Logistic distribution

Percentage Accurate: 99.6% → 99.6%

Time: 15.8s

Alternatives: 12

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: 99.6% 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, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{\left|x\right|}{-s}}\\ t_1 := t\_0 + 1\\ \frac{t\_0}{s \cdot \left(t\_1 \cdot t\_1\right)} \end{array} \end{array} \]

FPCore

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

C

float code(float x, float s) {
	float t_0 = expf((fabsf(x) / -s));
	float t_1 = t_0 + 1.0f;
	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 = t_0 + 1.0e0
    code = t_0 / (s * (t_1 * t_1))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.2%

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

   1. fabs-neg 99.2%

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

   2. distribute-frac-neg 99.2%

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

   3. distribute-frac-neg2 99.2%

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

   4. fabs-neg 99.2%

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

   5. *-commutative 99.2%

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

   6. fabs-neg 99.2%

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

   7. +-commutative 99.2%

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

   8. fabs-neg 99.2%

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

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. Final simplification 99.3%

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

Alternative 2: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 / ((t_0 + 1.0e0) * (s + (s / exp((abs(x) / s)))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.2%

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

3. Simplified 99.2%

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

5. Final simplification 99.2%

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

Alternative 3: 74.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.10000000149011612:\\ \;\;\;\;\frac{{e}^{\left(\frac{x}{s} + \mathsf{log1p}\left(e^{\frac{x}{s}}\right) \cdot -2\right)}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{e^{\frac{-x}{s}}}{s}}{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (fabs x) 0.10000000149011612)
   (/ (pow E (+ (/ x s) (* (log1p (exp (/ x s))) -2.0))) s)
   (/ (/ (exp (/ (- x) s)) s) 4.0)))

C

float code(float x, float s) {
	float tmp;
	if (fabsf(x) <= 0.10000000149011612f) {
		tmp = powf(((float) M_E), ((x / s) + (log1pf(expf((x / s))) * -2.0f))) / s;
	} else {
		tmp = (expf((-x / s)) / s) / 4.0f;
	}
	return tmp;
}

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (abs(x) <= Float32(0.10000000149011612))
		tmp = Float32((Float32(exp(1)) ^ Float32(Float32(x / s) + Float32(log1p(exp(Float32(x / s))) * Float32(-2.0)))) / s);
	else
		tmp = Float32(Float32(exp(Float32(Float32(-x) / s)) / s) / Float32(4.0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (fabs.f32 x) < 0.100000001

