Logistic function

Percentage Accurate: 99.8% → 99.8%

Time: 7.2s

Alternatives: 12

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 99.9%

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

2. Final simplification 99.9%

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

Alternative 2: 63.0% accurate, 5.6× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	float tmp;
	if (x <= -9.999999887266023e-27f) {
		tmp = 1.0f / (2.0f + ((x * (0.5f * (x / (s * s)))) - (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 <= (-9.999999887266023e-27)) then
        tmp = 1.0e0 / (2.0e0 + ((x * (0.5e0 * (x / (s * s)))) - (x / s)))
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-9.999999887266023e-27))
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(Float32(x * Float32(Float32(0.5) * Float32(x / Float32(s * s)))) - 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(-9.999999887266023e-27))
		tmp = single(1.0) / (single(2.0) + ((x * (single(0.5) * (x / (s * s)))) - (x / s)));
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -9.99999989e-27

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 80.9%

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

      1. mul-1-neg 80.9%

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

      2. unsub-neg 80.9%

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

      3. associate-*r/ 80.0%

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

      4. unpow2 80.0%

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

      5. unpow2 80.0%

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

   4. Simplified 80.0%

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

      1. associate-*r* 80.0%

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

      2. *-un-lft-identity 80.0%

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

      3. times-frac 82.5%

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

   6. Applied egg-rr 82.5%

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

      1. sub-neg 82.5%

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

      2. /-rgt-identity 82.5%

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

      3. *-commutative 82.5%

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

   8. Applied egg-rr 82.5%

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

      1. unsub-neg 82.5%

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

      2. associate-*l* 82.5%

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

   10. Applied egg-rr 82.5%

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

   if -9.99999989e-27 < x

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 48.2%

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

4. Final simplification 64.0%

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

Alternative 3: 62.9% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.9999999996399175 \cdot 10^{-23}:\\ \;\;\;\;\frac{1}{2 + \left(\frac{0.5}{s \cdot s} \cdot \left(x \cdot x\right) - \frac{x}{s}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -1.9999999996399175e-23)
   (/ 1.0 (+ 2.0 (- (* (/ 0.5 (* s s)) (* x x)) (/ x s))))
   0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -1.9999999996399175e-23f) {
		tmp = 1.0f / (2.0f + (((0.5f / (s * s)) * (x * x)) - (x / s)));
	} else {
		tmp = 0.5f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-1.9999999996399175e-23)) then
        tmp = 1.0e0 / (2.0e0 + (((0.5e0 / (s * s)) * (x * x)) - (x / s)))
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-1.9999999996399175e-23))
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(Float32(Float32(Float32(0.5) / Float32(s * s)) * Float32(x * x)) - Float32(x / s))));
	else
		tmp = Float32(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-1.9999999996399175e-23))
		tmp = single(1.0) / (single(2.0) + (((single(0.5) / (s * s)) * (x * x)) - (x / s)));
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -2e-23

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 82.0%

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

      1. mul-1-neg 82.0%

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

      2. unsub-neg 82.0%

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

      3. associate-*r/ 81.1%

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

      4. unpow2 81.1%

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

      5. unpow2 81.1%

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

   4. Simplified 81.1%

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

      1. associate-/l* 82.0%

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

      2. associate-/r/ 85.2%

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

   6. Applied egg-rr 85.2%

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

   if -2e-23 < x

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 48.4%

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

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.9999999996399175 \cdot 10^{-23}:\\ \;\;\;\;\frac{1}{2 + \left(\frac{0.5}{s \cdot s} \cdot \left(x \cdot x\right) - \frac{x}{s}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 4: 60.4% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.9999998413276127 \cdot 10^{-20}:\\ \;\;\;\;\frac{1}{2 + \frac{x}{s} \cdot \left(\frac{x}{\frac{s}{0.5}} + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 4.9999998413276127e-20)
   (/ 1.0 (+ 2.0 (* (/ x s) (+ (/ x (/ s 0.5)) -1.0))))
   0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= 4.9999998413276127e-20f) {
		tmp = 1.0f / (2.0f + ((x / s) * ((x / (s / 0.5f)) + -1.0f)));
	} 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 <= 4.9999998413276127e-20) then
        tmp = 1.0e0 / (2.0e0 + ((x / s) * ((x / (s / 0.5e0)) + (-1.0e0))))
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(4.9999998413276127e-20))
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(Float32(x / s) * Float32(Float32(x / Float32(s / Float32(0.5))) + Float32(-1.0)))));
	else
		tmp = Float32(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(4.9999998413276127e-20))
		tmp = single(1.0) / (single(2.0) + ((x / s) * ((x / (s / single(0.5))) + single(-1.0))));
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 4.99999984e-20

