Logistic distribution

Percentage Accurate: 99.5% → 99.6%

Time: 13.8s

Alternatives: 8

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: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.6% 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(Float32(-abs(x)) / 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 99.7%

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

   1. *-commutative 99.7%

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

   2. fabs-neg 99.7%

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

   3. +-commutative 99.7%

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

   4. fabs-neg 99.7%

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

   5. distribute-lft-in 99.7%

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

   6. *-rgt-identity 99.7%

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

   7. +-commutative 99.7%

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

3. Simplified 99.7%

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

5. Final simplification 99.7%

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

Alternative 2: 73.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.004999999888241291:\\ \;\;\;\;\frac{e^{\frac{x}{s} + -2 \cdot \mathsf{log1p}\left(e^{\frac{x}{s}}\right)}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{x}{-s}}}{s \cdot 4}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (fabs x) 0.004999999888241291)
   (/ (exp (+ (/ x s) (* -2.0 (log1p (exp (/ x s)))))) s)
   (/ (exp (/ x (- s))) (* s 4.0))))

C

float code(float x, float s) {
	float tmp;
	if (fabsf(x) <= 0.004999999888241291f) {
		tmp = expf(((x / s) + (-2.0f * log1pf(expf((x / s)))))) / s;
	} else {
		tmp = expf((x / -s)) / (s * 4.0f);
	}
	return tmp;
}

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (abs(x) <= Float32(0.004999999888241291))
		tmp = Float32(exp(Float32(Float32(x / s) + Float32(Float32(-2.0) * log1p(exp(Float32(x / s)))))) / s);
	else
		tmp = Float32(exp(Float32(x / Float32(-s))) / Float32(s * Float32(4.0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (fabs.f32 x) < 0.00499999989

   1. Initial program 99.3%

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

      1. fabs-neg 99.3%

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

      2. distribute-frac-neg 99.3%

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

      3. distribute-frac-neg2 99.3%

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

      4. fabs-neg 99.3%

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

      5. *-commutative 99.3%

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

      6. fabs-neg 99.3%

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

      7. +-commutative 99.3%

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

      8. fabs-neg 99.3%

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

   3. Simplified 99.2%

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

      1. add-exp-log 94.9%

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

      2. log-div 94.7%

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

      3. add-log-exp 94.7%

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

      4. add-sqr-sqrt -0.0%

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

      5. add-sqr-sqrt -0.0%

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

      6. fabs-sqr -0.0%

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

      7. add-sqr-sqrt -0.0%

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

      8. sqrt-unprod 61.3%

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

      9. sqr-neg 61.3%

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

      10. sqrt-unprod 71.7%

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

      11. add-sqr-sqrt 71.7%

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

      12. *-commutative 71.7%

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

   6. Applied egg-rr 94.3%

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

      1. associate--r+ 94.5%

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

      2. exp-diff 95.4%

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

      3. cancel-sign-sub-inv 95.4%

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

      4. metadata-eval 95.4%

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

      5. add-exp-log 99.3%

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

   8. Applied egg-rr 99.3%

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

   if 0.00499999989 < (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. fabs-neg 100.0%

