Logistic distribution

Percentage Accurate: 99.6% → 99.7%

Time: 12.9s

Alternatives: 7

Speedup: 1.2×

Specification

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

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{-\left|x\right|}{s}}\\ t_1 := 1 + t\_0\\ \frac{t\_0}{\left(s \cdot t\_1\right) \cdot t\_1} \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ (- (fabs x)) s))) (t_1 (+ 1.0 t_0)))
   (/ t_0 (* (* s t_1) t_1))))

C

float code(float x, float s) {
	float t_0 = expf((-fabsf(x) / s));
	float t_1 = 1.0f + t_0;
	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
    real(4) :: t_0
    real(4) :: t_1
    t_0 = exp((-abs(x) / s))
    t_1 = 1.0e0 + t_0
    code = t_0 / ((s * t_1) * t_1)
end function

Julia

function code(x, s)
	t_0 = exp(Float32(Float32(-abs(x)) / s))
	t_1 = Float32(Float32(1.0) + t_0)
	return Float32(t_0 / Float32(Float32(s * t_1) * t_1))
end

MATLAB

function tmp = code(x, s)
	t_0 = exp((-abs(x) / s));
	t_1 = single(1.0) + t_0;
	tmp = t_0 / ((s * t_1) * t_1);
end

Initial Program: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{-\left|x\right|}{s}}\\ t_1 := 1 + t\_0\\ \frac{t\_0}{\left(s \cdot t\_1\right) \cdot t\_1} \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ (- (fabs x)) s))) (t_1 (+ 1.0 t_0)))
   (/ t_0 (* (* s t_1) t_1))))

C

float code(float x, float s) {
	float t_0 = expf((-fabsf(x) / s));
	float t_1 = 1.0f + t_0;
	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
    real(4) :: t_0
    real(4) :: t_1
    t_0 = exp((-abs(x) / s))
    t_1 = 1.0e0 + t_0
    code = t_0 / ((s * t_1) * t_1)
end function

Julia

function code(x, s)
	t_0 = exp(Float32(Float32(-abs(x)) / s))
	t_1 = Float32(Float32(1.0) + t_0)
	return Float32(t_0 / Float32(Float32(s * t_1) * t_1))
end

MATLAB

function tmp = code(x, s)
	t_0 = exp((-abs(x) / s));
	t_1 = single(1.0) + t_0;
	tmp = t_0 / ((s * t_1) * t_1);
end

Alternative 1: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{-\left|x\right|}{s}}\\ \frac{t\_0}{\left(t\_0 + 1\right) \cdot \left(s + \frac{s}{e^{\frac{\left|x\right|}{s}}}\right)} \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ (- (fabs x)) s))))
   (/ t_0 (* (+ t_0 1.0) (+ s (/ s (exp (/ (fabs x) s))))))))

C

float code(float x, float s) {
	float t_0 = expf((-fabsf(x) / s));
	return t_0 / ((t_0 + 1.0f) * (s + (s / expf((fabsf(x) / s)))));
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    t_0 = exp((-abs(x) / s))
    code = t_0 / ((t_0 + 1.0e0) * (s + (s / exp((abs(x) / s)))))
end function

Julia

function code(x, s)
	t_0 = exp(Float32(Float32(-abs(x)) / s))
	return Float32(t_0 / Float32(Float32(t_0 + Float32(1.0)) * Float32(s + Float32(s / exp(Float32(abs(x) / s))))))
end

MATLAB

function tmp = code(x, s)
	t_0 = exp((-abs(x) / s));
	tmp = t_0 / ((t_0 + single(1.0)) * (s + (s / exp((abs(x) / s)))));
end

