Logistic distribution

Percentage Accurate: 99.6% → 99.6%

Time: 12.5s

Alternatives: 6

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, 2.0× speedup

Math

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

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (let* ((t_0 (exp (/ x_m (- s))))) (/ (/ t_0 s) (pow (+ t_0 1.0) 2.0))))

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

   1. fabs-neg 99.5%

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

   2. distribute-frac-neg 99.5%

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

   3. distribute-frac-neg2 99.5%

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

   4. fabs-neg 99.5%

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

   5. *-commutative 99.5%

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

   6. fabs-neg 99.5%

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

   7. +-commutative 99.5%

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

   8. fabs-neg 99.5%

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

3. Simplified 99.6%

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

5. Taylor expanded in x around 0 99.6%

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

   1. associate-/r* 99.6%

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

   2. mul-1-neg 99.6%

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

   3. distribute-neg-frac2 99.6%

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

   4. +-commutative 99.6%

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

   5. mul-1-neg 99.6%

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

   6. distribute-neg-frac2 99.6%

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

7. Simplified 99.6%

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

   1. distribute-frac-neg2 99.6%

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

   2. rec-exp 99.6%

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

   3. pow1 99.6%

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

   4. pow1 99.6%

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

   5. add-sqr-sqrt 56.5%

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

   6. fabs-sqr 56.5%

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

   7. add-sqr-sqrt 97.6%

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

9. Applied egg-rr 97.6%

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

    1. rec-exp 97.6%

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

    2. distribute-neg-frac2 97.6%

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

11. Simplified 97.6%

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

    1. distribute-frac-neg2 99.6%

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

    2. rec-exp 99.6%

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

    3. pow1 99.6%

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

    4. pow1 99.6%

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

    5. add-sqr-sqrt 56.5%

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

    6. fabs-sqr 56.5%

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

    7. add-sqr-sqrt 97.6%

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

13. Applied egg-rr 68.0%

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

    1. rec-exp 97.6%

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

    2. distribute-neg-frac2 97.6%

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

15. Simplified 68.1%

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

Alternative 2: 99.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\left|x\_m\right| \leq 24.5:\\ \;\;\;\;\frac{e^{\frac{x\_m}{s} + -2 \cdot \mathsf{log1p}\left(e^{\frac{x\_m}{s}}\right)}}{s}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (if (<= (fabs x_m) 24.5)
   (/ (exp (+ (/ x_m s) (* -2.0 (log1p (exp (/ x_m s)))))) s)
   0.0))

C

x_m = fabs(x);
float code(float x_m, float s) {
	float tmp;
	if (fabsf(x_m) <= 24.5f) {
		tmp = expf(((x_m / s) + (-2.0f * log1pf(expf((x_m / s)))))) / s;
	} else {
		tmp = 0.0f;
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m, s)
	tmp = Float32(0.0)
	if (abs(x_m) <= Float32(24.5))
		tmp = Float32(exp(Float32(Float32(x_m / s) + Float32(Float32(-2.0) * log1p(exp(Float32(x_m / s)))))) / s);
	else
		tmp = Float32(0.0);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (fabs.f32 x) < 24.5

   1. Initial program 99.2%

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

      1. fabs-neg 99.2%

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

      2. distribute-frac-neg 99.2%

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

      3. distribute-frac-neg2 99.2%

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

      4. fabs-neg 99.2%

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

      5. *-commutative 99.2%

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

      6. fabs-neg 99.2%

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

      7. +-commutative 99.2%

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

      8. fabs-neg 99.2%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in x around 0 99.3%

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

      1. associate-/r* 99.3%

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

      2. mul-1-neg 99.3%

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

      3. distribute-neg-frac2 99.3%

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

      4. +-commutative 99.3%

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

      5. mul-1-neg 99.3%

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

      6. distribute-neg-frac2 99.3%

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

   7. Simplified 99.3%

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

      1. div-inv 99.2%

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

      2. frac-times 99.3%

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

      3. add-sqr-sqrt -0.0%

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

      4. add-sqr-sqrt 99.3%

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

      5. add-sqr-sqrt 57.3%

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

      6. fabs-sqr 57.3%

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

      7. add-sqr-sqrt -0.0%

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

      8. sqrt-unprod 24.1%

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

      9. sqr-neg 24.1%

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

      10. sqrt-unprod 25.6%

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

      11. add-sqr-sqrt 67.5%

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

      12. add-sqr-sqrt 67.5%

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

   9. Applied egg-rr 94.8%

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

   10. Taylor expanded in x around inf 94.8%

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

       1. associate--r+ 94.8%

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

       2. log1p-define 94.8%

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

       3. unsub-neg 94.8%

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

       4. exp-sum 69.4%

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

       5. exp-diff 69.3%

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

       6. rem-exp-log 73.2%

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

       7. associate-*l/ 74.4%

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

       8. prod-exp 99.1%

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

       9. distribute-lft-neg-in 99.1%

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

       10. metadata-eval 99.1%

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

   12. Simplified 99.1%

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

   if 24.5 < (fabs.f32 x)

