Logistic function

Percentage Accurate: 99.8% → 99.8%

Time: 15.1s

Alternatives: 14

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{1}{{\left(e^{1.5}\right)}^{\left(\frac{x \cdot -0.6666666666666666}{s}\right)} + 1} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (/ 1.0 (+ (pow (exp 1.5) (/ (* x -0.6666666666666666) s)) 1.0)))

C

float code(float x, float s) {
	return 1.0f / (powf(expf(1.5f), ((x * -0.6666666666666666f) / s)) + 1.0f);
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 1.0e0 / ((exp(1.5e0) ** ((x * (-0.6666666666666666e0)) / s)) + 1.0e0)
end function

Julia

function code(x, s)
	return Float32(Float32(1.0) / Float32((exp(Float32(1.5)) ^ Float32(Float32(x * Float32(-0.6666666666666666)) / s)) + Float32(1.0)))
end

MATLAB

function tmp = code(x, s)
	tmp = single(1.0) / ((exp(single(1.5)) ^ ((x * single(-0.6666666666666666)) / s)) + single(1.0));
end

Derivation

1. Initial program 99.8%

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

   1. distribute-frac-neg 99.8%

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

   2. exp-neg 99.8%

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

   3. add-sqr-sqrt 51.9%

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

   4. sqrt-unprod 62.6%

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

   5. sqr-neg 62.6%

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

   6. sqrt-unprod 12.5%

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

   7. add-sqr-sqrt 23.1%

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

   8. add-cbrt-cube 23.1%

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

   9. pow1/3 23.1%

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

   10. pow-flip 23.1%

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

4. Applied egg-rr 99.5%

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

   1. add-sqr-sqrt 99.5%

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

   2. unpow-prod-down 99.5%

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

   3. sqrt-pow1 99.5%

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

   4. metadata-eval 99.5%

      \[\leadsto \frac{1}{1 + {\left({\left(e^{\frac{x}{s}}\right)}^{\color{blue}{1.5}}\right)}^{-0.3333333333333333} \cdot {\left(\sqrt{{\left(e^{\frac{x}{s}}\right)}^{3}}\right)}^{-0.3333333333333333}} \]

   5. sqrt-pow1 99.8%

      \[\leadsto \frac{1}{1 + {\left({\left(e^{\frac{x}{s}}\right)}^{1.5}\right)}^{-0.3333333333333333} \cdot {\color{blue}{\left({\left(e^{\frac{x}{s}}\right)}^{\left(\frac{3}{2}\right)}\right)}}^{-0.3333333333333333}} \]

   6. metadata-eval 99.8%

      \[\leadsto \frac{1}{1 + {\left({\left(e^{\frac{x}{s}}\right)}^{1.5}\right)}^{-0.3333333333333333} \cdot {\left({\left(e^{\frac{x}{s}}\right)}^{\color{blue}{1.5}}\right)}^{-0.3333333333333333}} \]

6. Applied egg-rr 99.8%

   \[\leadsto \frac{1}{1 + \color{blue}{{\left({\left(e^{\frac{x}{s}}\right)}^{1.5}\right)}^{-0.3333333333333333} \cdot {\left({\left(e^{\frac{x}{s}}\right)}^{1.5}\right)}^{-0.3333333333333333}}} \]
7. Step-by-step derivation

   1. pow-prod-up 99.8%

      \[\leadsto \frac{1}{1 + \color{blue}{{\left({\left(e^{\frac{x}{s}}\right)}^{1.5}\right)}^{\left(-0.3333333333333333 + -0.3333333333333333\right)}}} \]

   2. metadata-eval 99.8%

      \[\leadsto \frac{1}{1 + {\left({\left(e^{\frac{x}{s}}\right)}^{1.5}\right)}^{\color{blue}{-0.6666666666666666}}} \]

8. Applied egg-rr 99.8%

   \[\leadsto \frac{1}{1 + \color{blue}{{\left({\left(e^{\frac{x}{s}}\right)}^{1.5}\right)}^{-0.6666666666666666}}} \]
9. Step-by-step derivation