Derivation

1. Initial program 99.6%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
2. Step-by-step derivation

   1. *-commutative 99.6%

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

   2. distribute-lft-in 99.6%

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

   3. *-rgt-identity 99.6%

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

   4. fabs-neg 99.6%

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

   5. distribute-frac-neg 99.6%

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

   6. exp-neg 99.6%

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

   7. associate-*r/ 99.6%

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

   8. *-rgt-identity 99.6%

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

   9. *-lft-identity 99.6%

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

   10. metadata-eval 99.6%

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

   11. times-frac 99.6%

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

   12. neg-mul-1 99.6%

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

   13. neg-mul-1 99.6%

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

   14. fabs-neg 99.6%

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

3. Simplified 99.6%

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

5. Final simplification 99.6%

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

Alternative 2: 99.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{-\left|x\right|}{s}}\\ \frac{t\_0}{s \cdot {\left(t\_0 + 1\right)}^{2}} \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ (- (fabs x)) s)))) (/ t_0 (* s (pow (+ t_0 1.0) 2.0)))))

C

float code(float x, float s) {
	float t_0 = expf((-fabsf(x) / s));
	return t_0 / (s * powf((t_0 + 1.0f), 2.0f));
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    t_0 = exp((-abs(x) / s))
    code = t_0 / (s * ((t_0 + 1.0e0) ** 2.0e0))
end function

Julia

function code(x, s)
	t_0 = exp(Float32(Float32(-abs(x)) / s))
	return Float32(t_0 / Float32(s * (Float32(t_0 + Float32(1.0)) ^ Float32(2.0))))
end

MATLAB

function tmp = code(x, s)
	t_0 = exp((-abs(x) / s));
	tmp = t_0 / (s * ((t_0 + single(1.0)) ^ single(2.0)));
end

Derivation

1. Initial program 99.6%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
2. Step-by-step derivation

   1. associate-*l* 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in x around 0 99.6%

   \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{s \cdot \color{blue}{{\left(1 + e^{-1 \cdot \frac{\left|x\right|}{s}}\right)}^{2}}} \]
6. Step-by-step derivation

   1. associate-*r/ 99.6%

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

   2. mul-1-neg 99.6%

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

7. Simplified 99.6%

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

8. Final simplification 99.6%

   \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{s \cdot {\left(e^{\frac{-\left|x\right|}{s}} + 1\right)}^{2}} \]
9. Add Preprocessing

Alternative 3: 74.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.006000000052154064:\\ \;\;\;\;\frac{e^{\frac{x}{s} + -2 \cdot \mathsf{log1p}\left(e^{\frac{x}{s}}\right)}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{-x}{s}}}{s} \cdot 0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (fabs x) 0.006000000052154064)
   (/ (exp (+ (/ x s) (* -2.0 (log1p (exp (/ x s)))))) s)
   (* (/ (exp (/ (- x) s)) s) 0.25)))

C

float code(float x, float s) {
	float tmp;
	if (fabsf(x) <= 0.006000000052154064f) {
		tmp = expf(((x / s) + (-2.0f * log1pf(expf((x / s)))))) / s;
	} else {
		tmp = (expf((-x / s)) / s) * 0.25f;
	}
	return tmp;
}

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (abs(x) <= Float32(0.006000000052154064))
		tmp = Float32(exp(Float32(Float32(x / s) + Float32(Float32(-2.0) * log1p(exp(Float32(x / s)))))) / s);
	else
		tmp = Float32(Float32(exp(Float32(Float32(-x) / s)) / s) * Float32(0.25));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (fabs.f32 x) < 0.00600000005

   1. Initial program 99.2%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in x around 0 99.2%

      \[\leadsto \frac{e^{\frac{-\left|x\right|}{s}}}{s \cdot \color{blue}{{\left(1 + e^{-1 \cdot \frac{\left|x\right|}{s}}\right)}^{2}}} \]
   6. Step-by-step derivation

      1. associate-*r/ 99.2%

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

      2. mul-1-neg 99.2%

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

   7. Simplified 99.2%

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

   9. Applied egg-rr 94.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{e^{\frac{x}{s} + -2 \cdot \mathsf{log1p}\left(e^{\frac{x}{s}}\right)}}{s}\right)} - 1} \]
   10. Step-by-step derivation

