Logistic distribution

Percentage Accurate: 99.6% → 99.6%

Time: 12.6s

Alternatives: 16

Speedup: 1.2×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.6% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.8%

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

   1. *-lft-identity 99.8%

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

   2. associate-*r/ 99.8%

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

   3. associate-/l* 99.8%

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

   4. distribute-frac-neg 99.8%

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

   5. exp-neg 99.7%

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

   6. associate-/r/ 99.8%

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

   7. /-rgt-identity 99.8%

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

   8. associate-*l* 99.8%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 2: 99.4% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

   1. *-lft-identity 99.8%

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

   2. associate-*r/ 99.8%

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

   3. associate-*l* 99.8%

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

   4. times-frac 99.4%

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

   5. associate-*r/ 99.4%

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

   6. associate-/l* 99.4%

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

   7. distribute-frac-neg 99.4%

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

   8. exp-neg 99.5%

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

3. Simplified 99.5%

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

4. Final simplification 99.5%

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

Alternative 3: 97.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(x, s)
	tmp = (single(1.0) / s) / (exp((abs(x) / s)) + (single(2.0) + exp((x / s))));
end

Derivation

1. Initial program 99.8%

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

   1. *-lft-identity 99.8%

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

   2. associate-*r/ 99.8%

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

   3. associate-*l* 99.8%

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

   4. times-frac 99.4%

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

   5. associate-*r/ 99.4%

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

   6. associate-/l* 99.4%

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

   7. distribute-frac-neg 99.4%

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

   8. exp-neg 99.5%

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

3. Simplified 99.5%

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

   1. add-sqr-sqrt -0.0%

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

   2. sqrt-unprod 93.8%

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

   3. sqr-neg 93.8%

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

   4. sqrt-unprod 96.6%

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

   5. add-sqr-sqrt 96.6%

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

   6. *-un-lft-identity 96.6%

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

   7. div-inv 96.6%

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

   8. exp-prod 97.2%

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

   9. add-sqr-sqrt 97.2%

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

   10. sqrt-unprod 97.2%

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

   11. sqr-neg 97.2%

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

   12. sqrt-unprod -0.0%

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

   13. add-sqr-sqrt 97.6%

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

   14. exp-prod 99.5%

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

   15. div-inv 99.5%

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

5. Applied egg-rr 98.2%

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

   1. *-lft-identity 98.2%

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

7. Simplified 98.2%

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

8. Final simplification 98.2%

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

Alternative 4: 77.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{x}{s} \cdot \frac{x}{s}\right) \cdot -0.0625\\ \mathbf{if}\;\left|x\right| \leq 4.999999943633011 \cdot 10^{-27}:\\ \;\;\;\;\frac{\frac{0.0625 - t_0 \cdot t_0}{0.25 - t_0}}{s}\\ \mathbf{elif}\;\left|x\right| \leq 18000000319488:\\ \;\;\;\;\frac{1}{s \cdot \left(2 + 2 \cdot \left(1 + \left(\frac{x}{s} + \frac{0.5 \cdot \left(x \cdot x\right)}{s \cdot s}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{1}{x \cdot \frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (* (* (/ x s) (/ x s)) -0.0625)))
   (if (<= (fabs x) 4.999999943633011e-27)
     (/ (/ (- 0.0625 (* t_0 t_0)) (- 0.25 t_0)) s)
     (if (<= (fabs x) 18000000319488.0)
       (/
        1.0
        (* s (+ 2.0 (* 2.0 (+ 1.0 (+ (/ x s) (/ (* 0.5 (* x x)) (* s s))))))))
       (* 2.0 (/ 1.0 (* x (/ x s))))))))

C

float code(float x, float s) {
	float t_0 = ((x / s) * (x / s)) * -0.0625f;
	float tmp;
	if (fabsf(x) <= 4.999999943633011e-27f) {
		tmp = ((0.0625f - (t_0 * t_0)) / (0.25f - t_0)) / s;
	} else if (fabsf(x) <= 18000000319488.0f) {
		tmp = 1.0f / (s * (2.0f + (2.0f * (1.0f + ((x / s) + ((0.5f * (x * x)) / (s * s)))))));
	} else {
		tmp = 2.0f * (1.0f / (x * (x / s)));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: tmp
    t_0 = ((x / s) * (x / s)) * (-0.0625e0)
    if (abs(x) <= 4.999999943633011e-27) then
        tmp = ((0.0625e0 - (t_0 * t_0)) / (0.25e0 - t_0)) / s
    else if (abs(x) <= 18000000319488.0e0) then
        tmp = 1.0e0 / (s * (2.0e0 + (2.0e0 * (1.0e0 + ((x / s) + ((0.5e0 * (x * x)) / (s * s)))))))
    else
        tmp = 2.0e0 * (1.0e0 / (x * (x / s)))
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = Float32(Float32(Float32(x / s) * Float32(x / s)) * Float32(-0.0625))
	tmp = Float32(0.0)
	if (abs(x) <= Float32(4.999999943633011e-27))
		tmp = Float32(Float32(Float32(Float32(0.0625) - Float32(t_0 * t_0)) / Float32(Float32(0.25) - t_0)) / s);
	elseif (abs(x) <= Float32(18000000319488.0))
		tmp = Float32(Float32(1.0) / Float32(s * Float32(Float32(2.0) + Float32(Float32(2.0) * Float32(Float32(1.0) + Float32(Float32(x / s) + Float32(Float32(Float32(0.5) * Float32(x * x)) / Float32(s * s))))))));
	else
		tmp = Float32(Float32(2.0) * Float32(Float32(1.0) / Float32(x * Float32(x / s))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = ((x / s) * (x / s)) * single(-0.0625);
	tmp = single(0.0);
	if (abs(x) <= single(4.999999943633011e-27))
		tmp = ((single(0.0625) - (t_0 * t_0)) / (single(0.25) - t_0)) / s;
	elseif (abs(x) <= single(18000000319488.0))
		tmp = single(1.0) / (s * (single(2.0) + (single(2.0) * (single(1.0) + ((x / s) + ((single(0.5) * (x * x)) / (s * s)))))));
	else
		tmp = single(2.0) * (single(1.0) / (x * (x / s)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if (fabs.f32 x) < 4.99999994e-27

   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. associate-*l* 99.3%

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

      2. +-commutative 99.3%

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

      3. +-commutative 99.3%

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

   3. Simplified 99.3%

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

      1. add-exp-log 92.0%

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

      2. log-div 91.7%

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

      3. add-log-exp 91.7%

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

      4. add-sqr-sqrt -0.0%

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

      5. sqrt-unprod 73.8%

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

      6. sqr-neg 73.8%

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

      7. sqrt-unprod 73.1%

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

      8. add-sqr-sqrt 73.1%

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

      9. add-sqr-sqrt 53.7%

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

      10. fabs-sqr 53.7%

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

      11. add-sqr-sqrt 87.9%

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

      12. *-commutative 87.9%

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

   5. Applied egg-rr 91.1%

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

      1. associate--r+ 91.1%

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

      2. exp-diff 91.9%

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

      3. cancel-sign-sub-inv 91.9%

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

      4. metadata-eval 91.9%

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

      5. rem-exp-log 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in x around 0 62.8%

      \[\leadsto \frac{\color{blue}{0.25 + -0.0625 \cdot \frac{{x}^{2}}{{s}^{2}}}}{s} \]
   9. Step-by-step derivation

      1. unpow2 62.8%

         \[\leadsto \frac{0.25 + -0.0625 \cdot \frac{\color{blue}{x \cdot x}}{{s}^{2}}}{s} \]

      2. unpow2 62.8%

         \[\leadsto \frac{0.25 + -0.0625 \cdot \frac{x \cdot x}{\color{blue}{s \cdot s}}}{s} \]

   10. Simplified 62.8%

       \[\leadsto \frac{\color{blue}{0.25 + -0.0625 \cdot \frac{x \cdot x}{s \cdot s}}}{s} \]
   11. Step-by-step derivation

       1. flip-+ 62.8%

          \[\leadsto \frac{\color{blue}{\frac{0.25 \cdot 0.25 - \left(-0.0625 \cdot \frac{x \cdot x}{s \cdot s}\right) \cdot \left(-0.0625 \cdot \frac{x \cdot x}{s \cdot s}\right)}{0.25 - -0.0625 \cdot \frac{x \cdot x}{s \cdot s}}}}{s} \]

       2. metadata-eval 62.8%

          \[\leadsto \frac{\frac{\color{blue}{0.0625} - \left(-0.0625 \cdot \frac{x \cdot x}{s \cdot s}\right) \cdot \left(-0.0625 \cdot \frac{x \cdot x}{s \cdot s}\right)}{0.25 - -0.0625 \cdot \frac{x \cdot x}{s \cdot s}}}{s} \]

       3. *-commutative 62.8%

          \[\leadsto \frac{\frac{0.0625 - \color{blue}{\left(\frac{x \cdot x}{s \cdot s} \cdot -0.0625\right)} \cdot \left(-0.0625 \cdot \frac{x \cdot x}{s \cdot s}\right)}{0.25 - -0.0625 \cdot \frac{x \cdot x}{s \cdot s}}}{s} \]

       4. times-frac 62.8%

          \[\leadsto \frac{\frac{0.0625 - \left(\color{blue}{\left(\frac{x}{s} \cdot \frac{x}{s}\right)} \cdot -0.0625\right) \cdot \left(-0.0625 \cdot \frac{x \cdot x}{s \cdot s}\right)}{0.25 - -0.0625 \cdot \frac{x \cdot x}{s \cdot s}}}{s} \]

