Logistic distribution

Percentage Accurate: 99.5% → 99.5%

Time: 14.5s

Alternatives: 10

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: 99.5% 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: 99.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

   1. fabs-neg 99.5%

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

   2. distribute-frac-neg 99.5%

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

   3. distribute-frac-neg2 99.5%

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

   4. fabs-neg 99.5%

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

   5. *-commutative 99.5%

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

   6. fabs-neg 99.5%

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

   7. +-commutative 99.5%

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

   8. fabs-neg 99.5%

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

3. Simplified 99.6%

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

5. Taylor expanded in x around 0 99.6%

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

   1. unpow2 99.6%

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

   2. distribute-lft-in 99.6%

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

   3. mul-1-neg 99.6%

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

   4. rec-exp 99.6%

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

   5. distribute-lft-in 99.6%

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

   6. associate-*r/ 99.6%

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

   7. mul-1-neg 99.6%

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

7. Simplified 99.6%

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

   1. distribute-frac-neg 99.6%

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

   2. exp-neg 99.6%

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

   3. add-sqr-sqrt 49.3%

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

   4. fabs-sqr 49.3%

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

   5. add-sqr-sqrt 95.6%

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

   6. add-sqr-sqrt 49.3%

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

   7. sqrt-unprod 99.6%

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

   8. add-sqr-sqrt 49.3%

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

   9. fabs-sqr 49.3%

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

   10. add-sqr-sqrt 50.5%

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

   11. add-sqr-sqrt 49.3%

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

   12. fabs-sqr 49.3%

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

   13. add-sqr-sqrt 99.6%

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

   14. sqr-neg 99.6%

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

   15. distribute-frac-neg 99.6%

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

   16. distribute-frac-neg 99.6%

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

   17. sqrt-unprod -0.0%

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

   18. add-sqr-sqrt 91.5%

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

9. Applied egg-rr 95.6%

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

    1. rec-exp 95.7%

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

    2. distribute-frac-neg 95.7%

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

11. Simplified 95.7%

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

    1. distribute-frac-neg 99.6%

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

    2. exp-neg 99.6%

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

    3. add-sqr-sqrt 49.3%

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

    4. fabs-sqr 49.3%

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

    5. add-sqr-sqrt 95.6%

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

    6. add-sqr-sqrt 49.3%

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

    7. sqrt-unprod 99.6%

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

    8. add-sqr-sqrt 49.3%

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

    9. fabs-sqr 49.3%

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

    10. add-sqr-sqrt 50.5%

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

    11. add-sqr-sqrt 49.3%

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

    12. fabs-sqr 49.3%

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

    13. add-sqr-sqrt 99.6%

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

    14. sqr-neg 99.6%

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

    15. distribute-frac-neg 99.6%

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

    16. distribute-frac-neg 99.6%

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

    17. sqrt-unprod -0.0%

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

    18. add-sqr-sqrt 91.5%

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

13. Applied egg-rr 65.6%

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

    1. rec-exp 95.7%

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

    2. distribute-frac-neg 95.7%

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

15. Simplified 66.0%

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

16. Final simplification 66.0%

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

Alternative 2: 99.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\left|x\_m\right| \leq 1:\\ \;\;\;\;\frac{e^{\frac{x\_m}{s} + -2 \cdot \mathsf{log1p}\left(e^{\frac{x\_m}{s}}\right)}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{x\_m}{-s}}}{s \cdot 4}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (if (<= (fabs x_m) 1.0)
   (/ (exp (+ (/ x_m s) (* -2.0 (log1p (exp (/ x_m s)))))) s)
   (/ (exp (/ x_m (- s))) (* s 4.0))))

C

x_m = fabs(x);
float code(float x_m, float s) {
	float tmp;
	if (fabsf(x_m) <= 1.0f) {
		tmp = expf(((x_m / s) + (-2.0f * log1pf(expf((x_m / s)))))) / s;
	} else {
		tmp = expf((x_m / -s)) / (s * 4.0f);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m, s)
	tmp = Float32(0.0)
	if (abs(x_m) <= Float32(1.0))
		tmp = Float32(exp(Float32(Float32(x_m / s) + Float32(Float32(-2.0) * log1p(exp(Float32(x_m / s)))))) / s);
	else
		tmp = Float32(exp(Float32(x_m / Float32(-s))) / Float32(s * Float32(4.0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (fabs.f32 x) < 1

