Logistic distribution

Percentage Accurate: 99.6% → 99.6%

Time: 13.6s

Alternatives: 12

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%

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

8. Final simplification 63.2%

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

Alternative 2: 99.6% accurate, 2.0× speedup

Math

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

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 99.8%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
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. unpow2 99.8%

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

   2. distribute-rgt-in 99.9%

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

   3. mul-1-neg 99.9%

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

   4. rec-exp 99.8%

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

   5. distribute-rgt-in 99.8%

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

7. Simplified 63.1%

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

8. Final simplification 63.1%

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

Alternative 3: 99.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{e^{\mathsf{log1p}\left(e^{\frac{x\_m}{-s}}\right) \cdot -2 - \frac{x\_m}{s}}}{s} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (/ (exp (- (* (log1p (exp (/ x_m (- s)))) -2.0) (/ x_m s))) s))

C

x_m = fabs(x);
float code(float x_m, float s) {
	return expf(((log1pf(expf((x_m / -s))) * -2.0f) - (x_m / s))) / s;
}

Julia

x_m = abs(x)
function code(x_m, s)
	return Float32(exp(Float32(Float32(log1p(exp(Float32(x_m / Float32(-s)))) * Float32(-2.0)) - Float32(x_m / s))) / 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%

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

8. Taylor expanded in x around inf 63.2%

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

10. Simplified 86.9%

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

11. Taylor expanded in x around inf 86.8%

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

    1. distribute-neg-in 86.8%

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

    2. unsub-neg 86.8%

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

    3. *-commutative 86.8%

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

    4. distribute-rgt-neg-in 86.8%

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

    5. log1p-define 86.9%

       \[\leadsto \frac{e^{\color{blue}{\mathsf{log1p}\left(e^{-1 \cdot \frac{x}{s}}\right)} \cdot \left(-2\right) - \frac{x}{s}}}{s} \]

    6. neg-mul-1 86.9%

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

    7. distribute-neg-frac 86.9%

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

    8. metadata-eval 86.9%

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

13. Simplified 86.9%

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

14. Final simplification 86.9%

    \[\leadsto \frac{e^{\mathsf{log1p}\left(e^{\frac{x}{-s}}\right) \cdot -2 - \frac{x}{s}}}{s} \]
15. Add Preprocessing

Alternative 4: 96.9% accurate, 2.8× speedup

Math

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

FPCore

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

C

x_m = fabs(x);
float code(float x_m, float s) {
	return ((1.0f / expf((x_m / s))) / (expf((x_m / -s)) + 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
    code = ((1.0e0 / exp((x_m / s))) / (exp((x_m / -s)) + 1.0e0)) / (s + (s / (1.0e0 + (x_m / s))))
end function

Julia

x_m = abs(x)
function code(x_m, s)
	return Float32(Float32(Float32(Float32(1.0) / exp(Float32(x_m / s))) / Float32(exp(Float32(x_m / Float32(-s))) + 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)
	tmp = ((single(1.0) / exp((x_m / s))) / (exp((x_m / -s)) + 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%

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

8. Taylor expanded in x around 0 60.6%

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

   1. +-commutative 60.6%

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

10. Simplified 60.6%

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

    1. distribute-frac-neg2 60.6%

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

    2. rec-exp 60.6%

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

12. Applied egg-rr 60.6%

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

13. Final simplification 60.6%

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

Alternative 5: 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%

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

8. Taylor expanded in x around 0 60.6%

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

   1. +-commutative 60.6%

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

10. Simplified 60.6%

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

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 6: 96.8% accurate, 5.3× speedup

Math

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

FPCore

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

C

x_m = fabs(x);
float code(float x_m, float s) {
	return (1.0f / (1.0f + expf((x_m / s)))) / (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
    code = (1.0e0 / (1.0e0 + exp((x_m / s)))) / (s + (s / (1.0e0 + (x_m / s))))
end function

Julia

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

MATLAB

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

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

8. Taylor expanded in x around 0 60.6%

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

   1. +-commutative 60.6%

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

10. Simplified 60.6%

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

    1. distribute-frac-neg2 60.6%

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

    2. rec-exp 60.6%

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

12. Applied egg-rr 60.6%

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

13. Taylor expanded in x around inf 60.6%

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

    1. +-commutative 60.6%

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

    2. mul-1-neg 60.6%

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

    3. distribute-frac-neg2 60.6%

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

    4. distribute-lft-in 22.3%

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

    5. distribute-frac-neg2 22.3%

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

    6. rec-exp 22.3%

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

    7. rgt-mult-inverse 62.2%

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

    8. *-rgt-identity 62.2%

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

15. Simplified 62.2%

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

16. Final simplification 62.2%

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

Alternative 7: 88.0% accurate, 5.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.5000000601271012 \cdot 10^{-26}:\\ \;\;\;\;\frac{0.5 - \frac{x\_m \cdot 0.25}{s}}{s + \left(s + x\_m \cdot \left(\frac{x\_m}{s} \cdot 0.5 + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{x\_m}{-s}}}{s}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (if (<= x_m 1.5000000601271012e-26)
   (/
    (- 0.5 (/ (* x_m 0.25) s))
    (+ s (+ s (* x_m (+ (* (/ x_m s) 0.5) -1.0)))))
   (/ (exp (/ x_m (- s))) s)))