       5. *-commutative 62.8%

          \[\leadsto \frac{\frac{0.0625 - \left(\left(\frac{x}{s} \cdot \frac{x}{s}\right) \cdot -0.0625\right) \cdot \color{blue}{\left(\frac{x \cdot x}{s \cdot s} \cdot -0.0625\right)}}{0.25 - -0.0625 \cdot \frac{x \cdot x}{s \cdot s}}}{s} \]

       6. times-frac 62.8%

          \[\leadsto \frac{\frac{0.0625 - \left(\left(\frac{x}{s} \cdot \frac{x}{s}\right) \cdot -0.0625\right) \cdot \left(\color{blue}{\left(\frac{x}{s} \cdot \frac{x}{s}\right)} \cdot -0.0625\right)}{0.25 - -0.0625 \cdot \frac{x \cdot x}{s \cdot s}}}{s} \]

       7. *-commutative 62.8%

          \[\leadsto \frac{\frac{0.0625 - \left(\left(\frac{x}{s} \cdot \frac{x}{s}\right) \cdot -0.0625\right) \cdot \left(\left(\frac{x}{s} \cdot \frac{x}{s}\right) \cdot -0.0625\right)}{0.25 - \color{blue}{\frac{x \cdot x}{s \cdot s} \cdot -0.0625}}}{s} \]

       8. times-frac 84.4%

          \[\leadsto \frac{\frac{0.0625 - \left(\left(\frac{x}{s} \cdot \frac{x}{s}\right) \cdot -0.0625\right) \cdot \left(\left(\frac{x}{s} \cdot \frac{x}{s}\right) \cdot -0.0625\right)}{0.25 - \color{blue}{\left(\frac{x}{s} \cdot \frac{x}{s}\right)} \cdot -0.0625}}{s} \]

   12. Applied egg-rr 84.4%

       \[\leadsto \frac{\color{blue}{\frac{0.0625 - \left(\left(\frac{x}{s} \cdot \frac{x}{s}\right) \cdot -0.0625\right) \cdot \left(\left(\frac{x}{s} \cdot \frac{x}{s}\right) \cdot -0.0625\right)}{0.25 - \left(\frac{x}{s} \cdot \frac{x}{s}\right) \cdot -0.0625}}}{s} \]

   if 4.99999994e-27 < (fabs.f32 x) < 1.80000003e13

   1. Initial program 99.8%

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

      1. *-lft-identity 99.8%

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

      2. associate-*r/ 99.8%

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

      3. associate-*l* 99.9%

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

      4. times-frac 99.8%

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

      5. associate-*r/ 99.8%

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

      6. associate-/l* 99.8%

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

      7. distribute-frac-neg 99.8%

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

      8. exp-neg 99.9%

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

   3. Simplified 99.9%

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

      1. add-sqr-sqrt -0.0%

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

      2. sqrt-unprod 96.8%

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

      3. sqr-neg 96.8%

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

      4. sqrt-unprod 97.0%

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

      5. add-sqr-sqrt 97.0%

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

      6. *-un-lft-identity 97.0%

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

      7. div-inv 97.0%

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

      8. exp-prod 97.7%

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

      9. add-sqr-sqrt 97.7%

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

      10. sqrt-unprod 97.7%

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

      11. sqr-neg 97.7%

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

      12. sqrt-unprod -0.0%

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

      13. add-sqr-sqrt 98.5%

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

      14. exp-prod 99.9%

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

      15. div-inv 99.9%

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

   5. Applied egg-rr 98.0%

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

      1. *-lft-identity 98.0%

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

   7. Simplified 98.0%

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

      1. expm1-log1p-u 96.9%

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

      2. expm1-udef 96.9%

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

      3. add-sqr-sqrt 53.6%

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

      4. fabs-sqr 53.6%

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

      5. add-sqr-sqrt 62.4%

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

   9. Applied egg-rr 62.4%

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

       1. expm1-def 62.4%

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

       2. expm1-log1p 63.4%

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

       3. associate-/r* 63.4%

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

       4. associate-+r+ 63.4%

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

       5. +-commutative 63.4%

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

       6. count-2 63.4%

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

       7. *-commutative 63.4%

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

   11. Simplified 63.4%

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

   12. Taylor expanded in x around 0 67.9%

       \[\leadsto \frac{1}{s \cdot \left(2 + \color{blue}{\left(1 + \left(\frac{x}{s} + 0.5 \cdot \frac{{x}^{2}}{{s}^{2}}\right)\right)} \cdot 2\right)} \]
   13. Step-by-step derivation

       1. associate-*r/ 67.9%

          \[\leadsto \frac{1}{s \cdot \left(2 + \left(1 + \left(\frac{x}{s} + \color{blue}{\frac{0.5 \cdot {x}^{2}}{{s}^{2}}}\right)\right) \cdot 2\right)} \]

       2. unpow2 67.9%

          \[\leadsto \frac{1}{s \cdot \left(2 + \left(1 + \left(\frac{x}{s} + \frac{0.5 \cdot {x}^{2}}{\color{blue}{s \cdot s}}\right)\right) \cdot 2\right)} \]

       3. unpow2 67.9%

          \[\leadsto \frac{1}{s \cdot \left(2 + \left(1 + \left(\frac{x}{s} + \frac{0.5 \cdot \color{blue}{\left(x \cdot x\right)}}{s \cdot s}\right)\right) \cdot 2\right)} \]

   14. Simplified 67.9%

       \[\leadsto \frac{1}{s \cdot \left(2 + \color{blue}{\left(1 + \left(\frac{x}{s} + \frac{0.5 \cdot \left(x \cdot x\right)}{s \cdot s}\right)\right)} \cdot 2\right)} \]

   if 1.80000003e13 < (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. *-lft-identity 100.0%

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

      2. associate-*r/ 100.0%

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

      3. associate-*l* 100.0%

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

      4. times-frac 98.8%

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

      5. associate-*r/ 98.8%

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

      6. associate-/l* 98.8%

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

      7. distribute-frac-neg 98.8%

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

      8. exp-neg 98.8%

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

   3. Simplified 98.8%

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

      1. add-sqr-sqrt -0.0%

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

      2. sqrt-unprod 98.8%

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

      3. sqr-neg 98.8%

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

      4. sqrt-unprod 98.8%

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

      5. add-sqr-sqrt 98.8%

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

      6. *-un-lft-identity 98.8%

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

      7. div-inv 98.8%

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

      8. exp-prod 98.8%

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

      9. add-sqr-sqrt 98.8%

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

      10. sqrt-unprod 98.8%

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

      11. sqr-neg 98.8%

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

      12. sqrt-unprod -0.0%

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

      13. add-sqr-sqrt 98.8%

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

      14. exp-prod 98.8%

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

      15. div-inv 98.8%

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

   5. Applied egg-rr 98.8%

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

      1. *-lft-identity 98.8%

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

   7. Simplified 98.8%

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

   8. Taylor expanded in x around 0 63.5%

      \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \color{blue}{\left(3 + \left(\frac{x}{s} + 0.5 \cdot \frac{{x}^{2}}{{s}^{2}}\right)\right)}} \]
   9. Step-by-step derivation

      1. associate-*r/ 63.5%

         \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(3 + \left(\frac{x}{s} + \color{blue}{\frac{0.5 \cdot {x}^{2}}{{s}^{2}}}\right)\right)} \]

      2. unpow2 63.5%

         \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(3 + \left(\frac{x}{s} + \frac{0.5 \cdot {x}^{2}}{\color{blue}{s \cdot s}}\right)\right)} \]

      3. unpow2 63.5%

         \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(3 + \left(\frac{x}{s} + \frac{0.5 \cdot \color{blue}{\left(x \cdot x\right)}}{s \cdot s}\right)\right)} \]

   10. Simplified 63.5%

       \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \color{blue}{\left(3 + \left(\frac{x}{s} + \frac{0.5 \cdot \left(x \cdot x\right)}{s \cdot s}\right)\right)}} \]

   11. Taylor expanded in s around 0 92.4%

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

       1. unpow2 92.4%

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

   13. Simplified 92.4%

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

       1. clear-num 94.3%

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

       2. inv-pow 94.3%

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

   15. Applied egg-rr 94.3%

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

       1. unpow-1 94.3%

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

       2. associate-*r/ 94.3%

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

   17. Simplified 94.3%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 4.999999943633011 \cdot 10^{-27}:\\ \;\;\;\;\frac{\frac{0.0625 - \left(\left(\frac{x}{s} \cdot \frac{x}{s}\right) \cdot -0.0625\right) \cdot \left(\left(\frac{x}{s} \cdot \frac{x}{s}\right) \cdot -0.0625\right)}{0.25 - \left(\frac{x}{s} \cdot \frac{x}{s}\right) \cdot -0.0625}}{s}\\ \mathbf{elif}\;\left|x\right| \leq 18000000319488:\\ \;\;\;\;\frac{1}{s \cdot \left(2 + 2 \cdot \left(1 + \left(\frac{x}{s} + \frac{0.5 \cdot \left(x \cdot x\right)}{s \cdot s}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{1}{x \cdot \frac{x}{s}}\\ \end{array} \]

Alternative 5: 96.4% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

   1. *-lft-identity 99.8%

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

   2. associate-*r/ 99.8%

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

   3. associate-*l* 99.8%

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

   4. times-frac 99.4%

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

   5. associate-*r/ 99.4%

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

   6. associate-/l* 99.4%

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

   7. distribute-frac-neg 99.4%

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

   8. exp-neg 99.5%

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

3. Simplified 99.5%

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

   1. add-sqr-sqrt -0.0%

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

   2. sqrt-unprod 93.8%

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

   3. sqr-neg 93.8%

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

   4. sqrt-unprod 96.6%

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

   5. add-sqr-sqrt 96.6%

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

   6. *-un-lft-identity 96.6%

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

   7. div-inv 96.6%

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

   8. exp-prod 97.2%

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

   9. add-sqr-sqrt 97.2%

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

   10. sqrt-unprod 97.2%

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

   11. sqr-neg 97.2%

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

   12. sqrt-unprod -0.0%

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

   13. add-sqr-sqrt 97.6%

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

   14. exp-prod 99.5%

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

   15. div-inv 99.5%

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

5. Applied egg-rr 98.2%

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

   1. *-lft-identity 98.2%

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

7. Simplified 98.2%

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

8. Taylor expanded in x around 0 97.6%

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

9. Final simplification 97.6%

   \[\leadsto \frac{1}{s \cdot \left(e^{\frac{\left|x\right|}{s}} + 3\right)} \]

