Logistic function

Percentage Accurate: 99.8% → 99.8%

Time: 12.6s

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 99.9%

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

2. Final simplification 99.9%

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

Alternative 2: 61.0% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\frac{x}{s}\\ \mathbf{if}\;t_0 \leq -1:\\ \;\;\;\;0.5\\ \mathbf{elif}\;t_0 \leq 1.2000000427639215 \cdot 10^{+38}:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{s \cdot \frac{s}{x}}}{2 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + -0.3333333333333333 \cdot \left(\frac{x}{s} \cdot 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (- (/ x s))))
   (if (<= t_0 -1.0)
     0.5
     (if (<= t_0 1.2000000427639215e+38)
       (/ 1.0 (/ (- 4.0 (/ x (* s (/ s x)))) (+ 2.0 (/ x s))))
       (/ 1.0 (+ 2.0 (* -0.3333333333333333 (* (/ x s) 3.0))))))))

C

float code(float x, float s) {
	float t_0 = -(x / s);
	float tmp;
	if (t_0 <= -1.0f) {
		tmp = 0.5f;
	} else if (t_0 <= 1.2000000427639215e+38f) {
		tmp = 1.0f / ((4.0f - (x / (s * (s / x)))) / (2.0f + (x / s)));
	} else {
		tmp = 1.0f / (2.0f + (-0.3333333333333333f * ((x / s) * 3.0f)));
	}
	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 <= (-1.0e0)) then
        tmp = 0.5e0
    else if (t_0 <= 1.2000000427639215e+38) then
        tmp = 1.0e0 / ((4.0e0 - (x / (s * (s / x)))) / (2.0e0 + (x / s)))
    else
        tmp = 1.0e0 / (2.0e0 + ((-0.3333333333333333e0) * ((x / s) * 3.0e0)))
    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(-1.0))
		tmp = Float32(0.5);
	elseif (t_0 <= Float32(1.2000000427639215e+38))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(4.0) - Float32(x / Float32(s * Float32(s / x)))) / Float32(Float32(2.0) + Float32(x / s))));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(Float32(-0.3333333333333333) * Float32(Float32(x / s) * Float32(3.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = -(x / s);
	tmp = single(0.0);
	if (t_0 <= single(-1.0))
		tmp = single(0.5);
	elseif (t_0 <= single(1.2000000427639215e+38))
		tmp = single(1.0) / ((single(4.0) - (x / (s * (s / x)))) / (single(2.0) + (x / s)));
	else
		tmp = single(1.0) / (single(2.0) + (single(-0.3333333333333333) * ((x / s) * single(3.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 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 28.1%

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

   if -1 < (/.f32 (neg.f32 x) s) < 1.20000004e38

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 47.9%

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

      1. mul-1-neg 47.9%

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

      2. unsub-neg 47.9%

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

   4. Simplified 47.9%

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

      1. sub-neg 47.9%

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

      2. add-log-exp 95.1%

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

      3. neg-log 95.2%

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

      4. unpow-1 95.2%

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

      5. metadata-eval 95.2%

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

      6. pow-pow 95.2%

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

      7. pow1/3 95.2%

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

      8. flip-+ 41.5%

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

   6. Applied egg-rr 72.7%

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

      1. clear-num 72.7%

         \[\leadsto \frac{1}{\frac{4 - \color{blue}{\frac{1}{\frac{s}{-x}}} \cdot \frac{-x}{s}}{2 - \frac{-x}{s}}} \]

      2. frac-times 74.9%

         \[\leadsto \frac{1}{\frac{4 - \color{blue}{\frac{1 \cdot \left(-x\right)}{\frac{s}{-x} \cdot s}}}{2 - \frac{-x}{s}}} \]

      3. *-un-lft-identity 74.9%

         \[\leadsto \frac{1}{\frac{4 - \frac{\color{blue}{-x}}{\frac{s}{-x} \cdot s}}{2 - \frac{-x}{s}}} \]

      4. add-sqr-sqrt 58.3%

         \[\leadsto \frac{1}{\frac{4 - \frac{\color{blue}{\sqrt{-x} \cdot \sqrt{-x}}}{\frac{s}{-x} \cdot s}}{2 - \frac{-x}{s}}} \]

