Logistic distribution

Percentage Accurate: 99.4% → 99.4%

Time: 13.7s

Alternatives: 11

Speedup: 1.0×

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: 99.4% 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: 99.4% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if (fabs.f32 x) < 1

   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

   6. Applied egg-rr 76.6%

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

      1. *-lft-identity 76.6%

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

      2. *-commutative 76.6%

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

      3. exp-to-pow 76.5%

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

      4. log1p-undefine 76.7%

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

      5. *-commutative 76.7%

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

      6. rem-exp-log 72.2%

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

      7. exp-sum 71.9%

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

      8. exp-diff 94.6%

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

      9. associate--r+ 94.7%

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

      10. exp-diff 95.1%

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

      11. cancel-sign-sub-inv 95.1%

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

      12. metadata-eval 95.1%

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

   8. Simplified 99.6%

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

   if 1 < (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 s around inf 100.0%

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

      1. clear-num 100.0%

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

      2. inv-pow 100.0%

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

   7. Applied egg-rr 56.1%

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

      1. unpow-1 56.1%

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

      2. associate-*l* 56.1%

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

   9. Simplified 56.1%

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

       1. add-exp-log 56.1%

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

       2. log-rec 56.1%

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

       3. associate-*r* 56.1%

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

       4. log-prod 56.1%

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

       5. add-log-exp 56.1%

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

   11. Applied egg-rr 56.1%

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

   12. Taylor expanded in s around 0 56.1%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 1:\\ \;\;\;\;\frac{e^{\frac{x}{s} + -2 \cdot \mathsf{log1p}\left(e^{\frac{x}{s}}\right)}}{s}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{-s}}\\ \end{array} \]
5. 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 99.7%

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

3. Simplified 99.8%

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

5. Final simplification 99.8%

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

Alternative 3: 87.1% accurate, 5.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 7.199999771511762 \cdot 10^{-21}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x\_m}{-s}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (if (<= x_m 7.199999771511762e-21) (/ 0.25 s) (exp (/ x_m (- s)))))

C

x_m = fabs(x);
float code(float x_m, float s) {
	float tmp;
	if (x_m <= 7.199999771511762e-21f) {
		tmp = 0.25f / s;
	} else {
		tmp = 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 <= 7.199999771511762e-21) then
        tmp = 0.25e0 / s
    else
        tmp = 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(7.199999771511762e-21))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = exp(Float32(x_m / Float32(-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(7.199999771511762e-21))
		tmp = single(0.25) / s;
	else
		tmp = exp((x_m / -s));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 7.1999998e-21

   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 s around inf 42.8%

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

   if 7.1999998e-21 < 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 s around inf 93.5%

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

      1. clear-num 93.5%

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

      2. inv-pow 93.5%

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

   7. Applied egg-rr 93.5%

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

      1. unpow-1 93.5%

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

      2. associate-*l* 93.5%

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

   9. Simplified 93.5%

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

       1. add-exp-log 93.3%

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

       2. log-rec 93.3%

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

       3. associate-*r* 93.3%

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

       4. log-prod 93.3%

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

       5. add-log-exp 93.3%

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

   11. Applied egg-rr 93.3%

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

   12. Taylor expanded in s around 0 87.9%

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

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7.199999771511762 \cdot 10^{-21}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{-s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.8% accurate, 5.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{\frac{0.5}{s}}{1 + e^{\frac{x\_m}{s}}} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s) :precision binary32 (/ (/ 0.5 s) (+ 1.0 (exp (/ x_m s)))))

C

x_m = fabs(x);
float code(float x_m, float s) {
	return (0.5f / s) / (1.0f + 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
    code = (0.5e0 / s) / (1.0e0 + exp((x_m / s)))
end function

Julia

x_m = abs(x)
function code(x_m, s)
	return Float32(Float32(Float32(0.5) / s) / Float32(Float32(1.0) + exp(Float32(x_m / s))))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m, s)
	tmp = (single(0.5) / s) / (single(1.0) + 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. fabs-neg 99.7%

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

   2. distribute-frac-neg 99.7%

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

   3. distribute-frac-neg2 99.7%

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

   4. fabs-neg 99.7%

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

   5. *-commutative 99.7%

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

   6. fabs-neg 99.7%

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

   7. +-commutative 99.7%

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

   8. fabs-neg 99.7%

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

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

6. Applied egg-rr 63.8%

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

   1. associate-*l/ 63.9%

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

   2. *-lft-identity 63.9%

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

8. Simplified 63.9%

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

9. Taylor expanded in x around 0 64.5%

   \[\leadsto \frac{\color{blue}{\frac{0.5}{s}}}{1 + e^{\frac{x}{s}}} \]

10. Final simplification 64.5%

    \[\leadsto \frac{\frac{0.5}{s}}{1 + e^{\frac{x}{s}}} \]
11. Add Preprocessing

Alternative 5: 94.6% accurate, 5.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

x_m = abs(x);
function tmp = code(x_m, s)
	tmp = single(0.25) / (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. fabs-neg 99.7%

