Logistic distribution

Percentage Accurate: 99.5% → 99.0%

Time: 13.4s

Alternatives: 11

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: 99.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := e^{\frac{x\_m}{s}}\\ \mathbf{if}\;\left|x\_m\right| \leq 3.9999998989515007 \cdot 10^{-5}:\\ \;\;\;\;\frac{e^{\frac{x\_m}{s} - 2 \cdot \mathsf{log1p}\left(t\_0\right)}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.25}{x\_m \cdot t\_0}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (let* ((t_0 (exp (/ x_m s))))
   (if (<= (fabs x_m) 3.9999998989515007e-5)
     (/ (exp (- (/ x_m s) (* 2.0 (log1p t_0)))) s)
     (/ -0.25 (* x_m t_0)))))

C

x_m = fabs(x);
float code(float x_m, float s) {
	float t_0 = expf((x_m / s));
	float tmp;
	if (fabsf(x_m) <= 3.9999998989515007e-5f) {
		tmp = expf(((x_m / s) - (2.0f * log1pf(t_0)))) / s;
	} else {
		tmp = -0.25f / (x_m * t_0);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m, s)
	t_0 = exp(Float32(x_m / s))
	tmp = Float32(0.0)
	if (abs(x_m) <= Float32(3.9999998989515007e-5))
		tmp = Float32(exp(Float32(Float32(x_m / s) - Float32(Float32(2.0) * log1p(t_0)))) / s);
	else
		tmp = Float32(Float32(-0.25) / Float32(x_m * t_0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (fabs.f32 x) < 3.9999999e-5

   1. Initial program 99.1%

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

      1. fabs-neg 99.1%

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

      2. distribute-frac-neg 99.1%

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

      3. distribute-frac-neg2 99.1%

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

      4. fabs-neg 99.1%

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

      5. *-commutative 99.1%

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

      6. fabs-neg 99.1%

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

      7. +-commutative 99.1%

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

      8. fabs-neg 99.1%

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

   3. Simplified 99.2%

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

   6. Applied egg-rr 75.7%

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

      1. *-lft-identity 75.7%

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

      2. *-commutative 75.7%

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

      3. exp-to-pow 75.7%

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

      4. log1p-undefine 75.9%

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

      5. *-commutative 75.9%

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

      6. rem-exp-log 71.9%

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

      7. prod-exp 71.4%

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

      8. exp-diff 94.5%

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

      9. associate--r+ 94.8%

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

      10. exp-diff 95.2%

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

      11. rem-exp-log 99.4%

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

   8. Simplified 99.4%

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

   if 3.9999999e-5 < (fabs.f32 x)

   1. Initial program 100.0%

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

      1. fabs-neg 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. distribute-frac-neg2 100.0%

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

      4. fabs-neg 100.0%

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

      5. *-commutative 100.0%

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

      6. fabs-neg 100.0%

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

      7. +-commutative 100.0%

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

      8. fabs-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 100.0%

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

      1. associate-/r* 100.0%

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

      2. exp-prod 100.0%

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

      3. rem-square-sqrt 53.8%

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

      4. fabs-sqr 53.8%

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

      5. rem-square-sqrt 55.2%

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

      6. exp-prod 55.2%

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

      7. neg-mul-1 55.2%

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

      8. distribute-neg-frac2 55.2%

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

      9. +-commutative 55.2%

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

      10. exp-prod 55.2%

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

      11. rem-square-sqrt 53.8%

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

      12. fabs-sqr 53.8%

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

      13. rem-square-sqrt 53.8%

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

      14. exp-prod 53.8%

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

      15. neg-mul-1 53.8%

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

      16. distribute-neg-frac2 53.8%

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

   7. Simplified 53.8%

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

   8. Taylor expanded in x around 0 54.7%

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

      1. associate-*r/ 54.7%

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

      2. *-commutative 54.7%

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

   10. Simplified 54.7%

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

   11. Taylor expanded in x around inf 55.2%

       \[\leadsto \color{blue}{-0.25 \cdot \frac{e^{-1 \cdot \frac{x}{s}}}{x}} \]
   12. Step-by-step derivation

