Logistic distribution

Percentage Accurate: 99.5% → 99.2%

Time: 15.6s

Alternatives: 13

Speedup: 2.0×

Specification

\[0 \leq s \land s \leq 1.0651631\]

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \mathbf{if}\;\left|x\right| \leq 9.999999747378752 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{\frac{s}{e^{\frac{x}{s} - 2 \cdot \mathsf{log1p}\left(t_0\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{t_0 + 3}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ x s))))
   (if (<= (fabs x) 9.999999747378752e-5)
     (/ 1.0 (/ s (exp (- (/ x s) (* 2.0 (log1p t_0))))))
     (/ (/ 1.0 s) (+ t_0 3.0)))))

C

x = abs(x);
float code(float x, float s) {
	float t_0 = expf((x / s));
	float tmp;
	if (fabsf(x) <= 9.999999747378752e-5f) {
		tmp = 1.0f / (s / expf(((x / s) - (2.0f * log1pf(t_0)))));
	} else {
		tmp = (1.0f / s) / (t_0 + 3.0f);
	}
	return tmp;
}

Julia

x = abs(x)
function code(x, s)
	t_0 = exp(Float32(x / s))
	tmp = Float32(0.0)
	if (abs(x) <= Float32(9.999999747378752e-5))
		tmp = Float32(Float32(1.0) / Float32(s / exp(Float32(Float32(x / s) - Float32(Float32(2.0) * log1p(t_0))))));
	else
		tmp = Float32(Float32(Float32(1.0) / s) / Float32(t_0 + Float32(3.0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (fabs.f32 x) < 9.99999975e-5

   1. Initial program 99.2%

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

      1. *-un-lft-identity 99.2%

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

      2. associate-*l* 99.2%

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

      3. times-frac 99.2%

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

   3. Applied egg-rr 84.8%

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

      1. associate-*l/ 84.8%

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

      2. *-un-lft-identity 84.8%

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

      3. clear-num 84.8%

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

      4. add-exp-log 84.9%

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

      5. log-div 84.9%

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

      6. add-log-exp 99.2%

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

      7. log-pow 99.2%

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

      8. log1p-udef 99.4%

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

   5. Applied egg-rr 99.4%

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

   if 9.99999975e-5 < (fabs.f32 x)

   1. Initial program 100.0%

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

   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 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. expm1-udef 100.0%

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

      3. associate-/l/ 100.0%

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

      4. +-commutative 100.0%

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

      5. metadata-eval 100.0%

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

      6. add-sqr-sqrt 49.0%

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

      7. fabs-sqr 49.0%

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

      8. add-sqr-sqrt 51.3%

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

   6. Applied egg-rr 51.3%

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

      1. expm1-def 51.3%

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

      2. expm1-log1p 51.3%

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

      3. associate-/r* 51.3%

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

      4. *-lft-identity 51.3%

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

      5. associate-*l/ 51.3%

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

      6. associate-*r/ 51.3%

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

      7. *-rgt-identity 51.3%

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

   8. Simplified 51.3%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 9.999999747378752 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{\frac{s}{e^{\frac{x}{s} - 2 \cdot \mathsf{log1p}\left(e^{\frac{x}{s}}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{e^{\frac{x}{s}} + 3}\\ \end{array} \]

Alternative 2: 99.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \mathbf{if}\;\left|x\right| \leq 9.999999747378752 \cdot 10^{-5}:\\ \;\;\;\;\frac{e^{\frac{x}{s} - 2 \cdot \mathsf{log1p}\left(t_0\right)}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{t_0 + 3}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ x s))))
   (if (<= (fabs x) 9.999999747378752e-5)
     (/ (exp (- (/ x s) (* 2.0 (log1p t_0)))) s)
     (/ (/ 1.0 s) (+ t_0 3.0)))))

C

x = abs(x);
float code(float x, float s) {
	float t_0 = expf((x / s));
	float tmp;
	if (fabsf(x) <= 9.999999747378752e-5f) {
		tmp = expf(((x / s) - (2.0f * log1pf(t_0)))) / s;
	} else {
		tmp = (1.0f / s) / (t_0 + 3.0f);
	}
	return tmp;
}

Julia

x = abs(x)
function code(x, s)
	t_0 = exp(Float32(x / s))
	tmp = Float32(0.0)
	if (abs(x) <= Float32(9.999999747378752e-5))
		tmp = Float32(exp(Float32(Float32(x / s) - Float32(Float32(2.0) * log1p(t_0)))) / s);
	else
		tmp = Float32(Float32(Float32(1.0) / s) / Float32(t_0 + Float32(3.0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (fabs.f32 x) < 9.99999975e-5

