Logistic function

Percentage Accurate: 99.8% → 99.8%

Time: 8.8s

Alternatives: 11

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

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

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. Add Preprocessing

3. Final simplification 99.8%

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

Alternative 2: 56.8% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-x \leq 1.5999999674579037 \cdot 10^{-30}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;-x \leq 9:\\ \;\;\;\;\frac{1}{x \cdot \left(\left(s \cdot 2 - x\right) \cdot \frac{1}{x \cdot s}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x \cdot \left(2 - \frac{x}{s}\right)}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- x) 1.5999999674579037e-30)
   0.5
   (if (<= (- x) 9.0)
     (/ 1.0 (* x (* (- (* s 2.0) x) (/ 1.0 (* x s)))))
     (/ 1.0 (/ (* x (- 2.0 (/ x s))) x)))))

C

float code(float x, float s) {
	float tmp;
	if (-x <= 1.5999999674579037e-30f) {
		tmp = 0.5f;
	} else if (-x <= 9.0f) {
		tmp = 1.0f / (x * (((s * 2.0f) - x) * (1.0f / (x * s))));
	} else {
		tmp = 1.0f / ((x * (2.0f - (x / 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 <= 1.5999999674579037e-30) then
        tmp = 0.5e0
    else if (-x <= 9.0e0) then
        tmp = 1.0e0 / (x * (((s * 2.0e0) - x) * (1.0e0 / (x * s))))
    else
        tmp = 1.0e0 / ((x * (2.0e0 - (x / s))) / x)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-x) <= Float32(1.5999999674579037e-30))
		tmp = Float32(0.5);
	elseif (Float32(-x) <= Float32(9.0))
		tmp = Float32(Float32(1.0) / Float32(x * Float32(Float32(Float32(s * Float32(2.0)) - x) * Float32(Float32(1.0) / Float32(x * s)))));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(x * Float32(Float32(2.0) - Float32(x / s))) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (-x <= single(1.5999999674579037e-30))
		tmp = single(0.5);
	elseif (-x <= single(9.0))
		tmp = single(1.0) / (x * (((s * single(2.0)) - x) * (single(1.0) / (x * s))));
	else
		tmp = single(1.0) / ((x * (single(2.0) - (x / s))) / x);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if (neg.f32 x) < 1.59999997e-30

   1. Initial program 99.8%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 49.0%

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

   if 1.59999997e-30 < (neg.f32 x) < 9

   1. Initial program 99.5%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 41.9%

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

      1. mul-1-neg 41.9%

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

      2. unsub-neg 41.9%

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

   5. Simplified 41.9%

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

   6. Taylor expanded in x around inf 41.8%

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

      1. associate-*r/ 41.8%

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

      2. metadata-eval 41.8%

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

   8. Simplified 41.8%

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

      1. frac-sub 57.1%

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

      2. div-inv 64.0%

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

      3. *-rgt-identity 64.0%

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

   10. Applied egg-rr 64.0%

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

   if 9 < (neg.f32 x)

   1. Initial program 100.0%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 52.4%

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

      1. mul-1-neg 52.4%

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

      2. unsub-neg 52.4%

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

   5. Simplified 52.4%

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

   6. Taylor expanded in x around inf 52.4%

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

      1. associate-*r/ 52.4%

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

      2. metadata-eval 52.4%

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

   8. Simplified 52.4%

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

   9. Taylor expanded in x around 0 52.4%

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

       1. mul-1-neg 52.4%

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

       2. unsub-neg 52.4%

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

   11. Simplified 52.4%

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

       1. associate-*r/ 72.3%

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

   13. Applied egg-rr 72.3%

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

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-x \leq 1.5999999674579037 \cdot 10^{-30}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;-x \leq 9:\\ \;\;\;\;\frac{1}{x \cdot \left(\left(s \cdot 2 - x\right) \cdot \frac{1}{x \cdot s}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x \cdot \left(2 - \frac{x}{s}\right)}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 50.3% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-x \leq 4.999999999099794 \cdot 10^{-24}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;s \cdot \frac{x}{x \cdot \left(s \cdot 2 - x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- x) 4.999999999099794e-24) 0.5 (* s (/ x (* x (- (* s 2.0) x))))))

