Logistic function

Percentage Accurate: 99.8% → 99.8%

Time: 7.6s

Alternatives: 14

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: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	return 1.0f / (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 + (1.0e0 / exp((x / s))))
end function

Julia

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

MATLAB

function tmp = code(x, s)
	tmp = single(1.0) / (single(1.0) + (single(1.0) / exp((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 99.9%

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

3. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

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

Alternative 2: 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

Derivation

1. Initial program 99.8%

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

2. Final simplification 99.8%

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

Alternative 3: 89.0% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-x \leq 9.999999887266023 \cdot 10^{-27}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + \left(0.5 \cdot \frac{x}{\frac{s \cdot s}{x}} - \frac{x}{s}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- x) 9.999999887266023e-27)
   (/ 1.0 (+ 1.0 (/ 1.0 (+ 1.0 (/ x s)))))
   (/ 1.0 (+ 2.0 (- (* 0.5 (/ x (/ (* s s) x))) (/ x s))))))

C

float code(float x, float s) {
	float tmp;
	if (-x <= 9.999999887266023e-27f) {
		tmp = 1.0f / (1.0f + (1.0f / (1.0f + (x / s))));
	} else {
		tmp = 1.0f / (2.0f + ((0.5f * (x / ((s * s) / x))) - (x / s)));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (-x <= 9.999999887266023e-27) then
        tmp = 1.0e0 / (1.0e0 + (1.0e0 / (1.0e0 + (x / s))))
    else
        tmp = 1.0e0 / (2.0e0 + ((0.5e0 * (x / ((s * s) / x))) - (x / s)))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-x) <= Float32(9.999999887266023e-27))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(x / s)))));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(Float32(Float32(0.5) * Float32(x / Float32(Float32(s * s) / x))) - Float32(x / s))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (-x <= single(9.999999887266023e-27))
		tmp = single(1.0) / (single(1.0) + (single(1.0) / (single(1.0) + (x / s))));
	else
		tmp = single(1.0) / (single(2.0) + ((single(0.5) * (x / ((s * s) / x))) - (x / s)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (neg.f32 x) < 9.99999989e-27

   1. Initial program 99.9%

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

      1. distribute-frac-neg 99.9%

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

      2. exp-neg 99.9%

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

   3. Applied egg-rr 99.9%

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

   4. Taylor expanded in x around 0 93.8%

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

   if 9.99999989e-27 < (neg.f32 x)

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 83.7%

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

      1. mul-1-neg 83.7%

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

      2. unsub-neg 83.7%

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

      3. unpow2 83.7%

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

      4. unpow2 83.7%

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

      5. times-frac 75.4%

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

   4. Simplified 75.4%

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

      1. clear-num 75.4%

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

      2. frac-times 79.9%

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

      3. *-un-lft-identity 79.9%

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

   6. Applied egg-rr 79.9%

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

      1. associate-*l/ 87.3%

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

   8. Applied egg-rr 87.3%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-x \leq 9.999999887266023 \cdot 10^{-27}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + \left(0.5 \cdot \frac{x}{\frac{s \cdot s}{x}} - \frac{x}{s}\right)}\\ \end{array} \]

Alternative 4: 85.3% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq -20:\\ \;\;\;\;\frac{1}{1 + \frac{s}{x}}\\ \mathbf{elif}\;t_0 \leq 1:\\ \;\;\;\;0.5 + \frac{x}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{s}{x} \cdot \frac{s}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)))
   (if (<= t_0 -20.0)
     (/ 1.0 (+ 1.0 (/ s x)))
     (if (<= t_0 1.0) (+ 0.5 (* (/ x s) 0.25)) (* 2.0 (* (/ s x) (/ s x)))))))

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if (t_0 <= -20.0f) {
		tmp = 1.0f / (1.0f + (s / x));
	} else if (t_0 <= 1.0f) {
		tmp = 0.5f + ((x / s) * 0.25f);
	} else {
		tmp = 2.0f * ((s / x) * (s / x));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: tmp
    t_0 = -x / s
    if (t_0 <= (-20.0e0)) then
        tmp = 1.0e0 / (1.0e0 + (s / x))
    else if (t_0 <= 1.0e0) then
        tmp = 0.5e0 + ((x / s) * 0.25e0)
    else
        tmp = 2.0e0 * ((s / x) * (s / x))
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = Float32(Float32(-x) / s)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-20.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(s / x)));
	elseif (t_0 <= Float32(1.0))
		tmp = Float32(Float32(0.5) + Float32(Float32(x / s) * Float32(0.25)));
	else
		tmp = Float32(Float32(2.0) * Float32(Float32(s / x) * Float32(s / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = -x / s;
	tmp = single(0.0);
	if (t_0 <= single(-20.0))
		tmp = single(1.0) / (single(1.0) + (s / x));
	elseif (t_0 <= single(1.0))
		tmp = single(0.5) + ((x / s) * single(0.25));
	else
		tmp = single(2.0) * ((s / x) * (s / x));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if (/.f32 (neg.f32 x) s) < -20