   1. Initial program 98.2%

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

      1. fabs-neg 98.2%

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

      2. distribute-frac-neg 98.2%

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

      3. distribute-frac-neg2 98.2%

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

      4. fabs-neg 98.2%

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

      5. *-commutative 98.2%

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

      6. fabs-neg 98.2%

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

      7. +-commutative 98.2%

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

      8. fabs-neg 98.2%

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

   3. Simplified 98.3%

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

   6. Applied egg-rr 93.9%

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

      1. associate--r+ 94.1%

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

      2. exp-diff 94.6%

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

      3. add-exp-log 98.2%

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

   8. Applied egg-rr 98.2%

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

      1. *-un-lft-identity 98.2%

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

      2. exp-prod 98.3%

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

      3. cancel-sign-sub-inv 98.3%

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

      4. metadata-eval 98.3%

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

   10. Applied egg-rr 98.3%

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

       1. exp-1-e 98.3%

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

       2. *-commutative 98.3%

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

   12. Simplified 98.3%

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

   if 0.100000001 < (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

   5. Taylor expanded in x around 0 100.0%

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

      1. associate-/r* 100.0%

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

      2. mul-1-neg 100.0%

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

      3. distribute-neg-frac2 100.0%

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

      4. +-commutative 100.0%

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

      5. mul-1-neg 100.0%

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

      6. distribute-neg-frac2 100.0%

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

   7. Simplified 100.0%

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

      1. distribute-frac-neg2 100.0%

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

      2. rec-exp 100.0%

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

      3. pow1 100.0%

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

      4. pow1 100.0%

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

      5. add-sqr-sqrt 53.8%

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

      6. fabs-sqr 53.8%

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

      7. add-sqr-sqrt 100.0%

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

   9. Applied egg-rr 100.0%

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

       1. rec-exp 100.0%

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

       2. distribute-neg-frac2 100.0%

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

   11. Simplified 100.0%

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

       1. distribute-frac-neg2 100.0%

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

       2. rec-exp 100.0%

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

       3. pow1 100.0%

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

       4. pow1 100.0%

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

       5. add-sqr-sqrt 53.8%

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

       6. fabs-sqr 53.8%

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

       7. add-sqr-sqrt 100.0%

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

   13. Applied egg-rr 53.5%

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

       1. rec-exp 100.0%

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

       2. distribute-neg-frac2 100.0%

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

   15. Simplified 53.8%

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

   16. Taylor expanded in x around 0 55.0%

       \[\leadsto \frac{\frac{e^{\frac{x}{-s}}}{s}}{\color{blue}{4}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.10000000149011612:\\ \;\;\;\;\frac{{e}^{\left(\frac{x}{s} + \mathsf{log1p}\left(e^{\frac{x}{s}}\right) \cdot -2\right)}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{e^{\frac{-x}{s}}}{s}}{4}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.10000000149011612:\\ \;\;\;\;\frac{e^{\frac{x}{s} - 2 \cdot \mathsf{log1p}\left(e^{\frac{x}{s}}\right)}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{e^{\frac{-x}{s}}}{s}}{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (fabs x) 0.10000000149011612)
   (/ (exp (- (/ x s) (* 2.0 (log1p (exp (/ x s)))))) s)
   (/ (/ (exp (/ (- x) s)) s) 4.0)))

C

float code(float x, float s) {
	float tmp;
	if (fabsf(x) <= 0.10000000149011612f) {
		tmp = expf(((x / s) - (2.0f * log1pf(expf((x / s)))))) / s;
	} else {
		tmp = (expf((-x / s)) / s) / 4.0f;
	}
	return tmp;
}

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (abs(x) <= Float32(0.10000000149011612))
		tmp = Float32(exp(Float32(Float32(x / s) - Float32(Float32(2.0) * log1p(exp(Float32(x / s)))))) / s);
	else
		tmp = Float32(Float32(exp(Float32(Float32(-x) / s)) / s) / Float32(4.0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (fabs.f32 x) < 0.100000001

