Logistic distribution

Percentage Accurate: 99.5% → 99.6%

Time: 10.0s

Alternatives: 9

Speedup: 2.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.6% accurate, 2.0× speedup

Math

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

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (let* ((t_0 (exp (/ x_m (- s))))) (/ t_0 (* s (pow (+ t_0 1.0) 2.0)))))

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in s around 0 99.5%

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

   1. +-commutative 99.5%

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

   2. associate-*r/ 99.5%

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

   3. mul-1-neg 99.5%

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

5. Simplified 99.5%

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

   1. distribute-frac-neg 99.5%

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

   2. exp-neg 99.5%

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

   3. add-sqr-sqrt 99.5%

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

   4. sqrt-unprod 98.0%

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

   5. sqr-neg 98.0%

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

   6. sqrt-unprod -0.0%

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

   7. add-sqr-sqrt 94.4%

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

   8. distribute-frac-neg 94.4%

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

   9. distribute-frac-neg2 94.4%

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

   10. add-sqr-sqrt 48.5%

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

   11. fabs-sqr 48.5%

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

   12. add-sqr-sqrt 97.3%

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

   13. add-sqr-sqrt -0.0%

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

   14. sqrt-unprod 94.5%

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

   15. sqr-neg 94.5%

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

   16. sqrt-unprod 96.6%

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

   17. add-sqr-sqrt 96.6%

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

7. Applied egg-rr 96.6%

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

   1. rec-exp 96.6%

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

   2. distribute-neg-frac2 96.6%

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

9. Simplified 96.6%

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

    1. distribute-frac-neg 99.5%

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

    2. exp-neg 99.5%

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

    3. add-sqr-sqrt 99.5%

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

    4. sqrt-unprod 98.0%

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

    5. sqr-neg 98.0%

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

    6. sqrt-unprod -0.0%

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

    7. add-sqr-sqrt 94.4%

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

    8. distribute-frac-neg 94.4%

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

    9. distribute-frac-neg2 94.4%

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

    10. add-sqr-sqrt 48.5%

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

    11. fabs-sqr 48.5%

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

    12. add-sqr-sqrt 97.3%

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

    13. add-sqr-sqrt -0.0%

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

    14. sqrt-unprod 94.5%

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

    15. sqr-neg 94.5%

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

    16. sqrt-unprod 96.6%

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

    17. add-sqr-sqrt 96.6%

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

11. Applied egg-rr 63.8%

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

    1. rec-exp 96.6%

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

    2. distribute-neg-frac2 96.6%

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

13. Simplified 63.8%

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

Alternative 2: 99.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \frac{x\_m}{-s}\\ \mathbf{if}\;\left|x\_m\right| \leq 0.20000000298023224:\\ \;\;\;\;\frac{1}{s \cdot e^{t\_0 - \mathsf{log1p}\left(e^{\frac{x\_m}{s}}\right) \cdot -2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{t\_0}}{s \cdot 4}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (let* ((t_0 (/ x_m (- s))))
   (if (<= (fabs x_m) 0.20000000298023224)
     (/ 1.0 (* s (exp (- t_0 (* (log1p (exp (/ x_m s))) -2.0)))))
     (/ (exp t_0) (* s 4.0)))))

C

x_m = fabs(x);
float code(float x_m, float s) {
	float t_0 = x_m / -s;
	float tmp;
	if (fabsf(x_m) <= 0.20000000298023224f) {
		tmp = 1.0f / (s * expf((t_0 - (log1pf(expf((x_m / s))) * -2.0f))));
	} else {
		tmp = expf(t_0) / (s * 4.0f);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m, s)
	t_0 = Float32(x_m / Float32(-s))
	tmp = Float32(0.0)
	if (abs(x_m) <= Float32(0.20000000298023224))
		tmp = Float32(Float32(1.0) / Float32(s * exp(Float32(t_0 - Float32(log1p(exp(Float32(x_m / s))) * Float32(-2.0))))));
	else
		tmp = Float32(exp(t_0) / Float32(s * Float32(4.0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (fabs.f32 x) < 0.200000003

