Logistic distribution

Percentage Accurate: 99.5% → 99.4%

Time: 16.0s

Alternatives: 14

Speedup: 1.0×

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: 99.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\left|x\_m\right| \leq 0.05000000074505806:\\ \;\;\;\;\frac{e^{\frac{x\_m}{s} + -2 \cdot \mathsf{log1p}\left(e^{\frac{x\_m}{s}}\right)}}{s}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{-x\_m}{s} - \log \left(s \cdot 4\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (if (<= (fabs x_m) 0.05000000074505806)
   (/ (exp (+ (/ x_m s) (* -2.0 (log1p (exp (/ x_m s)))))) s)
   (exp (- (/ (- x_m) s) (log (* s 4.0))))))

C

x_m = fabs(x);
float code(float x_m, float s) {
	float tmp;
	if (fabsf(x_m) <= 0.05000000074505806f) {
		tmp = expf(((x_m / s) + (-2.0f * log1pf(expf((x_m / s)))))) / s;
	} else {
		tmp = expf(((-x_m / s) - logf((s * 4.0f))));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m, s)
	tmp = Float32(0.0)
	if (abs(x_m) <= Float32(0.05000000074505806))
		tmp = Float32(exp(Float32(Float32(x_m / s) + Float32(Float32(-2.0) * log1p(exp(Float32(x_m / s)))))) / s);
	else
		tmp = exp(Float32(Float32(Float32(-x_m) / s) - log(Float32(s * Float32(4.0)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (fabs.f32 x) < 0.0500000007

   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. associate-/r* 99.0%

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

      2. div-inv 99.0%

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

   6. Applied egg-rr 80.9%

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

      1. associate-*l/ 80.9%

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

      2. add-exp-log 80.9%

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

      3. prod-exp 99.0%

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

      4. pow-flip 99.0%

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

      5. log-pow 99.0%

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

      6. metadata-eval 99.0%

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

      7. log1p-define 99.1%

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

   8. Applied egg-rr 99.1%

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

   if 0.0500000007 < (fabs.f32 x)

   1. Initial program 99.9%

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

      1. fabs-neg 99.9%

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

      2. distribute-frac-neg 99.9%

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

      3. distribute-frac-neg2 99.9%

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

      4. fabs-neg 99.9%

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

      5. *-commutative 99.9%

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

      6. fabs-neg 99.9%

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

      7. +-commutative 99.9%

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

      8. fabs-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in s around inf 99.9%

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

      1. clear-num 99.9%

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

      2. inv-pow 99.9%

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

   7. Applied egg-rr 54.2%

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

      1. unpow-1 54.2%

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

      2. *-commutative 54.2%

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

   9. Simplified 54.2%

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

       1. inv-pow 54.2%

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

       2. pow-to-exp 54.2%

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

       3. log-prod 54.2%

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

       4. add-log-exp 54.2%

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

   11. Applied egg-rr 54.2%

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

   12. Taylor expanded in x around inf 54.2%

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

       1. +-commutative 54.2%

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

       2. *-commutative 54.2%

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

       3. neg-mul-1 54.2%

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

       4. neg-sub0 54.2%

          \[\leadsto e^{\color{blue}{0 - \left(\frac{x}{s} + \log \left(s \cdot 4\right)\right)}} \]

       5. associate--r+ 54.2%

          \[\leadsto e^{\color{blue}{\left(0 - \frac{x}{s}\right) - \log \left(s \cdot 4\right)}} \]

       6. neg-sub0 54.2%

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

       7. distribute-neg-frac2 54.2%

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

   14. Simplified 54.2%

       \[\leadsto \color{blue}{e^{\frac{x}{-s} - \log \left(s \cdot 4\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.05000000074505806:\\ \;\;\;\;\frac{e^{\frac{x}{s} + -2 \cdot \mathsf{log1p}\left(e^{\frac{x}{s}}\right)}}{s}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{-x}{s} - \log \left(s \cdot 4\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.5% accurate, 1.0× speedup

Math

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

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (let* ((t_0 (exp (/ (fabs x_m) (- s)))))
   (/ t_0 (* (+ t_0 1.0) (+ s (/ s (exp (/ (fabs x_m) s))))))))

