?

Average Error: 0.44% → 0.43%
Time: 16.0s
Precision: binary32
Cost: 6880

?

\[0 \leq s \land s \leq 1.0651631\]
\[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
↓
\[\begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \frac{1}{\left(s + \frac{s}{t_0}\right) \cdot \left(1 + t_0\right)} \end{array} \]
(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 (/ x s)))) (/ 1.0 (* (+ s (/ s t_0)) (+ 1.0 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 = expf((x / s));
	return 1.0f / ((s + (s / t_0)) * (1.0f + 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 = exp((x / s))
    code = 1.0e0 / ((s + (s / t_0)) * (1.0e0 + 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 = exp(Float32(x / s))
	return Float32(Float32(1.0) / Float32(Float32(s + Float32(s / t_0)) * Float32(Float32(1.0) + 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 = exp((x / s));
	tmp = single(1.0) / ((s + (s / t_0)) * (single(1.0) + 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 := e^{\frac{x}{s}}\\
\frac{1}{\left(s + \frac{s}{t_0}\right) \cdot \left(1 + t_0\right)}
\end{array}

Error?

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation?

  1. Initial program 0.44

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

  2. Simplified0.46

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

    [Start]0.44

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

    associate-/l/ [<=]0.42

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

    *-lft-identity [<=]0.42

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

    *-lft-identity [<=]0.42

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

    *-commutative [<=]0.42

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

    associate-*r/ [=>]0.42

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

    associate-/l* [=>]0.5

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

    associate-/l/ [=>]0.48

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

  3. Applied egg-rr36.01

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

  4. Simplified0.43

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

    [Start]36.01

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

    associate-+r+ [=>]36.05

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

    *-lft-identity [<=]36.05

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

    distribute-rgt-in [<=]0.43

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

  5. Final simplification0.43

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

Alternatives

Alternative 1
Error3.79%
Cost6688
\[\frac{1}{s \cdot \left(3 + e^{\frac{\left|x\right|}{s}}\right)} \]
Alternative 2
Error5.27%
Cost6656
\[\frac{e^{\frac{-\left|x\right|}{s}}}{s \cdot 4} \]
Alternative 3
Error6.52%
Cost3812
\[\begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \mathbf{if}\;x \leq -3.99999987306209 \cdot 10^{-19}:\\ \;\;\;\;\frac{1}{\frac{s + \frac{s}{t_0}}{0.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(1 + t_0\right) \cdot \left(s + \frac{s}{1 + \frac{x}{s}}\right)}\\ \end{array} \]
Alternative 4
Error4.6%
Cost3620
\[\begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \mathbf{if}\;x \leq -9.999999796611898 \cdot 10^{-32}:\\ \;\;\;\;\frac{1}{\frac{s + \frac{s}{t_0}}{0.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{t_0 + 3}\\ \end{array} \]
Alternative 5
Error4.74%
Cost3556
\[\begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \mathbf{if}\;x \leq -9.999999796611898 \cdot 10^{-32}:\\ \;\;\;\;\frac{\frac{t_0}{s}}{4}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{t_0 + 3}\\ \end{array} \]
Alternative 6
Error5.28%
Cost3524
\[\begin{array}{l} \mathbf{if}\;x \leq -9.999999350456404 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{e^{\frac{x}{s}}}{s}}{4}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{-x}{s}}}{s \cdot 4}\\ \end{array} \]
Alternative 7
Error13.51%
Cost3492
\[\begin{array}{l} \mathbf{if}\;x \leq -3.199999943845344 \cdot 10^{-12}:\\ \;\;\;\;\left(1 + \frac{s}{x \cdot x}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.25}{s}}{e^{\frac{x}{s}}}\\ \end{array} \]
Alternative 8
Error5.31%
Cost3492
\[\begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \mathbf{if}\;x \leq -9.999999350456404 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{t_0}{s}}{4}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.25}{s}}{t_0}\\ \end{array} \]
Alternative 9
Error15.4%
Cost1132
\[\begin{array}{l} t_0 := \left(1 + \frac{s}{x \cdot x}\right) + -1\\ \mathbf{if}\;x \leq -4.999999858590343 \cdot 10^{-10}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 9.999999998199587 \cdot 10^{-24}:\\ \;\;\;\;\frac{1}{s \cdot 4 + x \cdot \frac{x}{s}}\\ \mathbf{elif}\;x \leq 1.9999999494757503 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{\left(s + \frac{s}{1 + \frac{x}{s}}\right) \cdot \left(1 + \left(\frac{x}{s} + \left(1 + \frac{x \cdot \left(x \cdot 0.5\right)}{s \cdot s}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 10
Error18.69%
Cost489
\[\begin{array}{l} \mathbf{if}\;x \leq -4.999999858590343 \cdot 10^{-10} \lor \neg \left(x \leq 9.999999960041972 \cdot 10^{-12}\right):\\ \;\;\;\;\left(1 + \frac{s}{x \cdot x}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot 4 + x \cdot \frac{x}{s}}\\ \end{array} \]
Alternative 11
Error19.73%
Cost425
\[\begin{array}{l} \mathbf{if}\;x \leq -3.199999943845344 \cdot 10^{-12} \lor \neg \left(x \leq 9.999999960041972 \cdot 10^{-12}\right):\\ \;\;\;\;\left(1 + \frac{s}{x \cdot x}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]
Alternative 12
Error37.18%
Cost360
\[\begin{array}{l} \mathbf{if}\;x \leq -9.999999747378752 \cdot 10^{-6}:\\ \;\;\;\;\frac{s}{x \cdot x}\\ \mathbf{elif}\;x \leq 1.9999999494757503 \cdot 10^{-5}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{s}{x} \cdot \frac{1}{x}\\ \end{array} \]
Alternative 13
Error37.15%
Cost297
\[\begin{array}{l} \mathbf{if}\;x \leq -9.999999747378752 \cdot 10^{-6} \lor \neg \left(x \leq 1.9999999494757503 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{s}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]
Alternative 14
Error72.57%
Cost96
\[\frac{0.25}{s} \]

Error

Reproduce?

herbie shell --seed 2023115 
(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))))))