      5. sqrt-unprod 75.0%

         \[\leadsto \frac{1}{\frac{4 - \frac{\color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}}}{\frac{s}{-x} \cdot s}}{2 - \frac{-x}{s}}} \]

      6. sqr-neg 75.0%

         \[\leadsto \frac{1}{\frac{4 - \frac{\sqrt{\color{blue}{x \cdot x}}}{\frac{s}{-x} \cdot s}}{2 - \frac{-x}{s}}} \]

      7. sqrt-unprod 16.6%

         \[\leadsto \frac{1}{\frac{4 - \frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\frac{s}{-x} \cdot s}}{2 - \frac{-x}{s}}} \]

      8. add-sqr-sqrt 74.6%

         \[\leadsto \frac{1}{\frac{4 - \frac{\color{blue}{x}}{\frac{s}{-x} \cdot s}}{2 - \frac{-x}{s}}} \]

      9. add-sqr-sqrt 58.0%

         \[\leadsto \frac{1}{\frac{4 - \frac{x}{\frac{s}{\color{blue}{\sqrt{-x} \cdot \sqrt{-x}}} \cdot s}}{2 - \frac{-x}{s}}} \]

      10. sqrt-unprod 74.7%

          \[\leadsto \frac{1}{\frac{4 - \frac{x}{\frac{s}{\color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}}} \cdot s}}{2 - \frac{-x}{s}}} \]

      11. sqr-neg 74.7%

          \[\leadsto \frac{1}{\frac{4 - \frac{x}{\frac{s}{\sqrt{\color{blue}{x \cdot x}}} \cdot s}}{2 - \frac{-x}{s}}} \]

      12. sqrt-unprod 16.6%

          \[\leadsto \frac{1}{\frac{4 - \frac{x}{\frac{s}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}} \cdot s}}{2 - \frac{-x}{s}}} \]

      13. add-sqr-sqrt 74.9%

          \[\leadsto \frac{1}{\frac{4 - \frac{x}{\frac{s}{\color{blue}{x}} \cdot s}}{2 - \frac{-x}{s}}} \]

   8. Applied egg-rr 74.9%

      \[\leadsto \frac{1}{\frac{4 - \color{blue}{\frac{x}{\frac{s}{x} \cdot s}}}{2 - \frac{-x}{s}}} \]

   if 1.20000004e38 < (/.f32 (neg.f32 x) s)

   1. Initial program 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. exp-neg 100.0%

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

      3. add-sqr-sqrt -0.0%

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

      4. sqrt-unprod 6.3%

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

      5. sqr-neg 6.3%

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

      6. sqrt-unprod 6.3%

         \[\leadsto \frac{1}{1 + \frac{1}{e^{\frac{\color{blue}{\sqrt{-x} \cdot \sqrt{-x}}}{s}}}} \]

      7. add-sqr-sqrt 6.3%

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

      8. add-cbrt-cube 6.3%

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

      9. pow1/3 6.3%

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

      10. pow-flip 6.3%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in s around inf 100.0%

      \[\leadsto \frac{1}{1 + \color{blue}{\left(1 + -0.3333333333333333 \cdot \frac{x + 2 \cdot x}{s}\right)}} \]
   5. Step-by-step derivation

      1. distribute-rgt1-in 100.0%

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

      2. metadata-eval 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-/l* 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in x around 0 100.0%

      \[\leadsto \frac{1}{1 + \left(1 + -0.3333333333333333 \cdot \color{blue}{\left(3 \cdot \frac{x}{s}\right)}\right)} \]
   8. Step-by-step derivation

      1. *-commutative 100.0%

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

   9. Simplified 100.0%

      \[\leadsto \frac{1}{1 + \left(1 + -0.3333333333333333 \cdot \color{blue}{\left(\frac{x}{s} \cdot 3\right)}\right)} \]
   10. Step-by-step derivation

       1. expm1-log1p-u 100.0%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{1 + \left(1 + -0.3333333333333333 \cdot \left(\frac{x}{s} \cdot 3\right)\right)}\right)\right)} \]