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

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. *-commutative 99.4%

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

   2. fabs-neg 99.4%

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

   3. +-commutative 99.4%

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

   4. fabs-neg 99.4%

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

   5. distribute-lft-in 99.5%

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

   6. *-rgt-identity 99.5%

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

   7. +-commutative 99.5%

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

3. Simplified 99.5%

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

5. Final simplification 99.5%

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

Alternative 3: 89.3% accurate, 5.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 2.7999999628422514 \cdot 10^{-18}:\\ \;\;\;\;\frac{\left(0.25 + \frac{x\_m}{s} \cdot -0.25\right) + -0.25 \cdot \left(x\_m \cdot \frac{-1}{s}\right)}{s}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{-x\_m}{s}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (if (<= x_m 2.7999999628422514e-18)
   (/ (+ (+ 0.25 (* (/ x_m s) -0.25)) (* -0.25 (* x_m (/ -1.0 s)))) s)
   (exp (/ (- x_m) s))))

C

x_m = fabs(x);
float code(float x_m, float s) {
	float tmp;
	if (x_m <= 2.7999999628422514e-18f) {
		tmp = ((0.25f + ((x_m / s) * -0.25f)) + (-0.25f * (x_m * (-1.0f / s)))) / s;
	} else {
		tmp = expf((-x_m / s));
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(4) function code(x_m, s)
    real(4), intent (in) :: x_m
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x_m <= 2.7999999628422514e-18) then
        tmp = ((0.25e0 + ((x_m / s) * (-0.25e0))) + ((-0.25e0) * (x_m * ((-1.0e0) / s)))) / s
    else
        tmp = exp((-x_m / s))
    end if
    code = tmp
end function

Julia

x_m = abs(x)
function code(x_m, s)
	tmp = Float32(0.0)
	if (x_m <= Float32(2.7999999628422514e-18))
		tmp = Float32(Float32(Float32(Float32(0.25) + Float32(Float32(x_m / s) * Float32(-0.25))) + Float32(Float32(-0.25) * Float32(x_m * Float32(Float32(-1.0) / s)))) / s);
	else
		tmp = exp(Float32(Float32(-x_m) / s));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, s)
	tmp = single(0.0);
	if (x_m <= single(2.7999999628422514e-18))
		tmp = ((single(0.25) + ((x_m / s) * single(-0.25))) + (single(-0.25) * (x_m * (single(-1.0) / s)))) / s;
	else
		tmp = exp((-x_m / s));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 2.79999996e-18

   1. Initial program 99.3%

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

      1. fabs-neg 99.3%

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

      2. distribute-frac-neg 99.3%

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

      3. distribute-frac-neg2 99.3%

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

      4. fabs-neg 99.3%

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

      5. *-commutative 99.3%

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

      6. fabs-neg 99.3%

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

      7. +-commutative 99.3%

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

      8. fabs-neg 99.3%

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

   3. Simplified 99.3%

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

   6. Applied egg-rr 95.9%

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

      1. *-lft-identity 95.9%

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

   8. Simplified 95.9%

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

      1. *-un-lft-identity 39.6%

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

      2. exp-prod 39.6%

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

   10. Applied egg-rr 95.8%

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

       1. exp-1-e 39.6%

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

   12. Simplified 95.8%

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

   13. Taylor expanded in s around -inf 42.0%

       \[\leadsto \color{blue}{-1 \cdot \frac{-0.25 \cdot \frac{x \cdot \log e}{s} - \left(0.25 + -0.25 \cdot \frac{x}{s}\right)}{s}} \]
   14. Step-by-step derivation

       1. mul-1-neg 42.0%

          \[\leadsto \color{blue}{-\frac{-0.25 \cdot \frac{x \cdot \log e}{s} - \left(0.25 + -0.25 \cdot \frac{x}{s}\right)}{s}} \]

       2. log-E 71.1%

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

       3. associate-/l* 61.2%

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

   15. Simplified 61.2%

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

   if 2.79999996e-18 < x

   1. Initial program 99.7%

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

      1. fabs-neg 99.7%

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

      2. distribute-frac-neg 99.7%

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

      3. distribute-frac-neg2 99.7%

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

      4. fabs-neg 99.7%

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

      5. *-commutative 99.7%

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

      6. fabs-neg 99.7%

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

      7. +-commutative 99.7%

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

      8. fabs-neg 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in s around inf 95.5%