   1. exp-prod 99.8%

      \[\leadsto \frac{1}{1 + {\color{blue}{\left(e^{\frac{x}{s} \cdot 1.5}\right)}}^{-0.6666666666666666}} \]

   2. *-commutative 99.8%

      \[\leadsto \frac{1}{1 + {\left(e^{\color{blue}{1.5 \cdot \frac{x}{s}}}\right)}^{-0.6666666666666666}} \]

   3. exp-prod 99.8%

      \[\leadsto \frac{1}{1 + \color{blue}{e^{\left(1.5 \cdot \frac{x}{s}\right) \cdot -0.6666666666666666}}} \]

   4. associate-*l* 99.8%

      \[\leadsto \frac{1}{1 + e^{\color{blue}{1.5 \cdot \left(\frac{x}{s} \cdot -0.6666666666666666\right)}}} \]

   5. exp-prod 99.8%

      \[\leadsto \frac{1}{1 + \color{blue}{{\left(e^{1.5}\right)}^{\left(\frac{x}{s} \cdot -0.6666666666666666\right)}}} \]

   6. associate-*l/ 99.8%

      \[\leadsto \frac{1}{1 + {\left(e^{1.5}\right)}^{\color{blue}{\left(\frac{x \cdot -0.6666666666666666}{s}\right)}}} \]

10. Simplified 99.8%

    \[\leadsto \frac{1}{1 + \color{blue}{{\left(e^{1.5}\right)}^{\left(\frac{x \cdot -0.6666666666666666}{s}\right)}}} \]

11. Final simplification 99.8%

    \[\leadsto \frac{1}{{\left(e^{1.5}\right)}^{\left(\frac{x \cdot -0.6666666666666666}{s}\right)} + 1} \]
12. Add Preprocessing

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

   \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
2. Add Preprocessing

3. Final simplification 99.8%

   \[\leadsto \frac{1}{e^{\frac{-x}{s}} + 1} \]
4. Add Preprocessing

Alternative 3: 60.9% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq -10:\\ \;\;\;\;0.5\\ \mathbf{elif}\;t_0 \leq 9.999999680285692 \cdot 10^{+37}:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{s \cdot \frac{s}{x}}}{2 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)))
   (if (<= t_0 -10.0)
     0.5
     (if (<= t_0 9.999999680285692e+37)
       (/ 1.0 (/ (- 4.0 (/ x (* s (/ s x)))) (+ 2.0 (/ x s))))
       (/ -1.0 (/ x s))))))

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if (t_0 <= -10.0f) {
		tmp = 0.5f;
	} else if (t_0 <= 9.999999680285692e+37f) {
		tmp = 1.0f / ((4.0f - (x / (s * (s / x)))) / (2.0f + (x / s)));
	} else {
		tmp = -1.0f / (x / s);
	}
	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 <= (-10.0e0)) then
        tmp = 0.5e0
    else if (t_0 <= 9.999999680285692e+37) then
        tmp = 1.0e0 / ((4.0e0 - (x / (s * (s / x)))) / (2.0e0 + (x / s)))
    else
        tmp = (-1.0e0) / (x / s)
    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(-10.0))
		tmp = Float32(0.5);
	elseif (t_0 <= Float32(9.999999680285692e+37))
		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(x / s));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = -x / s;
	tmp = single(0.0);
	if (t_0 <= single(-10.0))
		tmp = single(0.5);
	elseif (t_0 <= single(9.999999680285692e+37))
		tmp = single(1.0) / ((single(4.0) - (x / (s * (s / x)))) / (single(2.0) + (x / s)));
	else
		tmp = single(-1.0) / (x / s);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if (/.f32 (neg.f32 x) s) < -10

