Logistic distribution

Percentage Accurate: 99.6% → 99.6%

Time: 13.6s

Alternatives: 9

Speedup: 2.0×

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: 99.6% accurate, 1.5× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s)
 :precision binary32
 (/ (pow (exp -1.0) (/ x s)) (* s (pow (+ (exp (/ (- x) s)) 1.0) 2.0))))

C

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

Fortran

NOTE: x should be positive before calling this function
real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = (exp((-1.0e0)) ** (x / s)) / (s * ((exp((-x / s)) + 1.0e0) ** 2.0e0))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

   1. /-rgt-identity 99.1%

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

   2. associate-/l/ 99.1%

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

   3. *-commutative 99.1%

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

   4. associate-*l* 99.1%

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

   5. associate-*l* 99.1%

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

   6. *-rgt-identity 99.1%

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

   7. +-commutative 99.1%

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

   8. +-commutative 99.1%

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

3. Simplified 99.1%

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

4. Taylor expanded in x around 0 99.1%

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

   1. +-commutative 99.1%

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

   2. associate-*r/ 99.1%

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

   3. mul-1-neg 99.1%

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

6. Simplified 99.1%

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

   1. distribute-frac-neg 99.1%

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

   2. exp-neg 99.1%

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

   3. add-sqr-sqrt 99.1%

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

   4. sqrt-unprod 96.4%

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

   5. sqr-neg 96.4%

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

   6. sqrt-unprod -0.0%

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

   7. add-sqr-sqrt 92.1%

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

   8. add-sqr-sqrt -0.0%

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

   9. add-sqr-sqrt 92.1%

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

   10. add-sqr-sqrt -0.0%

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

   11. sqrt-unprod 96.4%

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

   12. sqr-neg 96.4%

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

   13. sqrt-unprod 99.1%

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

   14. add-sqr-sqrt 99.1%

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

   15. add-sqr-sqrt 47.7%

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

   16. fabs-sqr 47.7%

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

   17. add-sqr-sqrt 95.1%

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

8. Applied egg-rr 95.1%

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

   1. rec-exp 95.2%

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

10. Simplified 95.2%

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

    1. distribute-frac-neg 99.1%

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

    2. exp-neg 99.1%

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

    3. add-sqr-sqrt 99.1%

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

    4. sqrt-unprod 96.4%

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

    5. sqr-neg 96.4%

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

    6. sqrt-unprod -0.0%

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

    7. add-sqr-sqrt 92.1%

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

    8. add-sqr-sqrt -0.0%

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

    9. add-sqr-sqrt 92.1%

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

    10. add-sqr-sqrt -0.0%

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

    11. sqrt-unprod 96.4%

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

    12. sqr-neg 96.4%

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

    13. sqrt-unprod 99.1%

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

    14. add-sqr-sqrt 99.1%

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

    15. add-sqr-sqrt 47.7%

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

    16. fabs-sqr 47.7%

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

    17. add-sqr-sqrt 95.1%

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

12. Applied egg-rr 63.9%

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

    1. rec-exp 95.2%

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

14. Simplified 64.0%

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

    1. neg-mul-1 64.0%

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

    2. metadata-eval 64.0%

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

    3. exp-prod 63.9%

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

    4. metadata-eval 63.9%

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

16. Applied egg-rr 63.9%

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

17. Final simplification 63.9%

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

Alternative 2: 99.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;-\left|x\right| \leq -200:\\ \;\;\;\;\frac{e^{\frac{-x}{s}}}{s \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{x}{s} + \mathsf{log1p}\left(e^{\frac{x}{s}}\right) \cdot -2}}{s}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s)
 :precision binary32
 (if (<= (- (fabs x)) -200.0)
   (/ (exp (/ (- x) s)) (* s 4.0))
   (/ (exp (+ (/ x s) (* (log1p (exp (/ x s))) -2.0))) s)))

C

x = abs(x);
float code(float x, float s) {
	float tmp;
	if (-fabsf(x) <= -200.0f) {
		tmp = expf((-x / s)) / (s * 4.0f);
	} else {
		tmp = expf(((x / s) + (log1pf(expf((x / s))) * -2.0f))) / s;
	}
	return tmp;
}

Julia

x = abs(x)
function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-abs(x)) <= Float32(-200.0))
		tmp = Float32(exp(Float32(Float32(-x) / s)) / Float32(s * Float32(4.0)));
	else
		tmp = Float32(exp(Float32(Float32(x / s) + Float32(log1p(exp(Float32(x / s))) * Float32(-2.0)))) / s);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (neg.f32 (fabs.f32 x)) < -200