   1. Initial program 98.2%

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

      1. fabs-neg 98.2%

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

      2. distribute-frac-neg 98.2%

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

      3. distribute-frac-neg2 98.2%

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

      4. fabs-neg 98.2%

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

      5. *-commutative 98.2%

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

      6. fabs-neg 98.2%

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

      7. +-commutative 98.2%

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

      8. fabs-neg 98.2%

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

   3. Simplified 98.3%

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

   6. Applied egg-rr 93.9%

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

      1. associate--r+ 94.1%

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

      2. exp-diff 94.6%

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

      3. add-exp-log 98.2%

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

   8. Applied egg-rr 98.2%

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

   if 0.100000001 < (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

   5. Taylor expanded in x around 0 100.0%

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

      1. associate-/r* 100.0%

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

      2. mul-1-neg 100.0%

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

      3. distribute-neg-frac2 100.0%

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

      4. +-commutative 100.0%

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

      5. mul-1-neg 100.0%

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

      6. distribute-neg-frac2 100.0%

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

   7. Simplified 100.0%

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

      1. distribute-frac-neg2 100.0%

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

      2. rec-exp 100.0%

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

      3. pow1 100.0%

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

      4. pow1 100.0%

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

      5. add-sqr-sqrt 53.8%

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

      6. fabs-sqr 53.8%

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

      7. add-sqr-sqrt 100.0%

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

   9. Applied egg-rr 100.0%

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

       1. rec-exp 100.0%

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

       2. distribute-neg-frac2 100.0%

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

   11. Simplified 100.0%

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

       1. distribute-frac-neg2 100.0%

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

       2. rec-exp 100.0%

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

       3. pow1 100.0%

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

       4. pow1 100.0%

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

       5. add-sqr-sqrt 53.8%

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

       6. fabs-sqr 53.8%

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

       7. add-sqr-sqrt 100.0%

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

   13. Applied egg-rr 53.5%

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

       1. rec-exp 100.0%

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

       2. distribute-neg-frac2 100.0%

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

   15. Simplified 53.8%

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

   16. Taylor expanded in x around 0 55.0%

       \[\leadsto \frac{\frac{e^{\frac{x}{-s}}}{s}}{\color{blue}{4}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.10000000149011612:\\ \;\;\;\;\frac{e^{\frac{x}{s} - 2 \cdot \mathsf{log1p}\left(e^{\frac{x}{s}}\right)}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{e^{\frac{-x}{s}}}{s}}{4}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 63.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	float t_0 = expf((-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((-x / s))
    code = (t_0 / s) / ((t_0 + 1.0e0) ** 2.0e0)
end function

Julia

function code(x, s)
	t_0 = exp(Float32(Float32(-x) / 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((-x / s));
	tmp = (t_0 / s) / ((t_0 + single(1.0)) ^ single(2.0));
end

Derivation

1. Initial program 99.2%

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

   1. fabs-neg 99.2%

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

   2. distribute-frac-neg 99.2%

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

   3. distribute-frac-neg2 99.2%

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

   4. fabs-neg 99.2%

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

   5. *-commutative 99.2%

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

   6. fabs-neg 99.2%

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

   7. +-commutative 99.2%

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

   8. fabs-neg 99.2%

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

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 x around 0 99.3%

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

   1. associate-/r* 99.2%

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

   2. mul-1-neg 99.2%

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

   3. distribute-neg-frac2 99.2%

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

   4. +-commutative 99.2%

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

   5. mul-1-neg 99.2%

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

   6. distribute-neg-frac2 99.2%

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

7. Simplified 99.2%

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

   1. distribute-frac-neg2 99.2%

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

   2. rec-exp 99.2%

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

   3. pow1 99.2%

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

   4. pow1 99.2%

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

   5. add-sqr-sqrt 53.3%

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

   6. fabs-sqr 53.3%

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

   7. add-sqr-sqrt 96.6%

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

9. Applied egg-rr 96.6%

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

    1. rec-exp 96.6%

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

    2. distribute-neg-frac2 96.6%

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

11. Simplified 96.6%

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

    1. distribute-frac-neg2 99.2%

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

    2. rec-exp 99.2%

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

    3. pow1 99.2%

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

    4. pow1 99.2%

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

    5. add-sqr-sqrt 53.3%

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

    6. fabs-sqr 53.3%

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

    7. add-sqr-sqrt 96.6%

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

13. Applied egg-rr 62.9%

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

    1. rec-exp 96.6%

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

    2. distribute-neg-frac2 96.6%

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

15. Simplified 63.0%

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

16. Final simplification 63.0%

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

Alternative 6: 60.1% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{-1 - e^{\frac{x}{s}}} \cdot \frac{0.5}{s} \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.2%

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

   1. fabs-neg 99.2%

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

   2. distribute-frac-neg 99.2%

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

   3. distribute-frac-neg2 99.2%

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

   4. fabs-neg 99.2%

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

   5. *-commutative 99.2%

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

   6. fabs-neg 99.2%

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

   7. +-commutative 99.2%

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

   8. fabs-neg 99.2%

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

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. Step-by-step derivation

   1. associate-*r* 99.2%

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

   2. associate-/r* 99.2%

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

   3. distribute-rgt-in 99.2%

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

   4. *-un-lft-identity 99.2%

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

   5. *-commutative 99.2%

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

   6. distribute-frac-neg2 99.2%

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

   7. rec-exp 99.2%

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

   8. div-inv 99.2%

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

   9. associate-/r* 99.2%

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

6. Applied egg-rr 59.4%

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

7. Taylor expanded in x around 0 60.8%

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

8. Final simplification 60.8%

   \[\leadsto \frac{-1}{-1 - e^{\frac{x}{s}}} \cdot \frac{0.5}{s} \]
9. Add Preprocessing