   1. Initial program 99.9%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 28.1%

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

   if -10 < (/.f32 (neg.f32 x) s) < 9.99999968e37

   1. Initial program 99.7%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 50.5%

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

      1. mul-1-neg 50.5%

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

      2. unsub-neg 50.5%

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

   5. Simplified 50.5%

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

      1. sub-neg 50.5%

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

      2. neg-mul-1 50.5%

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

      3. add-log-exp 94.4%

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

      4. log-pow 94.4%

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

      5. metadata-eval 94.4%

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

      6. pow-pow 94.4%

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

      7. flip-+ 44.7%

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

   7. Applied egg-rr 76.0%

      \[\leadsto \frac{1}{\color{blue}{\frac{4 - \left(-\frac{x}{s}\right) \cdot \left(-\frac{x}{s}\right)}{2 - \left(-\frac{x}{s}\right)}}} \]
   8. Step-by-step derivation

      1. sqr-neg 76.0%

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

      2. clear-num 76.0%

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

      3. frac-times 77.6%

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

      4. *-un-lft-identity 77.6%

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

   9. Applied egg-rr 77.6%

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

   if 9.99999968e37 < (/.f32 (neg.f32 x) s)

   1. Initial program 100.0%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf 100.0%

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

      1. neg-mul-1 100.0%

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

      2. distribute-neg-frac 100.0%

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

   8. Simplified 100.0%

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

      1. neg-mul-1 100.0%

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

      2. associate-/l* 100.0%

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

      3. metadata-eval 100.0%

         \[\leadsto \frac{1}{\frac{\color{blue}{\frac{0.3333333333333333}{-0.3333333333333333}}}{\frac{s}{x}}} \]

      4. associate-/r* 100.0%

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

      5. clear-num 91.4%

         \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot \frac{s}{x}}{0.3333333333333333}} \]

      6. div-inv 91.4%

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

      7. associate-*r/ 91.4%

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

      8. associate-/l* 100.0%

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

      9. metadata-eval 100.0%

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

   10. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{\frac{-0.3333333333333333}{\frac{x}{s}} \cdot 3} \]
   11. Step-by-step derivation

       1. associate-*l/ 100.0%

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

       2. metadata-eval 100.0%

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

   12. Simplified 100.0%

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

4. Final simplification 59.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -10:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\frac{-x}{s} \leq 9.999999680285692 \cdot 10^{+37}:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{s \cdot \frac{s}{x}}}{2 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{x}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 58.4% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq 2:\\ \;\;\;\;0.5\\ \mathbf{elif}\;t_0 \leq 9.999999680285692 \cdot 10^{+37}:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{s} \cdot \frac{x}{s}}{\frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)))
   (if (<= t_0 2.0)
     0.5
     (if (<= t_0 9.999999680285692e+37)
       (/ 1.0 (/ (- 4.0 (* (/ x s) (/ x s))) (/ x s)))
       (/ -1.0 (/ x s))))))

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 <= 9.999999680285692e+37f) {
		tmp = 1.0f / ((4.0f - ((x / s) * (x / s))) / (x / s));
	} else {
		tmp = -1.0f / (x / s);
	}
	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 <= 9.999999680285692e+37) then
        tmp = 1.0e0 / ((4.0e0 - ((x / s) * (x / s))) / (x / s))
    else
        tmp = (-1.0e0) / (x / s)
    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(9.999999680285692e+37))
		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(x / s));
	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(9.999999680285692e+37))
		tmp = single(1.0) / ((single(4.0) - ((x / s) * (x / s))) / (x / s));
	else
		tmp = single(-1.0) / (x / s);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.9%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 47.6%

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

   if 2 < (/.f32 (neg.f32 x) s) < 9.99999968e37

   1. Initial program 99.6%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 11.6%

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

      1. mul-1-neg 11.6%

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

      2. unsub-neg 11.6%

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

   5. Simplified 11.6%

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

      1. sub-neg 11.6%

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

      2. neg-mul-1 11.6%

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

      3. add-log-exp 94.5%

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

      4. log-pow 94.5%

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

      5. metadata-eval 94.5%

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

      6. pow-pow 94.5%

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

      7. flip-+ 0.9%

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

   7. Applied egg-rr 59.8%

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

   8. Taylor expanded in x around inf 59.7%

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

   if 9.99999968e37 < (/.f32 (neg.f32 x) s)

