Logistic distribution

Percentage Accurate: 99.5% → 99.4%

Time: 12.2s

Alternatives: 7

Speedup: N/A×

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, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.7%

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

   1. fabs-neg 99.7%

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

   2. distribute-frac-neg 99.7%

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

   3. distribute-frac-neg2 99.7%

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

   4. fabs-neg 99.7%

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

   5. *-commutative 99.7%

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

   6. fabs-neg 99.7%

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

   7. +-commutative 99.7%

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

   8. fabs-neg 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in x around 0 99.7%

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

   1. associate-/r* 99.8%

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

   2. mul-1-neg 99.8%

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

   3. rec-exp 99.8%

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

   4. rem-square-sqrt 42.4%

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

   5. fabs-sqr 42.4%

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

   6. rem-square-sqrt 57.9%

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

   7. rec-exp 57.9%

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

   8. distribute-neg-frac2 57.9%

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

7. Simplified 58.3%

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

   1. add-sqr-sqrt -0.0%

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

   2. sqrt-unprod 62.2%

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

   3. sqr-neg 62.2%

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

   4. sqrt-unprod 65.8%

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

   5. add-sqr-sqrt 65.8%

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

   6. clear-num 65.8%

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

   7. inv-pow 65.8%

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

   8. div-inv 65.8%

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

   9. exp-neg 65.8%

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

   10. distribute-frac-neg2 65.8%

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

   11. add-sqr-sqrt -0.0%

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

   12. sqrt-unprod 53.4%

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

   13. sqr-neg 53.4%

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

   14. sqrt-unprod 58.3%

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

   15. add-sqr-sqrt 58.3%

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

9. Applied egg-rr 58.3%

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

    1. unpow-1 58.3%

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

11. Simplified 58.3%

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

    1. expm1-log1p-u 58.3%

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

    2. expm1-undefine 40.7%

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

13. Applied egg-rr 40.7%

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

    1. expm1-define 58.3%

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

15. Simplified 58.3%

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

16. Final simplification 58.3%

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

Alternative 2: 99.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\left|x\_m\right| \leq 5:\\ \;\;\;\;\frac{e^{\frac{x\_m}{s} + -2 \cdot \mathsf{log1p}\left(e^{\frac{x\_m}{s}}\right)}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0}{s}}{s}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (if (<= (fabs x_m) 5.0)
   (/ (exp (+ (/ x_m s) (* -2.0 (log1p (exp (/ x_m s)))))) s)
   (/ (/ 0.0 s) s)))

C

x_m = fabs(x);
float code(float x_m, float s) {
	float tmp;
	if (fabsf(x_m) <= 5.0f) {
		tmp = expf(((x_m / s) + (-2.0f * log1pf(expf((x_m / s)))))) / s;
	} else {
		tmp = (0.0f / s) / s;
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m, s)
	tmp = Float32(0.0)
	if (abs(x_m) <= Float32(5.0))
		tmp = Float32(exp(Float32(Float32(x_m / s) + Float32(Float32(-2.0) * log1p(exp(Float32(x_m / s)))))) / s);
	else
		tmp = Float32(Float32(Float32(0.0) / s) / s);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (fabs.f32 x) < 5

   1. Initial program 99.4%

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

      1. fabs-neg 99.4%

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

      2. distribute-frac-neg 99.4%

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

      3. distribute-frac-neg2 99.4%

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

      4. fabs-neg 99.4%

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

      5. *-commutative 99.4%

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

      6. fabs-neg 99.4%

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

      7. +-commutative 99.4%

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

      8. fabs-neg 99.4%

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

   3. Simplified 99.5%

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

   6. Applied egg-rr 82.5%

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

      1. *-lft-identity 82.5%

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

      2. *-commutative 82.5%

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

      3. exp-to-pow 82.6%

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

      4. log1p-undefine 82.6%

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

      5. *-commutative 82.6%

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

      6. rem-exp-log 78.8%

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

      7. exp-sum 78.6%

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

      8. exp-diff 95.5%

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

      9. associate--r+ 95.7%

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

      10. exp-diff 95.9%

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

      11. cancel-sign-sub-inv 95.9%

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

      12. metadata-eval 95.9%

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

      13. rem-exp-log 99.6%

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

   8. Simplified 99.6%

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

   if 5 < (fabs.f32 x)