       2. expm1-udef 100.0%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{1 + \left(1 + -0.3333333333333333 \cdot \left(\frac{x}{s} \cdot 3\right)\right)}\right)} - 1} \]

       3. associate-+r+ 100.0%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{\color{blue}{\left(1 + 1\right) + -0.3333333333333333 \cdot \left(\frac{x}{s} \cdot 3\right)}}\right)} - 1 \]

       4. metadata-eval 100.0%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{\color{blue}{2} + -0.3333333333333333 \cdot \left(\frac{x}{s} \cdot 3\right)}\right)} - 1 \]

       5. associate-*l/ 100.0%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{2 + -0.3333333333333333 \cdot \color{blue}{\frac{x \cdot 3}{s}}}\right)} - 1 \]

   11. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{2 + -0.3333333333333333 \cdot \frac{x \cdot 3}{s}}\right)} - 1} \]
   12. Step-by-step derivation

       1. expm1-def 100.0%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{2 + -0.3333333333333333 \cdot \frac{x \cdot 3}{s}}\right)\right)} \]

       2. expm1-log1p 100.0%

          \[\leadsto \color{blue}{\frac{1}{2 + -0.3333333333333333 \cdot \frac{x \cdot 3}{s}}} \]

       3. *-commutative 100.0%

          \[\leadsto \frac{1}{2 + -0.3333333333333333 \cdot \frac{\color{blue}{3 \cdot x}}{s}} \]

       4. associate-*r/ 100.0%

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

       5. *-commutative 100.0%

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

   13. Simplified 100.0%

       \[\leadsto \color{blue}{\frac{1}{2 + -0.3333333333333333 \cdot \left(\frac{x}{s} \cdot 3\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 56.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq -1:\\ \;\;\;\;0.5\\ \mathbf{elif}\;-\frac{x}{s} \leq 1.2000000427639215 \cdot 10^{+38}:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{s \cdot \frac{s}{x}}}{2 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + -0.3333333333333333 \cdot \left(\frac{x}{s} \cdot 3\right)}\\ \end{array} \]

Alternative 3: 58.4% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\frac{x}{s}\\ \mathbf{if}\;t_0 \leq 2:\\ \;\;\;\;0.5\\ \mathbf{elif}\;t_0 \leq 1.2000000427639215 \cdot 10^{+38}:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{s} \cdot \frac{x}{s}}{\frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + -0.3333333333333333 \cdot \left(\frac{x}{s} \cdot 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (- (/ x s))))
   (if (<= t_0 2.0)
     0.5
     (if (<= t_0 1.2000000427639215e+38)
       (/ 1.0 (/ (- 4.0 (* (/ x s) (/ x s))) (/ x s)))
       (/ 1.0 (+ 2.0 (* -0.3333333333333333 (* (/ x s) 3.0))))))))

C

float code(float x, float s) {
	float t_0 = -(x / s);
	float tmp;
	if (t_0 <= 2.0f) {
		tmp = 0.5f;
	} else if (t_0 <= 1.2000000427639215e+38f) {
		tmp = 1.0f / ((4.0f - ((x / s) * (x / s))) / (x / s));
	} else {
		tmp = 1.0f / (2.0f + (-0.3333333333333333f * ((x / s) * 3.0f)));
	}
	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 if (t_0 <= 1.2000000427639215e+38) then
        tmp = 1.0e0 / ((4.0e0 - ((x / s) * (x / s))) / (x / s))
    else
        tmp = 1.0e0 / (2.0e0 + ((-0.3333333333333333e0) * ((x / s) * 3.0e0)))
    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);
	elseif (t_0 <= Float32(1.2000000427639215e+38))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(4.0) - Float32(Float32(x / s) * Float32(x / s))) / Float32(x / s)));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(Float32(-0.3333333333333333) * Float32(Float32(x / s) * Float32(3.0)))));
	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);
	elseif (t_0 <= single(1.2000000427639215e+38))
		tmp = single(1.0) / ((single(4.0) - ((x / s) * (x / s))) / (x / s));
	else
		tmp = single(1.0) / (single(2.0) + (single(-0.3333333333333333) * ((x / s) * single(3.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 47.8%

