Logistic distribution

Percentage Accurate: 99.5% → 99.5%

Time: 16.5s

Alternatives: 12

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: 99.5% accurate, 1.0× speedup

Math

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

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(Float32(t_0 / 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. Taylor expanded in x around 0 99.6%

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

   1. associate-/r* 99.6%

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

   2. associate-*r/ 99.6%

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

   3. mul-1-neg 99.6%

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

   4. +-commutative 99.6%

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

   5. associate-*r/ 99.6%

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

   6. mul-1-neg 99.6%

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

6. Simplified 99.6%

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

7. Final simplification 99.6%

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

Alternative 2: 96.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{e^{\frac{-\left|x\right|}{s}}}{s}}{{\left(1 + e^{\frac{-x}{s}}\right)}^{2}} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (/ (/ (exp (/ (- (fabs x)) s)) s) (pow (+ 1.0 (exp (/ (- x) s))) 2.0)))

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(x, s)
	tmp = (exp((-abs(x) / s)) / s) / ((single(1.0) + exp((-x / s))) ^ single(2.0));
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. *-commutative 99.6%

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

   3. fabs-neg 99.6%

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

   4. fabs-neg 99.6%

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

   5. associate-*r* 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. /-rgt-identity 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in x around 0 99.6%

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

   1. *-commutative 99.6%

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

   2. *-rgt-identity 99.6%

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

   3. distribute-lft-in 99.6%

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

   4. mul-1-neg 99.6%

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

   5. rec-exp 99.6%

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

   6. associate-*r* 99.6%

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

   7. associate-/r* 99.7%

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

   8. associate-*r/ 99.7%

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

   9. mul-1-neg 99.7%

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

6. Simplified 99.6%

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

   1. div-inv 99.6%

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

   2. exp-prod 95.5%

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

   3. neg-mul-1 95.5%

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

   4. exp-prod 95.5%

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

   5. add-sqr-sqrt 47.6%

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

   6. fabs-sqr 47.6%

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

   7. add-sqr-sqrt 95.0%

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

   8. exp-prod 95.0%

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

   9. neg-mul-1 95.0%

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

   10. exp-prod 97.4%

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

   11. div-inv 97.4%

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

   12. distribute-frac-neg 97.4%

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

   13. exp-neg 97.4%

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

8. Applied egg-rr 97.4%

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

   1. expm1-log1p-u 97.4%

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

   2. expm1-udef 97.4%

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

   3. rec-exp 97.4%

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

   4. distribute-neg-frac 97.4%

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

10. Applied egg-rr 97.4%

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

    1. expm1-def 97.4%

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

    2. expm1-log1p 97.4%

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

12. Simplified 97.4%

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

13. Final simplification 97.4%

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

Alternative 3: 99.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 1:\\ \;\;\;\;\frac{e^{\frac{x}{s} - 2 \cdot \mathsf{log1p}\left(e^{\frac{x}{s}}\right)}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{\left|x\right|}{-s}}}{s \cdot 4}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (fabs x) 1.0)
   (/ (exp (- (/ x s) (* 2.0 (log1p (exp (/ x s)))))) s)
   (/ (exp (/ (fabs x) (- s))) (* s 4.0))))

C

float code(float x, float s) {
	float tmp;
	if (fabsf(x) <= 1.0f) {
		tmp = expf(((x / s) - (2.0f * log1pf(expf((x / s)))))) / s;
	} else {
		tmp = expf((fabsf(x) / -s)) / (s * 4.0f);
	}
	return tmp;
}

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (abs(x) <= Float32(1.0))
		tmp = Float32(exp(Float32(Float32(x / s) - Float32(Float32(2.0) * log1p(exp(Float32(x / s)))))) / s);
	else
		tmp = Float32(exp(Float32(abs(x) / Float32(-s))) / Float32(s * Float32(4.0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (fabs.f32 x) < 1

   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. *-commutative 99.2%

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

      3. fabs-neg 99.2%

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

      4. fabs-neg 99.2%

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

      5. associate-*r* 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. /-rgt-identity 99.2%

