Logistic function

Percentage Accurate: 99.8% → 99.8%

Time: 10.9s

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, 0.3× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + e^{\frac{x}{s} \cdot -0.6666666666666666} \cdot {\left(\sqrt[3]{e^{-1}}\right)}^{\left(\frac{x}{s}\right)}} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (/
  1.0
  (+
   1.0
   (* (exp (* (/ x s) -0.6666666666666666)) (pow (cbrt (exp -1.0)) (/ x s))))))

C

float code(float x, float s) {
	return 1.0f / (1.0f + (expf(((x / s) * -0.6666666666666666f)) * powf(cbrtf(expf(-1.0f)), (x / s))));
}

Julia

function code(x, s)
	return Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(exp(Float32(Float32(x / s) * Float32(-0.6666666666666666))) * (cbrt(exp(Float32(-1.0))) ^ Float32(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 82.9%

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

   3. neg-mul-1 82.9%

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

   4. exp-prod 82.9%

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

   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. Step-by-step derivation

   1. add-cube-cbrt 99.8%

      \[\leadsto \frac{1}{1 + {\color{blue}{\left(\left(\sqrt[3]{e^{-1}} \cdot \sqrt[3]{e^{-1}}\right) \cdot \sqrt[3]{e^{-1}}\right)}}^{\left(\frac{x}{s}\right)}} \]

   2. unpow-prod-down 99.8%

      \[\leadsto \frac{1}{1 + \color{blue}{{\left(\sqrt[3]{e^{-1}} \cdot \sqrt[3]{e^{-1}}\right)}^{\left(\frac{x}{s}\right)} \cdot {\left(\sqrt[3]{e^{-1}}\right)}^{\left(\frac{x}{s}\right)}}} \]

   3. cbrt-unprod 99.9%

      \[\leadsto \frac{1}{1 + {\color{blue}{\left(\sqrt[3]{e^{-1} \cdot e^{-1}}\right)}}^{\left(\frac{x}{s}\right)} \cdot {\left(\sqrt[3]{e^{-1}}\right)}^{\left(\frac{x}{s}\right)}} \]

   4. prod-exp 99.9%

      \[\leadsto \frac{1}{1 + {\left(\sqrt[3]{\color{blue}{e^{-1 + -1}}}\right)}^{\left(\frac{x}{s}\right)} \cdot {\left(\sqrt[3]{e^{-1}}\right)}^{\left(\frac{x}{s}\right)}} \]

   5. metadata-eval 99.9%

      \[\leadsto \frac{1}{1 + {\left(\sqrt[3]{e^{\color{blue}{-2}}}\right)}^{\left(\frac{x}{s}\right)} \cdot {\left(\sqrt[3]{e^{-1}}\right)}^{\left(\frac{x}{s}\right)}} \]

5. Applied egg-rr 99.9%

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

   1. add-exp-log 99.9%

      \[\leadsto \frac{1}{1 + \color{blue}{e^{\log \left({\left(\sqrt[3]{e^{-2}}\right)}^{\left(\frac{x}{s}\right)}\right)}} \cdot {\left(\sqrt[3]{e^{-1}}\right)}^{\left(\frac{x}{s}\right)}} \]

   2. log-pow 99.9%

      \[\leadsto \frac{1}{1 + e^{\color{blue}{\frac{x}{s} \cdot \log \left(\sqrt[3]{e^{-2}}\right)}} \cdot {\left(\sqrt[3]{e^{-1}}\right)}^{\left(\frac{x}{s}\right)}} \]

   3. pow1/3 99.9%

      \[\leadsto \frac{1}{1 + e^{\frac{x}{s} \cdot \log \color{blue}{\left({\left(e^{-2}\right)}^{0.3333333333333333}\right)}} \cdot {\left(\sqrt[3]{e^{-1}}\right)}^{\left(\frac{x}{s}\right)}} \]

   4. log-pow 99.9%

      \[\leadsto \frac{1}{1 + e^{\frac{x}{s} \cdot \color{blue}{\left(0.3333333333333333 \cdot \log \left(e^{-2}\right)\right)}} \cdot {\left(\sqrt[3]{e^{-1}}\right)}^{\left(\frac{x}{s}\right)}} \]