C

float code(float x, float s) {
	float tmp;
	if (-x <= 4.999999999099794e-24f) {
		tmp = 0.5f;
	} else {
		tmp = s * (x / (x * ((s * 2.0f) - 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 <= 4.999999999099794e-24) then
        tmp = 0.5e0
    else
        tmp = s * (x / (x * ((s * 2.0e0) - x)))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-x) <= Float32(4.999999999099794e-24))
		tmp = Float32(0.5);
	else
		tmp = Float32(s * Float32(x / Float32(x * Float32(Float32(s * Float32(2.0)) - x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (-x <= single(4.999999999099794e-24))
		tmp = single(0.5);
	else
		tmp = s * (x / (x * ((s * single(2.0)) - x)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (neg.f32 x) < 5e-24

   1. Initial program 99.8%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 49.0%

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

   if 5e-24 < (neg.f32 x)

   1. Initial program 99.8%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 47.8%

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

      1. mul-1-neg 47.8%

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

      2. unsub-neg 47.8%

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

   5. Simplified 47.8%

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

   6. Taylor expanded in x around inf 47.8%

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

      1. associate-*r/ 47.8%

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

      2. metadata-eval 47.8%

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

   8. Simplified 47.8%

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

      1. associate-/r* 44.4%

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

      2. frac-sub 47.1%

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

      3. associate-/r/ 49.2%

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

      4. *-rgt-identity 49.2%

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

   10. Applied egg-rr 49.2%

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

       1. associate-*r* 47.3%

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

       2. *-commutative 47.3%

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

       3. associate-/l/ 50.6%

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

       4. associate-*l/ 51.5%

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

       5. *-commutative 51.5%

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

       6. *-rgt-identity 51.5%

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

       7. *-commutative 51.5%

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

   12. Simplified 51.5%

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

4. Final simplification 49.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-x \leq 4.999999999099794 \cdot 10^{-24}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;s \cdot \frac{x}{x \cdot \left(s \cdot 2 - x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 54.2% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-x \leq -1.9999999593223797 \cdot 10^{-31}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{x \cdot \left(2 - \frac{x}{s}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- x) -1.9999999593223797e-31)
   0.5
   (* x (/ 1.0 (* x (- 2.0 (/ x s)))))))

C

float code(float x, float s) {
	float tmp;
	if (-x <= -1.9999999593223797e-31f) {
		tmp = 0.5f;
	} else {
		tmp = x * (1.0f / (x * (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 <= (-1.9999999593223797e-31)) then
        tmp = 0.5e0
    else
        tmp = x * (1.0e0 / (x * (2.0e0 - (x / s))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-x) <= Float32(-1.9999999593223797e-31))
		tmp = Float32(0.5);
	else
		tmp = Float32(x * Float32(Float32(1.0) / Float32(x * Float32(Float32(2.0) - Float32(x / s)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (-x <= single(-1.9999999593223797e-31))
		tmp = single(0.5);
	else
		tmp = x * (single(1.0) / (x * (single(2.0) - (x / s))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (neg.f32 x) < -1.99999996e-31

   1. Initial program 99.9%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 38.7%

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

   if -1.99999996e-31 < (neg.f32 x)

   1. Initial program 99.7%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 57.8%

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

      1. mul-1-neg 57.8%

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

      2. unsub-neg 57.8%

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

   5. Simplified 57.8%

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

   6. Taylor expanded in x around inf 57.6%

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

      1. associate-*r/ 57.6%

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

      2. metadata-eval 57.6%

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

   8. Simplified 57.6%

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

   9. Taylor expanded in x around 0 57.6%

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

       1. mul-1-neg 57.6%

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

       2. unsub-neg 57.6%

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

   11. Simplified 57.6%

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

       1. associate-*r/ 66.7%

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

       2. associate-/r/ 66.5%

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

   13. Applied egg-rr 66.5%

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

4. Final simplification 53.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-x \leq -1.9999999593223797 \cdot 10^{-31}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{x \cdot \left(2 - \frac{x}{s}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 54.3% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-x \leq -1.9999999593223797 \cdot 10^{-31}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x \cdot \left(2 - \frac{x}{s}\right)}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- x) -1.9999999593223797e-31)
   0.5
   (/ 1.0 (/ (* x (- 2.0 (/ x s))) x))))