       1. neg-mul-1 55.2%

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

       2. distribute-neg-frac 55.2%

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

   13. Simplified 55.2%

       \[\leadsto \color{blue}{-0.25 \cdot \frac{e^{\frac{-x}{s}}}{x}} \]
   14. Step-by-step derivation

       1. clear-num 55.2%

          \[\leadsto -0.25 \cdot \color{blue}{\frac{1}{\frac{x}{e^{\frac{-x}{s}}}}} \]

       2. un-div-inv 55.2%

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

       3. div-inv 55.2%

          \[\leadsto \frac{-0.25}{\color{blue}{x \cdot \frac{1}{e^{\frac{-x}{s}}}}} \]

       4. add-sqr-sqrt 1.5%

          \[\leadsto \frac{-0.25}{x \cdot \frac{1}{e^{\color{blue}{\sqrt{\frac{-x}{s}} \cdot \sqrt{\frac{-x}{s}}}}}} \]

       5. sqrt-unprod 3.1%

          \[\leadsto \frac{-0.25}{x \cdot \frac{1}{e^{\color{blue}{\sqrt{\frac{-x}{s} \cdot \frac{-x}{s}}}}}} \]

       6. distribute-frac-neg 3.1%

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

       7. distribute-frac-neg 3.1%

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

       8. sqr-neg 3.1%

          \[\leadsto \frac{-0.25}{x \cdot \frac{1}{e^{\sqrt{\color{blue}{\frac{x}{s} \cdot \frac{x}{s}}}}}} \]

       9. sqrt-unprod 1.7%

          \[\leadsto \frac{-0.25}{x \cdot \frac{1}{e^{\color{blue}{\sqrt{\frac{x}{s}} \cdot \sqrt{\frac{x}{s}}}}}} \]

       10. add-sqr-sqrt 47.9%

           \[\leadsto \frac{-0.25}{x \cdot \frac{1}{e^{\color{blue}{\frac{x}{s}}}}} \]

       11. exp-neg 47.9%

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

       12. distribute-frac-neg 47.9%

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

       13. add-sqr-sqrt 46.2%

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

       14. sqrt-unprod 100.0%

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

       15. distribute-frac-neg 100.0%

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

       16. distribute-frac-neg 100.0%

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

       17. sqr-neg 100.0%

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

   15. Applied egg-rr 55.2%

       \[\leadsto \color{blue}{\frac{-0.25}{x \cdot e^{\frac{x}{s}}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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((-abs(x_m) / s))
    code = t_0 / ((t_0 + 1.0e0) * (s + (s / exp((abs(x_m) / s)))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

   1. *-commutative 99.6%

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

   2. fabs-neg 99.6%

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

   3. distribute-lft-in 99.7%

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

   4. *-rgt-identity 99.7%

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

   5. fabs-neg 99.7%

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

   6. distribute-rgt-in 99.7%

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

   7. cancel-sign-sub 99.7%

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

3. Simplified 99.7%

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

5. Final simplification 99.7%

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

Alternative 3: 99.5% accurate, 2.0× speedup

Math

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

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (let* ((t_0 (exp (/ x_m (- s))))) (/ (/ t_0 s) (pow (+ 1.0 t_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((1.0f + t_0), 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) / ((1.0e0 + t_0) ** 2.0e0)
end function