   5. add-log-exp 99.9%

      \[\leadsto \frac{1}{1 + e^{\frac{x}{s} \cdot \left(0.3333333333333333 \cdot \color{blue}{-2}\right)} \cdot {\left(\sqrt[3]{e^{-1}}\right)}^{\left(\frac{x}{s}\right)}} \]

   6. metadata-eval 99.9%

      \[\leadsto \frac{1}{1 + e^{\frac{x}{s} \cdot \color{blue}{-0.6666666666666666}} \cdot {\left(\sqrt[3]{e^{-1}}\right)}^{\left(\frac{x}{s}\right)}} \]

7. Applied egg-rr 99.9%

   \[\leadsto \frac{1}{1 + \color{blue}{e^{\frac{x}{s} \cdot -0.6666666666666666}} \cdot {\left(\sqrt[3]{e^{-1}}\right)}^{\left(\frac{x}{s}\right)}} \]

8. Final simplification 99.9%

   \[\leadsto \frac{1}{1 + e^{\frac{x}{s} \cdot -0.6666666666666666} \cdot {\left(\sqrt[3]{e^{-1}}\right)}^{\left(\frac{x}{s}\right)}} \]

Alternative 2: 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 82.9%

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

   3. neg-mul-1 82.9%

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

   4. exp-prod 82.9%

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

   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 3: 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 4: 63.2% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.999999943633011 \cdot 10^{-27}:\\ \;\;\;\;\frac{1}{2 + \left(0.5 \cdot \frac{x}{\frac{s \cdot s}{x}} - \frac{x}{s}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -4.999999943633011e-27)
   (/ 1.0 (+ 2.0 (- (* 0.5 (/ x (/ (* s s) x))) (/ x s))))
   0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -4.999999943633011e-27f) {
		tmp = 1.0f / (2.0f + ((0.5f * (x / ((s * s) / 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 <= (-4.999999943633011e-27)) then
        tmp = 1.0e0 / (2.0e0 + ((0.5e0 * (x / ((s * s) / 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(-4.999999943633011e-27))
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(Float32(Float32(0.5) * Float32(x / Float32(Float32(s * s) / 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(-4.999999943633011e-27))
		tmp = single(1.0) / (single(2.0) + ((single(0.5) * (x / ((s * s) / x))) - (x / s)));
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -4.99999994e-27

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 81.0%

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

      1. mul-1-neg 81.0%

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

      2. unsub-neg 81.0%

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

      3. unpow2 81.0%

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

      4. unpow2 81.0%

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

      5. times-frac 73.4%

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

   4. Simplified 73.4%

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

      1. frac-times 81.0%

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

      2. associate-/l* 85.1%

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

   6. Applied egg-rr 85.1%

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

   if -4.99999994e-27 < x

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 49.1%

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

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.999999943633011 \cdot 10^{-27}:\\ \;\;\;\;\frac{1}{2 + \left(0.5 \cdot \frac{x}{\frac{s \cdot s}{x}} - \frac{x}{s}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 5: 61.8% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.00000006274879 \cdot 10^{-22}:\\ \;\;\;\;\frac{1}{2 + 0.5 \cdot \frac{x \cdot x}{s \cdot s}}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -2.00000006274879e-22)
   (/ 1.0 (+ 2.0 (* 0.5 (/ (* x x) (* s s)))))
   0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -2.00000006274879e-22f) {
		tmp = 1.0f / (2.0f + (0.5f * ((x * x) / (s * 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 <= (-2.00000006274879e-22)) then
        tmp = 1.0e0 / (2.0e0 + (0.5e0 * ((x * x) / (s * s))))
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-2.00000006274879e-22))
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(Float32(0.5) * Float32(Float32(x * x) / Float32(s * s)))));
	else
		tmp = Float32(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-2.00000006274879e-22))
		tmp = single(1.0) / (single(2.0) + (single(0.5) * ((x * x) / (s * s))));
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -2.00000006e-22