C

float code(float x, float s) {
	float tmp;
	if (-x <= -1.9999999593223797e-31f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / ((x * (2.0f - (x / 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 <= (-1.9999999593223797e-31)) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 / ((x * (2.0e0 - (x / s))) / x)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-x) <= Float32(-1.9999999593223797e-31))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(x * Float32(Float32(2.0) - Float32(x / s))) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (-x <= single(-1.9999999593223797e-31))
		tmp = single(0.5);
	else
		tmp = single(1.0) / ((x * (single(2.0) - (x / s))) / x);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (neg.f32 x) < -1.99999996e-31

   1. Initial program 99.9%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 38.7%

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

   if -1.99999996e-31 < (neg.f32 x)

   1. Initial program 99.7%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 57.8%

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

      1. mul-1-neg 57.8%

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

      2. unsub-neg 57.8%

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

   5. Simplified 57.8%

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

   6. Taylor expanded in x around inf 57.6%

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

      1. associate-*r/ 57.6%

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

      2. metadata-eval 57.6%

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

   8. Simplified 57.6%

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

   9. Taylor expanded in x around 0 57.6%

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

       1. mul-1-neg 57.6%

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

       2. unsub-neg 57.6%

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

   11. Simplified 57.6%

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

       1. associate-*r/ 66.7%

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

   13. Applied egg-rr 66.7%

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

4. Final simplification 53.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-x \leq -1.9999999593223797 \cdot 10^{-31}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x \cdot \left(2 - \frac{x}{s}\right)}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 48.7% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-x \leq -2.0000000063421537 \cdot 10^{-28}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{s \cdot 2 - x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- x) -2.0000000063421537e-28) 0.5 (/ 1.0 (/ (- (* s 2.0) x) s))))

C

float code(float x, float s) {
	float tmp;
	if (-x <= -2.0000000063421537e-28f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / (((s * 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 <= (-2.0000000063421537e-28)) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 / (((s * 2.0e0) - x) / s)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-x) <= Float32(-2.0000000063421537e-28))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(s * Float32(2.0)) - x) / s));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (-x <= single(-2.0000000063421537e-28))
		tmp = single(0.5);
	else
		tmp = single(1.0) / (((s * single(2.0)) - x) / s);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (neg.f32 x) < -2.00000001e-28

   1. Initial program 99.9%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 37.1%

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

   if -2.00000001e-28 < (neg.f32 x)

   1. Initial program 99.8%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 58.7%

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

      1. mul-1-neg 58.7%

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

      2. unsub-neg 58.7%

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

   5. Simplified 58.7%

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

   6. Taylor expanded in s around 0 58.7%

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

      1. *-commutative 58.7%

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

   8. Simplified 58.7%

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

4. Final simplification 48.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-x \leq -2.0000000063421537 \cdot 10^{-28}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{s \cdot 2 - x}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 48.8% accurate, 8.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-x \leq -1.9999999593223797 \cdot 10^{-31}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- x) -1.9999999593223797e-31) 0.5 (/ 1.0 (- 2.0 (/ x s)))))

C

float code(float x, float s) {
	float tmp;
	if (-x <= -1.9999999593223797e-31f) {
		tmp = 0.5f;
	} 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 <= (-1.9999999593223797e-31)) then
        tmp = 0.5e0
    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(-x) <= Float32(-1.9999999593223797e-31))
		tmp = Float32(0.5);
	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 <= single(-1.9999999593223797e-31))
		tmp = single(0.5);
	else
		tmp = single(1.0) / (single(2.0) - (x / s));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (neg.f32 x) < -1.99999996e-31

   1. Initial program 99.9%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 38.7%

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

   if -1.99999996e-31 < (neg.f32 x)

   1. Initial program 99.7%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 57.8%

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

      1. mul-1-neg 57.8%

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

      2. unsub-neg 57.8%

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

   5. Simplified 57.8%

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

4. Final simplification 48.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-x \leq -1.9999999593223797 \cdot 10^{-31}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 46.3% accurate, 9.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-x \leq 9.999999747378752 \cdot 10^{-6}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;s \cdot \frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- x) 9.999999747378752e-6) 0.5 (* s (/ 1.0 x))))