       1. expm1-def 95.8%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{e^{\frac{x}{s} + -2 \cdot \mathsf{log1p}\left(e^{\frac{x}{s}}\right)}}{s}\right)\right)} \]

       2. expm1-log1p 99.5%

          \[\leadsto \color{blue}{\frac{e^{\frac{x}{s} + -2 \cdot \mathsf{log1p}\left(e^{\frac{x}{s}}\right)}}{s}} \]

   11. Simplified 99.5%

       \[\leadsto \color{blue}{\frac{e^{\frac{x}{s} + -2 \cdot \mathsf{log1p}\left(e^{\frac{x}{s}}\right)}}{s}} \]

   if 0.00600000005 < (fabs.f32 x)

   1. Initial program 100.0%

      \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{e^{\frac{-\left|x\right|}{s}}}{s \cdot \left(\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r* 100.0%

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

      2. div-inv 100.0%

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

   6. Applied egg-rr 52.1%

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

   7. Taylor expanded in x around 0 27.6%

      \[\leadsto \frac{e^{\frac{x}{s}}}{s} \cdot \color{blue}{\left(0.25 + -0.25 \cdot \frac{x}{s}\right)} \]
   8. Step-by-step derivation

      1. *-commutative 27.6%

         \[\leadsto \frac{e^{\frac{x}{s}}}{s} \cdot \left(0.25 + \color{blue}{\frac{x}{s} \cdot -0.25}\right) \]

   9. Simplified 27.6%

      \[\leadsto \frac{e^{\frac{x}{s}}}{s} \cdot \color{blue}{\left(0.25 + \frac{x}{s} \cdot -0.25\right)} \]
   10. Step-by-step derivation

       1. add-sqr-sqrt 1.5%

          \[\leadsto \frac{e^{\color{blue}{\sqrt{\frac{x}{s}} \cdot \sqrt{\frac{x}{s}}}}}{s} \cdot \left(0.25 + \frac{x}{s} \cdot -0.25\right) \]

       2. sqrt-unprod 3.1%

          \[\leadsto \frac{e^{\color{blue}{\sqrt{\frac{x}{s} \cdot \frac{x}{s}}}}}{s} \cdot \left(0.25 + \frac{x}{s} \cdot -0.25\right) \]

       3. add-sqr-sqrt 1.5%

          \[\leadsto \frac{e^{\sqrt{\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{s} \cdot \frac{x}{s}}}}{s} \cdot \left(0.25 + \frac{x}{s} \cdot -0.25\right) \]

       4. fabs-sqr 1.5%

          \[\leadsto \frac{e^{\sqrt{\frac{\color{blue}{\left|\sqrt{x} \cdot \sqrt{x}\right|}}{s} \cdot \frac{x}{s}}}}{s} \cdot \left(0.25 + \frac{x}{s} \cdot -0.25\right) \]

       5. add-sqr-sqrt 1.5%

          \[\leadsto \frac{e^{\sqrt{\frac{\left|\color{blue}{x}\right|}{s} \cdot \frac{x}{s}}}}{s} \cdot \left(0.25 + \frac{x}{s} \cdot -0.25\right) \]

       6. add-sqr-sqrt 1.5%

          \[\leadsto \frac{e^{\sqrt{\frac{\left|x\right|}{s} \cdot \frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{s}}}}{s} \cdot \left(0.25 + \frac{x}{s} \cdot -0.25\right) \]

       7. fabs-sqr 1.5%

          \[\leadsto \frac{e^{\sqrt{\frac{\left|x\right|}{s} \cdot \frac{\color{blue}{\left|\sqrt{x} \cdot \sqrt{x}\right|}}{s}}}}{s} \cdot \left(0.25 + \frac{x}{s} \cdot -0.25\right) \]