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

      3. +-commutative 100.0%

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

      4. fabs-neg 100.0%

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

      5. distribute-lft-in 100.0%

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

      6. *-rgt-identity 100.0%

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

      7. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in s around 0 100.0%

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

      1. rec-exp 100.0%

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

      2. mul-1-neg 100.0%

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

      3. unpow2 100.0%

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

      4. associate-*r/ 100.0%

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

      5. mul-1-neg 100.0%

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

   7. Simplified 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. exp-neg 100.0%

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

      3. add-sqr-sqrt 55.1%

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

      4. fabs-sqr 55.1%

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

      5. add-sqr-sqrt 100.0%

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

      6. add-sqr-sqrt 55.1%

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

      7. sqrt-unprod 100.0%

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

      8. add-sqr-sqrt 55.1%

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

      9. fabs-sqr 55.1%

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

      10. add-sqr-sqrt 55.1%

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

      11. add-sqr-sqrt 55.1%

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

      12. fabs-sqr 55.1%

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

      13. add-sqr-sqrt 100.0%

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

      14. sqr-neg 100.0%

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

      15. distribute-frac-neg 100.0%

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

      16. distribute-frac-neg 100.0%

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

      17. sqrt-unprod -0.0%

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

      18. add-sqr-sqrt 100.0%

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

   9. Applied egg-rr 100.0%

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

       1. rec-exp 100.0%

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

       2. mul-1-neg 100.0%

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

       3. associate-*r/ 100.0%

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

       4. mul-1-neg 100.0%

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

   11. Simplified 100.0%

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

       1. distribute-frac-neg 100.0%

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

       2. exp-neg 100.0%

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

       3. add-sqr-sqrt 55.1%

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

       4. fabs-sqr 55.1%

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

       5. add-sqr-sqrt 100.0%

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

       6. add-sqr-sqrt 55.1%

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

       7. sqrt-unprod 100.0%

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

       8. add-sqr-sqrt 55.1%

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

       9. fabs-sqr 55.1%

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

       10. add-sqr-sqrt 55.1%

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

       11. add-sqr-sqrt 55.1%

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

       12. fabs-sqr 55.1%

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

       13. add-sqr-sqrt 100.0%

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

       14. sqr-neg 100.0%

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

       15. distribute-frac-neg 100.0%

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

       16. distribute-frac-neg 100.0%

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

       17. sqrt-unprod -0.0%

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

       18. add-sqr-sqrt 100.0%

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

   13. Applied egg-rr 55.1%

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

       1. rec-exp 100.0%

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

       2. mul-1-neg 100.0%

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

       3. associate-*r/ 100.0%

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

       4. mul-1-neg 100.0%

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

   15. Simplified 55.1%

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

   16. Taylor expanded in x around 0 56.5%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.004999999888241291:\\ \;\;\;\;\frac{e^{\frac{x}{s} + -2 \cdot \mathsf{log1p}\left(e^{\frac{x}{s}}\right)}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{x}{-s}}}{s \cdot 4}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 63.1% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ x (- s))))) (/ t_0 (* s (pow (+ 1.0 t_0) 2.0)))))

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

   1. *-commutative 99.7%

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

   2. fabs-neg 99.7%

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

   3. +-commutative 99.7%

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

   4. fabs-neg 99.7%

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

   5. distribute-lft-in 99.7%

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

   6. *-rgt-identity 99.7%

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

   7. +-commutative 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in s around 0 99.7%

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

   1. rec-exp 99.7%

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

   2. mul-1-neg 99.7%

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

   3. unpow2 99.7%

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

   4. associate-*r/ 99.7%

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

   5. mul-1-neg 99.7%

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

7. Simplified 99.7%

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

   1. distribute-frac-neg 99.7%

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

   2. exp-neg 99.6%

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

   3. add-sqr-sqrt 52.6%

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

   4. fabs-sqr 52.6%

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

   5. add-sqr-sqrt 95.8%

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

   6. add-sqr-sqrt 52.6%

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

   7. sqrt-unprod 99.6%

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

   8. add-sqr-sqrt 52.6%

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

   9. fabs-sqr 52.6%

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

   10. add-sqr-sqrt 54.2%

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

   11. add-sqr-sqrt 52.6%

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

   12. fabs-sqr 52.6%

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

   13. add-sqr-sqrt 99.6%

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

   14. sqr-neg 99.6%

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

   15. distribute-frac-neg 99.6%

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

   16. distribute-frac-neg 99.6%

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

   17. sqrt-unprod -0.0%

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

   18. add-sqr-sqrt 93.9%

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

9. Applied egg-rr 95.8%

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

    1. rec-exp 95.8%

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

    2. mul-1-neg 95.8%

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

    3. associate-*r/ 95.8%

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

    4. mul-1-neg 95.8%

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

11. Simplified 95.8%

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

    1. distribute-frac-neg 99.7%

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

    2. exp-neg 99.6%

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

    3. add-sqr-sqrt 52.6%

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

    4. fabs-sqr 52.6%

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

    5. add-sqr-sqrt 95.8%

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

    6. add-sqr-sqrt 52.6%

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

    7. sqrt-unprod 99.6%

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

    8. add-sqr-sqrt 52.6%

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

    9. fabs-sqr 52.6%

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

    10. add-sqr-sqrt 54.2%

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

    11. add-sqr-sqrt 52.6%

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

    12. fabs-sqr 52.6%

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

    13. add-sqr-sqrt 99.6%

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

    14. sqr-neg 99.6%

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

    15. distribute-frac-neg 99.6%

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

    16. distribute-frac-neg 99.6%

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

    17. sqrt-unprod -0.0%

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

    18. add-sqr-sqrt 93.9%

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

13. Applied egg-rr 66.9%

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

    1. rec-exp 95.8%

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

    2. mul-1-neg 95.8%

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

    3. associate-*r/ 95.8%

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

    4. mul-1-neg 95.8%

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

15. Simplified 67.0%

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

16. Final simplification 67.0%

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

Alternative 4: 60.2% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\mathsf{fma}\left(s, e^{\frac{x}{s}}, s\right)} \cdot 0.5 \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (* (/ 1.0 (fma s (exp (/ x s)) s)) 0.5))