Alternative 6: 94.8% 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(Float32(-abs(x)) / 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 99.8%

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

   1. associate-*l* 99.8%

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

   2. +-commutative 99.8%

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

   3. +-commutative 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in s around inf 96.7%

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

5. Final simplification 96.7%

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

Alternative 7: 94.4% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.600000032318924 \cdot 10^{-27}:\\ \;\;\;\;\frac{1}{s \cdot \left(2 + 2 \cdot e^{-\frac{x}{s}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot 2 + s \cdot \left(2 \cdot e^{\frac{x}{s}}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 2.600000032318924e-27)
   (/ 1.0 (* s (+ 2.0 (* 2.0 (exp (- (/ x s)))))))
   (/ 1.0 (+ (* s 2.0) (* s (* 2.0 (exp (/ x s))))))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 2.600000032318924e-27f) {
		tmp = 1.0f / (s * (2.0f + (2.0f * expf(-(x / s)))));
	} else {
		tmp = 1.0f / ((s * 2.0f) + (s * (2.0f * expf((x / s)))));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 2.600000032318924e-27) then
        tmp = 1.0e0 / (s * (2.0e0 + (2.0e0 * exp(-(x / s)))))
    else
        tmp = 1.0e0 / ((s * 2.0e0) + (s * (2.0e0 * exp((x / s)))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(2.600000032318924e-27))
		tmp = Float32(Float32(1.0) / Float32(s * Float32(Float32(2.0) + Float32(Float32(2.0) * exp(Float32(-Float32(x / s)))))));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(s * Float32(2.0)) + Float32(s * Float32(Float32(2.0) * exp(Float32(x / s))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(2.600000032318924e-27))
		tmp = single(1.0) / (s * (single(2.0) + (single(2.0) * exp(-(x / s)))));
	else
		tmp = single(1.0) / ((s * single(2.0)) + (s * (single(2.0) * exp((x / s)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 2.60000003e-27

   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. *-lft-identity 99.7%

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

      2. associate-*r/ 99.7%

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

      3. associate-*l* 99.8%

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

      4. times-frac 99.8%

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

      5. associate-*r/ 99.8%

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

      6. associate-/l* 99.8%

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

      7. distribute-frac-neg 99.8%

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

      8. exp-neg 99.9%

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

   3. Simplified 99.9%

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

      1. add-sqr-sqrt -0.0%

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

      2. sqrt-unprod 91.6%

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

      3. sqr-neg 91.6%

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

      4. sqrt-unprod 96.4%

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

      5. add-sqr-sqrt 96.4%

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

      6. *-un-lft-identity 96.4%

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

      7. div-inv 96.4%

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

      8. exp-prod 96.9%

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

      9. add-sqr-sqrt 96.9%

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

      10. sqrt-unprod 96.9%

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

      11. sqr-neg 96.9%

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

      12. sqrt-unprod -0.0%

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

      13. add-sqr-sqrt 97.5%

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

      14. exp-prod 99.9%

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

      15. div-inv 99.9%

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

   5. Applied egg-rr 99.3%

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

      1. *-lft-identity 99.3%

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

   7. Simplified 99.3%

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

      1. expm1-log1p-u 96.9%

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

      2. expm1-udef 97.0%

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

      3. add-sqr-sqrt 15.5%

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

      4. fabs-sqr 15.5%

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

      5. add-sqr-sqrt 31.0%

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

   9. Applied egg-rr 31.0%

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

       1. expm1-def 31.0%

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

       2. expm1-log1p 32.6%

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

       3. associate-/r* 32.6%

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

       4. associate-+r+ 32.6%

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

       5. +-commutative 32.6%

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

       6. count-2 32.6%

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

       7. *-commutative 32.6%

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

   11. Simplified 32.6%

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

       1. div-inv 32.6%

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

       2. exp-prod 34.2%

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

       3. add-sqr-sqrt 17.3%

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

       4. fabs-sqr 17.3%

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

       5. add-sqr-sqrt 85.2%

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

       6. add-sqr-sqrt 85.2%

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

       7. sqrt-unprod 85.2%

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

       8. sqr-neg 85.2%

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

       9. sqrt-unprod -0.0%

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

       10. add-sqr-sqrt 34.2%

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

       11. exp-prod 32.7%

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

       12. div-inv 32.7%

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

       13. distribute-frac-neg 32.7%

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

       14. exp-neg 32.6%

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

       15. add-sqr-sqrt 16.7%

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

       16. fabs-sqr 16.7%

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

       17. add-sqr-sqrt 96.4%

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

       18. add-sqr-sqrt 16.7%

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

       19. add-sqr-sqrt 96.4%

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

   13. Applied egg-rr 96.4%

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

       1. rec-exp 96.4%

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

       2. distribute-frac-neg 96.4%

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

   15. Simplified 96.4%

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

   if 2.60000003e-27 < x

   1. Initial program 99.8%

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

      1. *-lft-identity 99.8%

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

      2. associate-*r/ 99.8%

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

      3. associate-*l* 99.9%

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

      4. times-frac 99.0%

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

      5. associate-*r/ 99.0%

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

      6. associate-/l* 98.9%

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

      7. distribute-frac-neg 98.9%

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

      8. exp-neg 99.0%

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

   3. Simplified 99.0%

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

      1. add-sqr-sqrt -0.0%

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

      2. sqrt-unprod 96.6%

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

      3. sqr-neg 96.6%

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

      4. sqrt-unprod 96.8%

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

      5. add-sqr-sqrt 96.8%

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

      6. *-un-lft-identity 96.8%

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

      7. div-inv 96.8%

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

      8. exp-prod 97.5%

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

      9. add-sqr-sqrt 97.5%

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

      10. sqrt-unprod 97.5%

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

      11. sqr-neg 97.5%

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

      12. sqrt-unprod -0.0%

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

      13. add-sqr-sqrt 97.7%

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

      14. exp-prod 99.0%

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

      15. div-inv 99.0%

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

   5. Applied egg-rr 96.8%

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

      1. *-lft-identity 96.8%

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

   7. Simplified 96.8%

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

      1. expm1-log1p-u 96.0%

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

      2. expm1-udef 96.0%

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

      3. add-sqr-sqrt 96.0%

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

      4. fabs-sqr 96.0%

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

      5. add-sqr-sqrt 96.0%

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

   9. Applied egg-rr 96.0%

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

       1. expm1-def 96.0%

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

       2. expm1-log1p 96.8%

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

       3. associate-/r* 97.7%

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

       4. associate-+r+ 97.7%

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

       5. +-commutative 97.7%

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

       6. count-2 97.7%

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

       7. *-commutative 97.7%

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

   11. Simplified 97.7%

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

       1. distribute-lft-in 97.7%

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

       2. *-commutative 97.7%

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

   13. Applied egg-rr 97.7%

       \[\leadsto \frac{1}{\color{blue}{s \cdot 2 + s \cdot \left(2 \cdot e^{\frac{x}{s}}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.600000032318924 \cdot 10^{-27}:\\ \;\;\;\;\frac{1}{s \cdot \left(2 + 2 \cdot e^{-\frac{x}{s}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot 2 + s \cdot \left(2 \cdot e^{\frac{x}{s}}\right)}\\ \end{array} \]

Alternative 8: 94.4% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.600000032318924 \cdot 10^{-27}:\\ \;\;\;\;\frac{1}{s \cdot \left(2 + 2 \cdot e^{-\frac{x}{s}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(2 + 2 \cdot e^{\frac{x}{s}}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 2.600000032318924e-27)
   (/ 1.0 (* s (+ 2.0 (* 2.0 (exp (- (/ x s)))))))
   (/ 1.0 (* s (+ 2.0 (* 2.0 (exp (/ x s))))))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 2.600000032318924e-27f) {
		tmp = 1.0f / (s * (2.0f + (2.0f * expf(-(x / s)))));
	} else {
		tmp = 1.0f / (s * (2.0f + (2.0f * expf((x / s)))));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 2.600000032318924e-27) then
        tmp = 1.0e0 / (s * (2.0e0 + (2.0e0 * exp(-(x / s)))))
    else
        tmp = 1.0e0 / (s * (2.0e0 + (2.0e0 * exp((x / s)))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(2.600000032318924e-27))
		tmp = Float32(Float32(1.0) / Float32(s * Float32(Float32(2.0) + Float32(Float32(2.0) * exp(Float32(-Float32(x / s)))))));
	else
		tmp = Float32(Float32(1.0) / Float32(s * Float32(Float32(2.0) + Float32(Float32(2.0) * exp(Float32(x / s))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(2.600000032318924e-27))
		tmp = single(1.0) / (s * (single(2.0) + (single(2.0) * exp(-(x / s)))));
	else
		tmp = single(1.0) / (s * (single(2.0) + (single(2.0) * exp((x / s)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 2.60000003e-27

