Logistic distribution

Percentage Accurate: 99.6% → 99.6%

Time: 12.8s

Alternatives: 8

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, 1.9× speedup

Math

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

FPCore

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

C

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

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 / (t_0 + 1.0e0)) / (s + (s / exp((x_m / s))))
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 / Float32(t_0 + Float32(1.0))) / Float32(s + Float32(s / exp(Float32(x_m / s)))))
end

MATLAB

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

Derivation

1. Initial program 99.8%

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

   1. *-commutative 99.8%

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

   2. fabs-neg 99.8%

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

   3. +-commutative 99.8%

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

   4. fabs-neg 99.8%

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

   5. distribute-lft-in 99.9%

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

   6. *-rgt-identity 99.9%

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

   7. +-commutative 99.9%

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

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 x around 0 99.8%

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

   1. associate-/r* 99.8%

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

7. Simplified 63.2%

   \[\leadsto \color{blue}{\frac{\frac{e^{\frac{x}{-s}}}{e^{\frac{x}{-s}} + 1}}{s + \frac{s}{e^{\frac{x}{s}}}}} \]
8. Add Preprocessing

Alternative 2: 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 99.8%

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

   1. fabs-neg 99.8%

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

   2. distribute-frac-neg 99.8%

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

   3. distribute-frac-neg2 99.8%

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

   4. fabs-neg 99.8%

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

   5. *-commutative 99.8%

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

   6. fabs-neg 99.8%

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

   7. +-commutative 99.8%

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

   8. fabs-neg 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in x around 0 99.8%

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

   1. associate-/r* 99.8%

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

   2. mul-1-neg 99.8%

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

   3. rec-exp 99.8%

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

   4. rem-square-sqrt 50.3%

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

   5. fabs-sqr 50.3%

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

   6. rem-square-sqrt 61.5%

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

   7. rec-exp 61.5%

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

   8. distribute-neg-frac2 61.5%

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

7. Simplified 63.1%

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

Alternative 3: 96.9% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

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 / (t_0 + 1.0e0)) / (s + (s / (1.0e0 + (x_m / s))))
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 / Float32(t_0 + Float32(1.0))) / Float32(s + Float32(s / Float32(Float32(1.0) + Float32(x_m / s)))))
end

MATLAB

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

Derivation

1. Initial program 99.8%

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

   1. *-commutative 99.8%

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

   2. fabs-neg 99.8%

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

   3. +-commutative 99.8%

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

   4. fabs-neg 99.8%

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

   5. distribute-lft-in 99.9%

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

   6. *-rgt-identity 99.9%

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

   7. +-commutative 99.9%

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

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 x around 0 99.8%

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

   1. associate-/r* 99.8%

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

7. Simplified 63.2%

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

8. Taylor expanded in x around 0 60.6%

   \[\leadsto \frac{\frac{e^{\frac{x}{-s}}}{e^{\frac{x}{-s}} + 1}}{s + \frac{s}{\color{blue}{1 + \frac{x}{s}}}} \]
9. Step-by-step derivation

   1. +-commutative 60.6%

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

10. Simplified 60.6%

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

11. Final simplification 60.6%

    \[\leadsto \frac{\frac{e^{\frac{x}{-s}}}{e^{\frac{x}{-s}} + 1}}{s + \frac{s}{1 + \frac{x}{s}}} \]
12. Add Preprocessing

Alternative 4: 95.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := e^{\frac{x\_m}{-s}}\\ \frac{t\_0 \cdot 0.5}{s \cdot \left(t\_0 + 1\right)} \end{array} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (let* ((t_0 (exp (/ x_m (- s))))) (/ (* t_0 0.5) (* s (+ t_0 1.0)))))

C

x_m = fabs(x);
float code(float x_m, float s) {
	float t_0 = expf((x_m / -s));
	return (t_0 * 0.5f) / (s * (t_0 + 1.0f));
}

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 * 0.5e0) / (s * (t_0 + 1.0e0))
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 * Float32(0.5)) / Float32(s * Float32(t_0 + Float32(1.0))))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m, s)
	t_0 = exp((x_m / -s));
	tmp = (t_0 * single(0.5)) / (s * (t_0 + single(1.0)));
end

