Logistic distribution

Percentage Accurate: 99.6% → 99.5%

Time: 15.0s

Alternatives: 9

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 5:\\ \;\;\;\;\frac{e^{\frac{x}{s} + \mathsf{log1p}\left(e^{\frac{x}{s}}\right) \cdot -2}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (fabs x) 5.0)
   (/ (exp (+ (/ x s) (* (log1p (exp (/ x s))) -2.0))) s)
   (/ 0.0 s)))

C

float code(float x, float s) {
	float tmp;
	if (fabsf(x) <= 5.0f) {
		tmp = expf(((x / s) + (log1pf(expf((x / s))) * -2.0f))) / s;
	} else {
		tmp = 0.0f / s;
	}
	return tmp;
}

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (abs(x) <= Float32(5.0))
		tmp = Float32(exp(Float32(Float32(x / s) + Float32(log1p(exp(Float32(x / s))) * Float32(-2.0)))) / s);
	else
		tmp = Float32(Float32(0.0) / s);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (fabs.f32 x) < 5

   1. Initial program 99.2%

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

      1. fabs-neg 99.2%

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

      2. distribute-frac-neg 99.2%

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

      3. distribute-frac-neg2 99.2%

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

      4. fabs-neg 99.2%

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

      5. *-commutative 99.2%

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

      6. fabs-neg 99.2%

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

      7. +-commutative 99.2%

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

      8. fabs-neg 99.2%

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

   3. Simplified 99.4%

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

   6. Applied egg-rr 77.5%

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

      1. *-lft-identity 77.5%

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

   8. Simplified 77.5%

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

      1. add-exp-log 73.5%

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

      2. log-div 73.4%

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

      3. add-log-exp 94.5%

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

      4. *-commutative 94.5%

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

      5. sum-log 93.8%

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

      6. log-pow 94.3%

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

      7. log1p-undefine 94.3%

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

      8. *-un-lft-identity 94.3%

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

      9. associate--r+ 94.5%

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

      10. exp-diff 95.2%

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

   10. Applied egg-rr 99.4%

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

       1. *-lft-identity 99.4%

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

   12. Simplified 99.4%

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

   if 5 < (fabs.f32 x)

   1. Initial program 100.0%

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

      1. fabs-neg 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. distribute-frac-neg2 100.0%

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

      4. fabs-neg 100.0%

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

      5. *-commutative 100.0%

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

      6. fabs-neg 100.0%

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

      7. +-commutative 100.0%

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

      8. fabs-neg 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 53.1%

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

      1. *-lft-identity 53.1%

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

   8. Simplified 53.1%

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

      1. *-un-lft-identity 53.1%

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

      2. exp-prod 53.1%

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

   10. Applied egg-rr 53.1%

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

       1. exp-1-e 53.1%

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

   12. Simplified 53.1%

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

   13. Taylor expanded in s around inf 7.2%

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

   14. Taylor expanded in x around inf 31.4%

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

       1. log-E 100.0%

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

       2. +-inverses 100.0%

          \[\leadsto \frac{x \cdot \color{blue}{0}}{s} \]

       3. metadata-eval 100.0%

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

       4. distribute-rgt-out-- 100.0%

          \[\leadsto \frac{\color{blue}{0.25 \cdot x - 0.25 \cdot x}}{s} \]

       5. +-inverses 100.0%

          \[\leadsto \frac{\color{blue}{0}}{s} \]

   16. Simplified 100.0%

       \[\leadsto \frac{\color{blue}{0}}{s} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 5:\\ \;\;\;\;\frac{e^{\frac{x}{s} + \mathsf{log1p}\left(e^{\frac{x}{s}}\right) \cdot -2}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	float t_0 = expf((fabsf(x) / -s));
	float t_1 = t_0 + 1.0f;
	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 = t_0 + 1.0e0
    code = t_0 / (s * (t_1 * t_1))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