   1. Initial program 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. exp-neg 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in x around 0 96.3%

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

   5. Taylor expanded in x around inf 96.3%

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

   if -20 < (/.f32 (neg.f32 x) s) < 1

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 95.6%

      \[\leadsto \color{blue}{0.5 + 0.25 \cdot \frac{x}{s}} \]
   3. Step-by-step derivation

      1. *-commutative 95.6%

         \[\leadsto 0.5 + \color{blue}{\frac{x}{s} \cdot 0.25} \]

   4. Simplified 95.6%

      \[\leadsto \color{blue}{0.5 + \frac{x}{s} \cdot 0.25} \]

   if 1 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 79.7%

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

      1. mul-1-neg 79.7%

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

      2. unsub-neg 79.7%

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

      3. unpow2 79.7%

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

      4. unpow2 79.7%

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

      5. times-frac 71.3%

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

   4. Simplified 71.3%

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

   5. Taylor expanded in x around inf 79.6%

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

      1. unpow2 79.6%

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

      2. unpow2 79.6%

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

   7. Simplified 79.6%

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

      1. times-frac 69.2%

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

   9. Applied egg-rr 69.2%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -20:\\ \;\;\;\;\frac{1}{1 + \frac{s}{x}}\\ \mathbf{elif}\;\frac{-x}{s} \leq 1:\\ \;\;\;\;0.5 + \frac{x}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{s}{x} \cdot \frac{s}{x}\right)\\ \end{array} \]

Alternative 5: 88.2% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq -20:\\ \;\;\;\;\frac{1}{1 + \frac{s}{x}}\\ \mathbf{elif}\;t_0 \leq 2.5:\\ \;\;\;\;0.5 + \frac{x}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{s \cdot s}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)))
   (if (<= t_0 -20.0)
     (/ 1.0 (+ 1.0 (/ s x)))
     (if (<= t_0 2.5) (+ 0.5 (* (/ x s) 0.25)) (* 2.0 (/ (* s s) (* x x)))))))

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if (t_0 <= -20.0f) {
		tmp = 1.0f / (1.0f + (s / x));
	} else if (t_0 <= 2.5f) {
		tmp = 0.5f + ((x / s) * 0.25f);
	} else {
		tmp = 2.0f * ((s * s) / (x * x));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: tmp
    t_0 = -x / s
    if (t_0 <= (-20.0e0)) then
        tmp = 1.0e0 / (1.0e0 + (s / x))
    else if (t_0 <= 2.5e0) then
        tmp = 0.5e0 + ((x / s) * 0.25e0)
    else
        tmp = 2.0e0 * ((s * s) / (x * x))
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = Float32(Float32(-x) / s)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-20.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(s / x)));
	elseif (t_0 <= Float32(2.5))
		tmp = Float32(Float32(0.5) + Float32(Float32(x / s) * Float32(0.25)));
	else
		tmp = Float32(Float32(2.0) * Float32(Float32(s * s) / Float32(x * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = -x / s;
	tmp = single(0.0);
	if (t_0 <= single(-20.0))
		tmp = single(1.0) / (single(1.0) + (s / x));
	elseif (t_0 <= single(2.5))
		tmp = single(0.5) + ((x / s) * single(0.25));
	else
		tmp = single(2.0) * ((s * s) / (x * x));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if (/.f32 (neg.f32 x) s) < -20

   1. Initial program 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. exp-neg 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in x around 0 96.3%

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

   5. Taylor expanded in x around inf 96.3%

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

   if -20 < (/.f32 (neg.f32 x) s) < 2.5

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 94.2%

      \[\leadsto \color{blue}{0.5 + 0.25 \cdot \frac{x}{s}} \]
   3. Step-by-step derivation

