Logistic function

Percentage Accurate: 99.8% → 99.8%

Time: 5.8s

Alternatives: 7

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, 0.5× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	return 1.0f / (1.0f + powf(expf(-1.0f), (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((-1.0e0)) ** (x / s)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.9%

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

   1. div-inv 99.9%

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

   2. exp-prod 85.6%

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

   3. neg-mul-1 85.6%

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

   4. exp-prod 85.6%

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

   5. pow-pow 99.9%

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

   6. div-inv 99.9%

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

3. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.9%

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

2. Final simplification 99.9%

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

Alternative 3: 61.7% accurate, 11.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.9999999920083944 \cdot 10^{-12}:\\ \;\;\;\;-1 + \left(1 + \frac{s}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -1.9999999920083944e-12) (+ -1.0 (+ 1.0 (/ s x))) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -1.9999999920083944e-12f) {
		tmp = -1.0f + (1.0f + (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 <= (-1.9999999920083944e-12)) then
        tmp = (-1.0e0) + (1.0e0 + (s / x))
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-1.9999999920083944e-12))
		tmp = Float32(Float32(-1.0) + Float32(Float32(1.0) + 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(-1.9999999920083944e-12))
		tmp = single(-1.0) + (single(1.0) + (s / x));
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -1.99999999e-12

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 42.2%

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

      1. mul-1-neg 42.2%

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

      2. unsub-neg 42.2%

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

   4. Simplified 42.2%

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

   5. Taylor expanded in x around inf 35.5%

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

      1. mul-1-neg 35.5%

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

      2. distribute-frac-neg 35.5%

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

   7. Simplified 35.5%

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

      1. clear-num 37.9%

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

      2. associate-/r/ 35.5%

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

      3. add-sqr-sqrt -0.0%

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

      4. sqrt-unprod 51.5%

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

      5. sqr-neg 51.5%

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

      6. sqrt-unprod 34.5%

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

      7. add-sqr-sqrt 34.5%

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

   9. Applied egg-rr 34.5%

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

       1. expm1-log1p-u 34.3%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{x} \cdot s\right)\right)} \]

       2. expm1-udef 87.1%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{x} \cdot s\right)} - 1} \]

       3. log1p-udef 87.1%

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

       4. rem-exp-log 87.2%

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

       5. associate-*l/ 87.2%

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

       6. *-un-lft-identity 87.2%

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

   11. Applied egg-rr 87.2%

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

   if -1.99999999e-12 < x

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 47.5%

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

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.9999999920083944 \cdot 10^{-12}:\\ \;\;\;\;-1 + \left(1 + \frac{s}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 4: 48.1% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.999999979801942 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -7.999999979801942e-6) (/ 1.0 (/ x s)) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -7.999999979801942e-6f) {
		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 <= (-7.999999979801942e-6)) 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(-7.999999979801942e-6))
		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(-7.999999979801942e-6))
		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 < -7.99999998e-6

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 44.7%

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

      1. mul-1-neg 44.7%

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

      2. unsub-neg 44.7%

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

   4. Simplified 44.7%

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

   5. Taylor expanded in x around inf 41.7%

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

      1. mul-1-neg 41.7%

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

      2. distribute-frac-neg 41.7%

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

   7. Simplified 41.7%

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

      1. clear-num 44.7%

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

      2. associate-/r/ 41.7%

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

      3. add-sqr-sqrt -0.0%

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

      4. sqrt-unprod 56.1%

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

      5. sqr-neg 56.1%

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

      6. sqrt-unprod 41.7%

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

      7. add-sqr-sqrt 41.7%

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

   9. Applied egg-rr 41.7%

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

       1. associate-*l/ 41.7%

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

       2. associate-/l* 44.7%

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

   11. Applied egg-rr 44.7%

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

   if -7.99999998e-6 < x

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 45.3%

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

4. Final simplification 45.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.999999979801942 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 5: 46.8% accurate, 17.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.000000058430487 \cdot 10^{-8}:\\ \;\;\;\;\frac{-s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -5.000000058430487e-8) (/ (- s) x) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -5.000000058430487e-8f) {
		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 <= (-5.000000058430487e-8)) 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(-5.000000058430487e-8))
		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(-5.000000058430487e-8))
		tmp = -s / x;
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -5.00000006e-8

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 43.1%

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

      1. mul-1-neg 43.1%

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

      2. unsub-neg 43.1%

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

   4. Simplified 43.1%

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

   5. Taylor expanded in x around inf 40.2%

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

      1. mul-1-neg 40.2%

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

      2. distribute-frac-neg 40.2%

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

   7. Simplified 40.2%

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

   if -5.00000006e-8 < x

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 46.1%

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

4. Final simplification 44.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.000000058430487 \cdot 10^{-8}:\\ \;\;\;\;\frac{-s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 6: 46.8% accurate, 21.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.999999979801942 \cdot 10^{-6}:\\ \;\;\;\;\frac{s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -7.999999979801942e-6) (/ s x) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -7.999999979801942e-6f) {
		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 <= (-7.999999979801942e-6)) 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(-7.999999979801942e-6))
		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(-7.999999979801942e-6))
		tmp = s / x;
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -7.99999998e-6

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 44.7%

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

      1. mul-1-neg 44.7%

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

      2. unsub-neg 44.7%

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

   4. Simplified 44.7%

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

   5. Taylor expanded in x around inf 41.7%

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

      1. mul-1-neg 41.7%

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

      2. distribute-frac-neg 41.7%

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

   7. Simplified 41.7%

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

      1. expm1-log1p-u 41.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-s}{x}\right)\right)} \]

      2. expm1-udef 100.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-s}{x}\right)} - 1} \]

      3. add-sqr-sqrt -0.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{-s} \cdot \sqrt{-s}}}{x}\right)} - 1 \]

      4. sqrt-unprod 97.6%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{\left(-s\right) \cdot \left(-s\right)}}}{x}\right)} - 1 \]

      5. sqr-neg 97.6%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\sqrt{\color{blue}{s \cdot s}}}{x}\right)} - 1 \]

      6. sqrt-unprod 97.6%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{s} \cdot \sqrt{s}}}{x}\right)} - 1 \]

      7. add-sqr-sqrt 97.6%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{s}}{x}\right)} - 1 \]

   9. Applied egg-rr 97.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{s}{x}\right)} - 1} \]
   10. Step-by-step derivation

       1. expm1-def 41.7%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{s}{x}\right)\right)} \]

       2. expm1-log1p 41.7%

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

   11. Simplified 41.7%

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

   if -7.99999998e-6 < x

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 45.3%

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

4. Final simplification 44.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.999999979801942 \cdot 10^{-6}:\\ \;\;\;\;\frac{s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 7: 35.4% 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 33.5%

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

3. Final simplification 33.5%

   \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023310 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))