Logistic distribution

Percentage Accurate: 99.5% → 99.5%

Time: 9.9s

Alternatives: 13

Speedup: 2.9×

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, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.8%

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

   1. *-lft-identity 99.8%

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

   2. associate-*r/ 99.8%

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

   3. associate-/l* 99.8%

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

   4. distribute-frac-neg 99.8%

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

   5. exp-neg 99.8%

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

   6. associate-/r/ 99.8%

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

   7. /-rgt-identity 99.8%

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

   8. associate-*l* 99.8%

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

3. Simplified 99.9%

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

4. Taylor expanded in x around 0 99.8%

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

   1. mul-1-neg 99.8%

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

   2. distribute-frac-neg 99.8%

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

   3. expm1-log1p-u 99.8%

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

   4. expm1-udef 99.8%

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

6. Applied egg-rr 99.8%

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

   1. expm1-def 99.8%

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

   2. expm1-log1p 99.8%

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

   3. unpow1 99.8%

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

   4. sqr-pow 49.9%

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

   5. fabs-sqr 49.9%

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

   6. sqr-pow 98.1%

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

   7. unpow1 98.1%

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

8. Simplified 98.1%

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

9. Taylor expanded in s around 0 98.1%

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

    1. +-commutative 98.1%

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

    2. distribute-lft-in 98.1%

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

    3. *-rgt-identity 98.1%

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

    4. fma-udef 98.1%

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

    5. unpow1 98.1%

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

    6. sqr-pow 49.9%

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

    7. fabs-sqr 49.9%

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

    8. sqr-pow 99.8%

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

    9. unpow1 99.8%

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

11. Simplified 99.8%

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

12. Final simplification 99.8%

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

Alternative 2: 97.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.4000000059604645:\\ \;\;\;\;{\left(\frac{3 + \left(e^{\frac{x}{s}} - \frac{x}{s}\right)}{\frac{1}{s}}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \frac{e^{\frac{-x}{s}}}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s)
 :precision binary32
 (if (<= (fabs x) 0.4000000059604645)
   (pow (/ (+ 3.0 (- (exp (/ x s)) (/ x s))) (/ 1.0 s)) -1.0)
   (* -0.25 (/ (exp (/ (- x) s)) x))))

C

x = abs(x);
float code(float x, float s) {
	float tmp;
	if (fabsf(x) <= 0.4000000059604645f) {
		tmp = powf(((3.0f + (expf((x / s)) - (x / s))) / (1.0f / s)), -1.0f);
	} else {
		tmp = -0.25f * (expf((-x / s)) / x);
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (abs(x) <= 0.4000000059604645e0) then
        tmp = ((3.0e0 + (exp((x / s)) - (x / s))) / (1.0e0 / s)) ** (-1.0e0)
    else
        tmp = (-0.25e0) * (exp((-x / s)) / x)
    end if
    code = tmp
end function

Julia

x = abs(x)
function code(x, s)
	tmp = Float32(0.0)
	if (abs(x) <= Float32(0.4000000059604645))
		tmp = Float32(Float32(Float32(3.0) + Float32(exp(Float32(x / s)) - Float32(x / s))) / Float32(Float32(1.0) / s)) ^ Float32(-1.0);
	else
		tmp = Float32(Float32(-0.25) * Float32(exp(Float32(Float32(-x) / s)) / x));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (abs(x) <= single(0.4000000059604645))
		tmp = ((single(3.0) + (exp((x / s)) - (x / s))) / (single(1.0) / s)) ^ single(-1.0);
	else
		tmp = single(-0.25) * (exp((-x / s)) / x);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (fabs.f32 x) < 0.400000006

   1. Initial program 99.6%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in s around inf 95.9%

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

      1. mul-1-neg 95.9%

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

      2. unsub-neg 95.9%

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

   6. Simplified 95.9%

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

   7. Taylor expanded in x around 0 95.8%

      \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\left(3 + e^{\frac{\left|x\right|}{s}}\right) - \frac{\left|x\right|}{s}}} \]
   8. Step-by-step derivation

      1. associate--l+ 95.9%

         \[\leadsto \frac{\frac{1}{s}}{\color{blue}{3 + \left(e^{\frac{\left|x\right|}{s}} - \frac{\left|x\right|}{s}\right)}} \]

      2. unpow1 95.9%

         \[\leadsto \frac{\frac{1}{s}}{3 + \left(e^{\frac{\left|\color{blue}{{x}^{1}}\right|}{s}} - \frac{\left|x\right|}{s}\right)} \]

      3. sqr-pow 55.0%

         \[\leadsto \frac{\frac{1}{s}}{3 + \left(e^{\frac{\left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|}{s}} - \frac{\left|x\right|}{s}\right)} \]

      4. fabs-sqr 55.0%

         \[\leadsto \frac{\frac{1}{s}}{3 + \left(e^{\frac{\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}}{s}} - \frac{\left|x\right|}{s}\right)} \]

      5. sqr-pow 67.9%

         \[\leadsto \frac{\frac{1}{s}}{3 + \left(e^{\frac{\color{blue}{{x}^{1}}}{s}} - \frac{\left|x\right|}{s}\right)} \]

      6. unpow1 67.9%

         \[\leadsto \frac{\frac{1}{s}}{3 + \left(e^{\frac{\color{blue}{x}}{s}} - \frac{\left|x\right|}{s}\right)} \]

      7. unpow1 67.9%

         \[\leadsto \frac{\frac{1}{s}}{3 + \left(e^{\frac{x}{s}} - \frac{\left|\color{blue}{{x}^{1}}\right|}{s}\right)} \]

      8. sqr-pow 55.1%

         \[\leadsto \frac{\frac{1}{s}}{3 + \left(e^{\frac{x}{s}} - \frac{\left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|}{s}\right)} \]

      9. fabs-sqr 55.1%

         \[\leadsto \frac{\frac{1}{s}}{3 + \left(e^{\frac{x}{s}} - \frac{\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}}{s}\right)} \]

      10. sqr-pow 68.9%

          \[\leadsto \frac{\frac{1}{s}}{3 + \left(e^{\frac{x}{s}} - \frac{\color{blue}{{x}^{1}}}{s}\right)} \]

      11. unpow1 68.9%

          \[\leadsto \frac{\frac{1}{s}}{3 + \left(e^{\frac{x}{s}} - \frac{\color{blue}{x}}{s}\right)} \]

   9. Simplified 68.9%

      \[\leadsto \frac{\frac{1}{s}}{\color{blue}{3 + \left(e^{\frac{x}{s}} - \frac{x}{s}\right)}} \]
   10. Step-by-step derivation

       1. associate-+r- 68.9%

          \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\left(3 + e^{\frac{x}{s}}\right) - \frac{x}{s}}} \]

   11. Applied egg-rr 68.9%

       \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\left(3 + e^{\frac{x}{s}}\right) - \frac{x}{s}}} \]
   12. Step-by-step derivation