   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. *-lft-identity 99.7%

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

      2. associate-*r/ 99.7%

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

      3. associate-*l* 99.8%

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

      4. times-frac 99.8%

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

      5. associate-*r/ 99.8%

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

      6. associate-/l* 99.8%

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

      7. distribute-frac-neg 99.8%

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

      8. exp-neg 99.9%

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

   3. Simplified 99.9%

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

      1. add-sqr-sqrt -0.0%

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

      2. sqrt-unprod 91.6%

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

      3. sqr-neg 91.6%

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

      4. sqrt-unprod 96.4%

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

      5. add-sqr-sqrt 96.4%

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

      6. *-un-lft-identity 96.4%

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

      7. div-inv 96.4%

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

      8. exp-prod 96.9%

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

      9. add-sqr-sqrt 96.9%

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

      10. sqrt-unprod 96.9%

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

      11. sqr-neg 96.9%

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

      12. sqrt-unprod -0.0%

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

      13. add-sqr-sqrt 97.5%

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

      14. exp-prod 99.9%

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

      15. div-inv 99.9%

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

   5. Applied egg-rr 99.3%

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

      1. *-lft-identity 99.3%

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

   7. Simplified 99.3%

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

      1. expm1-log1p-u 96.9%

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

      2. expm1-udef 97.0%

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

      3. add-sqr-sqrt 15.5%

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

      4. fabs-sqr 15.5%

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

      5. add-sqr-sqrt 31.0%

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

   9. Applied egg-rr 31.0%

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

       1. expm1-def 31.0%

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

       2. expm1-log1p 32.6%

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

       3. associate-/r* 32.6%

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

       4. associate-+r+ 32.6%

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

       5. +-commutative 32.6%

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

       6. count-2 32.6%

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

       7. *-commutative 32.6%

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

   11. Simplified 32.6%

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

       1. div-inv 32.6%

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

       2. exp-prod 34.2%

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

       3. add-sqr-sqrt 17.3%

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

       4. fabs-sqr 17.3%

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

       5. add-sqr-sqrt 85.2%

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

       6. add-sqr-sqrt 85.2%

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

       7. sqrt-unprod 85.2%

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

       8. sqr-neg 85.2%

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

       9. sqrt-unprod -0.0%

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

       10. add-sqr-sqrt 34.2%

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

       11. exp-prod 32.7%

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

       12. div-inv 32.7%

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

       13. distribute-frac-neg 32.7%

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

       14. exp-neg 32.6%

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

       15. add-sqr-sqrt 16.7%

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

       16. fabs-sqr 16.7%

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

       17. add-sqr-sqrt 96.4%

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

       18. add-sqr-sqrt 16.7%

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

       19. add-sqr-sqrt 96.4%

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

   13. Applied egg-rr 96.4%

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

       1. rec-exp 96.4%

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

       2. distribute-frac-neg 96.4%

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

   15. Simplified 96.4%

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

   if 2.60000003e-27 < x

   1. Initial program 99.8%

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

      1. *-lft-identity 99.8%

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

      2. associate-*r/ 99.8%

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

      3. associate-*l* 99.9%

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

      4. times-frac 99.0%

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

      5. associate-*r/ 99.0%

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

      6. associate-/l* 98.9%

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

      7. distribute-frac-neg 98.9%

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

      8. exp-neg 99.0%

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

   3. Simplified 99.0%

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

      1. expm1-log1p-u 97.9%

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

      2. expm1-udef 97.9%

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

   5. Applied egg-rr 96.9%

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

      1. expm1-def 96.9%

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

      2. expm1-log1p 97.7%

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

      3. *-commutative 97.7%

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

      4. +-commutative 97.7%

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

      5. +-commutative 97.7%

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

      6. associate-+l+ 97.7%

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

      7. count-2 97.7%

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

   7. Simplified 97.7%

      \[\leadsto \color{blue}{\frac{1}{s \cdot \left(2 + 2 \cdot e^{\frac{x}{s}}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.600000032318924 \cdot 10^{-27}:\\ \;\;\;\;\frac{1}{s \cdot \left(2 + 2 \cdot e^{-\frac{x}{s}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(2 + 2 \cdot e^{\frac{x}{s}}\right)}\\ \end{array} \]

Alternative 9: 95.0% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \mathbf{if}\;x \leq -1.000000046701102 \cdot 10^{-34}:\\ \;\;\;\;\frac{t_0}{s \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(2 + 2 \cdot t_0\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ x s))))
   (if (<= x -1.000000046701102e-34)
     (/ t_0 (* s 4.0))
     (/ 1.0 (* s (+ 2.0 (* 2.0 t_0)))))))

C

float code(float x, float s) {
	float t_0 = expf((x / s));
	float tmp;
	if (x <= -1.000000046701102e-34f) {
		tmp = t_0 / (s * 4.0f);
	} else {
		tmp = 1.0f / (s * (2.0f + (2.0f * t_0)));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: tmp
    t_0 = exp((x / s))
    if (x <= (-1.000000046701102e-34)) then
        tmp = t_0 / (s * 4.0e0)
    else
        tmp = 1.0e0 / (s * (2.0e0 + (2.0e0 * t_0)))
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = exp(Float32(x / s))
	tmp = Float32(0.0)
	if (x <= Float32(-1.000000046701102e-34))
		tmp = Float32(t_0 / Float32(s * Float32(4.0)));
	else
		tmp = Float32(Float32(1.0) / Float32(s * Float32(Float32(2.0) + Float32(Float32(2.0) * t_0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = exp((x / s));
	tmp = single(0.0);
	if (x <= single(-1.000000046701102e-34))
		tmp = t_0 / (s * single(4.0));
	else
		tmp = single(1.0) / (s * (single(2.0) + (single(2.0) * t_0)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -1.00000005e-34

   1. Initial program 99.8%

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

      1. associate-*l* 99.9%

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

      2. +-commutative 99.9%

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

      3. +-commutative 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in s around inf 96.6%

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

      1. expm1-log1p-u 96.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{e^{\frac{-\left|x\right|}{s}}}{s \cdot 4}\right)\right)} \]

      2. expm1-udef 96.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{e^{\frac{-\left|x\right|}{s}}}{s \cdot 4}\right)} - 1} \]

   6. Applied egg-rr 96.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{e^{\frac{x}{s}}}{s \cdot 4}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 96.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{e^{\frac{x}{s}}}{s \cdot 4}\right)\right)} \]

      2. expm1-log1p 96.6%

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

   8. Simplified 96.6%

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

   if -1.00000005e-34 < x

   1. Initial program 99.7%

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

      1. *-lft-identity 99.7%

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

      2. associate-*r/ 99.7%

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

      3. associate-*l* 99.8%

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

      4. times-frac 99.1%

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

      5. associate-*r/ 99.1%

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

      6. associate-/l* 99.1%

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

      7. distribute-frac-neg 99.1%

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

      8. exp-neg 99.1%

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

   3. Simplified 99.2%

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

      1. expm1-log1p-u 96.5%

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

      2. expm1-udef 96.4%

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

   5. Applied egg-rr 95.1%

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

      1. expm1-def 95.2%

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

      2. expm1-log1p 97.1%

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

      3. *-commutative 97.1%

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

      4. +-commutative 97.1%

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

      5. +-commutative 97.1%

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

      6. associate-+l+ 97.1%

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

      7. count-2 97.1%

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

   7. Simplified 97.1%

      \[\leadsto \color{blue}{\frac{1}{s \cdot \left(2 + 2 \cdot e^{\frac{x}{s}}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.000000046701102 \cdot 10^{-34}:\\ \;\;\;\;\frac{e^{\frac{x}{s}}}{s \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(2 + 2 \cdot e^{\frac{x}{s}}\right)}\\ \end{array} \]

Alternative 10: 86.9% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.999999943633011 \cdot 10^{-27}:\\ \;\;\;\;\frac{e^{\frac{x}{s}}}{s \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(2 + 2 \cdot \left(1 + \left(\frac{x}{s} + \frac{0.5 \cdot \left(x \cdot x\right)}{s \cdot s}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 4.999999943633011e-27)
   (/ (exp (/ x s)) (* s 4.0))
   (/
    1.0
    (* s (+ 2.0 (* 2.0 (+ 1.0 (+ (/ x s) (/ (* 0.5 (* x x)) (* s s))))))))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 4.999999943633011e-27f) {
		tmp = expf((x / s)) / (s * 4.0f);
	} else {
		tmp = 1.0f / (s * (2.0f + (2.0f * (1.0f + ((x / s) + ((0.5f * (x * x)) / (s * s)))))));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 4.999999943633011e-27) then
        tmp = exp((x / s)) / (s * 4.0e0)
    else
        tmp = 1.0e0 / (s * (2.0e0 + (2.0e0 * (1.0e0 + ((x / s) + ((0.5e0 * (x * x)) / (s * s)))))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(4.999999943633011e-27))
		tmp = Float32(exp(Float32(x / s)) / Float32(s * Float32(4.0)));
	else
		tmp = Float32(Float32(1.0) / Float32(s * Float32(Float32(2.0) + Float32(Float32(2.0) * Float32(Float32(1.0) + Float32(Float32(x / s) + Float32(Float32(Float32(0.5) * Float32(x * x)) / Float32(s * s))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(4.999999943633011e-27))
		tmp = exp((x / s)) / (s * single(4.0));
	else
		tmp = single(1.0) / (s * (single(2.0) + (single(2.0) * (single(1.0) + ((x / s) + ((single(0.5) * (x * x)) / (s * s)))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 4.99999994e-27

   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. associate-*l* 99.8%

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

      2. +-commutative 99.8%

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

      3. +-commutative 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in s around inf 96.1%

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

      1. expm1-log1p-u 94.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{e^{\frac{-\left|x\right|}{s}}}{s \cdot 4}\right)\right)} \]

      2. expm1-udef 94.5%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{e^{\frac{-\left|x\right|}{s}}}{s \cdot 4}\right)} - 1} \]

   6. Applied egg-rr 93.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{e^{\frac{x}{s}}}{s \cdot 4}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 93.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{e^{\frac{x}{s}}}{s \cdot 4}\right)\right)} \]

