Logistic distribution

Percentage Accurate: 99.6% → 99.6%

Time: 10.3s

Alternatives: 6

Speedup: 1.2×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.6% accurate, 1.5× speedup

Math

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

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (if (<= (fabs x_m) 7.0)
   (/ 1.0 (* s (exp (- (* 2.0 (log1p (exp (/ x_m s)))) (/ x_m s)))))
   (/ (exp (/ x_m (- s))) (* s 4.0))))

C

x_m = fabs(x);
float code(float x_m, float s) {
	float tmp;
	if (fabsf(x_m) <= 7.0f) {
		tmp = 1.0f / (s * expf(((2.0f * log1pf(expf((x_m / s)))) - (x_m / s))));
	} else {
		tmp = expf((x_m / -s)) / (s * 4.0f);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m, s)
	tmp = Float32(0.0)
	if (abs(x_m) <= Float32(7.0))
		tmp = Float32(Float32(1.0) / Float32(s * exp(Float32(Float32(Float32(2.0) * log1p(exp(Float32(x_m / s)))) - Float32(x_m / s)))));
	else
		tmp = Float32(exp(Float32(x_m / Float32(-s))) / Float32(s * Float32(4.0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (fabs.f32 x) < 7

   1. Initial program 99.1%

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

      1. fabs-neg 99.1%

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

      2. distribute-frac-neg 99.1%

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

      3. distribute-frac-neg2 99.1%

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

      4. fabs-neg 99.1%

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

      5. *-commutative 99.1%

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

      6. fabs-neg 99.1%

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

      7. +-commutative 99.1%

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

      8. fabs-neg 99.1%

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

   3. Simplified 99.1%

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

      1. add-exp-log 95.7%

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

      2. log-div 95.4%

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

      3. add-log-exp 95.4%

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

      4. add-sqr-sqrt 51.5%

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

      5. fabs-sqr 51.5%

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

      6. add-sqr-sqrt 73.0%

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

      7. add-sqr-sqrt -0.0%

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

      8. sqrt-unprod 62.7%

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

      9. sqr-neg 62.7%

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

      10. sqrt-unprod 67.9%

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

      11. add-sqr-sqrt 67.9%

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

      12. *-commutative 67.9%

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

   6. Applied egg-rr 94.9%

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

   7. Taylor expanded in x around -inf 95.1%

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

      1. exp-neg 95.1%

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

      2. exp-sum 94.9%

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

      3. rem-exp-log 99.0%

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

      4. +-commutative 99.0%

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

      5. log1p-define 99.3%

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

      6. mul-1-neg 99.3%

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

      7. unsub-neg 99.3%

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

   9. Simplified 99.3%

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

   if 7 < (fabs.f32 x)

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{e^{\frac{-\left|x\right|}{s}}}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(s + \frac{s}{e^{\frac{\left|x\right|}{s}}}\right)}} \]
   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. add-sqr-sqrt 47.0%

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

      2. fabs-sqr 47.0%

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

      3. add-sqr-sqrt 48.7%

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

      4. distribute-frac-neg 48.7%

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

      5. rec-exp 48.7%

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

   9. Applied egg-rr 48.7%

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

       1. rec-exp 48.7%

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

       2. distribute-frac-neg 48.7%

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

   11. Simplified 48.7%

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

4. Final simplification 72.8%

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

Alternative 2: 99.6% accurate, 1.0× speedup

Math

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

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (let* ((t_0 (exp (/ (fabs x_m) (- s)))) (t_1 (+ t_0 1.0)))
   (/ t_0 (* s (* t_1 t_1)))))

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

   1. fabs-neg 99.6%

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

   2. distribute-frac-neg 99.6%

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

   3. distribute-frac-neg2 99.6%

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

   4. fabs-neg 99.6%

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

   5. *-commutative 99.6%

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

   6. fabs-neg 99.6%

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

   7. +-commutative 99.6%

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

   8. fabs-neg 99.6%

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

3. Simplified 99.6%

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

5. Final simplification 99.6%

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

Alternative 3: 99.6% 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 99.6%

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

3. Simplified 99.6%

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

   1. add-cube-cbrt 99.4%

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

   2. associate-/l* 99.4%

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

   3. pow2 99.4%

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

   4. add-sqr-sqrt 50.1%

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

   5. fabs-sqr 50.1%

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

   6. add-sqr-sqrt 96.9%

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

6. Applied egg-rr 96.9%

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

   1. associate-*r/ 96.9%

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

   2. unpow2 96.9%

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

   3. rem-3cbrt-lft 97.0%

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

8. Simplified 97.0%

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

9. Final simplification 97.0%

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

Alternative 4: 99.6% accurate, 1.5× speedup

Math

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

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (if (<= (fabs x_m) 7.0)
   (/ (exp (+ (/ x_m s) (* (log1p (exp (/ x_m s))) -2.0))) s)
   (/ (exp (/ x_m (- s))) (* s 4.0))))