   1. Initial program 100.0%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf 100.0%

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

      1. neg-mul-1 100.0%

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

      2. distribute-neg-frac 100.0%

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

   8. Simplified 100.0%

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

      1. neg-mul-1 100.0%

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

      2. associate-/l* 100.0%

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

      3. metadata-eval 100.0%

         \[\leadsto \frac{1}{\frac{\color{blue}{\frac{0.3333333333333333}{-0.3333333333333333}}}{\frac{s}{x}}} \]

      4. associate-/r* 100.0%

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

      5. clear-num 91.4%

         \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot \frac{s}{x}}{0.3333333333333333}} \]

      6. div-inv 91.4%

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

      7. associate-*r/ 91.4%

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

      8. associate-/l* 100.0%

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

      9. metadata-eval 100.0%

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

   10. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{\frac{-0.3333333333333333}{\frac{x}{s}} \cdot 3} \]
   11. Step-by-step derivation

       1. associate-*l/ 100.0%

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

       2. metadata-eval 100.0%

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

   12. Simplified 100.0%

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

4. Final simplification 57.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 2:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\frac{-x}{s} \leq 9.999999680285692 \cdot 10^{+37}:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{s} \cdot \frac{x}{s}}{\frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{x}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 49.5% accurate, 5.4× speedup

Math

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

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) -10.0)
   0.5
   (/ 1.0 (+ (+ (* 0.3333333333333333 (* x (* -3.0 (/ 1.0 s)))) 1.0) 1.0))))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= -10.0f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / (((0.3333333333333333f * (x * (-3.0f * (1.0f / s)))) + 1.0f) + 1.0f);
	}
	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) <= (-10.0e0)) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 / (((0.3333333333333333e0 * (x * ((-3.0e0) * (1.0e0 / s)))) + 1.0e0) + 1.0e0)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(-x) / s) <= Float32(-10.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(Float32(0.3333333333333333) * Float32(x * Float32(Float32(-3.0) * Float32(Float32(1.0) / s)))) + Float32(1.0)) + Float32(1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((-x / s) <= single(-10.0))
		tmp = single(0.5);
	else
		tmp = single(1.0) / (((single(0.3333333333333333) * (x * (single(-3.0) * (single(1.0) / s)))) + single(1.0)) + single(1.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < -10

   1. Initial program 99.9%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 28.1%

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

   if -10 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.7%

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

      1. distribute-frac-neg 99.7%

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

      2. exp-neg 99.7%

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

      3. add-sqr-sqrt 16.8%

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

      4. sqrt-unprod 37.8%

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

      5. sqr-neg 37.8%

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

      6. sqrt-unprod 21.6%

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

      7. add-sqr-sqrt 35.3%

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

      8. add-cbrt-cube 35.3%

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

      9. pow1/3 35.3%

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

      10. pow-flip 35.3%

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

   4. Applied egg-rr 99.2%

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

   5. Taylor expanded in s around -inf 61.5%

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

      1. distribute-rgt-out 61.5%

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

      2. metadata-eval 61.5%

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

   7. Simplified 61.5%

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

      1. div-inv 62.2%

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

      2. associate-*l* 62.2%

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

   9. Applied egg-rr 62.2%

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

4. Final simplification 47.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -10:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(0.3333333333333333 \cdot \left(x \cdot \left(-3 \cdot \frac{1}{s}\right)\right) + 1\right) + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 49.3% accurate, 6.0× speedup

Math

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

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) -10.0)
   0.5
   (/ 1.0 (+ (+ (* 0.3333333333333333 (* (/ x s) -3.0)) 1.0) 1.0))))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= -10.0f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / (((0.3333333333333333f * ((x / s) * -3.0f)) + 1.0f) + 1.0f);
	}
	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) <= (-10.0e0)) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 / (((0.3333333333333333e0 * ((x / s) * (-3.0e0))) + 1.0e0) + 1.0e0)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(-x) / s) <= Float32(-10.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(Float32(0.3333333333333333) * Float32(Float32(x / s) * Float32(-3.0))) + Float32(1.0)) + Float32(1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((-x / s) <= single(-10.0))
		tmp = single(0.5);
	else
		tmp = single(1.0) / (((single(0.3333333333333333) * ((x / s) * single(-3.0))) + single(1.0)) + single(1.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < -10

