Logistic distribution

Percentage Accurate: 99.5% → 99.5%

Time: 11.3s

Alternatives: 14

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, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

   1. *-commutative 99.1%

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

   2. fabs-neg 99.1%

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

   3. +-commutative 99.1%

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

   4. fabs-neg 99.1%

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

   5. distribute-lft-in 99.1%

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

   6. *-rgt-identity 99.1%

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

   7. +-commutative 99.1%

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

3. Simplified 99.2%

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

5. Taylor expanded in x around 0 99.2%

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

   1. associate-/r* 99.2%

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

7. Simplified 61.2%

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

8. Final simplification 61.2%

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

Alternative 2: 99.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\left|x\_m\right| \leq 0.0011500000255182385:\\ \;\;\;\;\frac{e^{\frac{x\_m}{s} + -2 \cdot \mathsf{log1p}\left(e^{\frac{x\_m}{s}}\right)}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{e^{\frac{-x\_m}{s}}}{s}}{4}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (if (<= (fabs x_m) 0.0011500000255182385)
   (/ (exp (+ (/ x_m s) (* -2.0 (log1p (exp (/ x_m s)))))) s)
   (/ (/ (exp (/ (- x_m) s)) s) 4.0)))

C

x_m = fabs(x);
float code(float x_m, float s) {
	float tmp;
	if (fabsf(x_m) <= 0.0011500000255182385f) {
		tmp = expf(((x_m / s) + (-2.0f * log1pf(expf((x_m / s)))))) / s;
	} else {
		tmp = (expf((-x_m / s)) / s) / 4.0f;
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m, s)
	tmp = Float32(0.0)
	if (abs(x_m) <= Float32(0.0011500000255182385))
		tmp = Float32(exp(Float32(Float32(x_m / s) + Float32(Float32(-2.0) * log1p(exp(Float32(x_m / s)))))) / s);
	else
		tmp = Float32(Float32(exp(Float32(Float32(-x_m) / s)) / s) / Float32(4.0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (fabs.f32 x) < 0.00115000003

   1. Initial program 98.1%

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

      1. fabs-neg 98.1%

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

      2. distribute-frac-neg 98.1%

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

      3. distribute-frac-neg2 98.1%

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

      4. fabs-neg 98.1%

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

      5. *-commutative 98.1%

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

      6. fabs-neg 98.1%

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

      7. +-commutative 98.1%

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

      8. fabs-neg 98.1%

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

   3. Simplified 98.0%

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

   6. Applied egg-rr 79.4%

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

      1. associate-*r/ 79.4%

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

      2. *-rgt-identity 79.4%

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

      3. associate-/r* 78.8%

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

      4. +-commutative 78.8%

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

   8. Simplified 78.8%

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

      1. associate-/l/ 79.4%

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

      2. *-un-lft-identity 79.4%

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

      3. times-frac 78.7%

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

      4. pow-flip 78.7%

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

      5. +-commutative 78.7%

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

      6. metadata-eval 78.7%

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

   10. Applied egg-rr 78.7%

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

       1. associate-*r/ 79.4%

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

       2. pow-to-exp 79.4%

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

       3. prod-exp 97.7%

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

       4. rem-log-exp 97.7%

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

       5. pow-to-exp 97.7%

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

       6. log-pow 97.7%

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

       7. +-commutative 97.7%

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

       8. log1p-undefine 98.0%

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

   12. Applied egg-rr 98.0%

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

   if 0.00115000003 < (fabs.f32 x)

   1. Initial program 100.0%

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

      1. fabs-neg 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. distribute-frac-neg2 100.0%