       1. clear-num 69.0%

          \[\leadsto \color{blue}{\frac{1}{\frac{\left(3 + e^{\frac{x}{s}}\right) - \frac{x}{s}}{\frac{1}{s}}}} \]

       2. inv-pow 69.0%

          \[\leadsto \color{blue}{{\left(\frac{\left(3 + e^{\frac{x}{s}}\right) - \frac{x}{s}}{\frac{1}{s}}\right)}^{-1}} \]

       3. associate--l+ 69.0%

          \[\leadsto {\left(\frac{\color{blue}{3 + \left(e^{\frac{x}{s}} - \frac{x}{s}\right)}}{\frac{1}{s}}\right)}^{-1} \]

   13. Applied egg-rr 69.0%

       \[\leadsto \color{blue}{{\left(\frac{3 + \left(e^{\frac{x}{s}} - \frac{x}{s}\right)}{\frac{1}{s}}\right)}^{-1}} \]

   if 0.400000006 < (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. Taylor expanded in s around inf 100.0%

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

   3. Taylor expanded in s around 0 100.0%

      \[\leadsto \color{blue}{-0.25 \cdot \frac{e^{-1 \cdot \frac{\left|x\right|}{s}}}{\left|x\right|}} \]
   4. Step-by-step derivation

      1. mul-1-neg 100.0%

         \[\leadsto -0.25 \cdot \frac{e^{\color{blue}{-\frac{\left|x\right|}{s}}}}{\left|x\right|} \]

      2. distribute-frac-neg 100.0%

         \[\leadsto -0.25 \cdot \frac{e^{\color{blue}{\frac{-\left|x\right|}{s}}}}{\left|x\right|} \]

      3. distribute-frac-neg 100.0%

         \[\leadsto -0.25 \cdot \frac{e^{\color{blue}{-\frac{\left|x\right|}{s}}}}{\left|x\right|} \]

      4. mul-1-neg 100.0%

         \[\leadsto -0.25 \cdot \frac{e^{\color{blue}{-1 \cdot \frac{\left|x\right|}{s}}}}{\left|x\right|} \]

      5. associate-*r/ 100.0%

         \[\leadsto -0.25 \cdot \frac{e^{\color{blue}{\frac{-1 \cdot \left|x\right|}{s}}}}{\left|x\right|} \]

      6. neg-mul-1 100.0%

         \[\leadsto -0.25 \cdot \frac{e^{\frac{\color{blue}{-\left|x\right|}}{s}}}{\left|x\right|} \]

      7. unpow1 100.0%

         \[\leadsto -0.25 \cdot \frac{e^{\frac{-\left|\color{blue}{{x}^{1}}\right|}{s}}}{\left|x\right|} \]

      8. sqr-pow 43.1%

         \[\leadsto -0.25 \cdot \frac{e^{\frac{-\left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|}{s}}}{\left|x\right|} \]

      9. fabs-sqr 43.1%

         \[\leadsto -0.25 \cdot \frac{e^{\frac{-\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}}{s}}}{\left|x\right|} \]

      10. sqr-pow 44.9%

          \[\leadsto -0.25 \cdot \frac{e^{\frac{-\color{blue}{{x}^{1}}}{s}}}{\left|x\right|} \]

      11. unpow1 44.9%

          \[\leadsto -0.25 \cdot \frac{e^{\frac{-\color{blue}{x}}{s}}}{\left|x\right|} \]

      12. unpow1 44.9%

          \[\leadsto -0.25 \cdot \frac{e^{\frac{-x}{s}}}{\left|\color{blue}{{x}^{1}}\right|} \]

      13. sqr-pow 43.1%

          \[\leadsto -0.25 \cdot \frac{e^{\frac{-x}{s}}}{\left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|} \]

      14. fabs-sqr 43.1%

          \[\leadsto -0.25 \cdot \frac{e^{\frac{-x}{s}}}{\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}} \]

      15. sqr-pow 44.9%

          \[\leadsto -0.25 \cdot \frac{e^{\frac{-x}{s}}}{\color{blue}{{x}^{1}}} \]

      16. unpow1 44.9%

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

   5. Simplified 44.9%

      \[\leadsto \color{blue}{-0.25 \cdot \frac{e^{\frac{-x}{s}}}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.4000000059604645:\\ \;\;\;\;{\left(\frac{3 + \left(e^{\frac{x}{s}} - \frac{x}{s}\right)}{\frac{1}{s}}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \frac{e^{\frac{-x}{s}}}{x}\\ \end{array} \]

Alternative 3: 97.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.4000000059604645:\\ \;\;\;\;\frac{1}{s \cdot \left(2 + \left(e^{\frac{x}{s}} + \left(1 - \frac{x}{s}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \frac{e^{\frac{-x}{s}}}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s)
 :precision binary32
 (if (<= (fabs x) 0.4000000059604645)
   (/ 1.0 (* s (+ 2.0 (+ (exp (/ x s)) (- 1.0 (/ x s))))))
   (* -0.25 (/ (exp (/ (- x) s)) x))))

C

x = abs(x);
float code(float x, float s) {
	float tmp;
	if (fabsf(x) <= 0.4000000059604645f) {
		tmp = 1.0f / (s * (2.0f + (expf((x / s)) + (1.0f - (x / s)))));
	} else {
		tmp = -0.25f * (expf((-x / s)) / x);
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (abs(x) <= 0.4000000059604645e0) then
        tmp = 1.0e0 / (s * (2.0e0 + (exp((x / s)) + (1.0e0 - (x / s)))))
    else
        tmp = (-0.25e0) * (exp((-x / s)) / x)
    end if
    code = tmp
end function

Julia

x = abs(x)
function code(x, s)
	tmp = Float32(0.0)
	if (abs(x) <= Float32(0.4000000059604645))
		tmp = Float32(Float32(1.0) / Float32(s * Float32(Float32(2.0) + Float32(exp(Float32(x / s)) + Float32(Float32(1.0) - Float32(x / s))))));
	else
		tmp = Float32(Float32(-0.25) * Float32(exp(Float32(Float32(-x) / s)) / x));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (abs(x) <= single(0.4000000059604645))
		tmp = single(1.0) / (s * (single(2.0) + (exp((x / s)) + (single(1.0) - (x / s)))));
	else
		tmp = single(-0.25) * (exp((-x / s)) / x);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (fabs.f32 x) < 0.400000006

