Logistic distribution

Percentage Accurate: 99.6% → 99.5%

Time: 13.4s

Alternatives: 12

Speedup: 1.5×

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: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.5% accurate, 1.5× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s)
 :precision binary32
 (/ (/ 1.0 s) (+ (pow E (/ x s)) (+ (exp (/ (fabs x) (- s))) 2.0))))

C

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

Julia

x = abs(x)
function code(x, s)
	return Float32(Float32(Float32(1.0) / s) / Float32((Float32(exp(1)) ^ Float32(x / s)) + Float32(exp(Float32(abs(x) / Float32(-s))) + Float32(2.0))))
end

MATLAB

x = abs(x)
function tmp = code(x, s)
	tmp = (single(1.0) / s) / ((single(2.71828182845904523536) ^ (x / s)) + (exp((abs(x) / -s)) + single(2.0)));
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 99.8%

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

   1. expm1-log1p-u 99.8%

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

5. Applied egg-rr 99.8%

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

6. Taylor expanded in x around 0 99.8%

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

   1. log1p-def 99.8%

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

   2. *-rgt-identity 99.8%

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

   3. *-rgt-identity 99.8%

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

   4. unpow1 99.8%

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

   5. sqr-pow 56.1%

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

   6. fabs-sqr 56.1%

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

   7. sqr-pow 66.2%

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

   8. unpow1 66.2%

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

8. Simplified 66.2%

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

   1. expm1-log1p-u 66.2%

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

   2. *-un-lft-identity 66.2%

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

   3. exp-prod 66.2%

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

   4. exp-1-e 66.2%

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

10. Applied egg-rr 66.2%

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

11. Final simplification 66.2%

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

Alternative 2: 99.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

x = abs(x)
function tmp = code(x, s)
	tmp = single(1.0) / (s * (single(2.0) + (exp((abs(x) / -s)) + exp((abs(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 99.8%

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

   1. expm1-log1p-u 98.1%

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

   2. expm1-udef 98.1%

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

5. Applied egg-rr 98.1%

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

   1. expm1-def 98.1%

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

   2. expm1-log1p 99.8%

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

   3. associate-/r* 99.8%

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

   4. +-commutative 99.8%

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

   5. associate-+r+ 99.8%

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

   6. +-commutative 99.8%

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

   7. associate-+l+ 99.8%

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

7. Simplified 99.8%

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

8. Final simplification 99.8%

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

Alternative 3: 96.4% accurate, 5.6× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \frac{1}{\frac{e^{\frac{x}{s}} + 3}{\frac{1}{s}}} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s) :precision binary32 (/ 1.0 (/ (+ (exp (/ x s)) 3.0) (/ 1.0 s))))

C

x = abs(x);
float code(float x, float s) {
	return 1.0f / ((expf((x / s)) + 3.0f) / (1.0f / 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 / ((exp((x / s)) + 3.0e0) / (1.0e0 / s))
end function

Julia

x = abs(x)
function code(x, s)
	return Float32(Float32(1.0) / Float32(Float32(exp(Float32(x / s)) + Float32(3.0)) / Float32(Float32(1.0) / s)))
end

MATLAB

x = abs(x)
function tmp = code(x, s)
	tmp = single(1.0) / ((exp((x / s)) + single(3.0)) / (single(1.0) / 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 99.8%

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

4. Taylor expanded in s around inf 96.8%

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

   1. add-cube-cbrt 96.4%

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

   2. pow3 96.3%

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

   3. +-commutative 96.3%

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

   4. add-sqr-sqrt 96.3%

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

   5. add-sqr-sqrt 96.3%

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

   6. add-sqr-sqrt 54.5%

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

   7. fabs-sqr 54.5%

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

   8. add-sqr-sqrt 64.5%

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

6. Applied egg-rr 64.5%

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

   1. rem-cube-cbrt 64.9%

      \[\leadsto \color{blue}{\frac{\frac{1}{s}}{3 + e^{\frac{x}{s}}}} \]

   2. clear-num 64.9%

      \[\leadsto \color{blue}{\frac{1}{\frac{3 + e^{\frac{x}{s}}}{\frac{1}{s}}}} \]