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

      4. fabs-neg 100.0%

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

      5. *-commutative 100.0%

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

      6. fabs-neg 100.0%

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

      7. +-commutative 100.0%

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

      8. fabs-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 100.0%

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

      1. associate-/r* 100.0%

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

      2. exp-prod 100.0%

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

      3. rem-square-sqrt 48.9%

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

      4. fabs-sqr 48.9%

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

      5. rem-square-sqrt 50.5%

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

      6. exp-prod 50.5%

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

      7. neg-mul-1 50.5%

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

      8. distribute-neg-frac2 50.5%

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

   7. Simplified 48.9%

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

   8. Taylor expanded in x around 0 49.6%

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

      1. *-commutative 49.6%

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

   10. Simplified 49.6%

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

   11. Taylor expanded in x around 0 50.5%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.0011500000255182385:\\ \;\;\;\;\frac{e^{\frac{x}{s} + -2 \cdot \mathsf{log1p}\left(e^{\frac{x}{s}}\right)}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{e^{\frac{-x}{s}}}{s}}{4}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := e^{\frac{-x\_m}{s}}\\ \frac{t\_0}{s \cdot {\left(t\_0 + 1\right)}^{2}} \end{array} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (let* ((t_0 (exp (/ (- x_m) s)))) (/ t_0 (* s (pow (+ t_0 1.0) 2.0)))))

C

x_m = fabs(x);
float code(float x_m, float s) {
	float t_0 = expf((-x_m / s));
	return t_0 / (s * powf((t_0 + 1.0f), 2.0f));
}

Fortran

x_m = abs(x)
real(4) function code(x_m, s)
    real(4), intent (in) :: x_m
    real(4), intent (in) :: s
    real(4) :: t_0
    t_0 = exp((-x_m / s))
    code = t_0 / (s * ((t_0 + 1.0e0) ** 2.0e0))
end function

Julia

x_m = abs(x)
function code(x_m, s)
	t_0 = exp(Float32(Float32(-x_m) / s))
	return Float32(t_0 / Float32(s * (Float32(t_0 + Float32(1.0)) ^ Float32(2.0))))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m, s)
	t_0 = exp((-x_m / s));
	tmp = t_0 / (s * ((t_0 + single(1.0)) ^ single(2.0)));
end

