Logistic distribution

Percentage Accurate: 99.6% → 99.6%

Time: 14.9s

Alternatives: 10

Speedup: 1.2×

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: 99.6% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

   1. fabs-neg 99.6%

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

   2. distribute-frac-neg 99.6%

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

   3. distribute-frac-neg2 99.6%

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

   4. fabs-neg 99.6%

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

   5. *-commutative 99.6%

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

   6. fabs-neg 99.6%

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

   7. +-commutative 99.6%

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

   8. fabs-neg 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in x around 0 99.6%

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

   1. associate-/r* 99.6%

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

   2. mul-1-neg 99.6%

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

   3. distribute-neg-frac2 99.6%

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

   4. +-commutative 99.6%

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

   5. mul-1-neg 99.6%

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

   6. distribute-neg-frac2 99.6%

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

7. Simplified 99.6%

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

8. Final simplification 99.6%

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

Alternative 2: 72.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \mathbf{if}\;\left|x\right| \leq 9.999999747378752 \cdot 10^{-6}:\\ \;\;\;\;\frac{e^{\frac{x}{s} - 2 \cdot \mathsf{log1p}\left(t\_0\right)}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{s \cdot \left(1 + t\_0\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ x s))))
   (if (<= (fabs x) 9.999999747378752e-6)
     (/ (exp (- (/ x s) (* 2.0 (log1p t_0)))) s)
     (/ 0.5 (* s (+ 1.0 t_0))))))

C

float code(float x, float s) {
	float t_0 = expf((x / s));
	float tmp;
	if (fabsf(x) <= 9.999999747378752e-6f) {
		tmp = expf(((x / s) - (2.0f * log1pf(t_0)))) / s;
	} else {
		tmp = 0.5f / (s * (1.0f + t_0));
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = exp(Float32(x / s))
	tmp = Float32(0.0)
	if (abs(x) <= Float32(9.999999747378752e-6))
		tmp = Float32(exp(Float32(Float32(x / s) - Float32(Float32(2.0) * log1p(t_0)))) / s);
	else
		tmp = Float32(Float32(0.5) / Float32(s * Float32(Float32(1.0) + t_0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (fabs.f32 x) < 9.99999975e-6

   1. Initial program 99.0%

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

      1. fabs-neg 99.0%

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

      2. distribute-frac-neg 99.0%

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

      3. distribute-frac-neg2 99.0%

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

      4. fabs-neg 99.0%

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

      5. *-commutative 99.0%

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

      6. fabs-neg 99.0%

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

      7. +-commutative 99.0%

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

      8. fabs-neg 99.0%

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

   3. Simplified 99.0%

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

      1. distribute-frac-neg2 99.0%

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

      2. distribute-frac-neg 99.0%

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

      3. *-un-lft-identity 99.0%

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

      4. times-frac 98.2%

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

   6. Applied egg-rr 80.4%

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

      1. associate-*l/ 80.4%

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

      2. add-exp-log 80.2%

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

      3. *-un-lft-identity 80.2%

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

      4. log-div 80.2%

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

      5. add-log-exp 97.3%

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

      6. log-pow 98.8%

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

      7. log1p-define 98.9%

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

   8. Applied egg-rr 98.9%

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

   if 9.99999975e-6 < (fabs.f32 x)

   1. Initial program 100.0%

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

      1. fabs-neg 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. distribute-frac-neg2 100.0%

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

      4. fabs-neg 100.0%

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

      5. *-commutative 100.0%

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

      6. fabs-neg 100.0%

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

      7. +-commutative 100.0%

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

      8. fabs-neg 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 51.4%

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

      1. associate-*l/ 51.4%

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

      2. *-lft-identity 51.4%

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

   8. Simplified 51.4%

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

   9. Taylor expanded in x around 0 50.9%

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

   10. Taylor expanded in s around 0 50.9%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 9.999999747378752 \cdot 10^{-6}:\\ \;\;\;\;\frac{e^{\frac{x}{s} - 2 \cdot \mathsf{log1p}\left(e^{\frac{x}{s}}\right)}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{s \cdot \left(1 + e^{\frac{x}{s}}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 60.0% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{s \cdot \left(1 + {e}^{\left(\frac{x}{s}\right)}\right)} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (/ 0.5 (* s (+ 1.0 (pow E (/ x s))))))

C

float code(float x, float s) {
	return 0.5f / (s * (1.0f + powf(((float) M_E), (x / s))));
}

Julia

function code(x, s)
	return Float32(Float32(0.5) / Float32(s * Float32(Float32(1.0) + (Float32(exp(1)) ^ Float32(x / s)))))
end