   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. /-rgt-identity 100.0%

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

      2. associate-/l/ 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. associate-*l* 100.0%

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

      6. *-rgt-identity 100.0%

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

      7. +-commutative 100.0%

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

      8. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. associate-*r/ 100.0%

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

      3. mul-1-neg 100.0%

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

   6. Simplified 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. exp-neg 100.0%

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

      3. add-sqr-sqrt 100.0%

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

      4. sqrt-unprod 100.0%

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

      5. sqr-neg 100.0%

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

      6. sqrt-unprod -0.0%

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

      7. add-sqr-sqrt 100.0%

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

      8. add-sqr-sqrt -0.0%

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

      9. add-sqr-sqrt 100.0%

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

      10. add-sqr-sqrt -0.0%

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

      11. sqrt-unprod 100.0%

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

      12. sqr-neg 100.0%

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

      13. sqrt-unprod 100.0%

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

      14. add-sqr-sqrt 100.0%

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

      15. add-sqr-sqrt 48.7%

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

      16. fabs-sqr 48.7%

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

      17. add-sqr-sqrt 100.0%

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

   8. Applied egg-rr 100.0%

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

      1. rec-exp 100.0%

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

   10. Simplified 100.0%

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

       1. distribute-frac-neg 100.0%

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

       2. exp-neg 100.0%

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

       3. add-sqr-sqrt 100.0%

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

       4. sqrt-unprod 100.0%

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

       5. sqr-neg 100.0%

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

       6. sqrt-unprod -0.0%

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

       7. add-sqr-sqrt 100.0%

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

       8. add-sqr-sqrt -0.0%

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

       9. add-sqr-sqrt 100.0%

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

       10. add-sqr-sqrt -0.0%

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

       11. sqrt-unprod 100.0%

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

       12. sqr-neg 100.0%

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

       13. sqrt-unprod 100.0%

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

       14. add-sqr-sqrt 100.0%

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

       15. add-sqr-sqrt 48.7%

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

       16. fabs-sqr 48.7%

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

       17. add-sqr-sqrt 100.0%

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

   12. Applied egg-rr 48.7%

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

       1. rec-exp 100.0%

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

   14. Simplified 48.7%

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

   15. Taylor expanded in s around inf 50.3%

       \[\leadsto \frac{e^{-\frac{x}{s}}}{\color{blue}{4 \cdot s}} \]
   16. Step-by-step derivation

       1. *-commutative 50.3%

          \[\leadsto \frac{e^{-\frac{x}{s}}}{\color{blue}{s \cdot 4}} \]

   17. Simplified 50.3%

       \[\leadsto \frac{e^{-\frac{x}{s}}}{\color{blue}{s \cdot 4}} \]

   if -200 < (neg.f32 (fabs.f32 x))

   1. Initial program 98.3%

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

      1. /-rgt-identity 98.3%

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

      2. associate-/l/ 98.3%

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

      3. *-commutative 98.3%

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

      4. associate-*l* 98.3%

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

      5. associate-*l* 98.3%

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

      6. *-rgt-identity 98.3%

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

      7. +-commutative 98.3%

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

      8. +-commutative 98.3%

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

   3. Simplified 98.3%

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

      1. associate-/r* 98.4%

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

      2. div-inv 98.4%

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

      3. add-sqr-sqrt -0.0%

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

      4. sqrt-unprod 48.0%

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

      5. sqr-neg 48.0%

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

      6. sqrt-unprod 47.6%

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

      7. add-sqr-sqrt 47.6%

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

      8. add-sqr-sqrt 23.7%

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

      9. fabs-sqr 23.7%

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

      10. add-sqr-sqrt 75.1%

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

      11. pow2 75.1%

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

   5. Applied egg-rr 80.0%

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

      1. associate-*l/ 80.0%

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

      2. add-exp-log 79.9%

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

      3. prod-exp 98.1%

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

      4. pow-flip 98.1%

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

      5. log-pow 98.2%

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

      6. +-commutative 98.2%

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

      7. log1p-udef 98.3%

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

      8. metadata-eval 98.3%

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

   7. Applied egg-rr 98.3%

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

      1. *-commutative 98.3%

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

   9. Simplified 98.3%

      \[\leadsto \color{blue}{\frac{e^{\frac{x}{s} + \mathsf{log1p}\left(e^{\frac{x}{s}}\right) \cdot -2}}{s}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\left|x\right| \leq -200:\\ \;\;\;\;\frac{e^{\frac{-x}{s}}}{s \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{x}{s} + \mathsf{log1p}\left(e^{\frac{x}{s}}\right) \cdot -2}}{s}\\ \end{array} \]