   1. Initial program 99.6%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in s around inf 95.9%

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

      1. mul-1-neg 95.9%

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

      2. unsub-neg 95.9%

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

   6. Simplified 95.9%

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

      1. expm1-log1p-u 93.6%

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

      2. expm1-udef 93.5%

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

      3. associate-/l/ 93.5%

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

   8. Applied egg-rr 93.5%

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

      1. expm1-def 93.6%

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

      2. expm1-log1p 95.9%

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

      3. *-commutative 95.9%

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

      4. +-commutative 95.9%

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

      5. +-commutative 95.9%

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

      6. associate-+l+ 95.9%

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

      7. unpow1 95.9%

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

      8. sqr-pow 55.0%

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

      9. fabs-sqr 55.0%

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

      10. sqr-pow 67.9%

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

      11. unpow1 67.9%

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

      12. unpow1 67.9%

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

      13. sqr-pow 55.1%

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

      14. fabs-sqr 55.1%

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

      15. sqr-pow 69.0%

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

      16. unpow1 69.0%

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

   10. Simplified 69.0%

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

   if 0.400000006 < (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. Taylor expanded in s around inf 100.0%

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

   3. Taylor expanded in s around 0 100.0%

      \[\leadsto \color{blue}{-0.25 \cdot \frac{e^{-1 \cdot \frac{\left|x\right|}{s}}}{\left|x\right|}} \]
   4. Step-by-step derivation

      1. mul-1-neg 100.0%

         \[\leadsto -0.25 \cdot \frac{e^{\color{blue}{-\frac{\left|x\right|}{s}}}}{\left|x\right|} \]

      2. distribute-frac-neg 100.0%

         \[\leadsto -0.25 \cdot \frac{e^{\color{blue}{\frac{-\left|x\right|}{s}}}}{\left|x\right|} \]

      3. distribute-frac-neg 100.0%

         \[\leadsto -0.25 \cdot \frac{e^{\color{blue}{-\frac{\left|x\right|}{s}}}}{\left|x\right|} \]

      4. mul-1-neg 100.0%

         \[\leadsto -0.25 \cdot \frac{e^{\color{blue}{-1 \cdot \frac{\left|x\right|}{s}}}}{\left|x\right|} \]

      5. associate-*r/ 100.0%

         \[\leadsto -0.25 \cdot \frac{e^{\color{blue}{\frac{-1 \cdot \left|x\right|}{s}}}}{\left|x\right|} \]

      6. neg-mul-1 100.0%

         \[\leadsto -0.25 \cdot \frac{e^{\frac{\color{blue}{-\left|x\right|}}{s}}}{\left|x\right|} \]

      7. unpow1 100.0%

         \[\leadsto -0.25 \cdot \frac{e^{\frac{-\left|\color{blue}{{x}^{1}}\right|}{s}}}{\left|x\right|} \]

      8. sqr-pow 43.1%

         \[\leadsto -0.25 \cdot \frac{e^{\frac{-\left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|}{s}}}{\left|x\right|} \]

      9. fabs-sqr 43.1%

         \[\leadsto -0.25 \cdot \frac{e^{\frac{-\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}}{s}}}{\left|x\right|} \]

      10. sqr-pow 44.9%

          \[\leadsto -0.25 \cdot \frac{e^{\frac{-\color{blue}{{x}^{1}}}{s}}}{\left|x\right|} \]

      11. unpow1 44.9%

          \[\leadsto -0.25 \cdot \frac{e^{\frac{-\color{blue}{x}}{s}}}{\left|x\right|} \]

      12. unpow1 44.9%

          \[\leadsto -0.25 \cdot \frac{e^{\frac{-x}{s}}}{\left|\color{blue}{{x}^{1}}\right|} \]

      13. sqr-pow 43.1%

          \[\leadsto -0.25 \cdot \frac{e^{\frac{-x}{s}}}{\left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|} \]

      14. fabs-sqr 43.1%

          \[\leadsto -0.25 \cdot \frac{e^{\frac{-x}{s}}}{\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}} \]

      15. sqr-pow 44.9%

          \[\leadsto -0.25 \cdot \frac{e^{\frac{-x}{s}}}{\color{blue}{{x}^{1}}} \]

      16. unpow1 44.9%

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

   5. Simplified 44.9%

      \[\leadsto \color{blue}{-0.25 \cdot \frac{e^{\frac{-x}{s}}}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.4000000059604645:\\ \;\;\;\;\frac{1}{s \cdot \left(2 + \left(e^{\frac{x}{s}} + \left(1 - \frac{x}{s}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \frac{e^{\frac{-x}{s}}}{x}\\ \end{array} \]

Alternative 4: 99.5% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

   1. *-lft-identity 99.8%

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

   2. associate-*r/ 99.8%

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

   3. associate-/l* 99.8%

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

   4. distribute-frac-neg 99.8%

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

   5. exp-neg 99.8%

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

   6. associate-/r/ 99.8%

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

   7. /-rgt-identity 99.8%

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

   8. associate-*l* 99.8%

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

3. Simplified 99.9%

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

4. Taylor expanded in x around 0 99.8%

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

   1. mul-1-neg 99.8%

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

   2. distribute-frac-neg 99.8%

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

   3. expm1-log1p-u 99.8%

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

   4. expm1-udef 99.8%

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

6. Applied egg-rr 99.8%

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

   1. expm1-def 99.8%

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

   2. expm1-log1p 99.8%

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

   3. unpow1 99.8%

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

   4. sqr-pow 49.9%

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

   5. fabs-sqr 49.9%

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

   6. sqr-pow 98.1%

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

   7. unpow1 98.1%

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

8. Simplified 98.1%

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

   1. expm1-log1p-u 98.1%

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

   2. expm1-udef 87.2%

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

10. Applied egg-rr 87.2%

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

    1. expm1-def 98.1%

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

    2. expm1-log1p 98.1%

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

    3. unpow1 98.1%

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

    4. sqr-pow 49.9%

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

    5. fabs-sqr 49.9%

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

    6. sqr-pow 99.8%

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

    7. unpow1 99.8%

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

12. Simplified 99.8%

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

13. Final simplification 99.8%

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

Alternative 5: 99.5% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

   1. *-lft-identity 99.8%

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

   2. associate-*r/ 99.8%

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

   3. associate-/l* 99.8%

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

   4. distribute-frac-neg 99.8%

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

   5. exp-neg 99.8%

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

   6. associate-/r/ 99.8%

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

   7. /-rgt-identity 99.8%

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

   8. associate-*l* 99.8%

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

3. Simplified 99.9%

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

4. Taylor expanded in x around 0 99.8%

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

   1. mul-1-neg 99.8%

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

   2. distribute-frac-neg 99.8%

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

   3. expm1-log1p-u 99.8%

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

   4. expm1-udef 99.8%

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

6. Applied egg-rr 99.8%

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

   1. expm1-def 99.8%

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

   2. expm1-log1p 99.8%

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

   3. unpow1 99.8%

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

   4. sqr-pow 49.9%

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

   5. fabs-sqr 49.9%

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

   6. sqr-pow 98.1%

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

   7. unpow1 98.1%

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

8. Simplified 98.1%

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

9. Taylor expanded in s around 0 98.1%

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

    1. +-commutative 98.1%

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

    2. distribute-lft-in 98.1%

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

    3. *-rgt-identity 98.1%

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

    4. fma-udef 98.1%

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

    5. unpow1 98.1%

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

    6. sqr-pow 49.9%

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

    7. fabs-sqr 49.9%

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

    8. sqr-pow 99.8%

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

    9. unpow1 99.8%

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

11. Simplified 99.8%

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

12. Taylor expanded in s around 0 99.8%

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

13. Final simplification 99.8%

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

Alternative 6: 97.4% accurate, 5.4× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 0.4000000059604645:\\ \;\;\;\;\frac{1}{s \cdot \left(3 + \left(e^{\frac{x}{s}} - \frac{x}{s}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \frac{e^{\frac{-x}{s}}}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s)
 :precision binary32
 (if (<= x 0.4000000059604645)
   (/ 1.0 (* s (+ 3.0 (- (exp (/ x s)) (/ x s)))))
   (* -0.25 (/ (exp (/ (- x) s)) x))))

