Logistic distribution

Average Accuracy: 99.5% → 99.6%

Time: 18.5s

Precision: binary32

Cost: 13184

\[0 \leq s \land s \leq 1.0651631\]

Math

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

↓

\[\begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \frac{e^{-\mathsf{log1p}\left(t_0\right)}}{s + \frac{s}{t_0}} \end{array} \]

FPCore

(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)))) (/ (exp (- (log1p t_0))) (+ s (/ s t_0)))))

C

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 expf(-log1pf(t_0)) / (s + (s / t_0));
}

Julia

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(exp(Float32(-log1p(t_0))) / Float32(s + Float32(s / t_0)))
end

Derivation

1. Initial program 99.5%

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

2. Simplified 99.5%

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

   [Start] 99.5	\[ \frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   associate-/l/ [<=] 99.6	\[ \color{blue}{\frac{\frac{e^{\frac{-\left|x\right|}{s}}}{1 + e^{\frac{-\left|x\right|}{s}}}}{s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]
   *-lft-identity [<=] 99.6	\[ \frac{\color{blue}{1 \cdot \frac{e^{\frac{-\left|x\right|}{s}}}{1 + e^{\frac{-\left|x\right|}{s}}}}}{s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   *-lft-identity [<=] 99.6	\[ \frac{1 \cdot \frac{e^{\frac{-\left|x\right|}{s}}}{\color{blue}{1 \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}}}{s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   *-commutative [<=] 99.6	\[ \frac{1 \cdot \frac{e^{\frac{-\left|x\right|}{s}}}{\color{blue}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot 1}}}{s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   associate-*r/ [=>] 99.6	\[ \frac{\color{blue}{\frac{1 \cdot e^{\frac{-\left|x\right|}{s}}}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot 1}}}{s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   associate-/l* [=>] 99.5	\[ \frac{\color{blue}{\frac{1}{\frac{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot 1}{e^{\frac{-\left|x\right|}{s}}}}}}{s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   associate-/l/ [=>] 99.5	\[ \color{blue}{\frac{1}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \frac{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot 1}{e^{\frac{-\left|x\right|}{s}}}}} \]

3. Applied egg-rr 99.1%

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

   [Start] 99.5	\[ \frac{1}{\left(s + \frac{s}{e^{\frac{\left|x\right|}{s}}}\right) \cdot \left(1 + e^{\frac{\left|x\right|}{s}}\right)} \]
   inv-pow [=>] 99.5	\[ \color{blue}{{\left(\left(s + \frac{s}{e^{\frac{\left|x\right|}{s}}}\right) \cdot \left(1 + e^{\frac{\left|x\right|}{s}}\right)\right)}^{-1}} \]
   add-sqr-sqrt [=>] 99.2	\[ {\color{blue}{\left(\sqrt{\left(s + \frac{s}{e^{\frac{\left|x\right|}{s}}}\right) \cdot \left(1 + e^{\frac{\left|x\right|}{s}}\right)} \cdot \sqrt{\left(s + \frac{s}{e^{\frac{\left|x\right|}{s}}}\right) \cdot \left(1 + e^{\frac{\left|x\right|}{s}}\right)}\right)}}^{-1} \]
   metadata-eval [<=] 99.2	\[ {\left(\sqrt{\left(s + \frac{s}{e^{\frac{\left|x\right|}{s}}}\right) \cdot \left(1 + e^{\frac{\left|x\right|}{s}}\right)} \cdot \sqrt{\left(s + \frac{s}{e^{\frac{\left|x\right|}{s}}}\right) \cdot \left(1 + e^{\frac{\left|x\right|}{s}}\right)}\right)}^{\color{blue}{\left(-1\right)}} \]
   unpow-prod-down [=>] 99.1	\[ \color{blue}{{\left(\sqrt{\left(s + \frac{s}{e^{\frac{\left|x\right|}{s}}}\right) \cdot \left(1 + e^{\frac{\left|x\right|}{s}}\right)}\right)}^{\left(-1\right)} \cdot {\left(\sqrt{\left(s + \frac{s}{e^{\frac{\left|x\right|}{s}}}\right) \cdot \left(1 + e^{\frac{\left|x\right|}{s}}\right)}\right)}^{\left(-1\right)}} \]

4. Simplified 99.2%

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

   [Start] 99.1	\[ {\left(\sqrt{\left(s + \frac{s}{e^{\frac{x}{s}}}\right) \cdot \left(e^{\frac{x}{s}} + 1\right)}\right)}^{-1} \cdot {\left(\sqrt{\left(s + \frac{s}{e^{\frac{x}{s}}}\right) \cdot \left(e^{\frac{x}{s}} + 1\right)}\right)}^{-1} \]
   pow-sqr [=>] 99.2	\[ \color{blue}{{\left(\sqrt{\left(s + \frac{s}{e^{\frac{x}{s}}}\right) \cdot \left(e^{\frac{x}{s}} + 1\right)}\right)}^{\left(2 \cdot -1\right)}} \]
   *-commutative [=>] 99.2	\[ {\left(\sqrt{\color{blue}{\left(e^{\frac{x}{s}} + 1\right) \cdot \left(s + \frac{s}{e^{\frac{x}{s}}}\right)}}\right)}^{\left(2 \cdot -1\right)} \]
   +-commutative [=>] 99.2	\[ {\left(\sqrt{\color{blue}{\left(1 + e^{\frac{x}{s}}\right)} \cdot \left(s + \frac{s}{e^{\frac{x}{s}}}\right)}\right)}^{\left(2 \cdot -1\right)} \]
   metadata-eval [=>] 99.2	\[ {\left(\sqrt{\left(1 + e^{\frac{x}{s}}\right) \cdot \left(s + \frac{s}{e^{\frac{x}{s}}}\right)}\right)}^{\color{blue}{-2}} \]