   1. Initial program 99.9%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 28.1%

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

   if -10 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.7%

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

      1. distribute-frac-neg 99.7%

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

      2. exp-neg 99.7%

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

      3. add-sqr-sqrt 16.8%

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

      4. sqrt-unprod 37.8%

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

      5. sqr-neg 37.8%

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

      6. sqrt-unprod 21.6%

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

      7. add-sqr-sqrt 35.3%

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

      8. add-cbrt-cube 35.3%

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

      9. pow1/3 35.3%

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

      10. pow-flip 35.3%

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

   4. Applied egg-rr 99.2%

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

   5. Taylor expanded in s around -inf 61.5%

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

      1. distribute-rgt-out 61.5%

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

      2. metadata-eval 61.5%

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

   7. Simplified 61.5%

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

   8. Taylor expanded in x around 0 61.5%

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

4. Final simplification 47.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -10:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(0.3333333333333333 \cdot \left(\frac{x}{s} \cdot -3\right) + 1\right) + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 49.3% accurate, 6.0× speedup

Math

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

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) -10.0)
   0.5
   (/ 1.0 (+ (+ (* 0.3333333333333333 (/ (* x -3.0) s)) 1.0) 1.0))))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= -10.0f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / (((0.3333333333333333f * ((x * -3.0f) / s)) + 1.0f) + 1.0f);
	}
	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) <= (-10.0e0)) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 / (((0.3333333333333333e0 * ((x * (-3.0e0)) / s)) + 1.0e0) + 1.0e0)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(-x) / s) <= Float32(-10.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(Float32(0.3333333333333333) * Float32(Float32(x * Float32(-3.0)) / s)) + Float32(1.0)) + Float32(1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((-x / s) <= single(-10.0))
		tmp = single(0.5);
	else
		tmp = single(1.0) / (((single(0.3333333333333333) * ((x * single(-3.0)) / s)) + single(1.0)) + single(1.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < -10

   1. Initial program 99.9%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 28.1%

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

   if -10 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.7%

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

      1. distribute-frac-neg 99.7%

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

      2. exp-neg 99.7%

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

      3. add-sqr-sqrt 16.8%

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

      4. sqrt-unprod 37.8%

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

      5. sqr-neg 37.8%

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

      6. sqrt-unprod 21.6%

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

      7. add-sqr-sqrt 35.3%

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

      8. add-cbrt-cube 35.3%

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

      9. pow1/3 35.3%

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

      10. pow-flip 35.3%

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

   4. Applied egg-rr 99.2%

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

   5. Taylor expanded in s around -inf 61.5%

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

      1. distribute-rgt-out 61.5%

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

      2. metadata-eval 61.5%

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

   7. Simplified 61.5%

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

4. Final simplification 47.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -10:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(0.3333333333333333 \cdot \frac{x \cdot -3}{s} + 1\right) + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 49.2% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -10:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(x \cdot \frac{0.3333333333333333}{s \cdot -0.3333333333333333} + 1\right) + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) -10.0)
   0.5
   (/
    1.0
    (+ (+ (* x (/ 0.3333333333333333 (* s -0.3333333333333333))) 1.0) 1.0))))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= -10.0f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / (((x * (0.3333333333333333f / (s * -0.3333333333333333f))) + 1.0f) + 1.0f);
	}
	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) <= (-10.0e0)) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 / (((x * (0.3333333333333333e0 / (s * (-0.3333333333333333e0)))) + 1.0e0) + 1.0e0)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(-x) / s) <= Float32(-10.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(x * Float32(Float32(0.3333333333333333) / Float32(s * Float32(-0.3333333333333333)))) + Float32(1.0)) + Float32(1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((-x / s) <= single(-10.0))
		tmp = single(0.5);
	else
		tmp = single(1.0) / (((x * (single(0.3333333333333333) / (s * single(-0.3333333333333333)))) + single(1.0)) + single(1.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < -10

