Logistic function

Percentage Accurate: 99.8% → 99.8%

Time: 8.8s

Alternatives: 15

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + {\left(e^{-0.25}\right)}^{\left(x \cdot \frac{4}{s}\right)}} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (/ 1.0 (+ 1.0 (pow (exp -0.25) (* x (/ 4.0 s))))))

C

float code(float x, float s) {
	return 1.0f / (1.0f + powf(expf(-0.25f), (x * (4.0f / s))));
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 1.0e0 / (1.0e0 + (exp((-0.25e0)) ** (x * (4.0e0 / s))))
end function

Julia

function code(x, s)
	return Float32(Float32(1.0) / Float32(Float32(1.0) + (exp(Float32(-0.25)) ^ Float32(x * Float32(Float32(4.0) / s)))))
end

MATLAB

function tmp = code(x, s)
	tmp = single(1.0) / (single(1.0) + (exp(single(-0.25)) ^ (x * (single(4.0) / s))));
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%

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

4. Applied egg-rr 99.8%

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

   1. inv-pow 99.8%

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

   2. pow-exp 99.8%

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

   3. *-commutative 99.8%

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

   4. pow-exp 99.7%

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

   5. add-sqr-sqrt 99.7%

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

   6. unpow-prod-down 99.7%

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

6. Applied egg-rr 99.7%

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

   1. pow-sqr 99.7%

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

   2. associate-*r/ 99.7%

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

   3. remove-double-neg 99.7%

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

   4. neg-mul-1 99.7%

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

   5. distribute-neg-frac2 99.7%

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

   6. associate-*r/ 99.7%

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

   7. neg-mul-1 99.7%

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

   8. *-commutative 99.7%

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

   9. distribute-lft-neg-in 99.7%

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

   10. distribute-frac-neg2 99.7%

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

   11. remove-double-neg 99.7%

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

8. Simplified 99.7%

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

   1. add-sqr-sqrt 99.7%

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

   2. unpow-prod-down 99.8%

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

   3. pow1/2 99.8%

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

   4. pow-exp 99.8%

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

   5. metadata-eval 99.8%

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

   6. pow1/2 99.8%

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

   7. pow-exp 99.8%

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

   8. metadata-eval 99.8%

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

10. Applied egg-rr 99.8%

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

    1. pow-sqr 99.8%

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

    2. associate-*l/ 99.8%

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

12. Simplified 99.8%

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

    1. sqr-pow 99.8%

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

    2. pow1/2 99.8%

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

    3. pow-exp 99.8%

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

    4. metadata-eval 99.8%

       \[\leadsto \frac{1}{1 + {\left(e^{\color{blue}{-0.25}}\right)}^{\left(\frac{2 \cdot \frac{x \cdot 2}{s}}{2}\right)} \cdot {\left(\sqrt{e^{-0.5}}\right)}^{\left(\frac{2 \cdot \frac{x \cdot 2}{s}}{2}\right)}} \]

    5. *-commutative 99.8%

       \[\leadsto \frac{1}{1 + {\left(e^{-0.25}\right)}^{\left(\frac{\color{blue}{\frac{x \cdot 2}{s} \cdot 2}}{2}\right)} \cdot {\left(\sqrt{e^{-0.5}}\right)}^{\left(\frac{2 \cdot \frac{x \cdot 2}{s}}{2}\right)}} \]

    6. associate-/l* 99.8%

       \[\leadsto \frac{1}{1 + {\left(e^{-0.25}\right)}^{\color{blue}{\left(\frac{x \cdot 2}{s} \cdot \frac{2}{2}\right)}} \cdot {\left(\sqrt{e^{-0.5}}\right)}^{\left(\frac{2 \cdot \frac{x \cdot 2}{s}}{2}\right)}} \]

    7. metadata-eval 99.8%

       \[\leadsto \frac{1}{1 + {\left(e^{-0.25}\right)}^{\left(\frac{x \cdot 2}{s} \cdot \color{blue}{1}\right)} \cdot {\left(\sqrt{e^{-0.5}}\right)}^{\left(\frac{2 \cdot \frac{x \cdot 2}{s}}{2}\right)}} \]

    8. *-commutative 99.8%

       \[\leadsto \frac{1}{1 + {\left(e^{-0.25}\right)}^{\color{blue}{\left(1 \cdot \frac{x \cdot 2}{s}\right)}} \cdot {\left(\sqrt{e^{-0.5}}\right)}^{\left(\frac{2 \cdot \frac{x \cdot 2}{s}}{2}\right)}} \]

    9. *-un-lft-identity 99.8%

       \[\leadsto \frac{1}{1 + {\left(e^{-0.25}\right)}^{\color{blue}{\left(\frac{x \cdot 2}{s}\right)}} \cdot {\left(\sqrt{e^{-0.5}}\right)}^{\left(\frac{2 \cdot \frac{x \cdot 2}{s}}{2}\right)}} \]

    10. *-commutative 99.8%

        \[\leadsto \frac{1}{1 + {\left(e^{-0.25}\right)}^{\left(\frac{\color{blue}{2 \cdot x}}{s}\right)} \cdot {\left(\sqrt{e^{-0.5}}\right)}^{\left(\frac{2 \cdot \frac{x \cdot 2}{s}}{2}\right)}} \]