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

   3. Simplified 99.2%

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

   5. Applied egg-rr 82.1%

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

      1. associate-*r/ 82.0%

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

      2. *-commutative 82.0%

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

      3. fma-udef 82.0%

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

      4. +-commutative 82.0%

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

      5. *-rgt-identity 82.0%

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

      6. distribute-lft-in 81.9%

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

      7. distribute-lft-out 82.0%

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

      8. *-rgt-identity 82.0%

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

      9. associate-*l* 81.3%

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

      10. distribute-lft-out 81.2%

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

      11. *-rgt-identity 81.2%

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

      12. distribute-lft-in 81.1%

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

   7. Simplified 81.1%

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

   9. Applied egg-rr 81.0%

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

       1. expm1-log1p-u 78.0%

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

       2. expm1-udef 76.8%

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

   11. Applied egg-rr 94.7%

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

       1. expm1-def 95.8%

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

       2. expm1-log1p 99.2%

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

   13. Simplified 99.2%

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

   if 1 < (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. *-commutative 100.0%

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

      3. fabs-neg 100.0%

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

      4. fabs-neg 100.0%

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

      5. associate-*r* 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. /-rgt-identity 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in s around inf 100.0%

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

      1. *-commutative 100.0%

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

   6. Simplified 100.0%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 1:\\ \;\;\;\;\frac{e^{\frac{x}{s} - 2 \cdot \mathsf{log1p}\left(e^{\frac{x}{s}}\right)}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{\left|x\right|}{-s}}}{s \cdot 4}\\ \end{array} \]

Alternative 4: 97.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(x, s)
	tmp = single(1.0) / ((s * (single(1.0) + exp((-x / s)))) * (single(1.0) + 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 99.3%

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

   1. *-un-lft-identity 99.3%

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

   2. neg-mul-1 99.3%

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

   3. times-frac 99.3%

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

   4. metadata-eval 99.3%

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

   5. metadata-eval 99.3%

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

   6. times-frac 99.3%

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

   7. neg-mul-1 99.3%

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

   8. *-un-lft-identity 99.3%

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

   9. distribute-frac-neg 99.3%

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

   10. rec-exp 99.3%

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

   11. add-sqr-sqrt 50.2%

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

   12. fabs-sqr 50.2%

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

   13. add-sqr-sqrt 97.5%

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

5. Applied egg-rr 97.5%

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

   1. rec-exp 97.5%

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

   2. distribute-neg-frac 97.5%

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

7. Simplified 97.5%

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

8. Taylor expanded in s around 0 97.5%

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

   1. associate-*r* 97.5%

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

   2. mul-1-neg 97.5%

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

10. Simplified 97.5%

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

11. Final simplification 97.5%

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

Alternative 5: 61.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	float t_0 = 1.0f / 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 = 1.0e0 / exp((x / s))
    code = (t_0 / s) / ((1.0e0 + t_0) ** 2.0e0)
end function

Julia

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

MATLAB

function tmp = code(x, s)
	t_0 = single(1.0) / exp((x / s));
	tmp = (t_0 / s) / ((single(1.0) + t_0) ^ single(2.0));
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. *-commutative 99.6%

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

   3. fabs-neg 99.6%

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

   4. fabs-neg 99.6%

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

   5. associate-*r* 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. /-rgt-identity 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in x around 0 99.6%

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

   1. *-commutative 99.6%

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

   2. *-rgt-identity 99.6%

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

   3. distribute-lft-in 99.6%

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

   4. mul-1-neg 99.6%

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

   5. rec-exp 99.6%

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

   6. associate-*r* 99.6%

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

   7. associate-/r* 99.7%

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

   8. associate-*r/ 99.7%

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

   9. mul-1-neg 99.7%

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

6. Simplified 99.6%

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

   1. div-inv 99.6%

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

   2. exp-prod 95.5%

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

   3. neg-mul-1 95.5%

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

   4. exp-prod 95.5%

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

   5. add-sqr-sqrt 47.6%

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

   6. fabs-sqr 47.6%

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

   7. add-sqr-sqrt 95.0%

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

   8. exp-prod 95.0%

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

   9. neg-mul-1 95.0%

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

   10. exp-prod 97.4%

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

   11. div-inv 97.4%

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

   12. distribute-frac-neg 97.4%

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

   13. exp-neg 97.4%

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

8. Applied egg-rr 97.4%

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

   1. div-inv 99.6%

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

   2. exp-prod 95.5%

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

   3. neg-mul-1 95.5%

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

   4. exp-prod 95.5%

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

   5. add-sqr-sqrt 47.6%

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

   6. fabs-sqr 47.6%

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

   7. add-sqr-sqrt 95.0%

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

   8. exp-prod 95.0%

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

   9. neg-mul-1 95.0%

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

   10. exp-prod 97.4%

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

   11. div-inv 97.4%

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

   12. distribute-frac-neg 97.4%

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

   13. exp-neg 97.4%

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

10. Applied egg-rr 61.9%

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

11. Final simplification 61.9%

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

Alternative 6: 95.2% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(x, s)
	tmp = single(0.5) / (s * (single(1.0) + 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 99.3%

