Logistic distribution

Percentage Accurate: 99.5% → 99.5%

Time: 13.4s

Alternatives: 7

Speedup: 1.5×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.5% accurate, 1.2× speedup

Math

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

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 (/ 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((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((x_m / s)))))
end function

Julia

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

Derivation

1. Initial program 99.7%

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

   1. *-commutative 99.7%

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

3. Simplified 99.7%

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

   1. +-commutative 99.7%

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

   2. distribute-lft-in 99.7%

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

   3. *-rgt-identity 99.7%

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

   4. fma-define 99.8%

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

   5. distribute-frac-neg 99.8%

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

   6. rec-exp 99.8%

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

   7. fma-define 99.8%

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

   8. div-inv 99.8%

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

   9. add-cube-cbrt 99.5%

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

   10. associate-/l* 99.5%

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

   11. fma-define 99.5%

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

6. Applied egg-rr 96.9%

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

   1. fma-undefine 96.9%

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

   2. associate-*r/ 96.9%

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

   3. unpow2 96.9%

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

   4. rem-3cbrt-lft 97.2%

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

   5. +-commutative 97.2%

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

8. Simplified 97.2%

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

9. Final simplification 97.2%

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

Alternative 2: 98.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;-\left|x\_m\right| \leq -1500000:\\ \;\;\;\;\frac{e^{\frac{x\_m}{-s}}}{s \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{x\_m}{s} + \mathsf{log1p}\left(e^{\frac{x\_m}{s}}\right) \cdot -2}}{s}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (if (<= (- (fabs x_m)) -1500000.0)
   (/ (exp (/ x_m (- s))) (* s 4.0))
   (/ (exp (+ (/ x_m s) (* (log1p (exp (/ x_m s))) -2.0))) s)))

C

x_m = fabs(x);
float code(float x_m, float s) {
	float tmp;
	if (-fabsf(x_m) <= -1500000.0f) {
		tmp = expf((x_m / -s)) / (s * 4.0f);
	} else {
		tmp = expf(((x_m / s) + (log1pf(expf((x_m / s))) * -2.0f))) / s;
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m, s)
	tmp = Float32(0.0)
	if (Float32(-abs(x_m)) <= Float32(-1500000.0))
		tmp = Float32(exp(Float32(x_m / Float32(-s))) / Float32(s * Float32(4.0)));
	else
		tmp = Float32(exp(Float32(Float32(x_m / s) + Float32(log1p(exp(Float32(x_m / s))) * Float32(-2.0)))) / s);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (neg.f32 (fabs.f32 x)) < -1.5e6

   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. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in s around inf 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. rec-exp 100.0%

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

      3. pow1 100.0%

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

      4. pow1 100.0%

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

      5. frac-2neg 100.0%

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

      6. frac-2neg 100.0%

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

      7. add-sqr-sqrt 52.1%

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

      8. fabs-sqr 52.1%

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

      9. add-sqr-sqrt 53.6%

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

   9. Applied egg-rr 53.6%

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

       1. rec-exp 53.6%

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

       2. distribute-neg-frac2 53.6%

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

   11. Simplified 53.6%

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

   if -1.5e6 < (neg.f32 (fabs.f32 x))

   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. *-commutative 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in x around 0 99.4%

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

      1. associate-*r/ 99.4%

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

      2. mul-1-neg 99.4%

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

   7. Simplified 99.4%

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

   8. Taylor expanded in x around 0 99.4%

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

      1. associate-/r* 99.5%

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

      2. mul-1-neg 99.5%

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

      3. distribute-neg-frac2 99.5%

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

      4. mul-1-neg 99.5%

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

      5. distribute-neg-frac2 99.5%

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

   10. Simplified 99.5%

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

       1. *-un-lft-identity 99.5%

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

       2. associate-/l/ 99.4%

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

       3. associate-/r* 99.5%

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

   12. Applied egg-rr 99.7%

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

       1. *-lft-identity 99.7%

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

       2. sub-neg 99.7%

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

       3. *-commutative 99.7%

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

       4. distribute-rgt-neg-in 99.7%

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

       5. metadata-eval 99.7%

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

   14. Simplified 99.7%

       \[\leadsto \color{blue}{\frac{e^{\frac{x}{s} + \mathsf{log1p}\left(e^{\frac{x}{s}}\right) \cdot -2}}{s}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\left|x\right| \leq -1500000:\\ \;\;\;\;\frac{e^{\frac{x}{-s}}}{s \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{x}{s} + \mathsf{log1p}\left(e^{\frac{x}{s}}\right) \cdot -2}}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

