Logistic function

Percentage Accurate: 99.8% → 99.9%

Time: 15.4s

Alternatives: 12

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ e^{-\mathsf{log1p}\left(e^{\frac{-x}{s}}\right)} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (exp (- (log1p (exp (/ (- x) s))))))

C

float code(float x, float s) {
	return expf(-log1pf(expf((-x / s))));
}

Julia

function code(x, s)
	return exp(Float32(-log1p(exp(Float32(Float32(-x) / s)))))
end

Derivation

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%

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

   3. add-sqr-sqrt 53.0%

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

   4. sqrt-unprod 63.9%

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

   5. sqr-neg 63.9%

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

   6. sqrt-unprod 11.5%

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

   7. add-sqr-sqrt 25.1%

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

   8. add-sqr-sqrt 25.1%

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

   9. pow2 25.1%

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

   10. metadata-eval 25.1%

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

   11. pow-flip 25.1%

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

   12. add-sqr-sqrt 11.5%

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

   13. sqrt-unprod 63.9%

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

   14. sqr-neg 63.9%

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

   15. sqrt-unprod 53.1%

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

   16. add-sqr-sqrt 99.9%

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

   17. metadata-eval 99.9%

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

   18. metadata-eval 99.9%

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

4. Applied egg-rr 99.9%

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

   1. add-exp-log 99.9%

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

   2. log-rec 99.9%

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

   3. log1p-udef 99.9%

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

   4. add-exp-log 99.9%

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

   5. sqrt-pow2 99.9%

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

   6. metadata-eval 99.9%

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

   7. log-pow 99.9%

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

   8. add-log-exp 99.9%

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

   9. neg-mul-1 99.9%

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

   10. distribute-neg-frac 99.9%

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

6. Applied egg-rr 99.9%

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

7. Final simplification 99.9%

   \[\leadsto e^{-\mathsf{log1p}\left(e^{\frac{-x}{s}}\right)} \]
8. 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 99.9%

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

3. Final simplification 99.9%

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

Alternative 3: 88.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq -2:\\ \;\;\;\;1\\ \mathbf{elif}\;t_0 \leq 2:\\ \;\;\;\;0.5 + \frac{x}{s} \cdot 0.25\\ \mathbf{elif}\;t_0 \leq 4.999999840142846 \cdot 10^{+37}:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{s \cdot \frac{s}{x}}}{\frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)))
   (if (<= t_0 -2.0)
     1.0
     (if (<= t_0 2.0)
       (+ 0.5 (* (/ x s) 0.25))
       (if (<= t_0 4.999999840142846e+37)
         (/ 1.0 (/ (- 4.0 (/ x (* s (/ s x)))) (/ 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 = 1.0f;
	} else if (t_0 <= 2.0f) {
		tmp = 0.5f + ((x / s) * 0.25f);
	} else if (t_0 <= 4.999999840142846e+37f) {
		tmp = 1.0f / ((4.0f - (x / (s * (s / x)))) / (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 = 1.0e0
    else if (t_0 <= 2.0e0) then
        tmp = 0.5e0 + ((x / s) * 0.25e0)
    else if (t_0 <= 4.999999840142846e+37) then
        tmp = 1.0e0 / ((4.0e0 - (x / (s * (s / x)))) / (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(1.0);
	elseif (t_0 <= Float32(2.0))
		tmp = Float32(Float32(0.5) + Float32(Float32(x / s) * Float32(0.25)));
	elseif (t_0 <= Float32(4.999999840142846e+37))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(4.0) - Float32(x / Float32(s * Float32(s / x)))) / 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(1.0);
	elseif (t_0 <= single(2.0))
		tmp = single(0.5) + ((x / s) * single(0.25));
	elseif (t_0 <= single(4.999999840142846e+37))
		tmp = single(1.0) / ((single(4.0) - (x / (s * (s / x)))) / (x / s));
	else
		tmp = single(1.0) / (x / s);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 4 regimes

