Logistic distribution

Percentage Accurate: 99.5% → 99.5%

Time: 12.4s

Alternatives: 8

Speedup: 2.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \mathbf{if}\;\left|x\right| \leq 40:\\ \;\;\;\;\frac{e^{\frac{x}{s} + -2 \cdot \mathsf{log1p}\left(t_0\right)}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{s \cdot \left(1 + t_0\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ x s))))
   (if (<= (fabs x) 40.0)
     (/ (exp (+ (/ x s) (* -2.0 (log1p t_0)))) s)
     (/ 0.5 (* s (+ 1.0 t_0))))))

C

x = abs(x);
float code(float x, float s) {
	float t_0 = expf((x / s));
	float tmp;
	if (fabsf(x) <= 40.0f) {
		tmp = expf(((x / s) + (-2.0f * log1pf(t_0)))) / s;
	} else {
		tmp = 0.5f / (s * (1.0f + t_0));
	}
	return tmp;
}

Julia

x = abs(x)
function code(x, s)
	t_0 = exp(Float32(x / s))
	tmp = Float32(0.0)
	if (abs(x) <= Float32(40.0))
		tmp = Float32(exp(Float32(Float32(x / s) + Float32(Float32(-2.0) * log1p(t_0)))) / s);
	else
		tmp = Float32(Float32(0.5) / Float32(s * Float32(Float32(1.0) + t_0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (fabs.f32 x) < 40

   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. associate-/r* 99.2%

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

   3. Simplified 99.2%

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

      1. expm1-log1p-u 95.8%

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

      2. expm1-udef 94.1%

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

   5. Applied egg-rr 73.1%

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

      1. expm1-def 74.2%

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

      2. expm1-log1p 77.6%

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

   7. Simplified 77.6%

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

      1. *-un-lft-identity 77.6%

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

      2. *-commutative 77.6%

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

      3. times-frac 77.5%

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

      4. metadata-eval 77.5%

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

      5. unpow2 77.5%

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

      6. frac-times 77.7%

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

      7. inv-pow 77.7%

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

      8. inv-pow 77.7%

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

      9. pow-prod-up 77.8%

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

      10. +-commutative 77.8%

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

      11. metadata-eval 77.8%

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

   9. Applied egg-rr 77.8%

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

       1. associate-*r/ 77.9%

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

       2. pow-to-exp 77.8%

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

       3. prod-exp 99.1%

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

       4. rem-log-exp 98.9%

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

       5. pow-to-exp 98.9%

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

       6. log-pow 99.1%

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

       7. +-commutative 99.1%

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

       8. log1p-udef 99.3%

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

   11. Applied egg-rr 99.3%

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

   if 40 < (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. associate-/r* 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 55.0%

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

   6. Taylor expanded in x around 0 47.4%

      \[\leadsto \color{blue}{\frac{0.5}{s}} \cdot \frac{1}{1 + e^{\frac{x}{s}}} \]
   7. Step-by-step derivation

      1. *-commutative 47.4%

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

      2. +-commutative 47.4%

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

      3. frac-times 47.4%

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

      4. metadata-eval 47.4%

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

      5. +-commutative 47.4%

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

   8. Applied egg-rr 47.4%

      \[\leadsto \color{blue}{\frac{0.5}{\left(1 + e^{\frac{x}{s}}\right) \cdot s}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 40:\\ \;\;\;\;\frac{e^{\frac{x}{s} + -2 \cdot \mathsf{log1p}\left(e^{\frac{x}{s}}\right)}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{s \cdot \left(1 + e^{\frac{x}{s}}\right)}\\ \end{array} \]

Alternative 2: 99.5% accurate, 1.5× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s)
 :precision binary32
 (/ 1.0 (* (fma s (exp (/ (fabs x) (- s))) s) (+ 1.0 (exp (/ x s))))))

C

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

Julia

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.6%

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

   1. *-un-lft-identity 99.6%

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

   2. div-inv 99.6%

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

   3. exp-prod 81.5%

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

   4. add-sqr-sqrt 81.5%

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

   5. sqrt-unprod 81.5%

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

   6. sqr-neg 81.5%

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

   7. sqrt-unprod -0.0%

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

   8. add-sqr-sqrt 28.7%

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

   9. exp-prod 28.4%

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

   10. div-inv 28.4%

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

   11. frac-2neg 28.4%

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

   12. remove-double-neg 28.4%

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

   13. add-sqr-sqrt -0.0%

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

   14. sqrt-unprod 94.5%

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

   15. sqr-neg 94.5%

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

   16. sqrt-unprod 99.6%

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

   17. add-sqr-sqrt 99.6%

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

   18. add-sqr-sqrt 47.8%

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

   19. fabs-sqr 47.8%

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

   20. add-sqr-sqrt 61.9%

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

5. Applied egg-rr 61.9%

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

   1. *-lft-identity 61.9%

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

7. Simplified 61.9%

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

8. Final simplification 61.9%

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

Alternative 3: 99.6% accurate, 2.0× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s)
 :precision binary32
 (/ 1.0 (* s (* (+ 1.0 (exp (/ x s))) (+ 1.0 (exp (- (/ (fabs x) s))))))))