Alternative 3: 99.6% accurate, 2.0× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ (- x) s)))) (/ t_0 (* s (pow (+ t_0 1.0) 2.0)))))

C

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

Fortran

NOTE: x should be positive before calling this function
real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    t_0 = exp((-x / s))
    code = t_0 / (s * ((t_0 + 1.0e0) ** 2.0e0))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

   1. /-rgt-identity 99.1%

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

   2. associate-/l/ 99.1%

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

   3. *-commutative 99.1%

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

   4. associate-*l* 99.1%

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

   5. associate-*l* 99.1%

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

   6. *-rgt-identity 99.1%

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

   7. +-commutative 99.1%

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

   8. +-commutative 99.1%

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

3. Simplified 99.1%

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

4. Taylor expanded in x around 0 99.1%

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

   1. +-commutative 99.1%

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

   2. associate-*r/ 99.1%

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

   3. mul-1-neg 99.1%

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

6. Simplified 99.1%

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

   1. distribute-frac-neg 99.1%

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

   2. exp-neg 99.1%

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

   3. add-sqr-sqrt 99.1%

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

   4. sqrt-unprod 96.4%

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

   5. sqr-neg 96.4%

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

   6. sqrt-unprod -0.0%

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

   7. add-sqr-sqrt 92.1%

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

   8. add-sqr-sqrt -0.0%

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

   9. add-sqr-sqrt 92.1%

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

   10. add-sqr-sqrt -0.0%

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

   11. sqrt-unprod 96.4%

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

   12. sqr-neg 96.4%

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

   13. sqrt-unprod 99.1%

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

   14. add-sqr-sqrt 99.1%

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

   15. add-sqr-sqrt 47.7%

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

   16. fabs-sqr 47.7%

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

   17. add-sqr-sqrt 95.1%

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

8. Applied egg-rr 95.1%

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

   1. rec-exp 95.2%

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

10. Simplified 95.2%

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

    1. distribute-frac-neg 99.1%

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

    2. exp-neg 99.1%

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

    3. add-sqr-sqrt 99.1%

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

    4. sqrt-unprod 96.4%

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

    5. sqr-neg 96.4%

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

    6. sqrt-unprod -0.0%

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

    7. add-sqr-sqrt 92.1%

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

    8. add-sqr-sqrt -0.0%

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

    9. add-sqr-sqrt 92.1%

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

    10. add-sqr-sqrt -0.0%

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

    11. sqrt-unprod 96.4%

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

    12. sqr-neg 96.4%

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

    13. sqrt-unprod 99.1%

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

    14. add-sqr-sqrt 99.1%

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

    15. add-sqr-sqrt 47.7%

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

    16. fabs-sqr 47.7%

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

    17. add-sqr-sqrt 95.1%

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

12. Applied egg-rr 63.9%

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

    1. rec-exp 95.2%

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

14. Simplified 64.0%

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

15. Final simplification 64.0%

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

Alternative 4: 96.4% accurate, 2.9× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s)
 :precision binary32
 (/ (exp (/ (- x) s)) (* s (pow (+ 1.0 (- 1.0 (/ x s))) 2.0))))

C

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

Fortran

NOTE: x should be positive before calling this function
real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = exp((-x / s)) / (s * ((1.0e0 + (1.0e0 - (x / s))) ** 2.0e0))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

