Logistic distribution

Percentage Accurate: 99.5% → 99.6%

Time: 11.5s

Alternatives: 9

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: 99.6% accurate, 2.0× speedup

Math

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

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(x_m / Float32(-s)))
	return Float32(Float32(t_0 / 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 99.5%

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

   1. fabs-neg 99.5%

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

   2. distribute-frac-neg 99.5%

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

   3. distribute-frac-neg2 99.5%

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

   4. fabs-neg 99.5%

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

   5. *-commutative 99.5%

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

   6. fabs-neg 99.5%

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

   7. +-commutative 99.5%

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

   8. fabs-neg 99.5%

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

3. Simplified 99.5%

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

5. Taylor expanded in x around 0 99.5%

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

   1. associate-/r* 99.6%

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

   2. mul-1-neg 99.6%

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

   3. distribute-neg-frac2 99.6%

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

   4. +-commutative 99.6%

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

   5. mul-1-neg 99.6%

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

   6. distribute-neg-frac2 99.6%

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

7. Simplified 99.6%

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

   1. distribute-frac-neg2 99.6%

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

   2. rec-exp 99.5%

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

   3. pow1 99.5%

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

   4. pow1 99.5%

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

   5. add-sqr-sqrt 50.7%

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

   6. fabs-sqr 50.7%

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

   7. add-sqr-sqrt 96.6%

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

9. Applied egg-rr 96.6%

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

    1. rec-exp 96.6%

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

    2. distribute-neg-frac2 96.6%

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

11. Simplified 96.6%

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

    1. distribute-frac-neg2 99.6%

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

    2. rec-exp 99.5%

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

    3. pow1 99.5%

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

    4. pow1 99.5%

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

    5. add-sqr-sqrt 50.7%

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

    6. fabs-sqr 50.7%

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

    7. add-sqr-sqrt 96.6%

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

13. Applied egg-rr 63.8%

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

    1. rec-exp 96.6%

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

    2. distribute-neg-frac2 96.6%

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

15. Simplified 63.8%

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

Alternative 2: 99.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\left|x\_m\right| \leq 0.20000000298023224:\\ \;\;\;\;\frac{e^{\frac{x\_m}{s} + \mathsf{log1p}\left(e^{\frac{x\_m}{s}}\right) \cdot -2}}{s}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x\_m}{-s}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (if (<= (fabs x_m) 0.20000000298023224)
   (/ (exp (+ (/ x_m s) (* (log1p (exp (/ x_m s))) -2.0))) s)
   (exp (/ x_m (- s)))))

C

x_m = fabs(x);
float code(float x_m, float s) {
	float tmp;
	if (fabsf(x_m) <= 0.20000000298023224f) {
		tmp = expf(((x_m / s) + (log1pf(expf((x_m / s))) * -2.0f))) / s;
	} else {
		tmp = expf((x_m / -s));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m, s)
	tmp = Float32(0.0)
	if (abs(x_m) <= Float32(0.20000000298023224))
		tmp = Float32(exp(Float32(Float32(x_m / s) + Float32(log1p(exp(Float32(x_m / s))) * Float32(-2.0)))) / s);
	else
		tmp = exp(Float32(x_m / Float32(-s)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (fabs.f32 x) < 0.200000003

   1. Initial program 98.9%

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

      1. fabs-neg 98.9%

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

      2. distribute-frac-neg 98.9%

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

      3. distribute-frac-neg2 98.9%

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

      4. fabs-neg 98.9%

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

      5. *-commutative 98.9%

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

      6. fabs-neg 98.9%

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

      7. +-commutative 98.9%

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

      8. fabs-neg 98.9%

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

   3. Simplified 99.0%

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

   6. Applied egg-rr 71.2%

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

      1. *-lft-identity 71.2%

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

   8. Simplified 71.2%

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

      1. add-exp-log 67.9%

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

      2. log-div 67.7%

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

      3. add-log-exp 95.1%

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

      4. *-commutative 95.1%

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

      5. sum-log 94.6%

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

      6. log-pow 95.2%

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

      7. log1p-undefine 95.3%

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

      8. *-un-lft-identity 95.3%

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

      9. associate--r+ 95.4%

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

      10. exp-diff 95.4%

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

   10. Applied egg-rr 98.9%

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

       1. *-lft-identity 98.9%

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

       2. *-commutative 98.9%

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

   12. Simplified 98.9%

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

   if 0.200000003 < (fabs.f32 x)

   1. Initial program 100.0%

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

      1. fabs-neg 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. distribute-frac-neg2 100.0%