   1. fabs-neg 99.6%

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

   2. distribute-frac-neg 99.6%

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

   3. distribute-frac-neg2 99.6%

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

   4. fabs-neg 99.6%

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

   5. *-commutative 99.6%

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

   6. fabs-neg 99.6%

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

   7. +-commutative 99.6%

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

   8. fabs-neg 99.6%

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

3. Simplified 99.7%

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

5. Final simplification 99.7%

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

Alternative 3: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.6%

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

5. Final simplification 99.6%

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

Alternative 4: 97.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ e^{\mathsf{fma}\left(x, \frac{1}{s}, \left(-\log s\right) - 2 \cdot \mathsf{log1p}\left(e^{\frac{x}{s}}\right)\right)} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (exp (fma x (/ 1.0 s) (- (- (log s)) (* 2.0 (log1p (exp (/ x s))))))))

C

float code(float x, float s) {
	return expf(fmaf(x, (1.0f / s), (-logf(s) - (2.0f * log1pf(expf((x / s)))))));
}

Julia

function code(x, s)
	return exp(fma(x, Float32(Float32(1.0) / s), Float32(Float32(-log(s)) - Float32(Float32(2.0) * log1p(exp(Float32(x / s)))))))
end

Derivation

1. Initial program 99.6%

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

   1. fabs-neg 99.6%

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

   2. distribute-frac-neg 99.6%

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

   3. distribute-frac-neg2 99.6%

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

   4. fabs-neg 99.6%

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

   5. *-commutative 99.6%

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

   6. fabs-neg 99.6%

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

   7. +-commutative 99.6%

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

   8. fabs-neg 99.6%

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

3. Simplified 99.7%

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

6. Applied egg-rr 87.0%

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

   1. div-inv 86.9%

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

   2. fma-neg 97.1%

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

   3. *-un-lft-identity 97.1%

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

   4. *-commutative 97.1%

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

   5. *-commutative 97.1%

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

   6. *-un-lft-identity 97.1%

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

   7. fma-define 97.1%

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

8. Applied egg-rr 97.1%

   \[\leadsto e^{\color{blue}{\mathsf{fma}\left(x, \frac{1}{s}, -\mathsf{fma}\left(2, \mathsf{log1p}\left(e^{\frac{x}{s}}\right), \log s\right)\right)}} \]
9. Step-by-step derivation

   1. fma-undefine 97.1%

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

10. Applied egg-rr 97.1%

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

11. Final simplification 97.1%

    \[\leadsto e^{\mathsf{fma}\left(x, \frac{1}{s}, \left(-\log s\right) - 2 \cdot \mathsf{log1p}\left(e^{\frac{x}{s}}\right)\right)} \]
12. Add Preprocessing

Alternative 5: 94.9% accurate, 3.0× speedup

Math

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

FPCore

(FPCore (x s) :precision binary32 (/ (exp (/ (fabs x) (- s))) (* s 4.0)))

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

   1. fabs-neg 99.6%

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

   2. distribute-frac-neg 99.6%

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

   3. distribute-frac-neg2 99.6%

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

   4. fabs-neg 99.6%

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

   5. *-commutative 99.6%

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

   6. fabs-neg 99.6%

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

   7. +-commutative 99.6%

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

   8. fabs-neg 99.6%

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

3. Simplified 99.7%

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

5. Taylor expanded in s around inf 94.3%

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

6. Final simplification 94.3%

   \[\leadsto \frac{e^{\frac{\left|x\right|}{-s}}}{s \cdot 4} \]
7. Add Preprocessing

Alternative 6: 77.7% accurate, 31.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6.000000212225132 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{0.25 \cdot \left(x + s\right)}{s} - \frac{x}{s} \cdot 0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 6.000000212225132e-6)
   (/ (- (/ (* 0.25 (+ x s)) s) (* (/ x s) 0.25)) s)
   (/ 0.0 s)))

C

float code(float x, float s) {
	float tmp;
	if (x <= 6.000000212225132e-6f) {
		tmp = (((0.25f * (x + s)) / s) - ((x / s) * 0.25f)) / s;
	} else {
		tmp = 0.0f / s;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 6.000000212225132e-6) then
        tmp = (((0.25e0 * (x + s)) / s) - ((x / s) * 0.25e0)) / s
    else
        tmp = 0.0e0 / s
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(6.000000212225132e-6))
		tmp = Float32(Float32(Float32(Float32(Float32(0.25) * Float32(x + s)) / s) - Float32(Float32(x / s) * Float32(0.25))) / s);
	else
		tmp = Float32(Float32(0.0) / s);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(6.000000212225132e-6))
		tmp = (((single(0.25) * (x + s)) / s) - ((x / s) * single(0.25))) / s;
	else
		tmp = single(0.0) / s;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 6.00000021e-6

