Logistic function

Percentage Accurate: 99.8% → 99.9%

Time: 8.5s

Alternatives: 10

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

float code(float x, float s) {
	return 1.0f / (1.0f + expf((-x / 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)))
end function

Julia

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

MATLAB

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	return 1.0f / (1.0f + expf((-x / 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)))
end function

Julia

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

MATLAB

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

Alternative 1: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ e^{-\mathsf{log1p}\left(e^{\frac{-x}{s}}\right)} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (exp (- (log1p (exp (/ (- x) s))))))

C

float code(float x, float s) {
	return expf(-log1pf(expf((-x / s))));
}

Julia

function code(x, s)
	return exp(Float32(-log1p(exp(Float32(Float32(-x) / s)))))
end

Derivation

1. Initial program 99.8%

   \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
2. Step-by-step derivation

   1. distribute-frac-neg 99.8%

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

   2. exp-neg 99.8%

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

   3. add-sqr-sqrt 55.4%

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

   4. sqrt-unprod 64.7%

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

   5. sqr-neg 64.7%

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

   6. sqrt-unprod 11.9%

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

   7. add-sqr-sqrt 25.8%

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

   8. pow1 25.8%

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

   9. pow1 25.8%

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

   10. add-cube-cbrt 25.8%

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

   11. pow3 25.8%

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

   12. pow-flip 25.8%

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

3. Applied egg-rr 99.8%

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

   1. add-exp-log 99.8%

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

   2. log-rec 99.8%

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

   3. log1p-udef 99.9%

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

   4. pow1/3 99.9%

      \[\leadsto e^{-\mathsf{log1p}\left({\color{blue}{\left({\left(e^{\frac{x}{s}}\right)}^{0.3333333333333333}\right)}}^{-3}\right)} \]

   5. pow-pow 100.0%

      \[\leadsto e^{-\mathsf{log1p}\left(\color{blue}{{\left(e^{\frac{x}{s}}\right)}^{\left(0.3333333333333333 \cdot -3\right)}}\right)} \]

   6. metadata-eval 100.0%

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

5. Applied egg-rr 100.0%

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

   1. unpow-1 100.0%

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

   2. rec-exp 100.0%

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

   3. distribute-frac-neg 100.0%

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

7. Simplified 100.0%

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

8. Final simplification 100.0%

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

2. Final simplification 99.8%

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

Alternative 3: 65.3% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.000000015855384 \cdot 10^{-30}:\\ \;\;\;\;\frac{1}{2 + s \cdot \left(\left(x \cdot \left(0.5 - \frac{s}{x}\right)\right) \cdot \frac{1}{\frac{s}{x} \cdot \left(s \cdot s\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -5.000000015855384e-30)
   (/ 1.0 (+ 2.0 (* s (* (* x (- 0.5 (/ s x))) (/ 1.0 (* (/ s x) (* s s)))))))
   0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -5.000000015855384e-30f) {
		tmp = 1.0f / (2.0f + (s * ((x * (0.5f - (s / x))) * (1.0f / ((s / x) * (s * s))))));
	} else {
		tmp = 0.5f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-5.000000015855384e-30)) then
        tmp = 1.0e0 / (2.0e0 + (s * ((x * (0.5e0 - (s / x))) * (1.0e0 / ((s / x) * (s * s))))))
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-5.000000015855384e-30))
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(s * Float32(Float32(x * Float32(Float32(0.5) - Float32(s / x))) * Float32(Float32(1.0) / Float32(Float32(s / x) * Float32(s * s)))))));
	else
		tmp = Float32(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-5.000000015855384e-30))
		tmp = single(1.0) / (single(2.0) + (s * ((x * (single(0.5) - (s / x))) * (single(1.0) / ((s / x) * (s * s))))));
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -5.00000002e-30

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 82.2%

      \[\leadsto \frac{1}{\color{blue}{2 + \left(-1 \cdot \frac{x}{s} + 0.5 \cdot \frac{{x}^{2}}{{s}^{2}}\right)}} \]
   3. Step-by-step derivation

      1. +-commutative 82.2%

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

      2. mul-1-neg 82.2%

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

      3. unsub-neg 82.2%

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

      4. unpow2 82.2%

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

      5. unpow2 82.2%

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

      6. times-frac 77.1%

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

   4. Simplified 77.1%

      \[\leadsto \frac{1}{\color{blue}{2 + \left(0.5 \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right) - \frac{x}{s}\right)}} \]
   5. Step-by-step derivation