       8. add-sqr-sqrt 3.1%

          \[\leadsto \frac{e^{\sqrt{\frac{\left|x\right|}{s} \cdot \frac{\left|\color{blue}{x}\right|}{s}}}}{s} \cdot \left(0.25 + \frac{x}{s} \cdot -0.25\right) \]

       9. sqr-neg 3.1%

          \[\leadsto \frac{e^{\sqrt{\color{blue}{\left(-\frac{\left|x\right|}{s}\right) \cdot \left(-\frac{\left|x\right|}{s}\right)}}}}{s} \cdot \left(0.25 + \frac{x}{s} \cdot -0.25\right) \]

       10. distribute-frac-neg 3.1%

           \[\leadsto \frac{e^{\sqrt{\color{blue}{\frac{-\left|x\right|}{s}} \cdot \left(-\frac{\left|x\right|}{s}\right)}}}{s} \cdot \left(0.25 + \frac{x}{s} \cdot -0.25\right) \]

       11. distribute-frac-neg 3.1%

           \[\leadsto \frac{e^{\sqrt{\frac{-\left|x\right|}{s} \cdot \color{blue}{\frac{-\left|x\right|}{s}}}}}{s} \cdot \left(0.25 + \frac{x}{s} \cdot -0.25\right) \]

       12. sqrt-unprod -0.0%

           \[\leadsto \frac{e^{\color{blue}{\sqrt{\frac{-\left|x\right|}{s}} \cdot \sqrt{\frac{-\left|x\right|}{s}}}}}{s} \cdot \left(0.25 + \frac{x}{s} \cdot -0.25\right) \]

       13. add-sqr-sqrt 53.8%

           \[\leadsto \frac{e^{\color{blue}{\frac{-\left|x\right|}{s}}}}{s} \cdot \left(0.25 + \frac{x}{s} \cdot -0.25\right) \]

       14. distribute-frac-neg 53.8%

           \[\leadsto \frac{e^{\color{blue}{-\frac{\left|x\right|}{s}}}}{s} \cdot \left(0.25 + \frac{x}{s} \cdot -0.25\right) \]

       15. exp-neg 53.8%

           \[\leadsto \frac{\color{blue}{\frac{1}{e^{\frac{\left|x\right|}{s}}}}}{s} \cdot \left(0.25 + \frac{x}{s} \cdot -0.25\right) \]

   11. Applied egg-rr 29.4%

       \[\leadsto \frac{\color{blue}{\frac{1}{e^{\frac{x}{s}}}}}{s} \cdot \left(0.25 + \frac{x}{s} \cdot -0.25\right) \]
   12. Step-by-step derivation

       1. rec-exp 29.4%

          \[\leadsto \frac{\color{blue}{e^{-\frac{x}{s}}}}{s} \cdot \left(0.25 + \frac{x}{s} \cdot -0.25\right) \]

       2. distribute-neg-frac 29.4%

          \[\leadsto \frac{e^{\color{blue}{\frac{-x}{s}}}}{s} \cdot \left(0.25 + \frac{x}{s} \cdot -0.25\right) \]

   13. Simplified 29.4%

       \[\leadsto \frac{\color{blue}{e^{\frac{-x}{s}}}}{s} \cdot \left(0.25 + \frac{x}{s} \cdot -0.25\right) \]

   14. Taylor expanded in x around 0 49.5%

       \[\leadsto \frac{e^{\frac{-x}{s}}}{s} \cdot \color{blue}{0.25} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.006000000052154064:\\ \;\;\;\;\frac{e^{\frac{x}{s} + -2 \cdot \mathsf{log1p}\left(e^{\frac{x}{s}}\right)}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{-x}{s}}}{s} \cdot 0.25\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 62.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\left(e^{\frac{-\left|x\right|}{s}} + 1\right) \cdot \mathsf{fma}\left(s, {e}^{\left(\frac{x}{s}\right)}, s\right)} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (/ 1.0 (* (+ (exp (/ (- (fabs x)) s)) 1.0) (fma s (pow E (/ x s)) s))))