   1. *-commutative 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in x around 0 99.6%

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

   1. associate-*r/ 99.6%

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

   2. mul-1-neg 99.6%

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

7. Simplified 99.6%

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

   1. distribute-frac-neg 94.4%

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

   2. rec-exp 94.4%

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

   3. pow1 94.4%

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

   4. pow1 94.4%

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

   5. frac-2neg 94.4%

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

   6. frac-2neg 94.4%

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

   7. add-sqr-sqrt 46.3%

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

   8. fabs-sqr 46.3%

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

   9. add-sqr-sqrt 61.4%

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

9. Applied egg-rr 97.0%

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

    1. rec-exp 61.4%

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

    2. distribute-neg-frac2 61.4%

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

11. Simplified 97.0%

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

12. Final simplification 97.0%

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

Alternative 4: 99.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

x_m = fabs(x);
float code(float x_m, float s) {
	return (expf((x_m / -s)) / s) / powf((expf((fabsf(x_m) / -s)) + 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
    code = (exp((x_m / -s)) / s) / ((exp((abs(x_m) / -s)) + 1.0e0) ** 2.0e0)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

   1. *-commutative 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in x around 0 99.6%

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

   1. associate-*r/ 99.6%

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

   2. mul-1-neg 99.6%

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

7. Simplified 99.6%

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

8. Taylor expanded in x around 0 99.6%

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

   1. associate-/r* 99.7%

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

   2. mul-1-neg 99.7%

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

   3. distribute-neg-frac2 99.7%

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

   4. mul-1-neg 99.7%

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

   5. distribute-neg-frac2 99.7%

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

10. Simplified 99.7%

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

    1. add-sqr-sqrt 49.0%

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

    2. fabs-sqr 49.0%

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

    3. add-sqr-sqrt 64.1%

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

    4. distribute-frac-neg2 64.1%

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

    5. rem-log-exp 64.1%

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

    6. rec-exp 64.0%

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

    7. frac-2neg 64.0%

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

    8. metadata-eval 64.0%

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

    9. add-exp-log 64.0%

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

    10. div-inv 64.0%

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

12. Applied egg-rr 64.0%

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

    1. mul-1-neg 64.0%

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

    2. distribute-frac-neg2 64.0%

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

    3. remove-double-neg 64.0%

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

    4. rec-exp 64.1%

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

    5. distribute-frac-neg 64.1%

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

14. Simplified 64.1%

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

15. Final simplification 64.1%

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

Alternative 5: 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 99.7%

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

   1. *-commutative 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in s around inf 94.4%

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

   1. *-commutative 94.4%

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

7. Simplified 94.4%

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

   1. distribute-frac-neg 94.4%

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

   2. rec-exp 94.4%

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

   3. pow1 94.4%

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

   4. pow1 94.4%

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

   5. frac-2neg 94.4%

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

   6. frac-2neg 94.4%

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

   7. add-sqr-sqrt 46.3%

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

   8. fabs-sqr 46.3%

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

   9. add-sqr-sqrt 61.4%

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

9. Applied egg-rr 61.4%

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

    1. rec-exp 61.4%

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

    2. distribute-neg-frac2 61.4%

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

11. Simplified 61.4%

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

12. Final simplification 61.4%

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

Alternative 6: 77.1% accurate, 47.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

   1. *-commutative 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in x around 0 99.6%

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

   1. associate-*r/ 99.6%

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

   2. mul-1-neg 99.6%

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

7. Simplified 99.6%

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

   1. clear-num 99.7%

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

   2. inv-pow 99.7%

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

9. Applied egg-rr 67.7%

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

    1. unpow-1 67.7%

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

    2. associate-/l* 67.7%

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

11. Simplified 67.7%

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

12. Taylor expanded in x around 0 81.1%

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

    1. unpow2 81.1%

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

    2. unpow2 81.1%

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

    3. frac-times 80.0%

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

14. Applied egg-rr 80.0%

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

15. Final simplification 80.0%

    \[\leadsto \frac{1}{s \cdot \left(4 + \frac{x}{s} \cdot \frac{x}{s}\right)} \]
16. Add Preprocessing

Alternative 7: 27.9% 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 99.7%

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

   1. *-commutative 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in s around inf 94.4%

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

   1. *-commutative 94.4%

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

7. Simplified 94.4%

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

8. Taylor expanded in s around inf 33.9%

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

9. Final simplification 33.9%

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

Reproduce

herbie shell --seed 2024067 
(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))))))