   1. Initial program 99.5%

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

      1. fabs-neg 99.5%

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

      2. distribute-frac-neg 99.5%

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

      3. distribute-frac-neg2 99.5%

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

      4. fabs-neg 99.5%

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

      5. *-commutative 99.5%

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

      6. fabs-neg 99.5%

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

      7. +-commutative 99.5%

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

      8. fabs-neg 99.5%

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

   3. Simplified 99.6%

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

   6. Applied egg-rr 89.8%

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

      1. *-lft-identity 89.8%

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

   8. Simplified 89.8%

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

      1. *-un-lft-identity 89.8%

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

      2. exp-prod 89.7%

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

   10. Applied egg-rr 89.7%

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

       1. exp-1-e 89.7%

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

   12. Simplified 89.7%

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

   13. Taylor expanded in s around inf 37.5%

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

   14. Taylor expanded in s around 0 37.5%

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

       1. +-commutative 37.5%

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

       2. log-E 67.9%

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

       3. associate-*r* 67.9%

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

       4. *-rgt-identity 67.9%

          \[\leadsto \frac{\frac{\color{blue}{0.25 \cdot x} + 0.25 \cdot s}{s} - 0.25 \cdot \frac{x}{s}}{s} \]

       5. distribute-lft-out 67.9%

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

   16. Simplified 67.9%

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

   if 6.00000021e-6 < x

   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

   6. Applied egg-rr 2.8%

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

      1. *-lft-identity 2.8%

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

   8. Simplified 2.8%

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

      1. *-un-lft-identity 2.8%

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

      2. exp-prod 2.8%

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

   10. Applied egg-rr 2.8%

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

       1. exp-1-e 2.8%

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

   12. Simplified 2.8%

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

   13. Taylor expanded in s around inf 10.1%

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

   14. Taylor expanded in x around inf 31.8%

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

       1. log-E 96.3%

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

       2. +-inverses 96.3%

          \[\leadsto \frac{x \cdot \color{blue}{0}}{s} \]

       3. metadata-eval 96.3%

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

       4. distribute-rgt-out-- 96.3%

          \[\leadsto \frac{\color{blue}{0.25 \cdot x - 0.25 \cdot x}}{s} \]

       5. +-inverses 96.3%

          \[\leadsto \frac{\color{blue}{0}}{s} \]

   16. Simplified 96.3%

       \[\leadsto \frac{\color{blue}{0}}{s} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.000000212225132 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{0.25 \cdot \left(x + s\right)}{s} - \frac{x}{s} \cdot 0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 56.2% accurate, 77.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.999999936531045 \cdot 10^{-20}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 1.999999936531045e-20) (/ 0.25 s) (/ 0.0 s)))

C

float code(float x, float s) {
	float tmp;
	if (x <= 1.999999936531045e-20f) {
		tmp = 0.25f / s;
	} else {
		tmp = 0.0f / s;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 1.999999936531045e-20) then
        tmp = 0.25e0 / s
    else
        tmp = 0.0e0 / s
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(1.999999936531045e-20))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(0.0) / s);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(1.999999936531045e-20))
		tmp = single(0.25) / s;
	else
		tmp = single(0.0) / s;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 1.99999994e-20

   1. Initial program 99.5%

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

      1. fabs-neg 99.5%

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

      2. distribute-frac-neg 99.5%

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

      3. distribute-frac-neg2 99.5%

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

      4. fabs-neg 99.5%

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

      5. *-commutative 99.5%

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

      6. fabs-neg 99.5%

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

      7. +-commutative 99.5%

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

      8. fabs-neg 99.5%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in s around inf 34.9%