   1. Initial program 99.1%

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

      1. fabs-neg 99.1%

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

      2. distribute-frac-neg 99.1%

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

      3. distribute-frac-neg2 99.1%

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

      4. fabs-neg 99.1%

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

      5. *-commutative 99.1%

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

      6. fabs-neg 99.1%

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

      7. +-commutative 99.1%

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

      8. fabs-neg 99.1%

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

   3. Simplified 99.2%

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

   6. Applied egg-rr 71.9%

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

      1. *-lft-identity 71.9%

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

      2. *-commutative 71.9%

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

      3. exp-to-pow 71.9%

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

      4. log1p-undefine 72.0%

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

      5. *-commutative 72.0%

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

      6. rem-exp-log 68.2%

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

      7. prod-exp 67.4%

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

      8. exp-diff 94.1%

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

      9. associate--r+ 94.4%

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

      10. exp-diff 94.9%

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

      11. cancel-sign-sub-inv 94.9%

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

      12. metadata-eval 94.9%

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

   8. Simplified 98.8%

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

   if 1 < (fabs.f32 x)

   1. Initial program 100.0%

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

      1. fabs-neg 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. distribute-frac-neg2 100.0%

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

      4. fabs-neg 100.0%

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

      5. *-commutative 100.0%

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

      6. fabs-neg 100.0%

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

      7. +-commutative 100.0%

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

      8. fabs-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 100.0%

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

      1. unpow2 100.0%

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

      2. distribute-lft-in 100.0%

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

      3. mul-1-neg 100.0%

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

      4. rec-exp 100.0%

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

      5. distribute-lft-in 100.0%

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

      6. associate-*r/ 100.0%

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

      7. mul-1-neg 100.0%

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

   7. Simplified 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. exp-neg 100.0%

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

      3. add-sqr-sqrt 46.3%

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

      4. fabs-sqr 46.3%

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

      5. add-sqr-sqrt 100.0%

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

      6. add-sqr-sqrt 46.3%

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

      7. sqrt-unprod 100.0%

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

      8. add-sqr-sqrt 46.3%

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

      9. fabs-sqr 46.3%

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

      10. add-sqr-sqrt 46.3%

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

      11. add-sqr-sqrt 46.3%

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

      12. fabs-sqr 46.3%

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

      13. add-sqr-sqrt 100.0%

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

      14. sqr-neg 100.0%

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

      15. distribute-frac-neg 100.0%

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

      16. distribute-frac-neg 100.0%

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

      17. sqrt-unprod -0.0%

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

      18. add-sqr-sqrt 100.0%

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

   9. Applied egg-rr 100.0%

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

       1. rec-exp 100.0%

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

       2. distribute-frac-neg 100.0%

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

   11. Simplified 100.0%

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

       1. distribute-frac-neg 100.0%

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

       2. exp-neg 100.0%

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

       3. add-sqr-sqrt 46.3%

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

       4. fabs-sqr 46.3%

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

       5. add-sqr-sqrt 100.0%

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

       6. add-sqr-sqrt 46.3%

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

       7. sqrt-unprod 100.0%

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

       8. add-sqr-sqrt 46.3%

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

       9. fabs-sqr 46.3%

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

       10. add-sqr-sqrt 46.3%

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

       11. add-sqr-sqrt 46.3%

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

       12. fabs-sqr 46.3%

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

       13. add-sqr-sqrt 100.0%

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

       14. sqr-neg 100.0%

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

       15. distribute-frac-neg 100.0%

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

       16. distribute-frac-neg 100.0%

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

       17. sqrt-unprod -0.0%

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

       18. add-sqr-sqrt 100.0%

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

   13. Applied egg-rr 46.3%

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

       1. rec-exp 100.0%

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

       2. distribute-frac-neg 100.0%

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

   15. Simplified 46.3%

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

   16. Taylor expanded in s around inf 48.0%

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

       1. *-commutative 48.0%

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

   18. Simplified 48.0%

       \[\leadsto \frac{e^{\frac{-x}{s}}}{\color{blue}{s \cdot 4}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 1:\\ \;\;\;\;\frac{e^{\frac{x}{s} + -2 \cdot \mathsf{log1p}\left(e^{\frac{x}{s}}\right)}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{x}{-s}}}{s \cdot 4}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;-\left|x\_m\right| \leq -50000:\\ \;\;\;\;\frac{e^{\frac{x\_m}{-s}}}{s \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 + \frac{x\_m \cdot 0.25}{s}}{s \cdot \left(e^{\frac{x\_m}{s}} + 1\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (if (<= (- (fabs x_m)) -50000.0)
   (/ (exp (/ x_m (- s))) (* s 4.0))
   (/ (+ 0.5 (/ (* x_m 0.25) s)) (* s (+ (exp (/ x_m s)) 1.0)))))