      2. expm1-log1p 95.4%

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

   8. Simplified 95.4%

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

   if 4.99999994e-27 < x

   1. Initial program 99.8%

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

      1. *-lft-identity 99.8%

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

      2. associate-*r/ 99.8%

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

      3. associate-*l* 99.9%

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

      4. times-frac 98.9%

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

      5. associate-*r/ 98.9%

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

      6. associate-/l* 98.9%

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

      7. distribute-frac-neg 98.9%

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

      8. exp-neg 99.0%

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

   3. Simplified 99.0%

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

      1. add-sqr-sqrt -0.0%

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

      2. sqrt-unprod 96.6%

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

      3. sqr-neg 96.6%

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

      4. sqrt-unprod 96.8%

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

      5. add-sqr-sqrt 96.8%

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

      6. *-un-lft-identity 96.8%

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

      7. div-inv 96.8%

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

      8. exp-prod 97.5%

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

      9. add-sqr-sqrt 97.5%

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

      10. sqrt-unprod 97.5%

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

      11. sqr-neg 97.5%

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

      12. sqrt-unprod -0.0%

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

      13. add-sqr-sqrt 97.7%

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

      14. exp-prod 99.0%

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

      15. div-inv 99.0%

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

   5. Applied egg-rr 96.8%

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

      1. *-lft-identity 96.8%

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

   7. Simplified 96.8%

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

      1. expm1-log1p-u 96.0%

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

      2. expm1-udef 96.0%

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

      3. add-sqr-sqrt 96.0%

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

      4. fabs-sqr 96.0%

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

      5. add-sqr-sqrt 96.0%

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

   9. Applied egg-rr 96.0%

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

       1. expm1-def 96.0%

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

       2. expm1-log1p 96.8%

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

       3. associate-/r* 97.7%

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

       4. associate-+r+ 97.7%

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

       5. +-commutative 97.7%

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

       6. count-2 97.7%

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

       7. *-commutative 97.7%

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

   11. Simplified 97.7%

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

   12. Taylor expanded in x around 0 79.1%

       \[\leadsto \frac{1}{s \cdot \left(2 + \color{blue}{\left(1 + \left(\frac{x}{s} + 0.5 \cdot \frac{{x}^{2}}{{s}^{2}}\right)\right)} \cdot 2\right)} \]
   13. Step-by-step derivation

       1. associate-*r/ 79.1%

          \[\leadsto \frac{1}{s \cdot \left(2 + \left(1 + \left(\frac{x}{s} + \color{blue}{\frac{0.5 \cdot {x}^{2}}{{s}^{2}}}\right)\right) \cdot 2\right)} \]

       2. unpow2 79.1%

          \[\leadsto \frac{1}{s \cdot \left(2 + \left(1 + \left(\frac{x}{s} + \frac{0.5 \cdot {x}^{2}}{\color{blue}{s \cdot s}}\right)\right) \cdot 2\right)} \]

       3. unpow2 79.1%

          \[\leadsto \frac{1}{s \cdot \left(2 + \left(1 + \left(\frac{x}{s} + \frac{0.5 \cdot \color{blue}{\left(x \cdot x\right)}}{s \cdot s}\right)\right) \cdot 2\right)} \]

   14. Simplified 79.1%

       \[\leadsto \frac{1}{s \cdot \left(2 + \color{blue}{\left(1 + \left(\frac{x}{s} + \frac{0.5 \cdot \left(x \cdot x\right)}{s \cdot s}\right)\right)} \cdot 2\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.999999943633011 \cdot 10^{-27}:\\ \;\;\;\;\frac{e^{\frac{x}{s}}}{s \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(2 + 2 \cdot \left(1 + \left(\frac{x}{s} + \frac{0.5 \cdot \left(x \cdot x\right)}{s \cdot s}\right)\right)\right)}\\ \end{array} \]

Alternative 11: 77.3% accurate, 21.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9999999827968:\\ \;\;\;\;2 \cdot \frac{1}{x \cdot \frac{x}{s}}\\ \mathbf{elif}\;x \leq -1.000000045813705 \cdot 10^{-18} \lor \neg \left(x \leq 4.999999943633011 \cdot 10^{-27}\right):\\ \;\;\;\;\frac{1}{s \cdot \left(2 + 2 \cdot \left(1 + \left(\frac{x}{s} + \frac{0.5 \cdot \left(x \cdot x\right)}{s \cdot s}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{x}{s} \cdot \frac{x}{s}\right) \cdot -0.0625 + 0.25}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -9999999827968.0)
   (* 2.0 (/ 1.0 (* x (/ x s))))
   (if (or (<= x -1.000000045813705e-18) (not (<= x 4.999999943633011e-27)))
     (/
      1.0
      (* s (+ 2.0 (* 2.0 (+ 1.0 (+ (/ x s) (/ (* 0.5 (* x x)) (* s s))))))))
     (/ (+ (* (* (/ x s) (/ x s)) -0.0625) 0.25) s))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -9999999827968.0f) {
		tmp = 2.0f * (1.0f / (x * (x / s)));
	} else if ((x <= -1.000000045813705e-18f) || !(x <= 4.999999943633011e-27f)) {
		tmp = 1.0f / (s * (2.0f + (2.0f * (1.0f + ((x / s) + ((0.5f * (x * x)) / (s * s)))))));
	} else {
		tmp = ((((x / s) * (x / s)) * -0.0625f) + 0.25f) / s;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-9999999827968.0e0)) then
        tmp = 2.0e0 * (1.0e0 / (x * (x / s)))
    else if ((x <= (-1.000000045813705e-18)) .or. (.not. (x <= 4.999999943633011e-27))) then
        tmp = 1.0e0 / (s * (2.0e0 + (2.0e0 * (1.0e0 + ((x / s) + ((0.5e0 * (x * x)) / (s * s)))))))
    else
        tmp = ((((x / s) * (x / s)) * (-0.0625e0)) + 0.25e0) / s
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-9999999827968.0))
		tmp = Float32(Float32(2.0) * Float32(Float32(1.0) / Float32(x * Float32(x / s))));
	elseif ((x <= Float32(-1.000000045813705e-18)) || !(x <= Float32(4.999999943633011e-27)))
		tmp = Float32(Float32(1.0) / Float32(s * Float32(Float32(2.0) + Float32(Float32(2.0) * Float32(Float32(1.0) + Float32(Float32(x / s) + Float32(Float32(Float32(0.5) * Float32(x * x)) / Float32(s * s))))))));
	else
		tmp = Float32(Float32(Float32(Float32(Float32(x / s) * Float32(x / s)) * Float32(-0.0625)) + Float32(0.25)) / s);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-9999999827968.0))
		tmp = single(2.0) * (single(1.0) / (x * (x / s)));
	elseif ((x <= single(-1.000000045813705e-18)) || ~((x <= single(4.999999943633011e-27))))
		tmp = single(1.0) / (s * (single(2.0) + (single(2.0) * (single(1.0) + ((x / s) + ((single(0.5) * (x * x)) / (s * s)))))));
	else
		tmp = ((((x / s) * (x / s)) * single(-0.0625)) + single(0.25)) / s;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -9999999830000

   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. *-lft-identity 100.0%

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

      2. associate-*r/ 100.0%

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

      3. associate-*l* 100.0%

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

      4. times-frac 100.0%

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

      5. associate-*r/ 100.0%

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

      6. associate-/l* 100.0%

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

      7. distribute-frac-neg 100.0%

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

      8. exp-neg 100.0%

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

   3. Simplified 100.0%

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

      1. add-sqr-sqrt -0.0%

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

      2. sqrt-unprod 100.0%

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

      3. sqr-neg 100.0%

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

      4. sqrt-unprod 100.0%

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

      5. add-sqr-sqrt 100.0%

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

      6. *-un-lft-identity 100.0%

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

      7. div-inv 100.0%

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

      8. exp-prod 100.0%

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

      9. add-sqr-sqrt 100.0%

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

      10. sqrt-unprod 100.0%

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

      11. sqr-neg 100.0%

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

      12. sqrt-unprod -0.0%

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

      13. add-sqr-sqrt 100.0%

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

      14. exp-prod 100.0%

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

      15. div-inv 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. *-lft-identity 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around 0 36.2%

      \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \color{blue}{\left(3 + \left(\frac{x}{s} + 0.5 \cdot \frac{{x}^{2}}{{s}^{2}}\right)\right)}} \]
   9. Step-by-step derivation

      1. associate-*r/ 36.2%

         \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(3 + \left(\frac{x}{s} + \color{blue}{\frac{0.5 \cdot {x}^{2}}{{s}^{2}}}\right)\right)} \]

      2. unpow2 36.2%

         \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(3 + \left(\frac{x}{s} + \frac{0.5 \cdot {x}^{2}}{\color{blue}{s \cdot s}}\right)\right)} \]

      3. unpow2 36.2%

         \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(3 + \left(\frac{x}{s} + \frac{0.5 \cdot \color{blue}{\left(x \cdot x\right)}}{s \cdot s}\right)\right)} \]

   10. Simplified 36.2%

       \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \color{blue}{\left(3 + \left(\frac{x}{s} + \frac{0.5 \cdot \left(x \cdot x\right)}{s \cdot s}\right)\right)}} \]