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

   if 1.99999994e-20 < x

   1. Initial program 99.7%

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

      1. fabs-neg 99.7%

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

      2. distribute-frac-neg 99.7%

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

      3. distribute-frac-neg2 99.7%

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

      4. fabs-neg 99.7%

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

      5. *-commutative 99.7%

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

      6. fabs-neg 99.7%

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

      7. +-commutative 99.7%

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

      8. fabs-neg 99.7%

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

   3. Simplified 99.7%

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

   6. Applied egg-rr 12.7%

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

      1. *-lft-identity 12.7%

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

   8. Simplified 12.7%

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

      1. *-un-lft-identity 12.7%

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

      2. exp-prod 12.8%

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

   10. Applied egg-rr 12.8%

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

       1. exp-1-e 12.8%

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

   12. Simplified 12.8%

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

   13. Taylor expanded in s around inf 17.9%

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

   14. Taylor expanded in x around inf 33.7%

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

       1. log-E 87.0%

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

       2. +-inverses 87.0%

          \[\leadsto \frac{x \cdot \color{blue}{0}}{s} \]

       3. metadata-eval 87.0%

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

       4. distribute-rgt-out-- 87.0%

          \[\leadsto \frac{\color{blue}{0.25 \cdot x - 0.25 \cdot x}}{s} \]

       5. +-inverses 87.0%

          \[\leadsto \frac{\color{blue}{0}}{s} \]

   16. Simplified 87.0%

       \[\leadsto \frac{\color{blue}{0}}{s} \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.999999936531045 \cdot 10^{-20}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.3% accurate, 206.7× speedup

Math

\[\begin{array}{l} \\ \frac{0}{s} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (/ 0.0 s))

C

float code(float x, float s) {
	return 0.0f / s;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 0.0e0 / s
end function

Julia

function code(x, s)
	return Float32(Float32(0.0) / s)
end

MATLAB

function tmp = code(x, s)
	tmp = single(0.0) / s;
end

Derivation

1. Initial program 99.6%

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

   1. fabs-neg 99.6%

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

   2. distribute-frac-neg 99.6%

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

   3. distribute-frac-neg2 99.6%

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

   4. fabs-neg 99.6%

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

   5. *-commutative 99.6%

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

   6. fabs-neg 99.6%

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

   7. +-commutative 99.6%

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

   8. fabs-neg 99.6%

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

3. Simplified 99.7%

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

6. Applied egg-rr 65.3%

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

   1. *-lft-identity 65.3%

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

8. Simplified 65.3%

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

   1. *-un-lft-identity 65.3%

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

   2. exp-prod 65.3%

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

10. Applied egg-rr 65.3%

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

    1. exp-1-e 65.3%

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

12. Simplified 65.3%

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

13. Taylor expanded in s around inf 29.8%

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

14. Taylor expanded in x around inf 27.2%

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

    1. log-E 73.0%

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

    2. +-inverses 73.0%

       \[\leadsto \frac{x \cdot \color{blue}{0}}{s} \]

    3. metadata-eval 73.0%

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

    4. distribute-rgt-out-- 73.0%

       \[\leadsto \frac{\color{blue}{0.25 \cdot x - 0.25 \cdot x}}{s} \]

    5. +-inverses 73.0%

       \[\leadsto \frac{\color{blue}{0}}{s} \]

16. Simplified 73.0%

    \[\leadsto \frac{\color{blue}{0}}{s} \]

17. Final simplification 73.0%

    \[\leadsto \frac{0}{s} \]
18. Add Preprocessing

Alternative 9: 8.2% accurate, 620.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x s) :precision binary32 1.0)

C

float code(float x, float s) {
	return 1.0f;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 1.0e0
end function

Julia

function code(x, s)
	return Float32(1.0)
end

MATLAB

function tmp = code(x, s)
	tmp = single(1.0);
end

Derivation

1. Initial program 99.6%

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

   1. fabs-neg 99.6%

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

   2. distribute-frac-neg 99.6%

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

   3. distribute-frac-neg2 99.6%

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

   4. fabs-neg 99.6%

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

   5. *-commutative 99.6%

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

   6. fabs-neg 99.6%

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

   7. +-commutative 99.6%

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

   8. fabs-neg 99.6%

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

3. Simplified 99.7%

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

6. Applied egg-rr 87.0%

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

7. Taylor expanded in x around inf 42.2%

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

8. Taylor expanded in x around 0 8.2%

   \[\leadsto \color{blue}{1} \]

9. Final simplification 8.2%

   \[\leadsto 1 \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024055 
(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))))))