Alternative 7: 77.8% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.999999873689376 \cdot 10^{-5}:\\ \;\;\;\;\frac{\left(0.25 + \frac{x \cdot -0.125}{s}\right) - 0.5 \cdot \frac{x \cdot -0.25}{s}}{s}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{-x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 4.999999873689376e-5)
   (/ (- (+ 0.25 (/ (* x -0.125) s)) (* 0.5 (/ (* x -0.25) s))) s)
   (exp (/ (- x) s))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 4.999999873689376e-5f) {
		tmp = ((0.25f + ((x * -0.125f) / s)) - (0.5f * ((x * -0.25f) / s))) / s;
	} else {
		tmp = expf((-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 <= 4.999999873689376e-5) then
        tmp = ((0.25e0 + ((x * (-0.125e0)) / s)) - (0.5e0 * ((x * (-0.25e0)) / s))) / s
    else
        tmp = exp((-x / s))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(4.999999873689376e-5))
		tmp = Float32(Float32(Float32(Float32(0.25) + Float32(Float32(x * Float32(-0.125)) / s)) - Float32(Float32(0.5) * Float32(Float32(x * Float32(-0.25)) / s))) / s);
	else
		tmp = exp(Float32(Float32(-x) / s));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(4.999999873689376e-5))
		tmp = ((single(0.25) + ((x * single(-0.125)) / s)) - (single(0.5) * ((x * single(-0.25)) / s))) / s;
	else
		tmp = exp((-x / s));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 4.99999987e-5

   1. Initial program 98.9%

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

      1. fabs-neg 98.9%

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

      2. distribute-frac-neg 98.9%

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

      3. distribute-frac-neg2 98.9%

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

      4. fabs-neg 98.9%

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

      5. *-commutative 98.9%

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

      6. fabs-neg 98.9%

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

      7. +-commutative 98.9%

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

      8. fabs-neg 98.9%

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

   3. Simplified 99.0%

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

      1. associate-*r* 98.9%

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

      2. associate-/r* 98.9%

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

      3. distribute-rgt-in 98.9%

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

      4. *-un-lft-identity 98.9%

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

      5. *-commutative 98.9%

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

      6. distribute-frac-neg2 98.9%

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

      7. rec-exp 98.9%

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

      8. div-inv 98.9%

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

      9. associate-/r* 99.0%

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

   6. Applied egg-rr 85.4%

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

   7. Taylor expanded in s around inf 38.2%

      \[\leadsto \frac{1}{e^{\frac{x}{s}} + 1} \cdot \color{blue}{\frac{\left(0.5 + 0.5 \cdot \frac{x}{s}\right) - 0.25 \cdot \frac{x}{s}}{s}} \]

   8. Taylor expanded in s around -inf 72.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{0.5 \cdot \frac{-0.5 \cdot x - -0.25 \cdot x}{s} - \left(0.25 + -0.125 \cdot \frac{x}{s}\right)}{s}} \]
   9. Step-by-step derivation

      1. mul-1-neg 72.5%

         \[\leadsto \color{blue}{-\frac{0.5 \cdot \frac{-0.5 \cdot x - -0.25 \cdot x}{s} - \left(0.25 + -0.125 \cdot \frac{x}{s}\right)}{s}} \]

      2. distribute-rgt-out-- 72.5%

         \[\leadsto -\frac{0.5 \cdot \frac{\color{blue}{x \cdot \left(-0.5 - -0.25\right)}}{s} - \left(0.25 + -0.125 \cdot \frac{x}{s}\right)}{s} \]

      3. metadata-eval 72.5%

         \[\leadsto -\frac{0.5 \cdot \frac{x \cdot \color{blue}{-0.25}}{s} - \left(0.25 + -0.125 \cdot \frac{x}{s}\right)}{s} \]

      4. associate-*r/ 73.1%

         \[\leadsto -\frac{0.5 \cdot \frac{x \cdot -0.25}{s} - \left(0.25 + \color{blue}{\frac{-0.125 \cdot x}{s}}\right)}{s} \]