MATLAB

function tmp = code(x, s)
	tmp = single(0.5) / (s * (single(1.0) + (single(2.71828182845904523536) ^ (x / s))));
end

Derivation

1. Initial program 99.6%

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

   1. fabs-neg 99.6%

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

   2. distribute-frac-neg 99.6%

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

   3. distribute-frac-neg2 99.6%

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

   4. fabs-neg 99.6%

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

   5. *-commutative 99.6%

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

   6. fabs-neg 99.6%

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

   7. +-commutative 99.6%

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

   8. fabs-neg 99.6%

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

3. Simplified 99.6%

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

6. Applied egg-rr 64.7%

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

   1. associate-*l/ 64.8%

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

   2. *-lft-identity 64.8%

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

8. Simplified 64.8%

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

9. Taylor expanded in x around 0 59.5%

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

10. Taylor expanded in s around 0 59.5%

    \[\leadsto \color{blue}{\frac{0.5}{s \cdot \left(1 + e^{\frac{x}{s}}\right)}} \]
11. Step-by-step derivation

    1. *-un-lft-identity 59.5%

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

    2. exp-prod 59.5%

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

12. Applied egg-rr 59.5%

    \[\leadsto \frac{0.5}{s \cdot \left(1 + \color{blue}{{\left(e^{1}\right)}^{\left(\frac{x}{s}\right)}}\right)} \]
13. Step-by-step derivation

    1. exp-1-e 59.5%

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

14. Simplified 59.5%

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

15. Final simplification 59.5%

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

Alternative 4: 60.0% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{s \cdot \left(1 + e^{\frac{x}{s}}\right)} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (/ 0.5 (* s (+ 1.0 (exp (/ x s))))))

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

   1. fabs-neg 99.6%

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

   2. distribute-frac-neg 99.6%

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

   3. distribute-frac-neg2 99.6%

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

   4. fabs-neg 99.6%

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

   5. *-commutative 99.6%

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

   6. fabs-neg 99.6%

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

   7. +-commutative 99.6%

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

   8. fabs-neg 99.6%

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

3. Simplified 99.6%

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

6. Applied egg-rr 64.7%

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

   1. associate-*l/ 64.8%

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

   2. *-lft-identity 64.8%

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

8. Simplified 64.8%

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

9. Taylor expanded in x around 0 59.5%

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

10. Taylor expanded in s around 0 59.5%

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

11. Final simplification 59.5%

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

Alternative 5: 63.3% accurate, 41.3× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{x \cdot \left(1 + \frac{x}{s} \cdot 0.5\right) + s \cdot 2} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (/ 0.5 (+ (* x (+ 1.0 (* (/ x s) 0.5))) (* s 2.0))))

C

float code(float x, float s) {
	return 0.5f / ((x * (1.0f + ((x / s) * 0.5f))) + (s * 2.0f));
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 0.5e0 / ((x * (1.0e0 + ((x / s) * 0.5e0))) + (s * 2.0e0))
end function

Julia

function code(x, s)
	return Float32(Float32(0.5) / Float32(Float32(x * Float32(Float32(1.0) + Float32(Float32(x / s) * Float32(0.5)))) + Float32(s * Float32(2.0))))
end

MATLAB

function tmp = code(x, s)
	tmp = single(0.5) / ((x * (single(1.0) + ((x / s) * single(0.5)))) + (s * single(2.0)));
end

Derivation

1. Initial program 99.6%

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

   1. fabs-neg 99.6%

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

   2. distribute-frac-neg 99.6%

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

   3. distribute-frac-neg2 99.6%

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

   4. fabs-neg 99.6%

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

   5. *-commutative 99.6%

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

   6. fabs-neg 99.6%

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

   7. +-commutative 99.6%

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

   8. fabs-neg 99.6%

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

3. Simplified 99.6%

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

6. Applied egg-rr 64.7%

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

   1. associate-*l/ 64.8%

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

   2. *-lft-identity 64.8%

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

8. Simplified 64.8%

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

9. Taylor expanded in x around 0 59.5%

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

10. Taylor expanded in s around 0 59.5%

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

11. Taylor expanded in x around 0 64.6%

    \[\leadsto \frac{0.5}{\color{blue}{2 \cdot s + x \cdot \left(1 + 0.5 \cdot \frac{x}{s}\right)}} \]

12. Final simplification 64.6%

    \[\leadsto \frac{0.5}{x \cdot \left(1 + \frac{x}{s} \cdot 0.5\right) + s \cdot 2} \]
13. Add Preprocessing