C

x = abs(x);
float code(float x, float s) {
	float tmp;
	if (x <= 0.4000000059604645f) {
		tmp = 1.0f / (s * (3.0f + (expf((x / s)) - (x / s))));
	} else {
		tmp = -0.25f * (expf((-x / s)) / x);
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 0.4000000059604645e0) then
        tmp = 1.0e0 / (s * (3.0e0 + (exp((x / s)) - (x / s))))
    else
        tmp = (-0.25e0) * (exp((-x / s)) / x)
    end if
    code = tmp
end function

Julia

x = abs(x)
function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(0.4000000059604645))
		tmp = Float32(Float32(1.0) / Float32(s * Float32(Float32(3.0) + Float32(exp(Float32(x / s)) - Float32(x / s)))));
	else
		tmp = Float32(Float32(-0.25) * Float32(exp(Float32(Float32(-x) / s)) / x));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(0.4000000059604645))
		tmp = single(1.0) / (s * (single(3.0) + (exp((x / s)) - (x / s))));
	else
		tmp = single(-0.25) * (exp((-x / s)) / x);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 0.400000006

   1. Initial program 99.7%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in s around inf 83.4%

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

      1. mul-1-neg 83.4%

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

      2. unsub-neg 83.4%

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

   6. Simplified 83.4%

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

   7. Taylor expanded in x around 0 83.4%

      \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\left(3 + e^{\frac{\left|x\right|}{s}}\right) - \frac{\left|x\right|}{s}}} \]
   8. Step-by-step derivation

      1. associate--l+ 83.4%

         \[\leadsto \frac{\frac{1}{s}}{\color{blue}{3 + \left(e^{\frac{\left|x\right|}{s}} - \frac{\left|x\right|}{s}\right)}} \]

      2. unpow1 83.4%

         \[\leadsto \frac{\frac{1}{s}}{3 + \left(e^{\frac{\left|\color{blue}{{x}^{1}}\right|}{s}} - \frac{\left|x\right|}{s}\right)} \]

      3. sqr-pow 34.7%

         \[\leadsto \frac{\frac{1}{s}}{3 + \left(e^{\frac{\left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|}{s}} - \frac{\left|x\right|}{s}\right)} \]

      4. fabs-sqr 34.7%

         \[\leadsto \frac{\frac{1}{s}}{3 + \left(e^{\frac{\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}}{s}} - \frac{\left|x\right|}{s}\right)} \]

      5. sqr-pow 58.3%

         \[\leadsto \frac{\frac{1}{s}}{3 + \left(e^{\frac{\color{blue}{{x}^{1}}}{s}} - \frac{\left|x\right|}{s}\right)} \]

      6. unpow1 58.3%

         \[\leadsto \frac{\frac{1}{s}}{3 + \left(e^{\frac{\color{blue}{x}}{s}} - \frac{\left|x\right|}{s}\right)} \]

      7. unpow1 58.3%

         \[\leadsto \frac{\frac{1}{s}}{3 + \left(e^{\frac{x}{s}} - \frac{\left|\color{blue}{{x}^{1}}\right|}{s}\right)} \]

      8. sqr-pow 34.7%

         \[\leadsto \frac{\frac{1}{s}}{3 + \left(e^{\frac{x}{s}} - \frac{\left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|}{s}\right)} \]

      9. fabs-sqr 34.7%

         \[\leadsto \frac{\frac{1}{s}}{3 + \left(e^{\frac{x}{s}} - \frac{\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}}{s}\right)} \]

      10. sqr-pow 58.9%

          \[\leadsto \frac{\frac{1}{s}}{3 + \left(e^{\frac{x}{s}} - \frac{\color{blue}{{x}^{1}}}{s}\right)} \]

      11. unpow1 58.9%

          \[\leadsto \frac{\frac{1}{s}}{3 + \left(e^{\frac{x}{s}} - \frac{\color{blue}{x}}{s}\right)} \]

   9. Simplified 58.9%

      \[\leadsto \frac{\frac{1}{s}}{\color{blue}{3 + \left(e^{\frac{x}{s}} - \frac{x}{s}\right)}} \]
   10. Step-by-step derivation

       1. associate-+r- 58.9%

          \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\left(3 + e^{\frac{x}{s}}\right) - \frac{x}{s}}} \]

   11. Applied egg-rr 58.9%

       \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\left(3 + e^{\frac{x}{s}}\right) - \frac{x}{s}}} \]
   12. Step-by-step derivation

       1. expm1-log1p-u 57.4%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1}{s}}{\left(3 + e^{\frac{x}{s}}\right) - \frac{x}{s}}\right)\right)} \]

       2. expm1-udef 70.5%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{1}{s}}{\left(3 + e^{\frac{x}{s}}\right) - \frac{x}{s}}\right)} - 1} \]

       3. associate--l+ 70.5%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{1}{s}}{\color{blue}{3 + \left(e^{\frac{x}{s}} - \frac{x}{s}\right)}}\right)} - 1 \]

   13. Applied egg-rr 70.5%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{1}{s}}{3 + \left(e^{\frac{x}{s}} - \frac{x}{s}\right)}\right)} - 1} \]
   14. Step-by-step derivation

       1. expm1-def 57.4%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1}{s}}{3 + \left(e^{\frac{x}{s}} - \frac{x}{s}\right)}\right)\right)} \]

       2. expm1-log1p 58.9%

          \[\leadsto \color{blue}{\frac{\frac{1}{s}}{3 + \left(e^{\frac{x}{s}} - \frac{x}{s}\right)}} \]

       3. associate-/l/ 59.4%

          \[\leadsto \color{blue}{\frac{1}{\left(3 + \left(e^{\frac{x}{s}} - \frac{x}{s}\right)\right) \cdot s}} \]