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 77.5%

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

      1. mul-1-neg 77.5%

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

      2. unsub-neg 77.5%

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

      3. associate-*r/ 76.9%

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

      4. unpow2 76.9%

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

      5. unpow2 76.9%

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

   4. Simplified 76.9%

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

      1. associate-/l* 77.5%

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

      2. frac-sub 61.4%

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

      3. times-frac 69.8%

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

      4. times-frac 73.9%

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

   6. Applied egg-rr 73.9%

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

      1. associate-*l* 82.5%

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

      2. *-commutative 82.5%

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

   8. Simplified 82.5%

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

   9. Taylor expanded in s around 0 82.5%

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

   10. Taylor expanded in s around 0 77.5%

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

       1. unpow2 77.5%

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

       2. unpow2 77.5%

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

       3. associate-*r/ 76.9%

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

       4. associate-*l* 76.9%

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

       5. times-frac 77.2%

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

       6. distribute-rgt-out 77.2%

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

       7. *-commutative 77.2%

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

       8. associate-/l* 77.2%

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

   12. Simplified 77.2%

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

   if 4.99999984e-20 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 32.6%

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

4. Final simplification 61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.9999998413276127 \cdot 10^{-20}:\\ \;\;\;\;\frac{1}{2 + \frac{x}{s} \cdot \left(\frac{x}{\frac{s}{0.5}} + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 5: 61.6% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.9999998413276127 \cdot 10^{-20}:\\ \;\;\;\;\frac{1}{2 + \frac{x \cdot 0.5 - s}{s \cdot \frac{s}{x}}}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 4.9999998413276127e-20)
   (/ 1.0 (+ 2.0 (/ (- (* x 0.5) s) (* s (/ s x)))))
   0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= 4.9999998413276127e-20f) {
		tmp = 1.0f / (2.0f + (((x * 0.5f) - s) / (s * (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 <= 4.9999998413276127e-20) then
        tmp = 1.0e0 / (2.0e0 + (((x * 0.5e0) - s) / (s * (s / x))))
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(4.9999998413276127e-20))
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(Float32(Float32(x * Float32(0.5)) - s) / Float32(s * 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(4.9999998413276127e-20))
		tmp = single(1.0) / (single(2.0) + (((x * single(0.5)) - s) / (s * (s / x))));
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 4.99999984e-20