      5. *-commutative 73.1%

         \[\leadsto -\frac{0.5 \cdot \frac{x \cdot -0.25}{s} - \left(0.25 + \frac{\color{blue}{x \cdot -0.125}}{s}\right)}{s} \]

   10. Simplified 73.1%

       \[\leadsto \color{blue}{-\frac{0.5 \cdot \frac{x \cdot -0.25}{s} - \left(0.25 + \frac{x \cdot -0.125}{s}\right)}{s}} \]

   if 4.99999987e-5 < 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

   6. Applied egg-rr 60.3%

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

   7. Taylor expanded in x around inf 3.4%

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

      1. div-inv 3.4%

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

      2. exp-prod 3.4%

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

      3. add-sqr-sqrt 3.4%

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

      4. fabs-sqr 3.4%

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

      5. add-sqr-sqrt 3.4%

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

      6. add-sqr-sqrt 3.4%

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

      7. sqrt-unprod 3.4%

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

      8. sqr-neg 3.4%

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

      9. sqrt-unprod -0.0%

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

      10. add-sqr-sqrt 98.5%

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

      11. exp-prod 98.5%

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

      12. div-inv 98.5%

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

      13. distribute-frac-neg2 98.5%

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

      14. exp-neg 98.5%

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

      15. add-sqr-sqrt 98.5%

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

      16. sqrt-unprod 98.5%

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

      17. sqr-neg 98.5%

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

      18. sqrt-unprod -0.0%

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

      19. add-sqr-sqrt 3.4%

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

      20. div-inv 3.4%

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

      21. exp-prod 3.4%

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

   9. Applied egg-rr 98.5%

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

       1. rec-exp 98.5%

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

       2. mul-1-neg 98.5%

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

       3. associate-*r/ 98.5%

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

       4. neg-mul-1 98.5%

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

   11. Simplified 98.5%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.999999873689376 \cdot 10^{-5}:\\ \;\;\;\;\frac{\left(0.25 + \frac{x \cdot -0.125}{s}\right) - 0.5 \cdot \frac{x \cdot -0.25}{s}}{s}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{-x}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 59.3% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{s \cdot e^{\frac{x}{s}}}}{4} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (/ (/ 1.0 (* s (exp (/ x s)))) 4.0))

C

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

Fortran

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

Julia

function code(x, s)
	return Float32(Float32(Float32(1.0) / Float32(s * exp(Float32(x / s)))) / Float32(4.0))
end

MATLAB

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

Derivation

1. Initial program 99.2%

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

   1. fabs-neg 99.2%

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

   2. distribute-frac-neg 99.2%

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

   3. distribute-frac-neg2 99.2%

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

   4. fabs-neg 99.2%

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

   5. *-commutative 99.2%

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

   6. fabs-neg 99.2%

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

   7. +-commutative 99.2%

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

   8. fabs-neg 99.2%

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

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 x around 0 99.3%

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

   1. associate-/r* 99.2%

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

   2. mul-1-neg 99.2%

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

   3. distribute-neg-frac2 99.2%

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

   4. +-commutative 99.2%

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

   5. mul-1-neg 99.2%

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

   6. distribute-neg-frac2 99.2%

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

7. Simplified 99.2%

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

8. Taylor expanded in s around inf 94.6%

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

10. Applied egg-rr 60.0%

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

    1. unpow-1 60.0%

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

12. Simplified 60.0%

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

13. Final simplification 60.0%

    \[\leadsto \frac{\frac{1}{s \cdot e^{\frac{x}{s}}}}{4} \]
14. Add Preprocessing

Alternative 9: 59.3% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{e^{\frac{-x}{s}}}{s}}{4} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (/ (/ (exp (/ (- x) s)) s) 4.0))

C

float code(float x, float s) {
	return (expf((-x / s)) / s) / 4.0f;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = (exp((-x / s)) / s) / 4.0e0
end function

Julia

function code(x, s)
	return Float32(Float32(exp(Float32(Float32(-x) / s)) / s) / Float32(4.0))
end