C

x_m = fabs(x);
float code(float x_m, float s) {
	float tmp;
	if (-fabsf(x_m) <= -50000.0f) {
		tmp = expf((x_m / -s)) / (s * 4.0f);
	} else {
		tmp = (0.5f + ((x_m * 0.25f) / s)) / (s * (expf((x_m / s)) + 1.0f));
	}
	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 (-abs(x_m) <= (-50000.0e0)) then
        tmp = exp((x_m / -s)) / (s * 4.0e0)
    else
        tmp = (0.5e0 + ((x_m * 0.25e0) / s)) / (s * (exp((x_m / s)) + 1.0e0))
    end if
    code = tmp
end function

Julia

x_m = abs(x)
function code(x_m, s)
	tmp = Float32(0.0)
	if (Float32(-abs(x_m)) <= Float32(-50000.0))
		tmp = Float32(exp(Float32(x_m / Float32(-s))) / Float32(s * Float32(4.0)));
	else
		tmp = Float32(Float32(Float32(0.5) + Float32(Float32(x_m * Float32(0.25)) / s)) / Float32(s * Float32(exp(Float32(x_m / s)) + Float32(1.0))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, s)
	tmp = single(0.0);
	if (-abs(x_m) <= single(-50000.0))
		tmp = exp((x_m / -s)) / (s * single(4.0));
	else
		tmp = (single(0.5) + ((x_m * single(0.25)) / s)) / (s * (exp((x_m / s)) + single(1.0)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (neg.f32 (fabs.f32 x)) < -5e4

   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. fabs-neg 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. distribute-frac-neg2 100.0%

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

      4. fabs-neg 100.0%

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

      5. *-commutative 100.0%

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

      6. fabs-neg 100.0%

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

      7. +-commutative 100.0%

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

      8. fabs-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 100.0%

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

      1. unpow2 100.0%

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

      2. distribute-lft-in 100.0%

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

      3. mul-1-neg 100.0%

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

      4. rec-exp 100.0%

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

      5. distribute-lft-in 100.0%

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

      6. associate-*r/ 100.0%

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

      7. mul-1-neg 100.0%

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

   7. Simplified 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. exp-neg 100.0%

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

      3. add-sqr-sqrt 47.7%

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

      4. fabs-sqr 47.7%

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

      5. add-sqr-sqrt 100.0%

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

      6. add-sqr-sqrt 47.7%

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

      7. sqrt-unprod 100.0%

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

      8. add-sqr-sqrt 47.7%

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

      9. fabs-sqr 47.7%

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

      10. add-sqr-sqrt 47.7%

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

      11. add-sqr-sqrt 47.7%

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

      12. fabs-sqr 47.7%

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

      13. add-sqr-sqrt 100.0%

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

      14. sqr-neg 100.0%

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

      15. distribute-frac-neg 100.0%

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

      16. distribute-frac-neg 100.0%

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

      17. sqrt-unprod -0.0%

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

      18. add-sqr-sqrt 100.0%

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

   9. Applied egg-rr 100.0%

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

       1. rec-exp 100.0%

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

       2. distribute-frac-neg 100.0%

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

   11. Simplified 100.0%

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

       1. distribute-frac-neg 100.0%

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

       2. exp-neg 100.0%

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

       3. add-sqr-sqrt 47.7%

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

       4. fabs-sqr 47.7%

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

       5. add-sqr-sqrt 100.0%

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

       6. add-sqr-sqrt 47.7%

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

       7. sqrt-unprod 100.0%

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

       8. add-sqr-sqrt 47.7%

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

       9. fabs-sqr 47.7%

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

       10. add-sqr-sqrt 47.7%

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

       11. add-sqr-sqrt 47.7%

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

       12. fabs-sqr 47.7%

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

       13. add-sqr-sqrt 100.0%

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

       14. sqr-neg 100.0%

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

       15. distribute-frac-neg 100.0%

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

       16. distribute-frac-neg 100.0%

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

       17. sqrt-unprod -0.0%

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

       18. add-sqr-sqrt 100.0%

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

   13. Applied egg-rr 47.7%

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

       1. rec-exp 100.0%

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

       2. distribute-frac-neg 100.0%

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

   15. Simplified 47.7%

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

   16. Taylor expanded in s around inf 49.3%

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

       1. *-commutative 49.3%

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

   18. Simplified 49.3%

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

   if -5e4 < (neg.f32 (fabs.f32 x))

