Logistic distribution

Percentage Accurate: 99.6% → 99.6%

Time: 11.3s

Alternatives: 8

Speedup: 2.0×

Specification

\[0 \leq s \land s \leq 1.0651631\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{-\left|x\right|}{s}}\\ t_1 := 1 + t\_0\\ \frac{t\_0}{\left(s \cdot t\_1\right) \cdot t\_1} \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ (- (fabs x)) s))) (t_1 (+ 1.0 t_0)))
   (/ t_0 (* (* s t_1) t_1))))

C

float code(float x, float s) {
	float t_0 = expf((-fabsf(x) / s));
	float t_1 = 1.0f + t_0;
	return t_0 / ((s * t_1) * t_1);
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: t_1
    t_0 = exp((-abs(x) / s))
    t_1 = 1.0e0 + t_0
    code = t_0 / ((s * t_1) * t_1)
end function

Julia

function code(x, s)
	t_0 = exp(Float32(Float32(-abs(x)) / s))
	t_1 = Float32(Float32(1.0) + t_0)
	return Float32(t_0 / Float32(Float32(s * t_1) * t_1))
end

MATLAB

function tmp = code(x, s)
	t_0 = exp((-abs(x) / s));
	t_1 = single(1.0) + t_0;
	tmp = t_0 / ((s * t_1) * t_1);
end

Initial Program: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{-\left|x\right|}{s}}\\ t_1 := 1 + t\_0\\ \frac{t\_0}{\left(s \cdot t\_1\right) \cdot t\_1} \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ (- (fabs x)) s))) (t_1 (+ 1.0 t_0)))
   (/ t_0 (* (* s t_1) t_1))))

C

float code(float x, float s) {
	float t_0 = expf((-fabsf(x) / s));
	float t_1 = 1.0f + t_0;
	return t_0 / ((s * t_1) * t_1);
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: t_1
    t_0 = exp((-abs(x) / s))
    t_1 = 1.0e0 + t_0
    code = t_0 / ((s * t_1) * t_1)
end function

Julia

function code(x, s)
	t_0 = exp(Float32(Float32(-abs(x)) / s))
	t_1 = Float32(Float32(1.0) + t_0)
	return Float32(t_0 / Float32(Float32(s * t_1) * t_1))
end

MATLAB

function tmp = code(x, s)
	t_0 = exp((-abs(x) / s));
	t_1 = single(1.0) + t_0;
	tmp = t_0 / ((s * t_1) * t_1);
end

Alternative 1: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

   1. fabs-neg 99.7%

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

   2. distribute-frac-neg 99.7%

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

   3. distribute-frac-neg2 99.7%

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

   4. fabs-neg 99.7%

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

   5. *-commutative 99.7%

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

   6. fabs-neg 99.7%

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

   7. +-commutative 99.7%

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

   8. fabs-neg 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in x around 0 99.7%

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

   1. associate-/r* 99.7%

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

   2. exp-prod 99.6%

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

   3. rem-square-sqrt 44.5%

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

   4. fabs-sqr 44.5%

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

   5. rem-square-sqrt 57.2%

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

   6. exp-prod 57.2%

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

   7. neg-mul-1 57.2%

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

   8. distribute-neg-frac2 57.2%

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

   9. +-commutative 57.2%

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

   10. exp-prod 57.2%

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

   11. rem-square-sqrt 44.5%

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

   12. fabs-sqr 44.5%

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

   13. rem-square-sqrt 58.4%

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

   14. exp-prod 58.4%

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

   15. neg-mul-1 58.4%

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

   16. distribute-neg-frac2 58.4%

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

7. Simplified 58.4%

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

Alternative 2: 96.1% accurate, 5.4× speedup

Math

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

FPCore

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

C

x_m = fabs(x);
float code(float x_m, float s) {
	return (expf((x_m / -s)) / s) / (4.0f + ((x_m / 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 + ((x_m / s) * (-4.0e0)))
end function

Julia

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

Derivation

1. Initial program 99.7%

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

   1. fabs-neg 99.7%

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

   2. distribute-frac-neg 99.7%

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

   3. distribute-frac-neg2 99.7%

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

   4. fabs-neg 99.7%

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

   5. *-commutative 99.7%

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

   6. fabs-neg 99.7%

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

   7. +-commutative 99.7%

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

   8. fabs-neg 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in x around 0 99.7%

