?

Average Error: 0.1 → 0.2
Time: 16.5s
Precision: binary32
Cost: 13248

?

\[0 \leq s \land s \leq 1.0651631\]
\[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
↓
\[\frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{-s}} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)} \]
(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))) (+ (exp (/ (fabs x) s)) 2.0))))
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) {
	return (1.0f / s) / (expf((fabsf(x) / -s)) + (expf((fabsf(x) / s)) + 2.0f));
}

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)) + (exp((abs(x) / s)) + 2.0e0))
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(Float32(1.0) / s) / Float32(exp(Float32(abs(x) / Float32(-s))) + Float32(exp(Float32(abs(x) / s)) + Float32(2.0))))
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)
	tmp = (single(1.0) / s) / (exp((abs(x) / -s)) + (exp((abs(x) / s)) + single(2.0)));
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)}
↓
\frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{-s}} + \left(e^{\frac{\left|x\right|}{s}} + 2\right)}

Error?

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation?

  1. Initial program 0.1

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

  2. Simplified0.2

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

    [Start]0.1

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

    associate-*l* [=>]0.1

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

    associate-/r* [=>]0.1

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

    distribute-frac-neg [=>]0.1

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

    exp-neg [=>]0.2

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

    associate-/l/ [=>]0.2

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

    associate-/r* [=>]0.2

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

    associate-/r* [<=]0.2

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

  3. Final simplification0.2

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

Alternatives

Alternative 1
Error0.2
Cost13248
\[\frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(e^{\frac{\left|x\right|}{-s}} + 2\right)} \]
Alternative 2
Error0.1
Cost6912
\[\frac{1}{\left(s \cdot \left(1 + e^{\frac{-x}{s}}\right)\right) \cdot \left(1 + e^{\frac{x}{s}}\right)} \]
Alternative 3
Error0.1
Cost6880
\[\begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \frac{1}{\left(1 + t_0\right) \cdot \left(s + \frac{s}{t_0}\right)} \end{array} \]
Alternative 4
Error1.3
Cost6688
\[\frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + 3} \]
Alternative 5
Error1.8
Cost6656
\[\frac{e^{\frac{-\left|x\right|}{s}}}{s \cdot 4} \]
Alternative 6
Error4.4
Cost3556
\[\begin{array}{l} \mathbf{if}\;x \leq -2.000000093402204 \cdot 10^{-34}:\\ \;\;\;\;\frac{0.5}{s + \frac{s}{e^{\frac{x}{s}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \frac{\frac{x \cdot x}{s}}{s}}\\ \end{array} \]
Alternative 7
Error1.5
Cost3556
\[\begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \mathbf{if}\;x \leq -2.000000093402204 \cdot 10^{-34}:\\ \;\;\;\;\frac{0.5}{s + \frac{s}{t_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{t_0 + 3}\\ \end{array} \]
Alternative 8
Error7.4
Cost416
\[\frac{\frac{1}{s}}{4 + \frac{\frac{x \cdot x}{s}}{s}} \]
Alternative 9
Error22.2
Cost288
\[\frac{1}{x \cdot -2 + s \cdot 4} \]
Alternative 10
Error29.2
Cost96
\[\frac{-0.5}{x} \]
Alternative 11
Error22.8
Cost96
\[\frac{0.25}{s} \]

Error

Reproduce?

herbie shell --seed 2023054 
(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))))))