   1. Initial program 99.2%

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

      1. fabs-neg 99.2%

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

      2. distribute-frac-neg 99.2%

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

      3. distribute-frac-neg2 99.2%

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

      4. fabs-neg 99.2%

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

      5. *-commutative 99.2%

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

      6. fabs-neg 99.2%

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

      7. +-commutative 99.2%

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

      8. fabs-neg 99.2%

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

   3. Simplified 99.3%

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

   6. Applied egg-rr 71.4%

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

      1. associate-*l/ 71.4%

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

      2. *-lft-identity 71.4%

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

   8. Simplified 71.4%

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

   9. Taylor expanded in s around inf 59.1%

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

       1. *-un-lft-identity 59.1%

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

       2. associate-/l/ 72.5%

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

       3. associate--l+ 72.6%

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

       4. associate-*r/ 72.6%

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

       5. associate-*r/ 72.6%

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

       6. sub-div 72.6%

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

   11. Applied egg-rr 72.6%

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

       1. associate-*r/ 72.6%

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

       2. *-commutative 72.6%

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

       3. *-lft-identity 72.6%

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

       4. distribute-rgt-out-- 72.6%

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

       5. metadata-eval 72.6%

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

   13. Simplified 72.6%

       \[\leadsto \color{blue}{\frac{0.5 + \frac{x \cdot 0.25}{s}}{s \cdot \left(1 + e^{\frac{x}{s}}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\left|x\right| \leq -50000:\\ \;\;\;\;\frac{e^{\frac{x}{-s}}}{s \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 + \frac{x \cdot 0.25}{s}}{s \cdot \left(e^{\frac{x}{s}} + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.8% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

   1. fabs-neg 99.5%

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

   2. distribute-frac-neg 99.5%

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

   3. distribute-frac-neg2 99.5%

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

   4. fabs-neg 99.5%

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

   5. *-commutative 99.5%

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

   6. fabs-neg 99.5%

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

   7. +-commutative 99.5%

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

   8. fabs-neg 99.5%

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

3. Simplified 99.6%

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

6. Applied egg-rr 63.4%

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

   1. associate-*l/ 63.5%

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

   2. *-lft-identity 63.5%

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

8. Simplified 63.5%

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

9. Taylor expanded in x around 0 60.4%

   \[\leadsto \frac{\color{blue}{\frac{0.5}{s}}}{1 + e^{\frac{x}{s}}} \]