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 83.7%

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

      1. mul-1-neg 83.7%

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

      2. unsub-neg 83.7%

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

      3. unpow2 83.7%

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

      4. unpow2 83.7%

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

      5. times-frac 74.8%

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

   4. Simplified 74.8%

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

      1. associate-*r* 74.8%

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

      2. *-un-lft-identity 74.8%

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

      3. distribute-rgt-out-- 74.8%

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

      4. associate-*r/ 74.8%

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

   6. Applied egg-rr 74.8%

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

   7. Taylor expanded in x around inf 82.8%

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

      1. unpow2 82.8%

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

      2. unpow2 82.8%

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

   9. Simplified 82.8%

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

   if -2.00000006e-22 < x

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 49.3%

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

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.00000006274879 \cdot 10^{-22}:\\ \;\;\;\;\frac{1}{2 + 0.5 \cdot \frac{x \cdot x}{s \cdot s}}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 6: 57.4% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.9999999920083944 \cdot 10^{-11}:\\ \;\;\;\;2 \cdot \left(\frac{s}{x} \cdot \frac{s}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -1.9999999920083944e-11) (* 2.0 (* (/ s x) (/ s x))) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -1.9999999920083944e-11f) {
		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 <= (-1.9999999920083944e-11)) 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(-1.9999999920083944e-11))
		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(-1.9999999920083944e-11))
		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 < -1.99999999e-11

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 81.9%

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

      1. mul-1-neg 81.9%

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

      2. unsub-neg 81.9%

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

      3. unpow2 81.9%

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

      4. unpow2 81.9%

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

      5. times-frac 81.9%

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

   4. Simplified 81.9%

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

   5. Taylor expanded in x around inf 79.8%

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

      1. unpow2 79.8%

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

      2. unpow2 79.8%

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

      3. times-frac 78.8%

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

   7. Simplified 78.8%

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

   if -1.99999999e-11 < x

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 48.7%

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

4. Final simplification 56.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.9999999920083944 \cdot 10^{-11}:\\ \;\;\;\;2 \cdot \left(\frac{s}{x} \cdot \frac{s}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 7: 57.4% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.9999999920083944 \cdot 10^{-11}:\\ \;\;\;\;2 \cdot \frac{s}{\frac{x \cdot x}{s}}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -1.9999999920083944e-11) (* 2.0 (/ s (/ (* x x) s))) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -1.9999999920083944e-11f) {
		tmp = 2.0f * (s / ((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 <= (-1.9999999920083944e-11)) then
        tmp = 2.0e0 * (s / ((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(-1.9999999920083944e-11))
		tmp = Float32(Float32(2.0) * Float32(s / Float32(Float32(x * 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(-1.9999999920083944e-11))
		tmp = single(2.0) * (s / ((x * x) / s));
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -1.99999999e-11

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 81.9%

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

      1. mul-1-neg 81.9%

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

      2. unsub-neg 81.9%

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

      3. unpow2 81.9%

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

      4. unpow2 81.9%

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

      5. times-frac 81.9%

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

   4. Simplified 81.9%

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

   5. Taylor expanded in x around inf 79.8%

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

      1. unpow2 79.8%

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

      2. unpow2 79.8%

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

      3. times-frac 78.8%

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

   7. Simplified 78.8%

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

      1. frac-times 79.8%

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

      2. associate-/l* 78.8%

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

   9. Applied egg-rr 78.8%

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

   if -1.99999999e-11 < x

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 48.7%

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

4. Final simplification 56.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.9999999920083944 \cdot 10^{-11}:\\ \;\;\;\;2 \cdot \frac{s}{\frac{x \cdot x}{s}}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 8: 59.0% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.0999999650920748 \cdot 10^{-20}:\\ \;\;\;\;2 \cdot \frac{s \cdot s}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -1.0999999650920748e-20) (* 2.0 (/ (* s s) (* x x))) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -1.0999999650920748e-20f) {
		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 <= (-1.0999999650920748e-20)) 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(-1.0999999650920748e-20))
		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(-1.0999999650920748e-20))
		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 < -1.09999997e-20