       4. *-commutative 59.4%

          \[\leadsto \frac{1}{\color{blue}{s \cdot \left(3 + \left(e^{\frac{x}{s}} - \frac{x}{s}\right)\right)}} \]

   15. Simplified 59.4%

       \[\leadsto \color{blue}{\frac{1}{s \cdot \left(3 + \left(e^{\frac{x}{s}} - \frac{x}{s}\right)\right)}} \]

   if 0.400000006 < 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. Taylor expanded in s around inf 100.0%

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

   3. Taylor expanded in s around 0 100.0%

      \[\leadsto \color{blue}{-0.25 \cdot \frac{e^{-1 \cdot \frac{\left|x\right|}{s}}}{\left|x\right|}} \]
   4. Step-by-step derivation

      1. mul-1-neg 100.0%

         \[\leadsto -0.25 \cdot \frac{e^{\color{blue}{-\frac{\left|x\right|}{s}}}}{\left|x\right|} \]

      2. distribute-frac-neg 100.0%

         \[\leadsto -0.25 \cdot \frac{e^{\color{blue}{\frac{-\left|x\right|}{s}}}}{\left|x\right|} \]

      3. distribute-frac-neg 100.0%

         \[\leadsto -0.25 \cdot \frac{e^{\color{blue}{-\frac{\left|x\right|}{s}}}}{\left|x\right|} \]

      4. mul-1-neg 100.0%

         \[\leadsto -0.25 \cdot \frac{e^{\color{blue}{-1 \cdot \frac{\left|x\right|}{s}}}}{\left|x\right|} \]

      5. associate-*r/ 100.0%

         \[\leadsto -0.25 \cdot \frac{e^{\color{blue}{\frac{-1 \cdot \left|x\right|}{s}}}}{\left|x\right|} \]

      6. neg-mul-1 100.0%

         \[\leadsto -0.25 \cdot \frac{e^{\frac{\color{blue}{-\left|x\right|}}{s}}}{\left|x\right|} \]

      7. unpow1 100.0%

         \[\leadsto -0.25 \cdot \frac{e^{\frac{-\left|\color{blue}{{x}^{1}}\right|}{s}}}{\left|x\right|} \]

      8. sqr-pow 100.0%

         \[\leadsto -0.25 \cdot \frac{e^{\frac{-\left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|}{s}}}{\left|x\right|} \]

      9. fabs-sqr 100.0%

         \[\leadsto -0.25 \cdot \frac{e^{\frac{-\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}}{s}}}{\left|x\right|} \]

      10. sqr-pow 100.0%

          \[\leadsto -0.25 \cdot \frac{e^{\frac{-\color{blue}{{x}^{1}}}{s}}}{\left|x\right|} \]

      11. unpow1 100.0%

          \[\leadsto -0.25 \cdot \frac{e^{\frac{-\color{blue}{x}}{s}}}{\left|x\right|} \]

      12. unpow1 100.0%

          \[\leadsto -0.25 \cdot \frac{e^{\frac{-x}{s}}}{\left|\color{blue}{{x}^{1}}\right|} \]

      13. sqr-pow 100.0%

          \[\leadsto -0.25 \cdot \frac{e^{\frac{-x}{s}}}{\left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|} \]

      14. fabs-sqr 100.0%

          \[\leadsto -0.25 \cdot \frac{e^{\frac{-x}{s}}}{\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}} \]

      15. sqr-pow 100.0%

          \[\leadsto -0.25 \cdot \frac{e^{\frac{-x}{s}}}{\color{blue}{{x}^{1}}} \]

      16. unpow1 100.0%

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

   5. Simplified 100.0%

      \[\leadsto \color{blue}{-0.25 \cdot \frac{e^{\frac{-x}{s}}}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.4000000059604645:\\ \;\;\;\;\frac{1}{s \cdot \left(3 + \left(e^{\frac{x}{s}} - \frac{x}{s}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \frac{e^{\frac{-x}{s}}}{x}\\ \end{array} \]

Alternative 7: 85.7% accurate, 5.6× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 3.999999999279835 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{1}{s}}{\frac{1}{\frac{s}{x} \cdot \frac{s}{x}} + 4}\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \frac{e^{\frac{-x}{s}}}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s)
 :precision binary32
 (if (<= x 3.999999999279835e-23)
   (/ (/ 1.0 s) (+ (/ 1.0 (* (/ s x) (/ s x))) 4.0))
   (* -0.25 (/ (exp (/ (- x) s)) x))))

C

x = abs(x);
float code(float x, float s) {
	float tmp;
	if (x <= 3.999999999279835e-23f) {
		tmp = (1.0f / s) / ((1.0f / ((s / x) * (s / x))) + 4.0f);
	} else {
		tmp = -0.25f * (expf((-x / s)) / x);
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 3.999999999279835e-23) then
        tmp = (1.0e0 / s) / ((1.0e0 / ((s / x) * (s / x))) + 4.0e0)
    else
        tmp = (-0.25e0) * (exp((-x / s)) / x)
    end if
    code = tmp
end function

Julia

x = abs(x)
function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(3.999999999279835e-23))
		tmp = Float32(Float32(Float32(1.0) / s) / Float32(Float32(Float32(1.0) / Float32(Float32(s / x) * Float32(s / x))) + Float32(4.0)));
	else
		tmp = Float32(Float32(-0.25) * Float32(exp(Float32(Float32(-x) / s)) / x));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(3.999999999279835e-23))
		tmp = (single(1.0) / s) / ((single(1.0) / ((s / x) * (s / x))) + single(4.0));
	else
		tmp = single(-0.25) * (exp((-x / s)) / x);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 4e-23

   1. Initial program 99.8%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in s around inf 51.3%

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

      1. associate-+r+ 51.3%

         \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right) + \left(-1 \cdot \frac{\left|x\right|}{s} + \frac{\left|x\right|}{s}\right)}} \]

      2. distribute-lft1-in 51.3%

         \[\leadsto \frac{\frac{1}{s}}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right) + \color{blue}{\left(-1 + 1\right) \cdot \frac{\left|x\right|}{s}}} \]

      3. metadata-eval 51.3%

         \[\leadsto \frac{\frac{1}{s}}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right) + \color{blue}{0} \cdot \frac{\left|x\right|}{s}} \]

      4. mul0-lft 68.8%

         \[\leadsto \frac{\frac{1}{s}}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right) + \color{blue}{0}} \]

      5. associate-+r+ 68.8%

         \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + \left(4 + 0\right)}} \]

      6. metadata-eval 68.8%

         \[\leadsto \frac{\frac{1}{s}}{\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + \color{blue}{4}} \]

      7. unpow2 68.8%

         \[\leadsto \frac{\frac{1}{s}}{\frac{\color{blue}{\left|x\right| \cdot \left|x\right|}}{{s}^{2}} + 4} \]

      8. sqr-abs 68.8%

         \[\leadsto \frac{\frac{1}{s}}{\frac{\color{blue}{x \cdot x}}{{s}^{2}} + 4} \]

