?

Average Error: 0.52% → 0.47%
Time: 15.8s
Precision: binary32
Cost: 10080

?

\[0 \leq s \land s \leq 1.0651631\]
\[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
↓
\[\frac{1}{\frac{s}{{e}^{\left(\frac{x}{s}\right)}} + s \cdot \left(e^{\frac{x}{s}} + 2\right)} \]
(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 (pow E (/ x s))) (* s (+ (exp (/ 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 / powf(((float) M_E), (x / s))) + (s * (expf((x / s)) + 2.0f)));
}

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(1.0) / Float32(Float32(s / (Float32(exp(1)) ^ Float32(x / s))) + Float32(s * Float32(exp(Float32(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 / (single(2.71828182845904523536) ^ (x / s))) + (s * (exp((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{1}{\frac{s}{{e}^{\left(\frac{x}{s}\right)}} + s \cdot \left(e^{\frac{x}{s}} + 2\right)}

Error?

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation?

  1. Initial program 0.52

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

  2. Simplified0.53

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

    [Start]0.52

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

    associate-/l/ [<=]0.51

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

    *-lft-identity [<=]0.51

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

    *-lft-identity [<=]0.51

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

    *-commutative [<=]0.51

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

    associate-*r/ [=>]0.51

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

    associate-/l* [=>]0.58

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

    associate-/l/ [=>]0.56

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

  3. Applied egg-rr36.82

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

  4. Taylor expanded in s around 0 36.84

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

  5. Simplified0.46

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

    [Start]36.84

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

    *-commutative [=>]36.84

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

    +-commutative [=>]36.84

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

    distribute-lft-in [=>]72.13

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

    associate-+l+ [=>]72.13

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

    *-rgt-identity [=>]72.13

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

    rgt-mult-inverse [=>]0.46

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

    metadata-eval [=>]0.46

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

  6. Applied egg-rr0.47

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

  7. Final simplification0.47

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

Alternatives

Alternative 1
Error0.45%
Cost6880
\[\begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \frac{1}{s \cdot \left(t_0 + \left(2 + \frac{1}{t_0}\right)\right)} \end{array} \]
Alternative 2
Error0.46%
Cost6880
\[\begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \frac{1}{s \cdot \left(t_0 + 2\right) + \frac{s}{t_0}} \end{array} \]
Alternative 3
Error3.86%
Cost3620
\[\begin{array}{l} \mathbf{if}\;x \leq -1.0000000359391298 \cdot 10^{-36}:\\ \;\;\;\;\frac{\frac{1}{s}}{3 + e^{-\frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-3 - e^{\frac{x}{s}}} \cdot \frac{-1}{s}\\ \end{array} \]
Alternative 4
Error3.86%
Cost3588
\[\begin{array}{l} \mathbf{if}\;x \leq -1.0000000359391298 \cdot 10^{-36}:\\ \;\;\;\;\frac{\frac{1}{s}}{3 + e^{-\frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{e^{\frac{x}{s}} + 3}\\ \end{array} \]
Alternative 5
Error13.82%
Cost3556
\[\begin{array}{l} \mathbf{if}\;x \leq -3.999999984016789 \cdot 10^{-12}:\\ \;\;\;\;\frac{1}{x + \frac{s}{e^{\frac{x}{s}}}}\\ \mathbf{elif}\;x \leq 9.999999998199587 \cdot 10^{-24}:\\ \;\;\;\;\frac{1}{s \cdot 4 + \left(\frac{x}{s} \cdot \frac{x}{0.5} - \frac{x}{\frac{s}{x}}\right)}\\ \mathbf{elif}\;x \leq 5.5000000998006726 \cdot 10^{-8}:\\ \;\;\;\;\frac{\frac{1}{s}}{\left(\frac{x}{s} + 4\right) + 0.5 \cdot \frac{x \cdot x}{s \cdot s}}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(1 + \frac{s}{x \cdot x}\right)\\ \end{array} \]
Alternative 6
Error9.99%
Cost3556
\[\begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \mathbf{if}\;x \leq -3.999999984016789 \cdot 10^{-12}:\\ \;\;\;\;\frac{1}{x + \frac{s}{t_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{t_0 + 3}\\ \end{array} \]
Alternative 7
Error0.46%
Cost3552
\[\frac{-1}{s \cdot \left(-2 + -2 \cdot \cosh \left(\frac{x}{s}\right)\right)} \]
Alternative 8
Error16.3%
Cost812
\[\begin{array}{l} t_0 := -1 + \left(1 + \frac{s}{x \cdot x}\right)\\ \mathbf{if}\;x \leq -3.999999984016789 \cdot 10^{-12}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 9.999999998199587 \cdot 10^{-24}:\\ \;\;\;\;\frac{1}{s \cdot 4 + \left(\frac{x}{s} \cdot \frac{x}{0.5} - \frac{x}{\frac{s}{x}}\right)}\\ \mathbf{elif}\;x \leq 5.5000000998006726 \cdot 10^{-8}:\\ \;\;\;\;\frac{\frac{1}{s}}{\left(\frac{x}{s} + 4\right) + 0.5 \cdot \frac{x \cdot x}{s \cdot s}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 9
Error19.11%
Cost745
\[\begin{array}{l} \mathbf{if}\;x \leq -3.999999984016789 \cdot 10^{-12} \lor \neg \left(x \leq 5.60000010807471 \cdot 10^{-8}\right):\\ \;\;\;\;-1 + \left(1 + \frac{s}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot 4 + \left(\frac{x}{s} \cdot \frac{x}{0.5} - \frac{x}{\frac{s}{x}}\right)}\\ \end{array} \]
Alternative 10
Error19.11%
Cost489
\[\begin{array}{l} \mathbf{if}\;x \leq -3.999999984016789 \cdot 10^{-12} \lor \neg \left(x \leq 5.60000010807471 \cdot 10^{-8}\right):\\ \;\;\;\;-1 + \left(1 + \frac{s}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot 4 + \frac{x}{\frac{s}{x}}}\\ \end{array} \]
Alternative 11
Error20.09%
Cost425
\[\begin{array}{l} \mathbf{if}\;x \leq -3.999999984016789 \cdot 10^{-12} \lor \neg \left(x \leq 5.5000000998006726 \cdot 10^{-8}\right):\\ \;\;\;\;-1 + \left(1 + \frac{s}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]
Alternative 12
Error36.44%
Cost361
\[\begin{array}{l} \mathbf{if}\;x \leq -0.20000000298023224 \lor \neg \left(x \leq 5.5000000998006726 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{1}{x \cdot \frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]
Alternative 13
Error37.64%
Cost297
\[\begin{array}{l} \mathbf{if}\;x \leq -0.20000000298023224 \lor \neg \left(x \leq 5.5000000998006726 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{s}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]
Alternative 14
Error37.66%
Cost296
\[\begin{array}{l} \mathbf{if}\;x \leq -0.20000000298023224:\\ \;\;\;\;\frac{s}{x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq 5.5000000998006726 \cdot 10^{-8}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{s}{x \cdot x}\\ \end{array} \]
Alternative 15
Error72.34%
Cost96
\[\frac{0.25}{s} \]

Error

Reproduce?

herbie shell --seed 2023121 
(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))))))