MATLAB

function tmp = code(x, s)
	tmp = (exp((-x / s)) / s) / single(4.0);
end

Derivation

1. Initial program 99.2%

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

   1. fabs-neg 99.2%

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

   2. distribute-frac-neg 99.2%

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

   3. distribute-frac-neg2 99.2%

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

   4. fabs-neg 99.2%

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

   5. *-commutative 99.2%

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

   6. fabs-neg 99.2%

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

   7. +-commutative 99.2%

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

   8. fabs-neg 99.2%

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

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 x around 0 99.3%

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

   1. associate-/r* 99.2%

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

   2. mul-1-neg 99.2%

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

   3. distribute-neg-frac2 99.2%

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

   4. +-commutative 99.2%

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

   5. mul-1-neg 99.2%

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

   6. distribute-neg-frac2 99.2%

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

7. Simplified 99.2%

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

   1. distribute-frac-neg2 99.2%

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

   2. rec-exp 99.2%

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

   3. pow1 99.2%

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

   4. pow1 99.2%

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

   5. add-sqr-sqrt 53.3%

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

   6. fabs-sqr 53.3%

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

   7. add-sqr-sqrt 96.6%

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

9. Applied egg-rr 96.6%

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

    1. rec-exp 96.6%

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

    2. distribute-neg-frac2 96.6%

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

11. Simplified 96.6%

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

    1. distribute-frac-neg2 99.2%

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

    2. rec-exp 99.2%

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

    3. pow1 99.2%

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

    4. pow1 99.2%

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

    5. add-sqr-sqrt 53.3%

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

    6. fabs-sqr 53.3%

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

    7. add-sqr-sqrt 96.6%

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

13. Applied egg-rr 62.9%

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

    1. rec-exp 96.6%

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

    2. distribute-neg-frac2 96.6%

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

15. Simplified 63.0%

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

16. Taylor expanded in x around 0 60.0%

    \[\leadsto \frac{\frac{e^{\frac{x}{-s}}}{s}}{\color{blue}{4}} \]

17. Final simplification 60.0%

    \[\leadsto \frac{\frac{e^{\frac{-x}{s}}}{s}}{4} \]
18. Add Preprocessing

Alternative 10: 65.7% accurate, 36.5× speedup

Math

\[\begin{array}{l} \\ \frac{\left(0.25 + \frac{x \cdot -0.125}{s}\right) - 0.5 \cdot \frac{x \cdot -0.25}{s}}{s} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (/ (- (+ 0.25 (/ (* x -0.125) s)) (* 0.5 (/ (* x -0.25) s))) s))

C

float code(float x, float s) {
	return ((0.25f + ((x * -0.125f) / s)) - (0.5f * ((x * -0.25f) / s))) / s;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = ((0.25e0 + ((x * (-0.125e0)) / s)) - (0.5e0 * ((x * (-0.25e0)) / s))) / s
end function

Julia

function code(x, s)
	return Float32(Float32(Float32(Float32(0.25) + Float32(Float32(x * Float32(-0.125)) / s)) - Float32(Float32(0.5) * Float32(Float32(x * Float32(-0.25)) / s))) / s)
end

MATLAB

function tmp = code(x, s)
	tmp = ((single(0.25) + ((x * single(-0.125)) / s)) - (single(0.5) * ((x * single(-0.25)) / s))) / s;
end

Derivation

1. Initial program 99.2%

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

   1. fabs-neg 99.2%

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

   2. distribute-frac-neg 99.2%

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

   3. distribute-frac-neg2 99.2%

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

   4. fabs-neg 99.2%

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

   5. *-commutative 99.2%

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

   6. fabs-neg 99.2%

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

   7. +-commutative 99.2%

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

   8. fabs-neg 99.2%

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

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. Step-by-step derivation

   1. associate-*r* 99.2%

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

   2. associate-/r* 99.2%

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

   3. distribute-rgt-in 99.2%

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

   4. *-un-lft-identity 99.2%

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

   5. *-commutative 99.2%

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

   6. distribute-frac-neg2 99.2%

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

   7. rec-exp 99.2%

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

   8. div-inv 99.2%

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

   9. associate-/r* 99.2%

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

6. Applied egg-rr 59.4%

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

7. Taylor expanded in s around inf 35.5%

   \[\leadsto \frac{1}{e^{\frac{x}{s}} + 1} \cdot \color{blue}{\frac{\left(0.5 + 0.5 \cdot \frac{x}{s}\right) - 0.25 \cdot \frac{x}{s}}{s}} \]