Derivation

1. Initial program 99.1%

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

   1. fabs-neg 99.1%

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

   2. distribute-frac-neg 99.1%

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

   3. distribute-frac-neg2 99.1%

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

   4. fabs-neg 99.1%

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

   5. *-commutative 99.1%

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

   6. fabs-neg 99.1%

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

   7. +-commutative 99.1%

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

   8. fabs-neg 99.1%

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

3. Simplified 99.1%

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

5. Taylor expanded in x around 0 99.1%

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

   1. +-commutative 99.1%

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

   2. exp-prod 99.1%

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

   3. rem-square-sqrt 49.6%

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

   4. fabs-sqr 49.6%

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

   5. rem-square-sqrt 95.9%

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

   6. exp-prod 95.9%

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

   7. neg-mul-1 95.9%

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

   8. distribute-neg-frac2 95.9%

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

7. Simplified 95.9%

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

8. Taylor expanded in x around 0 95.9%

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

   1. exp-prod 95.9%

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

   2. rem-square-sqrt 49.6%

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

   3. fabs-sqr 49.6%

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

   4. rem-square-sqrt 60.8%

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

   5. exp-prod 60.8%

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

   6. neg-mul-1 60.8%

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

   7. distribute-neg-frac2 60.8%

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

10. Simplified 60.8%

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

11. Final simplification 60.8%

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

Alternative 4: 96.8% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

   1. *-commutative 99.1%

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

   2. fabs-neg 99.1%

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

   3. +-commutative 99.1%

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

   4. fabs-neg 99.1%

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

   5. distribute-lft-in 99.1%

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

   6. *-rgt-identity 99.1%

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

   7. +-commutative 99.1%

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

3. Simplified 99.2%

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

5. Taylor expanded in x around 0 99.2%

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

   1. associate-/r* 99.2%

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

7. Simplified 61.2%

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

8. Taylor expanded in x around 0 57.2%

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

   1. +-commutative 57.2%

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

10. Simplified 57.2%

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

11. Final simplification 57.2%

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

Alternative 5: 96.2% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

   1. fabs-neg 99.1%

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

   2. distribute-frac-neg 99.1%

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

   3. distribute-frac-neg2 99.1%

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

   4. fabs-neg 99.1%

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

   5. *-commutative 99.1%

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

   6. fabs-neg 99.1%

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

   7. +-commutative 99.1%

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

   8. fabs-neg 99.1%

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

3. Simplified 99.1%

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

5. Taylor expanded in x around 0 99.1%

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

   1. associate-/r* 99.1%

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

   2. exp-prod 99.1%

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

   3. rem-square-sqrt 49.6%

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

   4. fabs-sqr 49.6%

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

   5. rem-square-sqrt 59.2%

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

   6. exp-prod 59.2%

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

   7. neg-mul-1 59.2%

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

   8. distribute-neg-frac2 59.2%

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

7. Simplified 60.4%

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

8. Taylor expanded in x around 0 57.1%

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

   1. neg-mul-1 57.1%

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

   2. unsub-neg 57.1%

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

10. Simplified 57.1%

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

11. Final simplification 57.1%

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

Alternative 6: 95.6% accurate, 5.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

Derivation

1. Initial program 99.1%

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

   1. fabs-neg 99.1%

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

   2. distribute-frac-neg 99.1%

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

   3. distribute-frac-neg2 99.1%

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

   4. fabs-neg 99.1%

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

   5. *-commutative 99.1%

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

   6. fabs-neg 99.1%

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

   7. +-commutative 99.1%

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

   8. fabs-neg 99.1%

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

3. Simplified 99.1%

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

5. Taylor expanded in x around 0 99.1%

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

   1. associate-/r* 99.1%

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

   2. exp-prod 99.1%

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

   3. rem-square-sqrt 49.6%

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

   4. fabs-sqr 49.6%

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

   5. rem-square-sqrt 59.2%

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

   6. exp-prod 59.2%

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

   7. neg-mul-1 59.2%

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

   8. distribute-neg-frac2 59.2%

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

7. Simplified 60.4%

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

8. Taylor expanded in x around 0 57.0%

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

   1. *-commutative 57.0%

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

10. Simplified 57.0%

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

11. Final simplification 57.0%

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

Alternative 7: 94.8% accurate, 5.6× speedup

Math

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

FPCore

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

C

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

Fortran

x_m = abs(x)
real(4) function code(x_m, s)
    real(4), intent (in) :: x_m
    real(4), intent (in) :: s
    code = 0.5e0 * ((1.0e0 / s) / (1.0e0 + exp((x_m / s))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

