((-abs(x) / s)) + (2.0e0 + exp((abs(x) / s)))))
end function

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(s * Float32(exp(Float32(Float32(-abs(x)) / s)) + Float32(Float32(2.0) + exp(Float32(abs(x) / s))))))
end

function tmp = code(x, s)
	tmp = exp((-abs(x) / s)) / ((s * (single(1.0) + exp((-abs(x) / s)))) * (single(1.0) + exp((-abs(x) / s))));
end
↓
function tmp = code(x, s)
	tmp = single(1.0) / (s * (exp((-abs(x) / s)) + (single(2.0) + exp((abs(x) / 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}{s \cdot \left(e^{\frac{-\left|x\right|}{s}} + \left(2 + e^{\frac{\left|x\right|}{s}}\right)\right)}

Local Percentage Accuracy?

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.

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Bogosity

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation?

  1. Initial program 99.2%

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

  2. Simplified98.9%

    \[\leadsto \color{blue}{\frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(e^{\frac{\left|x\right|}{-s}} + 2\right)}} \]
    Step-by-step derivation

    [Start]99.2

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

    *-lft-identity [<=]99.2

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

    associate-*r/ [=>]99.2

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

    associate-*l* [=>]99.2

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

    times-frac [=>]98.8

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

    associate-*r/ [=>]98.9

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

    associate-/l* [=>]98.8

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

    distribute-frac-neg [=>]98.8

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

    exp-neg [=>]98.9

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

  3. Taylor expanded in s around 0 99.3%

    \[\leadsto \color{blue}{\frac{1}{s \cdot \left(e^{-1 \cdot \frac{\left|x\right|}{s}} + \left(2 + e^{\frac{\left|x\right|}{s}}\right)\right)}} \]

  4. Simplified99.3%

    \[\leadsto \color{blue}{\frac{1}{s \cdot \left(e^{\frac{-\left|x\right|}{s}} + \left(2 + e^{\frac{\left|x\right|}{s}}\right)\right)}} \]
    Step-by-step derivation

    [Start]99.3

    \[ \frac{1}{s \cdot \left(e^{-1 \cdot \frac{\left|x\right|}{s}} + \left(2 + e^{\frac{\left|x\right|}{s}}\right)\right)} \]

    associate-*r/ [=>]99.3

    \[ \frac{1}{s \cdot \left(e^{\color{blue}{\frac{-1 \cdot \left|x\right|}{s}}} + \left(2 + e^{\frac{\left|x\right|}{s}}\right)\right)} \]

    mul-1-neg [=>]99.3

    \[ \frac{1}{s \cdot \left(e^{\frac{\color{blue}{-\left|x\right|}}{s}} + \left(2 + e^{\frac{\left|x\right|}{s}}\right)\right)} \]

  5. Final simplification99.3%

    \[\leadsto \frac{1}{s \cdot \left(e^{\frac{-\left|x\right|}{s}} + \left(2 + e^{\frac{\left|x\right|}{s}}\right)\right)} \]