Julia

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

Derivation

1. Initial program 99.6%

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

   1. fabs-neg 99.6%

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

   2. distribute-frac-neg 99.6%

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

   3. distribute-frac-neg2 99.6%

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

   4. fabs-neg 99.6%

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

   5. *-commutative 99.6%

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

   6. fabs-neg 99.6%

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

   7. +-commutative 99.6%

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

   8. fabs-neg 99.6%

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

3. Simplified 99.7%

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

5. Taylor expanded in x around 0 99.7%

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

   1. associate-/r* 99.6%

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

   2. exp-prod 99.6%

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

   3. rem-square-sqrt 56.1%

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

   4. fabs-sqr 56.1%

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

   5. rem-square-sqrt 63.5%

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

   6. exp-prod 63.5%

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

   7. neg-mul-1 63.5%

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

   8. distribute-neg-frac2 63.5%

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

   9. +-commutative 63.5%

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

   10. exp-prod 63.5%

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

   11. rem-square-sqrt 56.1%

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

   12. fabs-sqr 56.1%

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

   13. rem-square-sqrt 65.2%

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

   14. exp-prod 65.3%

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

   15. neg-mul-1 65.3%

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

   16. distribute-neg-frac2 65.3%

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

7. Simplified 65.3%

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

8. Final simplification 65.3%

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

Alternative 4: 95.4% accurate, 5.4× speedup

Math

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

FPCore

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

C

x_m = fabs(x);
float code(float x_m, float s) {
	return (expf((x_m / -s)) / s) / (4.0f + ((x_m * -4.0f) / 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 = (exp((x_m / -s)) / s) / (4.0e0 + ((x_m * (-4.0e0)) / s))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

   1. fabs-neg 99.6%

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

   2. distribute-frac-neg 99.6%

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

   3. distribute-frac-neg2 99.6%

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

   4. fabs-neg 99.6%

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

   5. *-commutative 99.6%

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

   6. fabs-neg 99.6%

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

   7. +-commutative 99.6%

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

   8. fabs-neg 99.6%

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

3. Simplified 99.7%

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

5. Taylor expanded in x around 0 99.7%

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

   1. associate-/r* 99.6%

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

   2. exp-prod 99.6%

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

   3. rem-square-sqrt 56.1%

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

   4. fabs-sqr 56.1%

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

   5. rem-square-sqrt 63.5%

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

   6. exp-prod 63.5%

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

   7. neg-mul-1 63.5%

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

   8. distribute-neg-frac2 63.5%

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

   9. +-commutative 63.5%

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

   10. exp-prod 63.5%

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

   11. rem-square-sqrt 56.1%

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

   12. fabs-sqr 56.1%

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

   13. rem-square-sqrt 65.2%

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

   14. exp-prod 65.3%

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

   15. neg-mul-1 65.3%

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

   16. distribute-neg-frac2 65.3%

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

7. Simplified 65.3%

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

8. Taylor expanded in x around 0 62.8%

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

   1. associate-*r/ 62.8%

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

   2. *-commutative 62.8%

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

10. Simplified 62.8%

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

Alternative 5: 86.3% accurate, 5.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 2.00000009162741 \cdot 10^{-18}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.25}{x\_m \cdot e^{\frac{x\_m}{s}}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (if (<= x_m 2.00000009162741e-18)
   (/ 0.25 s)
   (/ -0.25 (* x_m (exp (/ x_m s))))))

C

x_m = fabs(x);
float code(float x_m, float s) {
	float tmp;
	if (x_m <= 2.00000009162741e-18f) {
		tmp = 0.25f / s;
	} else {
		tmp = -0.25f / (x_m * expf((x_m / s)));
	}
	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) :: tmp
    if (x_m <= 2.00000009162741e-18) then
        tmp = 0.25e0 / s
    else
        tmp = (-0.25e0) / (x_m * exp((x_m / s)))
    end if
    code = tmp
end function

Julia

x_m = abs(x)
function code(x_m, s)
	tmp = Float32(0.0)
	if (x_m <= Float32(2.00000009162741e-18))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(-0.25) / Float32(x_m * exp(Float32(x_m / s))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, s)
	tmp = single(0.0);
	if (x_m <= single(2.00000009162741e-18))
		tmp = single(0.25) / s;
	else
		tmp = single(-0.25) / (x_m * exp((x_m / s)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 2.00000009e-18

   1. Initial program 99.5%

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

      1. fabs-neg 99.5%

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

      2. distribute-frac-neg 99.5%

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

      3. distribute-frac-neg2 99.5%

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

      4. fabs-neg 99.5%

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

      5. *-commutative 99.5%

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

      6. fabs-neg 99.5%

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

      7. +-commutative 99.5%

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

      8. fabs-neg 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in s around inf 33.8%