   11. Taylor expanded in s around 0 90.9%

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

       1. unpow2 90.9%

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

   13. Simplified 90.9%

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

       1. clear-num 91.4%

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

       2. inv-pow 91.4%

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

   15. Applied egg-rr 91.4%

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

       1. unpow-1 91.4%

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

       2. associate-*r/ 91.4%

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

   17. Simplified 91.4%

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

   if -9999999830000 < x < -1.00000005e-18 or 4.99999994e-27 < x

   1. Initial program 99.8%

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

      1. *-lft-identity 99.8%

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

      2. associate-*r/ 99.8%

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

      3. associate-*l* 99.9%

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

      4. times-frac 99.2%

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

      5. associate-*r/ 99.2%

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

      6. associate-/l* 99.2%

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

      7. distribute-frac-neg 99.2%

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

      8. exp-neg 99.3%

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

   3. Simplified 99.3%

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

      1. add-sqr-sqrt -0.0%

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

      2. sqrt-unprod 97.0%

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

      3. sqr-neg 97.0%

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

      4. sqrt-unprod 97.1%

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

      5. add-sqr-sqrt 97.1%

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

      6. *-un-lft-identity 97.1%

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

      7. div-inv 97.1%

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

      8. exp-prod 97.7%

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

      9. add-sqr-sqrt 97.7%

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

      10. sqrt-unprod 97.7%

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

      11. sqr-neg 97.7%

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

      12. sqrt-unprod -0.0%

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

      13. add-sqr-sqrt 98.3%

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

      14. exp-prod 99.3%

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

      15. div-inv 99.3%

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

   5. Applied egg-rr 97.8%

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

      1. *-lft-identity 97.8%

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

   7. Simplified 97.8%

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

      1. expm1-log1p-u 97.1%

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

      2. expm1-udef 97.1%

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

      3. add-sqr-sqrt 66.8%

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

      4. fabs-sqr 66.8%

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

      5. add-sqr-sqrt 69.9%

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

   9. Applied egg-rr 69.9%

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

       1. expm1-def 69.9%

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

       2. expm1-log1p 70.5%

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

       3. associate-/r* 71.2%

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

       4. associate-+r+ 71.2%

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

       5. +-commutative 71.2%

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

       6. count-2 71.2%

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

       7. *-commutative 71.2%

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

   11. Simplified 71.2%

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

   12. Taylor expanded in x around 0 74.3%

       \[\leadsto \frac{1}{s \cdot \left(2 + \color{blue}{\left(1 + \left(\frac{x}{s} + 0.5 \cdot \frac{{x}^{2}}{{s}^{2}}\right)\right)} \cdot 2\right)} \]
   13. Step-by-step derivation

       1. associate-*r/ 74.3%

          \[\leadsto \frac{1}{s \cdot \left(2 + \left(1 + \left(\frac{x}{s} + \color{blue}{\frac{0.5 \cdot {x}^{2}}{{s}^{2}}}\right)\right) \cdot 2\right)} \]

       2. unpow2 74.3%

          \[\leadsto \frac{1}{s \cdot \left(2 + \left(1 + \left(\frac{x}{s} + \frac{0.5 \cdot {x}^{2}}{\color{blue}{s \cdot s}}\right)\right) \cdot 2\right)} \]

       3. unpow2 74.3%

          \[\leadsto \frac{1}{s \cdot \left(2 + \left(1 + \left(\frac{x}{s} + \frac{0.5 \cdot \color{blue}{\left(x \cdot x\right)}}{s \cdot s}\right)\right) \cdot 2\right)} \]

   14. Simplified 74.3%

       \[\leadsto \frac{1}{s \cdot \left(2 + \color{blue}{\left(1 + \left(\frac{x}{s} + \frac{0.5 \cdot \left(x \cdot x\right)}{s \cdot s}\right)\right)} \cdot 2\right)} \]

   if -1.00000005e-18 < x < 4.99999994e-27

   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. associate-*l* 99.4%

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

      2. +-commutative 99.4%

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

      3. +-commutative 99.4%

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

   3. Simplified 99.4%

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

      1. add-exp-log 92.4%

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

      2. log-div 92.1%

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

      3. add-log-exp 92.1%

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

      4. add-sqr-sqrt -0.0%

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

      5. sqrt-unprod 74.5%

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

      6. sqr-neg 74.5%

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

      7. sqrt-unprod 73.9%

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

      8. add-sqr-sqrt 73.9%

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

      9. add-sqr-sqrt 45.3%

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

      10. fabs-sqr 45.3%

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

      11. add-sqr-sqrt 88.9%

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

      12. *-commutative 88.9%

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

   5. Applied egg-rr 91.2%

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

      1. associate--r+ 91.3%

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

      2. exp-diff 92.1%

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

      3. cancel-sign-sub-inv 92.1%

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

      4. metadata-eval 92.1%

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

      5. rem-exp-log 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in x around 0 66.7%

      \[\leadsto \frac{\color{blue}{0.25 + -0.0625 \cdot \frac{{x}^{2}}{{s}^{2}}}}{s} \]
   9. Step-by-step derivation

      1. unpow2 66.7%

         \[\leadsto \frac{0.25 + -0.0625 \cdot \frac{\color{blue}{x \cdot x}}{{s}^{2}}}{s} \]

      2. unpow2 66.7%

         \[\leadsto \frac{0.25 + -0.0625 \cdot \frac{x \cdot x}{\color{blue}{s \cdot s}}}{s} \]

   10. Simplified 66.7%

       \[\leadsto \frac{\color{blue}{0.25 + -0.0625 \cdot \frac{x \cdot x}{s \cdot s}}}{s} \]
   11. Step-by-step derivation

       1. times-frac 85.0%

          \[\leadsto \frac{0.25 + -0.0625 \cdot \color{blue}{\left(\frac{x}{s} \cdot \frac{x}{s}\right)}}{s} \]

   12. Applied egg-rr 85.0%

       \[\leadsto \frac{0.25 + -0.0625 \cdot \color{blue}{\left(\frac{x}{s} \cdot \frac{x}{s}\right)}}{s} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9999999827968:\\ \;\;\;\;2 \cdot \frac{1}{x \cdot \frac{x}{s}}\\ \mathbf{elif}\;x \leq -1.000000045813705 \cdot 10^{-18} \lor \neg \left(x \leq 4.999999943633011 \cdot 10^{-27}\right):\\ \;\;\;\;\frac{1}{s \cdot \left(2 + 2 \cdot \left(1 + \left(\frac{x}{s} + \frac{0.5 \cdot \left(x \cdot x\right)}{s \cdot s}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{x}{s} \cdot \frac{x}{s}\right) \cdot -0.0625 + 0.25}{s}\\ \end{array} \]

Alternative 12: 63.2% accurate, 35.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.999999974752427 \cdot 10^{-7} \lor \neg \left(x \leq 1.9999999920083944 \cdot 10^{-11}\right):\\ \;\;\;\;2 \cdot \frac{1}{x \cdot \frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{x}{s} \cdot \frac{x}{s}\right) \cdot -0.0625 + 0.25}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (or (<= x -9.999999974752427e-7) (not (<= x 1.9999999920083944e-11)))
   (* 2.0 (/ 1.0 (* x (/ x s))))
   (/ (+ (* (* (/ x s) (/ x s)) -0.0625) 0.25) s)))

C

float code(float x, float s) {
	float tmp;
	if ((x <= -9.999999974752427e-7f) || !(x <= 1.9999999920083944e-11f)) {
		tmp = 2.0f * (1.0f / (x * (x / s)));
	} else {
		tmp = ((((x / s) * (x / s)) * -0.0625f) + 0.25f) / s;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((x <= (-9.999999974752427e-7)) .or. (.not. (x <= 1.9999999920083944e-11))) then
        tmp = 2.0e0 * (1.0e0 / (x * (x / s)))
    else
        tmp = ((((x / s) * (x / s)) * (-0.0625e0)) + 0.25e0) / s
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if ((x <= Float32(-9.999999974752427e-7)) || !(x <= Float32(1.9999999920083944e-11)))
		tmp = Float32(Float32(2.0) * Float32(Float32(1.0) / Float32(x * Float32(x / s))));
	else
		tmp = Float32(Float32(Float32(Float32(Float32(x / s) * Float32(x / s)) * Float32(-0.0625)) + Float32(0.25)) / s);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((x <= single(-9.999999974752427e-7)) || ~((x <= single(1.9999999920083944e-11))))
		tmp = single(2.0) * (single(1.0) / (x * (x / s)));
	else
		tmp = ((((x / s) * (x / s)) * single(-0.0625)) + single(0.25)) / s;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -9.99999997e-7 or 1.99999999e-11 < 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. *-lft-identity 100.0%

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

      2. associate-*r/ 100.0%

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

      3. associate-*l* 100.0%

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

      4. times-frac 99.3%

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

      5. associate-*r/ 99.3%

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

      6. associate-/l* 99.3%

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

      7. distribute-frac-neg 99.3%

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

      8. exp-neg 99.4%

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

   3. Simplified 99.3%

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

      1. add-sqr-sqrt -0.0%

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

      2. sqrt-unprod 98.0%

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

      3. sqr-neg 98.0%

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

      4. sqrt-unprod 98.0%

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

      5. add-sqr-sqrt 98.0%

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

      6. *-un-lft-identity 98.0%

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

      7. div-inv 98.0%

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

      8. exp-prod 98.2%

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

      9. add-sqr-sqrt 98.2%

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

      10. sqrt-unprod 98.2%

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

      11. sqr-neg 98.2%

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

      12. sqrt-unprod -0.0%

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

      13. add-sqr-sqrt 98.8%

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

      14. exp-prod 99.3%

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

      15. div-inv 99.3%

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

   5. Applied egg-rr 98.5%

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

      1. *-lft-identity 98.5%

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

   7. Simplified 98.5%

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

   8. Taylor expanded in x around 0 78.0%

      \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \color{blue}{\left(3 + \left(\frac{x}{s} + 0.5 \cdot \frac{{x}^{2}}{{s}^{2}}\right)\right)}} \]
   9. Step-by-step derivation

      1. associate-*r/ 78.0%

         \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(3 + \left(\frac{x}{s} + \color{blue}{\frac{0.5 \cdot {x}^{2}}{{s}^{2}}}\right)\right)} \]

      2. unpow2 78.0%

         \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(3 + \left(\frac{x}{s} + \frac{0.5 \cdot {x}^{2}}{\color{blue}{s \cdot s}}\right)\right)} \]