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

   2. distribute-frac-neg 99.7%

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

   3. distribute-frac-neg2 99.7%

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

   4. fabs-neg 99.7%

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

   5. *-commutative 99.7%

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

   6. fabs-neg 99.7%

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

   7. +-commutative 99.7%

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

   8. fabs-neg 99.7%

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

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 s around inf 93.7%

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

   1. clear-num 93.7%

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

   2. inv-pow 93.7%

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

7. Applied egg-rr 63.7%

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

   1. unpow-1 63.7%

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

   2. associate-*l* 63.7%

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

9. Simplified 63.7%

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

10. Taylor expanded in s around 0 63.7%

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

11. Final simplification 63.7%

    \[\leadsto \frac{0.25}{s \cdot e^{\frac{x}{s}}} \]
12. Add Preprocessing

Alternative 6: 62.7% accurate, 47.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{0.25}{s + x\_m \cdot \left(1 + \frac{x\_m}{s} \cdot 0.5\right)} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (/ 0.25 (+ s (* x_m (+ 1.0 (* (/ x_m s) 0.5))))))

C

x_m = fabs(x);
float code(float x_m, float s) {
	return 0.25f / (s + (x_m * (1.0f + ((x_m / s) * 0.5f))));
}

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 + (x_m * (1.0e0 + ((x_m / s) * 0.5e0))))
end function

Julia

x_m = abs(x)
function code(x_m, s)
	return Float32(Float32(0.25) / Float32(s + Float32(x_m * Float32(Float32(1.0) + Float32(Float32(x_m / s) * Float32(0.5))))))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m, s)
	tmp = single(0.25) / (s + (x_m * (single(1.0) + ((x_m / s) * single(0.5)))));
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. fabs-neg 99.7%

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

   2. distribute-frac-neg 99.7%

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

   3. distribute-frac-neg2 99.7%

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

   4. fabs-neg 99.7%

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

   5. *-commutative 99.7%

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

   6. fabs-neg 99.7%

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

   7. +-commutative 99.7%

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

   8. fabs-neg 99.7%

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

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 s around inf 93.7%

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

   1. clear-num 93.7%

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

   2. inv-pow 93.7%

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

7. Applied egg-rr 63.7%

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

   1. unpow-1 63.7%

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

   2. associate-*l* 63.7%

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

9. Simplified 63.7%

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

10. Taylor expanded in s around 0 63.7%

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

11. Taylor expanded in x around 0 61.3%

    \[\leadsto \frac{0.25}{\color{blue}{s + x \cdot \left(1 + 0.5 \cdot \frac{x}{s}\right)}} \]
12. Step-by-step derivation

    1. *-commutative 61.3%

       \[\leadsto \frac{0.25}{s + x \cdot \left(1 + \color{blue}{\frac{x}{s} \cdot 0.5}\right)} \]

13. Simplified 61.3%

    \[\leadsto \frac{0.25}{\color{blue}{s + x \cdot \left(1 + \frac{x}{s} \cdot 0.5\right)}} \]

14. Final simplification 61.3%

    \[\leadsto \frac{0.25}{s + x \cdot \left(1 + \frac{x}{s} \cdot 0.5\right)} \]
15. Add Preprocessing

Alternative 7: 51.3% accurate, 56.4× speedup

Math

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

FPCore

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

C

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

MATLAB

x_m = abs(x);
function tmp = code(x_m, s)
	tmp = single(1.0) / (s * (single(4.0) + ((x_m / 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. fabs-neg 99.7%

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

   2. distribute-frac-neg 99.7%

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

   3. distribute-frac-neg2 99.7%

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

   4. fabs-neg 99.7%

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

   5. *-commutative 99.7%

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

   6. fabs-neg 99.7%

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

   7. +-commutative 99.7%

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

   8. fabs-neg 99.7%

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

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 s around inf 93.7%

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

   1. clear-num 93.7%

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

   2. inv-pow 93.7%

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

7. Applied egg-rr 63.7%

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

   1. unpow-1 63.7%

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

   2. associate-*l* 63.7%

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

9. Simplified 63.7%

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

10. Taylor expanded in s around inf 51.8%

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

11. Final simplification 51.8%

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

Alternative 8: 50.6% accurate, 68.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{0.25}{s \cdot \left(1 + \frac{x\_m}{s}\right)} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s) :precision binary32 (/ 0.25 (* s (+ 1.0 (/ x_m s)))))

C

x_m = fabs(x);
float code(float x_m, float s) {
	return 0.25f / (s * (1.0f + (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 = 0.25e0 / (s * (1.0e0 + (x_m / s)))
end function

Julia

x_m = abs(x)
function code(x_m, s)
	return Float32(Float32(0.25) / Float32(s * Float32(Float32(1.0) + Float32(x_m / s))))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m, s)
	tmp = single(0.25) / (s * (single(1.0) + (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. fabs-neg 99.7%