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

   if 2 < (/.f32 (neg.f32 x) s) < 1.20000004e38

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 12.0%

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

      1. mul-1-neg 12.0%

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

      2. unsub-neg 12.0%

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

   4. Simplified 12.0%

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

      1. sub-neg 12.0%

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

      2. add-log-exp 95.1%

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

      3. neg-log 95.1%

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

      4. unpow-1 95.1%

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

      5. metadata-eval 95.1%

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

      6. pow-pow 95.1%

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

      7. pow1/3 95.1%

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

      8. flip-+ 0.8%

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

   6. Applied egg-rr 55.6%

      \[\leadsto \frac{1}{\color{blue}{\frac{4 - \frac{-x}{s} \cdot \frac{-x}{s}}{2 - \frac{-x}{s}}}} \]

   7. Taylor expanded in x around inf 55.6%

      \[\leadsto \frac{1}{\frac{4 - \frac{-x}{s} \cdot \frac{-x}{s}}{\color{blue}{\frac{x}{s}}}} \]

   if 1.20000004e38 < (/.f32 (neg.f32 x) s)

   1. Initial program 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. exp-neg 100.0%

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

      3. add-sqr-sqrt -0.0%

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

      4. sqrt-unprod 6.3%

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

      5. sqr-neg 6.3%

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

      6. sqrt-unprod 6.3%

         \[\leadsto \frac{1}{1 + \frac{1}{e^{\frac{\color{blue}{\sqrt{-x} \cdot \sqrt{-x}}}{s}}}} \]

      7. add-sqr-sqrt 6.3%

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

      8. add-cbrt-cube 6.3%

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

      9. pow1/3 6.3%

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

      10. pow-flip 6.3%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in s around inf 100.0%

      \[\leadsto \frac{1}{1 + \color{blue}{\left(1 + -0.3333333333333333 \cdot \frac{x + 2 \cdot x}{s}\right)}} \]
   5. Step-by-step derivation

      1. distribute-rgt1-in 100.0%

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

      2. metadata-eval 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-/l* 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in x around 0 100.0%

      \[\leadsto \frac{1}{1 + \left(1 + -0.3333333333333333 \cdot \color{blue}{\left(3 \cdot \frac{x}{s}\right)}\right)} \]
   8. Step-by-step derivation

      1. *-commutative 100.0%

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

   9. Simplified 100.0%

      \[\leadsto \frac{1}{1 + \left(1 + -0.3333333333333333 \cdot \color{blue}{\left(\frac{x}{s} \cdot 3\right)}\right)} \]
   10. Step-by-step derivation

       1. expm1-log1p-u 100.0%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{1 + \left(1 + -0.3333333333333333 \cdot \left(\frac{x}{s} \cdot 3\right)\right)}\right)\right)} \]

       2. expm1-udef 100.0%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{1 + \left(1 + -0.3333333333333333 \cdot \left(\frac{x}{s} \cdot 3\right)\right)}\right)} - 1} \]

       3. associate-+r+ 100.0%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{\color{blue}{\left(1 + 1\right) + -0.3333333333333333 \cdot \left(\frac{x}{s} \cdot 3\right)}}\right)} - 1 \]

       4. metadata-eval 100.0%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{\color{blue}{2} + -0.3333333333333333 \cdot \left(\frac{x}{s} \cdot 3\right)}\right)} - 1 \]

       5. associate-*l/ 100.0%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{2 + -0.3333333333333333 \cdot \color{blue}{\frac{x \cdot 3}{s}}}\right)} - 1 \]

   11. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{2 + -0.3333333333333333 \cdot \frac{x \cdot 3}{s}}\right)} - 1} \]
   12. Step-by-step derivation

       1. expm1-def 100.0%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{2 + -0.3333333333333333 \cdot \frac{x \cdot 3}{s}}\right)\right)} \]

       2. expm1-log1p 100.0%

          \[\leadsto \color{blue}{\frac{1}{2 + -0.3333333333333333 \cdot \frac{x \cdot 3}{s}}} \]