      1. *-commutative 94.2%

         \[\leadsto 0.5 + \color{blue}{\frac{x}{s} \cdot 0.25} \]

   4. Simplified 94.2%

      \[\leadsto \color{blue}{0.5 + \frac{x}{s} \cdot 0.25} \]

   if 2.5 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 80.5%

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

      1. mul-1-neg 80.5%

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

      2. unsub-neg 80.5%

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

      3. unpow2 80.5%

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

      4. unpow2 80.5%

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

      5. times-frac 71.7%

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

   4. Simplified 71.7%

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

   5. Taylor expanded in x around inf 80.4%

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

      1. unpow2 80.4%

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

      2. unpow2 80.4%

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

   7. Simplified 80.4%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -20:\\ \;\;\;\;\frac{1}{1 + \frac{s}{x}}\\ \mathbf{elif}\;\frac{-x}{s} \leq 2.5:\\ \;\;\;\;0.5 + \frac{x}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{s \cdot s}{x \cdot x}\\ \end{array} \]

Alternative 6: 89.7% accurate, 6.0× speedup

Math

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

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) 10.0)
   (/ 1.0 (+ 1.0 (/ 1.0 (+ 1.0 (/ x s)))))
   (/ 1.0 (+ 2.0 (* x (/ (* x 0.5) (* s s)))))))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= 10.0f) {
		tmp = 1.0f / (1.0f + (1.0f / (1.0f + (x / s))));
	} else {
		tmp = 1.0f / (2.0f + (x * ((x * 0.5f) / (s * 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 / s) <= 10.0e0) then
        tmp = 1.0e0 / (1.0e0 + (1.0e0 / (1.0e0 + (x / s))))
    else
        tmp = 1.0e0 / (2.0e0 + (x * ((x * 0.5e0) / (s * s))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(-x) / s) <= Float32(10.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(x / s)))));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(x * Float32(Float32(x * Float32(0.5)) / Float32(s * s)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((-x / s) <= single(10.0))
		tmp = single(1.0) / (single(1.0) + (single(1.0) / (single(1.0) + (x / s))));
	else
		tmp = single(1.0) / (single(2.0) + (x * ((x * single(0.5)) / (s * s))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 10

   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. Applied egg-rr 99.8%

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

   4. Taylor expanded in x around 0 93.3%

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

   if 10 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 82.7%

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

      1. mul-1-neg 82.7%

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

      2. unsub-neg 82.7%

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

      3. unpow2 82.7%

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

      4. unpow2 82.7%

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

      5. times-frac 73.2%

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

   4. Simplified 73.2%

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

      1. clear-num 73.2%

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

      2. frac-times 78.4%

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

      3. *-un-lft-identity 78.4%

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

   6. Applied egg-rr 78.4%

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

      1. associate-*l/ 86.9%

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

   8. Applied egg-rr 86.9%

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

   9. Taylor expanded in x around inf 82.7%

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

       1. unpow2 82.7%

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

       2. associate-/l* 86.9%

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

       3. unpow2 86.9%

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

       4. associate-*r/ 78.4%

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

       5. associate-*r/ 78.4%

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

       6. *-commutative 78.4%

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

       7. associate-*r/ 86.9%

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

       8. unpow2 86.9%

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

       9. associate-/r/ 86.9%

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

       10. unpow2 86.9%

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

   11. Simplified 86.9%

       \[\leadsto \frac{1}{2 + \color{blue}{\frac{x \cdot 0.5}{s \cdot s} \cdot x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.9%

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

Alternative 7: 74.9% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq -20:\\ \;\;\;\;1 - \frac{s}{x}\\ \mathbf{elif}\;t_0 \leq 1:\\ \;\;\;\;0.5 + \frac{x}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)))
   (if (<= t_0 -20.0)
     (- 1.0 (/ s x))
     (if (<= t_0 1.0) (+ 0.5 (* (/ x s) 0.25)) (/ 1.0 t_0)))))

