Logistic distribution

Percentage Accurate: 99.5% → 99.5%

Time: 12.5s

Alternatives: 14

Speedup: 1.9×

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, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

x_m = abs(x)
function code(x_m, s)
	t_0 = Float32(x_m / Float32(-s))
	return Float32((Float32(exp(1)) ^ t_0) / Float32(Float32(Float32(1.0) + exp(t_0)) * Float32(s + Float32(s / exp(Float32(x_m / s))))))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m, s)
	t_0 = x_m / -s;
	tmp = (single(2.71828182845904523536) ^ t_0) / ((single(1.0) + exp(t_0)) * (s + (s / exp((x_m / s)))));
end

Derivation

1. Initial program 99.3%

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

   1. *-commutative 99.3%

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

   2. fabs-neg 99.3%

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

   3. +-commutative 99.3%

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

   4. fabs-neg 99.3%

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

   5. distribute-lft-in 99.3%

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

   6. *-rgt-identity 99.3%

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

   7. +-commutative 99.3%

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

3. Simplified 99.3%

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

   1. add-sqr-sqrt 99.1%

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

   2. associate-/l* 99.1%

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

   3. remove-double-neg 99.1%

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

   4. add-sqr-sqrt 99.1%

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

   5. sqrt-unprod 95.6%

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

   6. sqr-neg 95.6%

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

   7. sqrt-unprod -0.0%

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

   8. add-sqr-sqrt 94.3%

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

   9. frac-2neg 94.3%

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

   10. add-sqr-sqrt -0.0%

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

   11. sqrt-unprod 97.4%

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

   12. sqr-neg 97.4%

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

   13. sqrt-unprod 99.1%

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

   14. add-sqr-sqrt 99.1%

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

   15. add-sqr-sqrt 49.0%

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

   16. fabs-sqr 49.0%

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

   17. add-sqr-sqrt 97.6%

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

6. Applied egg-rr 97.6%

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

   1. associate-*r/ 97.6%

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

   2. rem-square-sqrt 97.8%

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

8. Simplified 97.8%

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

   1. distribute-frac-neg 97.8%

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

   2. add-sqr-sqrt 49.1%

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

   3. fabs-sqr 49.1%

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

   4. add-sqr-sqrt 97.7%

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

   5. rec-exp 97.7%

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

10. Applied egg-rr 97.7%

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

    1. rec-exp 97.7%

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

    2. distribute-frac-neg 97.7%

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

12. Simplified 97.7%

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

    1. distribute-frac-neg 97.8%

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

    2. add-sqr-sqrt 49.1%

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

    3. fabs-sqr 49.1%

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

    4. add-sqr-sqrt 97.7%

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

    5. rec-exp 97.7%

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

14. Applied egg-rr 60.4%

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

    1. rec-exp 97.7%

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

    2. distribute-frac-neg 97.7%

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

16. Simplified 60.7%

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

    1. *-un-lft-identity 60.7%

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

    2. exp-prod 60.7%

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

18. Applied egg-rr 60.7%

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

    1. exp-1-e 60.7%

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

20. Simplified 60.7%

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

21. Final simplification 60.7%

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

Alternative 2: 99.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

   1. *-commutative 99.3%

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

   2. fabs-neg 99.3%

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

   3. +-commutative 99.3%

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

   4. fabs-neg 99.3%

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

   5. distribute-lft-in 99.3%

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

   6. *-rgt-identity 99.3%

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

   7. +-commutative 99.3%

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

3. Simplified 99.3%

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

   1. add-sqr-sqrt 99.1%

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

   2. associate-/l* 99.1%

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

   3. remove-double-neg 99.1%

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

   4. add-sqr-sqrt 99.1%

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

   5. sqrt-unprod 95.6%

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

   6. sqr-neg 95.6%

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

   7. sqrt-unprod -0.0%

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

   8. add-sqr-sqrt 94.3%

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

   9. frac-2neg 94.3%

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

   10. add-sqr-sqrt -0.0%

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

   11. sqrt-unprod 97.4%

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

   12. sqr-neg 97.4%

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

   13. sqrt-unprod 99.1%