5. Applied egg-rr 97.5%

   \[\leadsto \color{blue}{e^{-1 \cdot \left(\mathsf{log1p}\left(e^{\frac{x}{s}}\right) + \log \left(s + \frac{s}{e^{\frac{x}{s}}}\right)\right)}} \]
   Proof

   [Start] 99.2	\[ {\left(\sqrt{\left(1 + e^{\frac{x}{s}}\right) \cdot \left(s + \frac{s}{e^{\frac{x}{s}}}\right)}\right)}^{-2} \]
   add-exp-log [=>] 97.7	\[ \color{blue}{e^{\log \left({\left(\sqrt{\left(1 + e^{\frac{x}{s}}\right) \cdot \left(s + \frac{s}{e^{\frac{x}{s}}}\right)}\right)}^{-2}\right)}} \]
   sqrt-pow2 [=>] 97.7	\[ e^{\log \color{blue}{\left({\left(\left(1 + e^{\frac{x}{s}}\right) \cdot \left(s + \frac{s}{e^{\frac{x}{s}}}\right)\right)}^{\left(\frac{-2}{2}\right)}\right)}} \]
   log-pow [=>] 97.7	\[ e^{\color{blue}{\frac{-2}{2} \cdot \log \left(\left(1 + e^{\frac{x}{s}}\right) \cdot \left(s + \frac{s}{e^{\frac{x}{s}}}\right)\right)}} \]
   metadata-eval [=>] 97.7	\[ e^{\color{blue}{-1} \cdot \log \left(\left(1 + e^{\frac{x}{s}}\right) \cdot \left(s + \frac{s}{e^{\frac{x}{s}}}\right)\right)} \]
   log-prod [=>] 97.5	\[ e^{-1 \cdot \color{blue}{\left(\log \left(1 + e^{\frac{x}{s}}\right) + \log \left(s + \frac{s}{e^{\frac{x}{s}}}\right)\right)}} \]
   log1p-def [=>] 97.5	\[ e^{-1 \cdot \left(\color{blue}{\mathsf{log1p}\left(e^{\frac{x}{s}}\right)} + \log \left(s + \frac{s}{e^{\frac{x}{s}}}\right)\right)} \]

6. Simplified 99.6%

   \[\leadsto \color{blue}{\frac{e^{-\mathsf{log1p}\left(e^{\frac{x}{s}}\right)}}{s + \frac{s}{e^{\frac{x}{s}}}}} \]
   Proof

   [Start] 97.5	\[ e^{-1 \cdot \left(\mathsf{log1p}\left(e^{\frac{x}{s}}\right) + \log \left(s + \frac{s}{e^{\frac{x}{s}}}\right)\right)} \]
   mul-1-neg [=>] 97.5	\[ e^{\color{blue}{-\left(\mathsf{log1p}\left(e^{\frac{x}{s}}\right) + \log \left(s + \frac{s}{e^{\frac{x}{s}}}\right)\right)}} \]
   exp-neg [=>] 97.5	\[ \color{blue}{\frac{1}{e^{\mathsf{log1p}\left(e^{\frac{x}{s}}\right) + \log \left(s + \frac{s}{e^{\frac{x}{s}}}\right)}}} \]
   exp-sum [=>] 97.7	\[ \frac{1}{\color{blue}{e^{\mathsf{log1p}\left(e^{\frac{x}{s}}\right)} \cdot e^{\log \left(s + \frac{s}{e^{\frac{x}{s}}}\right)}}} \]
   associate-/r* [=>] 97.7	\[ \color{blue}{\frac{\frac{1}{e^{\mathsf{log1p}\left(e^{\frac{x}{s}}\right)}}}{e^{\log \left(s + \frac{s}{e^{\frac{x}{s}}}\right)}}} \]
   exp-neg [<=] 97.7	\[ \frac{\color{blue}{e^{-\mathsf{log1p}\left(e^{\frac{x}{s}}\right)}}}{e^{\log \left(s + \frac{s}{e^{\frac{x}{s}}}\right)}} \]
   rem-exp-log [=>] 99.6	\[ \frac{e^{-\mathsf{log1p}\left(e^{\frac{x}{s}}\right)}}{\color{blue}{s + \frac{s}{e^{\frac{x}{s}}}}} \]