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if (t_0 <= -20.0f) {
		tmp = 1.0f - (s / x);
	} else if (t_0 <= 1.0f) {
		tmp = 0.5f + ((x / s) * 0.25f);
	} else {
		tmp = 1.0f / t_0;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: tmp
    t_0 = -x / s
    if (t_0 <= (-20.0e0)) then
        tmp = 1.0e0 - (s / x)
    else if (t_0 <= 1.0e0) then
        tmp = 0.5e0 + ((x / s) * 0.25e0)
    else
        tmp = 1.0e0 / t_0
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = Float32(Float32(-x) / s)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-20.0))
		tmp = Float32(Float32(1.0) - Float32(s / x));
	elseif (t_0 <= Float32(1.0))
		tmp = Float32(Float32(0.5) + Float32(Float32(x / s) * Float32(0.25)));
	else
		tmp = Float32(Float32(1.0) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = -x / s;
	tmp = single(0.0);
	if (t_0 <= single(-20.0))
		tmp = single(1.0) - (s / x);
	elseif (t_0 <= single(1.0))
		tmp = single(0.5) + ((x / s) * single(0.25));
	else
		tmp = single(1.0) / t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if (/.f32 (neg.f32 x) s) < -20

   1. Initial program 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. exp-neg 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in x around 0 96.3%

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

   5. Taylor expanded in x around inf 96.3%

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

      1. +-commutative 96.3%

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

      2. mul-1-neg 96.3%

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

      3. unsub-neg 96.3%

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

   7. Simplified 96.3%

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

   if -20 < (/.f32 (neg.f32 x) s) < 1

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 95.6%

      \[\leadsto \color{blue}{0.5 + 0.25 \cdot \frac{x}{s}} \]
   3. Step-by-step derivation

      1. *-commutative 95.6%

         \[\leadsto 0.5 + \color{blue}{\frac{x}{s} \cdot 0.25} \]

   4. Simplified 95.6%

      \[\leadsto \color{blue}{0.5 + \frac{x}{s} \cdot 0.25} \]

   if 1 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 40.6%

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

      1. mul-1-neg 40.6%

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

      2. unsub-neg 40.6%

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

   4. Simplified 40.6%

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

   5. Taylor expanded in x around inf 40.6%

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

      1. neg-mul-1 40.6%

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

      2. distribute-neg-frac 40.6%

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

   7. Simplified 40.6%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -20:\\ \;\;\;\;1 - \frac{s}{x}\\ \mathbf{elif}\;\frac{-x}{s} \leq 1:\\ \;\;\;\;0.5 + \frac{x}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{-x}{s}}\\ \end{array} \]

Alternative 8: 74.9% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq -20:\\ \;\;\;\;\frac{1}{1 + \frac{s}{x}}\\ \mathbf{elif}\;t_0 \leq 1:\\ \;\;\;\;0.5 + \frac{x}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)))
   (if (<= t_0 -20.0)
     (/ 1.0 (+ 1.0 (/ s x)))
     (if (<= t_0 1.0) (+ 0.5 (* (/ x s) 0.25)) (/ 1.0 t_0)))))

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if (t_0 <= -20.0f) {
		tmp = 1.0f / (1.0f + (s / x));
	} else if (t_0 <= 1.0f) {
		tmp = 0.5f + ((x / s) * 0.25f);
	} else {
		tmp = 1.0f / t_0;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: tmp
    t_0 = -x / s
    if (t_0 <= (-20.0e0)) then
        tmp = 1.0e0 / (1.0e0 + (s / x))
    else if (t_0 <= 1.0e0) then
        tmp = 0.5e0 + ((x / s) * 0.25e0)
    else
        tmp = 1.0e0 / t_0
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = Float32(Float32(-x) / s)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-20.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(s / x)));
	elseif (t_0 <= Float32(1.0))
		tmp = Float32(Float32(0.5) + Float32(Float32(x / s) * Float32(0.25)));
	else
		tmp = Float32(Float32(1.0) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = -x / s;
	tmp = single(0.0);
	if (t_0 <= single(-20.0))
		tmp = single(1.0) / (single(1.0) + (s / x));
	elseif (t_0 <= single(1.0))
		tmp = single(0.5) + ((x / s) * single(0.25));
	else
		tmp = single(1.0) / t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if (/.f32 (neg.f32 x) s) < -20

   1. Initial program 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. exp-neg 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in x around 0 96.3%

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

   5. Taylor expanded in x around inf 96.3%

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

   if -20 < (/.f32 (neg.f32 x) s) < 1

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 95.6%

      \[\leadsto \color{blue}{0.5 + 0.25 \cdot \frac{x}{s}} \]
   3. Step-by-step derivation

      1. *-commutative 95.6%

         \[\leadsto 0.5 + \color{blue}{\frac{x}{s} \cdot 0.25} \]