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

   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%

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

      3. add-sqr-sqrt 100.0%

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

      4. sqrt-unprod 99.3%

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

      5. sqr-neg 99.3%

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

      6. sqrt-unprod -0.0%

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

      7. add-sqr-sqrt 6.5%

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

      8. add-sqr-sqrt 6.5%

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

      9. pow2 6.5%

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

      10. metadata-eval 6.5%

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

      11. pow-flip 6.5%

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

      12. add-sqr-sqrt -0.0%

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

      13. sqrt-unprod 99.3%

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

      14. sqr-neg 99.3%

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

      15. sqrt-unprod 100.0%

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

      16. add-sqr-sqrt 100.0%

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

      17. metadata-eval 100.0%

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

      18. metadata-eval 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. add-exp-log 100.0%

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

      2. log-rec 100.0%

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

      3. log1p-udef 100.0%

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

      4. add-exp-log 100.0%

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

      5. sqrt-pow2 100.0%

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

      6. metadata-eval 100.0%

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

      7. log-pow 100.0%

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

      8. add-log-exp 100.0%

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

      9. neg-mul-1 100.0%

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

      10. distribute-neg-frac 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. exp-neg 100.0%

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

      2. log1p-udef 100.0%

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

      3. +-commutative 100.0%

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

      4. add-exp-log 100.0%

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

      5. add-sqr-sqrt 100.0%

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

      6. associate-/r* 100.0%

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

   8. Applied egg-rr 2.3%

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

      1. *-inverses 99.4%

         \[\leadsto \color{blue}{1} \]

   10. Simplified 99.4%

       \[\leadsto \color{blue}{1} \]

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

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 96.9%

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

      1. *-commutative 96.9%

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

   5. Simplified 96.9%

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

   if 2 < (/.f32 (neg.f32 x) s) < 4.99999984e37

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 11.1%

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

      1. mul-1-neg 11.1%

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

      2. unsub-neg 11.1%

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

   5. Simplified 11.1%

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

      1. sub-neg 11.1%

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

      2. neg-mul-1 11.1%

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

      3. add-log-exp 98.3%

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

      4. log-pow 98.3%

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

      5. metadata-eval 98.3%

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

      6. sqrt-pow2 98.3%

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

      7. flip-+ 0.2%

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

   7. Applied egg-rr 60.4%

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

   8. Taylor expanded in x around inf 60.4%

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

      1. clear-num 60.4%

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

      2. frac-2neg 60.4%

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

      3. frac-times 62.1%

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

      4. *-un-lft-identity 62.1%

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

      5. remove-double-neg 62.1%

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

      6. add-sqr-sqrt 62.1%

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

      7. sqrt-unprod 62.1%

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

      8. sqr-neg 62.1%

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

      9. sqrt-unprod -0.0%

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

      10. add-sqr-sqrt 62.0%

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

      11. add-sqr-sqrt -0.0%

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

      12. sqrt-unprod 74.8%

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

      13. sqr-neg 74.8%

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

      14. sqrt-unprod 62.1%

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

      15. add-sqr-sqrt 62.1%

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

   10. Applied egg-rr 62.1%

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

   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. 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 92.7%

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

      1. associate-*r/ 92.7%

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

      2. neg-mul-1 92.7%

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

   8. Simplified 92.7%

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

      1. remove-double-neg 92.7%

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

      2. frac-2neg 92.7%

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

      3. clear-num 100.0%

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

      4. inv-pow 100.0%

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

      5. add-sqr-sqrt 100.0%

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

      6. sqrt-unprod 100.0%

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

      7. sqr-neg 100.0%

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

      8. sqrt-unprod -0.0%

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

      9. add-sqr-sqrt 100.0%

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

   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 4 regimes into one program.

