Logistic distribution

Percentage Accurate: 99.6% → 98.8%

Time: 12.6s

Alternatives: 6

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: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\left|x\_m\right| \leq 6.000000212225132 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{\frac{s}{e^{\frac{x\_m}{s} + -2 \cdot \mathsf{log1p}\left(e^{\frac{x\_m}{s}}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{-x\_m}{s}}}{s \cdot 4}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (if (<= (fabs x_m) 6.000000212225132e-7)
   (/ 1.0 (/ s (exp (+ (/ x_m s) (* -2.0 (log1p (exp (/ x_m s))))))))
   (/ (exp (/ (- x_m) s)) (* s 4.0))))

C

x_m = fabs(x);
float code(float x_m, float s) {
	float tmp;
	if (fabsf(x_m) <= 6.000000212225132e-7f) {
		tmp = 1.0f / (s / expf(((x_m / s) + (-2.0f * log1pf(expf((x_m / s)))))));
	} else {
		tmp = expf((-x_m / s)) / (s * 4.0f);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m, s)
	tmp = Float32(0.0)
	if (abs(x_m) <= Float32(6.000000212225132e-7))
		tmp = Float32(Float32(1.0) / Float32(s / exp(Float32(Float32(x_m / s) + Float32(Float32(-2.0) * log1p(exp(Float32(x_m / s))))))));
	else
		tmp = Float32(exp(Float32(Float32(-x_m) / s)) / Float32(s * Float32(4.0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (fabs.f32 x) < 6.0000002e-7

   1. Initial program 97.9%

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

      1. fabs-neg 97.9%

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

      2. distribute-frac-neg 97.9%

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

      3. distribute-frac-neg2 97.9%

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

      4. fabs-neg 97.9%

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

      5. *-commutative 97.9%

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

      6. fabs-neg 97.9%

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

      7. +-commutative 97.9%

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

      8. fabs-neg 97.9%

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

   3. Simplified 98.1%

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

   5. Taylor expanded in x around 0 98.1%

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

      1. unpow2 98.1%

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

      2. mul-1-neg 98.1%

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

      3. rec-exp 98.1%

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

      4. associate-*r/ 98.1%

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

      5. mul-1-neg 98.1%

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

      6. rec-exp 98.1%

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

      7. mul-1-neg 98.1%

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

   7. Simplified 98.1%

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

   9. Applied egg-rr 96.7%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{e^{\frac{x}{s} - 2 \cdot \mathsf{log1p}\left(e^{\frac{x}{s}}\right)}}{s}}\right)}^{3}} \]
   10. Step-by-step derivation

       1. rem-cube-cbrt 98.0%

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

       2. clear-num 98.1%

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

       3. cancel-sign-sub-inv 98.1%

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

       4. metadata-eval 98.1%

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

   11. Applied egg-rr 98.1%

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

   if 6.0000002e-7 < (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. unpow2 100.0%