   1. fabs-neg 99.1%

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

   2. distribute-frac-neg 99.1%

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

   3. distribute-frac-neg2 99.1%

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

   4. fabs-neg 99.1%

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

   5. *-commutative 99.1%

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

   6. fabs-neg 99.1%

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

   7. +-commutative 99.1%

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

   8. fabs-neg 99.1%

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

3. Simplified 99.1%

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

6. Applied egg-rr 64.0%

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

   1. associate-*r/ 64.0%

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

   2. *-rgt-identity 64.0%

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

   3. associate-/r* 63.7%

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

   4. +-commutative 63.7%

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

8. Simplified 63.7%

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

10. Applied egg-rr 86.6%

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

11. Taylor expanded in x around 0 57.1%

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

    1. neg-mul-1 57.1%

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

    2. *-commutative 57.1%

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

    3. exp-to-pow 57.1%

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

    4. metadata-eval 57.1%

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

13. Simplified 57.1%

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

14. Final simplification 57.1%

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

Alternative 8: 94.7% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

   1. fabs-neg 99.1%

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

   2. distribute-frac-neg 99.1%

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

   3. distribute-frac-neg2 99.1%

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

   4. fabs-neg 99.1%

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

   5. *-commutative 99.1%

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

   6. fabs-neg 99.1%

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

   7. +-commutative 99.1%

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

   8. fabs-neg 99.1%

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

3. Simplified 99.1%

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

5. Taylor expanded in x around 0 99.1%

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

   1. associate-/r* 99.1%

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

   2. exp-prod 99.1%

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

   3. rem-square-sqrt 49.6%

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

   4. fabs-sqr 49.6%

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

   5. rem-square-sqrt 59.2%

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

   6. exp-prod 59.2%

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

   7. neg-mul-1 59.2%

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

   8. distribute-neg-frac2 59.2%

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

7. Simplified 60.4%

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

8. Taylor expanded in x around 0 57.0%

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

   1. *-commutative 57.0%

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

10. Simplified 57.0%

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

11. Taylor expanded in x around 0 56.3%

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

12. Final simplification 56.3%

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

Alternative 9: 55.2% accurate, 23.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;s \leq 7.99999974612418 \cdot 10^{-19}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 - \frac{x\_m}{s} \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 - \frac{x\_m \cdot 0.25}{s}}{s + \left(s + x\_m \cdot \left(\frac{x\_m}{s} \cdot 0.5 + -1\right)\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (if (<= s 7.99999974612418e-19)
   (/ (/ 1.0 s) (- 4.0 (* (/ x_m s) -4.0)))
   (/
    (- 0.5 (/ (* x_m 0.25) s))
    (+ s (+ s (* x_m (+ (* (/ x_m s) 0.5) -1.0)))))))

C

x_m = fabs(x);
float code(float x_m, float s) {
	float tmp;
	if (s <= 7.99999974612418e-19f) {
		tmp = (1.0f / s) / (4.0f - ((x_m / s) * -4.0f));
	} else {
		tmp = (0.5f - ((x_m * 0.25f) / s)) / (s + (s + (x_m * (((x_m / s) * 0.5f) + -1.0f))));
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(4) function code(x_m, s)
    real(4), intent (in) :: x_m
    real(4), intent (in) :: s
    real(4) :: tmp
    if (s <= 7.99999974612418e-19) then
        tmp = (1.0e0 / s) / (4.0e0 - ((x_m / s) * (-4.0e0)))
    else
        tmp = (0.5e0 - ((x_m * 0.25e0) / s)) / (s + (s + (x_m * (((x_m / s) * 0.5e0) + (-1.0e0)))))
    end if
    code = tmp
end function

Julia

x_m = abs(x)
function code(x_m, s)
	tmp = Float32(0.0)
	if (s <= Float32(7.99999974612418e-19))
		tmp = Float32(Float32(Float32(1.0) / s) / Float32(Float32(4.0) - Float32(Float32(x_m / s) * Float32(-4.0))));
	else
		tmp = Float32(Float32(Float32(0.5) - Float32(Float32(x_m * Float32(0.25)) / s)) / Float32(s + Float32(s + Float32(x_m * Float32(Float32(Float32(x_m / s) * Float32(0.5)) + Float32(-1.0))))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, s)
	tmp = single(0.0);
	if (s <= single(7.99999974612418e-19))
		tmp = (single(1.0) / s) / (single(4.0) - ((x_m / s) * single(-4.0)));
	else
		tmp = (single(0.5) - ((x_m * single(0.25)) / s)) / (s + (s + (x_m * (((x_m / s) * single(0.5)) + single(-1.0)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if s < 7.99999975e-19

   1. Initial program 99.0%

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

      1. fabs-neg 99.0%

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

      2. distribute-frac-neg 99.0%

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

      3. distribute-frac-neg2 99.0%

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

      4. fabs-neg 99.0%

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

      5. *-commutative 99.0%

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

      6. fabs-neg 99.0%

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

      7. +-commutative 99.0%

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

      8. fabs-neg 99.0%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in x around 0 99.0%

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

      1. associate-/r* 98.9%

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

      2. exp-prod 98.9%

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

      3. rem-square-sqrt 47.7%

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

      4. fabs-sqr 47.7%

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

      5. rem-square-sqrt 53.6%

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

      6. exp-prod 53.6%

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

      7. neg-mul-1 53.6%

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

      8. distribute-neg-frac2 53.6%

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

   7. Simplified 54.6%

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

   8. Taylor expanded in x around 0 50.3%

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

      1. *-commutative 50.3%

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

   10. Simplified 50.3%

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

   11. Taylor expanded in x around 0 54.8%

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

       1. add-sqr-sqrt 54.8%

          \[\leadsto \frac{\frac{1}{s}}{4 + \frac{x}{\color{blue}{\sqrt{s} \cdot \sqrt{s}}} \cdot -4} \]