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

   1. *-un-lft-identity 99.3%

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

   2. neg-mul-1 99.3%

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

   3. times-frac 99.3%

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

   4. metadata-eval 99.3%

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

   5. metadata-eval 99.3%

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

   6. times-frac 99.3%

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

   7. neg-mul-1 99.3%

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

   8. *-un-lft-identity 99.3%

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

   9. distribute-frac-neg 99.3%

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

   10. rec-exp 99.3%

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

   11. add-sqr-sqrt 50.2%

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

   12. fabs-sqr 50.2%

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

   13. add-sqr-sqrt 97.5%

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

5. Applied egg-rr 97.5%

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

   1. rec-exp 97.5%

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

   2. distribute-neg-frac 97.5%

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

7. Simplified 97.5%

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

8. Taylor expanded in x around 0 95.2%

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

9. Final simplification 95.2%

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

Alternative 7: 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. *-commutative 99.6%

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

   3. fabs-neg 99.6%

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

   4. fabs-neg 99.6%

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

   5. associate-*r* 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. /-rgt-identity 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in s around inf 94.9%

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

   1. *-commutative 94.9%

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

6. Simplified 94.9%

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

7. Final simplification 94.9%

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

Alternative 8: 64.4% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \frac{1}{s \cdot 4 + \frac{1}{\frac{s}{{x}^{2}}}} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (/ 1.0 (+ (* s 4.0) (/ 1.0 (/ s (pow x 2.0))))))

C

float code(float x, float s) {
	return 1.0f / ((s * 4.0f) + (1.0f / (s / powf(x, 2.0f))));
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 1.0e0 / ((s * 4.0e0) + (1.0e0 / (s / (x ** 2.0e0))))
end function

Julia

function code(x, s)
	return Float32(Float32(1.0) / Float32(Float32(s * Float32(4.0)) + Float32(Float32(1.0) / Float32(s / (x ^ Float32(2.0))))))
end

MATLAB

function tmp = code(x, s)
	tmp = single(1.0) / ((s * single(4.0)) + (single(1.0) / (s / (x ^ single(2.0)))));
end

Derivation

1. Initial program 99.6%

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

3. Simplified 99.3%

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

4. Taylor expanded in s around -inf 43.0%

   \[\leadsto \frac{1}{\color{blue}{-2 \cdot \left|x\right| + \left(-1 \cdot \frac{-2 \cdot {\left(\left|x\right|\right)}^{2} + {\left(\left|x\right|\right)}^{2}}{s} + \left(2 \cdot \left|x\right| + 4 \cdot s\right)\right)}} \]
5. Step-by-step derivation

   1. +-commutative 43.0%

      \[\leadsto \frac{1}{-2 \cdot \left|x\right| + \color{blue}{\left(\left(2 \cdot \left|x\right| + 4 \cdot s\right) + -1 \cdot \frac{-2 \cdot {\left(\left|x\right|\right)}^{2} + {\left(\left|x\right|\right)}^{2}}{s}\right)}} \]

   2. mul-1-neg 43.0%

      \[\leadsto \frac{1}{-2 \cdot \left|x\right| + \left(\left(2 \cdot \left|x\right| + 4 \cdot s\right) + \color{blue}{\left(-\frac{-2 \cdot {\left(\left|x\right|\right)}^{2} + {\left(\left|x\right|\right)}^{2}}{s}\right)}\right)} \]

   3. distribute-lft1-in 64.9%

      \[\leadsto \frac{1}{-2 \cdot \left|x\right| + \left(\left(2 \cdot \left|x\right| + 4 \cdot s\right) + \left(-\frac{\color{blue}{\left(-2 + 1\right) \cdot {\left(\left|x\right|\right)}^{2}}}{s}\right)\right)} \]

