Logistic distribution

Percentage Accurate: 99.5% → 99.4%

Time: 17.2s

Alternatives: 8

Speedup: 1.5×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{1}{\left(1 + e^{\frac{-\left|x\_m\right|}{s}}\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(s \cdot \left(1 + {\left(e^{2}\right)}^{\left(\frac{x\_m}{s \cdot 2}\right)}\right)\right)\right)} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (/
  1.0
  (*
   (+ 1.0 (exp (/ (- (fabs x_m)) s)))
   (expm1 (log1p (* s (+ 1.0 (pow (exp 2.0) (/ x_m (* s 2.0))))))))))

C

x_m = fabs(x);
float code(float x_m, float s) {
	return 1.0f / ((1.0f + expf((-fabsf(x_m) / s))) * expm1f(log1pf((s * (1.0f + powf(expf(2.0f), (x_m / (s * 2.0f))))))));
}

Julia

x_m = abs(x)
function code(x_m, s)
	return Float32(Float32(1.0) / Float32(Float32(Float32(1.0) + exp(Float32(Float32(-abs(x_m)) / s))) * expm1(log1p(Float32(s * Float32(Float32(1.0) + (exp(Float32(2.0)) ^ Float32(x_m / Float32(s * Float32(2.0))))))))))
end

Derivation

1. Initial program 99.1%

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

3. Simplified 99.2%

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

   1. fma-udef 99.2%

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

   2. *-commutative 99.2%

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

   3. add-sqr-sqrt 99.1%

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

   4. sqrt-unprod 94.8%

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

   5. sqr-neg 94.8%

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

   6. sqrt-unprod -0.0%

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

   7. add-sqr-sqrt 21.1%

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

   8. frac-2neg 21.1%

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

   9. frac-2neg 21.1%

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

   10. +-commutative 21.1%

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

   11. distribute-rgt1-in 21.1%

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

   12. +-commutative 21.1%

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

   13. add-exp-log 20.1%

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

   14. log-prod 20.1%

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

6. Applied egg-rr 57.3%

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

   1. expm1-log1p-u 57.3%

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

   2. exp-sum 57.5%

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

   3. add-exp-log 58.6%

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

   4. *-commutative 58.6%

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

   5. log1p-udef 58.7%

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

   6. rem-exp-log 58.7%

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

8. Applied egg-rr 58.7%

   \[\leadsto \frac{1}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(s \cdot \left(1 + e^{\frac{x}{s}}\right)\right)\right)}} \]
9. Step-by-step derivation

   1. *-un-lft-identity 58.7%

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

   2. pow-exp 58.6%

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

   3. e-exp-1 58.6%

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

   4. sqr-pow 58.6%

      \[\leadsto \frac{1}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(s \cdot \left(1 + \color{blue}{{e}^{\left(\frac{\frac{x}{s}}{2}\right)} \cdot {e}^{\left(\frac{\frac{x}{s}}{2}\right)}}\right)\right)\right)} \]

   5. pow-prod-down 58.6%

      \[\leadsto \frac{1}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(s \cdot \left(1 + \color{blue}{{\left(e \cdot e\right)}^{\left(\frac{\frac{x}{s}}{2}\right)}}\right)\right)\right)} \]

   6. e-exp-1 58.6%

      \[\leadsto \frac{1}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(s \cdot \left(1 + {\left(\color{blue}{e^{1}} \cdot e\right)}^{\left(\frac{\frac{x}{s}}{2}\right)}\right)\right)\right)} \]

   7. e-exp-1 58.6%

      \[\leadsto \frac{1}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(s \cdot \left(1 + {\left(e^{1} \cdot \color{blue}{e^{1}}\right)}^{\left(\frac{\frac{x}{s}}{2}\right)}\right)\right)\right)} \]

   8. prod-exp 58.7%

      \[\leadsto \frac{1}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(s \cdot \left(1 + {\color{blue}{\left(e^{1 + 1}\right)}}^{\left(\frac{\frac{x}{s}}{2}\right)}\right)\right)\right)} \]

   9. metadata-eval 58.7%

      \[\leadsto \frac{1}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(s \cdot \left(1 + {\left(e^{\color{blue}{2}}\right)}^{\left(\frac{\frac{x}{s}}{2}\right)}\right)\right)\right)} \]