Derivation

1. Initial program 99.8%

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

   1. *-commutative 99.8%

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

   2. fabs-neg 99.8%

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

   3. +-commutative 99.8%

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

   4. fabs-neg 99.8%

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

   5. distribute-lft-in 99.9%

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

   6. *-rgt-identity 99.9%

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

   7. +-commutative 99.9%

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

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 x around 0 99.8%

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

   1. associate-/r* 99.8%

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

7. Simplified 63.2%

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

8. Taylor expanded in s around inf 59.1%

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

9. Taylor expanded in x around inf 59.1%

   \[\leadsto \color{blue}{0.5 \cdot \frac{e^{-1 \cdot \frac{x}{s}}}{s \cdot \left(1 + e^{-1 \cdot \frac{x}{s}}\right)}} \]
10. Step-by-step derivation

    1. associate-*r/ 59.1%

       \[\leadsto \color{blue}{\frac{0.5 \cdot e^{-1 \cdot \frac{x}{s}}}{s \cdot \left(1 + e^{-1 \cdot \frac{x}{s}}\right)}} \]

    2. neg-mul-1 59.1%

       \[\leadsto \frac{0.5 \cdot e^{\color{blue}{-\frac{x}{s}}}}{s \cdot \left(1 + e^{-1 \cdot \frac{x}{s}}\right)} \]

    3. distribute-neg-frac2 59.1%

       \[\leadsto \frac{0.5 \cdot e^{\color{blue}{\frac{x}{-s}}}}{s \cdot \left(1 + e^{-1 \cdot \frac{x}{s}}\right)} \]

    4. neg-mul-1 59.1%

       \[\leadsto \frac{0.5 \cdot e^{\frac{x}{-s}}}{s \cdot \left(1 + e^{\color{blue}{-\frac{x}{s}}}\right)} \]

    5. distribute-neg-frac2 59.1%

       \[\leadsto \frac{0.5 \cdot e^{\frac{x}{-s}}}{s \cdot \left(1 + e^{\color{blue}{\frac{x}{-s}}}\right)} \]

11. Simplified 59.1%

    \[\leadsto \color{blue}{\frac{0.5 \cdot e^{\frac{x}{-s}}}{s \cdot \left(1 + e^{\frac{x}{-s}}\right)}} \]

12. Final simplification 59.1%

    \[\leadsto \frac{e^{\frac{x}{-s}} \cdot 0.5}{s \cdot \left(e^{\frac{x}{-s}} + 1\right)} \]
13. Add Preprocessing

Alternative 5: 94.7% 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 99.8%

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

   1. fabs-neg 99.8%

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

   2. distribute-frac-neg 99.8%

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

   3. distribute-frac-neg2 99.8%

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

   4. fabs-neg 99.8%

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

   5. *-commutative 99.8%

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

   6. fabs-neg 99.8%

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

   7. +-commutative 99.8%

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

   8. fabs-neg 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in x around 0 99.8%

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

   1. associate-/r* 99.8%

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

   2. mul-1-neg 99.8%

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

   3. rec-exp 99.8%

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

   4. rem-square-sqrt 50.3%

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

   5. fabs-sqr 50.3%

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

   6. rem-square-sqrt 61.5%

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

   7. rec-exp 61.5%

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

   8. distribute-neg-frac2 61.5%

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

7. Simplified 63.1%

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

8. Taylor expanded in x around 0 59.9%

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

Alternative 6: 85.8% accurate, 23.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.000000031374395 \cdot 10^{-22}:\\ \;\;\;\;\frac{0.5 - \frac{x\_m}{s} \cdot 0.25}{s \cdot 2 + x\_m \cdot \left(\frac{x\_m}{s} \cdot 0.5 + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (if (<= x_m 1.000000031374395e-22)
   (/
    (- 0.5 (* (/ x_m s) 0.25))
    (+ (* s 2.0) (* x_m (+ (* (/ x_m s) 0.5) -1.0))))
   0.0))

C

x_m = fabs(x);
float code(float x_m, float s) {
	float tmp;
	if (x_m <= 1.000000031374395e-22f) {
		tmp = (0.5f - ((x_m / s) * 0.25f)) / ((s * 2.0f) + (x_m * (((x_m / s) * 0.5f) + -1.0f)));
	} 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 <= 1.000000031374395e-22) then
        tmp = (0.5e0 - ((x_m / s) * 0.25e0)) / ((s * 2.0e0) + (x_m * (((x_m / s) * 0.5e0) + (-1.0e0))))
    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(1.000000031374395e-22))
		tmp = Float32(Float32(Float32(0.5) - Float32(Float32(x_m / s) * Float32(0.25))) / Float32(Float32(s * Float32(2.0)) + Float32(x_m * Float32(Float32(Float32(x_m / s) * Float32(0.5)) + Float32(-1.0)))));
	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(1.000000031374395e-22))
		tmp = (single(0.5) - ((x_m / s) * single(0.25))) / ((s * single(2.0)) + (x_m * (((x_m / s) * single(0.5)) + single(-1.0))));
	else
		tmp = single(0.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 1.00000003e-22