      3. unpow2 78.0%

         \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(3 + \left(\frac{x}{s} + \frac{0.5 \cdot \color{blue}{\left(x \cdot x\right)}}{s \cdot s}\right)\right)} \]

   10. Simplified 78.0%

       \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \color{blue}{\left(3 + \left(\frac{x}{s} + \frac{0.5 \cdot \left(x \cdot x\right)}{s \cdot s}\right)\right)}} \]

   11. Taylor expanded in s around 0 63.2%

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

       1. unpow2 63.2%

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

   13. Simplified 63.2%

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

       1. clear-num 66.8%

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

       2. inv-pow 66.8%

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

   15. Applied egg-rr 66.8%

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

       1. unpow-1 66.8%

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

       2. associate-*r/ 66.8%

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

   17. Simplified 66.8%

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

   if -9.99999997e-7 < x < 1.99999999e-11

   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. associate-*l* 99.6%

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

      2. +-commutative 99.6%

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

      3. +-commutative 99.6%

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

   3. Simplified 99.6%

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

      1. add-exp-log 94.9%

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

      2. log-div 94.8%

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

      3. add-log-exp 94.8%

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

      4. add-sqr-sqrt -0.0%

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

      5. sqrt-unprod 58.0%

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

      6. sqr-neg 58.0%

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

      7. sqrt-unprod 57.7%

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

      8. add-sqr-sqrt 57.7%

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

      9. add-sqr-sqrt 40.4%

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

      10. fabs-sqr 40.4%

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

      11. add-sqr-sqrt 78.8%

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

      12. *-commutative 78.8%

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

   5. Applied egg-rr 94.1%

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

      1. associate--r+ 94.1%

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

      2. exp-diff 94.4%

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

      3. cancel-sign-sub-inv 94.4%

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

      4. metadata-eval 94.4%

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

      5. rem-exp-log 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in x around 0 55.9%

      \[\leadsto \frac{\color{blue}{0.25 + -0.0625 \cdot \frac{{x}^{2}}{{s}^{2}}}}{s} \]
   9. Step-by-step derivation

      1. unpow2 55.9%

         \[\leadsto \frac{0.25 + -0.0625 \cdot \frac{\color{blue}{x \cdot x}}{{s}^{2}}}{s} \]

      2. unpow2 55.9%

         \[\leadsto \frac{0.25 + -0.0625 \cdot \frac{x \cdot x}{\color{blue}{s \cdot s}}}{s} \]

   10. Simplified 55.9%

       \[\leadsto \frac{\color{blue}{0.25 + -0.0625 \cdot \frac{x \cdot x}{s \cdot s}}}{s} \]
   11. Step-by-step derivation

       1. times-frac 65.0%

          \[\leadsto \frac{0.25 + -0.0625 \cdot \color{blue}{\left(\frac{x}{s} \cdot \frac{x}{s}\right)}}{s} \]

   12. Applied egg-rr 65.0%

       \[\leadsto \frac{0.25 + -0.0625 \cdot \color{blue}{\left(\frac{x}{s} \cdot \frac{x}{s}\right)}}{s} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.999999974752427 \cdot 10^{-7} \lor \neg \left(x \leq 1.9999999920083944 \cdot 10^{-11}\right):\\ \;\;\;\;2 \cdot \frac{1}{x \cdot \frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{x}{s} \cdot \frac{x}{s}\right) \cdot -0.0625 + 0.25}{s}\\ \end{array} \]

Alternative 13: 62.8% accurate, 46.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.999999974752427 \cdot 10^{-7} \lor \neg \left(x \leq 1.9999999920083944 \cdot 10^{-11}\right):\\ \;\;\;\;2 \cdot \frac{1}{x \cdot \frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (or (<= x -9.999999974752427e-7) (not (<= x 1.9999999920083944e-11)))
   (* 2.0 (/ 1.0 (* x (/ x s))))
   (/ 0.25 s)))

C

float code(float x, float s) {
	float tmp;
	if ((x <= -9.999999974752427e-7f) || !(x <= 1.9999999920083944e-11f)) {
		tmp = 2.0f * (1.0f / (x * (x / s)));
	} else {
		tmp = 0.25f / s;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((x <= (-9.999999974752427e-7)) .or. (.not. (x <= 1.9999999920083944e-11))) then
        tmp = 2.0e0 * (1.0e0 / (x * (x / s)))
    else
        tmp = 0.25e0 / s
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if ((x <= Float32(-9.999999974752427e-7)) || !(x <= Float32(1.9999999920083944e-11)))
		tmp = Float32(Float32(2.0) * Float32(Float32(1.0) / Float32(x * Float32(x / s))));
	else
		tmp = Float32(Float32(0.25) / s);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((x <= single(-9.999999974752427e-7)) || ~((x <= single(1.9999999920083944e-11))))
		tmp = single(2.0) * (single(1.0) / (x * (x / s)));
	else
		tmp = single(0.25) / s;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -9.99999997e-7 or 1.99999999e-11 < 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. *-lft-identity 100.0%

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

      2. associate-*r/ 100.0%

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

      3. associate-*l* 100.0%

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

      4. times-frac 99.3%

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

      5. associate-*r/ 99.3%

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

      6. associate-/l* 99.3%

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

      7. distribute-frac-neg 99.3%

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

      8. exp-neg 99.4%

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

   3. Simplified 99.3%

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

      1. add-sqr-sqrt -0.0%

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

      2. sqrt-unprod 98.0%

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

      3. sqr-neg 98.0%

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

      4. sqrt-unprod 98.0%

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

      5. add-sqr-sqrt 98.0%

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

      6. *-un-lft-identity 98.0%

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

      7. div-inv 98.0%

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

      8. exp-prod 98.2%

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

      9. add-sqr-sqrt 98.2%

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

      10. sqrt-unprod 98.2%

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

      11. sqr-neg 98.2%

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

      12. sqrt-unprod -0.0%

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

      13. add-sqr-sqrt 98.8%

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

      14. exp-prod 99.3%

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

      15. div-inv 99.3%

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

   5. Applied egg-rr 98.5%

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

      1. *-lft-identity 98.5%

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

   7. Simplified 98.5%

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

   8. Taylor expanded in x around 0 78.0%

      \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \color{blue}{\left(3 + \left(\frac{x}{s} + 0.5 \cdot \frac{{x}^{2}}{{s}^{2}}\right)\right)}} \]
   9. Step-by-step derivation

      1. associate-*r/ 78.0%

         \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(3 + \left(\frac{x}{s} + \color{blue}{\frac{0.5 \cdot {x}^{2}}{{s}^{2}}}\right)\right)} \]

      2. unpow2 78.0%

         \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(3 + \left(\frac{x}{s} + \frac{0.5 \cdot {x}^{2}}{\color{blue}{s \cdot s}}\right)\right)} \]

      3. unpow2 78.0%

         \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(3 + \left(\frac{x}{s} + \frac{0.5 \cdot \color{blue}{\left(x \cdot x\right)}}{s \cdot s}\right)\right)} \]

   10. Simplified 78.0%

       \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \color{blue}{\left(3 + \left(\frac{x}{s} + \frac{0.5 \cdot \left(x \cdot x\right)}{s \cdot s}\right)\right)}} \]

   11. Taylor expanded in s around 0 63.2%

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

       1. unpow2 63.2%

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

   13. Simplified 63.2%

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

       1. clear-num 66.8%

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

       2. inv-pow 66.8%

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

   15. Applied egg-rr 66.8%

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

       1. unpow-1 66.8%

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

       2. associate-*r/ 66.8%

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

   17. Simplified 66.8%

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

   if -9.99999997e-7 < x < 1.99999999e-11

   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. associate-*l* 99.6%

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

      2. +-commutative 99.6%

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

      3. +-commutative 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in s around inf 65.0%

      \[\leadsto \color{blue}{\frac{0.25}{s}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.999999974752427 \cdot 10^{-7} \lor \neg \left(x \leq 1.9999999920083944 \cdot 10^{-11}\right):\\ \;\;\;\;2 \cdot \frac{1}{x \cdot \frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]

Alternative 14: 61.5% accurate, 55.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.999999974752427 \cdot 10^{-7} \lor \neg \left(x \leq 1.9999999920083944 \cdot 10^{-11}\right):\\ \;\;\;\;2 \cdot \frac{s}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (or (<= x -9.999999974752427e-7) (not (<= x 1.9999999920083944e-11)))
   (* 2.0 (/ s (* x x)))
   (/ 0.25 s)))

C

float code(float x, float s) {
	float tmp;
	if ((x <= -9.999999974752427e-7f) || !(x <= 1.9999999920083944e-11f)) {
		tmp = 2.0f * (s / (x * x));
	} else {
		tmp = 0.25f / s;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((x <= (-9.999999974752427e-7)) .or. (.not. (x <= 1.9999999920083944e-11))) then
        tmp = 2.0e0 * (s / (x * x))
    else
        tmp = 0.25e0 / s
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if ((x <= Float32(-9.999999974752427e-7)) || !(x <= Float32(1.9999999920083944e-11)))
		tmp = Float32(Float32(2.0) * Float32(s / Float32(x * x)));
	else
		tmp = Float32(Float32(0.25) / s);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((x <= single(-9.999999974752427e-7)) || ~((x <= single(1.9999999920083944e-11))))
		tmp = single(2.0) * (s / (x * x));
	else
		tmp = single(0.25) / s;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -9.99999997e-7 or 1.99999999e-11 < 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. *-lft-identity 100.0%

