Logistic distribution

Average Accuracy: 99.5% → 99.5%

Time: 17.9s

Precision: binary32

Cost: 19840

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

Math

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

↓

\[\begin{array}{l} t_0 := e^{\frac{-\left|x\right|}{s}}\\ t_1 := t_0 + 1\\ \frac{t_0}{s \cdot \left(t_1 \cdot t_1\right)} \end{array} \]

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 (/ (- (fabs x)) s))) (t_1 (+ t_0 1.0)))
   (/ t_0 (* s (* t_1 t_1)))))

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((-fabsf(x) / s));
	float t_1 = t_0 + 1.0f;
	return t_0 / (s * (t_1 * t_1));
}

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
    real(4) :: t_0
    real(4) :: t_1
    t_0 = exp((-abs(x) / s))
    t_1 = t_0 + 1.0e0
    code = t_0 / (s * (t_1 * t_1))
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)
	t_0 = exp(Float32(Float32(-abs(x)) / s))
	t_1 = Float32(t_0 + Float32(1.0))
	return Float32(t_0 / Float32(s * Float32(t_1 * t_1)))
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)
	t_0 = exp((-abs(x) / s));
	t_1 = t_0 + single(1.0);
	tmp = t_0 / (s * (t_1 * t_1));
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%

   \[\leadsto \color{blue}{\frac{e^{\frac{-\left|x\right|}{s}}}{s \cdot \left(\left(e^{\frac{-\left|x\right|}{s}} + 1\right) \cdot \left(e^{\frac{-\left|x\right|}{s}} + 1\right)\right)}} \]
   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* [=>] 99.5	\[ \frac{e^{\frac{-\left|x\right|}{s}}}{\color{blue}{s \cdot \left(\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right)}} \]
   +-commutative [=>] 99.5	\[ \frac{e^{\frac{-\left|x\right|}{s}}}{s \cdot \left(\color{blue}{\left(e^{\frac{-\left|x\right|}{s}} + 1\right)} \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right)} \]
   +-commutative [=>] 99.5	\[ \frac{e^{\frac{-\left|x\right|}{s}}}{s \cdot \left(\left(e^{\frac{-\left|x\right|}{s}} + 1\right) \cdot \color{blue}{\left(e^{\frac{-\left|x\right|}{s}} + 1\right)}\right)} \]

3. Final simplification 99.5%

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

Alternatives

Alternative 1
Accuracy	99.5%
Cost	13280

\[\begin{array}{l} t_0 := e^{\frac{\left|x\right|}{s}}\\ \frac{1}{\left(s + \frac{s}{t_0}\right) \cdot \left(1 + t_0\right)} \end{array} \]

Alternative 2
Accuracy	99.6%
Cost	6880

\[\begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \frac{1}{\left(s + \frac{s}{t_0}\right) \cdot \left(1 + t_0\right)} \end{array} \]

Alternative 3
Accuracy	96.1%
Cost	6688

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

Alternative 4
Accuracy	94.6%
Cost	6656

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

Alternative 5
Accuracy	87.1%
Cost	3556

\[\begin{array}{l} \mathbf{if}\;x \leq 9.999999960041972 \cdot 10^{-13}:\\ \;\;\;\;\frac{0.5}{s + \frac{s}{e^{\frac{x}{s}}}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{s}{x \cdot x}\right) + -1\\ \end{array} \]

Alternative 6
Accuracy	95.5%
Cost	3556

\[\begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \mathbf{if}\;x \leq -5.00000023350551 \cdot 10^{-35}:\\ \;\;\;\;\frac{0.5}{s + \frac{s}{t_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{t_0 + 3}\\ \end{array} \]

Alternative 7
Accuracy	81.4%
Cost	489

\[\begin{array}{l} \mathbf{if}\;x \leq -1.0000000116860974 \cdot 10^{-7} \lor \neg \left(x \leq 1.9999999920083944 \cdot 10^{-12}\right):\\ \;\;\;\;\left(1 + \frac{s}{x \cdot x}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot 4 + x \cdot \frac{x}{s}}\\ \end{array} \]

Alternative 8
Accuracy	81.4%
Cost	489

\[\begin{array}{l} \mathbf{if}\;x \leq -1.0000000116860974 \cdot 10^{-7} \lor \neg \left(x \leq 1.9999999920083944 \cdot 10^{-12}\right):\\ \;\;\;\;\left(1 + \frac{s}{x \cdot x}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot 4 + \frac{x}{\frac{s}{x}}}\\ \end{array} \]

Alternative 9
Accuracy	80.4%
Cost	425

\[\begin{array}{l} \mathbf{if}\;x \leq -1.0000000116860974 \cdot 10^{-7} \lor \neg \left(x \leq 9.999999960041972 \cdot 10^{-13}\right):\\ \;\;\;\;\left(1 + \frac{s}{x \cdot x}\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]

Alternative 10
Accuracy	62.9%
Cost	297

\[\begin{array}{l} \mathbf{if}\;x \leq -3.999999989900971 \cdot 10^{-6} \lor \neg \left(x \leq 0.00018000000272877514\right):\\ \;\;\;\;\frac{s}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]

Alternative 11
Accuracy	27.2%
Cost	96

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

Reproduce

herbie shell --seed 2023135 
(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))))))