Logistic function

Percentage Accurate: 99.8% → 99.8%

Time: 7.6s

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: 99.8% 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(x / Float32(-s)))))
end

MATLAB

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

Derivation

1. Initial program 99.9%

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

3. Final simplification 99.9%

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

Alternative 2: 59.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{2}{x}\\ \mathbf{if}\;-x \leq 2.0000000390829628 \cdot 10^{-24}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;-x \leq 6.000000212225132 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{x \cdot \left(\left(s \cdot 2 - x\right) \cdot \left(\frac{1}{x} \cdot \frac{1}{s}\right)\right)}\\ \mathbf{elif}\;-x \leq 19999999655936:\\ \;\;\;\;\frac{1}{\frac{t\_0 \cdot t\_0 - \frac{x}{s} \cdot \frac{x}{s}}{t\_0 - \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{x \cdot \left(2 - \frac{x}{s}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (* x (/ 2.0 x))))
   (if (<= (- x) 2.0000000390829628e-24)
     0.5
     (if (<= (- x) 6.000000212225132e-7)
       (/ 1.0 (* x (* (- (* s 2.0) x) (* (/ 1.0 x) (/ 1.0 s)))))
       (if (<= (- x) 19999999655936.0)
         (/ 1.0 (/ (- (* t_0 t_0) (* (/ x s) (/ x s))) (- t_0 (/ x s))))
         (* x (/ 1.0 (* x (- 2.0 (/ x s))))))))))

C

float code(float x, float s) {
	float t_0 = x * (2.0f / x);
	float tmp;
	if (-x <= 2.0000000390829628e-24f) {
		tmp = 0.5f;
	} else if (-x <= 6.000000212225132e-7f) {
		tmp = 1.0f / (x * (((s * 2.0f) - x) * ((1.0f / x) * (1.0f / s))));
	} else if (-x <= 19999999655936.0f) {
		tmp = 1.0f / (((t_0 * t_0) - ((x / s) * (x / s))) / (t_0 - (x / s)));
	} 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) :: t_0
    real(4) :: tmp
    t_0 = x * (2.0e0 / x)
    if (-x <= 2.0000000390829628e-24) then
        tmp = 0.5e0
    else if (-x <= 6.000000212225132e-7) then
        tmp = 1.0e0 / (x * (((s * 2.0e0) - x) * ((1.0e0 / x) * (1.0e0 / s))))
    else if (-x <= 19999999655936.0e0) then
        tmp = 1.0e0 / (((t_0 * t_0) - ((x / s) * (x / s))) / (t_0 - (x / s)))
    else
        tmp = x * (1.0e0 / (x * (2.0e0 - (x / s))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = Float32(x * Float32(Float32(2.0) / x))
	tmp = Float32(0.0)
	if (Float32(-x) <= Float32(2.0000000390829628e-24))
		tmp = Float32(0.5);
	elseif (Float32(-x) <= Float32(6.000000212225132e-7))
		tmp = Float32(Float32(1.0) / Float32(x * Float32(Float32(Float32(s * Float32(2.0)) - x) * Float32(Float32(Float32(1.0) / x) * Float32(Float32(1.0) / s)))));
	elseif (Float32(-x) <= Float32(19999999655936.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(t_0 * t_0) - Float32(Float32(x / s) * Float32(x / s))) / Float32(t_0 - Float32(x / s))));
	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)
	t_0 = x * (single(2.0) / x);
	tmp = single(0.0);
	if (-x <= single(2.0000000390829628e-24))
		tmp = single(0.5);
	elseif (-x <= single(6.000000212225132e-7))
		tmp = single(1.0) / (x * (((s * single(2.0)) - x) * ((single(1.0) / x) * (single(1.0) / s))));
	elseif (-x <= single(19999999655936.0))
		tmp = single(1.0) / (((t_0 * t_0) - ((x / s) * (x / s))) / (t_0 - (x / s)));
	else
		tmp = x * (single(1.0) / (x * (single(2.0) - (x / s))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 4 regimes