   3. +-commutative 64.9%

      \[\leadsto \frac{1}{\frac{\color{blue}{e^{\frac{x}{s}} + 3}}{\frac{1}{s}}} \]

8. Applied egg-rr 64.9%

   \[\leadsto \color{blue}{\frac{1}{\frac{e^{\frac{x}{s}} + 3}{\frac{1}{s}}}} \]

9. Final simplification 64.9%

   \[\leadsto \frac{1}{\frac{e^{\frac{x}{s}} + 3}{\frac{1}{s}}} \]

Alternative 4: 99.5% accurate, 5.6× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \frac{\frac{1}{s}}{2 + 2 \cdot \cosh \left(\frac{x}{s}\right)} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s) :precision binary32 (/ (/ 1.0 s) (+ 2.0 (* 2.0 (cosh (/ x s))))))

C

x = abs(x);
float code(float x, float s) {
	return (1.0f / s) / (2.0f + (2.0f * coshf((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) / (2.0e0 + (2.0e0 * cosh((x / s))))
end function

Julia

x = abs(x)
function code(x, s)
	return Float32(Float32(Float32(1.0) / s) / Float32(Float32(2.0) + Float32(Float32(2.0) * cosh(Float32(x / s)))))
end

MATLAB

x = abs(x)
function tmp = code(x, s)
	tmp = (single(1.0) / s) / (single(2.0) + (single(2.0) * cosh((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 99.8%

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

   1. expm1-log1p-u 98.1%

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

   2. expm1-udef 98.1%

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

5. Applied egg-rr 98.1%

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

   1. expm1-def 98.1%

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

   2. expm1-log1p 99.8%

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

   3. associate-/r* 99.8%

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

   4. +-commutative 99.8%

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

   5. associate-+r+ 99.8%

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

   6. +-commutative 99.8%

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

   7. associate-+l+ 99.8%

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

7. Simplified 99.8%

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

8. Taylor expanded in s around 0 99.8%

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

   1. associate-/r* 99.8%

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

   2. +-commutative 99.8%

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

   3. associate-+l+ 99.8%

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

   4. *-rgt-identity 99.8%

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

   5. *-rgt-identity 99.8%

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

   6. unpow1 99.8%

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

   7. sqr-pow 56.1%

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

   8. fabs-sqr 56.1%

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

   9. sqr-pow 66.2%

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

   10. unpow1 66.2%

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

   11. mul-1-neg 66.2%

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

   12. distribute-frac-neg 66.2%

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

   13. unpow1 66.2%

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

   14. sqr-pow 56.1%

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

   15. fabs-sqr 56.1%

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

   16. sqr-pow 99.8%

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

   17. unpow1 99.8%

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

10. Simplified 99.8%

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

    1. expm1-log1p-u 98.1%

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

    2. expm1-udef 98.1%

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

    3. distribute-frac-neg 98.1%

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

    4. cosh-undef 98.1%

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

12. Applied egg-rr 98.1%

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

    1. expm1-def 98.1%

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

    2. expm1-log1p 99.8%

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

14. Simplified 99.8%

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

15. Final simplification 99.8%

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

Alternative 5: 96.4% accurate, 5.7× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \frac{1}{s \cdot \left(e^{\frac{x}{s}} + 3\right)} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s) :precision binary32 (/ 1.0 (* s (+ (exp (/ x s)) 3.0))))

C

x = abs(x);
float code(float x, float s) {
	return 1.0f / (s * (expf((x / s)) + 3.0f));
}

Fortran

NOTE: x should be positive before calling this function
real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 1.0e0 / (s * (exp((x / s)) + 3.0e0))
end function

Julia

x = abs(x)
function code(x, s)
	return Float32(Float32(1.0) / Float32(s * Float32(exp(Float32(x / s)) + Float32(3.0))))
end

MATLAB

x = abs(x)
function tmp = code(x, s)
	tmp = single(1.0) / (s * (exp((x / s)) + single(3.0)));
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 99.8%