C

float code(float x, float s) {
	return 1.0f / ((expf((-fabsf(x) / s)) + 1.0f) * fmaf(s, powf(((float) M_E), (x / s)), s));
}

Julia

function code(x, s)
	return Float32(Float32(1.0) / Float32(Float32(exp(Float32(Float32(-abs(x)) / s)) + Float32(1.0)) * fma(s, (Float32(exp(1)) ^ Float32(x / s)), s)))
end

Derivation

1. Initial program 99.6%

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

3. Simplified 99.5%

   \[\leadsto \color{blue}{\frac{1}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \mathsf{fma}\left(s, e^{\frac{\left|x\right|}{s}}, s\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. frac-2neg 99.5%

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

   2. frac-2neg 99.5%

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

   3. add-sqr-sqrt 99.4%

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

   4. sqrt-unprod 95.5%

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

   5. sqr-neg 95.5%

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

   6. sqrt-unprod -0.0%

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

   7. add-sqr-sqrt 26.9%

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

   8. *-un-lft-identity 26.9%

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

   9. exp-prod 26.9%

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

   10. add-sqr-sqrt -0.0%

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

   11. sqrt-unprod 95.5%

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

   12. sqr-neg 95.5%

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

   13. sqrt-unprod 99.4%

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

   14. add-sqr-sqrt 99.5%

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

   15. add-sqr-sqrt 46.4%

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

   16. fabs-sqr 46.4%

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

   17. add-sqr-sqrt 62.5%

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

6. Applied egg-rr 62.5%

   \[\leadsto \frac{1}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \mathsf{fma}\left(s, \color{blue}{{\left(e^{1}\right)}^{\left(\frac{x}{s}\right)}}, s\right)} \]
7. Step-by-step derivation

   1. exp-1-e 62.5%

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

8. Simplified 62.5%

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

9. Final simplification 62.5%

   \[\leadsto \frac{1}{\left(e^{\frac{-\left|x\right|}{s}} + 1\right) \cdot \mathsf{fma}\left(s, {e}^{\left(\frac{x}{s}\right)}, s\right)} \]
10. Add Preprocessing

Alternative 5: 62.9% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (x s)
 :precision binary32
 (/ 1.0 (* (+ (exp (/ (- (fabs x)) s)) 1.0) (+ s (* s (exp (/ x s)))))))

C

float code(float x, float s) {
	return 1.0f / ((expf((-fabsf(x) / s)) + 1.0f) * (s + (s * expf((x / s)))));
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 1.0e0 / ((exp((-abs(x) / s)) + 1.0e0) * (s + (s * exp((x / s)))))
end function

Julia

function code(x, s)
	return Float32(Float32(1.0) / Float32(Float32(exp(Float32(Float32(-abs(x)) / s)) + Float32(1.0)) * Float32(s + Float32(s * exp(Float32(x / s))))))
end

MATLAB

function tmp = code(x, s)
	tmp = single(1.0) / ((exp((-abs(x) / s)) + single(1.0)) * (s + (s * exp((x / s)))));
end

Derivation

1. Initial program 99.6%

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

3. Simplified 99.5%

   \[\leadsto \color{blue}{\frac{1}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \mathsf{fma}\left(s, e^{\frac{\left|x\right|}{s}}, s\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. fma-udef 99.5%

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

   2. frac-2neg 99.5%

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

   3. frac-2neg 99.5%

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

   4. add-sqr-sqrt 46.4%

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

   5. fabs-sqr 46.4%

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

   6. add-sqr-sqrt 62.5%

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

6. Applied egg-rr 62.5%

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

7. Final simplification 62.5%

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

Alternative 6: 60.2% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \frac{e^{\frac{-x}{s}}}{s} \cdot 0.25 \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (* (/ (exp (/ (- x) s)) s) 0.25))

C

float code(float x, float s) {
	return (expf((-x / s)) / s) * 0.25f;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = (exp((-x / s)) / s) * 0.25e0
end function