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

      4. fabs-neg 100.0%

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

      5. *-commutative 100.0%

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

      6. fabs-neg 100.0%

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

      7. +-commutative 100.0%

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

      8. fabs-neg 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 73.7%

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

   7. Taylor expanded in x around inf 51.2%

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

      1. div-inv 51.2%

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

      2. exp-prod 51.2%

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

      3. add-sqr-sqrt 1.6%

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

      4. fabs-sqr 1.6%

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

      5. add-sqr-sqrt 3.1%

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

      6. add-sqr-sqrt 3.1%

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

      7. sqrt-unprod 3.1%

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

      8. sqr-neg 3.1%

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

      9. sqrt-unprod -0.0%

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

      10. add-sqr-sqrt 100.0%

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

      11. exp-prod 100.0%

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

      12. div-inv 100.0%

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

      13. distribute-frac-neg2 100.0%

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

      14. exp-neg 100.0%

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

      15. add-sqr-sqrt 50.4%

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

      16. fabs-sqr 50.4%

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

      17. add-sqr-sqrt 51.9%

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

   9. Applied egg-rr 51.9%

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

       1. rec-exp 51.9%

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

       2. distribute-frac-neg 51.9%

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

   11. Simplified 51.9%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.20000000298023224:\\ \;\;\;\;\frac{e^{\frac{x}{s} + \mathsf{log1p}\left(e^{\frac{x}{s}}\right) \cdot -2}}{s}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{-s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 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(x_m / Float32(-s))) / s) / Float32(4.0))
end

MATLAB

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

Derivation

1. Initial program 99.5%

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

   1. fabs-neg 99.5%

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

   2. distribute-frac-neg 99.5%

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

   3. distribute-frac-neg2 99.5%

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

   4. fabs-neg 99.5%

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

   5. *-commutative 99.5%

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

   6. fabs-neg 99.5%

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

   7. +-commutative 99.5%

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

   8. fabs-neg 99.5%

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

3. Simplified 99.5%

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

5. Taylor expanded in x around 0 99.5%

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

   1. associate-/r* 99.6%

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

   2. mul-1-neg 99.6%

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

   3. distribute-neg-frac2 99.6%

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

   4. +-commutative 99.6%

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

   5. mul-1-neg 99.6%

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

   6. distribute-neg-frac2 99.6%

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

7. Simplified 99.6%

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

8. Taylor expanded in s around inf 94.9%

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

   1. distribute-frac-neg2 99.6%

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

   2. rec-exp 99.5%

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

   3. pow1 99.5%

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

   4. pow1 99.5%

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

   5. add-sqr-sqrt 50.7%

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

   6. fabs-sqr 50.7%

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

   7. add-sqr-sqrt 96.6%

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

10. Applied egg-rr 60.2%

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

    1. rec-exp 96.6%

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

    2. distribute-neg-frac2 96.6%

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

12. Simplified 60.3%

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

Alternative 4: 76.4% accurate, 6.0× speedup

Math

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

FPCore

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

C

x_m = fabs(x);
float code(float x_m, float s) {
	return 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 = exp((x_m / -s))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

   1. fabs-neg 99.5%

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

   2. distribute-frac-neg 99.5%

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

   3. distribute-frac-neg2 99.5%

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

   4. fabs-neg 99.5%

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

   5. *-commutative 99.5%

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

   6. fabs-neg 99.5%

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

   7. +-commutative 99.5%

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

   8. fabs-neg 99.5%

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

3. Simplified 99.5%

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

6. Applied egg-rr 83.7%

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

7. Taylor expanded in x around inf 40.1%

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

   1. div-inv 40.1%

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

   2. exp-prod 34.9%

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

   3. add-sqr-sqrt 3.0%

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

   4. fabs-sqr 3.0%

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

   5. add-sqr-sqrt 6.1%

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

   6. add-sqr-sqrt 6.1%

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

   7. sqrt-unprod 6.1%

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

   8. sqr-neg 6.1%

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

   9. sqrt-unprod 3.9%

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

   10. add-sqr-sqrt 62.9%

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

   11. exp-prod 78.4%

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

   12. div-inv 78.4%

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

   13. distribute-frac-neg2 78.4%

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

   14. exp-neg 78.4%

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

   15. add-sqr-sqrt 40.9%

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

   16. fabs-sqr 40.9%

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

   17. add-sqr-sqrt 43.8%

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

9. Applied egg-rr 43.8%

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

    1. rec-exp 43.8%

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

    2. distribute-frac-neg 43.8%

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

11. Simplified 43.8%

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

12. Final simplification 43.8%

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

Alternative 5: 62.9% accurate, 41.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{\frac{1}{s + x\_m \cdot \left(1 + \frac{x\_m}{s} \cdot 0.5\right)}}{4} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (/ (/ 1.0 (+ s (* x_m (+ 1.0 (* (/ x_m s) 0.5))))) 4.0))

C

x_m = fabs(x);
float code(float x_m, float s) {
	return (1.0f / (s + (x_m * (1.0f + ((x_m / s) * 0.5f))))) / 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 + (x_m * (1.0e0 + ((x_m / s) * 0.5e0))))) / 4.0e0
end function

Julia

x_m = abs(x)
function code(x_m, s)
	return Float32(Float32(Float32(1.0) / Float32(s + Float32(x_m * Float32(Float32(1.0) + Float32(Float32(x_m / s) * Float32(0.5)))))) / Float32(4.0))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m, s)
	tmp = (single(1.0) / (s + (x_m * (single(1.0) + ((x_m / s) * single(0.5)))))) / single(4.0);
end