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

   1. associate-/r* 99.7%

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

   2. exp-prod 99.6%

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

   3. rem-square-sqrt 44.5%

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

   4. fabs-sqr 44.5%

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

   5. rem-square-sqrt 57.2%

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

   6. exp-prod 57.2%

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

   7. neg-mul-1 57.2%

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

   8. distribute-neg-frac2 57.2%

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

   9. +-commutative 57.2%

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

   10. exp-prod 57.2%

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

   11. rem-square-sqrt 44.5%

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

   12. fabs-sqr 44.5%

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

   13. rem-square-sqrt 58.4%

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

   14. exp-prod 58.4%

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

   15. neg-mul-1 58.4%

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

   16. distribute-neg-frac2 58.4%

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

7. Simplified 58.4%

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

8. Taylor expanded in x around 0 57.4%

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

   1. *-commutative 57.4%

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

10. Simplified 57.4%

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

Alternative 3: 95.4% accurate, 5.6× speedup

Math

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

FPCore

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

C

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

Fortran

x_m = abs(x)
real(4) function code(x_m, s)
    real(4), intent (in) :: x_m
    real(4), intent (in) :: s
    code = ((-1.0e0) / ((-1.0e0) - exp((x_m / s)))) * (0.5e0 / s)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

   1. fabs-neg 99.7%

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

   2. distribute-frac-neg 99.7%

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

   3. distribute-frac-neg2 99.7%

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

   4. fabs-neg 99.7%

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

   5. *-commutative 99.7%

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

   6. fabs-neg 99.7%

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

   7. +-commutative 99.7%

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

   8. fabs-neg 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in x around 0 99.7%

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

   1. +-commutative 99.7%

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

   2. exp-prod 99.7%

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

   3. rem-square-sqrt 44.5%

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

   4. fabs-sqr 44.5%

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

   5. rem-square-sqrt 96.2%

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

   6. exp-prod 96.2%

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

   7. neg-mul-1 96.2%

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

   8. distribute-neg-frac2 96.2%

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

7. Simplified 96.2%

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

   1. add-sqr-sqrt -0.0%

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

   2. add-sqr-sqrt 96.2%

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

   3. add-sqr-sqrt 44.4%

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

   4. fabs-sqr 44.4%

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

   5. add-sqr-sqrt 58.4%

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

   6. *-rgt-identity 58.4%

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

   7. frac-times 58.4%

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

   8. div-inv 58.4%

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

   9. div-inv 58.4%

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

   10. unpow2 58.4%

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

   11. times-frac 59.5%

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

9. Applied egg-rr 67.6%

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

10. Taylor expanded in x around 0 56.7%

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

    1. associate-*r/ 56.7%

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

    2. clear-num 56.8%

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

    3. +-commutative 56.8%

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

    4. un-div-inv 56.8%

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

12. Applied egg-rr 56.8%

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

    1. associate-/r/ 56.7%

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

14. Simplified 56.7%

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

15. Final simplification 56.7%

    \[\leadsto \frac{-1}{-1 - e^{\frac{x}{s}}} \cdot \frac{0.5}{s} \]
16. Add Preprocessing

Alternative 4: 95.4% accurate, 5.7× speedup

Math

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

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(Float32(0.5) / 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 99.7%

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

   1. fabs-neg 99.7%

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

   2. distribute-frac-neg 99.7%

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

   3. distribute-frac-neg2 99.7%

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

   4. fabs-neg 99.7%

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

   5. *-commutative 99.7%

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

   6. fabs-neg 99.7%

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

   7. +-commutative 99.7%

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

   8. fabs-neg 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in x around 0 99.7%