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

   if 2.00000009e-18 < x

   1. Initial program 99.8%

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

      1. fabs-neg 99.8%

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

      2. distribute-frac-neg 99.8%

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

      3. distribute-frac-neg2 99.8%

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

      4. fabs-neg 99.8%

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

      5. *-commutative 99.8%

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

      6. fabs-neg 99.8%

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

      7. +-commutative 99.8%

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

      8. fabs-neg 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 99.8%

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

      1. associate-/r* 99.8%

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

      2. exp-prod 99.8%

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

      3. rem-square-sqrt 99.8%

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

      4. fabs-sqr 99.8%

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

      5. rem-square-sqrt 99.8%

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

      6. exp-prod 99.8%

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

      7. neg-mul-1 99.8%

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

      8. distribute-neg-frac2 99.8%

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

      9. +-commutative 99.8%

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

      10. exp-prod 99.8%

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

      11. rem-square-sqrt 99.8%

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

      12. fabs-sqr 99.8%

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

      13. rem-square-sqrt 99.8%

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

      14. exp-prod 99.8%

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

      15. neg-mul-1 99.8%

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

      16. distribute-neg-frac2 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in x around 0 99.3%

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

      1. associate-*r/ 99.3%

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

      2. *-commutative 99.3%

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

   10. Simplified 99.3%

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

   11. Taylor expanded in x around inf 91.5%

       \[\leadsto \color{blue}{-0.25 \cdot \frac{e^{-1 \cdot \frac{x}{s}}}{x}} \]
   12. Step-by-step derivation

       1. neg-mul-1 91.5%

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

       2. distribute-neg-frac 91.5%

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

   13. Simplified 91.5%

       \[\leadsto \color{blue}{-0.25 \cdot \frac{e^{\frac{-x}{s}}}{x}} \]
   14. Step-by-step derivation

       1. clear-num 91.5%

          \[\leadsto -0.25 \cdot \color{blue}{\frac{1}{\frac{x}{e^{\frac{-x}{s}}}}} \]

       2. un-div-inv 91.5%

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

       3. div-inv 91.5%

          \[\leadsto \frac{-0.25}{\color{blue}{x \cdot \frac{1}{e^{\frac{-x}{s}}}}} \]

       4. add-sqr-sqrt -0.0%

          \[\leadsto \frac{-0.25}{x \cdot \frac{1}{e^{\color{blue}{\sqrt{\frac{-x}{s}} \cdot \sqrt{\frac{-x}{s}}}}}} \]

       5. sqrt-unprod 3.1%

          \[\leadsto \frac{-0.25}{x \cdot \frac{1}{e^{\color{blue}{\sqrt{\frac{-x}{s} \cdot \frac{-x}{s}}}}}} \]

       6. distribute-frac-neg 3.1%

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

       7. distribute-frac-neg 3.1%

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

       8. sqr-neg 3.1%

          \[\leadsto \frac{-0.25}{x \cdot \frac{1}{e^{\sqrt{\color{blue}{\frac{x}{s} \cdot \frac{x}{s}}}}}} \]

       9. sqrt-unprod 3.1%

          \[\leadsto \frac{-0.25}{x \cdot \frac{1}{e^{\color{blue}{\sqrt{\frac{x}{s}} \cdot \sqrt{\frac{x}{s}}}}}} \]

       10. add-sqr-sqrt 3.1%

           \[\leadsto \frac{-0.25}{x \cdot \frac{1}{e^{\color{blue}{\frac{x}{s}}}}} \]

       11. exp-neg 3.1%

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

       12. distribute-frac-neg 3.1%

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

       13. add-sqr-sqrt -0.0%

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

       14. sqrt-unprod 91.5%

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

       15. distribute-frac-neg 91.5%

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

       16. distribute-frac-neg 91.5%

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

       17. sqr-neg 91.5%

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

   15. Applied egg-rr 91.5%

       \[\leadsto \color{blue}{\frac{-0.25}{x \cdot e^{\frac{x}{s}}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 94.5% accurate, 5.7× speedup

Math

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

FPCore

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

C

x_m = fabs(x);
float code(float x_m, float s) {
	return (1.0f / 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 = (1.0e0 / exp((x_m / s))) / (s * 4.0e0)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

   1. *-commutative 99.6%

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

   2. distribute-lft-in 99.7%

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

   3. *-rgt-identity 99.7%

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

   4. fabs-neg 99.7%

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

   5. +-commutative 99.7%

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

   6. fma-define 99.7%

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

   7. fabs-neg 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in s around inf 94.7%