    11. associate-/l* 99.8%

        \[\leadsto \frac{1}{1 + {\left(e^{-0.25}\right)}^{\color{blue}{\left(2 \cdot \frac{x}{s}\right)}} \cdot {\left(\sqrt{e^{-0.5}}\right)}^{\left(\frac{2 \cdot \frac{x \cdot 2}{s}}{2}\right)}} \]

    12. pow1/2 99.8%

        \[\leadsto \frac{1}{1 + {\left(e^{-0.25}\right)}^{\left(2 \cdot \frac{x}{s}\right)} \cdot {\color{blue}{\left({\left(e^{-0.5}\right)}^{0.5}\right)}}^{\left(\frac{2 \cdot \frac{x \cdot 2}{s}}{2}\right)}} \]

    13. pow-exp 99.8%

        \[\leadsto \frac{1}{1 + {\left(e^{-0.25}\right)}^{\left(2 \cdot \frac{x}{s}\right)} \cdot {\color{blue}{\left(e^{-0.5 \cdot 0.5}\right)}}^{\left(\frac{2 \cdot \frac{x \cdot 2}{s}}{2}\right)}} \]

    14. metadata-eval 99.8%

        \[\leadsto \frac{1}{1 + {\left(e^{-0.25}\right)}^{\left(2 \cdot \frac{x}{s}\right)} \cdot {\left(e^{\color{blue}{-0.25}}\right)}^{\left(\frac{2 \cdot \frac{x \cdot 2}{s}}{2}\right)}} \]

    15. *-commutative 99.8%

        \[\leadsto \frac{1}{1 + {\left(e^{-0.25}\right)}^{\left(2 \cdot \frac{x}{s}\right)} \cdot {\left(e^{-0.25}\right)}^{\left(\frac{\color{blue}{\frac{x \cdot 2}{s} \cdot 2}}{2}\right)}} \]

    16. associate-/l* 99.8%

        \[\leadsto \frac{1}{1 + {\left(e^{-0.25}\right)}^{\left(2 \cdot \frac{x}{s}\right)} \cdot {\left(e^{-0.25}\right)}^{\color{blue}{\left(\frac{x \cdot 2}{s} \cdot \frac{2}{2}\right)}}} \]

    17. metadata-eval 99.8%

        \[\leadsto \frac{1}{1 + {\left(e^{-0.25}\right)}^{\left(2 \cdot \frac{x}{s}\right)} \cdot {\left(e^{-0.25}\right)}^{\left(\frac{x \cdot 2}{s} \cdot \color{blue}{1}\right)}} \]

    18. *-commutative 99.8%

        \[\leadsto \frac{1}{1 + {\left(e^{-0.25}\right)}^{\left(2 \cdot \frac{x}{s}\right)} \cdot {\left(e^{-0.25}\right)}^{\color{blue}{\left(1 \cdot \frac{x \cdot 2}{s}\right)}}} \]

    19. *-un-lft-identity 99.8%

        \[\leadsto \frac{1}{1 + {\left(e^{-0.25}\right)}^{\left(2 \cdot \frac{x}{s}\right)} \cdot {\left(e^{-0.25}\right)}^{\color{blue}{\left(\frac{x \cdot 2}{s}\right)}}} \]

    20. *-commutative 99.8%

        \[\leadsto \frac{1}{1 + {\left(e^{-0.25}\right)}^{\left(2 \cdot \frac{x}{s}\right)} \cdot {\left(e^{-0.25}\right)}^{\left(\frac{\color{blue}{2 \cdot x}}{s}\right)}} \]

    21. associate-/l* 99.8%

        \[\leadsto \frac{1}{1 + {\left(e^{-0.25}\right)}^{\left(2 \cdot \frac{x}{s}\right)} \cdot {\left(e^{-0.25}\right)}^{\color{blue}{\left(2 \cdot \frac{x}{s}\right)}}} \]

14. Applied egg-rr 99.8%

    \[\leadsto \frac{1}{1 + \color{blue}{{\left(e^{-0.25}\right)}^{\left(2 \cdot \frac{x}{s}\right)} \cdot {\left(e^{-0.25}\right)}^{\left(2 \cdot \frac{x}{s}\right)}}} \]
15. Step-by-step derivation

    1. pow-sqr 99.8%

       \[\leadsto \frac{1}{1 + \color{blue}{{\left(e^{-0.25}\right)}^{\left(2 \cdot \left(2 \cdot \frac{x}{s}\right)\right)}}} \]

    2. associate-*r* 99.8%

       \[\leadsto \frac{1}{1 + {\left(e^{-0.25}\right)}^{\color{blue}{\left(\left(2 \cdot 2\right) \cdot \frac{x}{s}\right)}}} \]

    3. metadata-eval 99.8%

       \[\leadsto \frac{1}{1 + {\left(e^{-0.25}\right)}^{\left(\color{blue}{4} \cdot \frac{x}{s}\right)}} \]

    4. associate-*r/ 99.8%

       \[\leadsto \frac{1}{1 + {\left(e^{-0.25}\right)}^{\color{blue}{\left(\frac{4 \cdot x}{s}\right)}}} \]