10. Taylor expanded in s around 0 60.4%

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

11. Final simplification 60.4%

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

Alternative 5: 94.4% 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 99.5%

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

   1. fabs-neg 99.5%

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

   2. distribute-frac-neg 99.5%

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

   3. distribute-frac-neg2 99.5%

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

   4. fabs-neg 99.5%

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

   5. *-commutative 99.5%

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

   6. fabs-neg 99.5%

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

   7. +-commutative 99.5%

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

   8. fabs-neg 99.5%

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

3. Simplified 99.6%

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

5. Taylor expanded in x around 0 99.6%

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

   1. unpow2 99.6%

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

   2. distribute-lft-in 99.6%

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

   3. mul-1-neg 99.6%

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

   4. rec-exp 99.6%

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

   5. distribute-lft-in 99.6%

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

   6. associate-*r/ 99.6%

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

   7. mul-1-neg 99.6%

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

7. Simplified 99.6%

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

   1. distribute-frac-neg 99.6%

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

   2. exp-neg 99.6%

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

   3. add-sqr-sqrt 49.3%

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

   4. fabs-sqr 49.3%

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

   5. add-sqr-sqrt 95.6%

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

   6. add-sqr-sqrt 49.3%

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

   7. sqrt-unprod 99.6%

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

   8. add-sqr-sqrt 49.3%

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

   9. fabs-sqr 49.3%

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

   10. add-sqr-sqrt 50.5%

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

   11. add-sqr-sqrt 49.3%

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

   12. fabs-sqr 49.3%

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

   13. add-sqr-sqrt 99.6%

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

   14. sqr-neg 99.6%

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

   15. distribute-frac-neg 99.6%

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

   16. distribute-frac-neg 99.6%

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

   17. sqrt-unprod -0.0%

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

   18. add-sqr-sqrt 91.5%

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

9. Applied egg-rr 95.6%

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

    1. rec-exp 95.7%

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

    2. distribute-frac-neg 95.7%

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

11. Simplified 95.7%

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

    1. distribute-frac-neg 99.6%

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

    2. exp-neg 99.6%

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

    3. add-sqr-sqrt 49.3%

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

    4. fabs-sqr 49.3%

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

    5. add-sqr-sqrt 95.6%

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

    6. add-sqr-sqrt 49.3%

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

    7. sqrt-unprod 99.6%

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

    8. add-sqr-sqrt 49.3%

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

    9. fabs-sqr 49.3%

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

    10. add-sqr-sqrt 50.5%

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

    11. add-sqr-sqrt 49.3%

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

    12. fabs-sqr 49.3%

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

    13. add-sqr-sqrt 99.6%

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

    14. sqr-neg 99.6%

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

    15. distribute-frac-neg 99.6%

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

    16. distribute-frac-neg 99.6%

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

    17. sqrt-unprod -0.0%

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

    18. add-sqr-sqrt 91.5%

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

13. Applied egg-rr 65.6%

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

    1. rec-exp 95.7%

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

    2. distribute-frac-neg 95.7%

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

15. Simplified 66.0%

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

16. Taylor expanded in s around inf 59.5%

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

    1. *-commutative 59.5%

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

18. Simplified 59.5%

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

19. Final simplification 59.5%

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

Alternative 6: 77.6% accurate, 28.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1600000006029312:\\ \;\;\;\;\frac{\left(0.25 + \frac{x\_m \cdot -0.125}{s}\right) - 0.5 \cdot \left(\frac{x\_m}{s} \cdot -0.25\right)}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \frac{2 + \frac{x\_m}{s}}{0.5}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (if (<= x_m 1600000006029312.0)
   (/ (- (+ 0.25 (/ (* x_m -0.125) s)) (* 0.5 (* (/ x_m s) -0.25))) s)
   (/ 1.0 (* s (/ (+ 2.0 (/ x_m s)) 0.5)))))

C

x_m = fabs(x);
float code(float x_m, float s) {
	float tmp;
	if (x_m <= 1600000006029312.0f) {
		tmp = ((0.25f + ((x_m * -0.125f) / s)) - (0.5f * ((x_m / s) * -0.25f))) / s;
	} else {
		tmp = 1.0f / (s * ((2.0f + (x_m / s)) / 0.5f));
	}
	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 <= 1600000006029312.0e0) then
        tmp = ((0.25e0 + ((x_m * (-0.125e0)) / s)) - (0.5e0 * ((x_m / s) * (-0.25e0)))) / s
    else
        tmp = 1.0e0 / (s * ((2.0e0 + (x_m / s)) / 0.5e0))
    end if
    code = tmp
end function

Julia

x_m = abs(x)
function code(x_m, s)
	tmp = Float32(0.0)
	if (x_m <= Float32(1600000006029312.0))
		tmp = Float32(Float32(Float32(Float32(0.25) + Float32(Float32(x_m * Float32(-0.125)) / s)) - Float32(Float32(0.5) * Float32(Float32(x_m / s) * Float32(-0.25)))) / s);
	else
		tmp = Float32(Float32(1.0) / Float32(s * Float32(Float32(Float32(2.0) + Float32(x_m / s)) / Float32(0.5))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, s)
	tmp = single(0.0);
	if (x_m <= single(1600000006029312.0))
		tmp = ((single(0.25) + ((x_m * single(-0.125)) / s)) - (single(0.5) * ((x_m / s) * single(-0.25)))) / s;
	else
		tmp = single(1.0) / (s * ((single(2.0) + (x_m / s)) / single(0.5)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 1.60000001e15