       3. *-commutative 100.0%

          \[\leadsto \frac{1}{2 + -0.3333333333333333 \cdot \frac{\color{blue}{3 \cdot x}}{s}} \]

       4. associate-*r/ 100.0%

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

       5. *-commutative 100.0%

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

   13. Simplified 100.0%

       \[\leadsto \color{blue}{\frac{1}{2 + -0.3333333333333333 \cdot \left(\frac{x}{s} \cdot 3\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 54.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq 2:\\ \;\;\;\;0.5\\ \mathbf{elif}\;-\frac{x}{s} \leq 1.2000000427639215 \cdot 10^{+38}:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{s} \cdot \frac{x}{s}}{\frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + -0.3333333333333333 \cdot \left(\frac{x}{s} \cdot 3\right)}\\ \end{array} \]

Alternative 4: 49.2% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq -1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \left(1 + -0.3333333333333333 \cdot \left(\frac{x}{s} \cdot 3\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- (/ x s)) -1.0)
   0.5
   (/ 1.0 (+ 1.0 (+ 1.0 (* -0.3333333333333333 (* (/ x s) 3.0)))))))

C

float code(float x, float s) {
	float tmp;
	if (-(x / s) <= -1.0f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / (1.0f + (1.0f + (-0.3333333333333333f * ((x / s) * 3.0f))));
	}
	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 / (1.0e0 + (1.0e0 + ((-0.3333333333333333e0) * ((x / s) * 3.0e0))))
    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(1.0) + Float32(Float32(1.0) + Float32(Float32(-0.3333333333333333) * Float32(Float32(x / s) * Float32(3.0))))));
	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(1.0) + (single(1.0) + (single(-0.3333333333333333) * ((x / s) * single(3.0)))));
	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 28.1%

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

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

   1. Initial program 99.8%

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

      1. distribute-frac-neg 99.8%

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

      2. exp-neg 99.7%

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

      3. add-sqr-sqrt 14.5%

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

      4. sqrt-unprod 37.7%

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

      5. sqr-neg 37.7%

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

      6. sqrt-unprod 23.7%

         \[\leadsto \frac{1}{1 + \frac{1}{e^{\frac{\color{blue}{\sqrt{-x} \cdot \sqrt{-x}}}{s}}}} \]

      7. add-sqr-sqrt 36.4%

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

      8. add-cbrt-cube 36.4%

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

      9. pow1/3 36.4%

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

      10. pow-flip 36.4%

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

   3. Applied egg-rr 98.6%

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

   4. Taylor expanded in s around inf 55.8%

      \[\leadsto \frac{1}{1 + \color{blue}{\left(1 + -0.3333333333333333 \cdot \frac{x + 2 \cdot x}{s}\right)}} \]
   5. Step-by-step derivation

      1. distribute-rgt1-in 55.8%

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

      2. metadata-eval 55.8%

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

      3. *-commutative 55.8%

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

      4. associate-/l* 55.8%

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

   6. Simplified 55.8%

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

   7. Taylor expanded in x around 0 55.8%

      \[\leadsto \frac{1}{1 + \left(1 + -0.3333333333333333 \cdot \color{blue}{\left(3 \cdot \frac{x}{s}\right)}\right)} \]
   8. Step-by-step derivation

      1. *-commutative 55.8%

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

   9. Simplified 55.8%

      \[\leadsto \frac{1}{1 + \left(1 + -0.3333333333333333 \cdot \color{blue}{\left(\frac{x}{s} \cdot 3\right)}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq -1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \left(1 + -0.3333333333333333 \cdot \left(\frac{x}{s} \cdot 3\right)\right)}\\ \end{array} \]

Alternative 5: 49.2% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq -1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \left(1 + -0.3333333333333333 \cdot \frac{x}{\frac{s}{3}}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- (/ x s)) -1.0)
   0.5
   (/ 1.0 (+ 1.0 (+ 1.0 (* -0.3333333333333333 (/ x (/ s 3.0))))))))