Alternative 6: 39.4% accurate, 51.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 9.999999747378752 \cdot 10^{-6}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{s}}{\frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 9.999999747378752e-6) (/ 0.25 s) (/ (/ 0.5 s) (/ x s))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 9.999999747378752e-6f) {
		tmp = 0.25f / s;
	} else {
		tmp = (0.5f / s) / (x / s);
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 9.999999747378752e-6) then
        tmp = 0.25e0 / s
    else
        tmp = (0.5e0 / s) / (x / s)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(9.999999747378752e-6))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(Float32(0.5) / s) / Float32(x / s));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(9.999999747378752e-6))
		tmp = single(0.25) / s;
	else
		tmp = (single(0.5) / s) / (x / s);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 9.99999975e-6

   1. Initial program 99.4%

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

      1. fabs-neg 99.4%

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

      2. distribute-frac-neg 99.4%

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

      3. distribute-frac-neg2 99.4%

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

      4. fabs-neg 99.4%

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

      5. *-commutative 99.4%

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

      6. fabs-neg 99.4%

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

      7. +-commutative 99.4%

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

      8. fabs-neg 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in s around inf 36.1%

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

   if 9.99999975e-6 < x

   1. Initial program 100.0%

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

      1. fabs-neg 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. distribute-frac-neg2 100.0%

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

      4. fabs-neg 100.0%

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

      5. *-commutative 100.0%

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

      6. fabs-neg 100.0%

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

      7. +-commutative 100.0%

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

      8. fabs-neg 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr -0.0%

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

      1. associate-*l/ -0.0%

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

      2. *-lft-identity -0.0%

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

   8. Simplified -0.0%

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

   9. Taylor expanded in x around 0 100.0%

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

   10. Taylor expanded in x around 0 42.2%

       \[\leadsto \frac{\frac{0.5}{s}}{\color{blue}{2 + \frac{x}{s}}} \]
   11. Step-by-step derivation

       1. +-commutative 42.2%

          \[\leadsto \frac{\frac{0.5}{s}}{\color{blue}{\frac{x}{s} + 2}} \]

   12. Simplified 42.2%

       \[\leadsto \frac{\frac{0.5}{s}}{\color{blue}{\frac{x}{s} + 2}} \]

   13. Taylor expanded in x around inf 42.2%

       \[\leadsto \frac{\frac{0.5}{s}}{\color{blue}{\frac{x}{s}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9.999999747378752 \cdot 10^{-6}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{s}}{\frac{x}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 50.3% accurate, 68.9× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{s \cdot \left(2 + \frac{x}{s}\right)} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (/ 0.5 (* s (+ 2.0 (/ x s)))))

C

float code(float x, float s) {
	return 0.5f / (s * (2.0f + (x / s)));
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 0.5e0 / (s * (2.0e0 + (x / s)))
end function

Julia

function code(x, s)
	return Float32(Float32(0.5) / Float32(s * Float32(Float32(2.0) + Float32(x / s))))
end

MATLAB

function tmp = code(x, s)
	tmp = single(0.5) / (s * (single(2.0) + (x / s)));
end

Derivation

1. Initial program 99.6%

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

   1. fabs-neg 99.6%

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

   2. distribute-frac-neg 99.6%

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

   3. distribute-frac-neg2 99.6%

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

   4. fabs-neg 99.6%

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

   5. *-commutative 99.6%

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

   6. fabs-neg 99.6%

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

   7. +-commutative 99.6%

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

   8. fabs-neg 99.6%

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

3. Simplified 99.6%

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

6. Applied egg-rr 64.7%

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

   1. associate-*l/ 64.8%

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

   2. *-lft-identity 64.8%

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

8. Simplified 64.8%

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

9. Taylor expanded in x around 0 59.5%

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

10. Taylor expanded in s around 0 59.5%

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

11. Taylor expanded in s around inf 50.6%

    \[\leadsto \frac{0.5}{\color{blue}{s \cdot \left(2 + \frac{x}{s}\right)}} \]
12. Step-by-step derivation

    1. +-commutative 50.6%

       \[\leadsto \frac{0.5}{s \cdot \color{blue}{\left(\frac{x}{s} + 2\right)}} \]

13. Simplified 50.6%

    \[\leadsto \frac{0.5}{\color{blue}{s \cdot \left(\frac{x}{s} + 2\right)}} \]

14. Final simplification 50.6%

    \[\leadsto \frac{0.5}{s \cdot \left(2 + \frac{x}{s}\right)} \]
15. Add Preprocessing

Alternative 8: 28.7% accurate, 77.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 9.999999747378752 \cdot 10^{-6}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 9.999999747378752e-6) (/ 0.25 s) (/ 0.5 x)))