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 83.0%

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

      1. mul-1-neg 83.0%

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

      2. unsub-neg 83.0%

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

      3. unpow2 83.0%

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

      4. unpow2 83.0%

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

      5. times-frac 75.6%

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

   4. Simplified 75.6%

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

      1. associate-*r* 75.6%

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

      2. *-un-lft-identity 75.6%

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

      3. distribute-rgt-out-- 75.6%

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

      4. associate-*r/ 75.6%

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

   6. Applied egg-rr 75.6%

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

   7. Taylor expanded in x around inf 76.8%

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

      1. unpow2 76.8%

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

      2. unpow2 76.8%

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

   9. Simplified 76.8%

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

   if -1.09999997e-20 < x

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 49.9%

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

4. Final simplification 58.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.0999999650920748 \cdot 10^{-20}:\\ \;\;\;\;2 \cdot \frac{s \cdot s}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 9: 59.0% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.0999999650920748 \cdot 10^{-20}:\\ \;\;\;\;2 \cdot \frac{\frac{s \cdot s}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -1.0999999650920748e-20) (* 2.0 (/ (/ (* s s) x) x)) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -1.0999999650920748e-20f) {
		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 <= (-1.0999999650920748e-20)) 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(-1.0999999650920748e-20))
		tmp = Float32(Float32(2.0) * Float32(Float32(Float32(s * s) / 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(-1.0999999650920748e-20))
		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 < -1.09999997e-20

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 83.0%

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

      1. mul-1-neg 83.0%

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

      2. unsub-neg 83.0%

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

      3. unpow2 83.0%

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

      4. unpow2 83.0%

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

      5. times-frac 75.6%

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

   4. Simplified 75.6%

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

   5. Taylor expanded in x around inf 76.8%

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

      1. unpow2 76.8%

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

      2. unpow2 76.8%

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

      3. times-frac 68.0%

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

   7. Simplified 68.0%

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

      1. frac-times 76.8%

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

      2. associate-/r* 76.8%

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

   9. Applied egg-rr 76.8%

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

   if -1.09999997e-20 < x

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 49.9%

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

4. Final simplification 58.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.0999999650920748 \cdot 10^{-20}:\\ \;\;\;\;2 \cdot \frac{\frac{s \cdot s}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 10: 48.2% accurate, 11.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.999999943633011 \cdot 10^{-27}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 4.999999943633011e-27) (/ 1.0 (- 2.0 (/ x s))) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= 4.999999943633011e-27f) {
		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 <= 4.999999943633011e-27) 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(4.999999943633011e-27))
		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(4.999999943633011e-27))
		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 < 4.99999994e-27

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 61.9%

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

      1. mul-1-neg 61.9%

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

      2. unsub-neg 61.9%

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

   4. Simplified 61.9%

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

   if 4.99999994e-27 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 36.0%

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

4. Final simplification 49.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.999999943633011 \cdot 10^{-27}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 11: 46.0% accurate, 17.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.999999873689376 \cdot 10^{-6}:\\ \;\;\;\;\frac{-s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -4.999999873689376e-6) (/ (- s) x) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -4.999999873689376e-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 <= (-4.999999873689376e-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(-4.999999873689376e-6))
		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(-4.999999873689376e-6))
		tmp = -s / x;
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -4.99999987e-6

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 52.9%

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

      1. mul-1-neg 52.9%

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

      2. unsub-neg 52.9%

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

   4. Simplified 52.9%

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

   5. Taylor expanded in x around inf 47.5%

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

      1. associate-*r/ 47.5%

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

      2. neg-mul-1 47.5%

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

   7. Simplified 47.5%

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

   if -4.99999987e-6 < x

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 47.9%

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

4. Final simplification 47.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.999999873689376 \cdot 10^{-6}:\\ \;\;\;\;\frac{-s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 12: 34.5% accurate, 108.0× speedup

Math

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

FPCore

(FPCore (x s) :precision binary32 0.5)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.9%

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

2. Taylor expanded in x around 0 37.7%

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

3. Final simplification 37.7%

   \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023258 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))