      1. clear-num 77.1%

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

      2. frac-times 78.9%

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

      3. *-un-lft-identity 78.9%

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

   6. Applied egg-rr 78.9%

      \[\leadsto \frac{1}{2 + \left(0.5 \cdot \color{blue}{\frac{x}{\frac{s}{x} \cdot s}} - \frac{x}{s}\right)} \]
   7. Step-by-step derivation

      1. associate-*r/ 78.9%

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

      2. frac-sub 80.0%

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

      3. *-commutative 80.0%

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

      4. *-commutative 80.0%

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

   8. Applied egg-rr 80.0%

      \[\leadsto \frac{1}{2 + \color{blue}{\frac{\left(0.5 \cdot x\right) \cdot s - \left(s \cdot \frac{s}{x}\right) \cdot x}{\left(s \cdot \frac{s}{x}\right) \cdot s}}} \]
   9. Step-by-step derivation

      1. *-commutative 80.0%

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

      2. associate-*l* 80.0%

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

      3. distribute-lft-out-- 80.0%

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

      4. *-commutative 80.0%

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

      5. *-commutative 80.0%

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

   10. Simplified 80.0%

       \[\leadsto \frac{1}{2 + \color{blue}{\frac{s \cdot \left(x \cdot 0.5 - \frac{s}{x} \cdot x\right)}{s \cdot \left(s \cdot \frac{s}{x}\right)}}} \]
   11. Step-by-step derivation

       1. div-inv 82.7%

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

       2. *-commutative 82.7%

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

   12. Applied egg-rr 82.7%

       \[\leadsto \frac{1}{2 + \color{blue}{\left(s \cdot \left(x \cdot 0.5 - x \cdot \frac{s}{x}\right)\right) \cdot \frac{1}{s \cdot \left(s \cdot \frac{s}{x}\right)}}} \]
   13. Step-by-step derivation

       1. associate-*l* 89.6%

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

       2. distribute-lft-out-- 89.6%

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

       3. associate-*r* 89.6%

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

   14. Simplified 89.6%

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

   if -5.00000002e-30 < x

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 46.3%

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

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.000000015855384 \cdot 10^{-30}:\\ \;\;\;\;\frac{1}{2 + s \cdot \left(\left(x \cdot \left(0.5 - \frac{s}{x}\right)\right) \cdot \frac{1}{\frac{s}{x} \cdot \left(s \cdot s\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 4: 62.8% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.000000015855384 \cdot 10^{-30}:\\ \;\;\;\;\frac{1}{2 + \left(0.5 \cdot \frac{x}{\frac{s \cdot s}{x}} - \frac{x}{s}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -5.000000015855384e-30)
   (/ 1.0 (+ 2.0 (- (* 0.5 (/ x (/ (* s s) x))) (/ x s))))
   0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -5.000000015855384e-30f) {
		tmp = 1.0f / (2.0f + ((0.5f * (x / ((s * s) / x))) - (x / s)));
	} else {
		tmp = 0.5f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-5.000000015855384e-30)) then
        tmp = 1.0e0 / (2.0e0 + ((0.5e0 * (x / ((s * s) / x))) - (x / s)))
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-5.000000015855384e-30))
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(Float32(Float32(0.5) * Float32(x / Float32(Float32(s * s) / x))) - Float32(x / s))));
	else
		tmp = Float32(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-5.000000015855384e-30))
		tmp = single(1.0) / (single(2.0) + ((single(0.5) * (x / ((s * s) / x))) - (x / s)));
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -5.00000002e-30

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 82.2%

      \[\leadsto \frac{1}{\color{blue}{2 + \left(-1 \cdot \frac{x}{s} + 0.5 \cdot \frac{{x}^{2}}{{s}^{2}}\right)}} \]
   3. Step-by-step derivation

      1. +-commutative 82.2%

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

      2. mul-1-neg 82.2%

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

      3. unsub-neg 82.2%

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

      4. unpow2 82.2%

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

      5. unpow2 82.2%

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

      6. times-frac 77.1%

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

   4. Simplified 77.1%

      \[\leadsto \frac{1}{\color{blue}{2 + \left(0.5 \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right) - \frac{x}{s}\right)}} \]
   5. Step-by-step derivation