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

      2. mul-1-neg 100.0%

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

      3. rec-exp 100.0%

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

      4. associate-*r/ 100.0%

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

      5. mul-1-neg 100.0%

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

      6. rec-exp 100.0%

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

      7. mul-1-neg 100.0%

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

   7. Simplified 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. exp-neg 100.0%

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

      3. add-sqr-sqrt 48.6%

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

      4. fabs-sqr 48.6%

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

      5. add-sqr-sqrt 100.0%

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

   9. Applied egg-rr 100.0%

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

       1. rec-exp 100.0%

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

       2. distribute-frac-neg 100.0%

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

   11. Simplified 100.0%

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

       1. distribute-frac-neg 100.0%

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

       2. exp-neg 100.0%

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

       3. add-sqr-sqrt 48.6%

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

       4. fabs-sqr 48.6%

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

       5. add-sqr-sqrt 100.0%

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

   13. Applied egg-rr 48.6%

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

       1. rec-exp 100.0%

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

       2. distribute-frac-neg 100.0%

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

   15. Simplified 48.6%

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

   16. Taylor expanded in s around inf 50.2%

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

       1. *-commutative 50.2%

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

   18. Simplified 50.2%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 6.000000212225132 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{\frac{s}{e^{\frac{x}{s} + -2 \cdot \mathsf{log1p}\left(e^{\frac{x}{s}}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{-x}{s}}}{s \cdot 4}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\left|x\_m\right| \leq 6.000000212225132 \cdot 10^{-7}:\\ \;\;\;\;\frac{e^{\frac{x\_m}{s} - 2 \cdot \mathsf{log1p}\left(e^{\frac{x\_m}{s}}\right)}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{-x\_m}{s}}}{s \cdot 4}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (if (<= (fabs x_m) 6.000000212225132e-7)
   (/ (exp (- (/ x_m s) (* 2.0 (log1p (exp (/ x_m s)))))) s)
   (/ (exp (/ (- x_m) s)) (* s 4.0))))

C

x_m = fabs(x);
float code(float x_m, float s) {
	float tmp;
	if (fabsf(x_m) <= 6.000000212225132e-7f) {
		tmp = expf(((x_m / s) - (2.0f * log1pf(expf((x_m / s)))))) / s;
	} else {
		tmp = expf((-x_m / s)) / (s * 4.0f);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m, s)
	tmp = Float32(0.0)
	if (abs(x_m) <= Float32(6.000000212225132e-7))
		tmp = Float32(exp(Float32(Float32(x_m / s) - Float32(Float32(2.0) * log1p(exp(Float32(x_m / s)))))) / s);
	else
		tmp = Float32(exp(Float32(Float32(-x_m) / s)) / Float32(s * Float32(4.0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (fabs.f32 x) < 6.0000002e-7

   1. Initial program 97.9%

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

      1. fabs-neg 97.9%

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

      2. distribute-frac-neg 97.9%

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

      3. distribute-frac-neg2 97.9%

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

      4. fabs-neg 97.9%

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

      5. *-commutative 97.9%

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

      6. fabs-neg 97.9%

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

      7. +-commutative 97.9%

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

      8. fabs-neg 97.9%

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

   3. Simplified 98.1%

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

   6. Applied egg-rr 82.9%

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

      1. add-exp-log 78.3%

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

      2. associate-*l/ 78.3%

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

      3. *-un-lft-identity 78.3%

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

      4. log-div 78.1%

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

      5. add-log-exp 93.0%

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

      6. *-commutative 93.0%

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

      7. sum-log 92.8%

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

      8. log-pow 93.4%

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

      9. +-commutative 93.4%

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

      10. log1p-undefine 93.4%

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

      11. associate--r+ 93.5%

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

   8. Applied egg-rr 98.0%

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

   if 6.0000002e-7 < (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. unpow2 100.0%

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

      2. mul-1-neg 100.0%

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

      3. rec-exp 100.0%

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

      4. associate-*r/ 100.0%

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

      5. mul-1-neg 100.0%

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

      6. rec-exp 100.0%

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

      7. mul-1-neg 100.0%

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

   7. Simplified 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. exp-neg 100.0%

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

      3. add-sqr-sqrt 48.6%

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

      4. fabs-sqr 48.6%

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

      5. add-sqr-sqrt 100.0%

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

   9. Applied egg-rr 100.0%

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

       1. rec-exp 100.0%

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

       2. distribute-frac-neg 100.0%

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

   11. Simplified 100.0%

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

       1. distribute-frac-neg 100.0%

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

       2. exp-neg 100.0%

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

       3. add-sqr-sqrt 48.6%

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

       4. fabs-sqr 48.6%

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

       5. add-sqr-sqrt 100.0%

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

   13. Applied egg-rr 48.6%

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

       1. rec-exp 100.0%

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

       2. distribute-frac-neg 100.0%

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

   15. Simplified 48.6%

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

   16. Taylor expanded in s around inf 50.2%

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

       1. *-commutative 50.2%

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

   18. Simplified 50.2%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 6.000000212225132 \cdot 10^{-7}:\\ \;\;\;\;\frac{e^{\frac{x}{s} - 2 \cdot \mathsf{log1p}\left(e^{\frac{x}{s}}\right)}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{-x}{s}}}{s \cdot 4}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
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 99.1%