   1. Initial program 99.9%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 28.1%

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

   if -10 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.7%

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

      1. distribute-frac-neg 99.7%

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

      2. exp-neg 99.7%

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

      3. add-sqr-sqrt 16.8%

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

      4. sqrt-unprod 37.8%

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

      5. sqr-neg 37.8%

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

      6. sqrt-unprod 21.6%

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

      7. add-sqr-sqrt 35.3%

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

      8. add-cbrt-cube 35.3%

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

      9. pow1/3 35.3%

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

      10. pow-flip 35.3%

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

   4. Applied egg-rr 99.2%

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

   5. Taylor expanded in s around -inf 61.5%

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

      1. distribute-rgt-out 61.5%

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

      2. metadata-eval 61.5%

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

   7. Simplified 61.5%

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

      1. clear-num 61.5%

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

      2. un-div-inv 61.5%

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

      3. *-un-lft-identity 61.5%

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

      4. *-commutative 61.5%

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

      5. times-frac 61.5%

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

      6. metadata-eval 61.5%

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

   9. Applied egg-rr 61.5%

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

       1. associate-/r* 61.5%

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

       2. metadata-eval 61.5%

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

   11. Simplified 61.5%

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

       1. metadata-eval 61.5%

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

       2. associate-/r* 61.5%

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

       3. associate-*r/ 61.5%

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

       4. associate-/r/ 62.2%

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

       5. *-commutative 62.2%

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

   13. Applied egg-rr 62.2%

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

4. Final simplification 47.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -10:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(x \cdot \frac{0.3333333333333333}{s \cdot -0.3333333333333333} + 1\right) + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 49.1% accurate, 6.7× speedup

Math

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

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) -10.0) 0.5 (/ 1.0 (+ (+ (/ -1.0 (/ s x)) 1.0) 1.0))))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= -10.0f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / (((-1.0f / (s / x)) + 1.0f) + 1.0f);
	}
	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) <= (-10.0e0)) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 / ((((-1.0e0) / (s / x)) + 1.0e0) + 1.0e0)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(-x) / s) <= Float32(-10.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(Float32(-1.0) / Float32(s / x)) + Float32(1.0)) + Float32(1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((-x / s) <= single(-10.0))
		tmp = single(0.5);
	else
		tmp = single(1.0) / (((single(-1.0) / (s / x)) + single(1.0)) + single(1.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < -10

   1. Initial program 99.9%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 28.1%

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

   if -10 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.7%

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

      1. distribute-frac-neg 99.7%

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

      2. exp-neg 99.7%

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

      3. add-sqr-sqrt 16.8%

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

      4. sqrt-unprod 37.8%

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

      5. sqr-neg 37.8%

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

      6. sqrt-unprod 21.6%

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

      7. add-sqr-sqrt 35.3%

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

      8. add-cbrt-cube 35.3%

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

      9. pow1/3 35.3%

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

      10. pow-flip 35.3%

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

   4. Applied egg-rr 99.2%

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

   5. Taylor expanded in s around -inf 61.5%

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

      1. distribute-rgt-out 61.5%

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

      2. metadata-eval 61.5%

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

   7. Simplified 61.5%

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

      1. clear-num 61.5%

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

      2. un-div-inv 61.5%

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

      3. *-un-lft-identity 61.5%

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

      4. *-commutative 61.5%

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

      5. times-frac 61.5%

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

      6. metadata-eval 61.5%

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

   9. Applied egg-rr 61.5%

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

       1. associate-/r* 61.5%

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

       2. metadata-eval 61.5%

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

   11. Simplified 61.5%

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

4. Final simplification 47.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -10:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\frac{-1}{\frac{s}{x}} + 1\right) + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 49.1% accurate, 8.9× speedup

Math

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

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) -10.0) 0.5 (/ 1.0 (- 2.0 (/ x s)))))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= -10.0f) {
		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) <= (-10.0e0)) 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(-10.0))
		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(-10.0))
		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) < -10

   1. Initial program 99.9%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 28.1%

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

   if -10 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.7%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 61.5%

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

      1. mul-1-neg 61.5%

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

      2. unsub-neg 61.5%

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

   5. Simplified 61.5%

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

4. Final simplification 47.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -10:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 47.8% accurate, 10.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 47.6%