   4. metadata-eval 64.9%

      \[\leadsto \frac{1}{-2 \cdot \left|x\right| + \left(\left(2 \cdot \left|x\right| + 4 \cdot s\right) + \left(-\frac{\color{blue}{-1} \cdot {\left(\left|x\right|\right)}^{2}}{s}\right)\right)} \]

   5. associate-*r/ 64.9%

      \[\leadsto \frac{1}{-2 \cdot \left|x\right| + \left(\left(2 \cdot \left|x\right| + 4 \cdot s\right) + \left(-\color{blue}{-1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}}\right)\right)} \]

   6. mul-1-neg 64.9%

      \[\leadsto \frac{1}{-2 \cdot \left|x\right| + \left(\left(2 \cdot \left|x\right| + 4 \cdot s\right) + \left(-\color{blue}{\left(-\frac{{\left(\left|x\right|\right)}^{2}}{s}\right)}\right)\right)} \]

   7. remove-double-neg 64.9%

      \[\leadsto \frac{1}{-2 \cdot \left|x\right| + \left(\left(2 \cdot \left|x\right| + 4 \cdot s\right) + \color{blue}{\frac{{\left(\left|x\right|\right)}^{2}}{s}}\right)} \]

   8. associate-+r+ 64.9%

      \[\leadsto \frac{1}{\color{blue}{\left(-2 \cdot \left|x\right| + \left(2 \cdot \left|x\right| + 4 \cdot s\right)\right) + \frac{{\left(\left|x\right|\right)}^{2}}{s}}} \]

6. Simplified 65.3%

   \[\leadsto \frac{1}{\color{blue}{\left(0 + s \cdot 4\right) + \frac{x \cdot x}{s}}} \]
7. Step-by-step derivation

   1. clear-num 65.3%

      \[\leadsto \frac{1}{\left(0 + s \cdot 4\right) + \color{blue}{\frac{1}{\frac{s}{x \cdot x}}}} \]

   2. inv-pow 65.3%

      \[\leadsto \frac{1}{\left(0 + s \cdot 4\right) + \color{blue}{{\left(\frac{s}{x \cdot x}\right)}^{-1}}} \]

   3. pow2 65.3%

      \[\leadsto \frac{1}{\left(0 + s \cdot 4\right) + {\left(\frac{s}{\color{blue}{{x}^{2}}}\right)}^{-1}} \]

8. Applied egg-rr 65.3%

   \[\leadsto \frac{1}{\left(0 + s \cdot 4\right) + \color{blue}{{\left(\frac{s}{{x}^{2}}\right)}^{-1}}} \]
9. Step-by-step derivation

   1. unpow-1 65.3%

      \[\leadsto \frac{1}{\left(0 + s \cdot 4\right) + \color{blue}{\frac{1}{\frac{s}{{x}^{2}}}}} \]

10. Simplified 65.3%

    \[\leadsto \frac{1}{\left(0 + s \cdot 4\right) + \color{blue}{\frac{1}{\frac{s}{{x}^{2}}}}} \]

11. Final simplification 65.3%

    \[\leadsto \frac{1}{s \cdot 4 + \frac{1}{\frac{s}{{x}^{2}}}} \]

Alternative 9: 64.4% accurate, 56.4× speedup

Math

\[\begin{array}{l} \\ \frac{1}{s \cdot 4 + \frac{x \cdot x}{s}} \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(x, s)
	tmp = single(1.0) / ((s * single(4.0)) + ((x * 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 99.3%

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

4. Taylor expanded in s around -inf 43.0%

   \[\leadsto \frac{1}{\color{blue}{-2 \cdot \left|x\right| + \left(-1 \cdot \frac{-2 \cdot {\left(\left|x\right|\right)}^{2} + {\left(\left|x\right|\right)}^{2}}{s} + \left(2 \cdot \left|x\right| + 4 \cdot s\right)\right)}} \]
5. Step-by-step derivation

   1. +-commutative 43.0%

      \[\leadsto \frac{1}{-2 \cdot \left|x\right| + \color{blue}{\left(\left(2 \cdot \left|x\right| + 4 \cdot s\right) + -1 \cdot \frac{-2 \cdot {\left(\left|x\right|\right)}^{2} + {\left(\left|x\right|\right)}^{2}}{s}\right)}} \]