      1. frac-times 82.2%

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

      2. associate-/l* 85.2%

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

   6. Applied egg-rr 85.2%

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

   if -5.00000002e-30 < x

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 46.3%

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

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.000000015855384 \cdot 10^{-30}:\\ \;\;\;\;\frac{1}{2 + \left(0.5 \cdot \frac{x}{\frac{s \cdot s}{x}} - \frac{x}{s}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 5: 60.6% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.999999841327613 \cdot 10^{-21}:\\ \;\;\;\;\frac{1}{2 + 0.5 \cdot \frac{x \cdot x}{s \cdot s}}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -4.999999841327613e-21)
   (/ 1.0 (+ 2.0 (* 0.5 (/ (* x x) (* s s)))))
   0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -4.999999841327613e-21f) {
		tmp = 1.0f / (2.0f + (0.5f * ((x * x) / (s * s))));
	} else {
		tmp = 0.5f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-4.999999841327613e-21)) then
        tmp = 1.0e0 / (2.0e0 + (0.5e0 * ((x * x) / (s * s))))
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-4.999999841327613e-21))
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(Float32(0.5) * Float32(Float32(x * x) / Float32(s * s)))));
	else
		tmp = Float32(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-4.999999841327613e-21))
		tmp = single(1.0) / (single(2.0) + (single(0.5) * ((x * x) / (s * s))));
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -4.99999984e-21

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 83.4%

      \[\leadsto \frac{1}{\color{blue}{2 + \left(-1 \cdot \frac{x}{s} + 0.5 \cdot \frac{{x}^{2}}{{s}^{2}}\right)}} \]
   3. Step-by-step derivation

      1. +-commutative 83.4%

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

      2. mul-1-neg 83.4%

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

      3. unsub-neg 83.4%

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

      4. unpow2 83.4%

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

      5. unpow2 83.4%

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

      6. times-frac 77.4%

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

   4. Simplified 77.4%

      \[\leadsto \frac{1}{\color{blue}{2 + \left(0.5 \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right) - \frac{x}{s}\right)}} \]
   5. Step-by-step derivation

      1. clear-num 77.4%

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

      2. frac-times 77.4%

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

      3. *-un-lft-identity 77.4%

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

   6. Applied egg-rr 77.4%

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

   7. Taylor expanded in x around inf 82.4%

      \[\leadsto \frac{1}{2 + \color{blue}{0.5 \cdot \frac{{x}^{2}}{{s}^{2}}}} \]
   8. Step-by-step derivation

      1. *-commutative 82.4%

         \[\leadsto \frac{1}{2 + \color{blue}{\frac{{x}^{2}}{{s}^{2}} \cdot 0.5}} \]

      2. unpow2 82.4%

         \[\leadsto \frac{1}{2 + \frac{\color{blue}{x \cdot x}}{{s}^{2}} \cdot 0.5} \]

      3. unpow2 82.4%

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

   9. Simplified 82.4%

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

   if -4.99999984e-21 < x

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 48.1%

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

4. Final simplification 59.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.999999841327613 \cdot 10^{-21}:\\ \;\;\;\;\frac{1}{2 + 0.5 \cdot \frac{x \cdot x}{s \cdot s}}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 6: 56.5% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.999999960041972 \cdot 10^{-13}:\\ \;\;\;\;2 \cdot \left(\frac{s}{x} \cdot \frac{s}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -9.999999960041972e-13) (* 2.0 (* (/ s x) (/ s x))) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -9.999999960041972e-13f) {
		tmp = 2.0f * ((s / x) * (s / x));
	} else {
		tmp = 0.5f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-9.999999960041972e-13)) then
        tmp = 2.0e0 * ((s / x) * (s / x))
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-9.999999960041972e-13))
		tmp = Float32(Float32(2.0) * Float32(Float32(s / x) * Float32(s / x)));
	else
		tmp = Float32(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-9.999999960041972e-13))
		tmp = single(2.0) * ((s / x) * (s / x));
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -9.99999996e-13

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 82.6%

      \[\leadsto \frac{1}{\color{blue}{2 + \left(-1 \cdot \frac{x}{s} + 0.5 \cdot \frac{{x}^{2}}{{s}^{2}}\right)}} \]
   3. Step-by-step derivation

      1. +-commutative 82.6%

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

      2. mul-1-neg 82.6%

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

      3. unsub-neg 82.6%

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

      4. unpow2 82.6%

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

      5. unpow2 82.6%

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

      6. times-frac 80.5%

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

   4. Simplified 80.5%

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

   5. Taylor expanded in x around inf 79.4%

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

      1. unpow2 79.4%

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

      2. unpow2 79.4%

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

      3. times-frac 76.2%

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

   7. Simplified 76.2%

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

   if -9.99999996e-13 < x

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 47.5%

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

4. Final simplification 56.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.999999960041972 \cdot 10^{-13}:\\ \;\;\;\;2 \cdot \left(\frac{s}{x} \cdot \frac{s}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 7: 57.6% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.999999841327613 \cdot 10^{-21}:\\ \;\;\;\;2 \cdot \frac{s \cdot s}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -4.999999841327613e-21) (* 2.0 (/ (* s s) (* x x))) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -4.999999841327613e-21f) {
		tmp = 2.0f * ((s * s) / (x * x));
	} else {
		tmp = 0.5f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-4.999999841327613e-21)) then
        tmp = 2.0e0 * ((s * s) / (x * x))
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-4.999999841327613e-21))
		tmp = Float32(Float32(2.0) * Float32(Float32(s * s) / Float32(x * x)));
	else
		tmp = Float32(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-4.999999841327613e-21))
		tmp = single(2.0) * ((s * s) / (x * x));
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -4.99999984e-21