      9. unpow2 68.8%

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

   6. Simplified 68.8%

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

      1. clear-num 68.8%

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

      2. inv-pow 68.8%

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

   8. Applied egg-rr 68.8%

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

      1. unpow-1 68.8%

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

      2. times-frac 70.8%

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

   10. Simplified 70.8%

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

   if 4e-23 < x

   1. Initial program 99.8%

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

   2. Taylor expanded in s around inf 97.1%

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

   3. Taylor expanded in s around 0 89.4%

      \[\leadsto \color{blue}{-0.25 \cdot \frac{e^{-1 \cdot \frac{\left|x\right|}{s}}}{\left|x\right|}} \]
   4. Step-by-step derivation

      1. mul-1-neg 89.4%

         \[\leadsto -0.25 \cdot \frac{e^{\color{blue}{-\frac{\left|x\right|}{s}}}}{\left|x\right|} \]

      2. distribute-frac-neg 89.4%

         \[\leadsto -0.25 \cdot \frac{e^{\color{blue}{\frac{-\left|x\right|}{s}}}}{\left|x\right|} \]

      3. distribute-frac-neg 89.4%

         \[\leadsto -0.25 \cdot \frac{e^{\color{blue}{-\frac{\left|x\right|}{s}}}}{\left|x\right|} \]

      4. mul-1-neg 89.4%

         \[\leadsto -0.25 \cdot \frac{e^{\color{blue}{-1 \cdot \frac{\left|x\right|}{s}}}}{\left|x\right|} \]

      5. associate-*r/ 89.4%

         \[\leadsto -0.25 \cdot \frac{e^{\color{blue}{\frac{-1 \cdot \left|x\right|}{s}}}}{\left|x\right|} \]

      6. neg-mul-1 89.4%

         \[\leadsto -0.25 \cdot \frac{e^{\frac{\color{blue}{-\left|x\right|}}{s}}}{\left|x\right|} \]

      7. unpow1 89.4%

         \[\leadsto -0.25 \cdot \frac{e^{\frac{-\left|\color{blue}{{x}^{1}}\right|}{s}}}{\left|x\right|} \]

      8. sqr-pow 89.4%

         \[\leadsto -0.25 \cdot \frac{e^{\frac{-\left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|}{s}}}{\left|x\right|} \]

      9. fabs-sqr 89.4%

         \[\leadsto -0.25 \cdot \frac{e^{\frac{-\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}}{s}}}{\left|x\right|} \]

      10. sqr-pow 89.4%

          \[\leadsto -0.25 \cdot \frac{e^{\frac{-\color{blue}{{x}^{1}}}{s}}}{\left|x\right|} \]

      11. unpow1 89.4%

          \[\leadsto -0.25 \cdot \frac{e^{\frac{-\color{blue}{x}}{s}}}{\left|x\right|} \]

      12. unpow1 89.4%

          \[\leadsto -0.25 \cdot \frac{e^{\frac{-x}{s}}}{\left|\color{blue}{{x}^{1}}\right|} \]

      13. sqr-pow 89.4%

          \[\leadsto -0.25 \cdot \frac{e^{\frac{-x}{s}}}{\left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|} \]

      14. fabs-sqr 89.4%

          \[\leadsto -0.25 \cdot \frac{e^{\frac{-x}{s}}}{\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}} \]

      15. sqr-pow 89.4%

          \[\leadsto -0.25 \cdot \frac{e^{\frac{-x}{s}}}{\color{blue}{{x}^{1}}} \]

      16. unpow1 89.4%

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

   5. Simplified 89.4%

      \[\leadsto \color{blue}{-0.25 \cdot \frac{e^{\frac{-x}{s}}}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.999999999279835 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{1}{s}}{\frac{1}{\frac{s}{x} \cdot \frac{s}{x}} + 4}\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \frac{e^{\frac{-x}{s}}}{x}\\ \end{array} \]

Alternative 8: 82.1% accurate, 36.2× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 1.2800000250130962 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{1}{s}}{\frac{1}{\frac{s}{x} \cdot \frac{s}{x}} + 4}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \frac{x \cdot x}{s \cdot s}}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s)
 :precision binary32
 (if (<= x 1.2800000250130962e-23)
   (/ (/ 1.0 s) (+ (/ 1.0 (* (/ s x) (/ s x))) 4.0))
   (/ (/ 1.0 s) (+ 4.0 (/ (* x x) (* s s))))))

C

x = abs(x);
float code(float x, float s) {
	float tmp;
	if (x <= 1.2800000250130962e-23f) {
		tmp = (1.0f / s) / ((1.0f / ((s / x) * (s / x))) + 4.0f);
	} else {
		tmp = (1.0f / s) / (4.0f + ((x * x) / (s * s)));
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 1.2800000250130962e-23) then
        tmp = (1.0e0 / s) / ((1.0e0 / ((s / x) * (s / x))) + 4.0e0)
    else
        tmp = (1.0e0 / s) / (4.0e0 + ((x * x) / (s * s)))
    end if
    code = tmp
end function

Julia

x = abs(x)
function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(1.2800000250130962e-23))
		tmp = Float32(Float32(Float32(1.0) / s) / Float32(Float32(Float32(1.0) / Float32(Float32(s / x) * Float32(s / x))) + Float32(4.0)));
	else
		tmp = Float32(Float32(Float32(1.0) / s) / Float32(Float32(4.0) + Float32(Float32(x * x) / Float32(s * s))));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(1.2800000250130962e-23))
		tmp = (single(1.0) / s) / ((single(1.0) / ((s / x) * (s / x))) + single(4.0));
	else
		tmp = (single(1.0) / s) / (single(4.0) + ((x * x) / (s * s)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 1.28000003e-23

   1. Initial program 99.8%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in s around inf 51.0%

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

      1. associate-+r+ 51.0%

         \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right) + \left(-1 \cdot \frac{\left|x\right|}{s} + \frac{\left|x\right|}{s}\right)}} \]

      2. distribute-lft1-in 51.0%

         \[\leadsto \frac{\frac{1}{s}}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right) + \color{blue}{\left(-1 + 1\right) \cdot \frac{\left|x\right|}{s}}} \]

      3. metadata-eval 51.0%

         \[\leadsto \frac{\frac{1}{s}}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right) + \color{blue}{0} \cdot \frac{\left|x\right|}{s}} \]

      4. mul0-lft 68.6%

         \[\leadsto \frac{\frac{1}{s}}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right) + \color{blue}{0}} \]

      5. associate-+r+ 68.6%

         \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + \left(4 + 0\right)}} \]

      6. metadata-eval 68.6%

         \[\leadsto \frac{\frac{1}{s}}{\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + \color{blue}{4}} \]