2. if (neg.f32 x) < 2.00000004e-24

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 50.7%

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

   if 2.00000004e-24 < (neg.f32 x) < 6.0000002e-7

   1. Initial program 99.1%

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

   3. Taylor expanded in x around 0 35.5%

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

      1. neg-mul-1 35.5%

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

      2. unsub-neg 35.5%

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

   5. Simplified 35.5%

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

   6. Taylor expanded in x around inf 35.5%

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

      1. sub-neg 35.5%

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

      2. associate-*r/ 35.5%

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

      3. metadata-eval 35.5%

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

      4. distribute-neg-frac 35.5%

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

      5. metadata-eval 35.5%

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

   8. Simplified 35.5%

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

      1. frac-add 60.1%

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

      2. associate-/r* 35.5%

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

      3. *-commutative 35.5%

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

      4. neg-mul-1 35.5%

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

      5. fma-define 35.5%

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

      6. fma-neg 35.5%

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

   10. Applied egg-rr 35.5%

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

       1. div-inv 35.5%

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

       2. div-inv 35.5%

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

       3. associate-*l* 69.1%

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

       4. *-commutative 69.1%

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

   12. Applied egg-rr 69.1%

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

   if 6.0000002e-7 < (neg.f32 x) < 1.99999997e13

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 13.8%

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

      1. neg-mul-1 13.8%

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

      2. unsub-neg 13.8%

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

   5. Simplified 13.8%

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

   6. Taylor expanded in x around inf 13.8%

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

      1. sub-neg 13.8%

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

      2. associate-*r/ 13.8%

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

      3. metadata-eval 13.8%

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

      4. distribute-neg-frac 13.8%

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

      5. metadata-eval 13.8%

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

   8. Simplified 13.8%

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

      1. distribute-lft-in 13.8%

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

      2. frac-2neg 13.8%

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

      3. metadata-eval 13.8%

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

      4. div-inv 13.8%

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

      5. flip-+ 52.6%

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

   10. Applied egg-rr 52.6%

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

   if 1.99999997e13 < (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 68.9%

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

      1. neg-mul-1 68.9%

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

      2. unsub-neg 68.9%

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

   5. Simplified 68.9%

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

   6. Taylor expanded in x around inf 68.9%

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

      1. sub-neg 68.9%

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

      2. associate-*r/ 68.9%

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

      3. metadata-eval 68.9%

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

      4. distribute-neg-frac 68.9%

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

      5. metadata-eval 68.9%

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

   8. Simplified 68.9%

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

   9. Taylor expanded in x around 0 68.9%

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

       1. mul-1-neg 68.9%

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

       2. sub-neg 68.9%

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

   11. Simplified 68.9%

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

       1. associate-*r/ 97.8%

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

       2. associate-/r/ 97.8%

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

   13. Applied egg-rr 97.8%

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

4. Final simplification 60.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-x \leq 2.0000000390829628 \cdot 10^{-24}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;-x \leq 6.000000212225132 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{x \cdot \left(\left(s \cdot 2 - x\right) \cdot \left(\frac{1}{x} \cdot \frac{1}{s}\right)\right)}\\ \mathbf{elif}\;-x \leq 19999999655936:\\ \;\;\;\;\frac{1}{\frac{\left(x \cdot \frac{2}{x}\right) \cdot \left(x \cdot \frac{2}{x}\right) - \frac{x}{s} \cdot \frac{x}{s}}{x \cdot \frac{2}{x} - \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{x \cdot \left(2 - \frac{x}{s}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 56.9% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-x \leq 2.0000000390829628 \cdot 10^{-24}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;-x \leq 0.0044999998062849045:\\ \;\;\;\;\frac{1}{x \cdot \left(\left(s \cdot 2 - x\right) \cdot \left(\frac{1}{x} \cdot \frac{1}{s}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{x \cdot \left(2 - \frac{x}{s}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- x) 2.0000000390829628e-24)
   0.5
   (if (<= (- x) 0.0044999998062849045)
     (/ 1.0 (* x (* (- (* s 2.0) x) (* (/ 1.0 x) (/ 1.0 s)))))
     (* x (/ 1.0 (* x (- 2.0 (/ x s))))))))