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

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

   1. Initial program 99.7%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 42.6%

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

      1. mul-1-neg 42.6%

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

      2. unsub-neg 42.6%

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

   5. Simplified 42.6%

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

   6. Taylor expanded in x around inf 42.6%

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

      1. neg-mul-1 42.6%

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

      2. distribute-neg-frac 42.6%

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

   8. Simplified 42.6%

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

      1. neg-mul-1 42.6%

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

      2. associate-/l* 42.6%

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

      3. metadata-eval 42.6%

         \[\leadsto \frac{1}{\frac{\color{blue}{\frac{0.3333333333333333}{-0.3333333333333333}}}{\frac{s}{x}}} \]

      4. associate-/r* 42.6%

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

      5. clear-num 39.6%

         \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot \frac{s}{x}}{0.3333333333333333}} \]

      6. div-inv 39.6%

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

      7. associate-*r/ 39.6%

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

      8. associate-/l* 42.6%

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

      9. metadata-eval 42.6%

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

   10. Applied egg-rr 42.6%

       \[\leadsto \color{blue}{\frac{-0.3333333333333333}{\frac{x}{s}} \cdot 3} \]
   11. Step-by-step derivation

       1. associate-*l/ 42.6%

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

       2. metadata-eval 42.6%

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

   12. Simplified 42.6%

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

4. Final simplification 45.8%

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

Alternative 12: 46.1% accurate, 17.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.00019999999494757503:\\ \;\;\;\;-\frac{s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -0.00019999999494757503) (- (/ s x)) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -0.00019999999494757503f) {
		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 <= (-0.00019999999494757503e0)) 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(-0.00019999999494757503))
		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(-0.00019999999494757503))
		tmp = -(s / x);
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -1.99999995e-4

   1. Initial program 100.0%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 51.8%

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

      1. mul-1-neg 51.8%

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

      2. unsub-neg 51.8%

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

   5. Simplified 51.8%

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

   6. Taylor expanded in x around inf 47.9%

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

      1. associate-*r/ 47.9%

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

      2. neg-mul-1 47.9%

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

   8. Simplified 47.9%

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

   if -1.99999995e-4 < x

   1. Initial program 99.8%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 43.1%

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

4. Final simplification 44.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00019999999494757503:\\ \;\;\;\;-\frac{s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 46.1% accurate, 21.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.004999999888241291:\\ \;\;\;\;\frac{s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -0.004999999888241291) (/ s x) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -0.004999999888241291f) {
		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 <= (-0.004999999888241291e0)) 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(-0.004999999888241291))
		tmp = Float32(s / x);
	else
		tmp = Float32(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-0.004999999888241291))
		tmp = s / x;
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -0.00499999989

   1. Initial program 100.0%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 53.3%

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

      1. mul-1-neg 53.3%

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

      2. unsub-neg 53.3%

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

   5. Simplified 53.3%

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

   6. Taylor expanded in x around inf 53.3%

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

      1. neg-mul-1 53.3%

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

      2. distribute-neg-frac 53.3%

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

   8. Simplified 53.3%

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

      1. clear-num 49.2%

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

      2. add-sqr-sqrt 49.2%

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

      3. sqrt-unprod 58.8%

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

      4. sqr-neg 58.8%

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

      5. sqrt-unprod -0.0%

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

      6. add-sqr-sqrt 49.2%

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

      7. expm1-log1p-u 49.2%

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

      8. expm1-udef 98.7%

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

   10. Applied egg-rr 98.7%

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

       1. expm1-def 49.2%

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

       2. expm1-log1p 49.2%

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

   12. Simplified 49.2%

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

   if -0.00499999989 < x

   1. Initial program 99.8%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 42.7%

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

4. Final simplification 44.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.004999999888241291:\\ \;\;\;\;\frac{s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 34.8% accurate, 108.0× speedup

Math

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

FPCore

(FPCore (x s) :precision binary32 0.5)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

   \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 32.6%

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

4. Final simplification 32.6%

   \[\leadsto 0.5 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024011 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))