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 83.4%

      \[\leadsto \frac{1}{\color{blue}{2 + \left(-1 \cdot \frac{x}{s} + 0.5 \cdot \frac{{x}^{2}}{{s}^{2}}\right)}} \]
   3. Step-by-step derivation

      1. +-commutative 83.4%

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

      2. mul-1-neg 83.4%

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

      3. unsub-neg 83.4%

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

      4. unpow2 83.4%

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

      5. unpow2 83.4%

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

      6. times-frac 77.4%

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

   4. Simplified 77.4%

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

   5. Taylor expanded in x around inf 77.3%

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

      1. unpow2 77.3%

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

      2. unpow2 77.3%

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

      3. times-frac 70.3%

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

   7. Simplified 70.3%

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

      1. frac-times 77.3%

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

   9. Applied egg-rr 77.3%

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

   if -4.99999984e-21 < x

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 48.1%

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

4. Final simplification 58.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.999999841327613 \cdot 10^{-21}:\\ \;\;\;\;2 \cdot \frac{s \cdot s}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 8: 48.4% accurate, 11.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.999999982195158 \cdot 10^{-37}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -1.999999982195158e-37) (/ 1.0 (- 2.0 (/ x s))) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -1.999999982195158e-37f) {
		tmp = 1.0f / (2.0f - (x / s));
	} else {
		tmp = 0.5f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-1.999999982195158e-37)) then
        tmp = 1.0e0 / (2.0e0 - (x / s))
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-1.999999982195158e-37))
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) - Float32(x / s)));
	else
		tmp = Float32(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-1.999999982195158e-37))
		tmp = single(1.0) / (single(2.0) - (x / s));
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -1.99999998e-37

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 51.5%

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

      1. mul-1-neg 51.5%

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

      2. unsub-neg 51.5%

         \[\leadsto \frac{1}{\color{blue}{2 - \frac{x}{s}}} \]

   4. Simplified 51.5%

      \[\leadsto \frac{1}{\color{blue}{2 - \frac{x}{s}}} \]

   if -1.99999998e-37 < x

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 42.8%

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

4. Final simplification 46.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.999999982195158 \cdot 10^{-37}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 9: 46.0% accurate, 17.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.999999987845058 \cdot 10^{-8}:\\ \;\;\;\;\frac{-s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -1.999999987845058e-8) (/ (- s) x) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -1.999999987845058e-8f) {
		tmp = -s / x;
	} else {
		tmp = 0.5f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-1.999999987845058e-8)) then
        tmp = -s / x
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-1.999999987845058e-8))
		tmp = Float32(Float32(-s) / x);
	else
		tmp = Float32(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-1.999999987845058e-8))
		tmp = -s / x;
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -1.99999999e-8

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 45.6%

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

      1. mul-1-neg 45.6%

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

      2. unsub-neg 45.6%

         \[\leadsto \frac{1}{\color{blue}{2 - \frac{x}{s}}} \]

   4. Simplified 45.6%

      \[\leadsto \frac{1}{\color{blue}{2 - \frac{x}{s}}} \]

   5. Taylor expanded in x around inf 39.6%

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

      1. associate-*r/ 39.6%

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

      2. neg-mul-1 39.6%

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

   7. Simplified 39.6%

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

   if -1.99999999e-8 < x

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 46.4%

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

4. Final simplification 44.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.999999987845058 \cdot 10^{-8}:\\ \;\;\;\;\frac{-s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 10: 35.0% accurate, 108.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \end{array} \]

FPCore

(FPCore (x s) :precision binary32 0.5)

C

float code(float x, float s) {
	return 0.5f;
}

Fortran

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

Julia

function code(x, s)
	return Float32(0.5)
end

MATLAB

function tmp = code(x, s)
	tmp = single(0.5);
end

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in x around 0 35.3%

   \[\leadsto \color{blue}{0.5} \]

3. Final simplification 35.3%

   \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023297 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))