Average Error: 0.1 → 0.1
Time: 6.0s
Precision: binary32
\[0 \leq s \land s \leq 1.0651631\]
\[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
↓
\[\begin{array}{l} t_0 := e^{\frac{-\left|x\right|}{s}}\\ \frac{t_0}{s \cdot \left(1 + \left(2 \cdot t_0 + {t_0}^{2}\right)\right)} \end{array} \]
(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 (exp (/ (- (fabs x)) s))))
   (/ t_0 (* s (+ 1.0 (+ (* 2.0 t_0) (pow t_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 * (1.0f + ((2.0f * t_0) + powf(t_0, 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 * (1.0e0 + ((2.0e0 * t_0) + (t_0 ** 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(Float32(1.0) + Float32(Float32(Float32(2.0) * t_0) + (t_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 * (single(1.0) + ((single(2.0) * t_0) + (t_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(1 + \left(2 \cdot t_0 + {t_0}^{2}\right)\right)}
\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 0.1

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

  2. Taylor expanded in s around 0 0.1

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

  3. Final simplification0.1

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

Reproduce

herbie shell --seed 2022160 
(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))))))