   10. add-sqr-sqrt 47.9%

       \[\leadsto \frac{1}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(s \cdot \left(1 + {\left(e^{2}\right)}^{\left(\frac{\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{s}}{2}\right)}\right)\right)\right)} \]

   11. fabs-sqr 47.9%

       \[\leadsto \frac{1}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(s \cdot \left(1 + {\left(e^{2}\right)}^{\left(\frac{\frac{\color{blue}{\left|\sqrt{x} \cdot \sqrt{x}\right|}}{s}}{2}\right)}\right)\right)\right)} \]

   12. add-sqr-sqrt 99.2%

       \[\leadsto \frac{1}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(s \cdot \left(1 + {\left(e^{2}\right)}^{\left(\frac{\frac{\left|\color{blue}{x}\right|}{s}}{2}\right)}\right)\right)\right)} \]

   13. associate-/l/ 99.2%

       \[\leadsto \frac{1}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(s \cdot \left(1 + {\left(e^{2}\right)}^{\color{blue}{\left(\frac{\left|x\right|}{2 \cdot s}\right)}}\right)\right)\right)} \]

   14. add-sqr-sqrt 47.9%

       \[\leadsto \frac{1}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(s \cdot \left(1 + {\left(e^{2}\right)}^{\left(\frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}{2 \cdot s}\right)}\right)\right)\right)} \]

   15. fabs-sqr 47.9%

       \[\leadsto \frac{1}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(s \cdot \left(1 + {\left(e^{2}\right)}^{\left(\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{2 \cdot s}\right)}\right)\right)\right)} \]

   16. add-sqr-sqrt 58.7%

       \[\leadsto \frac{1}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(s \cdot \left(1 + {\left(e^{2}\right)}^{\left(\frac{\color{blue}{x}}{2 \cdot s}\right)}\right)\right)\right)} \]

10. Applied egg-rr 58.7%

    \[\leadsto \frac{1}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(s \cdot \left(1 + \color{blue}{{\left(e^{2}\right)}^{\left(\frac{x}{2 \cdot s}\right)}}\right)\right)\right)} \]
11. Step-by-step derivation

    1. *-commutative 58.7%

       \[\leadsto \frac{1}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(s \cdot \left(1 + {\left(e^{2}\right)}^{\left(\frac{x}{\color{blue}{s \cdot 2}}\right)}\right)\right)\right)} \]

12. Simplified 58.7%

    \[\leadsto \frac{1}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(s \cdot \left(1 + \color{blue}{{\left(e^{2}\right)}^{\left(\frac{x}{s \cdot 2}\right)}}\right)\right)\right)} \]

13. Final simplification 58.7%

    \[\leadsto \frac{1}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(s \cdot \left(1 + {\left(e^{2}\right)}^{\left(\frac{x}{s \cdot 2}\right)}\right)\right)\right)} \]
14. Add Preprocessing

Alternative 2: 99.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in x around 0 99.1%

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

   1. associate-/r* 99.2%

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

   2. associate-*r/ 99.2%

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

   3. mul-1-neg 99.2%

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

   4. associate-*r/ 99.2%

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

   5. mul-1-neg 99.2%

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

5. Simplified 99.2%

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

   1. clear-num 99.1%

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

   2. inv-pow 99.1%

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

7. Applied egg-rr 57.8%

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

   1. unpow-1 57.8%

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

   2. *-commutative 57.8%

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

   3. associate-/r* 57.8%

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

   4. rec-exp 57.9%

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

   5. distribute-neg-frac 57.9%

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

9. Simplified 57.9%

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

10. Final simplification 57.9%

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

Alternative 3: 99.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

3. Simplified 99.2%

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

   1. fma-udef 93.6%

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

   2. +-commutative 93.6%

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

   3. *-commutative 93.6%

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

   4. add-sqr-sqrt 93.6%

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

   5. sqrt-unprod 92.2%

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

   6. sqr-neg 92.2%

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

   7. sqrt-unprod -0.0%

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

   8. add-sqr-sqrt 21.3%

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

   9. frac-2neg 21.3%

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

   10. frac-2neg 21.3%

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

   11. distribute-rgt1-in 21.3%

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

6. Applied egg-rr 58.6%

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

7. Final simplification 58.6%

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

Alternative 4: 63.0% accurate, 3.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\left|x\_m\right| \leq 9.999999717180685 \cdot 10^{-10}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{s}{{x\_m}^{2}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (if (<= (fabs x_m) 9.999999717180685e-10) (/ 0.25 s) (/ s (pow x_m 2.0))))