C

float code(float x, float s) {
	float tmp;
	if (-(x / s) <= -1.0f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / (1.0f + (1.0f + (-0.3333333333333333f * (x / (s / 3.0f)))));
	}
	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 / (1.0e0 + (1.0e0 + ((-0.3333333333333333e0) * (x / (s / 3.0e0)))))
    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(1.0) + Float32(Float32(1.0) + Float32(Float32(-0.3333333333333333) * Float32(x / Float32(s / Float32(3.0)))))));
	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(1.0) + (single(1.0) + (single(-0.3333333333333333) * (x / (s / single(3.0))))));
	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 28.1%

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

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

   1. Initial program 99.8%

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

      1. distribute-frac-neg 99.8%

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

      2. exp-neg 99.7%

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

      3. add-sqr-sqrt 14.5%

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

      4. sqrt-unprod 37.7%

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

      5. sqr-neg 37.7%

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

      6. sqrt-unprod 23.7%

         \[\leadsto \frac{1}{1 + \frac{1}{e^{\frac{\color{blue}{\sqrt{-x} \cdot \sqrt{-x}}}{s}}}} \]

      7. add-sqr-sqrt 36.4%

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

      8. add-cbrt-cube 36.4%

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

      9. pow1/3 36.4%

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

      10. pow-flip 36.4%

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

   3. Applied egg-rr 98.6%

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

   4. Taylor expanded in s around inf 55.8%

      \[\leadsto \frac{1}{1 + \color{blue}{\left(1 + -0.3333333333333333 \cdot \frac{x + 2 \cdot x}{s}\right)}} \]
   5. Step-by-step derivation

      1. distribute-rgt1-in 55.8%

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

      2. metadata-eval 55.8%

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

      3. *-commutative 55.8%

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

      4. associate-/l* 55.8%

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

   6. Simplified 55.8%

      \[\leadsto \frac{1}{1 + \color{blue}{\left(1 + -0.3333333333333333 \cdot \frac{x}{\frac{s}{3}}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq -1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \left(1 + -0.3333333333333333 \cdot \frac{x}{\frac{s}{3}}\right)}\\ \end{array} \]

Alternative 6: 49.2% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq -1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + -0.3333333333333333 \cdot \left(\frac{x}{s} \cdot 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- (/ x s)) -1.0)
   0.5
   (/ 1.0 (+ 2.0 (* -0.3333333333333333 (* (/ x s) 3.0))))))

C

float code(float x, float s) {
	float tmp;
	if (-(x / s) <= -1.0f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / (2.0f + (-0.3333333333333333f * ((x / s) * 3.0f)));
	}
	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 + ((-0.3333333333333333e0) * ((x / s) * 3.0e0)))
    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(Float32(-0.3333333333333333) * Float32(Float32(x / s) * Float32(3.0)))));
	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) + (single(-0.3333333333333333) * ((x / s) * single(3.0))));
	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 28.1%

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

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

   1. Initial program 99.8%

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

      1. distribute-frac-neg 99.8%

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

      2. exp-neg 99.7%

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

      3. add-sqr-sqrt 14.5%

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

      4. sqrt-unprod 37.7%

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

      5. sqr-neg 37.7%

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

      6. sqrt-unprod 23.7%

         \[\leadsto \frac{1}{1 + \frac{1}{e^{\frac{\color{blue}{\sqrt{-x} \cdot \sqrt{-x}}}{s}}}} \]

      7. add-sqr-sqrt 36.4%

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

      8. add-cbrt-cube 36.4%

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

      9. pow1/3 36.4%

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

      10. pow-flip 36.4%

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

   3. Applied egg-rr 98.6%

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

   4. Taylor expanded in s around inf 55.8%

      \[\leadsto \frac{1}{1 + \color{blue}{\left(1 + -0.3333333333333333 \cdot \frac{x + 2 \cdot x}{s}\right)}} \]
   5. Step-by-step derivation

      1. distribute-rgt1-in 55.8%

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

      2. metadata-eval 55.8%

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

      3. *-commutative 55.8%

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

      4. associate-/l* 55.8%

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

   6. Simplified 55.8%

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

   7. Taylor expanded in x around 0 55.8%

      \[\leadsto \frac{1}{1 + \left(1 + -0.3333333333333333 \cdot \color{blue}{\left(3 \cdot \frac{x}{s}\right)}\right)} \]
   8. Step-by-step derivation