Derivation

1. Initial program 99.5%

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

   1. fabs-neg 99.5%

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

   2. distribute-frac-neg 99.5%

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

   3. distribute-frac-neg2 99.5%

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

   4. fabs-neg 99.5%

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

   5. *-commutative 99.5%

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

   6. fabs-neg 99.5%

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

   7. +-commutative 99.5%

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

   8. fabs-neg 99.5%

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

3. Simplified 99.5%

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

5. Taylor expanded in x around 0 99.5%

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

   1. associate-/r* 99.6%

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

   2. mul-1-neg 99.6%

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

   3. distribute-neg-frac2 99.6%

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

   4. +-commutative 99.6%

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

   5. mul-1-neg 99.6%

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

   6. distribute-neg-frac2 99.6%

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

7. Simplified 99.6%

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

8. Taylor expanded in s around inf 94.9%

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

10. Applied egg-rr 60.2%

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

    1. unpow-1 60.2%

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

12. Simplified 60.2%

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

13. Taylor expanded in x around 0 61.2%

    \[\leadsto \frac{\frac{1}{\color{blue}{s + x \cdot \left(1 + 0.5 \cdot \frac{x}{s}\right)}}}{4} \]

14. Final simplification 61.2%

    \[\leadsto \frac{\frac{1}{s + x \cdot \left(1 + \frac{x}{s} \cdot 0.5\right)}}{4} \]
15. Add Preprocessing

Alternative 6: 50.4% accurate, 56.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.5%

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

   1. fabs-neg 99.5%

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

   2. distribute-frac-neg 99.5%

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

   3. distribute-frac-neg2 99.5%

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

   4. fabs-neg 99.5%

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

   5. *-commutative 99.5%

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

   6. fabs-neg 99.5%

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

   7. +-commutative 99.5%

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

   8. fabs-neg 99.5%

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

3. Simplified 99.5%

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

5. Taylor expanded in x around 0 99.5%

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

   1. associate-/r* 99.6%

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

   2. mul-1-neg 99.6%

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

   3. distribute-neg-frac2 99.6%

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

   4. +-commutative 99.6%

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

   5. mul-1-neg 99.6%

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

   6. distribute-neg-frac2 99.6%

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

7. Simplified 99.6%

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

8. Taylor expanded in s around inf 94.9%

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

10. Applied egg-rr 60.2%

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

    1. unpow-1 60.2%

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

12. Simplified 60.2%

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

13. Taylor expanded in s around inf 47.3%

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

Alternative 7: 28.8% accurate, 88.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

   1. fabs-neg 99.5%

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

   2. distribute-frac-neg 99.5%

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

   3. distribute-frac-neg2 99.5%

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

   4. fabs-neg 99.5%

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

   5. *-commutative 99.5%

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

   6. fabs-neg 99.5%

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

   7. +-commutative 99.5%

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

   8. fabs-neg 99.5%

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

3. Simplified 99.5%

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

5. Taylor expanded in x around 0 99.5%

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

   1. associate-/r* 99.6%

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

   2. mul-1-neg 99.6%

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

   3. distribute-neg-frac2 99.6%

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

   4. +-commutative 99.6%

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

   5. mul-1-neg 99.6%

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

   6. distribute-neg-frac2 99.6%

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

7. Simplified 99.6%

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

8. Taylor expanded in s around inf 94.9%

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

10. Applied egg-rr 60.2%

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

    1. unpow-1 60.2%

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

12. Simplified 60.2%

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

13. Taylor expanded in x around 0 26.4%

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

14. Final simplification 26.4%

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

Alternative 8: 26.7% accurate, 206.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

   1. fabs-neg 99.5%

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

   2. distribute-frac-neg 99.5%

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

   3. distribute-frac-neg2 99.5%

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

   4. fabs-neg 99.5%

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

   5. *-commutative 99.5%

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

   6. fabs-neg 99.5%

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

   7. +-commutative 99.5%

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

   8. fabs-neg 99.5%

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

3. Simplified 99.5%

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

5. Taylor expanded in s around inf 24.5%

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

Alternative 9: 8.3% accurate, 620.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ 1 \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s) :precision binary32 1.0)

C

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

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
end function

Julia

x_m = abs(x)
function code(x_m, s)
	return Float32(1.0)
end

MATLAB

x_m = abs(x);
function tmp = code(x_m, s)
	tmp = single(1.0);
end

Derivation

1. Initial program 99.5%

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

   1. fabs-neg 99.5%

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

   2. distribute-frac-neg 99.5%

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

   3. distribute-frac-neg2 99.5%

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

   4. fabs-neg 99.5%

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

   5. *-commutative 99.5%

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

   6. fabs-neg 99.5%

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

   7. +-commutative 99.5%

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

   8. fabs-neg 99.5%

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

3. Simplified 99.5%

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

6. Applied egg-rr 83.7%

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

7. Taylor expanded in x around inf 40.1%

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

8. Taylor expanded in x around 0 8.0%

   \[\leadsto \color{blue}{1} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024096 
(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))))))