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

   1. +-commutative 99.7%

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

   2. exp-prod 99.7%

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

   3. rem-square-sqrt 44.5%

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

   4. fabs-sqr 44.5%

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

   5. rem-square-sqrt 96.2%

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

   6. exp-prod 96.2%

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

   7. neg-mul-1 96.2%

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

   8. distribute-neg-frac2 96.2%

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

7. Simplified 96.2%

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

   1. add-sqr-sqrt -0.0%

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

   2. add-sqr-sqrt 96.2%

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

   3. add-sqr-sqrt 44.4%

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

   4. fabs-sqr 44.4%

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

   5. add-sqr-sqrt 58.4%

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

   6. *-rgt-identity 58.4%

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

   7. frac-times 58.4%

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

   8. div-inv 58.4%

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

   9. div-inv 58.4%

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

   10. unpow2 58.4%

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

   11. times-frac 59.5%

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

9. Applied egg-rr 67.6%

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

10. Taylor expanded in x around 0 56.7%

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

11. Taylor expanded in s around 0 56.7%

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

    1. associate-/r* 56.7%

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

13. Simplified 56.7%

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

14. Final simplification 56.7%

    \[\leadsto \frac{\frac{0.5}{s}}{e^{\frac{x}{s}} + 1} \]
15. Add Preprocessing

Alternative 5: 95.2% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

   1. fabs-neg 99.7%

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

   2. distribute-frac-neg 99.7%

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

   3. distribute-frac-neg2 99.7%

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

   4. fabs-neg 99.7%

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

   5. *-commutative 99.7%

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

   6. fabs-neg 99.7%

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

   7. +-commutative 99.7%

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

   8. fabs-neg 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in x around 0 99.7%

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

   1. associate-/r* 99.7%

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

   2. exp-prod 99.6%

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

   3. rem-square-sqrt 44.5%

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

   4. fabs-sqr 44.5%

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

   5. rem-square-sqrt 57.2%

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

   6. exp-prod 57.2%

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

   7. neg-mul-1 57.2%

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

   8. distribute-neg-frac2 57.2%

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

   9. +-commutative 57.2%

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

   10. exp-prod 57.2%

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

   11. rem-square-sqrt 44.5%

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

   12. fabs-sqr 44.5%

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

   13. rem-square-sqrt 58.4%

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

   14. exp-prod 58.4%

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

   15. neg-mul-1 58.4%

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

   16. distribute-neg-frac2 58.4%

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

7. Simplified 58.4%

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

8. Taylor expanded in x around 0 57.4%

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

   1. *-commutative 57.4%

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

10. Simplified 57.4%

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

11. Taylor expanded in x around 0 56.0%

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

Alternative 6: 77.9% accurate, 31.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.0000000138484279 \cdot 10^{+24}:\\ \;\;\;\;\frac{\left(0.25 + \frac{x\_m}{s} \cdot 0.25\right) - \frac{x\_m \cdot 0.25}{s}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \frac{x\_m}{s} \cdot -4}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (if (<= x_m 1.0000000138484279e+24)
   (/ (- (+ 0.25 (* (/ x_m s) 0.25)) (/ (* x_m 0.25) s)) s)
   (/ (/ 1.0 s) (+ 4.0 (* (/ x_m s) -4.0)))))

C

x_m = fabs(x);
float code(float x_m, float s) {
	float tmp;
	if (x_m <= 1.0000000138484279e+24f) {
		tmp = ((0.25f + ((x_m / s) * 0.25f)) - ((x_m * 0.25f) / s)) / s;
	} else {
		tmp = (1.0f / s) / (4.0f + ((x_m / s) * -4.0f));
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(4) function code(x_m, s)
    real(4), intent (in) :: x_m
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x_m <= 1.0000000138484279e+24) then
        tmp = ((0.25e0 + ((x_m / s) * 0.25e0)) - ((x_m * 0.25e0) / s)) / s
    else
        tmp = (1.0e0 / s) / (4.0e0 + ((x_m / s) * (-4.0e0)))
    end if
    code = tmp
end function

Julia

x_m = abs(x)
function code(x_m, s)
	tmp = Float32(0.0)
	if (x_m <= Float32(1.0000000138484279e+24))
		tmp = Float32(Float32(Float32(Float32(0.25) + Float32(Float32(x_m / s) * Float32(0.25))) - Float32(Float32(x_m * Float32(0.25)) / s)) / s);
	else
		tmp = Float32(Float32(Float32(1.0) / s) / Float32(Float32(4.0) + Float32(Float32(x_m / s) * Float32(-4.0))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, s)
	tmp = single(0.0);
	if (x_m <= single(1.0000000138484279e+24))
		tmp = ((single(0.25) + ((x_m / s) * single(0.25))) - ((x_m * single(0.25)) / s)) / s;
	else
		tmp = (single(1.0) / s) / (single(4.0) + ((x_m / s) * single(-4.0)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 1.00000001e24