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

      2. associate-*r/ 100.0%

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

      3. associate-*l* 100.0%

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

      4. times-frac 99.3%

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

      5. associate-*r/ 99.3%

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

      6. associate-/l* 99.3%

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

      7. distribute-frac-neg 99.3%

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

      8. exp-neg 99.4%

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

   3. Simplified 99.3%

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

      1. add-sqr-sqrt -0.0%

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

      2. sqrt-unprod 98.0%

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

      3. sqr-neg 98.0%

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

      4. sqrt-unprod 98.0%

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

      5. add-sqr-sqrt 98.0%

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

      6. *-un-lft-identity 98.0%

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

      7. div-inv 98.0%

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

      8. exp-prod 98.2%

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

      9. add-sqr-sqrt 98.2%

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

      10. sqrt-unprod 98.2%

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

      11. sqr-neg 98.2%

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

      12. sqrt-unprod -0.0%

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

      13. add-sqr-sqrt 98.8%

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

      14. exp-prod 99.3%

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

      15. div-inv 99.3%

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

   5. Applied egg-rr 98.5%

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

      1. *-lft-identity 98.5%

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

   7. Simplified 98.5%

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

   8. Taylor expanded in x around 0 78.0%

      \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \color{blue}{\left(3 + \left(\frac{x}{s} + 0.5 \cdot \frac{{x}^{2}}{{s}^{2}}\right)\right)}} \]
   9. Step-by-step derivation

      1. associate-*r/ 78.0%

         \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(3 + \left(\frac{x}{s} + \color{blue}{\frac{0.5 \cdot {x}^{2}}{{s}^{2}}}\right)\right)} \]

      2. unpow2 78.0%

         \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(3 + \left(\frac{x}{s} + \frac{0.5 \cdot {x}^{2}}{\color{blue}{s \cdot s}}\right)\right)} \]

      3. unpow2 78.0%

         \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(3 + \left(\frac{x}{s} + \frac{0.5 \cdot \color{blue}{\left(x \cdot x\right)}}{s \cdot s}\right)\right)} \]

   10. Simplified 78.0%

       \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \color{blue}{\left(3 + \left(\frac{x}{s} + \frac{0.5 \cdot \left(x \cdot x\right)}{s \cdot s}\right)\right)}} \]

   11. Taylor expanded in s around 0 63.2%

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

       1. unpow2 63.2%

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

   13. Simplified 63.2%

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

   if -9.99999997e-7 < x < 1.99999999e-11

   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. associate-*l* 99.6%

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

      2. +-commutative 99.6%

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

      3. +-commutative 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in s around inf 65.0%

      \[\leadsto \color{blue}{\frac{0.25}{s}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.999999974752427 \cdot 10^{-7} \lor \neg \left(x \leq 1.9999999920083944 \cdot 10^{-11}\right):\\ \;\;\;\;2 \cdot \frac{s}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]

Alternative 15: 61.4% accurate, 55.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.999999974752427 \cdot 10^{-7}:\\ \;\;\;\;2 \cdot \frac{\frac{s}{x}}{x}\\ \mathbf{elif}\;x \leq 1.9999999920083944 \cdot 10^{-11}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{s}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -9.999999974752427e-7)
   (* 2.0 (/ (/ s x) x))
   (if (<= x 1.9999999920083944e-11) (/ 0.25 s) (* 2.0 (/ s (* x x))))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -9.999999974752427e-7f) {
		tmp = 2.0f * ((s / x) / x);
	} else if (x <= 1.9999999920083944e-11f) {
		tmp = 0.25f / s;
	} else {
		tmp = 2.0f * (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 <= (-9.999999974752427e-7)) then
        tmp = 2.0e0 * ((s / x) / x)
    else if (x <= 1.9999999920083944e-11) then
        tmp = 0.25e0 / s
    else
        tmp = 2.0e0 * (s / (x * x))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-9.999999974752427e-7))
		tmp = Float32(Float32(2.0) * Float32(Float32(s / x) / x));
	elseif (x <= Float32(1.9999999920083944e-11))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(2.0) * Float32(s / Float32(x * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-9.999999974752427e-7))
		tmp = single(2.0) * ((s / x) / x);
	elseif (x <= single(1.9999999920083944e-11))
		tmp = single(0.25) / s;
	else
		tmp = single(2.0) * (s / (x * x));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -9.99999997e-7

   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. *-lft-identity 100.0%

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

      2. associate-*r/ 100.0%

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

      3. associate-*l* 100.0%

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

      4. times-frac 100.0%

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

      5. associate-*r/ 100.0%

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

      6. associate-/l* 100.0%

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

      7. distribute-frac-neg 100.0%

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

      8. exp-neg 100.0%

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

   3. Simplified 100.0%

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

      1. add-sqr-sqrt -0.0%

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

      2. sqrt-unprod 99.1%

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

      3. sqr-neg 99.1%

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

      4. sqrt-unprod 99.1%

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

      5. add-sqr-sqrt 99.1%

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

      6. *-un-lft-identity 99.1%

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

      7. div-inv 99.1%

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

      8. exp-prod 99.1%

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

      9. add-sqr-sqrt 99.1%

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

      10. sqrt-unprod 99.1%

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

      11. sqr-neg 99.1%

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

      12. sqrt-unprod -0.0%

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

      13. add-sqr-sqrt 99.9%

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

      14. exp-prod 100.0%

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

      15. div-inv 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. *-lft-identity 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around 0 59.5%

      \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \color{blue}{\left(3 + \left(\frac{x}{s} + 0.5 \cdot \frac{{x}^{2}}{{s}^{2}}\right)\right)}} \]
   9. Step-by-step derivation

      1. associate-*r/ 59.5%

         \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(3 + \left(\frac{x}{s} + \color{blue}{\frac{0.5 \cdot {x}^{2}}{{s}^{2}}}\right)\right)} \]

      2. unpow2 59.5%

         \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(3 + \left(\frac{x}{s} + \frac{0.5 \cdot {x}^{2}}{\color{blue}{s \cdot s}}\right)\right)} \]

      3. unpow2 59.5%

         \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(3 + \left(\frac{x}{s} + \frac{0.5 \cdot \color{blue}{\left(x \cdot x\right)}}{s \cdot s}\right)\right)} \]

   10. Simplified 59.5%

       \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \color{blue}{\left(3 + \left(\frac{x}{s} + \frac{0.5 \cdot \left(x \cdot x\right)}{s \cdot s}\right)\right)}} \]

   11. Taylor expanded in s around 0 66.2%

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

       1. unpow2 66.2%

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

   13. Simplified 66.2%

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

   14. Taylor expanded in s around 0 66.2%

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

       1. unpow2 66.2%

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

       2. associate-/r* 66.2%

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

   16. Simplified 66.2%

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

   if -9.99999997e-7 < x < 1.99999999e-11

   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. associate-*l* 99.6%

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

      2. +-commutative 99.6%

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

      3. +-commutative 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in s around inf 65.0%

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

   if 1.99999999e-11 < 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. *-lft-identity 100.0%

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

      2. associate-*r/ 100.0%

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

      3. associate-*l* 100.0%

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

      4. times-frac 98.6%

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

      5. associate-*r/ 98.6%

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

      6. associate-/l* 98.6%

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

      7. distribute-frac-neg 98.6%

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

      8. exp-neg 98.7%

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

   3. Simplified 98.6%

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

      1. add-sqr-sqrt -0.0%

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

      2. sqrt-unprod 97.0%

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

      3. sqr-neg 97.0%

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

      4. sqrt-unprod 97.0%

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

      5. add-sqr-sqrt 97.0%

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

      6. *-un-lft-identity 97.0%

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

      7. div-inv 97.0%

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

      8. exp-prod 97.3%

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

      9. add-sqr-sqrt 97.3%

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

      10. sqrt-unprod 97.3%

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

      11. sqr-neg 97.3%

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

      12. sqrt-unprod -0.0%

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

      13. add-sqr-sqrt 97.5%

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

      14. exp-prod 98.6%

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

      15. div-inv 98.6%

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

   5. Applied egg-rr 97.0%

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

      1. *-lft-identity 97.0%

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

   7. Simplified 97.0%

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

   8. Taylor expanded in x around 0 97.1%

      \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \color{blue}{\left(3 + \left(\frac{x}{s} + 0.5 \cdot \frac{{x}^{2}}{{s}^{2}}\right)\right)}} \]
   9. Step-by-step derivation

      1. associate-*r/ 97.1%

         \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(3 + \left(\frac{x}{s} + \color{blue}{\frac{0.5 \cdot {x}^{2}}{{s}^{2}}}\right)\right)} \]

      2. unpow2 97.1%

         \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(3 + \left(\frac{x}{s} + \frac{0.5 \cdot {x}^{2}}{\color{blue}{s \cdot s}}\right)\right)} \]

      3. unpow2 97.1%

         \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(3 + \left(\frac{x}{s} + \frac{0.5 \cdot \color{blue}{\left(x \cdot x\right)}}{s \cdot s}\right)\right)} \]

   10. Simplified 97.1%

       \[\leadsto \frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \color{blue}{\left(3 + \left(\frac{x}{s} + \frac{0.5 \cdot \left(x \cdot x\right)}{s \cdot s}\right)\right)}} \]

   11. Taylor expanded in s around 0 60.1%

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

       1. unpow2 60.1%

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

   13. Simplified 60.1%

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

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.999999974752427 \cdot 10^{-7}:\\ \;\;\;\;2 \cdot \frac{\frac{s}{x}}{x}\\ \mathbf{elif}\;x \leq 1.9999999920083944 \cdot 10^{-11}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{s}{x \cdot x}\\ \end{array} \]

Alternative 16: 26.7% 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 99.8%

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

   1. associate-*l* 99.8%

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

   2. +-commutative 99.8%

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

   3. +-commutative 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in s around inf 29.0%

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

5. Final simplification 29.0%

   \[\leadsto \frac{0.25}{s} \]

Reproduce

herbie shell --seed 2023175 
(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))))))