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

      1. clear-num 95.5%

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

      2. inv-pow 95.5%

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

   7. Applied egg-rr 95.5%

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

      1. unpow-1 95.5%

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

      2. *-commutative 95.5%

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

   9. Simplified 95.5%

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

       1. inv-pow 95.5%

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

       2. pow-to-exp 95.5%

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

       3. log-prod 95.5%

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

       4. add-log-exp 95.5%

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

   11. Applied egg-rr 95.5%

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

   12. Taylor expanded in x around inf 93.5%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.7999999628422514 \cdot 10^{-18}:\\ \;\;\;\;\frac{\left(0.25 + \frac{x}{s} \cdot -0.25\right) + -0.25 \cdot \left(x \cdot \frac{-1}{s}\right)}{s}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{-x}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.5% accurate, 5.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

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 92.9%

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

   1. clear-num 92.9%

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

   2. inv-pow 92.9%

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

7. Applied egg-rr 59.5%

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

   1. unpow-1 59.5%

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

   2. *-commutative 59.5%

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

9. Simplified 59.5%

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

    1. *-un-lft-identity 59.5%

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

    2. exp-prod 59.5%

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

11. Applied egg-rr 59.5%

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

    1. exp-1-e 59.5%

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

13. Simplified 59.5%

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

14. Taylor expanded in x around inf 59.5%

    \[\leadsto \color{blue}{\frac{0.25}{s \cdot e^{\frac{x \cdot \log e}{s}}}} \]
15. Step-by-step derivation

    1. associate-/r* 59.5%

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

    2. log-E 59.5%

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

    3. metadata-eval 59.5%

       \[\leadsto \frac{\frac{0.25}{s}}{e^{\frac{x \cdot \color{blue}{{1}^{2}}}{s}}} \]

    4. log-E 59.5%

       \[\leadsto \frac{\frac{0.25}{s}}{e^{\frac{x \cdot {\color{blue}{\log e}}^{2}}{s}}} \]

    5. log-E 59.5%

       \[\leadsto \frac{\frac{0.25}{s}}{e^{\frac{x \cdot {\color{blue}{1}}^{2}}{s}}} \]

    6. metadata-eval 59.5%

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

    7. *-rgt-identity 59.5%

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

16. Simplified 59.5%

    \[\leadsto \color{blue}{\frac{\frac{0.25}{s}}{e^{\frac{x}{s}}}} \]
17. Add Preprocessing

Alternative 5: 94.5% accurate, 5.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

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 92.9%

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

   1. clear-num 92.9%

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

   2. inv-pow 92.9%

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

7. Applied egg-rr 59.5%

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

   1. unpow-1 59.5%

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

   2. *-commutative 59.5%

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

9. Simplified 59.5%

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

10. Taylor expanded in x around inf 59.5%

    \[\leadsto \color{blue}{\frac{0.25}{s \cdot e^{\frac{x}{s}}}} \]
11. Add Preprocessing

Alternative 6: 79.6% accurate, 28.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 5000000000:\\ \;\;\;\;\frac{\left(0.25 + \frac{x\_m}{s} \cdot -0.25\right) + -0.25 \cdot \left(x\_m \cdot \frac{-1}{s}\right)}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot 4 + x\_m \cdot \left(4 + \frac{x\_m}{s} \cdot 2\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (if (<= x_m 5000000000.0)
   (/ (+ (+ 0.25 (* (/ x_m s) -0.25)) (* -0.25 (* x_m (/ -1.0 s)))) s)
   (/ 1.0 (+ (* s 4.0) (* x_m (+ 4.0 (* (/ x_m s) 2.0)))))))

C

x_m = fabs(x);
float code(float x_m, float s) {
	float tmp;
	if (x_m <= 5000000000.0f) {
		tmp = ((0.25f + ((x_m / s) * -0.25f)) + (-0.25f * (x_m * (-1.0f / s)))) / s;
	} else {
		tmp = 1.0f / ((s * 4.0f) + (x_m * (4.0f + ((x_m / s) * 2.0f))));
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(4) function code(x_m, s)
    real(4), intent (in) :: x_m
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x_m <= 5000000000.0e0) then
        tmp = ((0.25e0 + ((x_m / s) * (-0.25e0))) + ((-0.25e0) * (x_m * ((-1.0e0) / s)))) / s
    else
        tmp = 1.0e0 / ((s * 4.0e0) + (x_m * (4.0e0 + ((x_m / s) * 2.0e0))))
    end if
    code = tmp
end function