    5. *-commutative 99.8%

       \[\leadsto \frac{1}{1 + {\left(e^{-0.25}\right)}^{\left(\frac{\color{blue}{x \cdot 4}}{s}\right)}} \]

    6. associate-/l* 99.8%

       \[\leadsto \frac{1}{1 + {\left(e^{-0.25}\right)}^{\color{blue}{\left(x \cdot \frac{4}{s}\right)}}} \]

16. Simplified 99.8%

    \[\leadsto \frac{1}{1 + \color{blue}{{\left(e^{-0.25}\right)}^{\left(x \cdot \frac{4}{s}\right)}}} \]
17. Add Preprocessing

Alternative 2: 82.1% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 0.5

   1. Initial program 99.9%

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

      1. distribute-frac-neg 99.9%

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

      2. exp-neg 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around 0 96.1%

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

   if 0.5 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 39.5%

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

      1. neg-mul-1 39.5%

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

      2. unsub-neg 39.5%

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

   5. Simplified 39.5%

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

   6. Taylor expanded in x around inf 35.4%

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

      1. associate-*r/ 35.4%

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

      2. neg-mul-1 35.4%

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

   8. Simplified 35.4%

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

      1. neg-sub0 35.4%

         \[\leadsto \frac{\color{blue}{0 - s}}{x} \]

      2. flip-- 67.7%

         \[\leadsto \frac{\color{blue}{\frac{0 \cdot 0 - s \cdot s}{0 + s}}}{x} \]

      3. metadata-eval 67.7%

         \[\leadsto \frac{\frac{\color{blue}{0} - s \cdot s}{0 + s}}{x} \]

      4. pow2 67.7%

         \[\leadsto \frac{\frac{0 - \color{blue}{{s}^{2}}}{0 + s}}{x} \]

      5. add-sqr-sqrt 67.7%

         \[\leadsto \frac{\frac{0 - {s}^{2}}{0 + \color{blue}{\sqrt{s} \cdot \sqrt{s}}}}{x} \]

      6. sqrt-unprod 11.3%

         \[\leadsto \frac{\frac{0 - {s}^{2}}{0 + \color{blue}{\sqrt{s \cdot s}}}}{x} \]

      7. sqr-neg 11.3%

         \[\leadsto \frac{\frac{0 - {s}^{2}}{0 + \sqrt{\color{blue}{\left(-s\right) \cdot \left(-s\right)}}}}{x} \]

      8. sqrt-unprod -0.0%

         \[\leadsto \frac{\frac{0 - {s}^{2}}{0 + \color{blue}{\sqrt{-s} \cdot \sqrt{-s}}}}{x} \]

      9. add-sqr-sqrt 67.4%

         \[\leadsto \frac{\frac{0 - {s}^{2}}{0 + \color{blue}{\left(-s\right)}}}{x} \]

      10. sub-neg 67.4%

          \[\leadsto \frac{\frac{0 - {s}^{2}}{\color{blue}{0 - s}}}{x} \]

      11. neg-sub0 67.4%

          \[\leadsto \frac{\frac{0 - {s}^{2}}{\color{blue}{-s}}}{x} \]

      12. add-sqr-sqrt -0.0%

          \[\leadsto \frac{\frac{0 - {s}^{2}}{\color{blue}{\sqrt{-s} \cdot \sqrt{-s}}}}{x} \]

      13. sqrt-unprod 11.3%

          \[\leadsto \frac{\frac{0 - {s}^{2}}{\color{blue}{\sqrt{\left(-s\right) \cdot \left(-s\right)}}}}{x} \]

      14. sqr-neg 11.3%

          \[\leadsto \frac{\frac{0 - {s}^{2}}{\sqrt{\color{blue}{s \cdot s}}}}{x} \]

      15. sqrt-unprod 67.7%

          \[\leadsto \frac{\frac{0 - {s}^{2}}{\color{blue}{\sqrt{s} \cdot \sqrt{s}}}}{x} \]

      16. add-sqr-sqrt 67.7%

          \[\leadsto \frac{\frac{0 - {s}^{2}}{\color{blue}{s}}}{x} \]

   10. Applied egg-rr 67.7%

       \[\leadsto \frac{\color{blue}{\frac{0 - {s}^{2}}{s}}}{x} \]
   11. Step-by-step derivation

       1. sub0-neg 67.7%

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

   12. Simplified 67.7%

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

4. Final simplification 85.0%

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

Alternative 3: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

Alternative 4: 84.6% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t\_0 \leq -500:\\ \;\;\;\;\frac{1}{1 + \left(1 + \left(-1 + \frac{-1}{\frac{x}{s} + -1}\right)\right)}\\ \mathbf{elif}\;t\_0 \leq 4.999999840142846 \cdot 10^{+37}:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{s} \cdot \frac{x}{s}}{\frac{x}{s} + 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)))
   (if (<= t_0 -500.0)
     (/ 1.0 (+ 1.0 (+ 1.0 (+ -1.0 (/ -1.0 (+ (/ x s) -1.0))))))
     (if (<= t_0 4.999999840142846e+37)
       (/ 1.0 (/ (- 4.0 (* (/ x s) (/ x s))) (+ (/ x s) 2.0)))
       (/ 1.0 (/ x s))))))

