Logistic distribution

Percentage Accurate: 99.6% → 99.2%

Time: 13.2s

Alternatives: 10

Speedup: N/A×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := e^{\frac{x_m}{s}}\\ \mathbf{if}\;\left|x_m\right| \leq 0.00019999999494757503:\\ \;\;\;\;\frac{e^{\frac{x_m}{s} + -2 \cdot \mathsf{log1p}\left(t_0\right)}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{s}}{1 + t_0}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (let* ((t_0 (exp (/ x_m s))))
   (if (<= (fabs x_m) 0.00019999999494757503)
     (/ (exp (+ (/ x_m s) (* -2.0 (log1p t_0)))) s)
     (/ (/ 0.5 s) (+ 1.0 t_0)))))

C

x_m = fabs(x);
float code(float x_m, float s) {
	float t_0 = expf((x_m / s));
	float tmp;
	if (fabsf(x_m) <= 0.00019999999494757503f) {
		tmp = expf(((x_m / s) + (-2.0f * log1pf(t_0)))) / s;
	} else {
		tmp = (0.5f / s) / (1.0f + t_0);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m, s)
	t_0 = exp(Float32(x_m / s))
	tmp = Float32(0.0)
	if (abs(x_m) <= Float32(0.00019999999494757503))
		tmp = Float32(exp(Float32(Float32(x_m / s) + Float32(Float32(-2.0) * log1p(t_0)))) / s);
	else
		tmp = Float32(Float32(Float32(0.5) / s) / Float32(Float32(1.0) + t_0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (fabs.f32 x) < 1.99999995e-4

   1. Initial program 98.7%

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

      1. *-commutative 98.7%

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

      2. distribute-lft-in 98.8%

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

      3. *-rgt-identity 98.8%

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

      4. fabs-neg 98.8%

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

      5. distribute-frac-neg 98.8%

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

      6. exp-neg 98.8%

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

      7. associate-*r/ 98.8%

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

      8. *-rgt-identity 98.8%

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

      9. *-lft-identity 98.8%

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

      10. metadata-eval 98.8%

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

      11. times-frac 98.8%

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

      12. neg-mul-1 98.8%

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

      13. neg-mul-1 98.8%

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

      14. fabs-neg 98.8%

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

   3. Simplified 98.8%

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

   5. Applied egg-rr 73.2%

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

      1. expm1-log1p-u 69.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{e^{\frac{x}{s}}}{s} \cdot \left(\frac{1}{e^{\frac{x}{s}} + 1} \cdot \frac{1}{e^{\frac{x}{s}} + 1}\right)\right)\right)} \]

      2. expm1-udef 69.5%

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

      3. pow2 69.5%

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

      4. inv-pow 69.5%

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

      5. +-commutative 69.5%

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

      6. pow-pow 69.5%

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

      7. metadata-eval 69.5%

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

   7. Applied egg-rr 69.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{e^{\frac{x}{s}}}{s} \cdot {\left(1 + e^{\frac{x}{s}}\right)}^{-2}\right)} - 1} \]
   8. Step-by-step derivation

      1. expm1-def 69.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{e^{\frac{x}{s}}}{s} \cdot {\left(1 + e^{\frac{x}{s}}\right)}^{-2}\right)\right)} \]

      2. expm1-log1p 73.3%

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

      3. *-commutative 73.3%

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

      4. metadata-eval 73.3%

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

      5. pow-sqr 73.2%

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

      6. unpow-1 73.2%

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

      7. unpow-1 73.2%

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

      8. unpow2 73.2%

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

      9. rem-exp-log 73.2%

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

      10. log1p-def 73.2%

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

      11. exp-neg 73.4%

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

      12. exp-prod 73.5%

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

      13. *-commutative 73.5%

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

      14. rem-exp-log 69.4%

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

      15. exp-diff 69.9%

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

   9. Simplified 99.0%

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

   if 1.99999995e-4 < (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%

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

      2. distribute-lft-in 100.0%

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

      3. *-rgt-identity 100.0%

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

      4. fabs-neg 100.0%

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

      5. distribute-frac-neg 100.0%

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

      6. exp-neg 100.0%

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

      7. associate-*r/ 100.0%

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

      8. *-rgt-identity 100.0%

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

      9. *-lft-identity 100.0%

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

      10. metadata-eval 100.0%

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

      11. times-frac 100.0%

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

      12. neg-mul-1 100.0%

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

      13. neg-mul-1 100.0%

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

      14. fabs-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 79.1%

      \[\leadsto \color{blue}{\frac{e^{\frac{x}{s} - \mathsf{log1p}\left(e^{\frac{x}{s}}\right)}}{s} \cdot \frac{1}{e^{\frac{x}{s}} + 1}} \]
   6. Step-by-step derivation