   4. Simplified 95.6%

      \[\leadsto \color{blue}{0.5 + \frac{x}{s} \cdot 0.25} \]

   if 1 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 40.6%

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

      1. mul-1-neg 40.6%

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

      2. unsub-neg 40.6%

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

   4. Simplified 40.6%

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

   5. Taylor expanded in x around inf 40.6%

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

      1. neg-mul-1 40.6%

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

      2. distribute-neg-frac 40.6%

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

   7. Simplified 40.6%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -20:\\ \;\;\;\;\frac{1}{1 + \frac{s}{x}}\\ \mathbf{elif}\;\frac{-x}{s} \leq 1:\\ \;\;\;\;0.5 + \frac{x}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{-x}{s}}\\ \end{array} \]

Alternative 9: 72.8% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq -2:\\ \;\;\;\;1 - \frac{s}{x}\\ \mathbf{elif}\;t_0 \leq 1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)))
   (if (<= t_0 -2.0) (- 1.0 (/ s x)) (if (<= t_0 1.0) 0.5 (/ 1.0 t_0)))))

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if (t_0 <= -2.0f) {
		tmp = 1.0f - (s / x);
	} else if (t_0 <= 1.0f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / t_0;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: tmp
    t_0 = -x / s
    if (t_0 <= (-2.0e0)) then
        tmp = 1.0e0 - (s / x)
    else if (t_0 <= 1.0e0) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 / t_0
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = Float32(Float32(-x) / s)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-2.0))
		tmp = Float32(Float32(1.0) - Float32(s / x));
	elseif (t_0 <= Float32(1.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = -x / s;
	tmp = single(0.0);
	if (t_0 <= single(-2.0))
		tmp = single(1.0) - (s / x);
	elseif (t_0 <= single(1.0))
		tmp = single(0.5);
	else
		tmp = single(1.0) / t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if (/.f32 (neg.f32 x) s) < -2

   1. Initial program 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. exp-neg 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in x around 0 95.7%

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

   5. Taylor expanded in x around inf 95.6%

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

      1. +-commutative 95.6%

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

      2. mul-1-neg 95.6%

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

      3. unsub-neg 95.6%

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

   7. Simplified 95.6%

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

   if -2 < (/.f32 (neg.f32 x) s) < 1

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 89.6%

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

   if 1 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 40.6%

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

      1. mul-1-neg 40.6%

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

      2. unsub-neg 40.6%

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

   4. Simplified 40.6%

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

   5. Taylor expanded in x around inf 40.6%

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

      1. neg-mul-1 40.6%

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

      2. distribute-neg-frac 40.6%

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

   7. Simplified 40.6%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -2:\\ \;\;\;\;1 - \frac{s}{x}\\ \mathbf{elif}\;\frac{-x}{s} \leq 1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{-x}{s}}\\ \end{array} \]

Alternative 10: 87.4% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 20000:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{s \cdot s}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) 20000.0)
   (/ 1.0 (+ 1.0 (/ 1.0 (+ 1.0 (/ x s)))))
   (* 2.0 (/ (* s s) (* x x)))))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= 20000.0f) {
		tmp = 1.0f / (1.0f + (1.0f / (1.0f + (x / s))));
	} else {
		tmp = 2.0f * ((s * s) / (x * x));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((-x / s) <= 20000.0e0) then
        tmp = 1.0e0 / (1.0e0 + (1.0e0 / (1.0e0 + (x / s))))
    else
        tmp = 2.0e0 * ((s * s) / (x * x))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(-x) / s) <= Float32(20000.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(x / s)))));
	else
		tmp = Float32(Float32(2.0) * Float32(Float32(s * s) / Float32(x * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((-x / s) <= single(20000.0))
		tmp = single(1.0) / (single(1.0) + (single(1.0) / (single(1.0) + (x / s))));
	else
		tmp = single(2.0) * ((s * s) / (x * x));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 2e4

   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. Applied egg-rr 99.8%

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

   4. Taylor expanded in x around 0 90.7%

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

   if 2e4 < (/.f32 (neg.f32 x) s)

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 87.0%

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

      1. mul-1-neg 87.0%

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

      2. unsub-neg 87.0%

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

      3. unpow2 87.0%

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

      4. unpow2 87.0%

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

      5. times-frac 76.8%

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

   4. Simplified 76.8%

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

   5. Taylor expanded in x around inf 87.0%

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

      1. unpow2 87.0%

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

      2. unpow2 87.0%

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

   7. Simplified 87.0%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 20000:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{s \cdot s}{x \cdot x}\\ \end{array} \]