   1. Initial program 99.4%

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

      1. fabs-neg 99.4%

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

      2. distribute-frac-neg 99.4%

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

      3. distribute-frac-neg2 99.4%

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

      4. fabs-neg 99.4%

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

      5. *-commutative 99.4%

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

      6. fabs-neg 99.4%

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

      7. +-commutative 99.4%

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

      8. fabs-neg 99.4%

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

   3. Simplified 99.5%

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

   6. Applied egg-rr 73.8%

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

      1. associate-*l/ 73.8%

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

      2. *-lft-identity 73.8%

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

   8. Simplified 73.8%

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

   9. Taylor expanded in s around inf 42.3%

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

   10. Taylor expanded in s around -inf 69.7%

       \[\leadsto \color{blue}{-1 \cdot \frac{0.5 \cdot \frac{-0.5 \cdot x - -0.25 \cdot x}{s} - \left(0.25 + -0.125 \cdot \frac{x}{s}\right)}{s}} \]
   11. Step-by-step derivation

       1. mul-1-neg 69.7%

          \[\leadsto \color{blue}{-\frac{0.5 \cdot \frac{-0.5 \cdot x - -0.25 \cdot x}{s} - \left(0.25 + -0.125 \cdot \frac{x}{s}\right)}{s}} \]

       2. *-commutative 69.7%

          \[\leadsto -\frac{\color{blue}{\frac{-0.5 \cdot x - -0.25 \cdot x}{s} \cdot 0.5} - \left(0.25 + -0.125 \cdot \frac{x}{s}\right)}{s} \]

       3. distribute-rgt-out-- 69.7%

          \[\leadsto -\frac{\frac{\color{blue}{x \cdot \left(-0.5 - -0.25\right)}}{s} \cdot 0.5 - \left(0.25 + -0.125 \cdot \frac{x}{s}\right)}{s} \]

       4. metadata-eval 69.7%

          \[\leadsto -\frac{\frac{x \cdot \color{blue}{-0.25}}{s} \cdot 0.5 - \left(0.25 + -0.125 \cdot \frac{x}{s}\right)}{s} \]

       5. *-commutative 69.7%

          \[\leadsto -\frac{\frac{\color{blue}{-0.25 \cdot x}}{s} \cdot 0.5 - \left(0.25 + -0.125 \cdot \frac{x}{s}\right)}{s} \]

       6. associate-*r/ 69.6%

          \[\leadsto -\frac{\color{blue}{\left(-0.25 \cdot \frac{x}{s}\right)} \cdot 0.5 - \left(0.25 + -0.125 \cdot \frac{x}{s}\right)}{s} \]

       7. *-commutative 69.6%

          \[\leadsto -\frac{\color{blue}{\left(\frac{x}{s} \cdot -0.25\right)} \cdot 0.5 - \left(0.25 + -0.125 \cdot \frac{x}{s}\right)}{s} \]

       8. associate-*r/ 69.7%

          \[\leadsto -\frac{\left(\frac{x}{s} \cdot -0.25\right) \cdot 0.5 - \left(0.25 + \color{blue}{\frac{-0.125 \cdot x}{s}}\right)}{s} \]

       9. *-commutative 69.7%

          \[\leadsto -\frac{\left(\frac{x}{s} \cdot -0.25\right) \cdot 0.5 - \left(0.25 + \frac{\color{blue}{x \cdot -0.125}}{s}\right)}{s} \]

   12. Simplified 69.7%

       \[\leadsto \color{blue}{-\frac{\left(\frac{x}{s} \cdot -0.25\right) \cdot 0.5 - \left(0.25 + \frac{x \cdot -0.125}{s}\right)}{s}} \]

   if 1.60000001e15 < 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. fabs-neg 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. distribute-frac-neg2 100.0%

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

      4. fabs-neg 100.0%

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

      5. *-commutative 100.0%

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

      6. fabs-neg 100.0%

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

      7. +-commutative 100.0%

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

      8. fabs-neg 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr -0.0%

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

      1. associate-*l/ -0.0%

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

      2. *-lft-identity -0.0%

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

   8. Simplified -0.0%

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

   9. Taylor expanded in x around 0 100.0%

      \[\leadsto \frac{\color{blue}{\frac{0.5}{s}}}{1 + e^{\frac{x}{s}}} \]

   10. Taylor expanded in x around 0 80.7%

       \[\leadsto \frac{\frac{0.5}{s}}{\color{blue}{2 + \frac{x}{s}}} \]
   11. Step-by-step derivation

       1. +-commutative 80.7%

          \[\leadsto \frac{\frac{0.5}{s}}{\color{blue}{\frac{x}{s} + 2}} \]