   1. Initial program 100.0%

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

      1. fabs-neg 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. distribute-frac-neg2 100.0%

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

      4. fabs-neg 100.0%

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

      5. *-commutative 100.0%

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

      6. fabs-neg 100.0%

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

      7. +-commutative 100.0%

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

      8. fabs-neg 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 53.5%

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

      1. associate-*l/ 53.5%

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

      2. *-lft-identity 53.5%

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

   8. Simplified 53.5%

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

   9. Taylor expanded in s around inf 12.4%

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

   10. Taylor expanded in s around inf 44.3%

       \[\leadsto \color{blue}{\frac{\left(0.25 + 0.5 \cdot \frac{0.5 \cdot x - 0.25 \cdot x}{s}\right) - 0.125 \cdot \frac{x}{s}}{s}} \]
   11. Step-by-step derivation

       1. cancel-sign-sub-inv 44.3%

          \[\leadsto \frac{\color{blue}{\left(0.25 + 0.5 \cdot \frac{0.5 \cdot x - 0.25 \cdot x}{s}\right) + \left(-0.125\right) \cdot \frac{x}{s}}}{s} \]

       2. distribute-rgt-out-- 44.3%

          \[\leadsto \frac{\left(0.25 + 0.5 \cdot \frac{\color{blue}{x \cdot \left(0.5 - 0.25\right)}}{s}\right) + \left(-0.125\right) \cdot \frac{x}{s}}{s} \]

       3. metadata-eval 44.3%

          \[\leadsto \frac{\left(0.25 + 0.5 \cdot \frac{x \cdot \color{blue}{0.25}}{s}\right) + \left(-0.125\right) \cdot \frac{x}{s}}{s} \]

       4. associate-*l/ 44.2%

          \[\leadsto \frac{\left(0.25 + 0.5 \cdot \color{blue}{\left(\frac{x}{s} \cdot 0.25\right)}\right) + \left(-0.125\right) \cdot \frac{x}{s}}{s} \]

       5. metadata-eval 44.2%

          \[\leadsto \frac{\left(0.25 + 0.5 \cdot \left(\frac{x}{s} \cdot 0.25\right)\right) + \color{blue}{-0.125} \cdot \frac{x}{s}}{s} \]

       6. *-commutative 44.2%

          \[\leadsto \frac{\left(0.25 + 0.5 \cdot \left(\frac{x}{s} \cdot 0.25\right)\right) + \color{blue}{\frac{x}{s} \cdot -0.125}}{s} \]

   12. Simplified 44.2%

       \[\leadsto \color{blue}{\frac{\left(0.25 + 0.5 \cdot \left(\frac{x}{s} \cdot 0.25\right)\right) + \frac{x}{s} \cdot -0.125}{s}} \]

   13. Taylor expanded in s around 0 100.0%

       \[\leadsto \frac{\color{blue}{\frac{-0.125 \cdot x + 0.125 \cdot x}{s}}}{s} \]
   14. Step-by-step derivation

       1. distribute-rgt-out 100.0%

          \[\leadsto \frac{\frac{\color{blue}{x \cdot \left(-0.125 + 0.125\right)}}{s}}{s} \]

       2. metadata-eval 100.0%

          \[\leadsto \frac{\frac{x \cdot \color{blue}{0}}{s}}{s} \]

       3. mul0-rgt 100.0%

          \[\leadsto \frac{\frac{\color{blue}{0}}{s}}{s} \]

   15. Simplified 100.0%

       \[\leadsto \frac{\color{blue}{\frac{0}{s}}}{s} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 99.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

   1. fabs-neg 99.7%

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

   2. distribute-frac-neg 99.7%

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

   3. distribute-frac-neg2 99.7%

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

   4. fabs-neg 99.7%

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

   5. *-commutative 99.7%

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

   6. fabs-neg 99.7%

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

   7. +-commutative 99.7%

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

   8. fabs-neg 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in x around 0 99.7%