   1. Initial program 99.2%

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

      1. *-un-lft-identity 99.2%

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

      2. associate-*l* 99.2%

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

      3. times-frac 99.2%

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

   3. Applied egg-rr 84.8%

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

      1. associate-*l/ 84.8%

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

      2. *-un-lft-identity 84.8%

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

      3. add-exp-log 84.8%

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

      4. log-div 84.9%

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

      5. add-log-exp 99.2%

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

      6. log-pow 99.2%

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

      7. log1p-udef 99.4%

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

   5. Applied egg-rr 99.4%

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

   if 9.99999975e-5 < (fabs.f32 x)

   1. Initial program 100.0%

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

   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 100.0%

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

      1. expm1-log1p-u 100.0%

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

      2. expm1-udef 100.0%

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

      3. associate-/l/ 100.0%

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

      4. +-commutative 100.0%

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

      5. metadata-eval 100.0%

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

      6. add-sqr-sqrt 49.0%

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

      7. fabs-sqr 49.0%

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

      8. add-sqr-sqrt 51.3%

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

   6. Applied egg-rr 51.3%

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

      1. expm1-def 51.3%

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

      2. expm1-log1p 51.3%

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

      3. associate-/r* 51.3%

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

      4. *-lft-identity 51.3%

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

      5. associate-*l/ 51.3%

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

      6. associate-*r/ 51.3%

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

      7. *-rgt-identity 51.3%

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

   8. Simplified 51.3%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 9.999999747378752 \cdot 10^{-5}:\\ \;\;\;\;\frac{e^{\frac{x}{s} - 2 \cdot \mathsf{log1p}\left(e^{\frac{x}{s}}\right)}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{e^{\frac{x}{s}} + 3}\\ \end{array} \]

Alternative 3: 99.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

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((abs(x) / s)) + (exp((-x / s)) + 2.0e0))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

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. remove-double-neg 99.8%

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

   2. distribute-frac-neg 99.8%

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

   3. frac-2neg 99.8%

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

   4. exp-neg 99.8%

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

   5. div-inv 99.8%

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

   6. exp-prod 96.4%

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

   7. add-sqr-sqrt 96.4%

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

   8. sqrt-unprod 96.4%

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

   9. sqr-neg 96.4%

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

   10. sqrt-unprod 35.7%

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

   11. add-sqr-sqrt 95.8%

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

   12. exp-prod 94.7%

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

   13. div-inv 94.7%

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

   14. add-sqr-sqrt 44.0%

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

   15. fabs-sqr 44.0%

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

   16. add-sqr-sqrt 97.0%

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

   17. add-sqr-sqrt -0.0%

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

   18. sqrt-unprod 95.0%

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

   19. sqr-neg 95.0%

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

   20. sqrt-unprod 97.4%

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

   21. add-sqr-sqrt 97.4%

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

5. Applied egg-rr 97.4%

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

   1. rec-exp 97.4%

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

   2. distribute-neg-frac 97.4%

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

7. Simplified 97.4%

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

8. Final simplification 97.4%

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

Alternative 4: 99.5% accurate, 2.0× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ (- x) s)))) (/ t_0 (* s (pow (+ 1.0 t_0) 2.0)))))

C

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

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

2. Taylor expanded in s around 0 99.6%

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

   1. associate-*r/ 99.6%

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

   2. mul-1-neg 99.6%

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

4. Simplified 99.6%

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

   1. distribute-frac-neg 99.6%

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

   2. exp-neg 99.7%

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

   3. add-sqr-sqrt 46.7%

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

   4. fabs-sqr 46.7%

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

   5. add-sqr-sqrt 96.9%

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

6. Applied egg-rr 96.9%

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

   1. rec-exp 96.9%

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

   2. distribute-neg-frac 96.9%

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

8. Simplified 96.9%

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

   1. distribute-frac-neg 99.6%

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

   2. exp-neg 99.7%

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

   3. add-sqr-sqrt 46.7%

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

   4. fabs-sqr 46.7%

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

   5. add-sqr-sqrt 96.9%

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

10. Applied egg-rr 60.6%

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

    1. rec-exp 96.9%

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

    2. distribute-neg-frac 96.9%

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

12. Simplified 60.6%

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

13. Final simplification 60.6%

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

Alternative 5: 96.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \frac{\frac{1}{s}}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{x}{s}} + 3\right)\right)} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s)
 :precision binary32
 (/ (/ 1.0 s) (expm1 (log1p (+ (exp (/ x s)) 3.0)))))