Julia

function code(x, s)
	return Float32(Float32(exp(Float32(Float32(-x) / s)) / s) * Float32(0.25))
end

MATLAB

function tmp = code(x, s)
	tmp = (exp((-x / s)) / s) * single(0.25);
end

Derivation

1. Initial program 99.6%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
2. Step-by-step derivation

   1. associate-*l* 99.6%

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

3. Simplified 99.6%

   \[\leadsto \color{blue}{\frac{e^{\frac{-\left|x\right|}{s}}}{s \cdot \left(\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-/r* 99.6%

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

   2. div-inv 99.6%

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

6. Applied egg-rr 64.5%

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

7. Taylor expanded in x around 0 50.7%

   \[\leadsto \frac{e^{\frac{x}{s}}}{s} \cdot \color{blue}{\left(0.25 + -0.25 \cdot \frac{x}{s}\right)} \]
8. Step-by-step derivation

   1. *-commutative 50.7%

      \[\leadsto \frac{e^{\frac{x}{s}}}{s} \cdot \left(0.25 + \color{blue}{\frac{x}{s} \cdot -0.25}\right) \]

9. Simplified 50.7%

   \[\leadsto \frac{e^{\frac{x}{s}}}{s} \cdot \color{blue}{\left(0.25 + \frac{x}{s} \cdot -0.25\right)} \]
10. Step-by-step derivation

    1. add-sqr-sqrt 11.4%

       \[\leadsto \frac{e^{\color{blue}{\sqrt{\frac{x}{s}} \cdot \sqrt{\frac{x}{s}}}}}{s} \cdot \left(0.25 + \frac{x}{s} \cdot -0.25\right) \]

    2. sqrt-unprod 26.8%

       \[\leadsto \frac{e^{\color{blue}{\sqrt{\frac{x}{s} \cdot \frac{x}{s}}}}}{s} \cdot \left(0.25 + \frac{x}{s} \cdot -0.25\right) \]

    3. add-sqr-sqrt 11.4%

       \[\leadsto \frac{e^{\sqrt{\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{s} \cdot \frac{x}{s}}}}{s} \cdot \left(0.25 + \frac{x}{s} \cdot -0.25\right) \]

    4. fabs-sqr 11.4%

       \[\leadsto \frac{e^{\sqrt{\frac{\color{blue}{\left|\sqrt{x} \cdot \sqrt{x}\right|}}{s} \cdot \frac{x}{s}}}}{s} \cdot \left(0.25 + \frac{x}{s} \cdot -0.25\right) \]

    5. add-sqr-sqrt 16.1%

       \[\leadsto \frac{e^{\sqrt{\frac{\left|\color{blue}{x}\right|}{s} \cdot \frac{x}{s}}}}{s} \cdot \left(0.25 + \frac{x}{s} \cdot -0.25\right) \]

    6. add-sqr-sqrt 11.4%

       \[\leadsto \frac{e^{\sqrt{\frac{\left|x\right|}{s} \cdot \frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{s}}}}{s} \cdot \left(0.25 + \frac{x}{s} \cdot -0.25\right) \]

    7. fabs-sqr 11.4%

       \[\leadsto \frac{e^{\sqrt{\frac{\left|x\right|}{s} \cdot \frac{\color{blue}{\left|\sqrt{x} \cdot \sqrt{x}\right|}}{s}}}}{s} \cdot \left(0.25 + \frac{x}{s} \cdot -0.25\right) \]

    8. add-sqr-sqrt 26.8%

       \[\leadsto \frac{e^{\sqrt{\frac{\left|x\right|}{s} \cdot \frac{\left|\color{blue}{x}\right|}{s}}}}{s} \cdot \left(0.25 + \frac{x}{s} \cdot -0.25\right) \]

