Logistic function

Percentage Accurate: 99.8% → 99.8%

Time: 12.4s

Alternatives: 11

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

2. Final simplification 99.8%

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

Alternative 2: 63.5% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-x \leq 1.0000000031710769 \cdot 10^{-29}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + \left(0.5 \cdot \frac{x}{\frac{s \cdot s}{x}} - \frac{x}{s}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- x) 1.0000000031710769e-29)
   0.5
   (/ 1.0 (+ 2.0 (- (* 0.5 (/ x (/ (* s s) x))) (/ x s))))))

C

float code(float x, float s) {
	float tmp;
	if (-x <= 1.0000000031710769e-29f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / (2.0f + ((0.5f * (x / ((s * s) / x))) - (x / s)));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (-x <= 1.0000000031710769e-29) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 / (2.0e0 + ((0.5e0 * (x / ((s * s) / x))) - (x / s)))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-x) <= Float32(1.0000000031710769e-29))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(Float32(Float32(0.5) * Float32(x / Float32(Float32(s * s) / x))) - Float32(x / s))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (-x <= single(1.0000000031710769e-29))
		tmp = single(0.5);
	else
		tmp = single(1.0) / (single(2.0) + ((single(0.5) * (x / ((s * s) / x))) - (x / s)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (neg.f32 x) < 1e-29

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 48.9%

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

   if 1e-29 < (neg.f32 x)

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 78.4%

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

      1. +-commutative 78.4%

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

      2. mul-1-neg 78.4%

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

      3. unsub-neg 78.4%

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

      4. unpow2 78.4%

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

      5. unpow2 78.4%

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

      6. times-frac 69.3%

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

   4. Simplified 69.3%

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

      1. clear-num 69.3%

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

      2. frac-times 70.7%

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

      3. *-un-lft-identity 70.7%

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

   6. Applied egg-rr 70.7%

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

      1. associate-*l/ 80.1%

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

   8. Applied egg-rr 80.1%

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

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-x \leq 1.0000000031710769 \cdot 10^{-29}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + \left(0.5 \cdot \frac{x}{\frac{s \cdot s}{x}} - \frac{x}{s}\right)}\\ \end{array} \]

Alternative 3: 63.2% accurate, 6.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 53.4%

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

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

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 77.2%

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

      1. +-commutative 77.2%

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

      2. mul-1-neg 77.2%

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

      3. unsub-neg 77.2%

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

      4. unpow2 77.2%

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

      5. unpow2 77.2%

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

      6. times-frac 64.8%

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

   4. Simplified 64.8%

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

   5. Taylor expanded in x around inf 76.5%

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

      1. unpow2 76.5%

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

      2. unpow2 76.5%

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

      3. times-frac 64.1%

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

   7. Simplified 64.1%

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

      1. associate-*r* 64.1%

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

      2. associate-*r/ 65.8%

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

      3. *-commutative 65.8%

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

   9. Applied egg-rr 65.8%

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

       1. clear-num 66.5%

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

       2. inv-pow 66.5%

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

       3. *-commutative 66.5%

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

       4. associate-*l/ 66.5%

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

   11. Applied egg-rr 66.5%

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

       1. unpow-1 66.5%

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

       2. associate-*r/ 79.2%

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

   13. Simplified 79.2%

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

4. Final simplification 63.4%

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

Alternative 4: 58.6% accurate, 7.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 53.9%

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

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

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 75.7%

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

      1. +-commutative 75.7%

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

      2. mul-1-neg 75.7%

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

      3. unsub-neg 75.7%

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

      4. unpow2 75.7%

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

      5. unpow2 75.7%

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

      6. times-frac 63.9%

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

   4. Simplified 63.9%

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

   5. Taylor expanded in x around inf 75.0%

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

      1. unpow2 75.0%

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

      2. unpow2 75.0%

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

      3. times-frac 63.2%

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

   7. Simplified 63.2%

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

4. Final simplification 57.5%

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

Alternative 5: 58.7% accurate, 7.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 53.9%

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

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

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 75.7%

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

      1. +-commutative 75.7%

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

      2. mul-1-neg 75.7%

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

      3. unsub-neg 75.7%

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

      4. unpow2 75.7%

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

      5. unpow2 75.7%

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

      6. times-frac 63.9%

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

   4. Simplified 63.9%

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

   5. Taylor expanded in x around inf 75.0%

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

      1. unpow2 75.0%

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

      2. unpow2 75.0%

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

      3. times-frac 63.2%

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

   7. Simplified 63.2%

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

      1. associate-*l/ 64.8%

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

      2. associate-/l* 63.2%

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

      3. div-inv 63.2%

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

      4. clear-num 63.2%

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

   9. Applied egg-rr 63.2%

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

4. Final simplification 57.5%

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

Alternative 6: 61.3% accurate, 7.6× speedup

Math

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

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) 20.0) 0.5 (* 2.0 (/ (* s s) (* x x)))))