C

float code(float x, float s) {
	return (1.0f / fmaf(s, expf((x / s)), s)) * 0.5f;
}

Julia

function code(x, s)
	return Float32(Float32(Float32(1.0) / fma(s, exp(Float32(x / s)), s)) * Float32(0.5))
end

Derivation

1. Initial program 99.7%

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

   1. fabs-neg 99.7%

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

   2. distribute-frac-neg 99.7%

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

   3. distribute-frac-neg2 99.7%

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

   4. fabs-neg 99.7%

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

   5. *-commutative 99.7%

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

   6. fabs-neg 99.7%

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

   7. +-commutative 99.7%

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

   8. fabs-neg 99.7%

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

3. Simplified 99.7%

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

   1. *-un-lft-identity 99.7%

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

   2. associate-*r* 99.7%

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

   3. times-frac 99.6%

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

6. Applied egg-rr 60.6%

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

7. Taylor expanded in x around 0 64.0%

   \[\leadsto \frac{1}{\mathsf{fma}\left(s, e^{\frac{x}{s}}, s\right)} \cdot \color{blue}{0.5} \]
8. Add Preprocessing

Alternative 5: 59.5% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

   1. *-commutative 99.7%

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

   2. fabs-neg 99.7%

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

   3. +-commutative 99.7%

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

   4. fabs-neg 99.7%

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

   5. distribute-lft-in 99.7%

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

   6. *-rgt-identity 99.7%

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

   7. +-commutative 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in s around 0 99.7%

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

   1. rec-exp 99.7%

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

   2. mul-1-neg 99.7%

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

   3. unpow2 99.7%

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

   4. associate-*r/ 99.7%

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

   5. mul-1-neg 99.7%

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

7. Simplified 99.7%

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

   1. distribute-frac-neg 99.7%

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

   2. exp-neg 99.6%

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

   3. add-sqr-sqrt 52.6%

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

   4. fabs-sqr 52.6%

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

   5. add-sqr-sqrt 95.8%

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

   6. add-sqr-sqrt 52.6%

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

   7. sqrt-unprod 99.6%

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

   8. add-sqr-sqrt 52.6%

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

   9. fabs-sqr 52.6%

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

   10. add-sqr-sqrt 54.2%

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

   11. add-sqr-sqrt 52.6%

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

   12. fabs-sqr 52.6%

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

   13. add-sqr-sqrt 99.6%

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

   14. sqr-neg 99.6%

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

   15. distribute-frac-neg 99.6%

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

   16. distribute-frac-neg 99.6%

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

   17. sqrt-unprod -0.0%

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

   18. add-sqr-sqrt 93.9%

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

9. Applied egg-rr 95.8%

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

    1. rec-exp 95.8%

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

    2. mul-1-neg 95.8%

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

    3. associate-*r/ 95.8%

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

    4. mul-1-neg 95.8%

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

11. Simplified 95.8%

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

    1. distribute-frac-neg 99.7%

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

    2. exp-neg 99.6%

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

    3. add-sqr-sqrt 52.6%

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

    4. fabs-sqr 52.6%

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

    5. add-sqr-sqrt 95.8%

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

    6. add-sqr-sqrt 52.6%

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

    7. sqrt-unprod 99.6%

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

    8. add-sqr-sqrt 52.6%

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

    9. fabs-sqr 52.6%

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

    10. add-sqr-sqrt 54.2%

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

    11. add-sqr-sqrt 52.6%

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

    12. fabs-sqr 52.6%

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

    13. add-sqr-sqrt 99.6%

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

    14. sqr-neg 99.6%

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

    15. distribute-frac-neg 99.6%

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

    16. distribute-frac-neg 99.6%

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

    17. sqrt-unprod -0.0%

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

    18. add-sqr-sqrt 93.9%

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

13. Applied egg-rr 66.9%

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

    1. rec-exp 95.8%

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

    2. mul-1-neg 95.8%

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

    3. associate-*r/ 95.8%

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

    4. mul-1-neg 95.8%

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

15. Simplified 67.0%

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

16. Taylor expanded in x around 0 63.2%

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

17. Final simplification 63.2%

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

Alternative 6: 38.5% accurate, 32.6× speedup

Math

\[\begin{array}{l} \\ \frac{1 + \frac{x}{s}}{s \cdot 4 + x \cdot \left(4 + \frac{x}{s} \cdot 3\right)} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (/ (+ 1.0 (/ x s)) (+ (* s 4.0) (* x (+ 4.0 (* (/ x s) 3.0))))))