C

x = abs(x);
float code(float x, float s) {
	return (1.0f / s) / expm1f(log1pf((expf((x / s)) + 3.0f)));
}

Julia

x = abs(x)
function code(x, s)
	return Float32(Float32(Float32(1.0) / s) / expm1(log1p(Float32(exp(Float32(x / s)) + Float32(3.0)))))
end

Derivation

1. Initial program 99.7%

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

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 95.8%

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

   1. expm1-log1p-u 95.9%

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

   2. +-commutative 95.9%

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

   3. metadata-eval 95.9%

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

   4. add-sqr-sqrt 44.5%

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

   5. fabs-sqr 44.5%

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

   6. add-sqr-sqrt 57.8%

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

6. Applied egg-rr 57.8%

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

7. Final simplification 57.8%

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

Alternative 6: 96.1% accurate, 5.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

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 95.8%

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

   1. div-inv 95.8%

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

   2. +-commutative 95.8%

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

   3. metadata-eval 95.8%

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

   4. add-sqr-sqrt 44.5%

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

   5. fabs-sqr 44.5%

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

   6. add-sqr-sqrt 57.8%

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

6. Applied egg-rr 57.8%

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

7. Final simplification 57.8%

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

Alternative 7: 96.1% 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 99.7%

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

3. Simplified 99.8%

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

4. Taylor expanded in s around inf 95.8%

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

   1. expm1-log1p-u 94.6%

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

   2. expm1-udef 94.6%

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

   3. associate-/l/ 94.6%

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

   4. +-commutative 94.6%

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

   5. metadata-eval 94.6%

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

   6. add-sqr-sqrt 44.0%

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

   7. fabs-sqr 44.0%

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

   8. add-sqr-sqrt 56.6%

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

6. Applied egg-rr 56.6%

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

   1. expm1-def 56.6%

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

   2. expm1-log1p 57.8%

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

   3. associate-/r* 57.8%

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

   4. *-lft-identity 57.8%

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

   5. associate-*l/ 57.8%

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

   6. associate-*r/ 57.8%

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

   7. *-rgt-identity 57.8%

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

8. Simplified 57.8%

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

9. Final simplification 57.8%

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

Alternative 8: 96.2% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

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 95.8%

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

   1. div-inv 95.8%

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

   2. +-commutative 95.8%

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

   3. metadata-eval 95.8%

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

   4. add-sqr-sqrt 44.5%

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

   5. fabs-sqr 44.5%

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

   6. add-sqr-sqrt 57.8%

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

6. Applied egg-rr 57.8%

   \[\leadsto \color{blue}{\frac{1}{s} \cdot \frac{1}{3 + e^{\frac{x}{s}}}} \]
7. Step-by-step derivation

   1. associate-*l/ 57.8%

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

   2. *-un-lft-identity 57.8%

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

8. Applied egg-rr 57.8%

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

9. Final simplification 57.8%

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

Alternative 9: 82.4% accurate, 41.0× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 4.999999999099794 \cdot 10^{-24}:\\ \;\;\;\;\frac{1}{s \cdot \left(\frac{x}{s \cdot \frac{s}{x}} + 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \frac{x \cdot x}{s \cdot s}}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s)
 :precision binary32
 (if (<= x 4.999999999099794e-24)
   (/ 1.0 (* s (+ (/ x (* s (/ 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 <= 4.999999999099794e-24f) {
		tmp = 1.0f / (s * ((x / (s * (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 <= 4.999999999099794e-24) then
        tmp = 1.0e0 / (s * ((x / (s * (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(4.999999999099794e-24))
		tmp = Float32(Float32(1.0) / Float32(s * Float32(Float32(x / Float32(s * 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(4.999999999099794e-24))
		tmp = single(1.0) / (s * ((x / (s * (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 < 5e-24