   1. Initial program 99.7%

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

      1. *-commutative 99.7%

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

      2. fabs-neg 99.7%

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

      3. +-commutative 99.7%

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

      4. fabs-neg 99.7%

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

      5. distribute-lft-in 99.8%

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

      6. *-rgt-identity 99.8%

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

      7. +-commutative 99.8%

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

   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 x around 0 99.8%

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

      1. associate-/r* 99.8%

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

   7. Simplified 38.8%

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

   8. Taylor expanded in s around -inf 73.7%

      \[\leadsto \frac{\color{blue}{0.5 + -1 \cdot \frac{0.5 \cdot x - 0.25 \cdot x}{s}}}{s + \frac{s}{e^{\frac{x}{s}}}} \]
   9. Step-by-step derivation

      1. mul-1-neg 73.7%

         \[\leadsto \frac{0.5 + \color{blue}{\left(-\frac{0.5 \cdot x - 0.25 \cdot x}{s}\right)}}{s + \frac{s}{e^{\frac{x}{s}}}} \]

      2. distribute-rgt-out-- 73.7%

         \[\leadsto \frac{0.5 + \left(-\frac{\color{blue}{x \cdot \left(0.5 - 0.25\right)}}{s}\right)}{s + \frac{s}{e^{\frac{x}{s}}}} \]

      3. metadata-eval 73.7%

         \[\leadsto \frac{0.5 + \left(-\frac{x \cdot \color{blue}{0.25}}{s}\right)}{s + \frac{s}{e^{\frac{x}{s}}}} \]

      4. *-commutative 73.7%

         \[\leadsto \frac{0.5 + \left(-\frac{\color{blue}{0.25 \cdot x}}{s}\right)}{s + \frac{s}{e^{\frac{x}{s}}}} \]

      5. associate-*r/ 73.7%

         \[\leadsto \frac{0.5 + \left(-\color{blue}{0.25 \cdot \frac{x}{s}}\right)}{s + \frac{s}{e^{\frac{x}{s}}}} \]

      6. unsub-neg 73.7%

         \[\leadsto \frac{\color{blue}{0.5 - 0.25 \cdot \frac{x}{s}}}{s + \frac{s}{e^{\frac{x}{s}}}} \]

      7. *-commutative 73.7%

         \[\leadsto \frac{0.5 - \color{blue}{\frac{x}{s} \cdot 0.25}}{s + \frac{s}{e^{\frac{x}{s}}}} \]

   10. Simplified 73.7%

       \[\leadsto \frac{\color{blue}{0.5 - \frac{x}{s} \cdot 0.25}}{s + \frac{s}{e^{\frac{x}{s}}}} \]

   11. Taylor expanded in x around 0 48.9%

       \[\leadsto \frac{0.5 - \frac{x}{s} \cdot 0.25}{\color{blue}{2 \cdot s + x \cdot \left(0.5 \cdot \frac{x}{s} - 1\right)}} \]

   if 1.00000003e-22 < x

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. fabs-neg 100.0%

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

      3. +-commutative 100.0%

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

      4. fabs-neg 100.0%

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

      5. distribute-lft-in 99.9%

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

      6. *-rgt-identity 99.9%

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

      7. +-commutative 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 99.9%

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

      1. associate-/r* 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in s around inf 98.1%

      \[\leadsto \frac{\frac{e^{\frac{x}{-s}}}{e^{\frac{x}{-s}} + 1}}{s + \color{blue}{s}} \]
   9. Step-by-step derivation