      7. unpow2 68.6%

         \[\leadsto \frac{\frac{1}{s}}{\frac{\color{blue}{\left|x\right| \cdot \left|x\right|}}{{s}^{2}} + 4} \]

      8. sqr-abs 68.6%

         \[\leadsto \frac{\frac{1}{s}}{\frac{\color{blue}{x \cdot x}}{{s}^{2}} + 4} \]

      9. unpow2 68.6%

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

   6. Simplified 68.6%

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

      1. clear-num 68.6%

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

      2. inv-pow 68.6%

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

   8. Applied egg-rr 68.6%

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

      1. unpow-1 68.6%

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

      2. times-frac 70.6%

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

   10. Simplified 70.6%

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

   if 1.28000003e-23 < x

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in s around inf 52.3%

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

      1. associate-+r+ 52.3%

         \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right) + \left(-1 \cdot \frac{\left|x\right|}{s} + \frac{\left|x\right|}{s}\right)}} \]

      2. distribute-lft1-in 52.3%

         \[\leadsto \frac{\frac{1}{s}}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right) + \color{blue}{\left(-1 + 1\right) \cdot \frac{\left|x\right|}{s}}} \]

      3. metadata-eval 52.3%

         \[\leadsto \frac{\frac{1}{s}}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right) + \color{blue}{0} \cdot \frac{\left|x\right|}{s}} \]

      4. mul0-lft 82.4%

         \[\leadsto \frac{\frac{1}{s}}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right) + \color{blue}{0}} \]

      5. associate-+r+ 82.4%

         \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + \left(4 + 0\right)}} \]

      6. metadata-eval 82.4%

         \[\leadsto \frac{\frac{1}{s}}{\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + \color{blue}{4}} \]

      7. unpow2 82.4%

         \[\leadsto \frac{\frac{1}{s}}{\frac{\color{blue}{\left|x\right| \cdot \left|x\right|}}{{s}^{2}} + 4} \]

      8. sqr-abs 82.4%

         \[\leadsto \frac{\frac{1}{s}}{\frac{\color{blue}{x \cdot x}}{{s}^{2}} + 4} \]

      9. unpow2 82.4%

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

   6. Simplified 82.4%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.2800000250130962 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{1}{s}}{\frac{1}{\frac{s}{x} \cdot \frac{s}{x}} + 4}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \frac{x \cdot x}{s \cdot s}}\\ \end{array} \]

Alternative 9: 82.1% accurate, 41.0× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 1.2800000250130962 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \frac{x}{s} \cdot \frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \frac{x \cdot x}{s \cdot s}}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s)
 :precision binary32
 (if (<= x 1.2800000250130962e-23)
   (/ (/ 1.0 s) (+ 4.0 (* (/ x s) (/ x s))))
   (/ (/ 1.0 s) (+ 4.0 (/ (* x x) (* s s))))))

C

x = abs(x);
float code(float x, float s) {
	float tmp;
	if (x <= 1.2800000250130962e-23f) {
		tmp = (1.0f / s) / (4.0f + ((x / s) * (x / s)));
	} else {
		tmp = (1.0f / s) / (4.0f + ((x * x) / (s * s)));
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 1.2800000250130962e-23) then
        tmp = (1.0e0 / s) / (4.0e0 + ((x / s) * (x / s)))
    else
        tmp = (1.0e0 / s) / (4.0e0 + ((x * x) / (s * s)))
    end if
    code = tmp
end function

Julia

x = abs(x)
function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(1.2800000250130962e-23))
		tmp = Float32(Float32(Float32(1.0) / s) / Float32(Float32(4.0) + Float32(Float32(x / s) * Float32(x / s))));
	else
		tmp = Float32(Float32(Float32(1.0) / s) / Float32(Float32(4.0) + Float32(Float32(x * x) / Float32(s * s))));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(1.2800000250130962e-23))
		tmp = (single(1.0) / s) / (single(4.0) + ((x / s) * (x / s)));
	else
		tmp = (single(1.0) / s) / (single(4.0) + ((x * x) / (s * s)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 1.28000003e-23

   1. Initial program 99.8%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in s around inf 51.0%

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

      1. associate-+r+ 51.0%

         \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right) + \left(-1 \cdot \frac{\left|x\right|}{s} + \frac{\left|x\right|}{s}\right)}} \]

      2. distribute-lft1-in 51.0%

         \[\leadsto \frac{\frac{1}{s}}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right) + \color{blue}{\left(-1 + 1\right) \cdot \frac{\left|x\right|}{s}}} \]

      3. metadata-eval 51.0%

         \[\leadsto \frac{\frac{1}{s}}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right) + \color{blue}{0} \cdot \frac{\left|x\right|}{s}} \]

      4. mul0-lft 68.6%

         \[\leadsto \frac{\frac{1}{s}}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right) + \color{blue}{0}} \]

      5. associate-+r+ 68.6%

         \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + \left(4 + 0\right)}} \]

      6. metadata-eval 68.6%

         \[\leadsto \frac{\frac{1}{s}}{\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + \color{blue}{4}} \]

      7. unpow2 68.6%

         \[\leadsto \frac{\frac{1}{s}}{\frac{\color{blue}{\left|x\right| \cdot \left|x\right|}}{{s}^{2}} + 4} \]

      8. sqr-abs 68.6%

         \[\leadsto \frac{\frac{1}{s}}{\frac{\color{blue}{x \cdot x}}{{s}^{2}} + 4} \]

      9. unpow2 68.6%

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

   6. Simplified 68.6%

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

      1. times-frac 70.6%

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

   8. Applied egg-rr 70.6%

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

   if 1.28000003e-23 < x

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in s around inf 52.3%

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

      1. associate-+r+ 52.3%

         \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right) + \left(-1 \cdot \frac{\left|x\right|}{s} + \frac{\left|x\right|}{s}\right)}} \]

      2. distribute-lft1-in 52.3%

         \[\leadsto \frac{\frac{1}{s}}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right) + \color{blue}{\left(-1 + 1\right) \cdot \frac{\left|x\right|}{s}}} \]

      3. metadata-eval 52.3%

         \[\leadsto \frac{\frac{1}{s}}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right) + \color{blue}{0} \cdot \frac{\left|x\right|}{s}} \]

      4. mul0-lft 82.4%

         \[\leadsto \frac{\frac{1}{s}}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right) + \color{blue}{0}} \]

      5. associate-+r+ 82.4%

         \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + \left(4 + 0\right)}} \]

      6. metadata-eval 82.4%

         \[\leadsto \frac{\frac{1}{s}}{\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + \color{blue}{4}} \]

      7. unpow2 82.4%

         \[\leadsto \frac{\frac{1}{s}}{\frac{\color{blue}{\left|x\right| \cdot \left|x\right|}}{{s}^{2}} + 4} \]

