Logistic distribution

Percentage Accurate: 99.4% → 99.4%

Time: 13.0s

Alternatives: 7

Speedup: 1.2×

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: 99.4% 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: 99.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\left|x\_m\right| \leq 2.200000047683716:\\ \;\;\;\;\frac{e^{\frac{x\_m}{s} - 2 \cdot \mathsf{log1p}\left(e^{\frac{x\_m}{s}}\right)}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{x\_m}{-s}}}{s \cdot 4}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (if (<= (fabs x_m) 2.200000047683716)
   (/ (exp (- (/ x_m s) (* 2.0 (log1p (exp (/ x_m s)))))) s)
   (/ (exp (/ x_m (- s))) (* s 4.0))))

C

x_m = fabs(x);
float code(float x_m, float s) {
	float tmp;
	if (fabsf(x_m) <= 2.200000047683716f) {
		tmp = expf(((x_m / s) - (2.0f * log1pf(expf((x_m / s)))))) / s;
	} else {
		tmp = expf((x_m / -s)) / (s * 4.0f);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m, s)
	tmp = Float32(0.0)
	if (abs(x_m) <= Float32(2.200000047683716))
		tmp = Float32(exp(Float32(Float32(x_m / s) - Float32(Float32(2.0) * log1p(exp(Float32(x_m / s)))))) / s);
	else
		tmp = Float32(exp(Float32(x_m / Float32(-s))) / Float32(s * Float32(4.0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (fabs.f32 x) < 2.20000005

   1. Initial program 99.6%

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

      1. fabs-neg 99.6%

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

      2. distribute-frac-neg 99.6%

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

      3. distribute-frac-neg2 99.6%

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

      4. fabs-neg 99.6%

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

      5. *-commutative 99.6%

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

      6. fabs-neg 99.6%

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

      7. +-commutative 99.6%

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

      8. fabs-neg 99.6%

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

   3. Simplified 99.6%

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

   6. Applied egg-rr 76.9%

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

      1. *-lft-identity 76.9%

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

      2. *-commutative 76.9%

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

      3. exp-to-pow 76.9%

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

      4. log1p-undefine 76.9%

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

      5. *-commutative 76.9%

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

      6. rem-exp-log 72.4%

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

      7. prod-exp 72.2%

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

      8. exp-diff 94.7%

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

      9. associate--r+ 94.8%

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

      10. exp-diff 95.1%

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

      11. rem-exp-log 99.6%

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

   8. Simplified 99.6%

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

   if 2.20000005 < (fabs.f32 x)

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. exp-prod 100.0%

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

      3. rem-square-sqrt 45.3%

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

      4. fabs-sqr 45.3%

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

      5. rem-square-sqrt 100.0%

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

      6. exp-prod 100.0%

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

      7. neg-mul-1 100.0%

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

      8. distribute-neg-frac2 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in s around inf 100.0%

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

      1. *-commutative 100.0%

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

   10. Simplified 100.0%

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

       1. add-sqr-sqrt 45.3%

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

       2. fabs-sqr 45.3%

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

       3. sqrt-unprod 100.0%

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

       4. sqr-neg 100.0%

          \[\leadsto \frac{e^{\frac{-\sqrt{\color{blue}{\left(-x\right) \cdot \left(-x\right)}}}{s}}}{s \cdot 4} \]

       5. add-sqr-sqrt 45.3%

          \[\leadsto \frac{e^{\frac{-\sqrt{\left(-\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right) \cdot \left(-x\right)}}{s}}}{s \cdot 4} \]

       6. fabs-sqr 45.3%

          \[\leadsto \frac{e^{\frac{-\sqrt{\left(-\color{blue}{\left|\sqrt{x} \cdot \sqrt{x}\right|}\right) \cdot \left(-x\right)}}{s}}}{s \cdot 4} \]

       7. add-sqr-sqrt 45.3%

          \[\leadsto \frac{e^{\frac{-\sqrt{\left(-\left|\color{blue}{x}\right|\right) \cdot \left(-x\right)}}{s}}}{s \cdot 4} \]

       8. add-sqr-sqrt 45.3%

          \[\leadsto \frac{e^{\frac{-\sqrt{\left(-\left|x\right|\right) \cdot \left(-\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right)}}{s}}}{s \cdot 4} \]

       9. fabs-sqr 45.3%

          \[\leadsto \frac{e^{\frac{-\sqrt{\left(-\left|x\right|\right) \cdot \left(-\color{blue}{\left|\sqrt{x} \cdot \sqrt{x}\right|}\right)}}{s}}}{s \cdot 4} \]

       10. add-sqr-sqrt 100.0%

           \[\leadsto \frac{e^{\frac{-\sqrt{\left(-\left|x\right|\right) \cdot \left(-\left|\color{blue}{x}\right|\right)}}{s}}}{s \cdot 4} \]