Julia

x_m = abs(x)
function code(x_m, s)
	tmp = Float32(0.0)
	if (x_m <= Float32(5000000000.0))
		tmp = Float32(Float32(Float32(Float32(0.25) + Float32(Float32(x_m / s) * Float32(-0.25))) + Float32(Float32(-0.25) * Float32(x_m * Float32(Float32(-1.0) / s)))) / s);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(s * Float32(4.0)) + Float32(x_m * Float32(Float32(4.0) + Float32(Float32(x_m / s) * Float32(2.0))))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, s)
	tmp = single(0.0);
	if (x_m <= single(5000000000.0))
		tmp = ((single(0.25) + ((x_m / s) * single(-0.25))) + (single(-0.25) * (x_m * (single(-1.0) / s)))) / s;
	else
		tmp = single(1.0) / ((s * single(4.0)) + (x_m * (single(4.0) + ((x_m / s) * single(2.0)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 5e9

   1. Initial program 99.3%

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

      1. fabs-neg 99.3%

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

      2. distribute-frac-neg 99.3%

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

      3. distribute-frac-neg2 99.3%

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

      4. fabs-neg 99.3%

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

      5. *-commutative 99.3%

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

      6. fabs-neg 99.3%

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

      7. +-commutative 99.3%

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

      8. fabs-neg 99.3%

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

   3. Simplified 99.3%

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

   6. Applied egg-rr 81.7%

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

      1. *-lft-identity 81.7%

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

   8. Simplified 81.7%

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

      1. *-un-lft-identity 48.6%

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

      2. exp-prod 48.6%

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

   10. Applied egg-rr 81.6%

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

       1. exp-1-e 48.6%

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

   12. Simplified 81.6%

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

   13. Taylor expanded in s around -inf 39.1%

       \[\leadsto \color{blue}{-1 \cdot \frac{-0.25 \cdot \frac{x \cdot \log e}{s} - \left(0.25 + -0.25 \cdot \frac{x}{s}\right)}{s}} \]
   14. Step-by-step derivation

       1. mul-1-neg 39.1%

          \[\leadsto \color{blue}{-\frac{-0.25 \cdot \frac{x \cdot \log e}{s} - \left(0.25 + -0.25 \cdot \frac{x}{s}\right)}{s}} \]

       2. log-E 71.5%

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

       3. associate-/l* 60.5%

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

   15. Simplified 60.5%

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

   if 5e9 < 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

   5. Taylor expanded in s around inf 100.0%

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

      1. clear-num 100.0%

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

      2. inv-pow 100.0%

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

   7. Applied egg-rr 100.0%

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

      1. unpow-1 100.0%

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

      2. *-commutative 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in x around 0 95.6%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5000000000:\\ \;\;\;\;\frac{\left(0.25 + \frac{x}{s} \cdot -0.25\right) + -0.25 \cdot \left(x \cdot \frac{-1}{s}\right)}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot 4 + x \cdot \left(4 + \frac{x}{s} \cdot 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 63.9% accurate, 41.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{1}{s \cdot 4 + x\_m \cdot \left(4 + \frac{x\_m}{s} \cdot 2\right)} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (/ 1.0 (+ (* s 4.0) (* x_m (+ 4.0 (* (/ x_m s) 2.0))))))

C

x_m = fabs(x);
float code(float x_m, float s) {
	return 1.0f / ((s * 4.0f) + (x_m * (4.0f + ((x_m / s) * 2.0f))));
}

Fortran

x_m = abs(x)
real(4) function code(x_m, s)
    real(4), intent (in) :: x_m
    real(4), intent (in) :: s
    code = 1.0e0 / ((s * 4.0e0) + (x_m * (4.0e0 + ((x_m / s) * 2.0e0))))
end function

Julia

x_m = abs(x)
function code(x_m, s)
	return Float32(Float32(1.0) / Float32(Float32(s * Float32(4.0)) + Float32(x_m * Float32(Float32(4.0) + Float32(Float32(x_m / s) * Float32(2.0))))))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m, s)
	tmp = single(1.0) / ((s * single(4.0)) + (x_m * (single(4.0) + ((x_m / s) * single(2.0)))));
end