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if (t_0 <= -500.0f) {
		tmp = 1.0f / (1.0f + (1.0f + (-1.0f + (-1.0f / ((x / s) + -1.0f)))));
	} else if (t_0 <= 4.999999840142846e+37f) {
		tmp = 1.0f / ((4.0f - ((x / s) * (x / s))) / ((x / s) + 2.0f));
	} 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 <= (-500.0e0)) then
        tmp = 1.0e0 / (1.0e0 + (1.0e0 + ((-1.0e0) + ((-1.0e0) / ((x / s) + (-1.0e0))))))
    else if (t_0 <= 4.999999840142846e+37) then
        tmp = 1.0e0 / ((4.0e0 - ((x / s) * (x / s))) / ((x / s) + 2.0e0))
    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(-500.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) + Float32(Float32(-1.0) + Float32(Float32(-1.0) / Float32(Float32(x / s) + Float32(-1.0)))))));
	elseif (t_0 <= Float32(4.999999840142846e+37))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(4.0) - Float32(Float32(x / s) * Float32(x / s))) / Float32(Float32(x / s) + Float32(2.0))));
	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(-500.0))
		tmp = single(1.0) / (single(1.0) + (single(1.0) + (single(-1.0) + (single(-1.0) / ((x / s) + single(-1.0))))));
	elseif (t_0 <= single(4.999999840142846e+37))
		tmp = single(1.0) / ((single(4.0) - ((x / s) * (x / s))) / ((x / s) + single(2.0)));
	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) < -500

   1. Initial program 100.0%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing
   3. 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%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0 98.0%

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

      1. expm1-log1p-u 98.0%

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

      2. log1p-define 98.0%

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

      3. expm1-undefine 98.0%

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

      4. add-exp-log 98.0%

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

      5. frac-2neg 98.0%

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

      6. metadata-eval 98.0%

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

      7. distribute-neg-in 98.0%

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

      8. metadata-eval 98.0%

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

      9. distribute-frac-neg2 98.0%

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

      10. add-sqr-sqrt -0.0%

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

      11. sqrt-unprod 99.5%

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

      12. sqr-neg 99.5%

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

      13. sqrt-unprod 98.2%

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

      14. add-sqr-sqrt 98.2%

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

   7. Applied egg-rr 98.2%

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

      1. associate--l+ 98.2%

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

      2. sub-neg 98.2%

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

      3. metadata-eval 98.2%

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

   9. Simplified 98.2%

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

   if -500 < (/.f32 (neg.f32 x) s) < 4.99999984e37

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 54.3%

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

      1. neg-mul-1 54.3%

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

      2. unsub-neg 54.3%

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

   5. Simplified 54.3%

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

      1. *-un-lft-identity 54.3%

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

      2. cancel-sign-sub-inv 54.3%

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

      3. metadata-eval 54.3%

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

      4. add-log-exp 93.8%

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

      5. pow-exp 93.8%

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

      6. flip-+ 49.4%

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

      7. metadata-eval 49.4%

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

      8. pow-exp 49.4%

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

      9. add-log-exp 49.4%

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

      10. neg-mul-1 49.4%

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

      11. pow-exp 49.4%

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

      12. add-log-exp 50.0%

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

      13. neg-mul-1 50.0%

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

      14. distribute-neg-frac2 50.0%

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

      15. distribute-neg-frac2 50.0%

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

      16. pow-exp 50.0%

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

   7. Applied egg-rr 76.3%

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

   if 4.99999984e37 < (/.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. neg-mul-1 100.0%

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

      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 87.1%

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

      1. associate-*r/ 87.1%

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

      2. neg-mul-1 87.1%

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

   8. Simplified 87.1%

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

      1. clear-num 100.0%

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

      2. inv-pow 100.0%

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

      3. add-sqr-sqrt -0.0%

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

      4. sqrt-unprod 100.0%

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

      5. sqr-neg 100.0%

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

      6. sqrt-unprod 100.0%

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

      7. add-sqr-sqrt 100.0%

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

   10. Applied egg-rr 100.0%

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

       1. unpow-1 100.0%

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

   12. Simplified 100.0%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -500:\\ \;\;\;\;\frac{1}{1 + \left(1 + \left(-1 + \frac{-1}{\frac{x}{s} + -1}\right)\right)}\\ \mathbf{elif}\;\frac{-x}{s} \leq 4.999999840142846 \cdot 10^{+37}:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{s} \cdot \frac{x}{s}}{\frac{x}{s} + 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.0% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t\_0 \leq -500:\\ \;\;\;\;1 - \frac{s}{x}\\ \mathbf{elif}\;t\_0 \leq 0.5:\\ \;\;\;\;0.5 + \frac{x \cdot 0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)))
   (if (<= t_0 -500.0)
     (- 1.0 (/ s x))
     (if (<= t_0 0.5) (+ 0.5 (/ (* x 0.25) s)) (/ -1.0 (/ x s))))))

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if (t_0 <= -500.0f) {
		tmp = 1.0f - (s / x);
	} else if (t_0 <= 0.5f) {
		tmp = 0.5f + ((x * 0.25f) / 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 <= (-500.0e0)) then
        tmp = 1.0e0 - (s / x)
    else if (t_0 <= 0.5e0) then
        tmp = 0.5e0 + ((x * 0.25e0) / 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(-500.0))
		tmp = Float32(Float32(1.0) - Float32(s / x));
	elseif (t_0 <= Float32(0.5))
		tmp = Float32(Float32(0.5) + Float32(Float32(x * Float32(0.25)) / 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(-500.0))
		tmp = single(1.0) - (s / x);
	elseif (t_0 <= single(0.5))
		tmp = single(0.5) + ((x * single(0.25)) / 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) < -500