   12. Simplified 80.7%

       \[\leadsto \frac{\frac{0.5}{s}}{\color{blue}{\frac{x}{s} + 2}} \]
   13. Step-by-step derivation

       1. clear-num 80.7%

          \[\leadsto \color{blue}{\frac{1}{\frac{\frac{x}{s} + 2}{\frac{0.5}{s}}}} \]

       2. inv-pow 80.7%

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

       3. +-commutative 80.7%

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

   14. Applied egg-rr 80.7%

       \[\leadsto \color{blue}{{\left(\frac{2 + \frac{x}{s}}{\frac{0.5}{s}}\right)}^{-1}} \]
   15. Step-by-step derivation

       1. unpow-1 80.7%

          \[\leadsto \color{blue}{\frac{1}{\frac{2 + \frac{x}{s}}{\frac{0.5}{s}}}} \]

       2. associate-/r/ 80.7%

          \[\leadsto \frac{1}{\color{blue}{\frac{2 + \frac{x}{s}}{0.5} \cdot s}} \]

   16. Simplified 80.7%

       \[\leadsto \color{blue}{\frac{1}{\frac{2 + \frac{x}{s}}{0.5} \cdot s}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1600000006029312:\\ \;\;\;\;\frac{\left(0.25 + \frac{x \cdot -0.125}{s}\right) - 0.5 \cdot \left(\frac{x}{s} \cdot -0.25\right)}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \frac{2 + \frac{x}{s}}{0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 53.4% accurate, 51.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0020000000949949026:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{s}}{\frac{x\_m}{s}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (if (<= x_m 0.0020000000949949026) (/ 0.25 s) (/ (/ 0.5 s) (/ x_m s))))

C

x_m = fabs(x);
float code(float x_m, float s) {
	float tmp;
	if (x_m <= 0.0020000000949949026f) {
		tmp = 0.25f / s;
	} else {
		tmp = (0.5f / s) / (x_m / 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 <= 0.0020000000949949026e0) then
        tmp = 0.25e0 / s
    else
        tmp = (0.5e0 / s) / (x_m / 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(0.0020000000949949026))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(Float32(0.5) / s) / Float32(x_m / 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(0.0020000000949949026))
		tmp = single(0.25) / s;
	else
		tmp = (single(0.5) / s) / (x_m / s);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 0.00200000009

   1. Initial program 99.4%

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

      1. fabs-neg 99.4%

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

      2. distribute-frac-neg 99.4%

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

      3. distribute-frac-neg2 99.4%

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

      4. fabs-neg 99.4%

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

      5. *-commutative 99.4%

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

      6. fabs-neg 99.4%

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

      7. +-commutative 99.4%

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

      8. fabs-neg 99.4%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in s around inf 35.3%

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

   if 0.00200000009 < 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. fabs-neg 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. distribute-frac-neg2 100.0%

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

      4. fabs-neg 100.0%

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

      5. *-commutative 100.0%

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

      6. fabs-neg 100.0%

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

      7. +-commutative 100.0%

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

      8. fabs-neg 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr -0.0%

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

      1. associate-*l/ -0.0%

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

      2. *-lft-identity -0.0%

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

   8. Simplified -0.0%

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

   9. Taylor expanded in x around 0 100.0%

      \[\leadsto \frac{\color{blue}{\frac{0.5}{s}}}{1 + e^{\frac{x}{s}}} \]

   10. Taylor expanded in x around 0 58.3%

       \[\leadsto \frac{\frac{0.5}{s}}{\color{blue}{2 + \frac{x}{s}}} \]
   11. Step-by-step derivation

       1. +-commutative 58.3%

          \[\leadsto \frac{\frac{0.5}{s}}{\color{blue}{\frac{x}{s} + 2}} \]

   12. Simplified 58.3%

       \[\leadsto \frac{\frac{0.5}{s}}{\color{blue}{\frac{x}{s} + 2}} \]

   13. Taylor expanded in x around inf 58.3%

       \[\leadsto \frac{\frac{0.5}{s}}{\color{blue}{\frac{x}{s}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0020000000949949026:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{s}}{\frac{x}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 52.1% accurate, 56.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

x_m = abs(x);
function tmp = code(x_m, s)
	tmp = (single(0.5) / s) * (single(1.0) / (single(2.0) + (x_m / s)));
end