      1. associate-*r/ 79.1%

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

      2. *-rgt-identity 79.1%

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

      3. +-commutative 79.1%

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

   7. Simplified 79.1%

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

   8. Taylor expanded in x around 0 45.9%

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

      1. exp-neg 45.9%

         \[\leadsto \frac{\frac{\color{blue}{\frac{1}{e^{\log 2}}}}{s}}{1 + e^{\frac{x}{s}}} \]

      2. rem-exp-log 45.9%

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

      3. metadata-eval 45.9%

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

   10. Simplified 45.9%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.00019999999494757503:\\ \;\;\;\;\frac{e^{\frac{x}{s} + -2 \cdot \mathsf{log1p}\left(e^{\frac{x}{s}}\right)}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{s}}{1 + e^{\frac{x}{s}}}\\ \end{array} \]

Alternative 2: 99.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Simplified 99.4%

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

4. Taylor expanded in s around 0 99.4%

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

   1. associate-*r/ 99.4%

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

   2. mul-1-neg 99.4%

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

6. Simplified 99.4%

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

7. Final simplification 99.4%

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

Alternative 3: 99.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.4%

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

3. Simplified 99.4%

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

   1. add-sqr-sqrt 99.4%

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

   2. sqrt-unprod 91.6%

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

   3. sqr-neg 91.6%

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

   4. sqrt-unprod -0.0%

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

   5. add-sqr-sqrt 24.4%

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

   6. expm1-log1p-u 24.4%

      \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right)}} \]

   7. expm1-udef 24.4%

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

5. Applied egg-rr 59.3%

   \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(e^{\frac{x}{s}} + 1\right)} - 1\right)}} \]
6. Step-by-step derivation

   1. expm1-def 59.3%

      \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{-\left|x\right|}{s}}, s\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{x}{s}} + 1\right)\right)}} \]

   2. expm1-log1p 59.3%

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

   3. +-commutative 59.3%

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

7. Simplified 59.3%

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

8. Final simplification 59.3%

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

Alternative 4: 94.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

x_m = abs(x)
function code(x_m, s)
	return Float32(Float32(Float32(0.5) / s) / Float32(Float32(1.0) + exp(Float32(Float32(x_m * log(Float32(exp(1)))) / s))))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m, s)
	tmp = (single(0.5) / s) / (single(1.0) + exp(((x_m * log(single(2.71828182845904523536))) / s)));
end

Derivation

1. Initial program 99.4%

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

   1. *-commutative 99.4%

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

   2. distribute-lft-in 99.4%

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

   3. *-rgt-identity 99.4%

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

   4. fabs-neg 99.4%

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

   5. distribute-frac-neg 99.4%

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

   6. exp-neg 99.4%

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

   7. associate-*r/ 99.4%

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

   8. *-rgt-identity 99.4%

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

   9. *-lft-identity 99.4%

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

   10. metadata-eval 99.4%

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

   11. times-frac 99.4%

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

   12. neg-mul-1 99.4%

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

   13. neg-mul-1 99.4%

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

   14. fabs-neg 99.4%

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

3. Simplified 99.4%

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

5. Applied egg-rr 88.4%

   \[\leadsto \color{blue}{\frac{e^{\frac{x}{s} - \mathsf{log1p}\left(e^{\frac{x}{s}}\right)}}{s} \cdot \frac{1}{e^{\frac{x}{s}} + 1}} \]
6. Step-by-step derivation

   1. associate-*r/ 88.5%

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

   2. *-rgt-identity 88.5%

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

   3. +-commutative 88.5%

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

7. Simplified 88.5%

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

8. Taylor expanded in x around 0 58.1%

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

   1. exp-neg 58.1%

      \[\leadsto \frac{\frac{\color{blue}{\frac{1}{e^{\log 2}}}}{s}}{1 + e^{\frac{x}{s}}} \]