       2. sqrt-unprod 76.4%

          \[\leadsto \frac{\frac{1}{s}}{4 + \frac{x}{\color{blue}{\sqrt{s \cdot s}}} \cdot -4} \]

       3. sqr-neg 76.4%

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

       4. sqrt-unprod -0.0%

          \[\leadsto \frac{\frac{1}{s}}{4 + \frac{x}{\color{blue}{\sqrt{-s} \cdot \sqrt{-s}}} \cdot -4} \]

       5. add-sqr-sqrt 55.2%

          \[\leadsto \frac{\frac{1}{s}}{4 + \frac{x}{\color{blue}{-s}} \cdot -4} \]

       6. distribute-frac-neg2 55.2%

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

   13. Applied egg-rr 55.2%

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

       1. distribute-neg-frac2 55.2%

          \[\leadsto \frac{\frac{1}{s}}{4 + \color{blue}{\frac{x}{-s}} \cdot -4} \]

   15. Simplified 55.2%

       \[\leadsto \frac{\frac{1}{s}}{4 + \color{blue}{\frac{x}{-s}} \cdot -4} \]

   if 7.99999975e-19 < s

   1. Initial program 99.3%

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

      1. *-commutative 99.3%

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

      2. fabs-neg 99.3%

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

      3. +-commutative 99.3%

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

      4. fabs-neg 99.3%

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

      5. distribute-lft-in 99.3%

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

      6. *-rgt-identity 99.3%

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

      7. +-commutative 99.3%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around 0 99.4%

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

      1. associate-/r* 99.4%

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

   7. Simplified 68.1%

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

   8. Taylor expanded in x around 0 59.5%

      \[\leadsto \frac{\color{blue}{0.5 + -0.25 \cdot \frac{x}{s}}}{s + \frac{s}{e^{\frac{x}{s}}}} \]
   9. Step-by-step derivation

      1. metadata-eval 59.5%

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

      2. distribute-lft-neg-in 59.5%

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

      3. unsub-neg 59.5%

         \[\leadsto \frac{\color{blue}{0.5 - 0.25 \cdot \frac{x}{s}}}{s + \frac{s}{e^{\frac{x}{s}}}} \]

      4. associate-*r/ 62.2%

         \[\leadsto \frac{0.5 - \color{blue}{\frac{0.25 \cdot x}{s}}}{s + \frac{s}{e^{\frac{x}{s}}}} \]

      5. *-commutative 62.2%

         \[\leadsto \frac{0.5 - \frac{\color{blue}{x \cdot 0.25}}{s}}{s + \frac{s}{e^{\frac{x}{s}}}} \]

   10. Simplified 62.2%

       \[\leadsto \frac{\color{blue}{0.5 - \frac{x \cdot 0.25}{s}}}{s + \frac{s}{e^{\frac{x}{s}}}} \]

   11. Taylor expanded in x around 0 69.5%

       \[\leadsto \frac{0.5 - \frac{x \cdot 0.25}{s}}{s + \color{blue}{\left(s + x \cdot \left(0.5 \cdot \frac{x}{s} - 1\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 7.99999974612418 \cdot 10^{-19}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 - \frac{x}{s} \cdot -4}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 - \frac{x \cdot 0.25}{s}}{s + \left(s + x \cdot \left(\frac{x}{s} \cdot 0.5 + -1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 53.3% accurate, 44.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0011500000255182385:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{\frac{x\_m \cdot -4}{s}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (if (<= x_m 0.0011500000255182385)
   (/ 0.25 s)
   (/ (/ 1.0 s) (/ (* x_m -4.0) s))))