   1. Initial program 98.9%

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

      1. clear-num 98.9%

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

      2. associate-*l* 99.0%

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

      3. associate-/l* 99.0%

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

   4. Applied egg-rr 71.1%

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

      1. clear-num 71.1%

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

      2. pow-to-exp 71.1%

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

      3. div-exp 98.7%

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

      4. rem-log-exp 98.0%

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

      5. pow-to-exp 98.0%

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

      6. log-pow 98.7%

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

      7. +-commutative 98.7%

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

      8. log1p-define 98.9%

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

   6. Applied egg-rr 98.9%

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

      1. rec-exp 98.9%

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

      2. sub-neg 98.9%

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

      3. *-commutative 98.9%

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

      4. distribute-rgt-neg-in 98.9%

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

      5. metadata-eval 98.9%

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

   8. Simplified 98.9%

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

   if 0.200000003 < (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. Add Preprocessing

   3. Taylor expanded in s around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. associate-*r/ 100.0%

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

      3. mul-1-neg 100.0%

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

   5. Simplified 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. exp-neg 100.0%

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

      3. add-sqr-sqrt 100.0%

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

      4. sqrt-unprod 100.0%

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

      5. sqr-neg 100.0%

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

      6. sqrt-unprod -0.0%

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

      7. add-sqr-sqrt 100.0%

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

      8. distribute-frac-neg 100.0%

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

      9. distribute-frac-neg2 100.0%

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

      10. add-sqr-sqrt 50.4%

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

      11. fabs-sqr 50.4%

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

      12. add-sqr-sqrt 100.0%

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

      13. add-sqr-sqrt -0.0%

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

      14. sqrt-unprod 100.0%

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

      15. sqr-neg 100.0%

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

      16. sqrt-unprod 100.0%

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

      17. add-sqr-sqrt 100.0%

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

   7. Applied egg-rr 100.0%

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

      1. rec-exp 100.0%

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

      2. distribute-neg-frac2 100.0%

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

   9. Simplified 100.0%

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

       1. distribute-frac-neg 100.0%

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

       2. exp-neg 100.0%

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

       3. add-sqr-sqrt 100.0%

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

       4. sqrt-unprod 100.0%

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

       5. sqr-neg 100.0%

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

       6. sqrt-unprod -0.0%

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

       7. add-sqr-sqrt 100.0%

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

       8. distribute-frac-neg 100.0%

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

       9. distribute-frac-neg2 100.0%

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

       10. add-sqr-sqrt 50.4%

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

       11. fabs-sqr 50.4%

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

       12. add-sqr-sqrt 100.0%

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

       13. add-sqr-sqrt -0.0%

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

       14. sqrt-unprod 100.0%

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

       15. sqr-neg 100.0%

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

       16. sqrt-unprod 100.0%

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

       17. add-sqr-sqrt 100.0%

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

   11. Applied egg-rr 50.4%

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

       1. rec-exp 100.0%

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

       2. distribute-neg-frac2 100.0%

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

   13. Simplified 50.4%

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

   14. Taylor expanded in x around 0 51.9%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.20000000298023224:\\ \;\;\;\;\frac{1}{s \cdot e^{\frac{x}{-s} - \mathsf{log1p}\left(e^{\frac{x}{s}}\right) \cdot -2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{x}{-s}}}{s \cdot 4}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\left|x\_m\right| \leq 0.20000000298023224:\\ \;\;\;\;\frac{1}{s} \cdot e^{\frac{x\_m}{s} - 2 \cdot \mathsf{log1p}\left(e^{\frac{x\_m}{s}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{x\_m}{-s}}}{s \cdot 4}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (if (<= (fabs x_m) 0.20000000298023224)
   (* (/ 1.0 s) (exp (- (/ x_m s) (* 2.0 (log1p (exp (/ x_m s)))))))
   (/ (exp (/ x_m (- s))) (* s 4.0))))