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -2:\\ \;\;\;\;1\\ \mathbf{elif}\;\frac{-x}{s} \leq 2:\\ \;\;\;\;0.5 + \frac{x}{s} \cdot 0.25\\ \mathbf{elif}\;\frac{-x}{s} \leq 4.999999840142846 \cdot 10^{+37}:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{s \cdot \frac{s}{x}}}{\frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 88.2% accurate, 3.3× speedup

Math

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

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)))
   (if (<= t_0 -2.0)
     1.0
     (if (<= t_0 4.999999840142846e+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 <= -2.0f) {
		tmp = 1.0f;
	} else if (t_0 <= 4.999999840142846e+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 <= (-2.0e0)) then
        tmp = 1.0e0
    else if (t_0 <= 4.999999840142846e+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(-2.0))
		tmp = Float32(1.0);
	elseif (t_0 <= Float32(4.999999840142846e+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(-2.0))
		tmp = single(1.0);
	elseif (t_0 <= single(4.999999840142846e+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) < -2

   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%

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

      3. add-sqr-sqrt 100.0%

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

      4. sqrt-unprod 99.3%

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

      5. sqr-neg 99.3%

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

      6. sqrt-unprod -0.0%

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

      7. add-sqr-sqrt 6.5%

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

      8. add-sqr-sqrt 6.5%

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

      9. pow2 6.5%

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

      10. metadata-eval 6.5%

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

      11. pow-flip 6.5%

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

      12. add-sqr-sqrt -0.0%

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

      13. sqrt-unprod 99.3%

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

      14. sqr-neg 99.3%

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

      15. sqrt-unprod 100.0%

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

      16. add-sqr-sqrt 100.0%

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

      17. metadata-eval 100.0%

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

      18. metadata-eval 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. add-exp-log 100.0%

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

      2. log-rec 100.0%

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

      3. log1p-udef 100.0%

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

      4. add-exp-log 100.0%

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

      5. sqrt-pow2 100.0%

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

      6. metadata-eval 100.0%

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

      7. log-pow 100.0%

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

      8. add-log-exp 100.0%

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

      9. neg-mul-1 100.0%

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

      10. distribute-neg-frac 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. exp-neg 100.0%

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

      2. log1p-udef 100.0%

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

      3. +-commutative 100.0%

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

      4. add-exp-log 100.0%

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

      5. add-sqr-sqrt 100.0%

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

      6. associate-/r* 100.0%

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

   8. Applied egg-rr 2.3%

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

      1. *-inverses 99.4%

         \[\leadsto \color{blue}{1} \]

   10. Simplified 99.4%

       \[\leadsto \color{blue}{1} \]

   if -2 < (/.f32 (neg.f32 x) s) < 4.99999984e37

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 56.0%

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

      1. mul-1-neg 56.0%

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

      2. unsub-neg 56.0%

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

   5. Simplified 56.0%

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

      1. sub-neg 56.0%

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

      2. neg-mul-1 56.0%

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

      3. add-log-exp 96.1%

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

      4. log-pow 96.1%

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

      5. metadata-eval 96.1%

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

      6. sqrt-pow2 96.1%

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

      7. flip-+ 51.0%

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

   7. Applied egg-rr 78.7%

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

      1. clear-num 32.2%

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

      2. frac-2neg 32.2%

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

      3. frac-times 33.0%

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

      4. *-un-lft-identity 33.0%

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

      5. remove-double-neg 33.0%

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

      6. add-sqr-sqrt 29.6%

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

      7. sqrt-unprod 33.0%

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

      8. sqr-neg 33.0%

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

      9. sqrt-unprod 3.4%

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

      10. add-sqr-sqrt 33.0%

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

      11. add-sqr-sqrt -0.0%

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

      12. sqrt-unprod 38.9%

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

      13. sqr-neg 38.9%

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

      14. sqrt-unprod 33.0%

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

      15. add-sqr-sqrt 33.0%

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

   9. Applied egg-rr 79.5%

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

   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. 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 92.7%