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

   2. distribute-frac-neg 99.7%

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

   3. distribute-frac-neg2 99.7%

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

   4. fabs-neg 99.7%

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

   5. *-commutative 99.7%

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

   6. fabs-neg 99.7%

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

   7. +-commutative 99.7%

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

   8. fabs-neg 99.7%

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

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 s around inf 93.7%

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

   1. clear-num 93.7%

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

   2. inv-pow 93.7%

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

7. Applied egg-rr 63.7%

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

   1. unpow-1 63.7%

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

   2. associate-*l* 63.7%

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

9. Simplified 63.7%

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

10. Taylor expanded in s around 0 63.7%

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

11. Taylor expanded in s around inf 51.1%

    \[\leadsto \frac{0.25}{\color{blue}{s \cdot \left(1 + \frac{x}{s}\right)}} \]

12. Final simplification 51.1%

    \[\leadsto \frac{0.25}{s \cdot \left(1 + \frac{x}{s}\right)} \]
13. Add Preprocessing

Alternative 9: 28.6% accurate, 124.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

x_m = abs(x);
function tmp = code(x_m, s)
	tmp = single(0.25) / (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. fabs-neg 99.7%

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

   2. distribute-frac-neg 99.7%

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

   3. distribute-frac-neg2 99.7%

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

   4. fabs-neg 99.7%

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

   5. *-commutative 99.7%

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

   6. fabs-neg 99.7%

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

   7. +-commutative 99.7%

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

   8. fabs-neg 99.7%

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

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 s around inf 93.7%

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

   1. clear-num 93.7%

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

   2. inv-pow 93.7%

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

7. Applied egg-rr 63.7%

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

   1. unpow-1 63.7%

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

   2. associate-*l* 63.7%

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

9. Simplified 63.7%

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

10. Taylor expanded in s around 0 63.7%

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

11. Taylor expanded in x around 0 32.7%

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

12. Final simplification 32.7%

    \[\leadsto \frac{0.25}{x + s} \]
13. Add Preprocessing

Alternative 10: 26.6% 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. fabs-neg 99.7%

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

   2. distribute-frac-neg 99.7%

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

   3. distribute-frac-neg2 99.7%

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

   4. fabs-neg 99.7%

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

   5. *-commutative 99.7%

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

   6. fabs-neg 99.7%

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

   7. +-commutative 99.7%

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

   8. fabs-neg 99.7%

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

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 s around inf 31.6%

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

6. Final simplification 31.6%

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

Alternative 11: 8.2% 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 99.7%

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

   1. fabs-neg 99.7%

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

   2. distribute-frac-neg 99.7%

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

   3. distribute-frac-neg2 99.7%

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

   4. fabs-neg 99.7%

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

   5. *-commutative 99.7%

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

   6. fabs-neg 99.7%

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

   7. +-commutative 99.7%

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

   8. fabs-neg 99.7%

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

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 s around inf 93.7%

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

   1. clear-num 93.7%

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

   2. inv-pow 93.7%

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

7. Applied egg-rr 63.7%

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

   1. unpow-1 63.7%

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

   2. associate-*l* 63.7%

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

9. Simplified 63.7%

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

    1. add-exp-log 62.1%

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

    2. log-rec 62.1%

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

    3. associate-*r* 62.1%

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

    4. log-prod 62.1%

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

    5. add-log-exp 62.1%

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

11. Applied egg-rr 62.1%

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

    1. neg-sub0 62.1%

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

    2. metadata-eval 62.1%

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

    3. add-log-exp 62.1%

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

    4. sum-log 62.1%

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

    5. expm1-log1p-u 62.1%

       \[\leadsto e^{\log 1 - \log \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\left(s \cdot 4\right) \cdot e^{\frac{x}{s}}\right)\right)\right)}} \]

    6. log-div 62.1%

       \[\leadsto e^{\color{blue}{\log \left(\frac{1}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(s \cdot 4\right) \cdot e^{\frac{x}{s}}\right)\right)}\right)}} \]

    7. add-exp-log 63.7%

       \[\leadsto \color{blue}{\frac{1}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(s \cdot 4\right) \cdot e^{\frac{x}{s}}\right)\right)}} \]

    8. add-sqr-sqrt 63.3%

       \[\leadsto \frac{1}{\color{blue}{\sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(s \cdot 4\right) \cdot e^{\frac{x}{s}}\right)\right)} \cdot \sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(s \cdot 4\right) \cdot e^{\frac{x}{s}}\right)\right)}}} \]

    9. associate-/r* 63.3%

       \[\leadsto \color{blue}{\frac{\frac{1}{\sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(s \cdot 4\right) \cdot e^{\frac{x}{s}}\right)\right)}}}{\sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(s \cdot 4\right) \cdot e^{\frac{x}{s}}\right)\right)}}} \]

13. Applied egg-rr 4.3%

    \[\leadsto \color{blue}{\frac{\frac{0.5}{\sqrt{s \cdot e^{\frac{x}{s}}}}}{\frac{0.5}{\sqrt{s \cdot e^{\frac{x}{s}}}}}} \]
14. Step-by-step derivation

    1. *-inverses 8.5%

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

15. Simplified 8.5%

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

16. Final simplification 8.5%

    \[\leadsto 1 \]
17. Add Preprocessing

Reproduce

herbie shell --seed 2024077 
(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))))))