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

   14. add-sqr-sqrt 99.1%

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

   15. add-sqr-sqrt 49.0%

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

   16. fabs-sqr 49.0%

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

   17. add-sqr-sqrt 97.6%

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

6. Applied egg-rr 97.6%

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

   1. associate-*r/ 97.6%

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

   2. rem-square-sqrt 97.8%

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

8. Simplified 97.8%

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

   1. distribute-frac-neg 97.8%

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

   2. add-sqr-sqrt 49.1%

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

   3. fabs-sqr 49.1%

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

   4. add-sqr-sqrt 97.7%

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

   5. rec-exp 97.7%

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

10. Applied egg-rr 97.7%

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

    1. rec-exp 97.7%

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

    2. distribute-frac-neg 97.7%

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

12. Simplified 97.7%

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

    1. distribute-frac-neg 97.8%

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

    2. add-sqr-sqrt 49.1%

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

    3. fabs-sqr 49.1%

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

    4. add-sqr-sqrt 97.7%

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

    5. rec-exp 97.7%

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

14. Applied egg-rr 60.4%

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

    1. rec-exp 97.7%

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

    2. distribute-frac-neg 97.7%

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

16. Simplified 60.7%

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

17. Final simplification 60.7%

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

Alternative 3: 96.7% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

   1. *-commutative 99.3%

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

   2. fabs-neg 99.3%

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

   3. +-commutative 99.3%

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

   4. fabs-neg 99.3%

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

   5. distribute-lft-in 99.3%

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

   6. *-rgt-identity 99.3%

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

   7. +-commutative 99.3%

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

3. Simplified 99.3%

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

   1. add-sqr-sqrt 99.1%

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

   2. associate-/l* 99.1%

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

   3. remove-double-neg 99.1%

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

   4. add-sqr-sqrt 99.1%

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

   5. sqrt-unprod 95.6%

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

   6. sqr-neg 95.6%

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

   7. sqrt-unprod -0.0%

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

   8. add-sqr-sqrt 94.3%

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

   9. frac-2neg 94.3%

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

   10. add-sqr-sqrt -0.0%

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

   11. sqrt-unprod 97.4%

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

   12. sqr-neg 97.4%

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

   13. sqrt-unprod 99.1%

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

   14. add-sqr-sqrt 99.1%

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

   15. add-sqr-sqrt 49.0%

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

   16. fabs-sqr 49.0%

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

   17. add-sqr-sqrt 97.6%

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

6. Applied egg-rr 97.6%

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

   1. associate-*r/ 97.6%

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

   2. rem-square-sqrt 97.8%

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

8. Simplified 97.8%

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

   1. distribute-frac-neg 97.8%

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

   2. add-sqr-sqrt 49.1%

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

   3. fabs-sqr 49.1%

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

   4. add-sqr-sqrt 97.7%

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

   5. rec-exp 97.7%

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

10. Applied egg-rr 97.7%

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

    1. rec-exp 97.7%

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

    2. distribute-frac-neg 97.7%

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

12. Simplified 97.7%

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

    1. distribute-frac-neg 97.8%

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

    2. add-sqr-sqrt 49.1%

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

    3. fabs-sqr 49.1%

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

    4. add-sqr-sqrt 97.7%

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

    5. rec-exp 97.7%

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

14. Applied egg-rr 60.4%

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

    1. rec-exp 97.7%

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

    2. distribute-frac-neg 97.7%

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

16. Simplified 60.7%

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

17. Taylor expanded in x around 0 58.1%

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

18. Final simplification 58.1%

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

Alternative 4: 94.8% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

x_m = abs(x)
function code(x_m, s)
	return Float32((Float32(exp(1)) ^ Float32(x_m / Float32(-s))) / Float32(Float32(s + Float32(s / exp(Float32(x_m / s)))) * Float32(2.0)))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m, s)
	tmp = (single(2.71828182845904523536) ^ (x_m / -s)) / ((s + (s / exp((x_m / s)))) * single(2.0));
end