   1. /-rgt-identity 99.1%

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

   2. associate-/l/ 99.1%

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

   3. *-commutative 99.1%

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

   4. associate-*l* 99.1%

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

   5. associate-*l* 99.1%

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

   6. *-rgt-identity 99.1%

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

   7. +-commutative 99.1%

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

   8. +-commutative 99.1%

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

3. Simplified 99.1%

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

4. Taylor expanded in x around 0 99.1%

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

   1. +-commutative 99.1%

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

   2. associate-*r/ 99.1%

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

   3. mul-1-neg 99.1%

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

6. Simplified 99.1%

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

   1. distribute-frac-neg 99.1%

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

   2. exp-neg 99.1%

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

   3. add-sqr-sqrt 99.1%

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

   4. sqrt-unprod 96.4%

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

   5. sqr-neg 96.4%

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

   6. sqrt-unprod -0.0%

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

   7. add-sqr-sqrt 92.1%

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

   8. add-sqr-sqrt -0.0%

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

   9. add-sqr-sqrt 92.1%

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

   10. add-sqr-sqrt -0.0%

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

   11. sqrt-unprod 96.4%

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

   12. sqr-neg 96.4%

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

   13. sqrt-unprod 99.1%

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

   14. add-sqr-sqrt 99.1%

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

   15. add-sqr-sqrt 47.7%

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

   16. fabs-sqr 47.7%

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

   17. add-sqr-sqrt 95.1%

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

8. Applied egg-rr 95.1%

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

   1. rec-exp 95.2%

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

10. Simplified 95.2%

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

    1. distribute-frac-neg 99.1%

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

    2. exp-neg 99.1%

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

    3. add-sqr-sqrt 99.1%

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

    4. sqrt-unprod 96.4%

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

    5. sqr-neg 96.4%

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

    6. sqrt-unprod -0.0%

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

    7. add-sqr-sqrt 92.1%

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

    8. add-sqr-sqrt -0.0%

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

    9. add-sqr-sqrt 92.1%

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

    10. add-sqr-sqrt -0.0%

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

    11. sqrt-unprod 96.4%

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

    12. sqr-neg 96.4%

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

    13. sqrt-unprod 99.1%

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

    14. add-sqr-sqrt 99.1%

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

    15. add-sqr-sqrt 47.7%

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

    16. fabs-sqr 47.7%

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

    17. add-sqr-sqrt 95.1%

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

12. Applied egg-rr 63.9%

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

    1. rec-exp 95.2%

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

14. Simplified 64.0%

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

15. Taylor expanded in x around 0 60.6%

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

    1. mul-1-neg 60.6%

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

    2. neg-sub0 60.6%

       \[\leadsto \frac{e^{-\frac{x}{s}}}{s \cdot {\left(\left(1 + \color{blue}{\left(0 - \frac{x}{s}\right)}\right) + 1\right)}^{2}} \]

    3. metadata-eval 60.6%

       \[\leadsto \frac{e^{-\frac{x}{s}}}{s \cdot {\left(\left(1 + \left(\color{blue}{\left(-0.125 + 0.125\right)} - \frac{x}{s}\right)\right) + 1\right)}^{2}} \]

    4. associate-+r- 60.6%

       \[\leadsto \frac{e^{-\frac{x}{s}}}{s \cdot {\left(\color{blue}{\left(\left(1 + \left(-0.125 + 0.125\right)\right) - \frac{x}{s}\right)} + 1\right)}^{2}} \]

    5. metadata-eval 60.6%

       \[\leadsto \frac{e^{-\frac{x}{s}}}{s \cdot {\left(\left(\left(1 + \color{blue}{0}\right) - \frac{x}{s}\right) + 1\right)}^{2}} \]

    6. metadata-eval 60.6%

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

17. Simplified 60.6%

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

18. Final simplification 60.6%

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

Alternative 5: 94.9% accurate, 5.7× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \frac{\frac{1}{e^{\frac{x}{s}}}}{s \cdot 4} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s) :precision binary32 (/ (/ 1.0 (exp (/ x s))) (* s 4.0)))

C

x = abs(x);
float code(float x, float s) {
	return (1.0f / expf((x / s))) / (s * 4.0f);
}

Fortran

NOTE: x should be positive before calling this function
real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = (1.0e0 / exp((x / s))) / (s * 4.0e0)
end function

Julia

x = abs(x)
function code(x, s)
	return Float32(Float32(Float32(1.0) / exp(Float32(x / s))) / Float32(s * Float32(4.0)))
end

MATLAB

x = abs(x)
function tmp = code(x, s)
	tmp = (single(1.0) / exp((x / s))) / (s * single(4.0));
end