      1. *-commutative 55.8%

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

   9. Simplified 55.8%

      \[\leadsto \frac{1}{1 + \left(1 + -0.3333333333333333 \cdot \color{blue}{\left(\frac{x}{s} \cdot 3\right)}\right)} \]
   10. Step-by-step derivation

       1. expm1-log1p-u 55.8%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{1 + \left(1 + -0.3333333333333333 \cdot \left(\frac{x}{s} \cdot 3\right)\right)}\right)\right)} \]

       2. expm1-udef 88.1%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{1 + \left(1 + -0.3333333333333333 \cdot \left(\frac{x}{s} \cdot 3\right)\right)}\right)} - 1} \]

       3. associate-+r+ 88.1%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{\color{blue}{\left(1 + 1\right) + -0.3333333333333333 \cdot \left(\frac{x}{s} \cdot 3\right)}}\right)} - 1 \]

       4. metadata-eval 88.1%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{\color{blue}{2} + -0.3333333333333333 \cdot \left(\frac{x}{s} \cdot 3\right)}\right)} - 1 \]

       5. associate-*l/ 88.1%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{2 + -0.3333333333333333 \cdot \color{blue}{\frac{x \cdot 3}{s}}}\right)} - 1 \]

   11. Applied egg-rr 88.1%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{2 + -0.3333333333333333 \cdot \frac{x \cdot 3}{s}}\right)} - 1} \]
   12. Step-by-step derivation

       1. expm1-def 55.7%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{2 + -0.3333333333333333 \cdot \frac{x \cdot 3}{s}}\right)\right)} \]

       2. expm1-log1p 55.8%

          \[\leadsto \color{blue}{\frac{1}{2 + -0.3333333333333333 \cdot \frac{x \cdot 3}{s}}} \]

       3. *-commutative 55.8%

          \[\leadsto \frac{1}{2 + -0.3333333333333333 \cdot \frac{\color{blue}{3 \cdot x}}{s}} \]

       4. associate-*r/ 55.8%

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

       5. *-commutative 55.8%

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

   13. Simplified 55.8%

       \[\leadsto \color{blue}{\frac{1}{2 + -0.3333333333333333 \cdot \left(\frac{x}{s} \cdot 3\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq -1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + -0.3333333333333333 \cdot \left(\frac{x}{s} \cdot 3\right)}\\ \end{array} \]

Alternative 7: 49.2% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq -1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + -0.3333333333333333 \cdot \frac{x \cdot 3}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- (/ x s)) -1.0)
   0.5
   (/ 1.0 (+ 2.0 (* -0.3333333333333333 (/ (* x 3.0) s))))))

C

float code(float x, float s) {
	float tmp;
	if (-(x / s) <= -1.0f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / (2.0f + (-0.3333333333333333f * ((x * 3.0f) / 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 + ((-0.3333333333333333e0) * ((x * 3.0e0) / 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(Float32(-0.3333333333333333) * Float32(Float32(x * Float32(3.0)) / 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) + (single(-0.3333333333333333) * ((x * single(3.0)) / 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 28.1%

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

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

   1. Initial program 99.8%

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

      1. distribute-frac-neg 99.8%

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

      2. exp-neg 99.7%

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

      3. add-sqr-sqrt 14.5%

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

      4. sqrt-unprod 37.7%

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

      5. sqr-neg 37.7%

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

      6. sqrt-unprod 23.7%

         \[\leadsto \frac{1}{1 + \frac{1}{e^{\frac{\color{blue}{\sqrt{-x} \cdot \sqrt{-x}}}{s}}}} \]

      7. add-sqr-sqrt 36.4%

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

      8. add-cbrt-cube 36.4%

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

      9. pow1/3 36.4%

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

      10. pow-flip 36.4%

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

   3. Applied egg-rr 98.6%

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

   4. Taylor expanded in s around inf 55.8%

      \[\leadsto \frac{1}{1 + \color{blue}{\left(1 + -0.3333333333333333 \cdot \frac{x + 2 \cdot x}{s}\right)}} \]
   5. Step-by-step derivation