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

   1. *-commutative 94.7%

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

7. Simplified 94.7%

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

   1. div-inv 94.7%

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

   2. exp-prod 82.2%

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

   3. neg-mul-1 82.2%

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

   4. exp-prod 82.2%

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

   5. add-sqr-sqrt 47.1%

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

   6. fabs-sqr 47.1%

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

   7. add-sqr-sqrt 55.4%

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

   8. exp-prod 55.4%

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

   9. neg-mul-1 55.4%

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

   10. exp-prod 61.5%

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

   11. div-inv 61.5%

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

   12. distribute-frac-neg 61.5%

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

   13. exp-neg 61.5%

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

9. Applied egg-rr 61.5%

   \[\leadsto \frac{\color{blue}{\frac{1}{e^{\frac{x}{s}}}}}{s \cdot 4} \]
10. Add Preprocessing

Alternative 7: 94.5% 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 99.6%

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

   1. *-commutative 99.6%

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

   2. distribute-lft-in 99.7%

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

   3. *-rgt-identity 99.7%

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

   4. fabs-neg 99.7%

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

   5. +-commutative 99.7%

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

   6. fma-define 99.7%

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

   7. fabs-neg 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in s around inf 94.7%

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

   1. *-commutative 94.7%

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

7. Simplified 94.7%

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

   1. neg-fabs 94.7%

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

   2. add-sqr-sqrt 40.8%

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

   3. fabs-sqr 40.8%

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

   4. add-sqr-sqrt 55.3%

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

   5. neg-sub0 55.3%

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

   6. sub-neg 55.3%

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

   7. add-sqr-sqrt 40.8%

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

   8. sqrt-unprod 93.3%

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

   9. sqr-neg 93.3%

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

   10. sqrt-unprod 54.0%

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

   11. add-sqr-sqrt 61.5%

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

9. Applied egg-rr 61.5%

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

    1. +-lft-identity 61.5%

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

11. Simplified 61.5%

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

12. Final simplification 61.5%

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

Alternative 8: 52.9% accurate, 44.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 3.9999998989515007 \cdot 10^{-5}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{x\_m \cdot \frac{-4}{s}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (if (<= x_m 3.9999998989515007e-5)
   (/ 0.25 s)
   (/ (/ 1.0 s) (* x_m (/ -4.0 s)))))

C

x_m = fabs(x);
float code(float x_m, float s) {
	float tmp;
	if (x_m <= 3.9999998989515007e-5f) {
		tmp = 0.25f / s;
	} else {
		tmp = (1.0f / s) / (x_m * (-4.0f / s));
	}
	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) :: tmp
    if (x_m <= 3.9999998989515007e-5) then
        tmp = 0.25e0 / s
    else
        tmp = (1.0e0 / s) / (x_m * ((-4.0e0) / s))
    end if
    code = tmp
end function

Julia

x_m = abs(x)
function code(x_m, s)
	tmp = Float32(0.0)
	if (x_m <= Float32(3.9999998989515007e-5))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(Float32(1.0) / s) / Float32(x_m * Float32(Float32(-4.0) / s)));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, s)
	tmp = single(0.0);
	if (x_m <= single(3.9999998989515007e-5))
		tmp = single(0.25) / s;
	else
		tmp = (single(1.0) / s) / (x_m * (single(-4.0) / s));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 3.9999999e-5

   1. Initial program 99.5%

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

      1. fabs-neg 99.5%

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

      2. distribute-frac-neg 99.5%

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

      3. distribute-frac-neg2 99.5%

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

      4. fabs-neg 99.5%

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

      5. *-commutative 99.5%

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

      6. fabs-neg 99.5%

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

      7. +-commutative 99.5%

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

      8. fabs-neg 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in s around inf 34.2%

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

   if 3.9999999e-5 < x

   1. Initial program 100.0%

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

      1. fabs-neg 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. distribute-frac-neg2 100.0%