C

x_m = fabs(x);
float code(float x_m, float s) {
	float tmp;
	if (fabsf(x_m) <= 9.999999717180685e-10f) {
		tmp = 0.25f / s;
	} else {
		tmp = s / powf(x_m, 2.0f);
	}
	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 (abs(x_m) <= 9.999999717180685e-10) then
        tmp = 0.25e0 / s
    else
        tmp = s / (x_m ** 2.0e0)
    end if
    code = tmp
end function

Julia

x_m = abs(x)
function code(x_m, s)
	tmp = Float32(0.0)
	if (abs(x_m) <= Float32(9.999999717180685e-10))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(s / (x_m ^ Float32(2.0)));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, s)
	tmp = single(0.0);
	if (abs(x_m) <= single(9.999999717180685e-10))
		tmp = single(0.25) / s;
	else
		tmp = s / (x_m ^ single(2.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (fabs.f32 x) < 9.99999972e-10

   1. Initial program 98.1%

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

   3. Taylor expanded in s around inf 52.2%

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

   if 9.99999972e-10 < (fabs.f32 x)

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in s around inf 98.6%

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

   6. Taylor expanded in s around inf 66.2%

      \[\leadsto \frac{1}{2 \cdot \color{blue}{\left(\left|x\right| + \left(0.5 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s} + 2 \cdot s\right)\right)}} \]
   7. Step-by-step derivation

      1. metadata-eval 66.2%

         \[\leadsto \frac{1}{2 \cdot \left(\left|x\right| + \left(\color{blue}{\left(-0.5 + 1\right)} \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s} + 2 \cdot s\right)\right)} \]

      2. distribute-lft1-in 4.2%

         \[\leadsto \frac{1}{2 \cdot \left(\left|x\right| + \left(\color{blue}{\left(-0.5 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s} + \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)} + 2 \cdot s\right)\right)} \]

      3. +-commutative 4.2%

         \[\leadsto \frac{1}{2 \cdot \left(\left|x\right| + \color{blue}{\left(2 \cdot s + \left(-0.5 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s} + \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)\right)}\right)} \]

      4. *-commutative 4.2%

         \[\leadsto \frac{1}{2 \cdot \left(\left|x\right| + \left(\color{blue}{s \cdot 2} + \left(-0.5 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s} + \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)\right)\right)} \]

      5. fma-def 4.2%

         \[\leadsto \frac{1}{2 \cdot \left(\left|x\right| + \color{blue}{\mathsf{fma}\left(s, 2, -0.5 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s} + \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)}\right)} \]

      6. distribute-lft1-in 66.2%

         \[\leadsto \frac{1}{2 \cdot \left(\left|x\right| + \mathsf{fma}\left(s, 2, \color{blue}{\left(-0.5 + 1\right) \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}}\right)\right)} \]

      7. metadata-eval 66.2%

         \[\leadsto \frac{1}{2 \cdot \left(\left|x\right| + \mathsf{fma}\left(s, 2, \color{blue}{0.5} \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)\right)} \]

      8. associate-*r/ 66.2%

         \[\leadsto \frac{1}{2 \cdot \left(\left|x\right| + \mathsf{fma}\left(s, 2, \color{blue}{\frac{0.5 \cdot {\left(\left|x\right|\right)}^{2}}{s}}\right)\right)} \]

      9. unpow2 66.2%

         \[\leadsto \frac{1}{2 \cdot \left(\left|x\right| + \mathsf{fma}\left(s, 2, \frac{0.5 \cdot \color{blue}{\left(\left|x\right| \cdot \left|x\right|\right)}}{s}\right)\right)} \]

      10. sqr-abs 66.2%

          \[\leadsto \frac{1}{2 \cdot \left(\left|x\right| + \mathsf{fma}\left(s, 2, \frac{0.5 \cdot \color{blue}{\left(x \cdot x\right)}}{s}\right)\right)} \]

