Logistic distribution

Average Accuracy: 99.5% → 99.6%

Time: 15.6s

Precision: binary32

Cost: 3488

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

Math

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

↓

\[\frac{\frac{0.5}{1 + \cosh \left(\frac{x}{s}\right)}}{s} \]

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 (/ (/ 0.5 (+ 1.0 (cosh (/ x s)))) s))

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) {
	return (0.5f / (1.0f + coshf((x / s)))) / s;
}

Fortran

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
    code = (0.5e0 / (1.0e0 + cosh((x / s)))) / s
end function

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)
	return Float32(Float32(Float32(0.5) / Float32(Float32(1.0) + cosh(Float32(x / s)))) / s)
end

MATLAB

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(0.5) / (single(1.0) + cosh((x / s)))) / s;
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 99.4%

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

   [Start] 99.5	\[ \frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   *-lft-identity [<=] 99.5	\[ \color{blue}{1 \cdot \frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]
   associate-*r/ [=>] 99.5	\[ \color{blue}{\frac{1 \cdot e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]
   associate-*l* [=>] 99.5	\[ \frac{1 \cdot e^{\frac{-\left|x\right|}{s}}}{\color{blue}{s \cdot \left(\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right)}} \]
   times-frac [=>] 99.4	\[ \color{blue}{\frac{1}{s} \cdot \frac{e^{\frac{-\left|x\right|}{s}}}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]
   associate-*r/ [=>] 99.4	\[ \color{blue}{\frac{\frac{1}{s} \cdot e^{\frac{-\left|x\right|}{s}}}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}} \]
   associate-/l* [=>] 99.3	\[ \color{blue}{\frac{\frac{1}{s}}{\frac{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{e^{\frac{-\left|x\right|}{s}}}}} \]
   distribute-frac-neg [=>] 99.3	\[ \frac{\frac{1}{s}}{\frac{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{e^{\color{blue}{-\frac{\left|x\right|}{s}}}}} \]
   exp-neg [=>] 99.4	\[ \frac{\frac{1}{s}}{\frac{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}{\color{blue}{\frac{1}{e^{\frac{\left|x\right|}{s}}}}}} \]

3. Taylor expanded in s around 0 99.6%

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

4. Simplified 99.5%

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

   [Start] 99.6	\[ \frac{1}{s \cdot \left(e^{-1 \cdot \frac{\left|x\right|}{s}} + \left(2 + e^{\frac{\left|x\right|}{s}}\right)\right)} \]
   associate-/r* [=>] 99.4	\[ \color{blue}{\frac{\frac{1}{s}}{e^{-1 \cdot \frac{\left|x\right|}{s}} + \left(2 + e^{\frac{\left|x\right|}{s}}\right)}} \]
   +-commutative [=>] 99.4	\[ \frac{\frac{1}{s}}{\color{blue}{\left(2 + e^{\frac{\left|x\right|}{s}}\right) + e^{-1 \cdot \frac{\left|x\right|}{s}}}} \]
   associate-+l+ [=>] 99.5	\[ \frac{\frac{1}{s}}{\color{blue}{2 + \left(e^{\frac{\left|x\right|}{s}} + e^{-1 \cdot \frac{\left|x\right|}{s}}\right)}} \]
   associate-*r/ [=>] 99.5	\[ \frac{\frac{1}{s}}{2 + \left(e^{\frac{\left|x\right|}{s}} + e^{\color{blue}{\frac{-1 \cdot \left|x\right|}{s}}}\right)} \]
   mul-1-neg [=>] 99.5	\[ \frac{\frac{1}{s}}{2 + \left(e^{\frac{\left|x\right|}{s}} + e^{\frac{\color{blue}{-\left|x\right|}}{s}}\right)} \]

5. Applied egg-rr 99.6%

   \[\leadsto \color{blue}{0 + \frac{1}{s \cdot \mathsf{fma}\left(2, \cosh \left(\frac{x}{s}\right), 2\right)}} \]
   Proof

