Average Error: 0.2 → 0.2
Time: 3.7s
Precision: binary32
\[0 \leq s \land s \leq 1.0651631\]
\[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
↓
\[\begin{array}{l} t_0 := e^{\frac{-\left|x\right|}{s}}\\ \frac{t_0}{s \cdot {\left(t_0 + 1\right)}^{2}} \end{array} \]
(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 (exp (/ (- (fabs x)) s)))) (/ t_0 (* s (pow (+ t_0 1.0) 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) {
	float t_0 = expf((-fabsf(x) / s));
	return t_0 / (s * powf((t_0 + 1.0f), 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
    real(4) :: t_0
    t_0 = exp((-abs(x) / s))
    code = t_0 / (s * ((t_0 + 1.0e0) ** 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)
	t_0 = exp(Float32(Float32(-abs(x)) / s))
	return Float32(t_0 / Float32(s * (Float32(t_0 + Float32(1.0)) ^ 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)
	t_0 = exp((-abs(x) / s));
	tmp = t_0 / (s * ((t_0 + single(1.0)) ^ 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)}
↓
\begin{array}{l}
t_0 := e^{\frac{-\left|x\right|}{s}}\\
\frac{t_0}{s \cdot {\left(t_0 + 1\right)}^{2}}
\end{array}

Error

Bits error versus x

Bits error versus s

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.2

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

  2. Taylor expanded in x around 0 0.2

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

  3. Simplified0.2

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

  4. Final simplification0.2

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

Reproduce

herbie shell --seed 2022159 
(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))))))