      1. add-log-exp 95.7%

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

      2. div-inv 95.7%

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

      3. exp-prod 95.7%

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

      4. flip-+ 92.2%

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

      5. +-inverses 92.2%

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

      6. +-inverses 92.2%

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

      7. +-inverses 92.2%

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

      8. +-inverses 92.2%

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

      9. clear-num 92.2%

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

      10. flip-+ 93.5%

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

      11. add-sqr-sqrt 93.5%

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

      12. sqrt-unprod 93.5%

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

      13. sqr-neg 93.5%

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

      14. sqrt-unprod 92.2%

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

      15. add-sqr-sqrt 92.6%

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

      16. sub-neg 92.6%

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

      17. +-inverses 92.6%

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

      18. metadata-eval 92.6%

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

   10. Applied egg-rr 92.6%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.000000031374395 \cdot 10^{-22}:\\ \;\;\;\;\frac{0.5 - \frac{x}{s} \cdot 0.25}{s \cdot 2 + x \cdot \left(\frac{x}{s} \cdot 0.5 + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.7% accurate, 77.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.000000031374395 \cdot 10^{-22}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (if (<= x_m 1.000000031374395e-22) (/ 0.25 s) 0.0))

C

x_m = fabs(x);
float code(float x_m, float s) {
	float tmp;
	if (x_m <= 1.000000031374395e-22f) {
		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 <= 1.000000031374395e-22) 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(1.000000031374395e-22))
		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(1.000000031374395e-22))
		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 < 1.00000003e-22

   1. Initial program 99.7%

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

      1. fabs-neg 99.7%

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

      2. distribute-frac-neg 99.7%

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

      3. distribute-frac-neg2 99.7%

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

      4. fabs-neg 99.7%

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

      5. *-commutative 99.7%

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

      6. fabs-neg 99.7%

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

      7. +-commutative 99.7%

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

      8. fabs-neg 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in s around inf 35.8%

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

   if 1.00000003e-22 < x

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. fabs-neg 100.0%

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

      3. +-commutative 100.0%

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

      4. fabs-neg 100.0%

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

      5. distribute-lft-in 99.9%

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

      6. *-rgt-identity 99.9%

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

      7. +-commutative 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 99.9%

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

      1. associate-/r* 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in s around inf 98.1%

      \[\leadsto \frac{\frac{e^{\frac{x}{-s}}}{e^{\frac{x}{-s}} + 1}}{s + \color{blue}{s}} \]
   9. Step-by-step derivation

      1. add-log-exp 95.7%

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

      2. div-inv 95.7%

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

      3. exp-prod 95.7%

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

      4. flip-+ 92.2%

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

      5. +-inverses 92.2%

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

      6. +-inverses 92.2%

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

      7. +-inverses 92.2%

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

      8. +-inverses 92.2%

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

      9. clear-num 92.2%

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

      10. flip-+ 93.5%

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

      11. add-sqr-sqrt 93.5%

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

      12. sqrt-unprod 93.5%

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

      13. sqr-neg 93.5%

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

      14. sqrt-unprod 92.2%

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

      15. add-sqr-sqrt 92.6%

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

      16. sub-neg 92.6%

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

      17. +-inverses 92.6%

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

      18. metadata-eval 92.6%

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

   10. Applied egg-rr 92.6%

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

Alternative 8: 74.3% 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 99.8%

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

   1. *-commutative 99.8%

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

   2. fabs-neg 99.8%

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

   3. +-commutative 99.8%

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

   4. fabs-neg 99.8%

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

   5. distribute-lft-in 99.9%

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

   6. *-rgt-identity 99.9%

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

   7. +-commutative 99.9%

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

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 x around 0 99.8%

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

   1. associate-/r* 99.8%

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

7. Simplified 63.2%

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

8. Taylor expanded in s around inf 59.1%

   \[\leadsto \frac{\frac{e^{\frac{x}{-s}}}{e^{\frac{x}{-s}} + 1}}{s + \color{blue}{s}} \]
9. Step-by-step derivation

   1. add-log-exp 42.8%

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

   2. div-inv 42.8%

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

   3. exp-prod 42.8%

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

   4. flip-+ 38.3%

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

   5. +-inverses 38.3%

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

   6. +-inverses 38.3%

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

   7. +-inverses 38.3%

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

   8. +-inverses 38.3%

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

   9. clear-num 38.3%

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

   10. flip-+ 40.4%

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

   11. add-sqr-sqrt 40.4%

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

   12. sqrt-unprod 40.4%

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

   13. sqr-neg 40.4%

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

   14. sqrt-unprod 38.3%

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

   15. add-sqr-sqrt 76.3%

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

   16. sub-neg 76.3%

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

   17. +-inverses 76.3%

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

   18. metadata-eval 76.3%

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

10. Applied egg-rr 76.3%

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

Reproduce

herbie shell --seed 2024139 
(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))))))