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

4. Taylor expanded in s around inf 96.8%

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

   1. expm1-log1p-u 95.7%

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

   2. expm1-udef 95.6%

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

   3. +-commutative 95.6%

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

   4. add-sqr-sqrt 95.6%

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

   5. add-sqr-sqrt 95.6%

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

   6. add-sqr-sqrt 54.1%

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

   7. fabs-sqr 54.1%

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

   8. add-sqr-sqrt 63.8%

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

6. Applied egg-rr 63.8%

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

   1. expm1-def 63.8%

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

   2. expm1-log1p 64.9%

      \[\leadsto \color{blue}{\frac{\frac{1}{s}}{3 + e^{\frac{x}{s}}}} \]

   3. associate-/r* 64.9%

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

   4. +-commutative 64.9%

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

8. Simplified 64.9%

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

9. Final simplification 64.9%

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

Alternative 6: 96.4% accurate, 5.7× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \frac{\frac{1}{s}}{e^{\frac{x}{s}} + 3} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s) :precision binary32 (/ (/ 1.0 s) (+ (exp (/ x s)) 3.0)))

C

x = abs(x);
float code(float x, float s) {
	return (1.0f / s) / (expf((x / s)) + 3.0f);
}

Fortran

NOTE: x should be positive before calling this function
real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = (1.0e0 / s) / (exp((x / s)) + 3.0e0)
end function

Julia

x = abs(x)
function code(x, s)
	return Float32(Float32(Float32(1.0) / s) / Float32(exp(Float32(x / s)) + Float32(3.0)))
end

MATLAB

x = abs(x)
function tmp = code(x, s)
	tmp = (single(1.0) / s) / (exp((x / s)) + single(3.0));
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 99.8%

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

4. Taylor expanded in s around inf 96.8%

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

   1. expm1-log1p-u 95.7%

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

   2. expm1-udef 95.6%

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

   3. +-commutative 95.6%

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

   4. add-sqr-sqrt 95.6%

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

   5. add-sqr-sqrt 95.6%

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

   6. add-sqr-sqrt 54.1%

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

   7. fabs-sqr 54.1%

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

   8. add-sqr-sqrt 63.8%

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

6. Applied egg-rr 63.8%

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

   1. expm1-def 63.8%

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

   2. expm1-log1p 64.9%

      \[\leadsto \color{blue}{\frac{\frac{1}{s}}{3 + e^{\frac{x}{s}}}} \]

   3. +-commutative 64.9%

      \[\leadsto \frac{\frac{1}{s}}{\color{blue}{e^{\frac{x}{s}} + 3}} \]

8. Simplified 64.9%

   \[\leadsto \color{blue}{\frac{\frac{1}{s}}{e^{\frac{x}{s}} + 3}} \]

9. Final simplification 64.9%

   \[\leadsto \frac{\frac{1}{s}}{e^{\frac{x}{s}} + 3} \]

Alternative 7: 82.2% accurate, 41.0× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 9.999999998199587 \cdot 10^{-24}:\\ \;\;\;\;\frac{\frac{1}{s}}{\frac{x}{s} \cdot \frac{x}{s} + 4}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \frac{x \cdot x}{s \cdot s}}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s)
 :precision binary32
 (if (<= x 9.999999998199587e-24)
   (/ (/ 1.0 s) (+ (* (/ x s) (/ x s)) 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 <= 9.999999998199587e-24f) {
		tmp = (1.0f / s) / (((x / s) * (x / s)) + 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 <= 9.999999998199587e-24) then
        tmp = (1.0e0 / s) / (((x / s) * (x / s)) + 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(9.999999998199587e-24))
		tmp = Float32(Float32(Float32(1.0) / s) / Float32(Float32(Float32(x / s) * Float32(x / s)) + 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(9.999999998199587e-24))
		tmp = (single(1.0) / s) / (((x / s) * (x / s)) + 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 < 1e-23