C

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

Fortran

NOTE: x should be positive before calling this function
real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 1.0e0 / (s * ((1.0e0 + exp((x / s))) * (1.0e0 + exp(-(abs(x) / s)))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.6%

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

   1. *-un-lft-identity 99.6%

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

   2. div-inv 99.6%

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

   3. exp-prod 81.5%

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

   4. add-sqr-sqrt 81.5%

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

   5. sqrt-unprod 81.5%

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

   6. sqr-neg 81.5%

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

   7. sqrt-unprod -0.0%

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

   8. add-sqr-sqrt 28.7%

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

   9. exp-prod 28.4%

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

   10. div-inv 28.4%

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

   11. frac-2neg 28.4%

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

   12. remove-double-neg 28.4%

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

   13. add-sqr-sqrt -0.0%

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

   14. sqrt-unprod 94.5%

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

   15. sqr-neg 94.5%

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

   16. sqrt-unprod 99.6%

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

   17. add-sqr-sqrt 99.6%

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

   18. add-sqr-sqrt 47.8%

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

   19. fabs-sqr 47.8%

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

   20. add-sqr-sqrt 61.9%

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

5. Applied egg-rr 61.9%

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

   1. *-lft-identity 61.9%

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

7. Simplified 61.9%

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

8. Taylor expanded in s around 0 61.9%

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

   1. mul-1-neg 61.9%

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

10. Simplified 61.9%

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

11. Final simplification 61.9%

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

Alternative 4: 95.2% accurate, 5.7× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s) :precision binary32 (/ 0.5 (* s (+ 1.0 (exp (/ x s))))))

C

x = abs(x);
float code(float x, float s) {
	return 0.5f / (s * (1.0f + expf((x / s))));
}

Fortran

NOTE: x should be positive before calling this function
real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 0.5e0 / (s * (1.0e0 + exp((x / s))))
end function

Julia

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

MATLAB

x = abs(x)
function tmp = code(x, s)
	tmp = single(0.5) / (s * (single(1.0) + exp((x / s))));
end

Derivation

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. associate-/r* 99.6%

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

3. Simplified 99.6%

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

5. Applied egg-rr 68.3%

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

6. Taylor expanded in x around 0 58.9%

   \[\leadsto \color{blue}{\frac{0.5}{s}} \cdot \frac{1}{1 + e^{\frac{x}{s}}} \]
7. Step-by-step derivation

   1. *-commutative 58.9%

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

   2. +-commutative 58.9%

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

   3. frac-times 58.9%

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

   4. metadata-eval 58.9%

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

   5. +-commutative 58.9%

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

8. Applied egg-rr 58.9%

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

9. Final simplification 58.9%

   \[\leadsto \frac{0.5}{s \cdot \left(1 + e^{\frac{x}{s}}\right)} \]

Alternative 5: 51.7% accurate, 55.7× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 0.11999999731779099:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{s} \cdot \frac{1}{\frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s)
 :precision binary32
 (if (<= x 0.11999999731779099) (/ 0.25 s) (* (/ 0.5 s) (/ 1.0 (/ x s)))))

C

x = abs(x);
float code(float x, float s) {
	float tmp;
	if (x <= 0.11999999731779099f) {
		tmp = 0.25f / s;
	} else {
		tmp = (0.5f / s) * (1.0f / (x / s));
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 0.11999999731779099e0) then
        tmp = 0.25e0 / s
    else
        tmp = (0.5e0 / s) * (1.0e0 / (x / s))
    end if
    code = tmp
end function

Julia

x = abs(x)
function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(0.11999999731779099))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(Float32(0.5) / s) * Float32(Float32(1.0) / Float32(x / s)));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(0.11999999731779099))
		tmp = single(0.25) / s;
	else
		tmp = (single(0.5) / s) * (single(1.0) / (x / s));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 0.119999997

   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. associate-/r* 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in s around inf 38.5%

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

   if 0.119999997 < x

   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. associate-/r* 99.6%

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

   3. Simplified 99.6%

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

   5. Applied egg-rr 3.1%

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

   6. Taylor expanded in x around 0 97.5%

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

   7. Taylor expanded in x around 0 43.6%

      \[\leadsto \frac{0.5}{s} \cdot \frac{1}{\color{blue}{2 + \frac{x}{s}}} \]
   8. Step-by-step derivation