Alternatives

Alternative 1
Accuracy97.5%
Cost10016
\[\frac{1}{s \cdot \left(\left(2 + e^{\frac{\left|x\right|}{s}}\right) + e^{\frac{x}{s}}\right)} \]
Alternative 2
Accuracy96.9%
Cost6752
\[\frac{1}{x + s \cdot \left(e^{\frac{\left|x\right|}{s}} + 3\right)} \]
Alternative 3
Accuracy96.4%
Cost6688
\[\frac{1}{s \cdot \left(e^{\frac{\left|x\right|}{s}} + 3\right)} \]
Alternative 4
Accuracy94.9%
Cost6656
\[\frac{e^{\frac{-\left|x\right|}{s}}}{s \cdot 4} \]
Alternative 5
Accuracy81.2%
Cost3812
\[\begin{array}{l} \mathbf{if}\;\left|x\right| \leq 1.2000000234497777 \cdot 10^{-24}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(3 + \left(1 + 0.5 \cdot \frac{x \cdot x}{s \cdot s}\right)\right)}\\ \end{array} \]
Alternative 6
Accuracy94.4%
Cost3620
\[\begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \mathbf{if}\;x \leq -4.999999943633011 \cdot 10^{-27}:\\ \;\;\;\;t_0 \cdot \frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_0 + 3} \cdot \frac{1}{s}\\ \end{array} \]
Alternative 7
Accuracy95.5%
Cost3620
\[\begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \mathbf{if}\;x \leq -4.999999943633011 \cdot 10^{-27}:\\ \;\;\;\;t_0 \cdot \frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(t_0 + 3\right) - x}\\ \end{array} \]
Alternative 8
Accuracy94.4%
Cost3556
\[\begin{array}{l} \mathbf{if}\;x \leq -4.999999943633011 \cdot 10^{-27}:\\ \;\;\;\;e^{\frac{x}{s}} \cdot \frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \mathsf{expm1}\left(\frac{x}{s}\right)}\\ \end{array} \]
Alternative 9
Accuracy94.9%
Cost3524
\[\begin{array}{l} \mathbf{if}\;x \leq 5.0000000900125474 \cdot 10^{-36}:\\ \;\;\;\;e^{\frac{x}{s}} \cdot \frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{-x}{s}}}{s \cdot 4}\\ \end{array} \]
Alternative 10
Accuracy90.5%
Cost3492
\[\begin{array}{l} \mathbf{if}\;x \leq 9.999999682655225 \cdot 10^{-21}:\\ \;\;\;\;e^{\frac{x}{s}} \cdot \frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{-x}{s}}\\ \end{array} \]
Alternative 11
Accuracy94.9%
Cost3492
\[\begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \mathbf{if}\;x \leq 5.0000000900125474 \cdot 10^{-36}:\\ \;\;\;\;t_0 \cdot \frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.25}{s}}{t_0}\\ \end{array} \]
Alternative 12
Accuracy94.9%
Cost3492
\[\begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \mathbf{if}\;x \leq 5.0000000900125474 \cdot 10^{-36}:\\ \;\;\;\;t_0 \cdot \frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.25}{t_0}}{s}\\ \end{array} \]
Alternative 13
Accuracy87.4%
Cost3464
\[\begin{array}{l} \mathbf{if}\;x \leq -4.999999918875795 \cdot 10^{-18}:\\ \;\;\;\;e^{\frac{x}{s}}\\ \mathbf{elif}\;x \leq 9.999999682655225 \cdot 10^{-21}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{-x}{s}}\\ \end{array} \]
Alternative 14
Accuracy66.5%
Cost488
\[\begin{array}{l} \mathbf{if}\;x \leq -200000:\\ \;\;\;\;-1 + \left(1 + \frac{-1}{x}\right)\\ \mathbf{elif}\;x \leq 1000000:\\ \;\;\;\;\frac{1}{s \cdot \left(3 + \left(1 - \frac{x}{s}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(-1 + \frac{-1}{x}\right)\\ \end{array} \]
Alternative 15
Accuracy66.6%
Cost361
\[\begin{array}{l} \mathbf{if}\;x \leq -0.0020000000949949026 \lor \neg \left(x \leq 1.0000000116860974 \cdot 10^{-7}\right):\\ \;\;\;\;1 + \left(-1 + \frac{-1}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]
Alternative 16
Accuracy66.8%
Cost360
\[\begin{array}{l} \mathbf{if}\;x \leq -0.0020000000949949026:\\ \;\;\;\;-1 + \left(1 + \frac{-1}{x}\right)\\ \mathbf{elif}\;x \leq 1.0000000116860974 \cdot 10^{-7}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(-1 + \frac{-1}{x}\right)\\ \end{array} \]
Alternative 17
Accuracy8.6%
Cost96
\[\frac{-1}{x} \]
Alternative 18
Accuracy27.5%
Cost96
\[\frac{0.25}{s} \]
Alternative 19
Accuracy8.3%
Cost32
\[1 \]

Error

Reproduce?

herbie shell --seed 2023160 
(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))))))