C

x_m = fabs(x);
float code(float x_m, float s) {
	float tmp;
	if (x_m <= 1.5000000601271012e-26f) {
		tmp = (0.5f - ((x_m * 0.25f) / s)) / (s + (s + (x_m * (((x_m / s) * 0.5f) + -1.0f))));
	} else {
		tmp = expf((x_m / -s)) / s;
	}
	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 <= 1.5000000601271012e-26) then
        tmp = (0.5e0 - ((x_m * 0.25e0) / s)) / (s + (s + (x_m * (((x_m / s) * 0.5e0) + (-1.0e0)))))
    else
        tmp = exp((x_m / -s)) / s
    end if
    code = tmp
end function

Julia

x_m = abs(x)
function code(x_m, s)
	tmp = Float32(0.0)
	if (x_m <= Float32(1.5000000601271012e-26))
		tmp = Float32(Float32(Float32(0.5) - Float32(Float32(x_m * Float32(0.25)) / s)) / Float32(s + Float32(s + Float32(x_m * Float32(Float32(Float32(x_m / s) * Float32(0.5)) + Float32(-1.0))))));
	else
		tmp = Float32(exp(Float32(x_m / Float32(-s))) / s);
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, s)
	tmp = single(0.0);
	if (x_m <= single(1.5000000601271012e-26))
		tmp = (single(0.5) - ((x_m * single(0.25)) / s)) / (s + (s + (x_m * (((x_m / s) * single(0.5)) + single(-1.0)))));
	else
		tmp = exp((x_m / -s)) / s;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 1.50000006e-26

   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 37.2%

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

   8. Taylor expanded in s around -inf 74.3%

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

      1. mul-1-neg 74.3%

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

      2. distribute-rgt-out-- 74.3%

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

      3. metadata-eval 74.3%

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

      4. *-commutative 74.3%

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

      5. associate-*r/ 74.3%

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

      6. unsub-neg 74.3%

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

      7. associate-*r/ 74.3%

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

      8. *-commutative 74.3%

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

   10. Simplified 74.3%

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

   11. Taylor expanded in x around 0 48.9%

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

   if 1.50000006e-26 < 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%

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

   8. Taylor expanded in x around inf 99.9%

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

   10. Simplified 99.9%

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

   11. Taylor expanded in x around inf 93.0%

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

       1. neg-mul-1 93.0%

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

       2. distribute-neg-frac 93.0%

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

   13. Simplified 93.0%

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

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.5000000601271012 \cdot 10^{-26}:\\ \;\;\;\;\frac{0.5 - \frac{x \cdot 0.25}{s}}{s + \left(s + x \cdot \left(\frac{x}{s} \cdot 0.5 + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{x}{-s}}}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 94.7% accurate, 5.7× speedup

Math

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

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(exp(Float32(x_m / Float32(-s))) / Float32(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. unpow2 99.8%

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

   2. distribute-rgt-in 99.9%

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

   3. mul-1-neg 99.9%

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

   4. rec-exp 99.8%

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

   5. distribute-rgt-in 99.8%

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

7. Simplified 63.1%

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

8. Taylor expanded in x around 0 59.9%

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

Alternative 9: 39.3% accurate, 29.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{0.5 - \frac{x\_m \cdot 0.25}{s}}{s + \left(s + x\_m \cdot \left(\frac{x\_m}{s} \cdot 0.5 + -1\right)\right)} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (/ (- 0.5 (/ (* x_m 0.25) s)) (+ s (+ s (* x_m (+ (* (/ x_m s) 0.5) -1.0))))))

C

x_m = fabs(x);
float code(float x_m, float s) {
	return (0.5f - ((x_m * 0.25f) / s)) / (s + (s + (x_m * (((x_m / s) * 0.5f) + -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
    code = (0.5e0 - ((x_m * 0.25e0) / s)) / (s + (s + (x_m * (((x_m / s) * 0.5e0) + (-1.0e0)))))
end function

Julia

x_m = abs(x)
function code(x_m, s)
	return Float32(Float32(Float32(0.5) - Float32(Float32(x_m * Float32(0.25)) / s)) / Float32(s + Float32(s + Float32(x_m * Float32(Float32(Float32(x_m / s) * Float32(0.5)) + Float32(-1.0))))))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m, s)
	tmp = (single(0.5) - ((x_m * single(0.25)) / s)) / (s + (s + (x_m * (((x_m / s) * single(0.5)) + 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%