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

      4. fabs-neg 100.0%

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

      5. *-commutative 100.0%

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

      6. fabs-neg 100.0%

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

      7. +-commutative 100.0%

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

      8. fabs-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 100.0%

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

      1. associate-/r* 100.0%

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

      2. exp-prod 100.0%

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

      3. rem-square-sqrt 100.0%

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

      4. fabs-sqr 100.0%

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

      5. rem-square-sqrt 100.0%

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

      6. exp-prod 100.0%

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

      7. neg-mul-1 100.0%

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

      8. distribute-neg-frac2 100.0%

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

      9. +-commutative 100.0%

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

      10. exp-prod 100.0%

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

      11. rem-square-sqrt 100.0%

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

      12. fabs-sqr 100.0%

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

      13. rem-square-sqrt 100.0%

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

      14. exp-prod 100.0%

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

      15. neg-mul-1 100.0%

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

      16. distribute-neg-frac2 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around 0 100.0%

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

      1. associate-*r/ 100.0%

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

      2. *-commutative 100.0%

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

   10. Simplified 100.0%

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

   11. Taylor expanded in x around 0 49.3%

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

   12. Taylor expanded in x around inf 49.3%

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

       1. metadata-eval 49.3%

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

       2. distribute-lft-neg-in 49.3%

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

       3. associate-*r/ 49.3%

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

       4. *-commutative 49.3%

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

       5. associate-*r/ 50.5%

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

       6. metadata-eval 50.5%

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

       7. associate-*r/ 50.5%

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

       8. distribute-rgt-neg-in 50.5%

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

       9. associate-*r/ 50.5%

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

       10. metadata-eval 50.5%

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

       11. distribute-neg-frac 50.5%

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

       12. metadata-eval 50.5%

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

   14. Simplified 50.5%

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

Alternative 9: 51.0% accurate, 56.4× speedup

Math

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

FPCore

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

C

x_m = fabs(x);
float code(float x_m, float s) {
	return (1.0f / s) / (4.0f + ((x_m * -4.0f) / 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 = (1.0e0 / s) / (4.0e0 + ((x_m * (-4.0e0)) / s))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

   1. fabs-neg 99.6%

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

   2. distribute-frac-neg 99.6%

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

   3. distribute-frac-neg2 99.6%

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

   4. fabs-neg 99.6%

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

   5. *-commutative 99.6%

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

   6. fabs-neg 99.6%

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

   7. +-commutative 99.6%

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

   8. fabs-neg 99.6%

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

3. Simplified 99.7%

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

5. Taylor expanded in x around 0 99.7%

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

   1. associate-/r* 99.6%

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

   2. exp-prod 99.6%

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

   3. rem-square-sqrt 56.1%

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

   4. fabs-sqr 56.1%

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

   5. rem-square-sqrt 63.5%

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

   6. exp-prod 63.5%

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

   7. neg-mul-1 63.5%

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

   8. distribute-neg-frac2 63.5%

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

   9. +-commutative 63.5%

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

   10. exp-prod 63.5%

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

   11. rem-square-sqrt 56.1%

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

   12. fabs-sqr 56.1%

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

   13. rem-square-sqrt 65.2%

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

   14. exp-prod 65.3%

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

   15. neg-mul-1 65.3%

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

   16. distribute-neg-frac2 65.3%

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

7. Simplified 65.3%

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

8. Taylor expanded in x around 0 62.8%

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

   1. associate-*r/ 62.8%

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

   2. *-commutative 62.8%

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

10. Simplified 62.8%

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

11. Taylor expanded in x around 0 47.3%

    \[\leadsto \frac{\color{blue}{\frac{1}{s}}}{4 + \frac{x \cdot -4}{s}} \]
12. Add Preprocessing

Alternative 10: 30.2% accurate, 77.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 3.9999998989515007 \cdot 10^{-5}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.25}{x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (if (<= x_m 3.9999998989515007e-5) (/ 0.25 s) (/ -0.25 x_m)))

C

x_m = fabs(x);
float code(float x_m, float s) {
	float tmp;
	if (x_m <= 3.9999998989515007e-5f) {
		tmp = 0.25f / s;
	} else {
		tmp = -0.25f / 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) :: tmp
    if (x_m <= 3.9999998989515007e-5) then
        tmp = 0.25e0 / s
    else
        tmp = (-0.25e0) / x_m
    end if
    code = tmp
end function