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 53.4%

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

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

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 77.2%

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

      1. +-commutative 77.2%

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

      2. mul-1-neg 77.2%

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

      3. unsub-neg 77.2%

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

      4. unpow2 77.2%

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

      5. unpow2 77.2%

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

      6. times-frac 64.8%

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

   4. Simplified 64.8%

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

   5. Taylor expanded in x around inf 76.5%

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

      1. unpow2 76.5%

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

      2. unpow2 76.5%

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

   7. Simplified 76.5%

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

4. Final simplification 62.3%

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

Alternative 7: 62.8% accurate, 7.6× speedup

Math

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

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) 20.0) 0.5 (* (/ (* s s) x) (/ 2.0 x))))

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 53.4%

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

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

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 77.2%

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

      1. +-commutative 77.2%

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

      2. mul-1-neg 77.2%

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

      3. unsub-neg 77.2%

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

      4. unpow2 77.2%

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

      5. unpow2 77.2%

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

      6. times-frac 64.8%

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

   4. Simplified 64.8%

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

   5. Taylor expanded in x around inf 76.5%

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

      1. unpow2 76.5%

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

      2. unpow2 76.5%

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

      3. times-frac 64.1%

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

   7. Simplified 64.1%

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

      1. associate-*r* 64.1%

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

      2. associate-*r/ 65.8%

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

      3. *-commutative 65.8%

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

   9. Applied egg-rr 65.8%

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

   10. Taylor expanded in s around 0 76.5%

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

       1. associate-*r/ 76.5%

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

       2. unpow2 76.5%

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

       3. times-frac 78.5%

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

       4. unpow2 78.5%

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

   12. Simplified 78.5%

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

4. Final simplification 63.1%

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

Alternative 8: 49.3% accurate, 8.9× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= -1.0f) {
		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 / s) <= (-1.0e0)) 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(Float32(-x) / s) <= Float32(-1.0))
		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 / s) <= single(-1.0))
		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 (/.f32 (neg.f32 x) s) < -1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 28.2%

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

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

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 61.5%

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

      1. mul-1-neg 61.5%

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

      2. unsub-neg 61.5%

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

   4. Simplified 61.5%

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

4. Final simplification 50.0%

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

Alternative 9: 47.8% accurate, 9.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 53.9%

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

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

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 41.5%

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

      1. mul-1-neg 41.5%

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

      2. unsub-neg 41.5%

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

   4. Simplified 41.5%

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

   5. Taylor expanded in x around inf 41.5%

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

      1. neg-mul-1 41.5%

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

      2. distribute-neg-frac 41.5%

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

   7. Simplified 41.5%

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

4. Final simplification 49.0%

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

Alternative 10: 46.2% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-x \leq 3.5000000675466936 \cdot 10^{-9}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{-s}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- x) 3.5000000675466936e-9) 0.5 (/ (- s) x)))

C

float code(float x, float s) {
	float tmp;
	if (-x <= 3.5000000675466936e-9f) {
		tmp = 0.5f;
	} else {
		tmp = -s / x;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (-x <= 3.5000000675466936e-9) then
        tmp = 0.5e0
    else
        tmp = -s / x
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-x) <= Float32(3.5000000675466936e-9))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(-s) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (-x <= single(3.5000000675466936e-9))
		tmp = single(0.5);
	else
		tmp = -s / x;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (neg.f32 x) < 3.50000007e-9

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 48.2%

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

   if 3.50000007e-9 < (neg.f32 x)

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 50.6%

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

      1. mul-1-neg 50.6%

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

      2. unsub-neg 50.6%

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

   4. Simplified 50.6%

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

   5. Taylor expanded in x around inf 45.8%

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

      1. associate-*r/ 45.8%

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

      2. neg-mul-1 45.8%

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

   7. Simplified 45.8%

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

4. Final simplification 47.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-x \leq 3.5000000675466936 \cdot 10^{-9}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{-s}{x}\\ \end{array} \]

Alternative 11: 35.0% accurate, 108.0× speedup

Math

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

FPCore

(FPCore (x s) :precision binary32 0.5)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in x around 0 35.2%

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

3. Final simplification 35.2%

   \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023275 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))