Derivation

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 92.9%

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

   1. clear-num 92.9%

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

   2. inv-pow 92.9%

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

7. Applied egg-rr 59.5%

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

   1. unpow-1 59.5%

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

   2. *-commutative 59.5%

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

9. Simplified 59.5%

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

10. Taylor expanded in x around 0 64.1%

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

11. Final simplification 64.1%

    \[\leadsto \frac{1}{s \cdot 4 + x \cdot \left(4 + \frac{x}{s} \cdot 2\right)} \]
12. Add Preprocessing

Alternative 8: 50.5% accurate, 47.7× speedup

Math

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

FPCore

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

C

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

Fortran

x_m = abs(x)
real(4) function code(x_m, s)
    real(4), intent (in) :: x_m
    real(4), intent (in) :: s
    code = 1.0e0 / ((s * 4.0e0) * (1.0e0 + (x_m * (1.0e0 / s))))
end function

Julia

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

MATLAB

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

Derivation

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 92.9%

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

   1. clear-num 92.9%

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

   2. inv-pow 92.9%

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

7. Applied egg-rr 59.5%

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

   1. unpow-1 59.5%

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

   2. *-commutative 59.5%

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

9. Simplified 59.5%

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

    1. *-un-lft-identity 59.5%

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

    2. exp-prod 59.5%

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

11. Applied egg-rr 59.5%

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

    1. exp-1-e 59.5%

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

13. Simplified 59.5%

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

14. Taylor expanded in x around 0 52.9%

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

    1. log-E 52.9%

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

    2. metadata-eval 52.9%

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

    3. log-E 52.9%

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

    4. log-E 52.9%

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

    5. metadata-eval 52.9%

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

    6. associate-/l* 53.3%

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

16. Simplified 53.3%

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

17. Final simplification 53.3%

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

Alternative 9: 51.2% accurate, 56.4× speedup

Math

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

FPCore

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

C

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

Fortran

x_m = abs(x)
real(4) function code(x_m, s)
    real(4), intent (in) :: x_m
    real(4), intent (in) :: s
    code = 1.0e0 / (s * (4.0e0 + ((x_m / s) * 4.0e0)))
end function

Julia

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

MATLAB

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

Derivation

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 92.9%

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

   1. clear-num 92.9%

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

   2. inv-pow 92.9%

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

7. Applied egg-rr 59.5%

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

   1. unpow-1 59.5%

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

   2. *-commutative 59.5%

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

9. Simplified 59.5%

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

10. Taylor expanded in s around inf 53.3%

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

11. Final simplification 53.3%

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

Alternative 10: 30.3% accurate, 77.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.05000000074505806:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s)
 :precision binary32
 (if (<= x_m 0.05000000074505806) (/ 0.25 s) (/ 0.25 x_m)))

C

x_m = fabs(x);
float code(float x_m, float s) {
	float tmp;
	if (x_m <= 0.05000000074505806f) {
		tmp = 0.25f / s;
	} else {
		tmp = 0.25f / x_m;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(4) function code(x_m, s)
    real(4), intent (in) :: x_m
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x_m <= 0.05000000074505806e0) then
        tmp = 0.25e0 / s
    else
        tmp = 0.25e0 / x_m
    end if
    code = tmp
end function

Julia

x_m = abs(x)
function code(x_m, s)
	tmp = Float32(0.0)
	if (x_m <= Float32(0.05000000074505806))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(0.25) / x_m);
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, s)
	tmp = single(0.0);
	if (x_m <= single(0.05000000074505806))
		tmp = single(0.25) / s;
	else
		tmp = single(0.25) / x_m;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 0.0500000007

   1. Initial program 99.3%

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

      1. fabs-neg 99.3%

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

      2. distribute-frac-neg 99.3%

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

      3. distribute-frac-neg2 99.3%

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

      4. fabs-neg 99.3%

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

      5. *-commutative 99.3%

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

      6. fabs-neg 99.3%

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

      7. +-commutative 99.3%

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

      8. fabs-neg 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in s around inf 37.2%

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

   if 0.0500000007 < x

   1. Initial program 99.8%

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

      1. fabs-neg 99.8%

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

      2. distribute-frac-neg 99.8%

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

      3. distribute-frac-neg2 99.8%

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

      4. fabs-neg 99.8%

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

      5. *-commutative 99.8%

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

      6. fabs-neg 99.8%

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

      7. +-commutative 99.8%

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

      8. fabs-neg 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in s around inf 99.8%