Derivation

1. Initial program 99.3%

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

   1. *-commutative 99.3%

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

   2. fabs-neg 99.3%

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

   3. +-commutative 99.3%

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

   4. fabs-neg 99.3%

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

   5. distribute-lft-in 99.3%

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

   6. *-rgt-identity 99.3%

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

   7. +-commutative 99.3%

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

3. Simplified 99.3%

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

   1. add-sqr-sqrt 99.1%

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

   2. associate-/l* 99.1%

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

   3. remove-double-neg 99.1%

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

   4. add-sqr-sqrt 99.1%

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

   5. sqrt-unprod 95.6%

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

   6. sqr-neg 95.6%

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

   7. sqrt-unprod -0.0%

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

   8. add-sqr-sqrt 94.3%

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

   9. frac-2neg 94.3%

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

   10. add-sqr-sqrt -0.0%

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

   11. sqrt-unprod 97.4%

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

   12. sqr-neg 97.4%

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

   13. sqrt-unprod 99.1%

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

   14. add-sqr-sqrt 99.1%

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

   15. add-sqr-sqrt 49.0%

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

   16. fabs-sqr 49.0%

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

   17. add-sqr-sqrt 97.6%

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

6. Applied egg-rr 97.6%

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

   1. associate-*r/ 97.6%

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

   2. rem-square-sqrt 97.8%

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

8. Simplified 97.8%

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

9. Taylor expanded in s around inf 94.8%

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

    1. distribute-frac-neg 97.8%

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

    2. add-sqr-sqrt 49.1%

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

    3. fabs-sqr 49.1%

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

    4. add-sqr-sqrt 97.7%

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

    5. rec-exp 97.7%

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

11. Applied egg-rr 56.2%

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

    1. rec-exp 97.7%

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

    2. distribute-frac-neg 97.7%

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

13. Simplified 56.3%

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

    1. *-un-lft-identity 60.7%

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

    2. exp-prod 60.7%

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

15. Applied egg-rr 56.3%

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

    1. exp-1-e 60.7%

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

17. Simplified 56.3%

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

18. Final simplification 56.3%

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

Alternative 5: 94.8% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

   1. *-commutative 99.3%

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

   2. fabs-neg 99.3%

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

   3. +-commutative 99.3%

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

   4. fabs-neg 99.3%

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

   5. distribute-lft-in 99.3%

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

   6. *-rgt-identity 99.3%

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

   7. +-commutative 99.3%

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

3. Simplified 99.3%

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

   1. add-sqr-sqrt 99.1%

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

   2. associate-/l* 99.1%

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

   3. remove-double-neg 99.1%

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

   4. add-sqr-sqrt 99.1%

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

   5. sqrt-unprod 95.6%

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

   6. sqr-neg 95.6%

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

   7. sqrt-unprod -0.0%

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

   8. add-sqr-sqrt 94.3%

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

   9. frac-2neg 94.3%

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

   10. add-sqr-sqrt -0.0%

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

   11. sqrt-unprod 97.4%

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

   12. sqr-neg 97.4%

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

   13. sqrt-unprod 99.1%

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

   14. add-sqr-sqrt 99.1%

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

   15. add-sqr-sqrt 49.0%

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

   16. fabs-sqr 49.0%

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

   17. add-sqr-sqrt 97.6%

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

6. Applied egg-rr 97.6%

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

   1. associate-*r/ 97.6%

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

   2. rem-square-sqrt 97.8%

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

8. Simplified 97.8%

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

9. Taylor expanded in s around inf 94.8%

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

    1. distribute-frac-neg 97.8%

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

    2. add-sqr-sqrt 49.1%

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

    3. fabs-sqr 49.1%

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

    4. add-sqr-sqrt 97.7%

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

    5. rec-exp 97.7%

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

11. Applied egg-rr 56.2%

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

    1. rec-exp 97.7%

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

    2. distribute-frac-neg 97.7%

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

13. Simplified 56.3%

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

14. Taylor expanded in x around inf 56.3%

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

    1. associate-*r/ 56.3%

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

    2. mul-1-neg 56.3%

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

16. Simplified 56.3%

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

17. Final simplification 56.3%

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

Alternative 6: 94.9% accurate, 5.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