C

float code(float x, float s) {
	float tmp;
	if (x <= 9.999999747378752e-6f) {
		tmp = 0.25f / s;
	} else {
		tmp = 0.5f / x;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 9.999999747378752e-6) then
        tmp = 0.25e0 / s
    else
        tmp = 0.5e0 / x
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(9.999999747378752e-6))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(0.5) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(9.999999747378752e-6))
		tmp = single(0.25) / s;
	else
		tmp = single(0.5) / x;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 9.99999975e-6

   1. Initial program 99.4%

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

      1. fabs-neg 99.4%

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

      2. distribute-frac-neg 99.4%

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

      3. distribute-frac-neg2 99.4%

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

      4. fabs-neg 99.4%

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

      5. *-commutative 99.4%

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

      6. fabs-neg 99.4%

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

      7. +-commutative 99.4%

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

      8. fabs-neg 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in s around inf 36.1%

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

   if 9.99999975e-6 < x

   1. Initial program 100.0%

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

      1. fabs-neg 100.0%

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

      2. distribute-frac-neg 100.0%

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

      3. distribute-frac-neg2 100.0%

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

      4. fabs-neg 100.0%

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

      5. *-commutative 100.0%

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

      6. fabs-neg 100.0%

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

      7. +-commutative 100.0%

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

      8. fabs-neg 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr -0.0%

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

      1. associate-*l/ -0.0%

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

      2. *-lft-identity -0.0%

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

   8. Simplified -0.0%

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

   9. Taylor expanded in x around 0 100.0%

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

   10. Taylor expanded in x around 0 42.2%

       \[\leadsto \frac{\frac{0.5}{s}}{\color{blue}{2 + \frac{x}{s}}} \]
   11. Step-by-step derivation

       1. +-commutative 42.2%

          \[\leadsto \frac{\frac{0.5}{s}}{\color{blue}{\frac{x}{s} + 2}} \]

   12. Simplified 42.2%

       \[\leadsto \frac{\frac{0.5}{s}}{\color{blue}{\frac{x}{s} + 2}} \]

   13. Taylor expanded in s around 0 10.2%

       \[\leadsto \color{blue}{\frac{0.5}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 29.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9.999999747378752 \cdot 10^{-6}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 28.9% accurate, 88.6× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{x + s \cdot 2} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (/ 0.5 (+ x (* s 2.0))))

C

float code(float x, float s) {
	return 0.5f / (x + (s * 2.0f));
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 0.5e0 / (x + (s * 2.0e0))
end function

Julia

function code(x, s)
	return Float32(Float32(0.5) / Float32(x + Float32(s * Float32(2.0))))
end

MATLAB

function tmp = code(x, s)
	tmp = single(0.5) / (x + (s * single(2.0)));
end

Derivation

1. Initial program 99.6%

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

   1. fabs-neg 99.6%

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

   2. distribute-frac-neg 99.6%

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

   3. distribute-frac-neg2 99.6%

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

   4. fabs-neg 99.6%

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

   5. *-commutative 99.6%

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

   6. fabs-neg 99.6%

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

   7. +-commutative 99.6%

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

   8. fabs-neg 99.6%

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

3. Simplified 99.6%

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

6. Applied egg-rr 64.7%

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

   1. associate-*l/ 64.8%

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

   2. *-lft-identity 64.8%

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

8. Simplified 64.8%

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

9. Taylor expanded in x around 0 59.5%

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

10. Taylor expanded in s around 0 59.5%

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

11. Taylor expanded in x around 0 29.4%

    \[\leadsto \frac{0.5}{\color{blue}{x + 2 \cdot s}} \]
12. Step-by-step derivation

    1. *-commutative 29.4%

       \[\leadsto \frac{0.5}{x + \color{blue}{s \cdot 2}} \]

13. Simplified 29.4%

    \[\leadsto \frac{0.5}{\color{blue}{x + s \cdot 2}} \]

14. Final simplification 29.4%

    \[\leadsto \frac{0.5}{x + s \cdot 2} \]
15. Add Preprocessing

Alternative 10: 27.1% accurate, 206.7× speedup

Math

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

FPCore

(FPCore (x s) :precision binary32 (/ 0.25 s))

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

   1. fabs-neg 99.6%

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

   2. distribute-frac-neg 99.6%

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

   3. distribute-frac-neg2 99.6%

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

   4. fabs-neg 99.6%

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

   5. *-commutative 99.6%

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

   6. fabs-neg 99.6%

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

   7. +-commutative 99.6%

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

   8. fabs-neg 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in s around inf 27.5%

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

6. Final simplification 27.5%

   \[\leadsto \frac{0.25}{s} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024080 
(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))))))