      1. +-commutative 43.6%

         \[\leadsto \frac{0.5}{s} \cdot \frac{1}{\color{blue}{\frac{x}{s} + 2}} \]

   9. Simplified 43.6%

      \[\leadsto \frac{0.5}{s} \cdot \frac{1}{\color{blue}{\frac{x}{s} + 2}} \]

   10. Taylor expanded in x around inf 43.6%

       \[\leadsto \frac{0.5}{s} \cdot \frac{1}{\color{blue}{\frac{x}{s}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 39.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.11999999731779099:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{s} \cdot \frac{1}{\frac{x}{s}}\\ \end{array} \]

Alternative 6: 44.7% accurate, 67.9× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 0.11999999731779099:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{s} \cdot \frac{s}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s)
 :precision binary32
 (if (<= x 0.11999999731779099) (/ 0.25 s) (* (/ 0.5 s) (/ s x))))

C

x = abs(x);
float code(float x, float s) {
	float tmp;
	if (x <= 0.11999999731779099f) {
		tmp = 0.25f / s;
	} else {
		tmp = (0.5f / s) * (s / x);
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 0.11999999731779099e0) then
        tmp = 0.25e0 / s
    else
        tmp = (0.5e0 / s) * (s / x)
    end if
    code = tmp
end function

Julia

x = abs(x)
function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(0.11999999731779099))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(Float32(0.5) / s) * Float32(s / x));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(0.11999999731779099))
		tmp = single(0.25) / s;
	else
		tmp = (single(0.5) / s) * (s / x);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 0.119999997

   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. associate-/r* 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in s around inf 38.5%

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

   if 0.119999997 < x

   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. associate-/r* 99.6%

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

   3. Simplified 99.6%

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

   5. Applied egg-rr 3.1%

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

   6. Taylor expanded in x around 0 97.5%

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

   7. Taylor expanded in x around 0 43.6%

      \[\leadsto \frac{0.5}{s} \cdot \frac{1}{\color{blue}{2 + \frac{x}{s}}} \]
   8. Step-by-step derivation

      1. +-commutative 43.6%

         \[\leadsto \frac{0.5}{s} \cdot \frac{1}{\color{blue}{\frac{x}{s} + 2}} \]

   9. Simplified 43.6%

      \[\leadsto \frac{0.5}{s} \cdot \frac{1}{\color{blue}{\frac{x}{s} + 2}} \]

   10. Taylor expanded in x around inf 34.3%

       \[\leadsto \frac{0.5}{s} \cdot \color{blue}{\frac{s}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.11999999731779099:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{s} \cdot \frac{s}{x}\\ \end{array} \]

Alternative 7: 50.4% accurate, 68.9× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \frac{0.5}{s \cdot \left(\frac{x}{s} + 2\right)} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s) :precision binary32 (/ 0.5 (* s (+ (/ x s) 2.0))))

C

x = abs(x);
float code(float x, float s) {
	return 0.5f / (s * ((x / s) + 2.0f));
}

Fortran

NOTE: x should be positive before calling this function
real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 0.5e0 / (s * ((x / s) + 2.0e0))
end function

Julia

x = abs(x)
function code(x, s)
	return Float32(Float32(0.5) / Float32(s * Float32(Float32(x / s) + Float32(2.0))))
end

MATLAB

x = abs(x)
function tmp = code(x, s)
	tmp = single(0.5) / (s * ((x / s) + single(2.0)));
end

Derivation

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. associate-/r* 99.6%

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

3. Simplified 99.6%

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

5. Applied egg-rr 68.3%

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

6. Taylor expanded in x around 0 58.9%

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

7. Taylor expanded in x around 0 49.4%

   \[\leadsto \frac{0.5}{s} \cdot \frac{1}{\color{blue}{2 + \frac{x}{s}}} \]
8. Step-by-step derivation

   1. +-commutative 49.4%

      \[\leadsto \frac{0.5}{s} \cdot \frac{1}{\color{blue}{\frac{x}{s} + 2}} \]

9. Simplified 49.4%

   \[\leadsto \frac{0.5}{s} \cdot \frac{1}{\color{blue}{\frac{x}{s} + 2}} \]
10. Step-by-step derivation

    1. frac-times 49.4%

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

    2. metadata-eval 49.4%

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

11. Applied egg-rr 49.4%

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

12. Final simplification 49.4%

    \[\leadsto \frac{0.5}{s \cdot \left(\frac{x}{s} + 2\right)} \]

Alternative 8: 27.0% accurate, 206.7× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s) :precision binary32 (/ 0.25 s))

C

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

Fortran

NOTE: x should be positive before calling this function
real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 0.25e0 / s
end function

Julia

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

MATLAB

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

Derivation

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. associate-/r* 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in s around inf 31.0%

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

5. Final simplification 31.0%

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

Reproduce

herbie shell --seed 2023320 
(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))))))