   [Start] 99.5	\[ \frac{\frac{1}{s}}{2 + \left(e^{\frac{\left|x\right|}{s}} + e^{\frac{-\left|x\right|}{s}}\right)} \]
   add-log-exp [=>] 75.4	\[ \color{blue}{\log \left(e^{\frac{\frac{1}{s}}{2 + \left(e^{\frac{\left|x\right|}{s}} + e^{\frac{-\left|x\right|}{s}}\right)}}\right)} \]
   *-un-lft-identity [=>] 75.4	\[ \log \color{blue}{\left(1 \cdot e^{\frac{\frac{1}{s}}{2 + \left(e^{\frac{\left|x\right|}{s}} + e^{\frac{-\left|x\right|}{s}}\right)}}\right)} \]
   log-prod [=>] 75.4	\[ \color{blue}{\log 1 + \log \left(e^{\frac{\frac{1}{s}}{2 + \left(e^{\frac{\left|x\right|}{s}} + e^{\frac{-\left|x\right|}{s}}\right)}}\right)} \]
   metadata-eval [=>] 75.4	\[ \color{blue}{0} + \log \left(e^{\frac{\frac{1}{s}}{2 + \left(e^{\frac{\left|x\right|}{s}} + e^{\frac{-\left|x\right|}{s}}\right)}}\right) \]
   add-log-exp [<=] 99.5	\[ 0 + \color{blue}{\frac{\frac{1}{s}}{2 + \left(e^{\frac{\left|x\right|}{s}} + e^{\frac{-\left|x\right|}{s}}\right)}} \]
   associate-/l/ [=>] 99.6	\[ 0 + \color{blue}{\frac{1}{\left(2 + \left(e^{\frac{\left|x\right|}{s}} + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot s}} \]
   *-commutative [=>] 99.6	\[ 0 + \frac{1}{\color{blue}{s \cdot \left(2 + \left(e^{\frac{\left|x\right|}{s}} + e^{\frac{-\left|x\right|}{s}}\right)\right)}} \]
   +-commutative [=>] 99.6	\[ 0 + \frac{1}{s \cdot \color{blue}{\left(\left(e^{\frac{\left|x\right|}{s}} + e^{\frac{-\left|x\right|}{s}}\right) + 2\right)}} \]
   distribute-frac-neg [=>] 99.6	\[ 0 + \frac{1}{s \cdot \left(\left(e^{\frac{\left|x\right|}{s}} + e^{\color{blue}{-\frac{\left|x\right|}{s}}}\right) + 2\right)} \]
   cosh-undef [=>] 99.6	\[ 0 + \frac{1}{s \cdot \left(\color{blue}{2 \cdot \cosh \left(\frac{\left|x\right|}{s}\right)} + 2\right)} \]
   fma-def [=>] 99.6	\[ 0 + \frac{1}{s \cdot \color{blue}{\mathsf{fma}\left(2, \cosh \left(\frac{\left|x\right|}{s}\right), 2\right)}} \]

6. Simplified 99.5%

   \[\leadsto \color{blue}{\frac{\frac{1}{s}}{\mathsf{fma}\left(2, \cosh \left(\frac{x}{s}\right), 2\right)}} \]
   Proof

   [Start] 99.6	\[ 0 + \frac{1}{s \cdot \mathsf{fma}\left(2, \cosh \left(\frac{x}{s}\right), 2\right)} \]
   +-lft-identity [=>] 99.6	\[ \color{blue}{\frac{1}{s \cdot \mathsf{fma}\left(2, \cosh \left(\frac{x}{s}\right), 2\right)}} \]
   associate-/r* [=>] 99.5	\[ \color{blue}{\frac{\frac{1}{s}}{\mathsf{fma}\left(2, \cosh \left(\frac{x}{s}\right), 2\right)}} \]

7. Applied egg-rr 99.5%

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

   [Start] 99.5	\[ \frac{\frac{1}{s}}{\mathsf{fma}\left(2, \cosh \left(\frac{x}{s}\right), 2\right)} \]
   fma-udef [=>] 99.5	\[ \frac{\frac{1}{s}}{\color{blue}{2 \cdot \cosh \left(\frac{x}{s}\right) + 2}} \]
   *-commutative [=>] 99.5	\[ \frac{\frac{1}{s}}{\color{blue}{\cosh \left(\frac{x}{s}\right) \cdot 2} + 2} \]

8. Applied egg-rr 99.4%

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

   [Start] 99.5	\[ \frac{\frac{1}{s}}{\cosh \left(\frac{x}{s}\right) \cdot 2 + 2} \]
   frac-2neg [=>] 99.5	\[ \color{blue}{\frac{-\frac{1}{s}}{-\left(\cosh \left(\frac{x}{s}\right) \cdot 2 + 2\right)}} \]
   div-inv [=>] 99.4	\[ \color{blue}{\left(-\frac{1}{s}\right) \cdot \frac{1}{-\left(\cosh \left(\frac{x}{s}\right) \cdot 2 + 2\right)}} \]
   distribute-neg-frac [=>] 99.4	\[ \color{blue}{\frac{-1}{s}} \cdot \frac{1}{-\left(\cosh \left(\frac{x}{s}\right) \cdot 2 + 2\right)} \]
   metadata-eval [=>] 99.4	\[ \frac{\color{blue}{-1}}{s} \cdot \frac{1}{-\left(\cosh \left(\frac{x}{s}\right) \cdot 2 + 2\right)} \]
   distribute-lft1-in [=>] 99.4	\[ \frac{-1}{s} \cdot \frac{1}{-\color{blue}{\left(\cosh \left(\frac{x}{s}\right) + 1\right) \cdot 2}} \]
   distribute-rgt-neg-in [=>] 99.4	\[ \frac{-1}{s} \cdot \frac{1}{\color{blue}{\left(\cosh \left(\frac{x}{s}\right) + 1\right) \cdot \left(-2\right)}} \]
   +-commutative [=>] 99.4	\[ \frac{-1}{s} \cdot \frac{1}{\color{blue}{\left(1 + \cosh \left(\frac{x}{s}\right)\right)} \cdot \left(-2\right)} \]
   metadata-eval [=>] 99.4	\[ \frac{-1}{s} \cdot \frac{1}{\left(1 + \cosh \left(\frac{x}{s}\right)\right) \cdot \color{blue}{-2}} \]