    9. sqr-neg 26.8%

       \[\leadsto \frac{e^{\sqrt{\color{blue}{\left(-\frac{\left|x\right|}{s}\right) \cdot \left(-\frac{\left|x\right|}{s}\right)}}}}{s} \cdot \left(0.25 + \frac{x}{s} \cdot -0.25\right) \]

    10. distribute-frac-neg 26.8%

        \[\leadsto \frac{e^{\sqrt{\color{blue}{\frac{-\left|x\right|}{s}} \cdot \left(-\frac{\left|x\right|}{s}\right)}}}{s} \cdot \left(0.25 + \frac{x}{s} \cdot -0.25\right) \]

    11. distribute-frac-neg 26.8%

        \[\leadsto \frac{e^{\sqrt{\frac{-\left|x\right|}{s} \cdot \color{blue}{\frac{-\left|x\right|}{s}}}}}{s} \cdot \left(0.25 + \frac{x}{s} \cdot -0.25\right) \]

    12. sqrt-unprod -0.0%

        \[\leadsto \frac{e^{\color{blue}{\sqrt{\frac{-\left|x\right|}{s}} \cdot \sqrt{\frac{-\left|x\right|}{s}}}}}{s} \cdot \left(0.25 + \frac{x}{s} \cdot -0.25\right) \]

    13. add-sqr-sqrt 74.2%

        \[\leadsto \frac{e^{\color{blue}{\frac{-\left|x\right|}{s}}}}{s} \cdot \left(0.25 + \frac{x}{s} \cdot -0.25\right) \]

    14. distribute-frac-neg 74.2%

        \[\leadsto \frac{e^{\color{blue}{-\frac{\left|x\right|}{s}}}}{s} \cdot \left(0.25 + \frac{x}{s} \cdot -0.25\right) \]

    15. exp-neg 74.2%

        \[\leadsto \frac{\color{blue}{\frac{1}{e^{\frac{\left|x\right|}{s}}}}}{s} \cdot \left(0.25 + \frac{x}{s} \cdot -0.25\right) \]

11. Applied egg-rr 50.3%

    \[\leadsto \frac{\color{blue}{\frac{1}{e^{\frac{x}{s}}}}}{s} \cdot \left(0.25 + \frac{x}{s} \cdot -0.25\right) \]
12. Step-by-step derivation

    1. rec-exp 50.3%

       \[\leadsto \frac{\color{blue}{e^{-\frac{x}{s}}}}{s} \cdot \left(0.25 + \frac{x}{s} \cdot -0.25\right) \]

    2. distribute-neg-frac 50.3%

       \[\leadsto \frac{e^{\color{blue}{\frac{-x}{s}}}}{s} \cdot \left(0.25 + \frac{x}{s} \cdot -0.25\right) \]

13. Simplified 50.3%

    \[\leadsto \frac{\color{blue}{e^{\frac{-x}{s}}}}{s} \cdot \left(0.25 + \frac{x}{s} \cdot -0.25\right) \]

14. Taylor expanded in x around 0 60.1%

    \[\leadsto \frac{e^{\frac{-x}{s}}}{s} \cdot \color{blue}{0.25} \]

15. Final simplification 60.1%

    \[\leadsto \frac{e^{\frac{-x}{s}}}{s} \cdot 0.25 \]
16. Add Preprocessing

Alternative 7: 27.2% accurate, 206.7× speedup

Math

\[\begin{array}{l} \\ \frac{0.25}{s} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (/ 0.25 s))

C

float code(float x, float s) {
	return 0.25f / s;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 0.25e0 / s
end function

Julia

function code(x, s)
	return Float32(Float32(0.25) / s)
end

MATLAB

function tmp = code(x, s)
	tmp = single(0.25) / s;
end

Derivation

1. Initial program 99.6%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
2. Step-by-step derivation

   1. associate-*l* 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in s around inf 29.5%

   \[\leadsto \color{blue}{\frac{0.25}{s}} \]

6. Final simplification 29.5%

   \[\leadsto \frac{0.25}{s} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024031 
(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))))))