       11. sqrt-unprod -0.0%

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

       12. add-sqr-sqrt 3.1%

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

       13. neg-sub0 3.1%

           \[\leadsto \frac{e^{\frac{-\color{blue}{\left(0 - \left|x\right|\right)}}{s}}}{s \cdot 4} \]

       14. add-sqr-sqrt 1.4%

           \[\leadsto \frac{e^{\frac{-\left(0 - \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}{s}}}{s \cdot 4} \]

       15. fabs-sqr 1.4%

           \[\leadsto \frac{e^{\frac{-\left(0 - \color{blue}{\sqrt{x} \cdot \sqrt{x}}\right)}{s}}}{s \cdot 4} \]

       16. add-sqr-sqrt 56.2%

           \[\leadsto \frac{e^{\frac{-\left(0 - \color{blue}{x}\right)}{s}}}{s \cdot 4} \]

       17. sub-neg 56.2%

           \[\leadsto \frac{e^{\frac{-\color{blue}{\left(0 + \left(-x\right)\right)}}{s}}}{s \cdot 4} \]

       18. add-sqr-sqrt 1.4%

           \[\leadsto \frac{e^{\frac{-\left(0 + \left(-\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right)\right)}{s}}}{s \cdot 4} \]

       19. fabs-sqr 1.4%

           \[\leadsto \frac{e^{\frac{-\left(0 + \left(-\color{blue}{\left|\sqrt{x} \cdot \sqrt{x}\right|}\right)\right)}{s}}}{s \cdot 4} \]

       20. add-sqr-sqrt 3.1%

           \[\leadsto \frac{e^{\frac{-\left(0 + \left(-\left|\color{blue}{x}\right|\right)\right)}{s}}}{s \cdot 4} \]

       21. add-sqr-sqrt -0.0%

           \[\leadsto \frac{e^{\frac{-\left(0 + \color{blue}{\sqrt{-\left|x\right|} \cdot \sqrt{-\left|x\right|}}\right)}{s}}}{s \cdot 4} \]

       22. sqrt-unprod 100.0%

           \[\leadsto \frac{e^{\frac{-\left(0 + \color{blue}{\sqrt{\left(-\left|x\right|\right) \cdot \left(-\left|x\right|\right)}}\right)}{s}}}{s \cdot 4} \]

   12. Applied egg-rr 47.0%

       \[\leadsto \frac{e^{\frac{-\color{blue}{\left(0 + x\right)}}{s}}}{s \cdot 4} \]
   13. Step-by-step derivation

       1. +-lft-identity 47.0%

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

   14. Simplified 47.0%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2.200000047683716:\\ \;\;\;\;\frac{e^{\frac{x}{s} - 2 \cdot \mathsf{log1p}\left(e^{\frac{x}{s}}\right)}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{x}{-s}}}{s \cdot 4}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.5% accurate, 1.0× speedup

Math

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

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (let* ((t_0 (exp (/ (fabs x_m) (- s)))) (t_1 (+ 1.0 t_0)))
   (/ t_0 (* s (* t_1 t_1)))))

C

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

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

   1. fabs-neg 99.8%

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

   2. distribute-frac-neg 99.8%

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

   3. distribute-frac-neg2 99.8%

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

   4. fabs-neg 99.8%

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

   5. *-commutative 99.8%

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

   6. fabs-neg 99.8%

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

   7. +-commutative 99.8%

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

   8. fabs-neg 99.8%

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

3. Simplified 99.8%

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

Alternative 3: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

x_m = fabs(x);
float code(float x_m, float s) {
	float t_0 = expf((fabsf(x_m) / -s));
	return t_0 / ((1.0f + t_0) * (s + (s / expf((fabsf(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((abs(x_m) / -s))
    code = t_0 / ((1.0e0 + t_0) * (s + (s / exp((abs(x_m) / s)))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

   1. *-commutative 99.8%

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

   2. fabs-neg 99.8%

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

   3. +-commutative 99.8%

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

   4. fabs-neg 99.8%

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

   5. distribute-lft-in 99.8%

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

   6. *-rgt-identity 99.8%

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

   7. +-commutative 99.8%

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

3. Simplified 99.8%

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

5. Final simplification 99.8%

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

Alternative 4: 99.4% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

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((abs(x_m) / -s))
    code = t_0 / ((1.0e0 + exp((x_m / -s))) * (s * (1.0e0 + t_0)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

   1. *-commutative 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in x around 0 99.8%