   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 99.7%

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

      1. expm1-log1p-u 97.1%

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

      2. expm1-udef 97.1%

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

   5. Applied egg-rr 97.1%

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

      1. expm1-def 97.1%

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

      2. expm1-log1p 99.7%

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

      3. associate-/r* 99.7%

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

      4. +-commutative 99.7%

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

      5. associate-+r+ 99.7%

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

      6. +-commutative 99.7%

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

      7. associate-+l+ 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in s around 0 99.7%

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

      1. associate-/r* 99.7%

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

      2. +-commutative 99.7%

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

      3. associate-+l+ 99.7%

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

      4. *-rgt-identity 99.7%

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

      5. *-rgt-identity 99.7%

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

      6. unpow1 99.7%

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

      7. sqr-pow 18.7%

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

      8. fabs-sqr 18.7%

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

      9. sqr-pow 37.4%

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

      10. unpow1 37.4%

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

      11. mul-1-neg 37.4%

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

      12. distribute-frac-neg 37.4%

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

      13. unpow1 37.4%

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

      14. sqr-pow 18.7%

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

      15. fabs-sqr 18.7%

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

      16. sqr-pow 99.7%

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

      17. unpow1 99.7%

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

   10. Simplified 99.7%

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

   11. Taylor expanded in x around 0 73.7%

       \[\leadsto \frac{\frac{1}{s}}{\color{blue}{4 + \frac{{x}^{2}}{{s}^{2}}}} \]
   12. Step-by-step derivation

       1. +-commutative 73.7%

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

       2. unpow2 73.7%

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

       3. unpow2 73.7%

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

       4. times-frac 79.7%

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

   13. Simplified 79.7%

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

   if 1e-23 < x

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

      1. expm1-log1p-u 99.4%

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

      2. expm1-udef 99.3%

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

   5. Applied egg-rr 99.3%

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

      1. expm1-def 99.4%

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

      2. expm1-log1p 99.9%

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

      3. associate-/r* 99.9%

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

      4. +-commutative 99.9%

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

      5. associate-+r+ 99.9%

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

      6. +-commutative 99.9%

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

      7. associate-+l+ 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in s around 0 99.9%

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

      1. associate-/r* 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-+l+ 99.9%

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

      4. *-rgt-identity 99.9%

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

      5. *-rgt-identity 99.9%

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

      6. unpow1 99.9%

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

      7. sqr-pow 99.9%

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

      8. fabs-sqr 99.9%

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

      9. sqr-pow 99.9%

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

      10. unpow1 99.9%

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

      11. mul-1-neg 99.9%

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

      12. distribute-frac-neg 99.9%

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

      13. unpow1 99.9%

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

      14. sqr-pow 99.9%

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

      15. fabs-sqr 99.9%

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

      16. sqr-pow 99.9%

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

      17. unpow1 99.9%

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

   10. Simplified 99.9%

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

   11. Taylor expanded in x around 0 84.2%

       \[\leadsto \frac{\frac{1}{s}}{\color{blue}{4 + \frac{{x}^{2}}{{s}^{2}}}} \]
   12. Step-by-step derivation

       1. +-commutative 84.2%

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

       2. unpow2 84.2%

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

       3. unpow2 84.2%

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

   13. Simplified 84.2%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9.999999998199587 \cdot 10^{-24}:\\ \;\;\;\;\frac{\frac{1}{s}}{\frac{x}{s} \cdot \frac{x}{s} + 4}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \frac{x \cdot x}{s \cdot s}}\\ \end{array} \]

Alternative 8: 76.4% accurate, 47.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

x = abs(x)
function tmp = code(x, s)
	tmp = (single(1.0) / s) / (((x / s) * (x / s)) + single(4.0));
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 99.8%