   1. Initial program 100.0%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing
   3. 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%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0 98.0%

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

   6. Taylor expanded in x around inf 98.0%

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

      1. neg-mul-1 98.0%

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

      2. unsub-neg 98.0%

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

   8. Simplified 98.0%

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

   if -500 < (/.f32 (neg.f32 x) s) < 0.5

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 96.7%

      \[\leadsto \color{blue}{0.5 + 0.25 \cdot \frac{x}{s}} \]
   4. Step-by-step derivation

      1. associate-*r/ 96.7%

         \[\leadsto 0.5 + \color{blue}{\frac{0.25 \cdot x}{s}} \]

      2. *-commutative 96.7%

         \[\leadsto 0.5 + \frac{\color{blue}{x \cdot 0.25}}{s} \]

   5. Simplified 96.7%

      \[\leadsto \color{blue}{0.5 + \frac{x \cdot 0.25}{s}} \]

   if 0.5 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 39.5%

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

      1. neg-mul-1 39.5%

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

      2. unsub-neg 39.5%

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

   5. Simplified 39.5%

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

   6. Taylor expanded in x around inf 39.5%

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

      1. associate-*r/ 39.5%

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

      2. neg-mul-1 39.5%

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

   8. Simplified 39.5%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -500:\\ \;\;\;\;1 - \frac{s}{x}\\ \mathbf{elif}\;\frac{-x}{s} \leq 0.5:\\ \;\;\;\;0.5 + \frac{x \cdot 0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{x}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.1% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -500:\\ \;\;\;\;\frac{1}{1 + \left(1 + \left(-1 + \frac{-1}{\frac{x}{s} + -1}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-1 + \left(3 - \frac{x}{s}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) -500.0)
   (/ 1.0 (+ 1.0 (+ 1.0 (+ -1.0 (/ -1.0 (+ (/ x s) -1.0))))))
   (/ 1.0 (+ -1.0 (- 3.0 (/ x s))))))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= -500.0f) {
		tmp = 1.0f / (1.0f + (1.0f + (-1.0f + (-1.0f / ((x / s) + -1.0f)))));
	} else {
		tmp = 1.0f / (-1.0f + (3.0f - (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) <= (-500.0e0)) then
        tmp = 1.0e0 / (1.0e0 + (1.0e0 + ((-1.0e0) + ((-1.0e0) / ((x / s) + (-1.0e0))))))
    else
        tmp = 1.0e0 / ((-1.0e0) + (3.0e0 - (x / s)))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(-x) / s) <= Float32(-500.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) + Float32(Float32(-1.0) + Float32(Float32(-1.0) / Float32(Float32(x / s) + Float32(-1.0)))))));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(-1.0) + Float32(Float32(3.0) - Float32(x / s))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((-x / s) <= single(-500.0))
		tmp = single(1.0) / (single(1.0) + (single(1.0) + (single(-1.0) + (single(-1.0) / ((x / s) + single(-1.0))))));
	else
		tmp = single(1.0) / (single(-1.0) + (single(3.0) - (x / s)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < -500

   1. Initial program 100.0%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing
   3. 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%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0 98.0%

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

      1. expm1-log1p-u 98.0%

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

      2. log1p-define 98.0%

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

      3. expm1-undefine 98.0%

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

      4. add-exp-log 98.0%

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

      5. frac-2neg 98.0%

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

      6. metadata-eval 98.0%

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

      7. distribute-neg-in 98.0%

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

      8. metadata-eval 98.0%

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

      9. distribute-frac-neg2 98.0%

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

      10. add-sqr-sqrt -0.0%

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

      11. sqrt-unprod 99.5%

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

      12. sqr-neg 99.5%

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

      13. sqrt-unprod 98.2%

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

      14. add-sqr-sqrt 98.2%

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

   7. Applied egg-rr 98.2%

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

      1. associate--l+ 98.2%

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

      2. sub-neg 98.2%

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

      3. metadata-eval 98.2%

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

   9. Simplified 98.2%

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

   if -500 < (/.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 62.7%

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

      1. neg-mul-1 62.7%

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

      2. unsub-neg 62.7%

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

   5. Simplified 62.7%

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

      1. expm1-log1p-u 62.6%

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

   7. Applied egg-rr 62.6%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 - \frac{x}{s}\right)\right)}} \]
   8. Step-by-step derivation

      1. expm1-undefine 62.6%

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

      2. sub-neg 62.6%

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

      3. log1p-undefine 62.6%

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

      4. rem-exp-log 62.7%

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

      5. associate-+r- 62.7%

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

      6. metadata-eval 62.7%

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

      7. metadata-eval 62.7%

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

   9. Simplified 62.7%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -500:\\ \;\;\;\;\frac{1}{1 + \left(1 + \left(-1 + \frac{-1}{\frac{x}{s} + -1}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-1 + \left(3 - \frac{x}{s}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.0% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t\_0 \leq -500:\\ \;\;\;\;1 - \frac{s}{x}\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)))
   (if (<= t_0 -500.0)
     (- 1.0 (/ s x))
     (if (<= t_0 2.0) 0.5 (/ -1.0 (/ x s))))))