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 77.5%

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

      1. mul-1-neg 77.5%

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

      2. unsub-neg 77.5%

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

      3. associate-*r/ 76.9%

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

      4. unpow2 76.9%

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

      5. unpow2 76.9%

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

   4. Simplified 76.9%

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

      1. associate-/r* 76.6%

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

      2. clear-num 76.6%

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

      3. frac-sub 61.2%

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

      4. *-un-lft-identity 61.2%

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

      5. times-frac 61.2%

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

      6. metadata-eval 61.2%

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

      7. associate-/l* 61.7%

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

      8. *-commutative 61.7%

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

      9. *-un-lft-identity 61.7%

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

   6. Applied egg-rr 61.7%

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

      1. associate-*l* 61.7%

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

      2. associate-/r/ 61.7%

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

   8. Simplified 61.7%

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

   9. Taylor expanded in x around 0 77.6%

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

   if 4.99999984e-20 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 32.6%

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

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.9999998413276127 \cdot 10^{-20}:\\ \;\;\;\;\frac{1}{2 + \frac{x \cdot 0.5 - s}{s \cdot \frac{s}{x}}}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 6: 59.1% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.0000000180025095 \cdot 10^{-35}:\\ \;\;\;\;\frac{1}{2 + \frac{0.5}{s} \cdot \frac{x}{\frac{s}{x}}}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -1.0000000180025095e-35)
   (/ 1.0 (+ 2.0 (* (/ 0.5 s) (/ x (/ s x)))))
   0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -1.0000000180025095e-35f) {
		tmp = 1.0f / (2.0f + ((0.5f / s) * (x / (s / x))));
	} else {
		tmp = 0.5f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-1.0000000180025095e-35)) then
        tmp = 1.0e0 / (2.0e0 + ((0.5e0 / s) * (x / (s / x))))
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-1.0000000180025095e-35))
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(Float32(Float32(0.5) / s) * Float32(x / Float32(s / x)))));
	else
		tmp = Float32(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-1.0000000180025095e-35))
		tmp = single(1.0) / (single(2.0) + ((single(0.5) / s) * (x / (s / x))));
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -1.00000002e-35

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 80.3%

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

      1. mul-1-neg 80.3%

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

      2. unsub-neg 80.3%

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

      3. associate-*r/ 79.6%

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

      4. unpow2 79.6%

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

      5. unpow2 79.6%

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

   4. Simplified 79.6%

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

      1. associate-/l* 80.3%

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

      2. frac-sub 73.2%

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

      3. times-frac 76.9%

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

      4. times-frac 78.7%

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

   6. Applied egg-rr 78.7%

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

      1. associate-*l* 83.3%

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

      2. *-commutative 83.3%

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

   8. Simplified 83.3%

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

   9. Taylor expanded in s around 0 79.1%

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

       1. unpow2 79.1%

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

       2. associate-*r/ 78.3%

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

       3. times-frac 74.6%

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

       4. unpow2 74.6%

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

       5. associate-/l* 74.7%

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

   11. Simplified 74.7%

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

   if -1.00000002e-35 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 44.7%

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

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.0000000180025095 \cdot 10^{-35}:\\ \;\;\;\;\frac{1}{2 + \frac{0.5}{s} \cdot \frac{x}{\frac{s}{x}}}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 7: 57.0% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.999999941330714 \cdot 10^{-10}:\\ \;\;\;\;2 \cdot \left(\frac{s}{x} \cdot \frac{s}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -5.999999941330714e-10) (* 2.0 (* (/ s x) (/ s x))) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -5.999999941330714e-10f) {
		tmp = 2.0f * ((s / x) * (s / x));
	} else {
		tmp = 0.5f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-5.999999941330714e-10)) then
        tmp = 2.0e0 * ((s / x) * (s / x))
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-5.999999941330714e-10))
		tmp = Float32(Float32(2.0) * Float32(Float32(s / x) * Float32(s / x)));
	else
		tmp = Float32(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-5.999999941330714e-10))
		tmp = single(2.0) * ((s / x) * (s / x));
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -5.99999994e-10

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 83.4%

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

      1. mul-1-neg 83.4%

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

      2. unsub-neg 83.4%

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

      3. associate-*r/ 83.4%

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

      4. unpow2 83.4%

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

      5. unpow2 83.4%

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

   4. Simplified 83.4%

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

   5. Taylor expanded in x around inf 81.2%

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

      1. unpow2 81.2%

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

      2. unpow2 81.2%

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

   7. Simplified 81.2%

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

      1. frac-times 81.2%

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

   9. Applied egg-rr 81.2%

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

   if -5.99999994e-10 < x

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 47.4%

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

4. Final simplification 58.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.999999941330714 \cdot 10^{-10}:\\ \;\;\;\;2 \cdot \left(\frac{s}{x} \cdot \frac{s}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 8: 58.3% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.300000037447892 \cdot 10^{-18}:\\ \;\;\;\;2 \cdot \frac{s \cdot s}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -3.300000037447892e-18) (* 2.0 (/ (* s s) (* x x))) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -3.300000037447892e-18f) {
		tmp = 2.0f * ((s * s) / (x * x));
	} else {
		tmp = 0.5f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-3.300000037447892e-18)) then
        tmp = 2.0e0 * ((s * s) / (x * x))
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-3.300000037447892e-18))
		tmp = Float32(Float32(2.0) * Float32(Float32(s * s) / Float32(x * x)));
	else
		tmp = Float32(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-3.300000037447892e-18))
		tmp = single(2.0) * ((s * s) / (x * x));
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -3.30000004e-18