8. Taylor expanded in s around -inf 66.8%

   \[\leadsto \color{blue}{-1 \cdot \frac{0.5 \cdot \frac{-0.5 \cdot x - -0.25 \cdot x}{s} - \left(0.25 + -0.125 \cdot \frac{x}{s}\right)}{s}} \]
9. Step-by-step derivation

   1. mul-1-neg 66.8%

      \[\leadsto \color{blue}{-\frac{0.5 \cdot \frac{-0.5 \cdot x - -0.25 \cdot x}{s} - \left(0.25 + -0.125 \cdot \frac{x}{s}\right)}{s}} \]

   2. distribute-rgt-out-- 66.8%

      \[\leadsto -\frac{0.5 \cdot \frac{\color{blue}{x \cdot \left(-0.5 - -0.25\right)}}{s} - \left(0.25 + -0.125 \cdot \frac{x}{s}\right)}{s} \]

   3. metadata-eval 66.8%

      \[\leadsto -\frac{0.5 \cdot \frac{x \cdot \color{blue}{-0.25}}{s} - \left(0.25 + -0.125 \cdot \frac{x}{s}\right)}{s} \]

   4. associate-*r/ 67.6%

      \[\leadsto -\frac{0.5 \cdot \frac{x \cdot -0.25}{s} - \left(0.25 + \color{blue}{\frac{-0.125 \cdot x}{s}}\right)}{s} \]

   5. *-commutative 67.6%

      \[\leadsto -\frac{0.5 \cdot \frac{x \cdot -0.25}{s} - \left(0.25 + \frac{\color{blue}{x \cdot -0.125}}{s}\right)}{s} \]

10. Simplified 67.6%

    \[\leadsto \color{blue}{-\frac{0.5 \cdot \frac{x \cdot -0.25}{s} - \left(0.25 + \frac{x \cdot -0.125}{s}\right)}{s}} \]

11. Final simplification 67.6%

    \[\leadsto \frac{\left(0.25 + \frac{x \cdot -0.125}{s}\right) - 0.5 \cdot \frac{x \cdot -0.25}{s}}{s} \]
12. Add Preprocessing

Alternative 11: 28.1% 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 99.2%

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

   1. fabs-neg 99.2%

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

   2. distribute-frac-neg 99.2%

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

   3. distribute-frac-neg2 99.2%

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

   4. fabs-neg 99.2%

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

   5. *-commutative 99.2%

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

   6. fabs-neg 99.2%

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

   7. +-commutative 99.2%

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

   8. fabs-neg 99.2%

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

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 24.2%

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

6. Final simplification 24.2%

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

Alternative 12: 8.3% accurate, 620.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x s) :precision binary32 1.0)

C

float code(float x, float s) {
	return 1.0f;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 1.0e0
end function

Julia

function code(x, s)
	return Float32(1.0)
end

MATLAB

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

Derivation

1. Initial program 99.2%

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

   1. fabs-neg 99.2%

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

   2. distribute-frac-neg 99.2%

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

   3. distribute-frac-neg2 99.2%

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

   4. fabs-neg 99.2%

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

   5. *-commutative 99.2%

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

   6. fabs-neg 99.2%

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

   7. +-commutative 99.2%

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

   8. fabs-neg 99.2%

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

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 85.1%

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

7. Taylor expanded in x around inf 40.2%

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

8. Taylor expanded in x around 0 8.2%

   \[\leadsto \color{blue}{1} \]

9. Final simplification 8.2%

   \[\leadsto 1 \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024059 
(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))))))