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if (t_0 <= -500.0f) {
		tmp = 1.0f - (s / x);
	} else if (t_0 <= 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) :: t_0
    real(4) :: tmp
    t_0 = -x / s
    if (t_0 <= (-500.0e0)) then
        tmp = 1.0e0 - (s / x)
    else if (t_0 <= 2.0e0) then
        tmp = 0.5e0
    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(-500.0))
		tmp = Float32(Float32(1.0) - Float32(s / x));
	elseif (t_0 <= 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)
	t_0 = -x / s;
	tmp = single(0.0);
	if (t_0 <= single(-500.0))
		tmp = single(1.0) - (s / x);
	elseif (t_0 <= single(2.0))
		tmp = single(0.5);
	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) < -500

   1. Initial program 100.0%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing
   3. 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%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0 98.0%

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

   6. Taylor expanded in x around inf 98.0%

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

      1. neg-mul-1 98.0%

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

      2. unsub-neg 98.0%

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

   8. Simplified 98.0%

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

   if -500 < (/.f32 (neg.f32 x) s) < 2

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 87.1%

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

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

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 39.7%

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

      1. neg-mul-1 39.7%

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

      2. unsub-neg 39.7%

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

   5. Simplified 39.7%

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

   6. Taylor expanded in x around inf 39.7%

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

      1. associate-*r/ 39.7%

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

      2. neg-mul-1 39.7%

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

   8. Simplified 39.7%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -500:\\ \;\;\;\;1 - \frac{s}{x}\\ \mathbf{elif}\;\frac{-x}{s} \leq 2:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{x}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.1% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -500:\\ \;\;\;\;\frac{1}{1 + \frac{-1}{\frac{x}{s} + -1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-1 + \left(3 - \frac{x}{s}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) -500.0)
   (/ 1.0 (+ 1.0 (/ -1.0 (+ (/ x s) -1.0))))
   (/ 1.0 (+ -1.0 (- 3.0 (/ x s))))))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= -500.0f) {
		tmp = 1.0f / (1.0f + (-1.0f / ((x / s) + -1.0f)));
	} else {
		tmp = 1.0f / (-1.0f + (3.0f - (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) <= (-500.0e0)) then
        tmp = 1.0e0 / (1.0e0 + ((-1.0e0) / ((x / s) + (-1.0e0))))
    else
        tmp = 1.0e0 / ((-1.0e0) + (3.0e0 - (x / s)))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(-x) / s) <= Float32(-500.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(-1.0) / Float32(Float32(x / s) + Float32(-1.0)))));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(-1.0) + Float32(Float32(3.0) - Float32(x / s))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((-x / s) <= single(-500.0))
		tmp = single(1.0) / (single(1.0) + (single(-1.0) / ((x / s) + single(-1.0))));
	else
		tmp = single(1.0) / (single(-1.0) + (single(3.0) - (x / s)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < -500

   1. Initial program 100.0%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing
   3. 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%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0 98.0%

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

      1. frac-2neg 98.0%

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

      2. metadata-eval 98.0%

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

      3. div-inv 98.0%

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

      4. distribute-neg-in 98.0%

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

      5. metadata-eval 98.0%

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

      6. distribute-frac-neg2 98.0%

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

      7. add-sqr-sqrt -0.0%

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

      8. sqrt-unprod 99.5%

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

      9. sqr-neg 99.5%

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

      10. sqrt-unprod 98.2%

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

      11. add-sqr-sqrt 98.2%

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

   7. Applied egg-rr 98.2%

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

      1. associate-*r/ 98.2%

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

      2. metadata-eval 98.2%

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

   9. Simplified 98.2%

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

   if -500 < (/.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 62.7%

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

      1. neg-mul-1 62.7%

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

      2. unsub-neg 62.7%

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

   5. Simplified 62.7%

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

      1. expm1-log1p-u 62.6%

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

   7. Applied egg-rr 62.6%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 - \frac{x}{s}\right)\right)}} \]
   8. Step-by-step derivation

      1. expm1-undefine 62.6%

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

      2. sub-neg 62.6%

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

      3. log1p-undefine 62.6%

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

      4. rem-exp-log 62.7%

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

      5. associate-+r- 62.7%

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

      6. metadata-eval 62.7%

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

      7. metadata-eval 62.7%

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

   9. Simplified 62.7%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -500:\\ \;\;\;\;\frac{1}{1 + \frac{-1}{\frac{x}{s} + -1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-1 + \left(3 - \frac{x}{s}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 74.0% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -500:\\ \;\;\;\;1 - \frac{s}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-1 + \left(3 - \frac{x}{s}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) -500.0) (- 1.0 (/ s x)) (/ 1.0 (+ -1.0 (- 3.0 (/ x s))))))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= -500.0f) {
		tmp = 1.0f - (s / x);
	} else {
		tmp = 1.0f / (-1.0f + (3.0f - (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) <= (-500.0e0)) then
        tmp = 1.0e0 - (s / x)
    else
        tmp = 1.0e0 / ((-1.0e0) + (3.0e0 - (x / s)))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(-x) / s) <= Float32(-500.0))
		tmp = Float32(Float32(1.0) - Float32(s / x));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(-1.0) + Float32(Float32(3.0) - Float32(x / s))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((-x / s) <= single(-500.0))
		tmp = single(1.0) - (s / x);
	else
		tmp = single(1.0) / (single(-1.0) + (single(3.0) - (x / s)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < -500