   1. Initial program 99.6%

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

      1. fabs-neg 99.6%

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

      2. distribute-frac-neg 99.6%

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

      3. distribute-frac-neg2 99.6%

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

      4. fabs-neg 99.6%

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

      5. *-commutative 99.6%

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

      6. fabs-neg 99.6%

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

      7. +-commutative 99.6%

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

      8. fabs-neg 99.6%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 99.7%

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

      1. +-commutative 99.7%

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

      2. exp-prod 99.7%

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

      3. rem-square-sqrt 38.2%

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

      4. fabs-sqr 38.2%

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

      5. rem-square-sqrt 95.8%

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

      6. exp-prod 95.8%

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

      7. neg-mul-1 95.8%

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

      8. distribute-neg-frac2 95.8%

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

   7. Simplified 95.8%

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

   8. Taylor expanded in s around -inf 47.9%

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

      1. mul-1-neg 47.9%

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

      2. *-commutative 47.9%

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

   10. Simplified 47.9%

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

       1. *-commutative 47.9%

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

       2. add-sqr-sqrt 26.3%

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

       3. fabs-sqr 26.3%

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

       4. add-sqr-sqrt 72.7%

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

       5. associate-*l/ 72.7%

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

   12. Applied egg-rr 72.7%

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

   if 1.00000001e24 < 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. associate-/r* 100.0%

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

      2. exp-prod 100.0%

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

      3. rem-square-sqrt 100.0%

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

      4. fabs-sqr 100.0%

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

      5. rem-square-sqrt 100.0%

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

      6. exp-prod 100.0%

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

      7. neg-mul-1 100.0%

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

      8. distribute-neg-frac2 100.0%

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

      9. +-commutative 100.0%

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

      10. exp-prod 100.0%

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

      11. rem-square-sqrt 100.0%

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

      12. fabs-sqr 100.0%

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

      13. rem-square-sqrt 100.0%

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

      14. exp-prod 100.0%

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

      15. neg-mul-1 100.0%

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

      16. distribute-neg-frac2 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around 0 100.0%

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

      1. *-commutative 100.0%

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

   10. Simplified 100.0%

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

   11. Taylor expanded in x around 0 80.1%

       \[\leadsto \frac{\color{blue}{\frac{1}{s}}}{4 + \frac{x}{s} \cdot -4} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.0000000138484279 \cdot 10^{+24}:\\ \;\;\;\;\frac{\left(0.25 + \frac{x}{s} \cdot 0.25\right) - \frac{x \cdot 0.25}{s}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \frac{x}{s} \cdot -4}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 50.6% accurate, 56.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.7%

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

   1. fabs-neg 99.7%

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

   2. distribute-frac-neg 99.7%

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

   3. distribute-frac-neg2 99.7%

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

   4. fabs-neg 99.7%

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

   5. *-commutative 99.7%

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

   6. fabs-neg 99.7%

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

   7. +-commutative 99.7%

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

   8. fabs-neg 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in x around 0 99.7%

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

   1. +-commutative 99.7%

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

   2. exp-prod 99.7%

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

   3. rem-square-sqrt 44.5%

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

   4. fabs-sqr 44.5%

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

   5. rem-square-sqrt 96.2%

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

   6. exp-prod 96.2%

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

   7. neg-mul-1 96.2%

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

   8. distribute-neg-frac2 96.2%

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

7. Simplified 96.2%

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

   1. add-sqr-sqrt -0.0%

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

   2. add-sqr-sqrt 96.2%

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

   3. add-sqr-sqrt 44.4%

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

   4. fabs-sqr 44.4%

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

   5. add-sqr-sqrt 58.4%

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

   6. *-rgt-identity 58.4%

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

   7. frac-times 58.4%

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

   8. div-inv 58.4%

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

   9. div-inv 58.4%

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

   10. unpow2 58.4%

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

   11. times-frac 59.5%

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

9. Applied egg-rr 67.6%

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

10. Taylor expanded in x around 0 56.7%

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

11. Taylor expanded in x around 0 51.5%

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

Alternative 8: 26.8% 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 99.7%

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

   1. fabs-neg 99.7%

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

   2. distribute-frac-neg 99.7%

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

   3. distribute-frac-neg2 99.7%

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

   4. fabs-neg 99.7%

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

   5. *-commutative 99.7%

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

   6. fabs-neg 99.7%

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

   7. +-commutative 99.7%

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

   8. fabs-neg 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in s around inf 28.1%

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

Reproduce

herbie shell --seed 2024172 
(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))))))