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

   1. +-commutative 99.8%

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

   2. exp-prod 99.8%

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

   3. rem-square-sqrt 49.5%

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

   4. fabs-sqr 49.5%

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

   5. rem-square-sqrt 97.4%

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

   6. exp-prod 97.4%

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

   7. neg-mul-1 97.4%

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

   8. distribute-neg-frac2 97.4%

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

7. Simplified 97.4%

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

8. Final simplification 97.4%

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

Alternative 5: 94.4% accurate, 5.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{1}{s \cdot \left(e^{\frac{x\_m}{s}} \cdot 4\right)} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s) :precision binary32 (/ 1.0 (* s (* (exp (/ x_m s)) 4.0))))

C

x_m = fabs(x);
float code(float x_m, float s) {
	return 1.0f / (s * (expf((x_m / s)) * 4.0f));
}

Fortran

x_m = abs(x)
real(4) function code(x_m, s)
    real(4), intent (in) :: x_m
    real(4), intent (in) :: s
    code = 1.0e0 / (s * (exp((x_m / s)) * 4.0e0))
end function

Julia

x_m = abs(x)
function code(x_m, s)
	return Float32(Float32(1.0) / Float32(s * Float32(exp(Float32(x_m / s)) * Float32(4.0))))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m, s)
	tmp = single(1.0) / (s * (exp((x_m / s)) * single(4.0)));
end

Derivation

1. Initial program 99.8%

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

   1. *-commutative 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in x around 0 99.8%

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

   1. +-commutative 99.8%

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

   2. exp-prod 99.8%

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

   3. rem-square-sqrt 49.5%

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

   4. fabs-sqr 49.5%

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

   5. rem-square-sqrt 97.4%

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

   6. exp-prod 97.4%

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

   7. neg-mul-1 97.4%

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

   8. distribute-neg-frac2 97.4%

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

7. Simplified 97.4%

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

8. Taylor expanded in s around inf 96.0%

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

   1. *-commutative 96.0%

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

10. Simplified 96.0%

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

    1. frac-2neg 96.0%

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

    2. div-inv 96.0%

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

    3. add-log-exp 87.1%

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

    4. add-log-exp 96.0%

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

    5. add-sqr-sqrt 48.0%

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

    6. fabs-sqr 48.0%

       \[\leadsto \frac{e^{\left(-\left(-\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right)\right) \cdot \frac{1}{-s}}}{s \cdot 4} \]

    7. add-sqr-sqrt 58.4%

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

    8. remove-double-neg 58.4%

       \[\leadsto \frac{e^{\color{blue}{x} \cdot \frac{1}{-s}}}{s \cdot 4} \]

12. Applied egg-rr 58.4%

    \[\leadsto \frac{e^{\color{blue}{x \cdot \frac{1}{-s}}}}{s \cdot 4} \]
13. Step-by-step derivation

    1. clear-num 58.5%

       \[\leadsto \color{blue}{\frac{1}{\frac{s \cdot 4}{e^{x \cdot \frac{1}{-s}}}}} \]

    2. inv-pow 58.5%

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

14. Applied egg-rr 58.4%

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

    1. unpow-1 58.4%

       \[\leadsto \color{blue}{\frac{1}{\left(s \cdot 4\right) \cdot e^{\frac{x}{s}}}} \]

    2. associate-*l* 58.4%

       \[\leadsto \frac{1}{\color{blue}{s \cdot \left(4 \cdot e^{\frac{x}{s}}\right)}} \]

16. Simplified 58.4%

    \[\leadsto \color{blue}{\frac{1}{s \cdot \left(4 \cdot e^{\frac{x}{s}}\right)}} \]

17. Final simplification 58.4%

    \[\leadsto \frac{1}{s \cdot \left(e^{\frac{x}{s}} \cdot 4\right)} \]
18. Add Preprocessing

Alternative 6: 94.4% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

   1. *-commutative 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in x around 0 99.8%

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

   1. +-commutative 99.8%

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

   2. exp-prod 99.8%

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

   3. rem-square-sqrt 49.5%

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

   4. fabs-sqr 49.5%

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

   5. rem-square-sqrt 97.4%

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

   6. exp-prod 97.4%

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

   7. neg-mul-1 97.4%

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

   8. distribute-neg-frac2 97.4%

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

7. Simplified 97.4%

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

8. Taylor expanded in s around inf 96.0%

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

   1. *-commutative 96.0%

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

10. Simplified 96.0%

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

    1. add-sqr-sqrt 48.0%

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

    2. fabs-sqr 48.0%

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

    3. sqrt-unprod 95.3%

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

    4. sqr-neg 95.3%

       \[\leadsto \frac{e^{\frac{-\sqrt{\color{blue}{\left(-x\right) \cdot \left(-x\right)}}}{s}}}{s \cdot 4} \]