   8. Simplified 66.2%

      \[\leadsto \frac{1}{2 \cdot \color{blue}{\left(\left|x\right| + \mathsf{fma}\left(s, 2, \frac{0.5 \cdot \left(x \cdot x\right)}{s}\right)\right)}} \]

   9. Taylor expanded in x around inf 64.6%

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

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 9.999999717180685 \cdot 10^{-10}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{s}{{x}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 29.8% accurate, 3.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\left|x\_m\right| \leq 9.999999717180685 \cdot 10^{-10}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\left|x\_m\right|}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (if (<= (fabs x_m) 9.999999717180685e-10) (/ 0.25 s) (/ 0.5 (fabs x_m))))

C

x_m = fabs(x);
float code(float x_m, float s) {
	float tmp;
	if (fabsf(x_m) <= 9.999999717180685e-10f) {
		tmp = 0.25f / s;
	} else {
		tmp = 0.5f / fabsf(x_m);
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(4) function code(x_m, s)
    real(4), intent (in) :: x_m
    real(4), intent (in) :: s
    real(4) :: tmp
    if (abs(x_m) <= 9.999999717180685e-10) then
        tmp = 0.25e0 / s
    else
        tmp = 0.5e0 / abs(x_m)
    end if
    code = tmp
end function

Julia

x_m = abs(x)
function code(x_m, s)
	tmp = Float32(0.0)
	if (abs(x_m) <= Float32(9.999999717180685e-10))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(0.5) / abs(x_m));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, s)
	tmp = single(0.0);
	if (abs(x_m) <= single(9.999999717180685e-10))
		tmp = single(0.25) / s;
	else
		tmp = single(0.5) / abs(x_m);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (fabs.f32 x) < 9.99999972e-10

   1. Initial program 98.1%

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

   3. Taylor expanded in s around inf 52.2%

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

   if 9.99999972e-10 < (fabs.f32 x)

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in s around inf 98.6%

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

   6. Taylor expanded in s around inf 10.3%

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

      1. *-commutative 10.3%

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

   8. Simplified 10.3%

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

   9. Taylor expanded in s around 0 10.2%

      \[\leadsto \color{blue}{\frac{0.5}{\left|x\right|}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 26.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 9.999999717180685 \cdot 10^{-10}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\left|x\right|}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 95.2% accurate, 5.6× speedup

Math

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

FPCore

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

C

x_m = fabs(x);
float code(float x_m, float s) {
	return 1.0f / (2.0f * (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 = 1.0e0 / (2.0e0 * (s * (1.0e0 + exp((x_m / s)))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

3. Simplified 99.2%

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

5. Taylor expanded in s around inf 93.6%

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

   1. fma-udef 93.6%

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

   2. +-commutative 93.6%

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

   3. *-commutative 93.6%

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

   4. add-sqr-sqrt 93.6%

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

   5. sqrt-unprod 92.2%

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

   6. sqr-neg 92.2%

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

   7. sqrt-unprod -0.0%

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

   8. add-sqr-sqrt 21.3%

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

   9. frac-2neg 21.3%

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

   10. frac-2neg 21.3%

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

   11. distribute-rgt1-in 21.3%

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

7. Applied egg-rr 56.4%

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

8. Final simplification 56.4%

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

Alternative 7: 94.8% accurate, 5.7× speedup

Math

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

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(Float32(exp(Float32(Float32(-x_m) / s)) / s) / Float32(4.0))
end

MATLAB

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in x around 0 99.1%

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

   1. associate-/r* 99.2%

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

   2. associate-*r/ 99.2%

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

   3. mul-1-neg 99.2%

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

   4. associate-*r/ 99.2%

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

   5. mul-1-neg 99.2%

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

5. Simplified 99.2%

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

   1. clear-num 99.1%

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

   2. inv-pow 99.1%

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

7. Applied egg-rr 57.8%

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

   1. unpow-1 57.8%

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

   2. *-commutative 57.8%

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

   3. associate-/r* 57.8%

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

   4. rec-exp 57.9%

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

   5. distribute-neg-frac 57.9%

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

9. Simplified 57.9%

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

10. Taylor expanded in s around inf 55.5%

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

11. Final simplification 55.5%

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

Alternative 8: 27.2% accurate, 206.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in s around inf 23.0%

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

4. Final simplification 23.0%

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

Reproduce

herbie shell --seed 2024040 
(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))))))