C

float code(float x, float s) {
	return (1.0f + (x / s)) / ((s * 4.0f) + (x * (4.0f + ((x / s) * 3.0f))));
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = (1.0e0 + (x / s)) / ((s * 4.0e0) + (x * (4.0e0 + ((x / s) * 3.0e0))))
end function

Julia

function code(x, s)
	return Float32(Float32(Float32(1.0) + Float32(x / s)) / Float32(Float32(s * Float32(4.0)) + Float32(x * Float32(Float32(4.0) + Float32(Float32(x / s) * Float32(3.0))))))
end

MATLAB

function tmp = code(x, s)
	tmp = (single(1.0) + (x / s)) / ((s * single(4.0)) + (x * (single(4.0) + ((x / s) * single(3.0)))));
end

Derivation

1. Initial program 99.7%

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

   1. fabs-neg 99.7%

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

   2. distribute-frac-neg 99.7%

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

   3. distribute-frac-neg2 99.7%

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

   4. fabs-neg 99.7%

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

   5. *-commutative 99.7%

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

   6. fabs-neg 99.7%

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

   7. +-commutative 99.7%

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

   8. fabs-neg 99.7%

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

3. Simplified 99.7%

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

6. Applied egg-rr 60.3%

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

   1. *-lft-identity 60.3%

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

   2. associate-/r* 60.6%

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

8. Simplified 60.6%

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

9. Taylor expanded in x around 0 57.8%

   \[\leadsto \frac{e^{\frac{x}{s}}}{\color{blue}{4 \cdot s + x \cdot \left(4 + 3 \cdot \frac{x}{s}\right)}} \]

10. Taylor expanded in x around 0 40.1%

    \[\leadsto \frac{\color{blue}{1 + \frac{x}{s}}}{4 \cdot s + x \cdot \left(4 + 3 \cdot \frac{x}{s}\right)} \]

11. Final simplification 40.1%

    \[\leadsto \frac{1 + \frac{x}{s}}{s \cdot 4 + x \cdot \left(4 + \frac{x}{s} \cdot 3\right)} \]
12. Add Preprocessing

Alternative 7: 27.2% accurate, 206.7× speedup

Math

\[\begin{array}{l} \\ \frac{0.25}{s} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (/ 0.25 s))

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

   1. fabs-neg 99.7%

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

   2. distribute-frac-neg 99.7%

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

   3. distribute-frac-neg2 99.7%

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

   4. fabs-neg 99.7%

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

   5. *-commutative 99.7%

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

   6. fabs-neg 99.7%

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

   7. +-commutative 99.7%

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

   8. fabs-neg 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in s around inf 28.1%

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

Alternative 8: 8.3% 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 99.7%

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

   1. fabs-neg 99.7%

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

   2. distribute-frac-neg 99.7%

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

   3. distribute-frac-neg2 99.7%

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

   4. fabs-neg 99.7%

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

   5. *-commutative 99.7%

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

   6. fabs-neg 99.7%

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

   7. +-commutative 99.7%

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

   8. fabs-neg 99.7%

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

3. Simplified 99.7%

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

   1. add-exp-log 97.6%

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

   2. log-div 97.6%

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

   3. add-log-exp 97.6%

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

   4. add-sqr-sqrt -0.0%

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

   5. add-sqr-sqrt -0.0%

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

   6. fabs-sqr -0.0%

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

   7. add-sqr-sqrt -0.0%

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

   8. sqrt-unprod 53.4%

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

   9. sqr-neg 53.4%

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

   10. sqrt-unprod 58.2%

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

   11. add-sqr-sqrt 58.2%

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

   12. *-commutative 58.2%

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

6. Applied egg-rr 81.4%

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

7. Taylor expanded in x around inf 37.2%

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

8. Taylor expanded in x around 0 6.6%

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

9. Taylor expanded in x around 0 8.5%

   \[\leadsto \color{blue}{1} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024083 
(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))))))