   2. rem-exp-log 58.1%

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

   3. metadata-eval 58.1%

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

10. Simplified 58.1%

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

    1. *-un-lft-identity 58.1%

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

    2. exp-prod 58.1%

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

12. Applied egg-rr 58.1%

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

    1. exp-1-e 58.1%

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

14. Simplified 58.1%

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

15. Taylor expanded in x around inf 58.1%

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

16. Final simplification 58.1%

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

Alternative 5: 94.8% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

x_m = abs(x)
function code(x_m, s)
	return Float32(Float32(Float32(0.5) / s) / Float32(Float32(1.0) + (Float32(exp(1)) ^ Float32(x_m / s))))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m, s)
	tmp = (single(0.5) / s) / (single(1.0) + (single(2.71828182845904523536) ^ (x_m / s)));
end

Derivation

1. Initial program 99.4%

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

   1. *-commutative 99.4%

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

   2. distribute-lft-in 99.4%

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

   3. *-rgt-identity 99.4%

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

   4. fabs-neg 99.4%

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

   5. distribute-frac-neg 99.4%

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

   6. exp-neg 99.4%

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

   7. associate-*r/ 99.4%

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

   8. *-rgt-identity 99.4%

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

   9. *-lft-identity 99.4%

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

   10. metadata-eval 99.4%

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

   11. times-frac 99.4%

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

   12. neg-mul-1 99.4%

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

   13. neg-mul-1 99.4%

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

   14. fabs-neg 99.4%

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

3. Simplified 99.4%

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

5. Applied egg-rr 88.4%

   \[\leadsto \color{blue}{\frac{e^{\frac{x}{s} - \mathsf{log1p}\left(e^{\frac{x}{s}}\right)}}{s} \cdot \frac{1}{e^{\frac{x}{s}} + 1}} \]
6. Step-by-step derivation

   1. associate-*r/ 88.5%

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

   2. *-rgt-identity 88.5%

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

   3. +-commutative 88.5%

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

7. Simplified 88.5%

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

8. Taylor expanded in x around 0 58.1%

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

   1. exp-neg 58.1%

      \[\leadsto \frac{\frac{\color{blue}{\frac{1}{e^{\log 2}}}}{s}}{1 + e^{\frac{x}{s}}} \]

   2. rem-exp-log 58.1%

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

   3. metadata-eval 58.1%

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

10. Simplified 58.1%

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

    1. *-un-lft-identity 58.1%

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

    2. exp-prod 58.1%

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

12. Applied egg-rr 58.1%

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

    1. exp-1-e 58.1%

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

14. Simplified 58.1%

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

15. Final simplification 58.1%

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

Alternative 6: 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 99.4%

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

   1. *-commutative 99.4%

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

   2. distribute-lft-in 99.4%

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

   3. *-rgt-identity 99.4%

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

   4. fabs-neg 99.4%

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

   5. distribute-frac-neg 99.4%

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

   6. exp-neg 99.4%

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

   7. associate-*r/ 99.4%

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

   8. *-rgt-identity 99.4%

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

   9. *-lft-identity 99.4%

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

   10. metadata-eval 99.4%

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

   11. times-frac 99.4%

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

   12. neg-mul-1 99.4%

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

   13. neg-mul-1 99.4%

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

   14. fabs-neg 99.4%

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

3. Simplified 99.4%

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

5. Applied egg-rr 88.4%

   \[\leadsto \color{blue}{\frac{e^{\frac{x}{s} - \mathsf{log1p}\left(e^{\frac{x}{s}}\right)}}{s} \cdot \frac{1}{e^{\frac{x}{s}} + 1}} \]
6. Step-by-step derivation

   1. associate-*r/ 88.5%

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

   2. *-rgt-identity 88.5%

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

   3. +-commutative 88.5%

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

7. Simplified 88.5%

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

8. Taylor expanded in x around 0 58.1%

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

   1. exp-neg 58.1%

      \[\leadsto \frac{\frac{\color{blue}{\frac{1}{e^{\log 2}}}}{s}}{1 + e^{\frac{x}{s}}} \]