C

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

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (-x <= 9.999999747378752e-6) then
        tmp = 0.5e0
    else
        tmp = s * (1.0e0 / x)
    end if
    code = tmp
end function

Julia

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

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (-x <= single(9.999999747378752e-6))
		tmp = single(0.5);
	else
		tmp = s * (single(1.0) / x);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.7%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 47.4%

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

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

   1. Initial program 100.0%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 49.3%

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

      1. mul-1-neg 49.3%

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

      2. unsub-neg 49.3%

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

   5. Simplified 49.3%

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

   6. Taylor expanded in x around inf 49.3%

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

      1. neg-mul-1 49.3%

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

      2. distribute-neg-frac2 49.3%

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

   8. Simplified 49.3%

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

      1. associate-/r/ 44.6%

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

      2. add-sqr-sqrt -0.0%

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

      3. sqrt-unprod 57.8%

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

      4. sqr-neg 57.8%

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

      5. sqrt-unprod 44.6%

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

      6. add-sqr-sqrt 44.6%

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

   10. Applied egg-rr 44.6%

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

4. Final simplification 46.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-x \leq 9.999999747378752 \cdot 10^{-6}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;s \cdot \frac{1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 47.7% accurate, 9.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-x \leq 9.999999747378752 \cdot 10^{-6}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- x) 9.999999747378752e-6) 0.5 (/ -1.0 (/ x s))))

C

float code(float x, float s) {
	float tmp;
	if (-x <= 9.999999747378752e-6f) {
		tmp = 0.5f;
	} else {
		tmp = -1.0f / (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 <= 9.999999747378752e-6) then
        tmp = 0.5e0
    else
        tmp = (-1.0e0) / (x / s)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-x) <= Float32(9.999999747378752e-6))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(-1.0) / Float32(x / s));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (-x <= single(9.999999747378752e-6))
		tmp = single(0.5);
	else
		tmp = single(-1.0) / (x / s);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.7%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 47.4%

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

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

   1. Initial program 100.0%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 49.3%

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

      1. mul-1-neg 49.3%

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

      2. unsub-neg 49.3%

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

   5. Simplified 49.3%

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

   6. Taylor expanded in x around inf 49.3%

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

      1. neg-mul-1 49.3%

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

      2. distribute-neg-frac2 49.3%

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

   8. Simplified 49.3%

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

4. Final simplification 47.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-x \leq 9.999999747378752 \cdot 10^{-6}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{x}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 46.4% accurate, 13.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.00019999999494757503:\\ \;\;\;\;\frac{s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -0.00019999999494757503) (/ s x) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -0.00019999999494757503f) {
		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 <= (-0.00019999999494757503e0)) 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(-0.00019999999494757503))
		tmp = 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(-0.00019999999494757503))
		tmp = s / x;
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -1.99999995e-4

   1. Initial program 100.0%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 49.3%

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

      1. mul-1-neg 49.3%

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

      2. unsub-neg 49.3%

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

   5. Simplified 49.3%

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

   6. Taylor expanded in x around inf 49.3%

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

      1. neg-mul-1 49.3%

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

      2. distribute-neg-frac2 49.3%

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

   8. Simplified 49.3%

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

      1. clear-num 44.6%

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

      2. add-sqr-sqrt -0.0%

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

      3. sqrt-unprod 57.8%

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

      4. sqr-neg 57.8%

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

      5. sqrt-unprod 44.6%

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

      6. add-sqr-sqrt 44.6%

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

      7. *-un-lft-identity 44.6%

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

   10. Applied egg-rr 44.6%

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

       1. *-lft-identity 44.6%

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

   12. Simplified 44.6%

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

   if -1.99999995e-4 < x

   1. Initial program 99.7%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 47.4%

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

4. Final simplification 46.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00019999999494757503:\\ \;\;\;\;\frac{s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 35.1% 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. Add Preprocessing

3. Taylor expanded in x around 0 36.7%

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

4. Final simplification 36.7%

   \[\leadsto 0.5 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024112 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))