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in s around inf 100.0%

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

      1. add-log-exp 100.0%

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

      2. div-inv 100.0%

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

      3. distribute-frac-neg 100.0%

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

      4. distribute-frac-neg2 100.0%

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

      5. exp-prod 100.0%

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

      6. add-log-exp 100.0%

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

      7. exp-prod 100.0%

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

      8. add-sqr-sqrt 100.0%

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

      9. sqrt-unprod 100.0%

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

      10. sqr-neg 100.0%

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

      11. sqrt-unprod 100.0%

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

      12. add-sqr-sqrt 100.0%

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

   7. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 99.6% accurate, 2.0× speedup

Math

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

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (let* ((t_0 (exp (/ x_m (- s))))) (/ t_0 (* s (pow (+ t_0 1.0) 2.0)))))

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

   1. fabs-neg 99.5%

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

   2. distribute-frac-neg 99.5%

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

   3. distribute-frac-neg2 99.5%

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

   4. fabs-neg 99.5%

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

   5. *-commutative 99.5%

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

   6. fabs-neg 99.5%

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

   7. +-commutative 99.5%

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

   8. fabs-neg 99.5%

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

3. Simplified 99.6%

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

5. Taylor expanded in x around 0 99.6%

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

   1. associate-/r* 99.6%

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

   2. mul-1-neg 99.6%

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

   3. distribute-neg-frac2 99.6%

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

   4. +-commutative 99.6%

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

   5. mul-1-neg 99.6%

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

   6. distribute-neg-frac2 99.6%

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

7. Simplified 99.6%

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

   1. distribute-frac-neg2 99.6%

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

   2. rec-exp 99.6%

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

   3. pow1 99.6%

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

   4. pow1 99.6%

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

   5. add-sqr-sqrt 56.5%

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

   6. fabs-sqr 56.5%

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

   7. add-sqr-sqrt 97.6%

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

9. Applied egg-rr 97.6%

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

    1. rec-exp 97.6%

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

    2. distribute-neg-frac2 97.6%

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

11. Simplified 97.6%

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

    1. distribute-frac-neg2 99.6%

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

    2. rec-exp 99.6%

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

    3. pow1 99.6%

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

    4. pow1 99.6%

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

    5. add-sqr-sqrt 56.5%

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

    6. fabs-sqr 56.5%

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

    7. add-sqr-sqrt 97.6%

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

13. Applied egg-rr 68.0%

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

    1. rec-exp 97.6%

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

    2. distribute-neg-frac2 97.6%

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

15. Simplified 68.1%

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

16. Taylor expanded in x around inf 68.1%

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

    1. associate-*r/ 68.1%

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

    2. mul-1-neg 68.1%

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

    3. mul-1-neg 68.1%

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

    4. distribute-frac-neg2 68.1%

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

    5. distribute-frac-neg2 68.1%

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

    6. distribute-frac-neg 68.1%

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

18. Simplified 68.1%

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

19. Final simplification 68.1%

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

Alternative 4: 94.6% accurate, 5.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{\frac{e^{\frac{x\_m}{-s}}}{s}}{4} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s) :precision binary32 (/ (/ (exp (/ x_m (- s))) s) 4.0))

C

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

Fortran

x_m = abs(x)
real(4) function code(x_m, s)
    real(4), intent (in) :: x_m
    real(4), intent (in) :: s
    code = (exp((x_m / -s)) / s) / 4.0e0
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