   1. fabs-neg 99.3%

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

   2. distribute-frac-neg 99.3%

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

   3. distribute-frac-neg2 99.3%

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

   4. fabs-neg 99.3%

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

   5. *-commutative 99.3%

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

   6. fabs-neg 99.3%

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

   7. +-commutative 99.3%

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

   8. fabs-neg 99.3%

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

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 66.1%

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

7. Taylor expanded in x around 0 58.0%

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

   1. associate-*r/ 58.0%

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

   2. +-commutative 58.0%

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

9. Applied egg-rr 58.0%

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

10. Final simplification 58.0%

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

Alternative 7: 94.8% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Fortran

x_m = abs(x)
real(4) function code(x_m, s)
    real(4), intent (in) :: x_m
    real(4), intent (in) :: s
    code = (0.5e0 / s) / (1.0e0 + exp((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) + exp(Float32(x_m / s))))
end

MATLAB

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

Derivation

1. Initial program 99.3%

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

   1. fabs-neg 99.3%

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

   2. distribute-frac-neg 99.3%

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

   3. distribute-frac-neg2 99.3%

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

   4. fabs-neg 99.3%

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

   5. *-commutative 99.3%

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

   6. fabs-neg 99.3%

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

   7. +-commutative 99.3%

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

   8. fabs-neg 99.3%

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

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 66.1%

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

7. Taylor expanded in x around 0 58.0%

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

8. Taylor expanded in x around inf 58.0%

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

   1. associate-/r* 58.0%

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

10. Simplified 58.0%

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

11. Final simplification 58.0%

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

Alternative 8: 52.1% accurate, 44.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.00019999999494757503:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{s} \cdot \frac{1}{\frac{x\_m}{s}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (if (<= x_m 0.00019999999494757503)
   (/ 0.25 s)
   (* (/ 0.5 s) (/ 1.0 (/ x_m s)))))

C

x_m = fabs(x);
float code(float x_m, float s) {
	float tmp;
	if (x_m <= 0.00019999999494757503f) {
		tmp = 0.25f / s;
	} else {
		tmp = (0.5f / s) * (1.0f / (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 <= 0.00019999999494757503e0) then
        tmp = 0.25e0 / s
    else
        tmp = (0.5e0 / s) * (1.0e0 / (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(0.00019999999494757503))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(Float32(0.5) / s) * Float32(Float32(1.0) / 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(0.00019999999494757503))
		tmp = single(0.25) / s;
	else
		tmp = (single(0.5) / s) * (single(1.0) / (x_m / s));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 1.99999995e-4

   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 99.0%

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

   5. Taylor expanded in s around inf 35.5%

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

   if 1.99999995e-4 < 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 1.6%

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

   7. Taylor expanded in x around 0 98.9%

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

   8. Taylor expanded in x around 0 43.7%

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

   9. Taylor expanded in x around inf 43.6%

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

4. Final simplification 37.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.00019999999494757503:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{s} \cdot \frac{1}{\frac{x}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 45.1% accurate, 51.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.00019999999494757503:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{s} \cdot \frac{s}{x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (if (<= x_m 0.00019999999494757503) (/ 0.25 s) (* (/ 0.5 s) (/ s x_m))))

C

x_m = fabs(x);
float code(float x_m, float s) {
	float tmp;
	if (x_m <= 0.00019999999494757503f) {
		tmp = 0.25f / s;
	} else {
		tmp = (0.5f / s) * (s / x_m);
	}
	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 <= 0.00019999999494757503e0) then
        tmp = 0.25e0 / s
    else
        tmp = (0.5e0 / s) * (s / x_m)
    end if
    code = tmp
end function

Julia

x_m = abs(x)
function code(x_m, s)
	tmp = Float32(0.0)
	if (x_m <= Float32(0.00019999999494757503))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(Float32(0.5) / s) * Float32(s / x_m));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, s)
	tmp = single(0.0);
	if (x_m <= single(0.00019999999494757503))
		tmp = single(0.25) / s;
	else
		tmp = (single(0.5) / s) * (s / x_m);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 1.99999995e-4

   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 99.0%

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

   5. Taylor expanded in s around inf 35.5%

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

   if 1.99999995e-4 < 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 1.6%

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

   7. Taylor expanded in x around 0 98.9%

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

   8. Taylor expanded in x around 0 43.7%

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

   9. Taylor expanded in x around inf 36.3%

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

4. Final simplification 35.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.00019999999494757503:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{s} \cdot \frac{s}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 50.8% accurate, 56.4× speedup