C

x_m = fabs(x);
float code(float x_m, float s) {
	float tmp;
	if (fabsf(x_m) <= 0.20000000298023224f) {
		tmp = (1.0f / s) * expf(((x_m / s) - (2.0f * log1pf(expf((x_m / s))))));
	} else {
		tmp = expf((x_m / -s)) / (s * 4.0f);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m, s)
	tmp = Float32(0.0)
	if (abs(x_m) <= Float32(0.20000000298023224))
		tmp = Float32(Float32(Float32(1.0) / s) * exp(Float32(Float32(x_m / s) - Float32(Float32(2.0) * log1p(exp(Float32(x_m / s)))))));
	else
		tmp = Float32(exp(Float32(x_m / Float32(-s))) / Float32(s * Float32(4.0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (fabs.f32 x) < 0.200000003

   1. Initial program 98.9%

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

   3. Taylor expanded in s around 0 99.0%

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

      1. +-commutative 99.0%

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

      2. associate-*r/ 99.0%

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

      3. mul-1-neg 99.0%

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

   5. Simplified 99.0%

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

      1. *-un-lft-identity 99.0%

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

      2. times-frac 98.9%

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

      3. distribute-frac-neg 98.9%

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

      4. distribute-frac-neg2 98.9%

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

      5. div-inv 98.8%

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

      6. exp-prod 56.2%

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

      7. add-sqr-sqrt 26.1%

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

      8. fabs-sqr 26.1%

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

      9. add-sqr-sqrt 49.0%

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

      10. add-sqr-sqrt 41.8%

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

      11. sqrt-unprod 51.7%

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

      12. sqr-neg 51.7%

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

      13. sqrt-prod 51.6%

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

      14. add-sqr-sqrt 51.7%

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

      15. exp-prod 68.1%

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

      16. div-inv 68.1%

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

   7. Applied egg-rr 98.9%

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

   if 0.200000003 < (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. Add Preprocessing

   3. Taylor expanded in s around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. associate-*r/ 100.0%

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

      3. mul-1-neg 100.0%

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

   5. Simplified 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. exp-neg 100.0%

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

      3. add-sqr-sqrt 100.0%

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

      4. sqrt-unprod 100.0%

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

      5. sqr-neg 100.0%

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

      6. sqrt-unprod -0.0%

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

      7. add-sqr-sqrt 100.0%

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

      8. distribute-frac-neg 100.0%

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

      9. distribute-frac-neg2 100.0%

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

      10. add-sqr-sqrt 50.4%

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

      11. fabs-sqr 50.4%

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

      12. add-sqr-sqrt 100.0%

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

      13. add-sqr-sqrt -0.0%

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

      14. sqrt-unprod 100.0%

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

      15. sqr-neg 100.0%

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

      16. sqrt-unprod 100.0%

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

      17. add-sqr-sqrt 100.0%

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

   7. Applied egg-rr 100.0%

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

      1. rec-exp 100.0%

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

      2. distribute-neg-frac2 100.0%

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

   9. Simplified 100.0%

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

       1. distribute-frac-neg 100.0%

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

       2. exp-neg 100.0%

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

       3. add-sqr-sqrt 100.0%

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

       4. sqrt-unprod 100.0%

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

       5. sqr-neg 100.0%

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

       6. sqrt-unprod -0.0%

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

       7. add-sqr-sqrt 100.0%

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

       8. distribute-frac-neg 100.0%

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

       9. distribute-frac-neg2 100.0%

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

       10. add-sqr-sqrt 50.4%

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

       11. fabs-sqr 50.4%

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

       12. add-sqr-sqrt 100.0%

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

       13. add-sqr-sqrt -0.0%

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

       14. sqrt-unprod 100.0%

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

       15. sqr-neg 100.0%

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

       16. sqrt-unprod 100.0%

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

       17. add-sqr-sqrt 100.0%

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

   11. Applied egg-rr 50.4%

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

       1. rec-exp 100.0%

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

       2. distribute-neg-frac2 100.0%

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

   13. Simplified 50.4%

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

   14. Taylor expanded in x around 0 51.9%

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

Alternative 4: 96.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{e^{\frac{x\_m}{-s}}}{s \cdot {\left(2 - \frac{x\_m}{s}\right)}^{2}} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (/ (exp (/ x_m (- s))) (* s (pow (- 2.0 (/ x_m s)) 2.0))))