C

float code(float x, float s) {
	float tmp;
	if (-x <= 2.0000000390829628e-24f) {
		tmp = 0.5f;
	} else if (-x <= 0.0044999998062849045f) {
		tmp = 1.0f / (x * (((s * 2.0f) - x) * ((1.0f / x) * (1.0f / s))));
	} 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 <= 2.0000000390829628e-24) then
        tmp = 0.5e0
    else if (-x <= 0.0044999998062849045e0) then
        tmp = 1.0e0 / (x * (((s * 2.0e0) - x) * ((1.0e0 / x) * (1.0e0 / s))))
    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(2.0000000390829628e-24))
		tmp = Float32(0.5);
	elseif (Float32(-x) <= Float32(0.0044999998062849045))
		tmp = Float32(Float32(1.0) / Float32(x * Float32(Float32(Float32(s * Float32(2.0)) - x) * Float32(Float32(Float32(1.0) / x) * Float32(Float32(1.0) / s)))));
	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(2.0000000390829628e-24))
		tmp = single(0.5);
	elseif (-x <= single(0.0044999998062849045))
		tmp = single(1.0) / (x * (((s * single(2.0)) - x) * ((single(1.0) / x) * (single(1.0) / s))));
	else
		tmp = x * (single(1.0) / (x * (single(2.0) - (x / s))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if (neg.f32 x) < 2.00000004e-24

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 50.7%

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

   if 2.00000004e-24 < (neg.f32 x) < 0.00449999981

   1. Initial program 99.3%

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

   3. Taylor expanded in x around 0 31.7%

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

      1. neg-mul-1 31.7%

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

      2. unsub-neg 31.7%

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

   5. Simplified 31.7%

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

   6. Taylor expanded in x around inf 31.7%

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

      1. sub-neg 31.7%

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

      2. associate-*r/ 31.7%

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

      3. metadata-eval 31.7%

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

      4. distribute-neg-frac 31.7%

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

      5. metadata-eval 31.7%

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

   8. Simplified 31.7%

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

      1. frac-add 52.7%

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

      2. associate-/r* 31.7%

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

      3. *-commutative 31.7%

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

      4. neg-mul-1 31.7%

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

      5. fma-define 31.7%

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

      6. fma-neg 31.7%

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

   10. Applied egg-rr 31.7%

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

       1. div-inv 31.7%

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

       2. div-inv 31.7%

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

       3. associate-*l* 62.9%

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

       4. *-commutative 62.9%

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

   12. Applied egg-rr 62.9%

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

   if 0.00449999981 < (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 45.9%

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

      1. neg-mul-1 45.9%

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

      2. unsub-neg 45.9%

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

   5. Simplified 45.9%

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

   6. Taylor expanded in x around inf 45.9%

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

      1. sub-neg 45.9%

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

      2. associate-*r/ 45.9%

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

      3. metadata-eval 45.9%

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

      4. distribute-neg-frac 45.9%

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

      5. metadata-eval 45.9%

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

   8. Simplified 45.9%

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

   9. Taylor expanded in x around 0 45.9%

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

       1. mul-1-neg 45.9%

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

       2. sub-neg 45.9%

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

   11. Simplified 45.9%

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

       1. associate-*r/ 68.4%

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

       2. associate-/r/ 68.4%

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

   13. Applied egg-rr 68.4%

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

4. Final simplification 57.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-x \leq 2.0000000390829628 \cdot 10^{-24}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;-x \leq 0.0044999998062849045:\\ \;\;\;\;\frac{1}{x \cdot \left(\left(s \cdot 2 - x\right) \cdot \left(\frac{1}{x} \cdot \frac{1}{s}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{x \cdot \left(2 - \frac{x}{s}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 56.9% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-x \leq 2.0000000390829628 \cdot 10^{-24}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;-x \leq 0.0044999998062849045:\\ \;\;\;\;\frac{1}{x \cdot \left(\left(s \cdot 2 - x\right) \cdot \frac{\frac{1}{x}}{s}\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{x \cdot \left(2 - \frac{x}{s}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- x) 2.0000000390829628e-24)
   0.5
   (if (<= (- x) 0.0044999998062849045)
     (/ 1.0 (* x (* (- (* s 2.0) x) (/ (/ 1.0 x) s))))
     (* x (/ 1.0 (* x (- 2.0 (/ x s))))))))