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

   1. associate-/r* 99.8%

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

   2. mul-1-neg 99.8%

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

   3. rec-exp 99.8%

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

   4. rem-square-sqrt 42.4%

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

   5. fabs-sqr 42.4%

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

   6. rem-square-sqrt 57.9%

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

   7. rec-exp 57.9%

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

   8. distribute-neg-frac2 57.9%

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

7. Simplified 58.3%

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

   1. add-sqr-sqrt -0.0%

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

   2. sqrt-unprod 62.2%

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

   3. sqr-neg 62.2%

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

   4. sqrt-unprod 65.8%

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

   5. add-sqr-sqrt 65.8%

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

   6. clear-num 65.8%

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

   7. inv-pow 65.8%

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

   8. div-inv 65.8%

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

   9. exp-neg 65.8%

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

   10. distribute-frac-neg2 65.8%

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

   11. add-sqr-sqrt -0.0%

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

   12. sqrt-unprod 53.4%

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

   13. sqr-neg 53.4%

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

   14. sqrt-unprod 58.3%

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

   15. add-sqr-sqrt 58.3%

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

9. Applied egg-rr 58.3%

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

    1. unpow-1 58.3%

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

11. Simplified 58.3%

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

    1. expm1-log1p-u 58.3%

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

    2. expm1-undefine 40.7%

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

13. Applied egg-rr 40.7%

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

    1. expm1-define 58.3%

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

15. Simplified 58.3%

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

    1. *-un-lft-identity 58.3%

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

    2. div-inv 58.3%

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

    3. expm1-log1p-u 58.3%

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

    4. pow-flip 58.3%

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

    5. +-commutative 58.3%

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

    6. metadata-eval 58.3%

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

17. Applied egg-rr 58.3%

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

    1. *-lft-identity 58.3%

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

    2. associate-*l/ 58.3%

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

    3. *-lft-identity 58.3%

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

19. Simplified 58.3%

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

Alternative 4: 94.5% 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(x_m / Float32(-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 99.7%

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

   1. fabs-neg 99.7%

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

   2. distribute-frac-neg 99.7%

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

   3. distribute-frac-neg2 99.7%

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

   4. fabs-neg 99.7%

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

   5. *-commutative 99.7%

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

   6. fabs-neg 99.7%

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

   7. +-commutative 99.7%

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

   8. fabs-neg 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in x around 0 99.7%

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

   1. associate-/r* 99.8%

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

   2. mul-1-neg 99.8%

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

   3. rec-exp 99.8%

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

   4. rem-square-sqrt 42.4%

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

   5. fabs-sqr 42.4%

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

   6. rem-square-sqrt 57.9%

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

   7. rec-exp 57.9%

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

   8. distribute-neg-frac2 57.9%

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

7. Simplified 58.3%

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

8. Taylor expanded in x around 0 56.3%

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

Alternative 5: 89.8% accurate, 31.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;s \leq 2.3200000374495537 \cdot 10^{-30}:\\ \;\;\;\;\frac{\frac{0}{s}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m \cdot -0.125 + \left(x\_m \cdot 0.125 + s \cdot 0.25\right)}{s}}{s}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (if (<= s 2.3200000374495537e-30)
   (/ (/ 0.0 s) s)
   (/ (/ (+ (* x_m -0.125) (+ (* x_m 0.125) (* s 0.25))) s) s)))

C

x_m = fabs(x);
float code(float x_m, float s) {
	float tmp;
	if (s <= 2.3200000374495537e-30f) {
		tmp = (0.0f / s) / s;
	} else {
		tmp = (((x_m * -0.125f) + ((x_m * 0.125f) + (s * 0.25f))) / s) / s;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(4) function code(x_m, s)
    real(4), intent (in) :: x_m
    real(4), intent (in) :: s
    real(4) :: tmp
    if (s <= 2.3200000374495537e-30) then
        tmp = (0.0e0 / s) / s
    else
        tmp = (((x_m * (-0.125e0)) + ((x_m * 0.125e0) + (s * 0.25e0))) / s) / s
    end if
    code = tmp
end function

Julia

x_m = abs(x)
function code(x_m, s)
	tmp = Float32(0.0)
	if (s <= Float32(2.3200000374495537e-30))
		tmp = Float32(Float32(Float32(0.0) / s) / s);
	else
		tmp = Float32(Float32(Float32(Float32(x_m * Float32(-0.125)) + Float32(Float32(x_m * Float32(0.125)) + Float32(s * Float32(0.25)))) / s) / s);
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, s)
	tmp = single(0.0);
	if (s <= single(2.3200000374495537e-30))
		tmp = (single(0.0) / s) / s;
	else
		tmp = (((x_m * single(-0.125)) + ((x_m * single(0.125)) + (s * single(0.25)))) / s) / s;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if s < 2.32000004e-30