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

8. Taylor expanded in s around -inf 48.3%

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

   1. mul-1-neg 48.3%

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

   2. distribute-rgt-out-- 48.3%

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

   3. metadata-eval 48.3%

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

   4. *-commutative 48.3%

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

   5. associate-*r/ 48.3%

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

   6. unsub-neg 48.3%

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

   7. associate-*r/ 48.3%

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

   8. *-commutative 48.3%

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

10. Simplified 48.3%

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

11. Taylor expanded in x around 0 39.9%

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

12. Final simplification 39.9%

    \[\leadsto \frac{0.5 - \frac{x \cdot 0.25}{s}}{s + \left(s + x \cdot \left(\frac{x}{s} \cdot 0.5 + -1\right)\right)} \]
13. Add Preprocessing

Alternative 10: 38.9% accurate, 32.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{0.5 - \frac{x\_m \cdot 0.25}{s}}{s \cdot 2 + x\_m \cdot \left(\frac{x\_m}{s} + -1\right)} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (/ (- 0.5 (/ (* x_m 0.25) s)) (+ (* s 2.0) (* x_m (+ (/ x_m s) -1.0)))))

C

x_m = fabs(x);
float code(float x_m, float s) {
	return (0.5f - ((x_m * 0.25f) / s)) / ((s * 2.0f) + (x_m * ((x_m / s) + -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
    code = (0.5e0 - ((x_m * 0.25e0) / s)) / ((s * 2.0e0) + (x_m * ((x_m / s) + (-1.0e0))))
end function

Julia

x_m = abs(x)
function code(x_m, s)
	return Float32(Float32(Float32(0.5) - Float32(Float32(x_m * Float32(0.25)) / s)) / Float32(Float32(s * Float32(2.0)) + Float32(x_m * Float32(Float32(x_m / s) + Float32(-1.0)))))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m, s)
	tmp = (single(0.5) - ((x_m * single(0.25)) / s)) / ((s * single(2.0)) + (x_m * ((x_m / s) + 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%

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

8. Taylor expanded in s around -inf 48.3%

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

   1. mul-1-neg 48.3%

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

   2. distribute-rgt-out-- 48.3%

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

   3. metadata-eval 48.3%

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

   4. *-commutative 48.3%

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

   5. associate-*r/ 48.3%

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

   6. unsub-neg 48.3%

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

   7. associate-*r/ 48.3%

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

   8. *-commutative 48.3%

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

10. Simplified 48.3%

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

11. Taylor expanded in x around 0 24.9%

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

    1. +-commutative 60.6%

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

13. Simplified 24.9%

    \[\leadsto \frac{0.5 - \frac{x \cdot 0.25}{s}}{s + \frac{s}{\color{blue}{\frac{x}{s} + 1}}} \]

14. Taylor expanded in x around 0 39.8%

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

15. Final simplification 39.8%

    \[\leadsto \frac{0.5 - \frac{x \cdot 0.25}{s}}{s \cdot 2 + x \cdot \left(\frac{x}{s} + -1\right)} \]
16. Add Preprocessing

Alternative 11: 28.3% accurate, 88.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{0.5}{s + \left(s - x\_m\right)} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s) :precision binary32 (/ 0.5 (+ s (- s x_m))))

C

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

Fortran

x_m = abs(x)
real(4) function code(x_m, s)
    real(4), intent (in) :: x_m
    real(4), intent (in) :: s
    code = 0.5e0 / (s + (s - x_m))
end function

Julia

x_m = abs(x)
function code(x_m, s)
	return Float32(Float32(0.5) / Float32(s + Float32(s - x_m)))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m, s)
	tmp = single(0.5) / (s + (s - x_m));
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%

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

8. Taylor expanded in s around -inf 48.3%

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

   1. mul-1-neg 48.3%

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

   2. distribute-rgt-out-- 48.3%

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

   3. metadata-eval 48.3%

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

   4. *-commutative 48.3%

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

   5. associate-*r/ 48.3%

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

   6. unsub-neg 48.3%

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

   7. associate-*r/ 48.3%

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

   8. *-commutative 48.3%

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

10. Simplified 48.3%

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

11. Taylor expanded in x around 0 25.5%

    \[\leadsto \frac{0.5 - \frac{x \cdot 0.25}{s}}{s + \color{blue}{\left(s + -1 \cdot x\right)}} \]
12. Step-by-step derivation

    1. neg-mul-1 25.5%

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

    2. unsub-neg 25.5%

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

13. Simplified 25.5%

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

14. Taylor expanded in x around 0 27.7%

    \[\leadsto \frac{\color{blue}{0.5}}{s + \left(s - x\right)} \]
15. Add Preprocessing

Alternative 12: 26.6% accurate, 206.7× speedup

Math

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

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s) :precision binary32 (/ 0.25 s))

C

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

Fortran

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

Julia

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

MATLAB

x_m = abs(x);
function tmp = code(x_m, s)
	tmp = single(0.25) / 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. 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 s around inf 25.7%

   \[\leadsto \color{blue}{\frac{0.25}{s}} \]
6. 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))))))