?

Average Error: 0.2 → 0.1
Time: 15.4s
Precision: binary32
Cost: 6944

?

\[0 \leq s \land s \leq 1.0651631\]
\[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
↓
\[\begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \frac{1}{s + \left(\frac{s}{t_0} + s \cdot \left(1 + t_0\right)\right)} \end{array} \]
(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 (exp (/ x s)))) (/ 1.0 (+ s (+ (/ s t_0) (* s (+ 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) + (s * (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) + (s * (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(s + Float32(Float32(s / t_0) + Float32(s * 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) + (s * (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}{s + \left(\frac{s}{t_0} + s \cdot \left(1 + 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 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. Simplified0.2

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

    [Start]0.2

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

    associate-/l/ [<=]0.2

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

    *-lft-identity [<=]0.2

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

    *-lft-identity [<=]0.2

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

    *-commutative [<=]0.2

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

    associate-*r/ [=>]0.2

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

    associate-/l* [=>]0.2

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

    associate-/l/ [=>]0.2

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

  3. Applied egg-rr11.5

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

  4. Taylor expanded in s around 0 11.6

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

  5. Simplified0.1

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

    [Start]11.6

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

    *-commutative [=>]11.6

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

    associate-*l* [=>]11.6

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

    *-commutative [<=]11.6

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

    distribute-lft-in [=>]23.3

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

    *-rgt-identity [=>]23.3

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

    rgt-mult-inverse [=>]0.1

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

  6. Final simplification0.1

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

Alternatives

Alternative 1
Error0.2
Cost6880
\[\begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \frac{1}{\left(s + \frac{s}{t_0}\right) \cdot \left(1 + t_0\right)} \end{array} \]
Alternative 2
Error0.2
Cost6848
\[\frac{\frac{1}{s}}{e^{\frac{x}{s}} + \left(2 + e^{\frac{-x}{s}}\right)} \]
Alternative 3
Error1.3
Cost6688
\[\frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + 3} \]
Alternative 4
Error1.7
Cost6656
\[\frac{e^{\frac{-\left|x\right|}{s}}}{s \cdot 4} \]
Alternative 5
Error1.2
Cost3684
\[\begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \mathbf{if}\;x \leq -5.000000179695649 \cdot 10^{-37}:\\ \;\;\;\;\frac{1}{s + \left(\frac{s}{t_0} + s \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{t_0 + 3}\\ \end{array} \]
Alternative 6
Error1.3
Cost3588
\[\begin{array}{l} \mathbf{if}\;x \leq -5.000000179695649 \cdot 10^{-37}:\\ \;\;\;\;\frac{\frac{1}{s}}{e^{\frac{-x}{s}} + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{e^{\frac{x}{s}} + 3}\\ \end{array} \]
Alternative 7
Error3.8
Cost3556
\[\begin{array}{l} \mathbf{if}\;x \leq -4.0000001089808046 \cdot 10^{-27}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \frac{x \cdot x}{s \cdot s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{e^{\frac{x}{s}} + 3}\\ \end{array} \]
Alternative 8
Error7.8
Cost416
\[\frac{\frac{1}{s}}{4 + \frac{x}{s} \cdot \frac{x}{s}} \]
Alternative 9
Error7.3
Cost416
\[\frac{\frac{1}{s}}{4 + \frac{x \cdot x}{s \cdot s}} \]
Alternative 10
Error11.5
Cost361
\[\begin{array}{l} \mathbf{if}\;x \leq -4.999999969612645 \cdot 10^{-9} \lor \neg \left(x \leq 1.9999999494757503 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{1}{x \cdot \frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]
Alternative 11
Error12.0
Cost360
\[\begin{array}{l} \mathbf{if}\;x \leq -4.999999969612645 \cdot 10^{-9}:\\ \;\;\;\;\frac{s}{x \cdot x}\\ \mathbf{elif}\;x \leq 1.9999999494757503 \cdot 10^{-5}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{s}{x} \cdot \frac{1}{x}\\ \end{array} \]
Alternative 12
Error11.5
Cost360
\[\begin{array}{l} \mathbf{if}\;x \leq -4.999999969612645 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{s + \frac{x}{\frac{s}{x}}}\\ \mathbf{elif}\;x \leq 1.9999999494757503 \cdot 10^{-5}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \frac{x}{s}}\\ \end{array} \]
Alternative 13
Error11.0
Cost352
\[\frac{1}{s \cdot 4 + x \cdot \frac{x}{s}} \]
Alternative 14
Error11.9
Cost297
\[\begin{array}{l} \mathbf{if}\;x \leq -4.999999969612645 \cdot 10^{-9} \lor \neg \left(x \leq 1.9999999494757503 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{s}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]
Alternative 15
Error12.0
Cost296
\[\begin{array}{l} \mathbf{if}\;x \leq -4.999999969612645 \cdot 10^{-9}:\\ \;\;\;\;\frac{s}{x \cdot x}\\ \mathbf{elif}\;x \leq 1.9999999494757503 \cdot 10^{-5}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{s}{x}}{x}\\ \end{array} \]
Alternative 16
Error23.3
Cost96
\[\frac{0.25}{s} \]

Error

Reproduce?

herbie shell --seed 2023034 
(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))))))