   1. Initial program 99.9%

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

      1. fabs-neg 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. distribute-frac-neg2 99.9%

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

      4. fabs-neg 99.9%

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

      5. *-commutative 99.9%

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

      6. fabs-neg 99.9%

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

      7. +-commutative 99.9%

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

      8. fabs-neg 99.9%

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

   3. Simplified 99.9%

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

   6. Applied egg-rr 65.5%

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

      1. associate-*l/ 65.5%

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

      2. *-lft-identity 65.5%

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

   8. Simplified 65.5%

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

   9. Taylor expanded in s around inf 7.1%

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

   10. Taylor expanded in s around inf 36.0%

       \[\leadsto \color{blue}{\frac{\left(0.25 + 0.5 \cdot \frac{0.5 \cdot x - 0.25 \cdot x}{s}\right) - 0.125 \cdot \frac{x}{s}}{s}} \]
   11. Step-by-step derivation

       1. cancel-sign-sub-inv 36.0%

          \[\leadsto \frac{\color{blue}{\left(0.25 + 0.5 \cdot \frac{0.5 \cdot x - 0.25 \cdot x}{s}\right) + \left(-0.125\right) \cdot \frac{x}{s}}}{s} \]

       2. distribute-rgt-out-- 36.0%

          \[\leadsto \frac{\left(0.25 + 0.5 \cdot \frac{\color{blue}{x \cdot \left(0.5 - 0.25\right)}}{s}\right) + \left(-0.125\right) \cdot \frac{x}{s}}{s} \]

       3. metadata-eval 36.0%

          \[\leadsto \frac{\left(0.25 + 0.5 \cdot \frac{x \cdot \color{blue}{0.25}}{s}\right) + \left(-0.125\right) \cdot \frac{x}{s}}{s} \]

       4. associate-*l/ 36.0%

          \[\leadsto \frac{\left(0.25 + 0.5 \cdot \color{blue}{\left(\frac{x}{s} \cdot 0.25\right)}\right) + \left(-0.125\right) \cdot \frac{x}{s}}{s} \]

       5. metadata-eval 36.0%

          \[\leadsto \frac{\left(0.25 + 0.5 \cdot \left(\frac{x}{s} \cdot 0.25\right)\right) + \color{blue}{-0.125} \cdot \frac{x}{s}}{s} \]

       6. *-commutative 36.0%

          \[\leadsto \frac{\left(0.25 + 0.5 \cdot \left(\frac{x}{s} \cdot 0.25\right)\right) + \color{blue}{\frac{x}{s} \cdot -0.125}}{s} \]

   12. Simplified 36.0%

       \[\leadsto \color{blue}{\frac{\left(0.25 + 0.5 \cdot \left(\frac{x}{s} \cdot 0.25\right)\right) + \frac{x}{s} \cdot -0.125}{s}} \]

   13. Taylor expanded in s around 0 96.8%

       \[\leadsto \frac{\color{blue}{\frac{-0.125 \cdot x + 0.125 \cdot x}{s}}}{s} \]
   14. Step-by-step derivation

       1. distribute-rgt-out 96.8%

          \[\leadsto \frac{\frac{\color{blue}{x \cdot \left(-0.125 + 0.125\right)}}{s}}{s} \]

       2. metadata-eval 96.8%

          \[\leadsto \frac{\frac{x \cdot \color{blue}{0}}{s}}{s} \]

       3. mul0-rgt 96.8%

          \[\leadsto \frac{\frac{\color{blue}{0}}{s}}{s} \]