C

float code(float x, float s) {
	float tmp;
	if (-x <= 2.0000000390829628e-24f) {
		tmp = 0.5f;
	} else if (-x <= 0.0044999998062849045f) {
		tmp = 1.0f / (x * (((s * 2.0f) - x) * ((1.0f / x) / s)));
	} 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 <= 2.0000000390829628e-24) then
        tmp = 0.5e0
    else if (-x <= 0.0044999998062849045e0) then
        tmp = 1.0e0 / (x * (((s * 2.0e0) - x) * ((1.0e0 / x) / s)))
    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(2.0000000390829628e-24))
		tmp = Float32(0.5);
	elseif (Float32(-x) <= Float32(0.0044999998062849045))
		tmp = Float32(Float32(1.0) / Float32(x * Float32(Float32(Float32(s * Float32(2.0)) - x) * Float32(Float32(Float32(1.0) / x) / s))));
	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(2.0000000390829628e-24))
		tmp = single(0.5);
	elseif (-x <= single(0.0044999998062849045))
		tmp = single(1.0) / (x * (((s * single(2.0)) - x) * ((single(1.0) / x) / s)));
	else
		tmp = x * (single(1.0) / (x * (single(2.0) - (x / s))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if (neg.f32 x) < 2.00000004e-24

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 50.7%

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

   if 2.00000004e-24 < (neg.f32 x) < 0.00449999981

   1. Initial program 99.3%

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

   3. Taylor expanded in x around 0 31.7%

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

      1. neg-mul-1 31.7%

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

      2. unsub-neg 31.7%

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

   5. Simplified 31.7%

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

   6. Taylor expanded in x around inf 31.7%

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

      1. sub-neg 31.7%

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

      2. associate-*r/ 31.7%

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

      3. metadata-eval 31.7%

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

      4. distribute-neg-frac 31.7%

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

      5. metadata-eval 31.7%

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

   8. Simplified 31.7%

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

      1. frac-add 52.7%

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

      2. associate-/r* 31.7%

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

      3. *-commutative 31.7%

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

      4. neg-mul-1 31.7%

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

      5. fma-define 31.7%

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

      6. fma-neg 31.7%

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

   10. Applied egg-rr 31.7%

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

       1. div-inv 31.7%

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

       2. associate-/l* 62.9%

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

       3. *-commutative 62.9%

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

   12. Applied egg-rr 62.9%

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

   if 0.00449999981 < (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 45.9%

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

      1. neg-mul-1 45.9%

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

      2. unsub-neg 45.9%

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

   5. Simplified 45.9%

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

   6. Taylor expanded in x around inf 45.9%

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

      1. sub-neg 45.9%

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

      2. associate-*r/ 45.9%

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

      3. metadata-eval 45.9%

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

      4. distribute-neg-frac 45.9%

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

      5. metadata-eval 45.9%

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

   8. Simplified 45.9%

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

   9. Taylor expanded in x around 0 45.9%

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

       1. mul-1-neg 45.9%

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

       2. sub-neg 45.9%

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

   11. Simplified 45.9%

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

       1. associate-*r/ 68.4%

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

       2. associate-/r/ 68.4%

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

   13. Applied egg-rr 68.4%

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

4. Final simplification 57.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-x \leq 2.0000000390829628 \cdot 10^{-24}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;-x \leq 0.0044999998062849045:\\ \;\;\;\;\frac{1}{x \cdot \left(\left(s \cdot 2 - x\right) \cdot \frac{\frac{1}{x}}{s}\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{x \cdot \left(2 - \frac{x}{s}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 56.9% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-x \leq 2.0000000390829628 \cdot 10^{-24}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;-x \leq 0.03999999910593033:\\ \;\;\;\;\frac{1}{x \cdot \left(\left(s \cdot 2 - x\right) \cdot \frac{1}{x \cdot s}\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{x \cdot \left(2 - \frac{x}{s}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- x) 2.0000000390829628e-24)
   0.5
   (if (<= (- x) 0.03999999910593033)
     (/ 1.0 (* x (* (- (* s 2.0) x) (/ 1.0 (* x s)))))
     (* x (/ 1.0 (* x (- 2.0 (/ x s))))))))