      1. distribute-rgt1-in 55.8%

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

      2. metadata-eval 55.8%

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

      3. *-commutative 55.8%

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

      4. associate-/l* 55.8%

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

   6. Simplified 55.8%

      \[\leadsto \frac{1}{1 + \color{blue}{\left(1 + -0.3333333333333333 \cdot \frac{x}{\frac{s}{3}}\right)}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 55.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{1 + \left(1 + -0.3333333333333333 \cdot \frac{x}{\frac{s}{3}}\right)}\right)\right)} \]

      2. expm1-udef 88.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{1 + \left(1 + -0.3333333333333333 \cdot \frac{x}{\frac{s}{3}}\right)}\right)} - 1} \]

      3. associate-+r+ 88.1%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{\color{blue}{\left(1 + 1\right) + -0.3333333333333333 \cdot \frac{x}{\frac{s}{3}}}}\right)} - 1 \]

      4. metadata-eval 88.1%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{\color{blue}{2} + -0.3333333333333333 \cdot \frac{x}{\frac{s}{3}}}\right)} - 1 \]

      5. associate-/r/ 88.1%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{2 + -0.3333333333333333 \cdot \color{blue}{\left(\frac{x}{s} \cdot 3\right)}}\right)} - 1 \]

   8. Applied egg-rr 88.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{2 + -0.3333333333333333 \cdot \left(\frac{x}{s} \cdot 3\right)}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 55.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{2 + -0.3333333333333333 \cdot \left(\frac{x}{s} \cdot 3\right)}\right)\right)} \]

      2. expm1-log1p 55.8%

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

      3. associate-*l/ 55.8%

         \[\leadsto \frac{1}{2 + -0.3333333333333333 \cdot \color{blue}{\frac{x \cdot 3}{s}}} \]

   10. Simplified 55.8%

       \[\leadsto \color{blue}{\frac{1}{2 + -0.3333333333333333 \cdot \frac{x \cdot 3}{s}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq -1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + -0.3333333333333333 \cdot \frac{x \cdot 3}{s}}\\ \end{array} \]

Alternative 8: 49.0% 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 28.1%

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

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

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 55.3%

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

      1. mul-1-neg 55.3%

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

      2. unsub-neg 55.3%

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

   4. Simplified 55.3%

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

4. Final simplification 43.5%

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

Alternative 9: 47.5% 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 100.0%

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

   2. Taylor expanded in x around 0 47.8%

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

   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 32.3%

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

      1. mul-1-neg 32.3%

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

      2. unsub-neg 32.3%

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

   4. Simplified 32.3%

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

   5. Taylor expanded in x around inf 32.3%

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

      1. mul-1-neg 32.3%

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

      2. distribute-frac-neg 32.3%

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

   7. Simplified 32.3%

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

4. Final simplification 42.3%

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

Alternative 10: 45.6% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-x \leq 9.999999960041972 \cdot 10^{-13}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{-s}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- x) 9.999999960041972e-13) 0.5 (/ (- s) x)))

C

float code(float x, float s) {
	float tmp;
	if (-x <= 9.999999960041972e-13f) {
		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 <= 9.999999960041972e-13) 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(9.999999960041972e-13))
		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(9.999999960041972e-13))
		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) < 9.99999996e-13

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 43.6%

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

   if 9.99999996e-13 < (neg.f32 x)

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 38.6%

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

      1. mul-1-neg 38.6%

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

      2. unsub-neg 38.6%

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

   4. Simplified 38.6%

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

   5. Taylor expanded in x around inf 34.8%

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

      1. associate-*r/ 34.8%

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

      2. neg-mul-1 34.8%

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

   7. Simplified 34.8%

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

4. Final simplification 41.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-x \leq 9.999999960041972 \cdot 10^{-13}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{-s}{x}\\ \end{array} \]

Alternative 11: 35.3% 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 33.0%

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

3. Final simplification 33.0%

   \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023336 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))