   1. fabs-neg 99.5%

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

   2. distribute-frac-neg 99.5%

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

   3. distribute-frac-neg2 99.5%

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

   4. fabs-neg 99.5%

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

   5. *-commutative 99.5%

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

   6. fabs-neg 99.5%

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

   7. +-commutative 99.5%

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

   8. fabs-neg 99.5%

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

3. Simplified 99.6%

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

5. Taylor expanded in x around 0 99.6%

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

   1. associate-/r* 99.6%

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

   2. mul-1-neg 99.6%

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

   3. distribute-neg-frac2 99.6%

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

   4. +-commutative 99.6%

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

   5. mul-1-neg 99.6%

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

   6. distribute-neg-frac2 99.6%

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

7. Simplified 99.6%

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

   1. distribute-frac-neg2 99.6%

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

   2. rec-exp 99.6%

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

   3. pow1 99.6%

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

   4. pow1 99.6%

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

   5. add-sqr-sqrt 56.5%

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

   6. fabs-sqr 56.5%

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

   7. add-sqr-sqrt 97.6%

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

9. Applied egg-rr 97.6%

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

    1. rec-exp 97.6%

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

    2. distribute-neg-frac2 97.6%

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

11. Simplified 97.6%

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

    1. distribute-frac-neg2 99.6%

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

    2. rec-exp 99.6%

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

    3. pow1 99.6%

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

    4. pow1 99.6%

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

    5. add-sqr-sqrt 56.5%

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

    6. fabs-sqr 56.5%

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

    7. add-sqr-sqrt 97.6%

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

13. Applied egg-rr 68.0%

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

    1. rec-exp 97.6%

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

    2. distribute-neg-frac2 97.6%

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

15. Simplified 68.1%

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

16. Taylor expanded in x around 0 63.7%

    \[\leadsto \frac{\frac{e^{\frac{x}{-s}}}{s}}{\color{blue}{4}} \]
17. Add Preprocessing

Alternative 5: 86.4% accurate, 77.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 4.999999918875795 \cdot 10^{-18}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (if (<= x_m 4.999999918875795e-18) (/ 0.25 s) 0.0))

C

x_m = fabs(x);
float code(float x_m, float s) {
	float tmp;
	if (x_m <= 4.999999918875795e-18f) {
		tmp = 0.25f / s;
	} else {
		tmp = 0.0f;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(4) function code(x_m, s)
    real(4), intent (in) :: x_m
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x_m <= 4.999999918875795e-18) then
        tmp = 0.25e0 / s
    else
        tmp = 0.0e0
    end if
    code = tmp
end function

Julia

x_m = abs(x)
function code(x_m, s)
	tmp = Float32(0.0)
	if (x_m <= Float32(4.999999918875795e-18))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(0.0);
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, s)
	tmp = single(0.0);
	if (x_m <= single(4.999999918875795e-18))
		tmp = single(0.25) / s;
	else
		tmp = single(0.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 4.99999992e-18

   1. Initial program 99.4%

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

      1. fabs-neg 99.4%

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

      2. distribute-frac-neg 99.4%

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

      3. distribute-frac-neg2 99.4%

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

      4. fabs-neg 99.4%

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

      5. *-commutative 99.4%

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

      6. fabs-neg 99.4%

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

      7. +-commutative 99.4%

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

      8. fabs-neg 99.4%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in s around inf 39.9%

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

   if 4.99999992e-18 < x

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in s around inf 96.9%

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

      1. add-log-exp 93.3%

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

      2. div-inv 93.3%

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

      3. distribute-frac-neg 93.3%

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

      4. distribute-frac-neg2 93.3%

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

      5. exp-prod 93.3%

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

      6. add-log-exp 93.2%

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

      7. exp-prod 93.2%

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

      8. add-sqr-sqrt 93.2%

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

      9. sqrt-unprod 93.2%

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

      10. sqr-neg 93.2%

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

      11. sqrt-unprod 91.8%

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

      12. add-sqr-sqrt 92.3%

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

   7. Applied egg-rr 92.3%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 73.6% accurate, 620.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ 0 \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s) :precision binary32 0.0)

C

x_m = fabs(x);
float code(float x_m, float s) {
	return 0.0f;
}

Fortran

x_m = abs(x)
real(4) function code(x_m, s)
    real(4), intent (in) :: x_m
    real(4), intent (in) :: s
    code = 0.0e0
end function

Julia

x_m = abs(x)
function code(x_m, s)
	return Float32(0.0)
end

MATLAB

x_m = abs(x);
function tmp = code(x_m, s)
	tmp = single(0.0);
end

Derivation

1. Initial program 99.5%

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

3. Simplified 99.6%

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

5. Taylor expanded in s around inf 94.4%

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

   1. add-log-exp 75.1%

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

   2. div-inv 75.1%

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

   3. distribute-frac-neg 75.1%

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

   4. distribute-frac-neg2 75.1%

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

   5. exp-prod 75.1%

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

   6. add-log-exp 75.0%

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

   7. exp-prod 75.0%

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

   8. add-sqr-sqrt 75.0%

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

   9. sqrt-unprod 75.0%

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

   10. sqr-neg 75.0%

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

   11. sqrt-unprod 70.4%

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

   12. add-sqr-sqrt 72.8%

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

7. Applied egg-rr 71.9%

   \[\leadsto \color{blue}{0} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024090 
(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))))))