      8. sqr-abs 82.4%

         \[\leadsto \frac{\frac{1}{s}}{\frac{\color{blue}{x \cdot x}}{{s}^{2}} + 4} \]

      9. unpow2 82.4%

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

   6. Simplified 82.4%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.2800000250130962 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \frac{x}{s} \cdot \frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \frac{x \cdot x}{s \cdot s}}\\ \end{array} \]

Alternative 10: 76.6% accurate, 47.7× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \frac{\frac{1}{s}}{4 + \frac{x}{s} \cdot \frac{x}{s}} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s) :precision binary32 (/ (/ 1.0 s) (+ 4.0 (* (/ x s) (/ x s)))))

C

x = abs(x);
float code(float x, float s) {
	return (1.0f / s) / (4.0f + ((x / s) * (x / s)));
}

Fortran

NOTE: x should be positive before calling this function
real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = (1.0e0 / s) / (4.0e0 + ((x / s) * (x / s)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.5%

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

4. Taylor expanded in s around inf 51.5%

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

   1. associate-+r+ 51.5%

      \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right) + \left(-1 \cdot \frac{\left|x\right|}{s} + \frac{\left|x\right|}{s}\right)}} \]

   2. distribute-lft1-in 51.5%

      \[\leadsto \frac{\frac{1}{s}}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right) + \color{blue}{\left(-1 + 1\right) \cdot \frac{\left|x\right|}{s}}} \]

   3. metadata-eval 51.5%

      \[\leadsto \frac{\frac{1}{s}}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right) + \color{blue}{0} \cdot \frac{\left|x\right|}{s}} \]

   4. mul0-lft 74.1%

      \[\leadsto \frac{\frac{1}{s}}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right) + \color{blue}{0}} \]

   5. associate-+r+ 74.1%

      \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + \left(4 + 0\right)}} \]

   6. metadata-eval 74.1%

      \[\leadsto \frac{\frac{1}{s}}{\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + \color{blue}{4}} \]

   7. unpow2 74.1%

      \[\leadsto \frac{\frac{1}{s}}{\frac{\color{blue}{\left|x\right| \cdot \left|x\right|}}{{s}^{2}} + 4} \]

   8. sqr-abs 74.1%

      \[\leadsto \frac{\frac{1}{s}}{\frac{\color{blue}{x \cdot x}}{{s}^{2}} + 4} \]

   9. unpow2 74.1%

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

6. Simplified 74.1%

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

   1. times-frac 71.7%

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

8. Applied egg-rr 71.7%

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

9. Final simplification 71.7%

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

Alternative 11: 63.3% accurate, 87.0× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 2.499999993688107 \cdot 10^{-7}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{s}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s)
 :precision binary32
 (if (<= x 2.499999993688107e-7) (/ 0.25 s) (/ s (* x x))))

C

x = abs(x);
float code(float x, float s) {
	float tmp;
	if (x <= 2.499999993688107e-7f) {
		tmp = 0.25f / s;
	} else {
		tmp = s / (x * x);
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 2.499999993688107e-7) then
        tmp = 0.25e0 / s
    else
        tmp = s / (x * x)
    end if
    code = tmp
end function

Julia

x = abs(x)
function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(2.499999993688107e-7))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(s / Float32(x * x));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(2.499999993688107e-7))
		tmp = single(0.25) / s;
	else
		tmp = s / (x * x);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 2.49999999e-7

   1. Initial program 99.7%

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

      1. *-lft-identity 99.7%

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

      2. associate-*r/ 99.7%

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

      3. associate-/l* 99.8%

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

      4. distribute-frac-neg 99.8%

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

      5. exp-neg 99.7%

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

      6. associate-/r/ 99.7%

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

      7. /-rgt-identity 99.7%

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

      8. associate-*l* 99.7%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in s around inf 27.7%

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

   if 2.49999999e-7 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in s around inf 43.1%

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

      1. associate-+r+ 43.1%

         \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right) + \left(-1 \cdot \frac{\left|x\right|}{s} + \frac{\left|x\right|}{s}\right)}} \]

      2. distribute-lft1-in 43.1%

         \[\leadsto \frac{\frac{1}{s}}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right) + \color{blue}{\left(-1 + 1\right) \cdot \frac{\left|x\right|}{s}}} \]

      3. metadata-eval 43.1%

         \[\leadsto \frac{\frac{1}{s}}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right) + \color{blue}{0} \cdot \frac{\left|x\right|}{s}} \]

      4. mul0-lft 88.0%

         \[\leadsto \frac{\frac{1}{s}}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right) + \color{blue}{0}} \]

      5. associate-+r+ 88.0%

         \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + \left(4 + 0\right)}} \]

      6. metadata-eval 88.0%

         \[\leadsto \frac{\frac{1}{s}}{\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + \color{blue}{4}} \]

      7. unpow2 88.0%

         \[\leadsto \frac{\frac{1}{s}}{\frac{\color{blue}{\left|x\right| \cdot \left|x\right|}}{{s}^{2}} + 4} \]

      8. sqr-abs 88.0%

         \[\leadsto \frac{\frac{1}{s}}{\frac{\color{blue}{x \cdot x}}{{s}^{2}} + 4} \]

      9. unpow2 88.0%

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

   6. Simplified 88.0%

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

   7. Taylor expanded in s around 0 61.3%

      \[\leadsto \color{blue}{\frac{s}{{x}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 61.3%

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

   9. Simplified 61.3%

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

4. Final simplification 36.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.499999993688107 \cdot 10^{-7}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{s}{x \cdot x}\\ \end{array} \]

Alternative 12: 7.1% accurate, 206.7× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \frac{-1}{x} \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.5%

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

4. Taylor expanded in s around inf 74.9%

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

   1. mul-1-neg 74.9%

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

   2. unsub-neg 74.9%

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

6. Simplified 74.9%

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

7. Taylor expanded in s around 0 6.9%

   \[\leadsto \color{blue}{\frac{-1}{\left|x\right|}} \]
8. Step-by-step derivation

   1. unpow1 6.9%

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

   2. sqr-pow 3.2%

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

   3. fabs-sqr 3.2%

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

   4. sqr-pow 7.8%

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

   5. unpow1 7.8%

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

9. Simplified 7.8%

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

10. Final simplification 7.8%

    \[\leadsto \frac{-1}{x} \]

Alternative 13: 27.6% accurate, 206.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

   1. *-lft-identity 99.8%

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

   2. associate-*r/ 99.8%

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

   3. associate-/l* 99.8%

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

   4. distribute-frac-neg 99.8%

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

   5. exp-neg 99.8%

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

   6. associate-/r/ 99.8%

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

   7. /-rgt-identity 99.8%

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

   8. associate-*l* 99.8%

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

3. Simplified 99.9%

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

4. Taylor expanded in s around inf 21.4%

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

5. Final simplification 21.4%

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

Reproduce

herbie shell --seed 2023238 
(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))))))