7. Final simplification 99.6%

   \[\leadsto \frac{e^{-\mathsf{log1p}\left(e^{\frac{x}{s}}\right)}}{s + \frac{s}{e^{\frac{x}{s}}}} \]

Alternatives

Alternative 1
Accuracy	99.6%
Cost	6848

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

Alternative 2
Accuracy	96.1%
Cost	6688

\[\frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + 3} \]

Alternative 3
Accuracy	94.7%
Cost	6656

\[\frac{e^{\frac{-\left|x\right|}{s}}}{s \cdot 4} \]

Alternative 4
Accuracy	96.1%
Cost	3812

\[\begin{array}{l} \mathbf{if}\;x \leq -1.9999999593223797 \cdot 10^{-31}:\\ \;\;\;\;\frac{\frac{1}{s}}{e^{-\frac{x}{s}} + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(e^{\frac{x}{s}} + 1\right) \cdot \left(s + \frac{s}{\frac{x}{s} + 1}\right)}\\ \end{array} \]

Alternative 5
Accuracy	95.7%
Cost	3684

\[\begin{array}{l} \mathbf{if}\;x \leq -1.9999999593223797 \cdot 10^{-31}:\\ \;\;\;\;\frac{\frac{1}{s}}{e^{-\frac{x}{s}} + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(e^{\frac{x}{s}} + 1\right) \cdot \left(s \cdot 2 - x\right)}\\ \end{array} \]

Alternative 6
Accuracy	95.7%
Cost	3620

\[\begin{array}{l} \mathbf{if}\;x \leq -1.9999999593223797 \cdot 10^{-31}:\\ \;\;\;\;\frac{\frac{1}{s}}{e^{-\frac{x}{s}} + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{1 + \left(e^{\frac{x}{s}} + 2\right)}\\ \end{array} \]

Alternative 7
Accuracy	95.0%
Cost	3588

\[\begin{array}{l} \mathbf{if}\;x \leq 1.0000000031710769 \cdot 10^{-29}:\\ \;\;\;\;\frac{\frac{1}{s}}{e^{-\frac{x}{s}} + 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.25}{s}}{e^{\frac{x}{s}}}\\ \end{array} \]

Alternative 8
Accuracy	88.0%
Cost	3492

\[\begin{array}{l} \mathbf{if}\;x \leq 9.999999998199587 \cdot 10^{-24}:\\ \;\;\;\;0.25 \cdot \frac{e^{\frac{x}{s}}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{2 + \left(\frac{x}{s} + 0.5 \cdot \frac{x \cdot x}{s \cdot s}\right)}}{s \cdot 2 - x}\\ \end{array} \]

Alternative 9
Accuracy	94.5%
Cost	3492

\[\begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \mathbf{if}\;x \leq -1.000000023742228 \cdot 10^{-33}:\\ \;\;\;\;0.25 \cdot \frac{t_0}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.25}{s}}{t_0}\\ \end{array} \]

Alternative 10
Accuracy	88.2%
Cost	688

\[\begin{array}{l} t_0 := \frac{\frac{1}{s}}{4 + \frac{x \cdot x}{s \cdot s}}\\ t_1 := -1 + \left(1 + \frac{s}{x \cdot x}\right)\\ \mathbf{if}\;x \leq -3.999999989900971 \cdot 10^{-6}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -9.999999682655225 \cdot 10^{-22}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 9.999999998199587 \cdot 10^{-24}:\\ \;\;\;\;\frac{1}{s \cdot 4 + x \cdot \frac{x}{s}}\\ \mathbf{elif}\;x \leq 9.999999747378752 \cdot 10^{-5}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Accuracy	81.0%
Cost	489

\[\begin{array}{l} \mathbf{if}\;x \leq -3.99999992980668 \cdot 10^{-13} \lor \neg \left(x \leq 4.999999987376214 \cdot 10^{-7}\right):\\ \;\;\;\;-1 + \left(1 + \frac{s}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot 4 + x \cdot \frac{x}{s}}\\ \end{array} \]

Alternative 12
Accuracy	80.4%
Cost	425

\[\begin{array}{l} \mathbf{if}\;x \leq -3.99999992980668 \cdot 10^{-13} \lor \neg \left(x \leq 9.99999993922529 \cdot 10^{-9}\right):\\ \;\;\;\;-1 + \left(1 + \frac{s}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]

Alternative 13
Accuracy	63.1%
Cost	297

\[\begin{array}{l} \mathbf{if}\;x \leq -0.009999999776482582 \lor \neg \left(x \leq 9.999999747378752 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{s}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]

Alternative 14
Accuracy	63.1%
Cost	296

\[\begin{array}{l} \mathbf{if}\;x \leq -0.009999999776482582:\\ \;\;\;\;\frac{\frac{s}{x}}{x}\\ \mathbf{elif}\;x \leq 9.999999747378752 \cdot 10^{-5}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{s}{x \cdot x}\\ \end{array} \]

Alternative 15
Accuracy	27.4%
Cost	96

\[\frac{0.25}{s} \]

Reproduce

herbie shell --seed 2023136 
(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))))))