C

x_m = fabs(x);
float code(float x_m, float s) {
	return expf((x_m / -s)) / (s * powf((2.0f - (x_m / s)), 2.0f));
}

Fortran

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

Julia

x_m = abs(x)
function code(x_m, s)
	return Float32(exp(Float32(x_m / Float32(-s))) / Float32(s * (Float32(Float32(2.0) - Float32(x_m / s)) ^ Float32(2.0))))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m, s)
	tmp = exp((x_m / -s)) / (s * ((single(2.0) - (x_m / s)) ^ single(2.0)));
end

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in s around 0 99.5%

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

   1. +-commutative 99.5%

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

   2. associate-*r/ 99.5%

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

   3. mul-1-neg 99.5%

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

5. Simplified 99.5%

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

   1. distribute-frac-neg 99.5%

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

   2. exp-neg 99.5%

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

   3. add-sqr-sqrt 99.5%

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

   4. sqrt-unprod 98.0%

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

   5. sqr-neg 98.0%

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

   6. sqrt-unprod -0.0%

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

   7. add-sqr-sqrt 94.4%

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

   8. distribute-frac-neg 94.4%

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

   9. distribute-frac-neg2 94.4%

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

   10. add-sqr-sqrt 48.5%

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

   11. fabs-sqr 48.5%

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

   12. add-sqr-sqrt 97.3%

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

   13. add-sqr-sqrt -0.0%

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

   14. sqrt-unprod 94.5%

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

   15. sqr-neg 94.5%

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

   16. sqrt-unprod 96.6%

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

   17. add-sqr-sqrt 96.6%

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

7. Applied egg-rr 96.6%

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

   1. rec-exp 96.6%

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

   2. distribute-neg-frac2 96.6%

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

9. Simplified 96.6%

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

    1. distribute-frac-neg 99.5%

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

    2. exp-neg 99.5%

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

    3. add-sqr-sqrt 99.5%

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

    4. sqrt-unprod 98.0%

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

    5. sqr-neg 98.0%

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

    6. sqrt-unprod -0.0%

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

    7. add-sqr-sqrt 94.4%

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

    8. distribute-frac-neg 94.4%

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

    9. distribute-frac-neg2 94.4%

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

    10. add-sqr-sqrt 48.5%

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

    11. fabs-sqr 48.5%

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

    12. add-sqr-sqrt 97.3%

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

    13. add-sqr-sqrt -0.0%

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

    14. sqrt-unprod 94.5%

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

    15. sqr-neg 94.5%

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

    16. sqrt-unprod 96.6%

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

    17. add-sqr-sqrt 96.6%

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

11. Applied egg-rr 63.8%

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

    1. rec-exp 96.6%

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

    2. distribute-neg-frac2 96.6%

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

13. Simplified 63.8%

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

14. Taylor expanded in x around 0 60.7%

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

    1. mul-1-neg 60.7%

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

    2. unsub-neg 60.7%

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

16. Simplified 60.7%

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

Alternative 5: 94.7% accurate, 5.7× speedup

Math

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

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s) :precision binary32 (/ (exp (/ x_m (- s))) (* s 4.0)))