Julia

x_m = abs(x)
function code(x_m, s)
	tmp = Float32(0.0)
	if (x_m <= Float32(3.9999998989515007e-5))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(-0.25) / x_m);
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, s)
	tmp = single(0.0);
	if (x_m <= single(3.9999998989515007e-5))
		tmp = single(0.25) / s;
	else
		tmp = single(-0.25) / x_m;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 3.9999999e-5

   1. Initial program 99.5%

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

      1. fabs-neg 99.5%

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

      2. distribute-frac-neg 99.5%

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

      3. distribute-frac-neg2 99.5%

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

      4. fabs-neg 99.5%

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

      5. *-commutative 99.5%

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

      6. fabs-neg 99.5%

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

      7. +-commutative 99.5%

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

      8. fabs-neg 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in s around inf 34.2%

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

   if 3.9999999e-5 < x

   1. Initial program 100.0%

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

      1. fabs-neg 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. distribute-frac-neg2 100.0%

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

      4. fabs-neg 100.0%

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

      5. *-commutative 100.0%

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

      6. fabs-neg 100.0%

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

      7. +-commutative 100.0%

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

      8. fabs-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 100.0%

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

      1. associate-/r* 100.0%

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

      2. exp-prod 100.0%

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

      3. rem-square-sqrt 100.0%

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

      4. fabs-sqr 100.0%

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

      5. rem-square-sqrt 100.0%

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

      6. exp-prod 100.0%

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

      7. neg-mul-1 100.0%

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

      8. distribute-neg-frac2 100.0%

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

      9. +-commutative 100.0%

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

      10. exp-prod 100.0%

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

      11. rem-square-sqrt 100.0%

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

      12. fabs-sqr 100.0%

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

      13. rem-square-sqrt 100.0%

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

      14. exp-prod 100.0%

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

      15. neg-mul-1 100.0%

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

      16. distribute-neg-frac2 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around 0 100.0%

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

      1. associate-*r/ 100.0%

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

      2. *-commutative 100.0%

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

   10. Simplified 100.0%

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

   11. Taylor expanded in x around 0 49.3%

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

   12. Taylor expanded in s around 0 11.4%

       \[\leadsto \color{blue}{\frac{-0.25}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 7.4% accurate, 206.7× speedup

Math

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

FPCore

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

C

x_m = fabs(x);
float code(float x_m, float s) {
	return -0.25f / 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 = (-0.25e0) / x_m
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

   1. fabs-neg 99.6%

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

   2. distribute-frac-neg 99.6%

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

   3. distribute-frac-neg2 99.6%

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

   4. fabs-neg 99.6%

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

   5. *-commutative 99.6%

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

   6. fabs-neg 99.6%

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

   7. +-commutative 99.6%

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

   8. fabs-neg 99.6%

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

3. Simplified 99.7%

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

5. Taylor expanded in x around 0 99.7%

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

   1. associate-/r* 99.6%

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

   2. exp-prod 99.6%

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

   3. rem-square-sqrt 56.1%

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

   4. fabs-sqr 56.1%

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

   5. rem-square-sqrt 63.5%

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

   6. exp-prod 63.5%

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

   7. neg-mul-1 63.5%

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

   8. distribute-neg-frac2 63.5%

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

   9. +-commutative 63.5%

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

   10. exp-prod 63.5%

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

   11. rem-square-sqrt 56.1%

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

   12. fabs-sqr 56.1%

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

   13. rem-square-sqrt 65.2%

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

   14. exp-prod 65.3%

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

   15. neg-mul-1 65.3%

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

   16. distribute-neg-frac2 65.3%

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

7. Simplified 65.3%

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

8. Taylor expanded in x around 0 62.8%

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

   1. associate-*r/ 62.8%

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

   2. *-commutative 62.8%

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

10. Simplified 62.8%

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

11. Taylor expanded in x around 0 47.3%

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

12. Taylor expanded in s around 0 8.9%

    \[\leadsto \color{blue}{\frac{-0.25}{x}} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024144 
(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))))))