C

x_m = fabs(x);
float code(float x_m, float s) {
	float tmp;
	if (fabsf(x_m) <= 7.0f) {
		tmp = expf(((x_m / s) + (log1pf(expf((x_m / s))) * -2.0f))) / s;
	} else {
		tmp = expf((x_m / -s)) / (s * 4.0f);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m, s)
	tmp = Float32(0.0)
	if (abs(x_m) <= Float32(7.0))
		tmp = Float32(exp(Float32(Float32(x_m / s) + Float32(log1p(exp(Float32(x_m / s))) * Float32(-2.0)))) / s);
	else
		tmp = Float32(exp(Float32(x_m / Float32(-s))) / Float32(s * Float32(4.0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (fabs.f32 x) < 7

   1. Initial program 99.1%

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

      1. fabs-neg 99.1%

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

      2. distribute-frac-neg 99.1%

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

      3. distribute-frac-neg2 99.1%

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

      4. fabs-neg 99.1%

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

      5. *-commutative 99.1%

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

      6. fabs-neg 99.1%

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

      7. +-commutative 99.1%

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

      8. fabs-neg 99.1%

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

   3. Simplified 99.1%

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

      1. div-inv 99.1%

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

      2. distribute-frac-neg2 99.1%

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

      3. distribute-frac-neg 99.1%

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

      4. add-sqr-sqrt -0.0%

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

      5. sqrt-unprod 47.8%

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

      6. sqr-neg 47.8%

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

      7. sqrt-unprod 47.4%

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

      8. add-sqr-sqrt 47.4%

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

      9. add-sqr-sqrt 25.0%

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

      10. fabs-sqr 25.0%

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

      11. add-sqr-sqrt 70.5%

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

      12. add-sqr-sqrt 69.9%

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

   6. Applied egg-rr 76.2%

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

      1. associate-*r/ 76.3%

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

      2. *-rgt-identity 76.3%

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

      3. associate-/r* 76.2%

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

      4. +-commutative 76.2%

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

   8. Simplified 76.2%

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

      1. div-inv 76.2%

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

      2. div-inv 76.2%

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

      3. associate-*l* 76.2%

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

      4. pow-flip 76.2%

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

      5. +-commutative 76.2%

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

      6. metadata-eval 76.2%

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

   10. Applied egg-rr 76.2%

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

       1. associate-*r* 76.2%

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

       2. associate-*r/ 76.3%

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

       3. *-rgt-identity 76.3%

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

       4. associate-*l/ 76.4%

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

       5. exp-to-pow 76.4%

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

       6. +-commutative 76.4%

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

       7. log1p-define 76.6%

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

       8. *-commutative 76.6%

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

       9. exp-sum 99.3%

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

   12. Simplified 99.3%

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

   if 7 < (fabs.f32 x)

   1. Initial program 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{e^{\frac{-\left|x\right|}{s}}}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \left(s + \frac{s}{e^{\frac{\left|x\right|}{s}}}\right)}} \]
   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. add-sqr-sqrt 47.0%

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

      2. fabs-sqr 47.0%

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

      3. add-sqr-sqrt 48.7%

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

      4. distribute-frac-neg 48.7%

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

      5. rec-exp 48.7%

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

   9. Applied egg-rr 48.7%

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

       1. rec-exp 48.7%

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

       2. distribute-frac-neg 48.7%

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

   11. Simplified 48.7%

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

4. Final simplification 72.8%

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

Alternative 5: 95.1% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in s around inf 94.2%

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

   1. *-commutative 94.2%

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

7. Simplified 94.2%

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

   1. add-sqr-sqrt 47.1%

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

   2. fabs-sqr 47.1%

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

   3. add-sqr-sqrt 58.8%

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

   4. distribute-frac-neg 58.8%

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

   5. rec-exp 58.8%

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

9. Applied egg-rr 58.8%

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

    1. rec-exp 58.8%

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

    2. distribute-frac-neg 58.8%

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

11. Simplified 58.8%

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

12. Final simplification 58.8%

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

Alternative 6: 26.4% 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 99.6%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
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 99.6%

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

5. Taylor expanded in s around inf 27.2%

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

Reproduce

herbie shell --seed 2024106 
(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))))))