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

   1. expm1-log1p-u 98.1%

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

   2. expm1-udef 98.1%

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

5. Applied egg-rr 98.1%

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

   1. expm1-def 98.1%

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

   2. expm1-log1p 99.8%

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

   3. associate-/r* 99.8%

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

   4. +-commutative 99.8%

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

   5. associate-+r+ 99.8%

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

   6. +-commutative 99.8%

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

   7. associate-+l+ 99.8%

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

7. Simplified 99.8%

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

8. Taylor expanded in s around 0 99.8%

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

   1. associate-/r* 99.8%

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

   2. +-commutative 99.8%

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

   3. associate-+l+ 99.8%

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

   4. *-rgt-identity 99.8%

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

   5. *-rgt-identity 99.8%

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

   6. unpow1 99.8%

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

   7. sqr-pow 56.1%

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

   8. fabs-sqr 56.1%

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

   9. sqr-pow 66.2%

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

   10. unpow1 66.2%

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

   11. mul-1-neg 66.2%

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

   12. distribute-frac-neg 66.2%

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

   13. unpow1 66.2%

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

   14. sqr-pow 56.1%

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

   15. fabs-sqr 56.1%

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

   16. sqr-pow 99.8%

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

   17. unpow1 99.8%

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

10. Simplified 99.8%

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

11. Taylor expanded in x around 0 78.5%

    \[\leadsto \frac{\frac{1}{s}}{\color{blue}{4 + \frac{{x}^{2}}{{s}^{2}}}} \]
12. Step-by-step derivation

    1. +-commutative 78.5%

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

    2. unpow2 78.5%

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

    3. unpow2 78.5%

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

    4. times-frac 80.0%

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

13. Simplified 80.0%

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

14. Final simplification 80.0%

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

Alternative 9: 65.8% accurate, 56.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

x = abs(x)
function tmp = code(x, s)
	tmp = single(1.0) / ((s * single(4.0)) + (x * (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 99.8%

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

   1. expm1-log1p-u 98.1%

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

   2. expm1-udef 98.1%

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

5. Applied egg-rr 98.1%

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

   1. expm1-def 98.1%

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

   2. expm1-log1p 99.8%

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

   3. associate-/r* 99.8%

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

   4. +-commutative 99.8%

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

   5. associate-+r+ 99.8%

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

   6. +-commutative 99.8%

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

   7. associate-+l+ 99.8%

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

7. Simplified 99.8%

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

8. Taylor expanded in s around inf 66.4%

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

   1. associate-+r+ 66.4%

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

   2. distribute-lft1-in 66.4%

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

   3. metadata-eval 66.4%

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

   4. mul0-lft 66.4%

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

   5. +-lft-identity 66.4%

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

   6. *-commutative 66.4%

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

   7. fma-def 66.4%

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

   8. unpow2 66.4%

      \[\leadsto \frac{1}{\mathsf{fma}\left(s, 4, \frac{\color{blue}{\left|x\right| \cdot \left|x\right|}}{s}\right)} \]

   9. sqr-abs 66.4%

      \[\leadsto \frac{1}{\mathsf{fma}\left(s, 4, \frac{\color{blue}{x \cdot x}}{s}\right)} \]

10. Simplified 66.4%

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

    1. fma-udef 66.4%

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

    2. associate-/l* 66.8%

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

    3. div-inv 66.8%

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

    4. clear-num 66.8%

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

12. Applied egg-rr 66.8%

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

13. Final simplification 66.8%

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

Alternative 10: 64.3% accurate, 67.9× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 2.9999999242136255 \cdot 10^{-5}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{\frac{s}{x}}}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s)
 :precision binary32
 (if (<= x 2.9999999242136255e-5) (/ 0.25 s) (/ 1.0 (/ x (/ s x)))))

C

x = abs(x);
float code(float x, float s) {
	float tmp;
	if (x <= 2.9999999242136255e-5f) {
		tmp = 0.25f / s;
	} else {
		tmp = 1.0f / (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 <= 2.9999999242136255e-5) then
        tmp = 0.25e0 / s
    else
        tmp = 1.0e0 / (x / (s / x))
    end if
    code = tmp
end function

Julia

x = abs(x)
function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(2.9999999242136255e-5))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(1.0) / Float32(x / Float32(s / x)));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(2.9999999242136255e-5))
		tmp = single(0.25) / s;
	else
		tmp = single(1.0) / (x / (s / x));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 2.99999992e-5

   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 99.7%

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

   4. Taylor expanded in s around inf 36.5%

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

   if 2.99999992e-5 < 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. Step-by-step derivation