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in s around 0 99.5%

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

   1. +-commutative 99.5%

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

   2. associate-*r/ 99.5%

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

   3. mul-1-neg 99.5%

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

5. Simplified 99.5%

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

   1. distribute-frac-neg 99.5%

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

   2. exp-neg 99.5%

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

   3. add-sqr-sqrt 99.5%

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

   4. sqrt-unprod 98.0%

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

   5. sqr-neg 98.0%

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

   6. sqrt-unprod -0.0%

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

   7. add-sqr-sqrt 94.4%

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

   8. distribute-frac-neg 94.4%

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

   9. distribute-frac-neg2 94.4%

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

   10. add-sqr-sqrt 48.5%

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

   11. fabs-sqr 48.5%

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

   12. add-sqr-sqrt 97.3%

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

   13. add-sqr-sqrt -0.0%

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

   14. sqrt-unprod 94.5%

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

   15. sqr-neg 94.5%

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

   16. sqrt-unprod 96.6%

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

   17. add-sqr-sqrt 96.6%

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

7. Applied egg-rr 96.6%

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

   1. rec-exp 96.6%

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

   2. distribute-neg-frac2 96.6%

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

9. Simplified 96.6%

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

    1. distribute-frac-neg 99.5%

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

    2. exp-neg 99.5%

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

    3. add-sqr-sqrt 99.5%

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

    4. sqrt-unprod 98.0%

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

    5. sqr-neg 98.0%

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

    6. sqrt-unprod -0.0%

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

    7. add-sqr-sqrt 94.4%

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

    8. distribute-frac-neg 94.4%

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

    9. distribute-frac-neg2 94.4%

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

    10. add-sqr-sqrt 48.5%

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

    11. fabs-sqr 48.5%

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

    12. add-sqr-sqrt 97.3%

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

    13. add-sqr-sqrt -0.0%

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

    14. sqrt-unprod 94.5%

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

    15. sqr-neg 94.5%

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

    16. sqrt-unprod 96.6%

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

    17. add-sqr-sqrt 96.6%

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

11. Applied egg-rr 63.8%

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

    1. rec-exp 96.6%

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

    2. distribute-neg-frac2 96.6%

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

13. Simplified 63.8%

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

14. Taylor expanded in x around 0 60.3%

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

Alternative 6: 86.4% accurate, 31.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \frac{x\_m}{s} \cdot -0.25\\ \mathbf{if}\;x\_m \leq 30000001024:\\ \;\;\;\;\frac{\left(0.25 + t\_0\right) - t\_0}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot 4 + \frac{1}{\frac{\frac{s}{x\_m}}{x\_m}}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (let* ((t_0 (* (/ x_m s) -0.25)))
   (if (<= x_m 30000001024.0)
     (/ (- (+ 0.25 t_0) t_0) s)
     (/ 1.0 (+ (* s 4.0) (/ 1.0 (/ (/ s x_m) x_m)))))))

C

x_m = fabs(x);
float code(float x_m, float s) {
	float t_0 = (x_m / s) * -0.25f;
	float tmp;
	if (x_m <= 30000001024.0f) {
		tmp = ((0.25f + t_0) - t_0) / s;
	} else {
		tmp = 1.0f / ((s * 4.0f) + (1.0f / ((s / x_m) / x_m)));
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(4) function code(x_m, s)
    real(4), intent (in) :: x_m
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: tmp
    t_0 = (x_m / s) * (-0.25e0)
    if (x_m <= 30000001024.0e0) then
        tmp = ((0.25e0 + t_0) - t_0) / s
    else
        tmp = 1.0e0 / ((s * 4.0e0) + (1.0e0 / ((s / x_m) / x_m)))
    end if
    code = tmp
end function

Julia

x_m = abs(x)
function code(x_m, s)
	t_0 = Float32(Float32(x_m / s) * Float32(-0.25))
	tmp = Float32(0.0)
	if (x_m <= Float32(30000001024.0))
		tmp = Float32(Float32(Float32(Float32(0.25) + t_0) - t_0) / s);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(s * Float32(4.0)) + Float32(Float32(1.0) / Float32(Float32(s / x_m) / x_m))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, s)
	t_0 = (x_m / s) * single(-0.25);
	tmp = single(0.0);
	if (x_m <= single(30000001024.0))
		tmp = ((single(0.25) + t_0) - t_0) / s;
	else
		tmp = single(1.0) / ((s * single(4.0)) + (single(1.0) / ((s / x_m) / x_m)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 30000001000