C

float code(float x, float s) {
	float tmp;
	if (-x <= 2.0000000390829628e-24f) {
		tmp = 0.5f;
	} else if (-x <= 0.03999999910593033f) {
		tmp = 1.0f / (x * (((s * 2.0f) - x) * (1.0f / (x * s))));
	} 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 <= 2.0000000390829628e-24) then
        tmp = 0.5e0
    else if (-x <= 0.03999999910593033e0) then
        tmp = 1.0e0 / (x * (((s * 2.0e0) - x) * (1.0e0 / (x * s))))
    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(2.0000000390829628e-24))
		tmp = Float32(0.5);
	elseif (Float32(-x) <= Float32(0.03999999910593033))
		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(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(2.0000000390829628e-24))
		tmp = single(0.5);
	elseif (-x <= single(0.03999999910593033))
		tmp = single(1.0) / (x * (((s * single(2.0)) - x) * (single(1.0) / (x * s))));
	else
		tmp = x * (single(1.0) / (x * (single(2.0) - (x / s))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if (neg.f32 x) < 2.00000004e-24

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 50.7%

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

   if 2.00000004e-24 < (neg.f32 x) < 0.0399999991

   1. Initial program 99.3%

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

   3. Taylor expanded in x around 0 29.8%

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

      1. neg-mul-1 29.8%

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

      2. unsub-neg 29.8%

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

   5. Simplified 29.8%

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

   6. Taylor expanded in x around inf 29.8%

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

      1. sub-neg 29.8%

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

      2. associate-*r/ 29.8%

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

      3. metadata-eval 29.8%

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

      4. distribute-neg-frac 29.8%

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

      5. metadata-eval 29.8%

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

   8. Simplified 29.8%

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

      1. frac-add 49.2%

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

      2. div-inv 58.4%

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

      3. *-commutative 58.4%

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

      4. neg-mul-1 58.4%

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

      5. fma-define 58.4%

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

      6. fma-neg 58.4%

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

   10. Applied egg-rr 58.4%

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

   if 0.0399999991 < (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 47.5%

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

      1. neg-mul-1 47.5%

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

      2. unsub-neg 47.5%

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

   5. Simplified 47.5%

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

   6. Taylor expanded in x around inf 47.5%

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

      1. sub-neg 47.5%

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

      2. associate-*r/ 47.5%

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

      3. metadata-eval 47.5%

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

      4. distribute-neg-frac 47.5%

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

      5. metadata-eval 47.5%

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

   8. Simplified 47.5%

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

   9. Taylor expanded in x around 0 47.5%

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

       1. mul-1-neg 47.5%

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

       2. sub-neg 47.5%

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

   11. Simplified 47.5%

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

       1. associate-*r/ 71.0%

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

       2. associate-/r/ 71.0%

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

   13. Applied egg-rr 71.0%

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

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-x \leq 2.0000000390829628 \cdot 10^{-24}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;-x \leq 0.03999999910593033:\\ \;\;\;\;\frac{1}{x \cdot \left(\left(s \cdot 2 - x\right) \cdot \frac{1}{x \cdot s}\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{x \cdot \left(2 - \frac{x}{s}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 53.8% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-x \leq -5.000000097707407 \cdot 10^{-26}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{x \cdot \left(2 - \frac{x}{s}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- x) -5.000000097707407e-26)
   0.5
   (* x (/ 1.0 (* x (- 2.0 (/ x s)))))))

C

float code(float x, float s) {
	float tmp;
	if (-x <= -5.000000097707407e-26f) {
		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 <= (-5.000000097707407e-26)) 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(-5.000000097707407e-26))
		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(-5.000000097707407e-26))
		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) < -5.0000001e-26