C

x_m = fabs(x);
float code(float x_m, float s) {
	float tmp;
	if (x_m <= 0.0011500000255182385f) {
		tmp = 0.25f / s;
	} else {
		tmp = (1.0f / s) / ((x_m * -4.0f) / 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 <= 0.0011500000255182385e0) then
        tmp = 0.25e0 / s
    else
        tmp = (1.0e0 / s) / ((x_m * (-4.0e0)) / 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(0.0011500000255182385))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(Float32(1.0) / s) / Float32(Float32(x_m * Float32(-4.0)) / 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(0.0011500000255182385))
		tmp = single(0.25) / s;
	else
		tmp = (single(1.0) / s) / ((x_m * single(-4.0)) / s);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 0.00115000003

   1. Initial program 98.8%

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

      1. fabs-neg 98.8%

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

      2. distribute-frac-neg 98.8%

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

      3. distribute-frac-neg2 98.8%

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

      4. fabs-neg 98.8%

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

      5. *-commutative 98.8%

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

      6. fabs-neg 98.8%

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

      7. +-commutative 98.8%

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

      8. fabs-neg 98.8%

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

   3. Simplified 98.8%

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

   5. Taylor expanded in s around inf 32.7%

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

   if 0.00115000003 < x

   1. Initial program 100.0%

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

      1. fabs-neg 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. distribute-frac-neg2 100.0%

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

      4. fabs-neg 100.0%

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

      5. *-commutative 100.0%

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

      6. fabs-neg 100.0%

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

      7. +-commutative 100.0%

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

      8. fabs-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 100.0%

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

      1. associate-/r* 100.0%

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

      2. exp-prod 100.0%

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

      3. rem-square-sqrt 100.0%

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

      4. fabs-sqr 100.0%

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

      5. rem-square-sqrt 100.0%

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

      6. exp-prod 100.0%

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

      7. neg-mul-1 100.0%

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

      8. distribute-neg-frac2 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around 0 100.0%

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

      1. *-commutative 100.0%

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

   10. Simplified 100.0%

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

   11. Taylor expanded in x around 0 53.3%

       \[\leadsto \frac{\frac{\color{blue}{1}}{s}}{4 + \frac{x}{s} \cdot -4} \]

   12. Taylor expanded in x around inf 53.3%

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

       1. associate-*r/ 53.3%

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

       2. *-commutative 53.3%

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

   14. Simplified 53.3%

       \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\frac{x \cdot -4}{s}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 51.9% accurate, 56.4× speedup

Math

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

FPCore

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

C

x_m = fabs(x);
float code(float x_m, float s) {
	return (1.0f / s) / (4.0f - ((x_m / s) * -4.0f));
}

Fortran

x_m = abs(x)
real(4) function code(x_m, s)
    real(4), intent (in) :: x_m
    real(4), intent (in) :: s
    code = (1.0e0 / s) / (4.0e0 - ((x_m / s) * (-4.0e0)))
end function

Julia

x_m = abs(x)
function code(x_m, s)
	return Float32(Float32(Float32(1.0) / s) / Float32(Float32(4.0) - Float32(Float32(x_m / s) * Float32(-4.0))))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m, s)
	tmp = (single(1.0) / s) / (single(4.0) - ((x_m / s) * single(-4.0)));
end

Derivation

1. Initial program 99.1%

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

   1. fabs-neg 99.1%

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

   2. distribute-frac-neg 99.1%

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

   3. distribute-frac-neg2 99.1%

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

   4. fabs-neg 99.1%

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

   5. *-commutative 99.1%

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

   6. fabs-neg 99.1%

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

   7. +-commutative 99.1%

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

   8. fabs-neg 99.1%

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

3. Simplified 99.1%

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

5. Taylor expanded in x around 0 99.1%

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

   1. associate-/r* 99.1%

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

   2. exp-prod 99.1%

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

   3. rem-square-sqrt 49.6%

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

   4. fabs-sqr 49.6%

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

   5. rem-square-sqrt 59.2%

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

   6. exp-prod 59.2%

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

   7. neg-mul-1 59.2%

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

   8. distribute-neg-frac2 59.2%

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

7. Simplified 60.4%

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

8. Taylor expanded in x around 0 57.0%

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

   1. *-commutative 57.0%

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

10. Simplified 57.0%

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

11. Taylor expanded in x around 0 52.6%

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

    1. add-sqr-sqrt 52.6%

       \[\leadsto \frac{\frac{1}{s}}{4 + \frac{x}{\color{blue}{\sqrt{s} \cdot \sqrt{s}}} \cdot -4} \]