   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. Add Preprocessing

   3. Taylor expanded in s around 0 99.4%

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

      1. +-commutative 99.4%

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

      2. associate-*r/ 99.4%

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

      3. mul-1-neg 99.4%

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

   5. Simplified 99.4%

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

      1. distribute-frac-neg 99.4%

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

      2. exp-neg 99.4%

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

      3. add-sqr-sqrt 99.4%

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

      4. sqrt-unprod 97.5%

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

      5. sqr-neg 97.5%

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

      6. sqrt-unprod -0.0%

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

      7. add-sqr-sqrt 93.1%

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

      8. distribute-frac-neg 93.1%

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

      9. distribute-frac-neg2 93.1%

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

      10. add-sqr-sqrt 36.9%

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

      11. fabs-sqr 36.9%

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

      12. add-sqr-sqrt 96.7%

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

      13. add-sqr-sqrt -0.0%

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

      14. sqrt-unprod 93.3%

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

      15. sqr-neg 93.3%

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

      16. sqrt-unprod 95.8%

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

      17. add-sqr-sqrt 95.8%

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

   7. Applied egg-rr 95.8%

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

      1. rec-exp 95.8%

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

      2. distribute-neg-frac2 95.8%

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

   9. Simplified 95.8%

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

   10. Taylor expanded in s around inf 45.6%

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

       1. +-commutative 45.6%

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

       2. associate-*r/ 45.6%

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

       3. add-sqr-sqrt 23.5%

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

       4. fabs-sqr 23.5%

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

       5. add-sqr-sqrt 71.3%

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

       6. associate-*r/ 71.3%

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

       7. *-commutative 71.3%

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

   12. Applied egg-rr 71.3%

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

   if 30000001000 < 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. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num 100.0%

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

      2. associate-*l* 100.0%

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

      3. associate-/l* 100.0%

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

   4. Applied egg-rr -0.0%

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

   5. Taylor expanded in x around 0 96.6%

      \[\leadsto \frac{1}{\color{blue}{4 \cdot s + \frac{{x}^{2}}{s}}} \]
   6. Step-by-step derivation

      1. +-commutative 96.6%

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

      2. add-sqr-sqrt 96.6%

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

      3. fma-define 96.6%

         \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\sqrt{\frac{{x}^{2}}{s}}, \sqrt{\frac{{x}^{2}}{s}}, 4 \cdot s\right)}} \]

      4. sqrt-div 96.6%

         \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{\frac{\sqrt{{x}^{2}}}{\sqrt{s}}}, \sqrt{\frac{{x}^{2}}{s}}, 4 \cdot s\right)} \]

      5. sqrt-pow1 96.6%

         \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{\color{blue}{{x}^{\left(\frac{2}{2}\right)}}}{\sqrt{s}}, \sqrt{\frac{{x}^{2}}{s}}, 4 \cdot s\right)} \]

      6. metadata-eval 96.6%

         \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{{x}^{\color{blue}{1}}}{\sqrt{s}}, \sqrt{\frac{{x}^{2}}{s}}, 4 \cdot s\right)} \]

      7. pow1 96.6%

         \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{\color{blue}{x}}{\sqrt{s}}, \sqrt{\frac{{x}^{2}}{s}}, 4 \cdot s\right)} \]

      8. sqrt-div 96.6%

         \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{x}{\sqrt{s}}, \color{blue}{\frac{\sqrt{{x}^{2}}}{\sqrt{s}}}, 4 \cdot s\right)} \]

      9. sqrt-pow1 96.6%

         \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{x}{\sqrt{s}}, \frac{\color{blue}{{x}^{\left(\frac{2}{2}\right)}}}{\sqrt{s}}, 4 \cdot s\right)} \]

      10. metadata-eval 96.6%

          \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{x}{\sqrt{s}}, \frac{{x}^{\color{blue}{1}}}{\sqrt{s}}, 4 \cdot s\right)} \]

      11. pow1 96.6%

          \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{x}{\sqrt{s}}, \frac{\color{blue}{x}}{\sqrt{s}}, 4 \cdot s\right)} \]

      12. *-commutative 96.6%

          \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{x}{\sqrt{s}}, \frac{x}{\sqrt{s}}, \color{blue}{s \cdot 4}\right)} \]

   7. Applied egg-rr 96.6%

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

      1. fma-undefine 96.6%

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

      2. unpow2 96.6%

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

   9. Simplified 96.6%

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

       1. unpow2 96.6%

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

       2. clear-num 96.6%

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

       3. clear-num 96.6%

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

       4. frac-times 96.6%

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

       5. metadata-eval 96.6%

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

   11. Applied egg-rr 96.6%

       \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{\sqrt{s}}{x} \cdot \frac{\sqrt{s}}{x}}} + s \cdot 4} \]
   12. Step-by-step derivation