Derivation

1. Initial program 99.1%

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

   1. /-rgt-identity 99.1%

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

   2. associate-/l/ 99.1%

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

   3. *-commutative 99.1%

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

   4. associate-*l* 99.1%

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

   5. associate-*l* 99.1%

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

   6. *-rgt-identity 99.1%

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

   7. +-commutative 99.1%

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

   8. +-commutative 99.1%

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

3. Simplified 99.1%

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

4. Taylor expanded in x around 0 99.1%

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

   1. +-commutative 99.1%

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

   2. associate-*r/ 99.1%

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

   3. mul-1-neg 99.1%

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

6. Simplified 99.1%

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

   1. distribute-frac-neg 99.1%

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

   2. exp-neg 99.1%

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

   3. add-sqr-sqrt 99.1%

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

   4. sqrt-unprod 96.4%

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

   5. sqr-neg 96.4%

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

   6. sqrt-unprod -0.0%

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

   7. add-sqr-sqrt 92.1%

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

   8. add-sqr-sqrt -0.0%

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

   9. add-sqr-sqrt 92.1%

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

   10. add-sqr-sqrt -0.0%

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

   11. sqrt-unprod 96.4%

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

   12. sqr-neg 96.4%

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

   13. sqrt-unprod 99.1%

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

   14. add-sqr-sqrt 99.1%

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

   15. add-sqr-sqrt 47.7%

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

   16. fabs-sqr 47.7%

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

   17. add-sqr-sqrt 95.1%

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

8. Applied egg-rr 61.5%

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

9. Taylor expanded in s around inf 58.8%

   \[\leadsto \frac{\frac{1}{e^{\frac{x}{s}}}}{s \cdot \color{blue}{4}} \]

10. Final simplification 58.8%

    \[\leadsto \frac{\frac{1}{e^{\frac{x}{s}}}}{s \cdot 4} \]

Alternative 6: 94.9% accurate, 5.7× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \frac{e^{\frac{-x}{s}}}{s \cdot 4} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s) :precision binary32 (/ (exp (/ (- x) s)) (* s 4.0)))

C

x = abs(x);
float code(float x, float s) {
	return expf((-x / s)) / (s * 4.0f);
}

Fortran

NOTE: x should be positive before calling this function
real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = exp((-x / s)) / (s * 4.0e0)
end function

Julia

x = abs(x)
function code(x, s)
	return Float32(exp(Float32(Float32(-x) / s)) / Float32(s * Float32(4.0)))
end

MATLAB

x = abs(x)
function tmp = code(x, s)
	tmp = exp((-x / s)) / (s * single(4.0));
end

Derivation

1. Initial program 99.1%

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

   1. /-rgt-identity 99.1%

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

   2. associate-/l/ 99.1%

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

   3. *-commutative 99.1%

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

   4. associate-*l* 99.1%

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

   5. associate-*l* 99.1%

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

   6. *-rgt-identity 99.1%

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

   7. +-commutative 99.1%

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

   8. +-commutative 99.1%

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

3. Simplified 99.1%

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

4. Taylor expanded in x around 0 99.1%

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

   1. +-commutative 99.1%

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

   2. associate-*r/ 99.1%

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

   3. mul-1-neg 99.1%

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

6. Simplified 99.1%

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

   1. distribute-frac-neg 99.1%

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

   2. exp-neg 99.1%

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

   3. add-sqr-sqrt 99.1%

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

   4. sqrt-unprod 96.4%

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

   5. sqr-neg 96.4%

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

   6. sqrt-unprod -0.0%

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

   7. add-sqr-sqrt 92.1%

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

   8. add-sqr-sqrt -0.0%

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

   9. add-sqr-sqrt 92.1%

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

   10. add-sqr-sqrt -0.0%

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

   11. sqrt-unprod 96.4%

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

   12. sqr-neg 96.4%

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

   13. sqrt-unprod 99.1%

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

   14. add-sqr-sqrt 99.1%

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

   15. add-sqr-sqrt 47.7%

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

   16. fabs-sqr 47.7%

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

   17. add-sqr-sqrt 95.1%

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

8. Applied egg-rr 95.1%

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

   1. rec-exp 95.2%

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

10. Simplified 95.2%

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

    1. distribute-frac-neg 99.1%

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

    2. exp-neg 99.1%

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

    3. add-sqr-sqrt 99.1%

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

    4. sqrt-unprod 96.4%

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

    5. sqr-neg 96.4%

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

    6. sqrt-unprod -0.0%

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

    7. add-sqr-sqrt 92.1%

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

    8. add-sqr-sqrt -0.0%

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

    9. add-sqr-sqrt 92.1%

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

    10. add-sqr-sqrt -0.0%

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

    11. sqrt-unprod 96.4%

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

    12. sqr-neg 96.4%

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

    13. sqrt-unprod 99.1%

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

    14. add-sqr-sqrt 99.1%

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

    15. add-sqr-sqrt 47.7%

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

    16. fabs-sqr 47.7%

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

    17. add-sqr-sqrt 95.1%

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

12. Applied egg-rr 63.9%

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

    1. rec-exp 95.2%

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

14. Simplified 64.0%

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

15. Taylor expanded in s around inf 58.8%

    \[\leadsto \frac{e^{-\frac{x}{s}}}{\color{blue}{4 \cdot s}} \]
16. Step-by-step derivation