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 35.1%

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

   if -5.0000001e-26 < (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 56.7%

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

      1. neg-mul-1 56.7%

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

      2. unsub-neg 56.7%

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

   5. Simplified 56.7%

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

   6. Taylor expanded in x around inf 56.5%

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

      1. sub-neg 56.5%

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

      2. associate-*r/ 56.5%

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

      3. metadata-eval 56.5%

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

      4. distribute-neg-frac 56.5%

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

      5. metadata-eval 56.5%

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

   8. Simplified 56.5%

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

   9. Taylor expanded in x around 0 56.5%

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

       1. mul-1-neg 56.5%

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

       2. sub-neg 56.5%

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

   11. Simplified 56.5%

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

       1. associate-*r/ 67.2%

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

       2. associate-/r/ 67.1%

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

   13. Applied egg-rr 67.1%

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

4. Final simplification 54.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-x \leq -5.000000097707407 \cdot 10^{-26}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{x \cdot \left(2 - \frac{x}{s}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 48.6% accurate, 8.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-x \leq -5.000000097707407 \cdot 10^{-26}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- x) -5.000000097707407e-26) 0.5 (/ 1.0 (- 2.0 (/ x s)))))

C

float code(float x, float s) {
	float tmp;
	if (-x <= -5.000000097707407e-26f) {
		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 <= (-5.000000097707407e-26)) 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(-5.000000097707407e-26))
		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(-5.000000097707407e-26))
		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) < -5.0000001e-26

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 35.1%

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

   if -5.0000001e-26 < (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 56.7%

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

      1. neg-mul-1 56.7%

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

      2. unsub-neg 56.7%

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

   5. Simplified 56.7%

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

Alternative 8: 47.5% accurate, 10.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.999999987376214 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -4.999999987376214e-7) (/ 1.0 (/ x s)) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -4.999999987376214e-7f) {
		tmp = 1.0f / (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 <= (-4.999999987376214e-7)) then
        tmp = 1.0e0 / (x / s)
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-4.999999987376214e-7))
		tmp = Float32(Float32(1.0) / 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(-4.999999987376214e-7))
		tmp = single(1.0) / (x / s);
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -4.99999999e-7

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 43.1%

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

      1. neg-mul-1 43.1%

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

      2. unsub-neg 43.1%

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

   5. Simplified 43.1%

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

   6. Taylor expanded in x around inf 41.5%

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

      1. associate-*r/ 41.5%

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

      2. neg-mul-1 41.5%

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

   8. Simplified 41.5%

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

      1. clear-num 43.1%

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

      2. inv-pow 43.1%

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

      3. add-sqr-sqrt -0.0%

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

      4. sqrt-unprod 54.1%

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

      5. sqr-neg 54.1%

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

      6. sqrt-unprod 43.1%

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

      7. add-sqr-sqrt 43.1%

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

   10. Applied egg-rr 43.1%

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

       1. unpow-1 43.1%

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

   12. Simplified 43.1%

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

   if -4.99999999e-7 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 47.9%

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

Alternative 9: 46.4% accurate, 13.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.999999987376214 \cdot 10^{-7}:\\ \;\;\;\;\frac{s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -4.999999987376214e-7) (/ s x) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -4.999999987376214e-7f) {
		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 <= (-4.999999987376214e-7)) 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(-4.999999987376214e-7))
		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(-4.999999987376214e-7))
		tmp = s / x;
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -4.99999999e-7

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 43.1%

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

      1. neg-mul-1 43.1%

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

      2. unsub-neg 43.1%

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

   5. Simplified 43.1%

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

   6. Taylor expanded in x around inf 41.5%

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

      1. associate-*r/ 41.5%

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

      2. neg-mul-1 41.5%

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

   8. Simplified 41.5%

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

      1. add-sqr-sqrt -0.0%

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

      2. sqrt-unprod 53.2%

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

      3. sqr-neg 53.2%

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

      4. sqrt-unprod 41.5%

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

      5. add-sqr-sqrt 41.5%

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

      6. *-un-lft-identity 41.5%

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

   10. Applied egg-rr 41.5%

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

       1. *-lft-identity 41.5%

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

   12. Simplified 41.5%

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

   if -4.99999999e-7 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 47.9%

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

Alternative 10: 35.5% 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 99.9%

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

3. Taylor expanded in x around 0 35.4%

   \[\leadsto \color{blue}{0.5} \]
4. Add Preprocessing

Reproduce

herbie shell --seed 2024141 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))