    2. sqrt-unprod 64.7%

       \[\leadsto \frac{\frac{1}{s}}{4 + \frac{x}{\color{blue}{\sqrt{s \cdot s}}} \cdot -4} \]

    3. sqr-neg 64.7%

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

    4. sqrt-unprod -0.0%

       \[\leadsto \frac{\frac{1}{s}}{4 + \frac{x}{\color{blue}{\sqrt{-s} \cdot \sqrt{-s}}} \cdot -4} \]

    5. add-sqr-sqrt 52.8%

       \[\leadsto \frac{\frac{1}{s}}{4 + \frac{x}{\color{blue}{-s}} \cdot -4} \]

    6. distribute-frac-neg2 52.8%

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

13. Applied egg-rr 52.8%

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

    1. distribute-neg-frac2 52.8%

       \[\leadsto \frac{\frac{1}{s}}{4 + \color{blue}{\frac{x}{-s}} \cdot -4} \]

15. Simplified 52.8%

    \[\leadsto \frac{\frac{1}{s}}{4 + \color{blue}{\frac{x}{-s}} \cdot -4} \]

16. Final simplification 52.8%

    \[\leadsto \frac{\frac{1}{s}}{4 - \frac{x}{s} \cdot -4} \]
17. Add Preprocessing

Alternative 12: 51.6% accurate, 56.4× speedup

Math

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

FPCore

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

C

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

Julia

x_m = abs(x)
function code(x_m, s)
	return Float32(Float32(Float32(1.0) / s) / Float32(Float32(4.0) + Float32(x_m * Float32(Float32(-4.0) / s))))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m, s)
	tmp = (single(1.0) / s) / (single(4.0) + (x_m * (single(-4.0) / s)));
end

Derivation

1. Initial program 99.1%

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

   1. fabs-neg 99.1%

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

   2. distribute-frac-neg 99.1%

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

   3. distribute-frac-neg2 99.1%

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

   4. fabs-neg 99.1%

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

   5. *-commutative 99.1%

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

   6. fabs-neg 99.1%

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

   7. +-commutative 99.1%

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

   8. fabs-neg 99.1%

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

3. Simplified 99.1%

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

5. Taylor expanded in x around 0 99.1%

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

   1. associate-/r* 99.1%

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

   2. exp-prod 99.1%

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

   3. rem-square-sqrt 49.6%

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

   4. fabs-sqr 49.6%

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

   5. rem-square-sqrt 59.2%

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

   6. exp-prod 59.2%

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

   7. neg-mul-1 59.2%

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

   8. distribute-neg-frac2 59.2%

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

7. Simplified 60.4%

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

8. Taylor expanded in x around 0 57.0%

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

   1. *-commutative 57.0%

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

10. Simplified 57.0%

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

11. Taylor expanded in x around 0 52.6%

    \[\leadsto \frac{\frac{\color{blue}{1}}{s}}{4 + \frac{x}{s} \cdot -4} \]

12. Taylor expanded in x around 0 52.6%

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

    1. *-commutative 52.6%

       \[\leadsto \frac{\frac{1}{s}}{4 + \color{blue}{\frac{x}{s} \cdot -4}} \]

    2. associate-*l/ 52.6%

       \[\leadsto \frac{\frac{1}{s}}{4 + \color{blue}{\frac{x \cdot -4}{s}}} \]

    3. associate-/l* 52.6%

       \[\leadsto \frac{\frac{1}{s}}{4 + \color{blue}{x \cdot \frac{-4}{s}}} \]

14. Simplified 52.6%

    \[\leadsto \frac{\frac{1}{s}}{4 + \color{blue}{x \cdot \frac{-4}{s}}} \]
15. Add Preprocessing