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

5. Taylor expanded in x around 0 99.1%

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

   1. unpow2 99.1%

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

   2. mul-1-neg 99.1%

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

   3. rec-exp 99.1%

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

   4. associate-*r/ 99.1%

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

   5. mul-1-neg 99.1%

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

   6. rec-exp 99.1%

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

   7. mul-1-neg 99.1%

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

7. Simplified 99.1%

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

   1. distribute-frac-neg 99.1%

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

   2. exp-neg 99.1%

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

   3. add-sqr-sqrt 49.5%

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

   4. fabs-sqr 49.5%

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

   5. add-sqr-sqrt 96.3%

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

9. Applied egg-rr 96.3%

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

    1. rec-exp 96.4%

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

    2. distribute-frac-neg 96.4%

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

11. Simplified 96.4%

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

    1. distribute-frac-neg 99.1%

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

    2. exp-neg 99.1%

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

    3. add-sqr-sqrt 49.5%

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

    4. fabs-sqr 49.5%

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

    5. add-sqr-sqrt 96.3%

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

13. Applied egg-rr 62.2%

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

    1. rec-exp 96.4%

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

    2. distribute-frac-neg 96.4%

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

15. Simplified 62.3%

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

16. Final simplification 62.3%

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

Alternative 4: 94.6% 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(Float32(-x_m) / 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 99.1%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
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 99.1%

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

5. Taylor expanded in x around 0 99.1%

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

   1. unpow2 99.1%

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

   2. mul-1-neg 99.1%

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

   3. rec-exp 99.1%

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

   4. associate-*r/ 99.1%

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

   5. mul-1-neg 99.1%

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

   6. rec-exp 99.1%

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

   7. mul-1-neg 99.1%

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

7. Simplified 99.1%

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

   1. distribute-frac-neg 99.1%

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

   2. exp-neg 99.1%

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

   3. add-sqr-sqrt 49.5%

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

   4. fabs-sqr 49.5%

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

   5. add-sqr-sqrt 96.3%

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

9. Applied egg-rr 96.3%

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

    1. rec-exp 96.4%

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

    2. distribute-frac-neg 96.4%

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

11. Simplified 96.4%

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

    1. distribute-frac-neg 99.1%

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

    2. exp-neg 99.1%

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

    3. add-sqr-sqrt 49.5%

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

    4. fabs-sqr 49.5%

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

    5. add-sqr-sqrt 96.3%

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

13. Applied egg-rr 62.2%

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

    1. rec-exp 96.4%

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

    2. distribute-frac-neg 96.4%

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

15. Simplified 62.3%

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

16. Taylor expanded in s around inf 59.1%

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

    1. *-commutative 59.1%

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

18. Simplified 59.1%

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

19. Final simplification 59.1%

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

Alternative 5: 26.9% accurate, 206.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
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 99.1%

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

5. Taylor expanded in s around inf 29.5%

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

6. Final simplification 29.5%

   \[\leadsto \frac{0.25}{s} \]
7. Add Preprocessing

Alternative 6: 8.3% accurate, 620.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ 1 \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s) :precision binary32 1.0)

C

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

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
end function

Julia

x_m = abs(x)
function code(x_m, s)
	return Float32(1.0)
end

MATLAB

x_m = abs(x);
function tmp = code(x_m, s)
	tmp = single(1.0);
end

Derivation

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 99.1%

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

6. Applied egg-rr 83.4%

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

7. Taylor expanded in x around inf 41.5%

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

8. Taylor expanded in x around 0 6.5%

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

9. Taylor expanded in x around 0 8.2%

   \[\leadsto \color{blue}{1} \]

10. Final simplification 8.2%

    \[\leadsto 1 \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024074 
(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))))))