?

Average Error: 99.5% → 99.6%
Time: 29.4s
Precision: binary32
Cost: 13248.00

?

\[0 \leq s \land s \leq 1.0651631\]
\[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
↓
\[\begin{array}{l} t_0 := \frac{\left|x\right|}{s}\\ \frac{1}{s \cdot \left(e^{-t_0} + \left(2 + e^{t_0}\right)\right)} \end{array} \]
(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
 (let* ((t_0 (/ (fabs x) s)))
   (/ 1.0 (* s (+ (exp (- t_0)) (+ 2.0 (exp t_0)))))))
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) {
	float t_0 = fabsf(x) / s;
	return 1.0f / (s * (expf(-t_0) + (2.0f + expf(t_0))));
}

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
    real(4) :: t_0
    t_0 = abs(x) / s
    code = 1.0e0 / (s * (exp(-t_0) + (2.0e0 + exp(t_0))))
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)
	t_0 = Float32(abs(x) / s)
	return Float32(Float32(1.0) / Float32(s * Float32(exp(Float32(-t_0)) + Float32(Float32(2.0) + exp(t_0)))))
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)
	t_0 = abs(x) / s;
	tmp = single(1.0) / (s * (exp(-t_0) + (single(2.0) + exp(t_0))));
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)}
↓
\begin{array}{l}
t_0 := \frac{\left|x\right|}{s}\\
\frac{1}{s \cdot \left(e^{-t_0} + \left(2 + e^{t_0}\right)\right)}
\end{array}

Error?

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation?

  1. Initial program 99.5

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

  2. Simplified99.4

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

    [Start]99.5

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

    associate-*l* [=>]99.5

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

    associate-/r* [=>]99.5

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

    distribute-frac-neg [=>]99.5

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

    exp-neg [=>]99.5

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

    associate-/l/ [=>]99.5

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

    associate-/r* [=>]99.4

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

    associate-/r* [<=]99.4

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

  3. Taylor expanded in s around 0 99.6

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

  4. Final simplification99.6

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

Alternatives

Alternative 1
Error99.4%
Cost13248.00
\[\frac{\frac{1}{s}}{\left(2 + e^{\frac{\left|x\right|}{s}}\right) + e^{\frac{\left|x\right|}{-s}}} \]
Alternative 2
Error99.5%
Cost6944.00
\[\begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \frac{1}{\frac{1 + t_0}{\frac{1}{s + \frac{s}{t_0}}}} \end{array} \]
Alternative 3
Error99.5%
Cost6912.00
\[\frac{1}{\left(1 + e^{\frac{-x}{s}}\right) \cdot \left(s \cdot \left(1 + e^{\frac{x}{s}}\right)\right)} \]
Alternative 4
Error99.5%
Cost6912.00
\[\frac{1}{\left(1 + e^{\frac{x}{s}}\right) \cdot \left(s \cdot \left(1 + e^{\frac{-x}{s}}\right)\right)} \]
Alternative 5
Error99.5%
Cost6880.00
\[\begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \frac{1}{\left(1 + t_0\right) \cdot \left(s + \frac{s}{t_0}\right)} \end{array} \]
Alternative 6
Error96.2%
Cost6688.00
\[\frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + 3} \]
Alternative 7
Error94.7%
Cost6656.00
\[\frac{e^{-\frac{\left|x\right|}{s}}}{s \cdot 4} \]
Alternative 8
Error95.1%
Cost3620.00
\[\begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \mathbf{if}\;x \leq -5.000000015855384 \cdot 10^{-30}:\\ \;\;\;\;\frac{1}{2 \cdot \left(s + \frac{s}{t_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{t_0 + 3}\\ \end{array} \]
Alternative 9
Error80.3%
Cost3556.00
\[\begin{array}{l} \mathbf{if}\;x \leq -5.000000015855384 \cdot 10^{-30}:\\ \;\;\;\;e^{\frac{x}{s}} \cdot \frac{1}{s \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot 4 + x \cdot \frac{x}{s}}\\ \end{array} \]
Alternative 10
Error95.0%
Cost3556.00
\[\begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \mathbf{if}\;x \leq -5.000000015855384 \cdot 10^{-30}:\\ \;\;\;\;t_0 \cdot \frac{1}{s \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(t_0 + 3\right)}\\ \end{array} \]
Alternative 11
Error95.0%
Cost3556.00
\[\begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \mathbf{if}\;x \leq -5.000000015855384 \cdot 10^{-30}:\\ \;\;\;\;t_0 \cdot \frac{1}{s \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{t_0 + 3}\\ \end{array} \]
Alternative 12
Error66.1%
Cost352.00
\[\frac{1}{s \cdot 4 + x \cdot \frac{x}{s}} \]
Alternative 13
Error63.2%
Cost297.00
\[\begin{array}{l} \mathbf{if}\;x \leq -1.999999987845058 \cdot 10^{-8} \lor \neg \left(x \leq 4.5000000682193786 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{s}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]
Alternative 14
Error27.4%
Cost96.00
\[\frac{0.25}{s} \]

Error

Reproduce?

herbie shell --seed 2023093 
(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))))))