   1. Initial program 100.0%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing
   3. 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%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0 98.0%

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

   6. Taylor expanded in x around inf 98.0%

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

      1. neg-mul-1 98.0%

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

      2. unsub-neg 98.0%

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

   8. Simplified 98.0%

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

   if -500 < (/.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 62.7%

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

      1. neg-mul-1 62.7%

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

      2. unsub-neg 62.7%

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

   5. Simplified 62.7%

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

      1. expm1-log1p-u 62.6%

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

   7. Applied egg-rr 62.6%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 - \frac{x}{s}\right)\right)}} \]
   8. Step-by-step derivation

      1. expm1-undefine 62.6%

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

      2. sub-neg 62.6%

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

      3. log1p-undefine 62.6%

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

      4. rem-exp-log 62.7%

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

      5. associate-+r- 62.7%

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

      6. metadata-eval 62.7%

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

      7. metadata-eval 62.7%

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

   9. Simplified 62.7%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -500:\\ \;\;\;\;1 - \frac{s}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-1 + \left(3 - \frac{x}{s}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 69.2% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.999999747378752 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \mathbf{elif}\;x \leq 1.0000000036274937 \cdot 10^{-15}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{s}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -9.999999747378752e-5)
   (/ 1.0 (/ x s))
   (if (<= x 1.0000000036274937e-15) 0.5 (- 1.0 (/ s x)))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -9.999999747378752e-5f) {
		tmp = 1.0f / (x / s);
	} else if (x <= 1.0000000036274937e-15f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f - (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 <= (-9.999999747378752e-5)) then
        tmp = 1.0e0 / (x / s)
    else if (x <= 1.0000000036274937e-15) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 - (s / x)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-9.999999747378752e-5))
		tmp = Float32(Float32(1.0) / Float32(x / s));
	elseif (x <= Float32(1.0000000036274937e-15))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) - Float32(s / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-9.999999747378752e-5))
		tmp = single(1.0) / (x / s);
	elseif (x <= single(1.0000000036274937e-15))
		tmp = single(0.5);
	else
		tmp = single(1.0) - (s / x);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -9.99999975e-5

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 50.7%

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

      1. neg-mul-1 50.7%

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

      2. unsub-neg 50.7%

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

   5. Simplified 50.7%

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

   6. Taylor expanded in x around inf 45.1%

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

      1. associate-*r/ 45.1%

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

      2. neg-mul-1 45.1%

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

   8. Simplified 45.1%

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

      1. clear-num 50.7%

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

      2. inv-pow 50.7%

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

      3. add-sqr-sqrt -0.0%

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

      4. sqrt-unprod 67.3%

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

      5. sqr-neg 67.3%

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

      6. sqrt-unprod 50.7%

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

      7. add-sqr-sqrt 50.7%

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

   10. Applied egg-rr 50.7%

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

       1. unpow-1 50.7%

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

   12. Simplified 50.7%

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

   if -9.99999975e-5 < x < 1e-15

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0 63.1%

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

   if 1e-15 < x

   1. Initial program 99.9%

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

      1. distribute-frac-neg 99.9%

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

      2. exp-neg 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0 98.2%

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

   6. Taylor expanded in x around inf 91.0%

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

      1. neg-mul-1 91.0%

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

      2. unsub-neg 91.0%

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

   8. Simplified 91.0%

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

Alternative 11: 68.0% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.999999747378752 \cdot 10^{-5}:\\ \;\;\;\;s \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq 1.0000000036274937 \cdot 10^{-15}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{s}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -9.999999747378752e-5)
   (* s (/ 1.0 x))
   (if (<= x 1.0000000036274937e-15) 0.5 (- 1.0 (/ s x)))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -9.999999747378752e-5f) {
		tmp = s * (1.0f / x);
	} else if (x <= 1.0000000036274937e-15f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f - (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 <= (-9.999999747378752e-5)) then
        tmp = s * (1.0e0 / x)
    else if (x <= 1.0000000036274937e-15) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 - (s / x)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-9.999999747378752e-5))
		tmp = Float32(s * Float32(Float32(1.0) / x));
	elseif (x <= Float32(1.0000000036274937e-15))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) - Float32(s / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-9.999999747378752e-5))
		tmp = s * (single(1.0) / x);
	elseif (x <= single(1.0000000036274937e-15))
		tmp = single(0.5);
	else
		tmp = single(1.0) - (s / x);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -9.99999975e-5