   2. rem-exp-log 58.1%

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

   3. metadata-eval 58.1%

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

10. Simplified 58.1%

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

11. Final simplification 58.1%

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

Alternative 7: 51.5% accurate, 67.9× speedup

Math

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

FPCore

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

C

x_m = fabs(x);
float code(float x_m, float s) {
	float tmp;
	if (x_m <= 0.00019999999494757503f) {
		tmp = 0.25f / s;
	} else {
		tmp = (0.5f / s) / (x_m / s);
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(4) function code(x_m, s)
    real(4), intent (in) :: x_m
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x_m <= 0.00019999999494757503e0) then
        tmp = 0.25e0 / s
    else
        tmp = (0.5e0 / s) / (x_m / s)
    end if
    code = tmp
end function

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if x < 1.99999995e-4

   1. Initial program 99.2%

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

      1. *-commutative 99.2%

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

      2. distribute-lft-in 99.3%

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

      3. *-rgt-identity 99.3%

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

      4. fabs-neg 99.3%

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

      5. distribute-frac-neg 99.3%

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

      6. exp-neg 99.3%

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

      7. associate-*r/ 99.2%

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

      8. *-rgt-identity 99.2%

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

      9. *-lft-identity 99.2%

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

      10. metadata-eval 99.2%

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

      11. times-frac 99.2%

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

      12. neg-mul-1 99.2%

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

      13. neg-mul-1 99.2%

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

      14. fabs-neg 99.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in s around inf 33.2%

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

   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. *-commutative 100.0%

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

      2. distribute-lft-in 100.0%

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

      3. *-rgt-identity 100.0%

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

      4. fabs-neg 100.0%

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

      5. distribute-frac-neg 100.0%

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

      6. exp-neg 100.0%

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

      7. associate-*r/ 100.0%

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

      8. *-rgt-identity 100.0%

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

      9. *-lft-identity 100.0%

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

      10. metadata-eval 100.0%

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

      11. times-frac 100.0%

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

      12. neg-mul-1 100.0%

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

      13. neg-mul-1 100.0%

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

      14. fabs-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 51.7%

      \[\leadsto \color{blue}{\frac{e^{\frac{x}{s} - \mathsf{log1p}\left(e^{\frac{x}{s}}\right)}}{s} \cdot \frac{1}{e^{\frac{x}{s}} + 1}} \]
   6. Step-by-step derivation

      1. associate-*r/ 51.7%

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

      2. *-rgt-identity 51.7%

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

      3. +-commutative 51.7%

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

   7. Simplified 51.7%

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

   8. Taylor expanded in x around 0 100.0%

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

      1. exp-neg 100.0%

         \[\leadsto \frac{\frac{\color{blue}{\frac{1}{e^{\log 2}}}}{s}}{1 + e^{\frac{x}{s}}} \]

      2. rem-exp-log 100.0%

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

      3. metadata-eval 100.0%

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

   10. Simplified 100.0%

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

   11. Taylor expanded in x around 0 52.3%

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

       1. +-commutative 52.3%

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

   13. Simplified 52.3%

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

   14. Taylor expanded in x around inf 52.3%

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

4. Final simplification 37.5%

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

Alternative 8: 50.2% 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 99.4%

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

   1. *-commutative 99.4%

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

   2. distribute-lft-in 99.4%

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

   3. *-rgt-identity 99.4%

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

   4. fabs-neg 99.4%

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

   5. distribute-frac-neg 99.4%

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

   6. exp-neg 99.4%

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

   7. associate-*r/ 99.4%

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

   8. *-rgt-identity 99.4%

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

   9. *-lft-identity 99.4%

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

   10. metadata-eval 99.4%

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

   11. times-frac 99.4%

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

   12. neg-mul-1 99.4%

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

   13. neg-mul-1 99.4%

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

   14. fabs-neg 99.4%

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

3. Simplified 99.4%

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

5. Applied egg-rr 88.4%

   \[\leadsto \color{blue}{\frac{e^{\frac{x}{s} - \mathsf{log1p}\left(e^{\frac{x}{s}}\right)}}{s} \cdot \frac{1}{e^{\frac{x}{s}} + 1}} \]
6. Step-by-step derivation