       1. associate-*l/ 96.6%

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

       2. associate-*r/ 96.6%

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

       3. rem-square-sqrt 96.6%

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

   13. Simplified 96.6%

       \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{\frac{s}{x}}{x}}} + s \cdot 4} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 30000001024:\\ \;\;\;\;\frac{\left(0.25 + \frac{x}{s} \cdot -0.25\right) - \frac{x}{s} \cdot -0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot 4 + \frac{1}{\frac{\frac{s}{x}}{x}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 65.6% accurate, 47.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{1}{s \cdot 4 + \frac{1}{\frac{\frac{s}{x\_m}}{x\_m}}} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (/ 1.0 (+ (* s 4.0) (/ 1.0 (/ (/ s x_m) x_m)))))

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

   1. clear-num 99.5%

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

   2. associate-*l* 99.5%

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

   3. associate-/l* 99.5%

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

4. Applied egg-rr 59.6%

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

5. Taylor expanded in x around 0 63.3%

   \[\leadsto \frac{1}{\color{blue}{4 \cdot s + \frac{{x}^{2}}{s}}} \]
6. Step-by-step derivation

   1. +-commutative 63.3%

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

   2. add-sqr-sqrt 63.3%

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

   3. fma-define 63.3%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\sqrt{\frac{{x}^{2}}{s}}, \sqrt{\frac{{x}^{2}}{s}}, 4 \cdot s\right)}} \]

   4. sqrt-div 63.3%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{\frac{\sqrt{{x}^{2}}}{\sqrt{s}}}, \sqrt{\frac{{x}^{2}}{s}}, 4 \cdot s\right)} \]

   5. sqrt-pow1 62.4%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{\color{blue}{{x}^{\left(\frac{2}{2}\right)}}}{\sqrt{s}}, \sqrt{\frac{{x}^{2}}{s}}, 4 \cdot s\right)} \]

   6. metadata-eval 62.4%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{{x}^{\color{blue}{1}}}{\sqrt{s}}, \sqrt{\frac{{x}^{2}}{s}}, 4 \cdot s\right)} \]

   7. pow1 62.4%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{\color{blue}{x}}{\sqrt{s}}, \sqrt{\frac{{x}^{2}}{s}}, 4 \cdot s\right)} \]

   8. sqrt-div 62.4%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{x}{\sqrt{s}}, \color{blue}{\frac{\sqrt{{x}^{2}}}{\sqrt{s}}}, 4 \cdot s\right)} \]

   9. sqrt-pow1 63.6%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{x}{\sqrt{s}}, \frac{\color{blue}{{x}^{\left(\frac{2}{2}\right)}}}{\sqrt{s}}, 4 \cdot s\right)} \]

   10. metadata-eval 63.6%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{x}{\sqrt{s}}, \frac{{x}^{\color{blue}{1}}}{\sqrt{s}}, 4 \cdot s\right)} \]

   11. pow1 63.6%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{x}{\sqrt{s}}, \frac{\color{blue}{x}}{\sqrt{s}}, 4 \cdot s\right)} \]

   12. *-commutative 63.6%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{x}{\sqrt{s}}, \frac{x}{\sqrt{s}}, \color{blue}{s \cdot 4}\right)} \]

7. Applied egg-rr 63.6%

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

   1. fma-undefine 63.6%

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

   2. unpow2 63.6%

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

9. Simplified 63.6%

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

    1. unpow2 63.6%

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

    2. clear-num 63.6%

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

    3. clear-num 63.6%

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

    4. frac-times 63.6%

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

    5. metadata-eval 63.6%

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

11. Applied egg-rr 63.6%

    \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{\sqrt{s}}{x} \cdot \frac{\sqrt{s}}{x}}} + s \cdot 4} \]
12. Step-by-step derivation