   15. Simplified 96.8%

       \[\leadsto \frac{\color{blue}{\frac{0}{s}}}{s} \]

   if 2.32000004e-30 < s

   1. Initial program 99.7%

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

      1. fabs-neg 99.7%

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

      2. distribute-frac-neg 99.7%

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

      3. distribute-frac-neg2 99.7%

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

      4. fabs-neg 99.7%

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

      5. *-commutative 99.7%

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

      6. fabs-neg 99.7%

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

      7. +-commutative 99.7%

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

      8. fabs-neg 99.7%

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

   3. Simplified 99.7%

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

   6. Applied egg-rr 69.4%

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

      1. associate-*l/ 69.4%

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

      2. *-lft-identity 69.4%

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

   8. Simplified 69.4%

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

   9. Taylor expanded in s around inf 43.8%

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

   10. Taylor expanded in s around inf 67.6%

       \[\leadsto \color{blue}{\frac{\left(0.25 + 0.5 \cdot \frac{0.5 \cdot x - 0.25 \cdot x}{s}\right) - 0.125 \cdot \frac{x}{s}}{s}} \]
   11. Step-by-step derivation

       1. cancel-sign-sub-inv 67.6%

          \[\leadsto \frac{\color{blue}{\left(0.25 + 0.5 \cdot \frac{0.5 \cdot x - 0.25 \cdot x}{s}\right) + \left(-0.125\right) \cdot \frac{x}{s}}}{s} \]

       2. distribute-rgt-out-- 67.6%

          \[\leadsto \frac{\left(0.25 + 0.5 \cdot \frac{\color{blue}{x \cdot \left(0.5 - 0.25\right)}}{s}\right) + \left(-0.125\right) \cdot \frac{x}{s}}{s} \]

       3. metadata-eval 67.6%

          \[\leadsto \frac{\left(0.25 + 0.5 \cdot \frac{x \cdot \color{blue}{0.25}}{s}\right) + \left(-0.125\right) \cdot \frac{x}{s}}{s} \]

       4. associate-*l/ 67.6%

          \[\leadsto \frac{\left(0.25 + 0.5 \cdot \color{blue}{\left(\frac{x}{s} \cdot 0.25\right)}\right) + \left(-0.125\right) \cdot \frac{x}{s}}{s} \]

       5. metadata-eval 67.6%

          \[\leadsto \frac{\left(0.25 + 0.5 \cdot \left(\frac{x}{s} \cdot 0.25\right)\right) + \color{blue}{-0.125} \cdot \frac{x}{s}}{s} \]

       6. *-commutative 67.6%

          \[\leadsto \frac{\left(0.25 + 0.5 \cdot \left(\frac{x}{s} \cdot 0.25\right)\right) + \color{blue}{\frac{x}{s} \cdot -0.125}}{s} \]

   12. Simplified 67.6%

       \[\leadsto \color{blue}{\frac{\left(0.25 + 0.5 \cdot \left(\frac{x}{s} \cdot 0.25\right)\right) + \frac{x}{s} \cdot -0.125}{s}} \]

   13. Taylor expanded in s around 0 89.1%

       \[\leadsto \frac{\color{blue}{\frac{-0.125 \cdot x + \left(0.125 \cdot x + 0.25 \cdot s\right)}{s}}}{s} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 2.3200000374495537 \cdot 10^{-30}:\\ \;\;\;\;\frac{\frac{0}{s}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot -0.125 + \left(x \cdot 0.125 + s \cdot 0.25\right)}{s}}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.0% accurate, 61.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.999999936531045 \cdot 10^{-20}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0}{s}}{s}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (if (<= x_m 1.999999936531045e-20) (/ 0.25 s) (/ (/ 0.0 s) s)))

C

x_m = fabs(x);
float code(float x_m, float s) {
	float tmp;
	if (x_m <= 1.999999936531045e-20f) {
		tmp = 0.25f / s;
	} else {
		tmp = (0.0f / s) / s;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(4) function code(x_m, s)
    real(4), intent (in) :: x_m
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x_m <= 1.999999936531045e-20) then
        tmp = 0.25e0 / s
    else
        tmp = (0.0e0 / s) / s
    end if
    code = tmp
end function

Julia

x_m = abs(x)
function code(x_m, s)
	tmp = Float32(0.0)
	if (x_m <= Float32(1.999999936531045e-20))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(Float32(0.0) / s) / s);
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, s)
	tmp = single(0.0);
	if (x_m <= single(1.999999936531045e-20))
		tmp = single(0.25) / s;
	else
		tmp = (single(0.0) / s) / s;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 1.99999994e-20