   2. mul-1-neg 43.0%

      \[\leadsto \frac{1}{-2 \cdot \left|x\right| + \left(\left(2 \cdot \left|x\right| + 4 \cdot s\right) + \color{blue}{\left(-\frac{-2 \cdot {\left(\left|x\right|\right)}^{2} + {\left(\left|x\right|\right)}^{2}}{s}\right)}\right)} \]

   3. distribute-lft1-in 64.9%

      \[\leadsto \frac{1}{-2 \cdot \left|x\right| + \left(\left(2 \cdot \left|x\right| + 4 \cdot s\right) + \left(-\frac{\color{blue}{\left(-2 + 1\right) \cdot {\left(\left|x\right|\right)}^{2}}}{s}\right)\right)} \]

   4. metadata-eval 64.9%

      \[\leadsto \frac{1}{-2 \cdot \left|x\right| + \left(\left(2 \cdot \left|x\right| + 4 \cdot s\right) + \left(-\frac{\color{blue}{-1} \cdot {\left(\left|x\right|\right)}^{2}}{s}\right)\right)} \]

   5. associate-*r/ 64.9%

      \[\leadsto \frac{1}{-2 \cdot \left|x\right| + \left(\left(2 \cdot \left|x\right| + 4 \cdot s\right) + \left(-\color{blue}{-1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}}\right)\right)} \]

   6. mul-1-neg 64.9%

      \[\leadsto \frac{1}{-2 \cdot \left|x\right| + \left(\left(2 \cdot \left|x\right| + 4 \cdot s\right) + \left(-\color{blue}{\left(-\frac{{\left(\left|x\right|\right)}^{2}}{s}\right)}\right)\right)} \]

   7. remove-double-neg 64.9%

      \[\leadsto \frac{1}{-2 \cdot \left|x\right| + \left(\left(2 \cdot \left|x\right| + 4 \cdot s\right) + \color{blue}{\frac{{\left(\left|x\right|\right)}^{2}}{s}}\right)} \]

   8. associate-+r+ 64.9%

      \[\leadsto \frac{1}{\color{blue}{\left(-2 \cdot \left|x\right| + \left(2 \cdot \left|x\right| + 4 \cdot s\right)\right) + \frac{{\left(\left|x\right|\right)}^{2}}{s}}} \]

6. Simplified 65.3%

   \[\leadsto \frac{1}{\color{blue}{\left(0 + s \cdot 4\right) + \frac{x \cdot x}{s}}} \]

7. Final simplification 65.3%

   \[\leadsto \frac{1}{s \cdot 4 + \frac{x \cdot x}{s}} \]

Alternative 10: 43.6% accurate, 67.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.9999999494757503 \cdot 10^{-5}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{s}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 1.9999999494757503e-5) (/ 0.25 s) (* (/ 1.0 x) (/ s x))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 1.9999999494757503e-5f) {
		tmp = 0.25f / s;
	} else {
		tmp = (1.0f / x) * (s / x);
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 1.9999999494757503e-5) then
        tmp = 0.25e0 / s
    else
        tmp = (1.0e0 / x) * (s / x)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(1.9999999494757503e-5))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(Float32(1.0) / x) * Float32(s / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(1.9999999494757503e-5))
		tmp = single(0.25) / s;
	else
		tmp = (single(1.0) / x) * (s / x);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 1.99999995e-5

   1. Initial program 99.4%

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

      1. fabs-neg 99.4%

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

      2. *-commutative 99.4%

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

      3. fabs-neg 99.4%

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

      4. fabs-neg 99.4%

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

      5. associate-*r* 99.4%

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

      6. /-rgt-identity 99.4%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in s around inf 35.3%

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

   if 1.99999995e-5 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in s around -inf 36.2%

      \[\leadsto \frac{1}{\color{blue}{-2 \cdot \left|x\right| + \left(-1 \cdot \frac{-2 \cdot {\left(\left|x\right|\right)}^{2} + {\left(\left|x\right|\right)}^{2}}{s} + \left(2 \cdot \left|x\right| + 4 \cdot s\right)\right)}} \]
   5. Step-by-step derivation