    1. *-commutative 58.8%

       \[\leadsto \frac{e^{-\frac{x}{s}}}{\color{blue}{s \cdot 4}} \]

17. Simplified 58.8%

    \[\leadsto \frac{e^{-\frac{x}{s}}}{\color{blue}{s \cdot 4}} \]

18. Final simplification 58.8%

    \[\leadsto \frac{e^{\frac{-x}{s}}}{s \cdot 4} \]

Alternative 7: 26.6% accurate, 36.5× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ -0.0625 \cdot \frac{\frac{x}{s}}{\frac{s}{\frac{x}{s}}} + 0.25 \cdot \frac{1}{s} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s)
 :precision binary32
 (+ (* -0.0625 (/ (/ x s) (/ s (/ x s)))) (* 0.25 (/ 1.0 s))))

C

x = abs(x);
float code(float x, float s) {
	return (-0.0625f * ((x / s) / (s / (x / s)))) + (0.25f * (1.0f / s));
}

Fortran

NOTE: x should be positive before calling this function
real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = ((-0.0625e0) * ((x / s) / (s / (x / s)))) + (0.25e0 * (1.0e0 / s))
end function

Julia

x = abs(x)
function code(x, s)
	return Float32(Float32(Float32(-0.0625) * Float32(Float32(x / s) / Float32(s / Float32(x / s)))) + Float32(Float32(0.25) * Float32(Float32(1.0) / s)))
end

MATLAB

x = abs(x)
function tmp = code(x, s)
	tmp = (single(-0.0625) * ((x / s) / (s / (x / s)))) + (single(0.25) * (single(1.0) / s));
end

Derivation

1. Initial program 99.1%

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

   1. /-rgt-identity 99.1%

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

   2. associate-/l/ 99.1%

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

   3. *-commutative 99.1%

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

   4. associate-*l* 99.1%

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

   5. associate-*l* 99.1%

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

   6. *-rgt-identity 99.1%

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

   7. +-commutative 99.1%

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

   8. +-commutative 99.1%

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

3. Simplified 99.1%

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

   1. associate-/r* 99.1%

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

   2. div-inv 99.1%

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

   3. add-sqr-sqrt -0.0%

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

   4. sqrt-unprod 27.5%

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

   5. sqr-neg 27.5%

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

   6. sqrt-unprod 27.3%

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

   7. add-sqr-sqrt 27.3%

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

   8. add-sqr-sqrt 13.5%

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

   9. fabs-sqr 13.5%

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

   10. add-sqr-sqrt 64.9%

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

   11. pow2 64.9%

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

5. Applied egg-rr 66.9%

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

6. Taylor expanded in x around 0 20.0%

   \[\leadsto \color{blue}{-0.0625 \cdot \frac{{x}^{2}}{{s}^{3}} + 0.25 \cdot \frac{1}{s}} \]
7. Step-by-step derivation

   1. *-un-lft-identity 20.0%

      \[\leadsto -0.0625 \cdot \frac{\color{blue}{1 \cdot {x}^{2}}}{{s}^{3}} + 0.25 \cdot \frac{1}{s} \]

   2. cube-mult 20.0%

      \[\leadsto -0.0625 \cdot \frac{1 \cdot {x}^{2}}{\color{blue}{s \cdot \left(s \cdot s\right)}} + 0.25 \cdot \frac{1}{s} \]

   3. times-frac 26.1%

      \[\leadsto -0.0625 \cdot \color{blue}{\left(\frac{1}{s} \cdot \frac{{x}^{2}}{s \cdot s}\right)} + 0.25 \cdot \frac{1}{s} \]

   4. unpow2 26.1%

      \[\leadsto -0.0625 \cdot \left(\frac{1}{s} \cdot \frac{\color{blue}{x \cdot x}}{s \cdot s}\right) + 0.25 \cdot \frac{1}{s} \]