   1. Initial program 99.5%

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

   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 55.2%

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

      1. unpow2 55.2%

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

      2. sqr-abs 55.2%

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

      3. unpow2 55.2%

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

      4. distribute-lft1-in 55.2%

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

      5. metadata-eval 55.2%

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

      6. mul0-lft 77.0%

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

      7. metadata-eval 77.0%

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

   6. Simplified 77.0%

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

      1. clear-num 77.0%

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

      2. inv-pow 77.0%

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

   8. Applied egg-rr 77.0%

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

      1. unpow-1 77.0%

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

      2. unpow2 77.0%

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

      3. associate-/l* 75.3%

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

      4. unpow2 75.3%

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

   10. Simplified 75.3%

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

       1. expm1-log1p-u 73.1%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1}{s}}{\frac{1}{\frac{s}{\frac{x \cdot x}{s}}} + 4}\right)\right)} \]

       2. expm1-udef 78.5%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{1}{s}}{\frac{1}{\frac{s}{\frac{x \cdot x}{s}}} + 4}\right)} - 1} \]

       3. associate-/r/ 78.5%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{1}{s}}{\color{blue}{\frac{1}{s} \cdot \frac{x \cdot x}{s}} + 4}\right)} - 1 \]

       4. fma-def 78.5%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{1}{s}}{\color{blue}{\mathsf{fma}\left(\frac{1}{s}, \frac{x \cdot x}{s}, 4\right)}}\right)} - 1 \]

       5. associate-/l* 79.1%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{1}{s}}{\mathsf{fma}\left(\frac{1}{s}, \color{blue}{\frac{x}{\frac{s}{x}}}, 4\right)}\right)} - 1 \]

   12. Applied egg-rr 79.1%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{1}{s}}{\mathsf{fma}\left(\frac{1}{s}, \frac{x}{\frac{s}{x}}, 4\right)}\right)} - 1} \]
   13. Step-by-step derivation

       1. expm1-def 73.7%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1}{s}}{\mathsf{fma}\left(\frac{1}{s}, \frac{x}{\frac{s}{x}}, 4\right)}\right)\right)} \]

       2. expm1-log1p 76.1%

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

       3. associate-/l/ 76.1%

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

       4. *-commutative 76.1%

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

       5. fma-udef 76.1%

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

       6. associate-*l/ 76.1%

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

       7. *-lft-identity 76.1%

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

       8. associate-/l/ 79.8%

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

   14. Simplified 79.8%

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

   if 5e-24 < 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 99.7%

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

   4. Taylor expanded in s around inf 59.3%

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

      1. unpow2 59.3%

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

      2. sqr-abs 59.3%

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

      3. unpow2 59.3%

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

      4. distribute-lft1-in 59.3%

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

      5. metadata-eval 59.3%

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

      6. mul0-lft 85.0%

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

      7. metadata-eval 85.0%

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

   6. Simplified 85.0%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.999999999099794 \cdot 10^{-24}:\\ \;\;\;\;\frac{1}{s \cdot \left(\frac{x}{s \cdot \frac{s}{x}} + 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \frac{x \cdot x}{s \cdot s}}\\ \end{array} \]

Alternative 10: 78.5% accurate, 47.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

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 56.9%

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

   1. unpow2 56.9%

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

   2. sqr-abs 56.9%

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

   3. unpow2 56.9%

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

   4. distribute-lft1-in 56.9%

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

   5. metadata-eval 56.9%

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

   6. mul0-lft 80.3%

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

   7. metadata-eval 80.3%

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

6. Simplified 80.3%

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

   1. clear-num 80.3%

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

   2. inv-pow 80.3%

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

8. Applied egg-rr 80.3%

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

   1. unpow-1 80.3%

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

   2. unpow2 80.3%

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

   3. associate-/l* 76.7%

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

   4. unpow2 76.7%

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

10. Simplified 76.7%

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

    1. expm1-log1p-u 75.2%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1}{s}}{\frac{1}{\frac{s}{\frac{x \cdot x}{s}}} + 4}\right)\right)} \]

    2. expm1-udef 81.8%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{1}{s}}{\frac{1}{\frac{s}{\frac{x \cdot x}{s}}} + 4}\right)} - 1} \]

    3. associate-/r/ 81.8%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{1}{s}}{\color{blue}{\frac{1}{s} \cdot \frac{x \cdot x}{s}} + 4}\right)} - 1 \]

    4. fma-def 81.8%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{1}{s}}{\color{blue}{\mathsf{fma}\left(\frac{1}{s}, \frac{x \cdot x}{s}, 4\right)}}\right)} - 1 \]