   1. Initial program 99.8%

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

      1. fabs-neg 99.8%

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

      2. distribute-frac-neg 99.8%

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

      3. distribute-frac-neg2 99.8%

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

      4. fabs-neg 99.8%

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

      5. *-commutative 99.8%

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

      6. fabs-neg 99.8%

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

      7. +-commutative 99.8%

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

      8. fabs-neg 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in s around inf 37.3%

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

   if 1.99999994e-20 < 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. 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

   6. Applied egg-rr 6.8%

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

      1. associate-*l/ 6.8%

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

      2. *-lft-identity 6.8%

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

   8. Simplified 6.8%

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

   9. Taylor expanded in s around inf 30.6%

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

   10. Taylor expanded in s around inf 55.6%

       \[\leadsto \color{blue}{\frac{\left(0.25 + 0.5 \cdot \frac{0.5 \cdot x - 0.25 \cdot x}{s}\right) - 0.125 \cdot \frac{x}{s}}{s}} \]
   11. Step-by-step derivation

       1. cancel-sign-sub-inv 55.6%

          \[\leadsto \frac{\color{blue}{\left(0.25 + 0.5 \cdot \frac{0.5 \cdot x - 0.25 \cdot x}{s}\right) + \left(-0.125\right) \cdot \frac{x}{s}}}{s} \]

       2. distribute-rgt-out-- 55.6%

          \[\leadsto \frac{\left(0.25 + 0.5 \cdot \frac{\color{blue}{x \cdot \left(0.5 - 0.25\right)}}{s}\right) + \left(-0.125\right) \cdot \frac{x}{s}}{s} \]

       3. metadata-eval 55.6%

          \[\leadsto \frac{\left(0.25 + 0.5 \cdot \frac{x \cdot \color{blue}{0.25}}{s}\right) + \left(-0.125\right) \cdot \frac{x}{s}}{s} \]

       4. associate-*l/ 55.6%

          \[\leadsto \frac{\left(0.25 + 0.5 \cdot \color{blue}{\left(\frac{x}{s} \cdot 0.25\right)}\right) + \left(-0.125\right) \cdot \frac{x}{s}}{s} \]

       5. metadata-eval 55.6%

          \[\leadsto \frac{\left(0.25 + 0.5 \cdot \left(\frac{x}{s} \cdot 0.25\right)\right) + \color{blue}{-0.125} \cdot \frac{x}{s}}{s} \]

       6. *-commutative 55.6%

          \[\leadsto \frac{\left(0.25 + 0.5 \cdot \left(\frac{x}{s} \cdot 0.25\right)\right) + \color{blue}{\frac{x}{s} \cdot -0.125}}{s} \]

   12. Simplified 55.6%

       \[\leadsto \color{blue}{\frac{\left(0.25 + 0.5 \cdot \left(\frac{x}{s} \cdot 0.25\right)\right) + \frac{x}{s} \cdot -0.125}{s}} \]

   13. Taylor expanded in s around 0 93.3%

       \[\leadsto \frac{\color{blue}{\frac{-0.125 \cdot x + 0.125 \cdot x}{s}}}{s} \]
   14. Step-by-step derivation

       1. distribute-rgt-out 93.3%

          \[\leadsto \frac{\frac{\color{blue}{x \cdot \left(-0.125 + 0.125\right)}}{s}}{s} \]

       2. metadata-eval 93.3%

          \[\leadsto \frac{\frac{x \cdot \color{blue}{0}}{s}}{s} \]

       3. mul0-rgt 93.3%

          \[\leadsto \frac{\frac{\color{blue}{0}}{s}}{s} \]

   15. Simplified 93.3%

       \[\leadsto \frac{\color{blue}{\frac{0}{s}}}{s} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 27.3% accurate, 206.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

   1. fabs-neg 99.7%

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

   2. distribute-frac-neg 99.7%

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

   3. distribute-frac-neg2 99.7%

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

   4. fabs-neg 99.7%

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

   5. *-commutative 99.7%

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

   6. fabs-neg 99.7%

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

   7. +-commutative 99.7%

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

   8. fabs-neg 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in s around inf 27.7%

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

Reproduce

herbie shell --seed 2024170 
(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))))))