Math

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

FPCore

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

C

x_m = fabs(x);
float code(float x_m, float s) {
	return (0.5f / s) * (1.0f / ((x_m / s) + 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
    code = (0.5e0 / s) * (1.0e0 / ((x_m / s) + 2.0e0))
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(x_m / s) + Float32(2.0))))
end

MATLAB

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

Derivation

1. Initial program 99.3%

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

   1. fabs-neg 99.3%

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

   2. distribute-frac-neg 99.3%

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

   3. distribute-frac-neg2 99.3%

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

   4. fabs-neg 99.3%

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

   5. *-commutative 99.3%

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

   6. fabs-neg 99.3%

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

   7. +-commutative 99.3%

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

   8. fabs-neg 99.3%

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

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 66.1%

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

7. Taylor expanded in x around 0 58.0%

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

8. Taylor expanded in x around 0 47.3%

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

9. Final simplification 47.3%

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

Alternative 11: 50.8% accurate, 56.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

   1. fabs-neg 99.3%

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

   2. distribute-frac-neg 99.3%

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

   3. distribute-frac-neg2 99.3%

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

   4. fabs-neg 99.3%

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

   5. *-commutative 99.3%

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

   6. fabs-neg 99.3%

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

   7. +-commutative 99.3%

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

   8. fabs-neg 99.3%

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

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 66.1%

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

7. Taylor expanded in x around 0 58.0%

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

8. Taylor expanded in x around 0 47.3%

   \[\leadsto \frac{1}{\color{blue}{2 + \frac{x}{s}}} \cdot \frac{0.5}{s} \]
9. Step-by-step derivation

   1. associate-*l/ 47.3%

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

   2. *-un-lft-identity 47.3%

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

   3. clear-num 47.3%

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

   4. +-commutative 47.3%

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

10. Applied egg-rr 47.3%

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

11. Final simplification 47.3%

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

Alternative 12: 50.8% accurate, 68.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

   1. fabs-neg 99.3%

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

   2. distribute-frac-neg 99.3%

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

   3. distribute-frac-neg2 99.3%

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

   4. fabs-neg 99.3%

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

   5. *-commutative 99.3%

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

   6. fabs-neg 99.3%

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

   7. +-commutative 99.3%

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

   8. fabs-neg 99.3%

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

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 66.1%

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

7. Taylor expanded in x around 0 58.0%

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

8. Taylor expanded in x around 0 47.3%

   \[\leadsto \frac{1}{\color{blue}{2 + \frac{x}{s}}} \cdot \frac{0.5}{s} \]
9. Step-by-step derivation

   1. frac-times 47.3%

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

   2. metadata-eval 47.3%

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

   3. +-commutative 47.3%

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

10. Applied egg-rr 47.3%

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

11. Final simplification 47.3%

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

Alternative 13: 50.8% accurate, 68.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

   1. fabs-neg 99.3%

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

   2. distribute-frac-neg 99.3%

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

   3. distribute-frac-neg2 99.3%

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

   4. fabs-neg 99.3%

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

   5. *-commutative 99.3%

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

   6. fabs-neg 99.3%

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

   7. +-commutative 99.3%

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

   8. fabs-neg 99.3%

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

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 66.1%

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

7. Taylor expanded in x around 0 58.0%

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

8. Taylor expanded in x around 0 47.3%

   \[\leadsto \frac{1}{\color{blue}{2 + \frac{x}{s}}} \cdot \frac{0.5}{s} \]
9. Step-by-step derivation

   1. associate-*l/ 47.3%

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

   2. *-un-lft-identity 47.3%

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

   3. +-commutative 47.3%

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

10. Applied egg-rr 47.3%

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

11. Final simplification 47.3%

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

Alternative 14: 27.0% 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 99.3%

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

   1. fabs-neg 99.3%

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

   2. distribute-frac-neg 99.3%

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

   3. distribute-frac-neg2 99.3%

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

   4. fabs-neg 99.3%

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

   5. *-commutative 99.3%

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

   6. fabs-neg 99.3%

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

   7. +-commutative 99.3%

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

   8. fabs-neg 99.3%

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

3. Simplified 99.3%

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

5. Taylor expanded in s around inf 28.1%

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

6. Final simplification 28.1%

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

Reproduce

herbie shell --seed 2024053 
(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))))))