9. Simplified 99.6%

   \[\leadsto \color{blue}{\frac{\frac{0.5}{1 + \cosh \left(\frac{x}{s}\right)}}{s}} \]
   Proof

   [Start] 99.4	\[ \frac{-1}{s} \cdot \frac{1}{\left(1 + \cosh \left(\frac{x}{s}\right)\right) \cdot -2} \]
   associate-*l/ [=>] 99.6	\[ \color{blue}{\frac{-1 \cdot \frac{1}{\left(1 + \cosh \left(\frac{x}{s}\right)\right) \cdot -2}}{s}} \]
   mul-1-neg [=>] 99.6	\[ \frac{\color{blue}{-\frac{1}{\left(1 + \cosh \left(\frac{x}{s}\right)\right) \cdot -2}}}{s} \]
   *-commutative [=>] 99.6	\[ \frac{-\frac{1}{\color{blue}{-2 \cdot \left(1 + \cosh \left(\frac{x}{s}\right)\right)}}}{s} \]
   associate-/r* [=>] 99.6	\[ \frac{-\color{blue}{\frac{\frac{1}{-2}}{1 + \cosh \left(\frac{x}{s}\right)}}}{s} \]
   metadata-eval [=>] 99.6	\[ \frac{-\frac{\color{blue}{-0.5}}{1 + \cosh \left(\frac{x}{s}\right)}}{s} \]
   distribute-neg-frac [=>] 99.6	\[ \frac{\color{blue}{\frac{--0.5}{1 + \cosh \left(\frac{x}{s}\right)}}}{s} \]
   metadata-eval [=>] 99.6	\[ \frac{\frac{\color{blue}{0.5}}{1 + \cosh \left(\frac{x}{s}\right)}}{s} \]

10. Final simplification 99.6%

    \[\leadsto \frac{\frac{0.5}{1 + \cosh \left(\frac{x}{s}\right)}}{s} \]

Alternatives

Alternative 1
Accuracy	88.6%
Cost	3492

\[\begin{array}{l} \mathbf{if}\;x \leq 5.00000011871114 \cdot 10^{-33}:\\ \;\;\;\;\frac{\frac{e^{\frac{x}{s}}}{s}}{4}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \frac{x}{\frac{s \cdot s}{x}}}\\ \end{array} \]

Alternative 2
Accuracy	99.5%
Cost	3488

\[\frac{\frac{0.5}{s}}{1 + \cosh \left(\frac{x}{s}\right)} \]

Alternative 3
Accuracy	86.7%
Cost	3364

\[\begin{array}{l} \mathbf{if}\;x \leq -2.000000033724767 \cdot 10^{-16}:\\ \;\;\;\;e^{\frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + x \cdot \frac{1}{s \cdot \frac{s}{x}}}\\ \end{array} \]

Alternative 4
Accuracy	82.6%
Cost	553

\[\begin{array}{l} \mathbf{if}\;x \leq -2.0000000390829628 \cdot 10^{-24} \lor \neg \left(x \leq 5.00000011871114 \cdot 10^{-33}\right):\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \frac{x}{\frac{s \cdot s}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]

Alternative 5
Accuracy	82.8%
Cost	553

\[\begin{array}{l} \mathbf{if}\;x \leq -2.0000000390829628 \cdot 10^{-24} \lor \neg \left(x \leq 5.00000011871114 \cdot 10^{-33}\right):\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \frac{x}{\frac{s \cdot s}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \frac{x}{s} \cdot \frac{x}{s}}\\ \end{array} \]

Alternative 6
Accuracy	82.9%
Cost	552

\[\begin{array}{l} \mathbf{if}\;x \leq -2.0000000390829628 \cdot 10^{-24}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + x \cdot \frac{x}{s \cdot s}}\\ \mathbf{elif}\;x \leq 5.00000011871114 \cdot 10^{-33}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \frac{x}{s} \cdot \frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \frac{x}{\frac{s \cdot s}{x}}}\\ \end{array} \]

Alternative 7
Accuracy	82.2%
Cost	480

\[\frac{\frac{1}{s}}{4 + x \cdot \frac{1}{s \cdot \frac{s}{x}}} \]

Alternative 8
Accuracy	62.4%
Cost	297

\[\begin{array}{l} \mathbf{if}\;x \leq -2.0000000233721948 \cdot 10^{-7} \lor \neg \left(x \leq 9.999999974752427 \cdot 10^{-7}\right):\\ \;\;\;\;\frac{s}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]

Alternative 9
Accuracy	27.2%
Cost	96

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

Alternative 10
Accuracy	8.3%
Cost	32

\[1 \]

Reproduce

herbie shell --seed 2023146 
(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))))))