   5. frac-times 31.4%

      \[\leadsto -0.0625 \cdot \left(\frac{1}{s} \cdot \color{blue}{\left(\frac{x}{s} \cdot \frac{x}{s}\right)}\right) + 0.25 \cdot \frac{1}{s} \]

   6. pow2 31.4%

      \[\leadsto -0.0625 \cdot \left(\frac{1}{s} \cdot \color{blue}{{\left(\frac{x}{s}\right)}^{2}}\right) + 0.25 \cdot \frac{1}{s} \]

8. Applied egg-rr 31.4%

   \[\leadsto -0.0625 \cdot \color{blue}{\left(\frac{1}{s} \cdot {\left(\frac{x}{s}\right)}^{2}\right)} + 0.25 \cdot \frac{1}{s} \]
9. Step-by-step derivation

   1. associate-*l/ 31.4%

      \[\leadsto -0.0625 \cdot \color{blue}{\frac{1 \cdot {\left(\frac{x}{s}\right)}^{2}}{s}} + 0.25 \cdot \frac{1}{s} \]

   2. *-un-lft-identity 31.4%

      \[\leadsto -0.0625 \cdot \frac{\color{blue}{{\left(\frac{x}{s}\right)}^{2}}}{s} + 0.25 \cdot \frac{1}{s} \]

   3. unpow2 31.4%

      \[\leadsto -0.0625 \cdot \frac{\color{blue}{\frac{x}{s} \cdot \frac{x}{s}}}{s} + 0.25 \cdot \frac{1}{s} \]

   4. associate-/l* 31.4%

      \[\leadsto -0.0625 \cdot \color{blue}{\frac{\frac{x}{s}}{\frac{s}{\frac{x}{s}}}} + 0.25 \cdot \frac{1}{s} \]

10. Applied egg-rr 31.4%

    \[\leadsto -0.0625 \cdot \color{blue}{\frac{\frac{x}{s}}{\frac{s}{\frac{x}{s}}}} + 0.25 \cdot \frac{1}{s} \]

11. Final simplification 31.4%

    \[\leadsto -0.0625 \cdot \frac{\frac{x}{s}}{\frac{s}{\frac{x}{s}}} + 0.25 \cdot \frac{1}{s} \]

Alternative 8: 27.0% accurate, 206.7× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s) :precision binary32 (/ 0.25 s))

C

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

Fortran

NOTE: x should be positive before calling this function
real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 0.25e0 / s
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

3. Simplified 99.1%

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

4. Taylor expanded in s around inf 31.4%

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

5. Final simplification 31.4%

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

Alternative 9: 8.2% accurate, 620.0× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ 1 \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s) :precision binary32 1.0)

C

x = abs(x);
float code(float x, float s) {
	return 1.0f;
}

Fortran

NOTE: x should be positive before calling this function
real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 1.0e0
end function

Julia

x = abs(x)
function code(x, s)
	return Float32(1.0)
end

MATLAB

x = abs(x)
function tmp = code(x, s)
	tmp = single(1.0);
end

Derivation

1. Initial program 99.1%

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

   1. /-rgt-identity 99.1%

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

   2. associate-/l/ 99.1%

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

   3. *-commutative 99.1%

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

   4. associate-*l* 99.1%

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

   5. associate-*l* 99.1%

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

   6. *-rgt-identity 99.1%

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

   7. +-commutative 99.1%

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

   8. +-commutative 99.1%

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

3. Simplified 99.1%

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

   1. add-exp-log 97.0%

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

   2. log-div 96.7%

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

   3. add-log-exp 97.2%

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

   4. add-sqr-sqrt -0.0%

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

   5. sqrt-unprod 26.4%

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

   6. sqr-neg 26.4%

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

   7. sqrt-unprod 26.2%

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

   8. add-sqr-sqrt 26.2%

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

   9. add-sqr-sqrt 12.9%

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

   10. fabs-sqr 12.9%

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

   11. add-sqr-sqrt 63.1%

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

   12. log-prod 62.9%

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

5. Applied egg-rr 84.0%

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

6. Taylor expanded in x around inf 40.5%

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

7. Taylor expanded in x around 0 7.1%

   \[\leadsto \color{blue}{1 + \frac{x}{s}} \]

8. Taylor expanded in x around 0 9.1%

   \[\leadsto \color{blue}{1} \]

9. Final simplification 9.1%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023301 
(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))))))