      1. +-commutative 36.2%

         \[\leadsto \frac{1}{-2 \cdot \left|x\right| + \color{blue}{\left(\left(2 \cdot \left|x\right| + 4 \cdot s\right) + -1 \cdot \frac{-2 \cdot {\left(\left|x\right|\right)}^{2} + {\left(\left|x\right|\right)}^{2}}{s}\right)}} \]

      2. mul-1-neg 36.2%

         \[\leadsto \frac{1}{-2 \cdot \left|x\right| + \left(\left(2 \cdot \left|x\right| + 4 \cdot s\right) + \color{blue}{\left(-\frac{-2 \cdot {\left(\left|x\right|\right)}^{2} + {\left(\left|x\right|\right)}^{2}}{s}\right)}\right)} \]

      3. distribute-lft1-in 72.7%

         \[\leadsto \frac{1}{-2 \cdot \left|x\right| + \left(\left(2 \cdot \left|x\right| + 4 \cdot s\right) + \left(-\frac{\color{blue}{\left(-2 + 1\right) \cdot {\left(\left|x\right|\right)}^{2}}}{s}\right)\right)} \]

      4. metadata-eval 72.7%

         \[\leadsto \frac{1}{-2 \cdot \left|x\right| + \left(\left(2 \cdot \left|x\right| + 4 \cdot s\right) + \left(-\frac{\color{blue}{-1} \cdot {\left(\left|x\right|\right)}^{2}}{s}\right)\right)} \]

      5. associate-*r/ 72.7%

         \[\leadsto \frac{1}{-2 \cdot \left|x\right| + \left(\left(2 \cdot \left|x\right| + 4 \cdot s\right) + \left(-\color{blue}{-1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}}\right)\right)} \]

      6. mul-1-neg 72.7%

         \[\leadsto \frac{1}{-2 \cdot \left|x\right| + \left(\left(2 \cdot \left|x\right| + 4 \cdot s\right) + \left(-\color{blue}{\left(-\frac{{\left(\left|x\right|\right)}^{2}}{s}\right)}\right)\right)} \]

      7. remove-double-neg 72.7%

         \[\leadsto \frac{1}{-2 \cdot \left|x\right| + \left(\left(2 \cdot \left|x\right| + 4 \cdot s\right) + \color{blue}{\frac{{\left(\left|x\right|\right)}^{2}}{s}}\right)} \]

      8. associate-+r+ 72.7%

         \[\leadsto \frac{1}{\color{blue}{\left(-2 \cdot \left|x\right| + \left(2 \cdot \left|x\right| + 4 \cdot s\right)\right) + \frac{{\left(\left|x\right|\right)}^{2}}{s}}} \]

   6. Simplified 74.1%

      \[\leadsto \frac{1}{\color{blue}{\left(0 + s \cdot 4\right) + \frac{x \cdot x}{s}}} \]

   7. Taylor expanded in s around 0 71.1%

      \[\leadsto \color{blue}{\frac{s}{{x}^{2}}} \]
   8. Step-by-step derivation

      1. *-un-lft-identity 71.1%

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

      2. unpow2 71.1%

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

      3. times-frac 71.1%

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

   9. Applied egg-rr 71.1%

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

4. Final simplification 45.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.9999999494757503 \cdot 10^{-5}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{s}{x}\\ \end{array} \]

Alternative 11: 43.6% accurate, 87.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.9999999494757503 \cdot 10^{-5}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{s}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 1.9999999494757503e-5) (/ 0.25 s) (/ (/ s x) x)))

C

float code(float x, float s) {
	float tmp;
	if (x <= 1.9999999494757503e-5f) {
		tmp = 0.25f / s;
	} else {
		tmp = (s / x) / x;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 1.9999999494757503e-5) then
        tmp = 0.25e0 / s
    else
        tmp = (s / x) / x
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(1.9999999494757503e-5))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(s / x) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(1.9999999494757503e-5))
		tmp = single(0.25) / s;
	else
		tmp = (s / x) / x;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 1.99999995e-5

   1. Initial program 99.4%

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

      1. fabs-neg 99.4%

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

      2. *-commutative 99.4%

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

      3. fabs-neg 99.4%

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

      4. fabs-neg 99.4%

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

      5. associate-*r* 99.4%

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

      6. /-rgt-identity 99.4%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in s around inf 35.3%