Alternative 11: 74.2% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -2:\\ \;\;\;\;\frac{1}{1 + \frac{s}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) -2.0) (/ 1.0 (+ 1.0 (/ s x))) (/ 1.0 (- 2.0 (/ x s)))))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= -2.0f) {
		tmp = 1.0f / (1.0f + (s / x));
	} else {
		tmp = 1.0f / (2.0f - (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 / s) <= (-2.0e0)) then
        tmp = 1.0e0 / (1.0e0 + (s / x))
    else
        tmp = 1.0e0 / (2.0e0 - (x / s))
    end if
    code = tmp
end function

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < -2

   1. Initial program 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. exp-neg 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in x around 0 95.7%

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

   5. Taylor expanded in x around inf 95.6%

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

   if -2 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 61.3%

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

      1. mul-1-neg 61.3%

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

      2. unsub-neg 61.3%

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

   4. Simplified 61.3%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -2:\\ \;\;\;\;\frac{1}{1 + \frac{s}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \end{array} \]

Alternative 12: 67.5% accurate, 11.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.99999993922529 \cdot 10^{-9}:\\ \;\;\;\;\frac{-s}{x}\\ \mathbf{elif}\;x \leq 2.0000000072549875 \cdot 10^{-15}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{s}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -9.99999993922529e-9)
   (/ (- s) x)
   (if (<= x 2.0000000072549875e-15) 0.5 (- 1.0 (/ s x)))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -9.99999993922529e-9f) {
		tmp = -s / x;
	} else if (x <= 2.0000000072549875e-15f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f - (s / x);
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-9.99999993922529e-9)) then
        tmp = -s / x
    else if (x <= 2.0000000072549875e-15) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 - (s / x)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-9.99999993922529e-9))
		tmp = Float32(Float32(-s) / x);
	elseif (x <= Float32(2.0000000072549875e-15))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) - Float32(s / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-9.99999993922529e-9))
		tmp = -s / x;
	elseif (x <= single(2.0000000072549875e-15))
		tmp = single(0.5);
	else
		tmp = single(1.0) - (s / x);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -9.99999994e-9

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 50.1%

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

      1. mul-1-neg 50.1%

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

      2. unsub-neg 50.1%

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

   4. Simplified 50.1%

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

   5. Taylor expanded in x around inf 46.3%

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

      1. mul-1-neg 46.3%

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

      2. distribute-neg-frac 46.3%

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

   7. Simplified 46.3%

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

   if -9.99999994e-9 < x < 2.00000001e-15

   1. Initial program 99.5%

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

   2. Taylor expanded in x around 0 63.8%

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

   if 2.00000001e-15 < x

   1. Initial program 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. exp-neg 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in x around 0 97.0%

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

   5. Taylor expanded in x around inf 92.5%

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

      1. +-commutative 92.5%

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

      2. mul-1-neg 92.5%

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

      3. unsub-neg 92.5%

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

   7. Simplified 92.5%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.99999993922529 \cdot 10^{-9}:\\ \;\;\;\;\frac{-s}{x}\\ \mathbf{elif}\;x \leq 2.0000000072549875 \cdot 10^{-15}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{s}{x}\\ \end{array} \]

Alternative 13: 46.1% accurate, 17.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.99999993922529 \cdot 10^{-9}:\\ \;\;\;\;\frac{-s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -9.99999993922529e-9) (/ (- s) x) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -9.99999993922529e-9f) {
		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 <= (-9.99999993922529e-9)) 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(-9.99999993922529e-9))
		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(-9.99999993922529e-9))
		tmp = -s / x;
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -9.99999994e-9

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 50.1%

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

      1. mul-1-neg 50.1%

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

      2. unsub-neg 50.1%

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

   4. Simplified 50.1%

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

   5. Taylor expanded in x around inf 46.3%

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

      1. mul-1-neg 46.3%

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

      2. distribute-neg-frac 46.3%

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

   7. Simplified 46.3%

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

   if -9.99999994e-9 < 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 46.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.99999993922529 \cdot 10^{-9}:\\ \;\;\;\;\frac{-s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 14: 34.8% 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 34.6%

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

3. Final simplification 34.6%

   \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023228 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))