    5. associate-/l* 82.2%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{1}{s}}{\mathsf{fma}\left(\frac{1}{s}, \color{blue}{\frac{x}{\frac{s}{x}}}, 4\right)}\right)} - 1 \]

12. Applied egg-rr 82.2%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{1}{s}}{\mathsf{fma}\left(\frac{1}{s}, \frac{x}{\frac{s}{x}}, 4\right)}\right)} - 1} \]
13. Step-by-step derivation

    1. expm1-def 75.5%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1}{s}}{\mathsf{fma}\left(\frac{1}{s}, \frac{x}{\frac{s}{x}}, 4\right)}\right)\right)} \]

    2. expm1-log1p 77.2%

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

    3. associate-/l/ 77.2%

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

    4. *-commutative 77.2%

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

    5. fma-udef 77.2%

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

    6. associate-*l/ 77.2%

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

    7. *-lft-identity 77.2%

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

    8. associate-/l/ 79.4%

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

14. Simplified 79.4%

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

15. Final simplification 79.4%

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

Alternative 11: 63.4% accurate, 67.9× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 9.999999747378752 \cdot 10^{-5}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s)
 :precision binary32
 (if (<= x 9.999999747378752e-5) (/ 0.25 s) (/ 1.0 (* x (/ x s)))))

C

x = abs(x);
float code(float x, float s) {
	float tmp;
	if (x <= 9.999999747378752e-5f) {
		tmp = 0.25f / s;
	} else {
		tmp = 1.0f / (x * (x / 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 <= 9.999999747378752e-5) then
        tmp = 0.25e0 / s
    else
        tmp = 1.0e0 / (x * (x / s))
    end if
    code = tmp
end function

Julia

x = abs(x)
function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(9.999999747378752e-5))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(1.0) / Float32(x * Float32(x / s)));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(9.999999747378752e-5))
		tmp = single(0.25) / s;
	else
		tmp = single(1.0) / (x * (x / s));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 9.99999975e-5

   1. Initial program 99.5%

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

   2. Taylor expanded in s around inf 35.4%

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

   if 9.99999975e-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. Taylor expanded in s around inf 46.8%

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

      1. unpow2 46.8%

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

      2. sqr-abs 46.8%

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

      3. unpow2 46.8%

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

      4. distribute-lft1-in 46.8%

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

      5. metadata-eval 46.8%

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

      6. mul0-lft 84.8%

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

      7. metadata-eval 84.8%

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

   6. Simplified 84.8%

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

   7. Taylor expanded in s around 0 69.6%

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

      1. unpow2 69.6%

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

   9. Simplified 69.6%

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

       1. clear-num 74.3%

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

       2. inv-pow 74.3%

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

       3. associate-/l* 74.3%

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

   11. Applied egg-rr 74.3%

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

       1. unpow-1 74.3%

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

       2. associate-/r/ 74.3%

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

   13. Simplified 74.3%

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

4. Final simplification 46.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9.999999747378752 \cdot 10^{-5}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \frac{x}{s}}\\ \end{array} \]

Alternative 12: 62.2% accurate, 87.0× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 9.999999747378752 \cdot 10^{-5}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{s}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s)
 :precision binary32
 (if (<= x 9.999999747378752e-5) (/ 0.25 s) (/ s (* x x))))

C

x = abs(x);
float code(float x, float s) {
	float tmp;
	if (x <= 9.999999747378752e-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 <= 9.999999747378752e-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(9.999999747378752e-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(9.999999747378752e-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 < 9.99999975e-5

   1. Initial program 99.5%

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

   2. Taylor expanded in s around inf 35.4%

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

   if 9.99999975e-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. Taylor expanded in s around inf 46.8%

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

      1. unpow2 46.8%

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

      2. sqr-abs 46.8%

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

      3. unpow2 46.8%

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

      4. distribute-lft1-in 46.8%

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

      5. metadata-eval 46.8%

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

      6. mul0-lft 84.8%

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

      7. metadata-eval 84.8%

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

   6. Simplified 84.8%

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

   7. Taylor expanded in s around 0 69.6%

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

      1. unpow2 69.6%

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

   9. Simplified 69.6%

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

4. Final simplification 44.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9.999999747378752 \cdot 10^{-5}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{s}{x \cdot x}\\ \end{array} \]

Alternative 13: 27.4% 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 99.7%

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

2. Taylor expanded in s around inf 26.8%

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

3. Final simplification 26.8%

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

Reproduce

herbie shell --seed 2023252 
(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))))))