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 82.2%

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

      1. mul-1-neg 82.2%

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

      2. unsub-neg 82.2%

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

      3. associate-*r/ 82.2%

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

      4. unpow2 82.2%

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

      5. unpow2 82.2%

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

   4. Simplified 82.2%

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

   5. Taylor expanded in x around inf 73.8%

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

      1. unpow2 73.8%

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

      2. unpow2 73.8%

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

   7. Simplified 73.8%

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

   if -3.30000004e-18 < x

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 48.6%

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

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.300000037447892 \cdot 10^{-18}:\\ \;\;\;\;2 \cdot \frac{s \cdot s}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 9: 58.3% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.300000037447892 \cdot 10^{-18}:\\ \;\;\;\;\frac{2 \cdot \left(s \cdot s\right)}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -3.300000037447892e-18) (/ (* 2.0 (* s s)) (* x x)) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -3.300000037447892e-18f) {
		tmp = (2.0f * (s * s)) / (x * x);
	} else {
		tmp = 0.5f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-3.300000037447892e-18)) then
        tmp = (2.0e0 * (s * s)) / (x * x)
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-3.300000037447892e-18))
		tmp = Float32(Float32(Float32(2.0) * Float32(s * s)) / Float32(x * x));
	else
		tmp = Float32(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-3.300000037447892e-18))
		tmp = (single(2.0) * (s * s)) / (x * x);
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -3.30000004e-18

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 82.2%

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

      1. mul-1-neg 82.2%

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

      2. unsub-neg 82.2%

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

      3. associate-*r/ 82.2%

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

      4. unpow2 82.2%

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

      5. unpow2 82.2%

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

   4. Simplified 82.2%

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

   5. Taylor expanded in x around inf 73.8%

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

      1. unpow2 73.8%

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

      2. unpow2 73.8%

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

   7. Simplified 73.8%

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

      1. associate-*r/ 73.8%

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

   9. Applied egg-rr 73.8%

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

   if -3.30000004e-18 < x

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 48.6%

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

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.300000037447892 \cdot 10^{-18}:\\ \;\;\;\;\frac{2 \cdot \left(s \cdot s\right)}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 10: 48.9% accurate, 11.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.0000000180025095 \cdot 10^{-35}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -1.0000000180025095e-35) (/ 1.0 (- 2.0 (/ x s))) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -1.0000000180025095e-35f) {
		tmp = 1.0f / (2.0f - (x / s));
	} else {
		tmp = 0.5f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-1.0000000180025095e-35)) then
        tmp = 1.0e0 / (2.0e0 - (x / s))
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-1.0000000180025095e-35))
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) - Float32(x / s)));
	else
		tmp = Float32(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-1.0000000180025095e-35))
		tmp = single(1.0) / (single(2.0) - (x / s));
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -1.00000002e-35

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 56.1%

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

      1. mul-1-neg 56.1%

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

      2. unsub-neg 56.1%

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

   4. Simplified 56.1%

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

   if -1.00000002e-35 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 44.7%

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

4. Final simplification 50.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.0000000180025095 \cdot 10^{-35}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 11: 46.4% accurate, 17.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.999999773744548 \cdot 10^{-9}:\\ \;\;\;\;\frac{-s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -7.999999773744548e-9) (/ (- s) x) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -7.999999773744548e-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 <= (-7.999999773744548e-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(-7.999999773744548e-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(-7.999999773744548e-9))
		tmp = -s / x;
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -7.99999977e-9

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 55.0%

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

      1. mul-1-neg 55.0%

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

      2. unsub-neg 55.0%

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

   4. Simplified 55.0%

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

   5. Taylor expanded in x around inf 51.5%

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

      1. associate-*r/ 51.5%

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

      2. neg-mul-1 51.5%

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

   7. Simplified 51.5%

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

   if -7.99999977e-9 < x

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 47.0%

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

4. Final simplification 48.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.999999773744548 \cdot 10^{-9}:\\ \;\;\;\;\frac{-s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 12: 35.6% 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. Taylor expanded in x around 0 34.2%

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

3. Final simplification 34.2%

   \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023221 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))