Derivation

1. Initial program 99.5%

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

   1. fabs-neg 99.5%

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

   2. distribute-frac-neg 99.5%

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

   3. distribute-frac-neg2 99.5%

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

   4. fabs-neg 99.5%

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

   5. *-commutative 99.5%

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

   6. fabs-neg 99.5%

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

   7. +-commutative 99.5%

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

   8. fabs-neg 99.5%

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

3. Simplified 99.6%

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

6. Applied egg-rr 63.4%

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

   1. associate-*l/ 63.5%

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

   2. *-lft-identity 63.5%

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

8. Simplified 63.5%

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

9. Taylor expanded in x around 0 60.4%

   \[\leadsto \frac{\color{blue}{\frac{0.5}{s}}}{1 + e^{\frac{x}{s}}} \]

10. Taylor expanded in x around 0 50.5%

    \[\leadsto \frac{\frac{0.5}{s}}{\color{blue}{2 + \frac{x}{s}}} \]
11. Step-by-step derivation

    1. +-commutative 50.5%

       \[\leadsto \frac{\frac{0.5}{s}}{\color{blue}{\frac{x}{s} + 2}} \]

12. Simplified 50.5%

    \[\leadsto \frac{\frac{0.5}{s}}{\color{blue}{\frac{x}{s} + 2}} \]
13. Step-by-step derivation

    1. div-inv 50.5%

       \[\leadsto \color{blue}{\frac{0.5}{s} \cdot \frac{1}{\frac{x}{s} + 2}} \]

    2. +-commutative 50.5%

       \[\leadsto \frac{0.5}{s} \cdot \frac{1}{\color{blue}{2 + \frac{x}{s}}} \]

14. Applied egg-rr 50.5%

    \[\leadsto \color{blue}{\frac{0.5}{s} \cdot \frac{1}{2 + \frac{x}{s}}} \]

15. Final simplification 50.5%

    \[\leadsto \frac{0.5}{s} \cdot \frac{1}{2 + \frac{x}{s}} \]
16. Add Preprocessing

Alternative 9: 52.1% accurate, 68.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{\frac{0.5}{s}}{2 + \frac{x\_m}{s}} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

x_m = abs(x);
function tmp = code(x_m, s)
	tmp = (single(0.5) / s) / (single(2.0) + (x_m / s));
end

Derivation

1. Initial program 99.5%

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

   1. fabs-neg 99.5%

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

   2. distribute-frac-neg 99.5%

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

   3. distribute-frac-neg2 99.5%

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

   4. fabs-neg 99.5%

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

   5. *-commutative 99.5%

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

   6. fabs-neg 99.5%

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

   7. +-commutative 99.5%

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

   8. fabs-neg 99.5%

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

3. Simplified 99.6%

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

6. Applied egg-rr 63.4%

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

   1. associate-*l/ 63.5%

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

   2. *-lft-identity 63.5%

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

8. Simplified 63.5%

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

9. Taylor expanded in x around 0 60.4%

   \[\leadsto \frac{\color{blue}{\frac{0.5}{s}}}{1 + e^{\frac{x}{s}}} \]

10. Taylor expanded in x around 0 50.5%

    \[\leadsto \frac{\frac{0.5}{s}}{\color{blue}{2 + \frac{x}{s}}} \]
11. Step-by-step derivation

    1. +-commutative 50.5%

       \[\leadsto \frac{\frac{0.5}{s}}{\color{blue}{\frac{x}{s} + 2}} \]

12. Simplified 50.5%

    \[\leadsto \frac{\frac{0.5}{s}}{\color{blue}{\frac{x}{s} + 2}} \]

13. Final simplification 50.5%

    \[\leadsto \frac{\frac{0.5}{s}}{2 + \frac{x}{s}} \]
14. Add Preprocessing

Alternative 10: 27.8% 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 99.5%

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

   1. fabs-neg 99.5%

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

   2. distribute-frac-neg 99.5%

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

   3. distribute-frac-neg2 99.5%

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

   4. fabs-neg 99.5%

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

   5. *-commutative 99.5%

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

   6. fabs-neg 99.5%

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

   7. +-commutative 99.5%

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

   8. fabs-neg 99.5%

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

3. Simplified 99.6%

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

5. Taylor expanded in s around inf 28.2%

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

6. Final simplification 28.2%

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

Reproduce

herbie shell --seed 2024078 
(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))))))