   1. associate-*r/ 88.5%

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

   2. *-rgt-identity 88.5%

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

   3. +-commutative 88.5%

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

7. Simplified 88.5%

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

8. Taylor expanded in x around 0 58.1%

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

   1. exp-neg 58.1%

      \[\leadsto \frac{\frac{\color{blue}{\frac{1}{e^{\log 2}}}}{s}}{1 + e^{\frac{x}{s}}} \]

   2. rem-exp-log 58.1%

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

   3. metadata-eval 58.1%

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

10. Simplified 58.1%

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

11. Taylor expanded in x around 0 51.1%

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

    1. +-commutative 51.1%

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

13. Simplified 51.1%

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

14. Final simplification 51.1%

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

Alternative 9: 30.1% accurate, 121.0× speedup

Math

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

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (if (<= x_m 0.00019999999494757503) (/ 0.25 s) (/ 0.5 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 / 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 / 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(0.5) / 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) / x_m;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 1.99999995e-4

   1. Initial program 99.2%

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

      1. *-commutative 99.2%

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

      2. distribute-lft-in 99.3%

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

      3. *-rgt-identity 99.3%

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

      4. fabs-neg 99.3%

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

      5. distribute-frac-neg 99.3%

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

      6. exp-neg 99.3%

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

      7. associate-*r/ 99.2%

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

      8. *-rgt-identity 99.2%

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

      9. *-lft-identity 99.2%

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

      10. metadata-eval 99.2%

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

      11. times-frac 99.2%

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

      12. neg-mul-1 99.2%

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

      13. neg-mul-1 99.2%

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

      14. fabs-neg 99.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in s around inf 33.2%

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

   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. *-commutative 100.0%

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

      2. distribute-lft-in 100.0%

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

      3. *-rgt-identity 100.0%

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

      4. fabs-neg 100.0%

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

      5. distribute-frac-neg 100.0%

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

      6. exp-neg 100.0%

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

      7. associate-*r/ 100.0%

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

      8. *-rgt-identity 100.0%

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

      9. *-lft-identity 100.0%

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

      10. metadata-eval 100.0%

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

      11. times-frac 100.0%

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

      12. neg-mul-1 100.0%

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

      13. neg-mul-1 100.0%

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

      14. fabs-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 51.7%

      \[\leadsto \color{blue}{\frac{e^{\frac{x}{s} - \mathsf{log1p}\left(e^{\frac{x}{s}}\right)}}{s} \cdot \frac{1}{e^{\frac{x}{s}} + 1}} \]
   6. Step-by-step derivation

      1. associate-*r/ 51.7%

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

      2. *-rgt-identity 51.7%

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

      3. +-commutative 51.7%

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

   7. Simplified 51.7%

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

   8. Taylor expanded in x around 0 100.0%

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

      1. exp-neg 100.0%

         \[\leadsto \frac{\frac{\color{blue}{\frac{1}{e^{\log 2}}}}{s}}{1 + e^{\frac{x}{s}}} \]

      2. rem-exp-log 100.0%

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

      3. metadata-eval 100.0%

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

   10. Simplified 100.0%

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

   11. Taylor expanded in x around 0 52.3%

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

       1. +-commutative 52.3%

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

   13. Simplified 52.3%

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

   14. Taylor expanded in s around 0 11.0%

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

4. Final simplification 28.2%

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

Alternative 10: 26.9% 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 99.4%

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

   1. *-commutative 99.4%

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

   2. distribute-lft-in 99.4%

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

   3. *-rgt-identity 99.4%

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

   4. fabs-neg 99.4%

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

   5. distribute-frac-neg 99.4%

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

   6. exp-neg 99.4%

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

   7. associate-*r/ 99.4%

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

   8. *-rgt-identity 99.4%

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

   9. *-lft-identity 99.4%

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

   10. metadata-eval 99.4%

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

   11. times-frac 99.4%

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

   12. neg-mul-1 99.4%

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

   13. neg-mul-1 99.4%

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

   14. fabs-neg 99.4%

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

3. Simplified 99.4%

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

4. Taylor expanded in s around inf 26.7%

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

5. Final simplification 26.7%

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

Reproduce

herbie shell --seed 2023322 
(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))))))