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 50.7%

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

      1. neg-mul-1 50.7%

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

      2. unsub-neg 50.7%

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

   5. Simplified 50.7%

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

   6. Taylor expanded in x around inf 45.1%

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

      1. associate-*r/ 45.1%

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

      2. neg-mul-1 45.1%

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

   8. Simplified 45.1%

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

      1. div-inv 45.1%

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

      2. add-sqr-sqrt -0.0%

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

      3. sqrt-unprod 64.2%

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

      4. sqr-neg 64.2%

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

      5. sqrt-unprod 45.1%

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

      6. add-sqr-sqrt 45.1%

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

   10. Applied egg-rr 45.1%

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

   if -9.99999975e-5 < x < 1e-15

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0 63.1%

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

   if 1e-15 < x

   1. Initial program 99.9%

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

      1. distribute-frac-neg 99.9%

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

      2. exp-neg 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0 98.2%

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

   6. Taylor expanded in x around inf 91.0%

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

      1. neg-mul-1 91.0%

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

      2. unsub-neg 91.0%

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

   8. Simplified 91.0%

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

Alternative 12: 74.0% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -500:\\ \;\;\;\;1 - \frac{s}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) -500.0) (- 1.0 (/ s x)) (/ 1.0 (- 2.0 (/ x s)))))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= -500.0f) {
		tmp = 1.0f - (s / x);
	} 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) <= (-500.0e0)) then
        tmp = 1.0e0 - (s / x)
    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(-500.0))
		tmp = Float32(Float32(1.0) - Float32(s / x));
	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(-500.0))
		tmp = single(1.0) - (s / x);
	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) < -500

   1. Initial program 100.0%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing
   3. 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%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0 98.0%

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

   6. Taylor expanded in x around inf 98.0%

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

      1. neg-mul-1 98.0%

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

      2. unsub-neg 98.0%

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

   8. Simplified 98.0%

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

   if -500 < (/.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 62.7%

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

      1. neg-mul-1 62.7%

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

      2. unsub-neg 62.7%

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

   5. Simplified 62.7%

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

Alternative 13: 46.7% accurate, 10.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.999999747378752 \cdot 10^{-5}:\\ \;\;\;\;s \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -9.999999747378752e-5) (* s (/ 1.0 x)) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -9.999999747378752e-5f) {
		tmp = s * (1.0f / 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 <= (-9.999999747378752e-5)) then
        tmp = s * (1.0e0 / x)
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-9.999999747378752e-5))
		tmp = Float32(s * Float32(Float32(1.0) / x));
	else
		tmp = Float32(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-9.999999747378752e-5))
		tmp = s * (single(1.0) / x);
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -9.99999975e-5

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 50.7%

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

      1. neg-mul-1 50.7%

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

      2. unsub-neg 50.7%

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

   5. Simplified 50.7%

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

   6. Taylor expanded in x around inf 45.1%

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

      1. associate-*r/ 45.1%

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

      2. neg-mul-1 45.1%

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

   8. Simplified 45.1%

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

      1. div-inv 45.1%

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

      2. add-sqr-sqrt -0.0%

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

      3. sqrt-unprod 64.2%

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

      4. sqr-neg 64.2%

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

      5. sqrt-unprod 45.1%

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

      6. add-sqr-sqrt 45.1%

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

   10. Applied egg-rr 45.1%

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

   if -9.99999975e-5 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 49.3%

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

Alternative 14: 46.7% accurate, 13.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.999999747378752 \cdot 10^{-5}:\\ \;\;\;\;\frac{s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -9.999999747378752e-5) (/ s x) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -9.999999747378752e-5f) {
		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 <= (-9.999999747378752e-5)) 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(-9.999999747378752e-5))
		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(-9.999999747378752e-5))
		tmp = s / x;
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -9.99999975e-5

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 50.7%

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

      1. neg-mul-1 50.7%

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

      2. unsub-neg 50.7%

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

   5. Simplified 50.7%

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

   6. Taylor expanded in x around inf 45.1%

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

      1. associate-*r/ 45.1%

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

      2. neg-mul-1 45.1%

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

   8. Simplified 45.1%

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

      1. neg-sub0 45.1%

         \[\leadsto \frac{\color{blue}{0 - s}}{x} \]

      2. sub-neg 45.1%

         \[\leadsto \frac{\color{blue}{0 + \left(-s\right)}}{x} \]

      3. add-sqr-sqrt -0.0%

         \[\leadsto \frac{0 + \color{blue}{\sqrt{-s} \cdot \sqrt{-s}}}{x} \]

      4. sqrt-unprod 64.2%

         \[\leadsto \frac{0 + \color{blue}{\sqrt{\left(-s\right) \cdot \left(-s\right)}}}{x} \]

      5. sqr-neg 64.2%

         \[\leadsto \frac{0 + \sqrt{\color{blue}{s \cdot s}}}{x} \]

      6. sqrt-unprod 45.1%

         \[\leadsto \frac{0 + \color{blue}{\sqrt{s} \cdot \sqrt{s}}}{x} \]

      7. add-sqr-sqrt 45.1%

         \[\leadsto \frac{0 + \color{blue}{s}}{x} \]

   10. Applied egg-rr 45.1%

       \[\leadsto \frac{\color{blue}{0 + s}}{x} \]
   11. Step-by-step derivation

       1. +-lft-identity 45.1%

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

   12. Simplified 45.1%

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

   if -9.99999975e-5 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 49.3%

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

Alternative 15: 35.1% 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 37.0%

   \[\leadsto \color{blue}{0.5} \]
4. Add Preprocessing

Reproduce

herbie shell --seed 2024117 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))