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

   if 1.99999995e-5 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in s around -inf 36.2%

      \[\leadsto \frac{1}{\color{blue}{-2 \cdot \left|x\right| + \left(-1 \cdot \frac{-2 \cdot {\left(\left|x\right|\right)}^{2} + {\left(\left|x\right|\right)}^{2}}{s} + \left(2 \cdot \left|x\right| + 4 \cdot s\right)\right)}} \]
   5. Step-by-step derivation

      1. +-commutative 36.2%

         \[\leadsto \frac{1}{-2 \cdot \left|x\right| + \color{blue}{\left(\left(2 \cdot \left|x\right| + 4 \cdot s\right) + -1 \cdot \frac{-2 \cdot {\left(\left|x\right|\right)}^{2} + {\left(\left|x\right|\right)}^{2}}{s}\right)}} \]

      2. mul-1-neg 36.2%

         \[\leadsto \frac{1}{-2 \cdot \left|x\right| + \left(\left(2 \cdot \left|x\right| + 4 \cdot s\right) + \color{blue}{\left(-\frac{-2 \cdot {\left(\left|x\right|\right)}^{2} + {\left(\left|x\right|\right)}^{2}}{s}\right)}\right)} \]

      3. distribute-lft1-in 72.7%

         \[\leadsto \frac{1}{-2 \cdot \left|x\right| + \left(\left(2 \cdot \left|x\right| + 4 \cdot s\right) + \left(-\frac{\color{blue}{\left(-2 + 1\right) \cdot {\left(\left|x\right|\right)}^{2}}}{s}\right)\right)} \]

      4. metadata-eval 72.7%

         \[\leadsto \frac{1}{-2 \cdot \left|x\right| + \left(\left(2 \cdot \left|x\right| + 4 \cdot s\right) + \left(-\frac{\color{blue}{-1} \cdot {\left(\left|x\right|\right)}^{2}}{s}\right)\right)} \]

      5. associate-*r/ 72.7%

         \[\leadsto \frac{1}{-2 \cdot \left|x\right| + \left(\left(2 \cdot \left|x\right| + 4 \cdot s\right) + \left(-\color{blue}{-1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}}\right)\right)} \]

      6. mul-1-neg 72.7%

         \[\leadsto \frac{1}{-2 \cdot \left|x\right| + \left(\left(2 \cdot \left|x\right| + 4 \cdot s\right) + \left(-\color{blue}{\left(-\frac{{\left(\left|x\right|\right)}^{2}}{s}\right)}\right)\right)} \]

      7. remove-double-neg 72.7%

         \[\leadsto \frac{1}{-2 \cdot \left|x\right| + \left(\left(2 \cdot \left|x\right| + 4 \cdot s\right) + \color{blue}{\frac{{\left(\left|x\right|\right)}^{2}}{s}}\right)} \]

      8. associate-+r+ 72.7%

         \[\leadsto \frac{1}{\color{blue}{\left(-2 \cdot \left|x\right| + \left(2 \cdot \left|x\right| + 4 \cdot s\right)\right) + \frac{{\left(\left|x\right|\right)}^{2}}{s}}} \]

   6. Simplified 74.1%

      \[\leadsto \frac{1}{\color{blue}{\left(0 + s \cdot 4\right) + \frac{x \cdot x}{s}}} \]

   7. Taylor expanded in s around 0 71.1%

      \[\leadsto \color{blue}{\frac{s}{{x}^{2}}} \]
   8. Step-by-step derivation

      1. *-un-lft-identity 71.1%

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

      2. unpow2 71.1%

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

      3. times-frac 71.1%

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

   9. Applied egg-rr 71.1%

      \[\leadsto \color{blue}{\frac{1}{x} \cdot \frac{s}{x}} \]
   10. Step-by-step derivation

       1. associate-*l/ 71.1%

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

       2. *-lft-identity 71.1%

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

   11. Simplified 71.1%

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

4. Final simplification 45.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.9999999494757503 \cdot 10^{-5}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{s}{x}}{x}\\ \end{array} \]

Alternative 12: 26.4% 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 99.6%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
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. *-commutative 99.6%

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

   3. fabs-neg 99.6%

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

   4. fabs-neg 99.6%

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

   5. associate-*r* 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. /-rgt-identity 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in s around inf 26.3%

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

5. Final simplification 26.3%

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

Reproduce

herbie shell --seed 2023319 
(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))))))