Alternative 13: 30.6% accurate, 77.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0011500000255182385:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.25}{x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (if (<= x_m 0.0011500000255182385) (/ 0.25 s) (/ -0.25 x_m)))

C

x_m = fabs(x);
float code(float x_m, float s) {
	float tmp;
	if (x_m <= 0.0011500000255182385f) {
		tmp = 0.25f / s;
	} else {
		tmp = -0.25f / x_m;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(4) function code(x_m, s)
    real(4), intent (in) :: x_m
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x_m <= 0.0011500000255182385e0) then
        tmp = 0.25e0 / s
    else
        tmp = (-0.25e0) / x_m
    end if
    code = tmp
end function

Julia

x_m = abs(x)
function code(x_m, s)
	tmp = Float32(0.0)
	if (x_m <= Float32(0.0011500000255182385))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(-0.25) / x_m);
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, s)
	tmp = single(0.0);
	if (x_m <= single(0.0011500000255182385))
		tmp = single(0.25) / s;
	else
		tmp = single(-0.25) / x_m;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 0.00115000003

   1. Initial program 98.8%

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

      1. fabs-neg 98.8%

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

      2. distribute-frac-neg 98.8%

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

      3. distribute-frac-neg2 98.8%

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

      4. fabs-neg 98.8%

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

      5. *-commutative 98.8%

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

      6. fabs-neg 98.8%

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

      7. +-commutative 98.8%

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

      8. fabs-neg 98.8%

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

   3. Simplified 98.8%

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

   5. Taylor expanded in s around inf 32.7%

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

   if 0.00115000003 < x

   1. Initial program 100.0%

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

      1. fabs-neg 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. distribute-frac-neg2 100.0%

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

      4. fabs-neg 100.0%

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

      5. *-commutative 100.0%

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

      6. fabs-neg 100.0%

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

      7. +-commutative 100.0%

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

      8. fabs-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 100.0%

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

      1. associate-/r* 100.0%

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

      2. exp-prod 100.0%

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

      3. rem-square-sqrt 100.0%

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

      4. fabs-sqr 100.0%

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

      5. rem-square-sqrt 100.0%

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

      6. exp-prod 100.0%

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

      7. neg-mul-1 100.0%

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

      8. distribute-neg-frac2 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around 0 100.0%

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

      1. *-commutative 100.0%

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

   10. Simplified 100.0%

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

   11. Taylor expanded in x around 0 53.3%

       \[\leadsto \frac{\frac{\color{blue}{1}}{s}}{4 + \frac{x}{s} \cdot -4} \]

   12. Taylor expanded in s around 0 11.5%

       \[\leadsto \color{blue}{\frac{-0.25}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 7.3% accurate, 206.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

   1. fabs-neg 99.1%

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

   2. distribute-frac-neg 99.1%

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

   3. distribute-frac-neg2 99.1%

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

   4. fabs-neg 99.1%

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

   5. *-commutative 99.1%

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

   6. fabs-neg 99.1%

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

   7. +-commutative 99.1%

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

   8. fabs-neg 99.1%

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

3. Simplified 99.1%

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

5. Taylor expanded in x around 0 99.1%

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

   1. associate-/r* 99.1%

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

   2. exp-prod 99.1%

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

   3. rem-square-sqrt 49.6%

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

   4. fabs-sqr 49.6%

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

   5. rem-square-sqrt 59.2%

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

   6. exp-prod 59.2%

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

   7. neg-mul-1 59.2%

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

   8. distribute-neg-frac2 59.2%

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

7. Simplified 60.4%

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

8. Taylor expanded in x around 0 57.0%

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

   1. *-commutative 57.0%

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

10. Simplified 57.0%

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

11. Taylor expanded in x around 0 52.6%

    \[\leadsto \frac{\frac{\color{blue}{1}}{s}}{4 + \frac{x}{s} \cdot -4} \]

12. Taylor expanded in s around 0 9.4%

    \[\leadsto \color{blue}{\frac{-0.25}{x}} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024145 
(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))))))