    5. add-sqr-sqrt 47.5%

       \[\leadsto \frac{e^{\frac{-\sqrt{\left(-\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right) \cdot \left(-x\right)}}{s}}}{s \cdot 4} \]

    6. fabs-sqr 47.5%

       \[\leadsto \frac{e^{\frac{-\sqrt{\left(-\color{blue}{\left|\sqrt{x} \cdot \sqrt{x}\right|}\right) \cdot \left(-x\right)}}{s}}}{s \cdot 4} \]

    7. add-sqr-sqrt 54.3%

       \[\leadsto \frac{e^{\frac{-\sqrt{\left(-\left|\color{blue}{x}\right|\right) \cdot \left(-x\right)}}{s}}}{s \cdot 4} \]

    8. add-sqr-sqrt 47.5%

       \[\leadsto \frac{e^{\frac{-\sqrt{\left(-\left|x\right|\right) \cdot \left(-\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right)}}{s}}}{s \cdot 4} \]

    9. fabs-sqr 47.5%

       \[\leadsto \frac{e^{\frac{-\sqrt{\left(-\left|x\right|\right) \cdot \left(-\color{blue}{\left|\sqrt{x} \cdot \sqrt{x}\right|}\right)}}{s}}}{s \cdot 4} \]

    10. add-sqr-sqrt 95.3%

        \[\leadsto \frac{e^{\frac{-\sqrt{\left(-\left|x\right|\right) \cdot \left(-\left|\color{blue}{x}\right|\right)}}{s}}}{s \cdot 4} \]

    11. sqrt-unprod -0.0%

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

    12. add-sqr-sqrt 24.6%

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

    13. neg-sub0 24.6%

        \[\leadsto \frac{e^{\frac{-\color{blue}{\left(0 - \left|x\right|\right)}}{s}}}{s \cdot 4} \]

    14. add-sqr-sqrt 14.2%

        \[\leadsto \frac{e^{\frac{-\left(0 - \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}{s}}}{s \cdot 4} \]

    15. fabs-sqr 14.2%

        \[\leadsto \frac{e^{\frac{-\left(0 - \color{blue}{\sqrt{x} \cdot \sqrt{x}}\right)}{s}}}{s \cdot 4} \]

    16. add-sqr-sqrt 62.1%

        \[\leadsto \frac{e^{\frac{-\left(0 - \color{blue}{x}\right)}{s}}}{s \cdot 4} \]

    17. sub-neg 62.1%

        \[\leadsto \frac{e^{\frac{-\color{blue}{\left(0 + \left(-x\right)\right)}}{s}}}{s \cdot 4} \]

    18. add-sqr-sqrt 14.2%

        \[\leadsto \frac{e^{\frac{-\left(0 + \left(-\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right)\right)}{s}}}{s \cdot 4} \]

    19. fabs-sqr 14.2%

        \[\leadsto \frac{e^{\frac{-\left(0 + \left(-\color{blue}{\left|\sqrt{x} \cdot \sqrt{x}\right|}\right)\right)}{s}}}{s \cdot 4} \]

    20. add-sqr-sqrt 24.6%

        \[\leadsto \frac{e^{\frac{-\left(0 + \left(-\left|\color{blue}{x}\right|\right)\right)}{s}}}{s \cdot 4} \]

    21. add-sqr-sqrt -0.0%

        \[\leadsto \frac{e^{\frac{-\left(0 + \color{blue}{\sqrt{-\left|x\right|} \cdot \sqrt{-\left|x\right|}}\right)}{s}}}{s \cdot 4} \]

    22. sqrt-unprod 95.3%

        \[\leadsto \frac{e^{\frac{-\left(0 + \color{blue}{\sqrt{\left(-\left|x\right|\right) \cdot \left(-\left|x\right|\right)}}\right)}{s}}}{s \cdot 4} \]

12. Applied egg-rr 58.4%

    \[\leadsto \frac{e^{\frac{-\color{blue}{\left(0 + x\right)}}{s}}}{s \cdot 4} \]
13. Step-by-step derivation

    1. +-lft-identity 58.4%

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

14. Simplified 58.4%

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

15. Final simplification 58.4%

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

Alternative 7: 26.3% accurate, 206.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

x_m = abs(x);
function tmp = code(x_m, s)
	tmp = single(0.25) / s;
end

Derivation

1. Initial program 99.8%

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

   1. fabs-neg 99.8%

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

   2. distribute-frac-neg 99.8%

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

   3. distribute-frac-neg2 99.8%

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

   4. fabs-neg 99.8%

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

   5. *-commutative 99.8%

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

   6. fabs-neg 99.8%

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

   7. +-commutative 99.8%

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

   8. fabs-neg 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in s around inf 26.7%

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

Reproduce

herbie shell --seed 2024185 
(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))))))