    1. associate-*l/ 63.6%

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

    2. associate-*r/ 63.6%

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

    3. rem-square-sqrt 63.6%

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

13. Simplified 63.6%

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

14. Final simplification 63.6%

    \[\leadsto \frac{1}{s \cdot 4 + \frac{1}{\frac{\frac{s}{x}}{x}}} \]
15. Add Preprocessing

Alternative 8: 65.6% accurate, 56.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{1}{s \cdot 4 + \frac{x\_m}{\frac{s}{x\_m}}} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s) :precision binary32 (/ 1.0 (+ (* s 4.0) (/ x_m (/ s x_m)))))

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

   1. clear-num 99.5%

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

   2. associate-*l* 99.5%

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

   3. associate-/l* 99.5%

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

4. Applied egg-rr 59.6%

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

5. Taylor expanded in x around 0 63.3%

   \[\leadsto \frac{1}{\color{blue}{4 \cdot s + \frac{{x}^{2}}{s}}} \]
6. Step-by-step derivation

   1. +-commutative 63.3%

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

   2. add-sqr-sqrt 63.3%

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

   3. fma-define 63.3%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\sqrt{\frac{{x}^{2}}{s}}, \sqrt{\frac{{x}^{2}}{s}}, 4 \cdot s\right)}} \]

   4. sqrt-div 63.3%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{\frac{\sqrt{{x}^{2}}}{\sqrt{s}}}, \sqrt{\frac{{x}^{2}}{s}}, 4 \cdot s\right)} \]

   5. sqrt-pow1 62.4%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{\color{blue}{{x}^{\left(\frac{2}{2}\right)}}}{\sqrt{s}}, \sqrt{\frac{{x}^{2}}{s}}, 4 \cdot s\right)} \]

   6. metadata-eval 62.4%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{{x}^{\color{blue}{1}}}{\sqrt{s}}, \sqrt{\frac{{x}^{2}}{s}}, 4 \cdot s\right)} \]

   7. pow1 62.4%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{\color{blue}{x}}{\sqrt{s}}, \sqrt{\frac{{x}^{2}}{s}}, 4 \cdot s\right)} \]

   8. sqrt-div 62.4%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{x}{\sqrt{s}}, \color{blue}{\frac{\sqrt{{x}^{2}}}{\sqrt{s}}}, 4 \cdot s\right)} \]

   9. sqrt-pow1 63.6%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{x}{\sqrt{s}}, \frac{\color{blue}{{x}^{\left(\frac{2}{2}\right)}}}{\sqrt{s}}, 4 \cdot s\right)} \]

   10. metadata-eval 63.6%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{x}{\sqrt{s}}, \frac{{x}^{\color{blue}{1}}}{\sqrt{s}}, 4 \cdot s\right)} \]

   11. pow1 63.6%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{x}{\sqrt{s}}, \frac{\color{blue}{x}}{\sqrt{s}}, 4 \cdot s\right)} \]

   12. *-commutative 63.6%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{x}{\sqrt{s}}, \frac{x}{\sqrt{s}}, \color{blue}{s \cdot 4}\right)} \]

7. Applied egg-rr 63.6%

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

   1. fma-undefine 63.6%

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

   2. unpow2 63.6%

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

9. Simplified 63.6%

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

    1. unpow2 63.6%

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

    2. clear-num 63.6%

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

    3. frac-times 63.6%

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

    4. *-un-lft-identity 63.6%

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

11. Applied egg-rr 63.6%

    \[\leadsto \frac{1}{\color{blue}{\frac{x}{\frac{\sqrt{s}}{x} \cdot \sqrt{s}}} + s \cdot 4} \]
12. Step-by-step derivation

    1. associate-*l/ 63.6%

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

    2. rem-square-sqrt 63.6%

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

13. Simplified 63.6%

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

14. Final simplification 63.6%

    \[\leadsto \frac{1}{s \cdot 4 + \frac{x}{\frac{s}{x}}} \]
15. Add Preprocessing

Alternative 9: 26.7% accurate, 206.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{0.25}{s} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s) :precision binary32 (/ 0.25 s))

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in s around inf 24.5%

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

Reproduce

herbie shell --seed 2024096 
(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))))))