      1. expm1-log1p-u 100.0%

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

      2. expm1-udef 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. expm1-def 100.0%

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

      2. expm1-log1p 100.0%

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

      3. associate-/r* 100.0%

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

      4. +-commutative 100.0%

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

      5. associate-+r+ 100.0%

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

      6. +-commutative 100.0%

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

      7. associate-+l+ 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in s around inf 68.5%

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

      1. associate-+r+ 68.5%

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

      2. distribute-lft1-in 68.5%

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

      3. metadata-eval 68.5%

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

      4. mul0-lft 68.5%

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

      5. +-lft-identity 68.5%

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

      6. *-commutative 68.5%

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

      7. fma-def 68.5%

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

      8. unpow2 68.5%

         \[\leadsto \frac{1}{\mathsf{fma}\left(s, 4, \frac{\color{blue}{\left|x\right| \cdot \left|x\right|}}{s}\right)} \]

      9. sqr-abs 68.5%

         \[\leadsto \frac{1}{\mathsf{fma}\left(s, 4, \frac{\color{blue}{x \cdot x}}{s}\right)} \]

   10. Simplified 68.5%

       \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(s, 4, \frac{x \cdot x}{s}\right)}} \]

   11. Taylor expanded in s around 0 67.0%

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

       1. unpow2 67.0%

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

   13. Simplified 67.0%

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

       1. clear-num 68.5%

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

       2. inv-pow 68.5%

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

   15. Applied egg-rr 68.5%

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

       1. unpow-1 68.5%

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

       2. associate-/l* 68.5%

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

   17. Simplified 68.5%

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

4. Final simplification 47.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.9999999242136255 \cdot 10^{-5}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{\frac{s}{x}}}\\ \end{array} \]

Alternative 11: 63.0% accurate, 87.0× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 2.9999999242136255 \cdot 10^{-5}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{s}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s)
 :precision binary32
 (if (<= x 2.9999999242136255e-5) (/ 0.25 s) (/ s (* x x))))

C

x = abs(x);
float code(float x, float s) {
	float tmp;
	if (x <= 2.9999999242136255e-5f) {
		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 <= 2.9999999242136255e-5) 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(2.9999999242136255e-5))
		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(2.9999999242136255e-5))
		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 < 2.99999992e-5

   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 99.7%

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

   4. Taylor expanded in s around inf 36.5%

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

   if 2.99999992e-5 < 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. Step-by-step derivation

      1. expm1-log1p-u 100.0%

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

      2. expm1-udef 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. expm1-def 100.0%

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

      2. expm1-log1p 100.0%

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

      3. associate-/r* 100.0%

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

      4. +-commutative 100.0%

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

      5. associate-+r+ 100.0%

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

      6. +-commutative 100.0%

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

      7. associate-+l+ 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in s around inf 68.5%

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

      1. associate-+r+ 68.5%

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

      2. distribute-lft1-in 68.5%

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

      3. metadata-eval 68.5%

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

      4. mul0-lft 68.5%

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

      5. +-lft-identity 68.5%

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

      6. *-commutative 68.5%

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

      7. fma-def 68.5%

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

      8. unpow2 68.5%

         \[\leadsto \frac{1}{\mathsf{fma}\left(s, 4, \frac{\color{blue}{\left|x\right| \cdot \left|x\right|}}{s}\right)} \]

      9. sqr-abs 68.5%

         \[\leadsto \frac{1}{\mathsf{fma}\left(s, 4, \frac{\color{blue}{x \cdot x}}{s}\right)} \]

   10. Simplified 68.5%

       \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(s, 4, \frac{x \cdot x}{s}\right)}} \]

   11. Taylor expanded in s around 0 67.0%

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

       1. unpow2 67.0%

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

   13. Simplified 67.0%

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

4. Final simplification 47.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.9999999242136255 \cdot 10^{-5}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{s}{x \cdot x}\\ \end{array} \]

Alternative 12: 27.2% 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%

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

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 25.3%

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

5. Final simplification 25.3%

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

Reproduce

herbie shell --seed 2023229 
(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))))))