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

      1. clear-num 99.8%

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

      2. inv-pow 99.8%

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

   7. Applied egg-rr 99.8%

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

      1. unpow-1 99.8%

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

      2. *-commutative 99.8%

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

   9. Simplified 99.8%

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

   10. Taylor expanded in x around 0 15.8%

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

       1. distribute-lft-out 15.8%

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

   12. Simplified 15.8%

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

   13. Taylor expanded in s around 0 12.9%

       \[\leadsto \color{blue}{\frac{0.25}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 29.2% accurate, 88.6× speedup

Math

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

FPCore

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

C

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

Fortran

x_m = abs(x)
real(4) function code(x_m, s)
    real(4), intent (in) :: x_m
    real(4), intent (in) :: s
    code = 1.0e0 / (4.0e0 * (x_m + s))
end function

Julia

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

MATLAB

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

Derivation

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 92.9%

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

   1. clear-num 92.9%

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

   2. inv-pow 92.9%

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

7. Applied egg-rr 59.5%

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

   1. unpow-1 59.5%

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

   2. *-commutative 59.5%

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

9. Simplified 59.5%

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

10. Taylor expanded in x around 0 32.2%

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

    1. distribute-lft-out 32.2%

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

12. Simplified 32.2%

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

13. Final simplification 32.2%

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

Alternative 12: 28.8% accurate, 88.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ 0.25 \cdot \frac{1}{x\_m + s} \end{array} \]

FPCore

x_m = (fabs.f32 x)
(FPCore (x_m s) :precision binary32 (* 0.25 (/ 1.0 (+ x_m s))))

C

x_m = fabs(x);
float code(float x_m, float s) {
	return 0.25f * (1.0f / (x_m + s));
}

Fortran

x_m = abs(x)
real(4) function code(x_m, s)
    real(4), intent (in) :: x_m
    real(4), intent (in) :: s
    code = 0.25e0 * (1.0e0 / (x_m + s))
end function

Julia

x_m = abs(x)
function code(x_m, s)
	return Float32(Float32(0.25) * Float32(Float32(1.0) / Float32(x_m + s)))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m, s)
	tmp = single(0.25) * (single(1.0) / (x_m + s));
end

Derivation

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 92.9%

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

   1. clear-num 92.9%

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

   2. inv-pow 92.9%

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

7. Applied egg-rr 59.5%

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

   1. unpow-1 59.5%

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

   2. *-commutative 59.5%

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

9. Simplified 59.5%

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

10. Taylor expanded in x around 0 32.2%

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

    1. distribute-lft-out 32.2%

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

12. Simplified 32.2%

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

    1. associate-/r* 30.7%

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

    2. metadata-eval 30.7%

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

    3. div-inv 30.7%

       \[\leadsto \color{blue}{0.25 \cdot \frac{1}{s + x}} \]

    4. +-commutative 30.7%

       \[\leadsto 0.25 \cdot \frac{1}{\color{blue}{x + s}} \]

14. Applied egg-rr 30.7%

    \[\leadsto \color{blue}{0.25 \cdot \frac{1}{x + s}} \]
15. Add Preprocessing

Alternative 13: 28.8% accurate, 124.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

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 92.9%

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

   1. clear-num 92.9%

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

   2. inv-pow 92.9%

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

7. Applied egg-rr 59.5%

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

   1. unpow-1 59.5%

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

   2. *-commutative 59.5%

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

9. Simplified 59.5%

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

10. Taylor expanded in x around 0 32.2%

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

    1. distribute-lft-out 32.2%

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

12. Simplified 32.2%

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

    1. *-un-lft-identity 32.2%

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

    2. associate-/r* 30.7%

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

    3. metadata-eval 30.7%

       \[\leadsto 1 \cdot \frac{\color{blue}{0.25}}{s + x} \]

    4. +-commutative 30.7%

       \[\leadsto 1 \cdot \frac{0.25}{\color{blue}{x + s}} \]

14. Applied egg-rr 30.7%

    \[\leadsto \color{blue}{1 \cdot \frac{0.25}{x + s}} \]
15. Step-by-step derivation

    1. *-lft-identity 30.7%

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

    2. +-commutative 30.7%

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

16. Simplified 30.7%

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

17. Final simplification 30.7%

    \[\leadsto \frac{0.25}{x + s} \]
18. Add Preprocessing

Alternative 14: 26.8% accurate, 206.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
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 28.8%

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

Reproduce

herbie shell --seed 2024081 
(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))))))