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

      1. associate-*r/ 92.7%

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

      2. neg-mul-1 92.7%

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

   8. Simplified 92.7%

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

      1. remove-double-neg 92.7%

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

      2. frac-2neg 92.7%

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

      3. clear-num 100.0%

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

      4. inv-pow 100.0%

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

      5. add-sqr-sqrt 100.0%

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

      6. sqrt-unprod 100.0%

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

      7. sqr-neg 100.0%

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

      8. sqrt-unprod -0.0%

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

      9. add-sqr-sqrt 100.0%

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

   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 90.9%

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

Alternative 5: 77.1% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq -2:\\ \;\;\;\;1\\ \mathbf{elif}\;t_0 \leq 2:\\ \;\;\;\;0.5 + \frac{x}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)))
   (if (<= t_0 -2.0)
     1.0
     (if (<= t_0 2.0) (+ 0.5 (* (/ x s) 0.25)) (/ 1.0 t_0)))))

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if (t_0 <= -2.0f) {
		tmp = 1.0f;
	} else if (t_0 <= 2.0f) {
		tmp = 0.5f + ((x / s) * 0.25f);
	} else {
		tmp = 1.0f / t_0;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: tmp
    t_0 = -x / s
    if (t_0 <= (-2.0e0)) then
        tmp = 1.0e0
    else if (t_0 <= 2.0e0) then
        tmp = 0.5e0 + ((x / s) * 0.25e0)
    else
        tmp = 1.0e0 / t_0
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = Float32(Float32(-x) / s)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-2.0))
		tmp = Float32(1.0);
	elseif (t_0 <= Float32(2.0))
		tmp = Float32(Float32(0.5) + Float32(Float32(x / s) * Float32(0.25)));
	else
		tmp = Float32(Float32(1.0) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = -x / s;
	tmp = single(0.0);
	if (t_0 <= single(-2.0))
		tmp = single(1.0);
	elseif (t_0 <= single(2.0))
		tmp = single(0.5) + ((x / s) * single(0.25));
	else
		tmp = single(1.0) / t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

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

      3. add-sqr-sqrt 100.0%

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

      4. sqrt-unprod 99.3%

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

      5. sqr-neg 99.3%

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

      6. sqrt-unprod -0.0%

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

      7. add-sqr-sqrt 6.5%

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

      8. add-sqr-sqrt 6.5%

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

      9. pow2 6.5%

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

      10. metadata-eval 6.5%

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

      11. pow-flip 6.5%

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

      12. add-sqr-sqrt -0.0%

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

      13. sqrt-unprod 99.3%

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

      14. sqr-neg 99.3%

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

      15. sqrt-unprod 100.0%

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

      16. add-sqr-sqrt 100.0%

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

      17. metadata-eval 100.0%

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

      18. metadata-eval 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. add-exp-log 100.0%

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

      2. log-rec 100.0%

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

      3. log1p-udef 100.0%

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

      4. add-exp-log 100.0%

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

      5. sqrt-pow2 100.0%

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

      6. metadata-eval 100.0%

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

      7. log-pow 100.0%

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

      8. add-log-exp 100.0%

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

      9. neg-mul-1 100.0%

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

      10. distribute-neg-frac 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. exp-neg 100.0%

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

      2. log1p-udef 100.0%

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

      3. +-commutative 100.0%

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

      4. add-exp-log 100.0%

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

      5. add-sqr-sqrt 100.0%

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

      6. associate-/r* 100.0%

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

   8. Applied egg-rr 2.3%

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

      1. *-inverses 99.4%

         \[\leadsto \color{blue}{1} \]

   10. Simplified 99.4%

       \[\leadsto \color{blue}{1} \]

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

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 96.9%

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

      1. *-commutative 96.9%

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

   5. Simplified 96.9%

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

   if 2 < (/.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 50.2%

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

      1. mul-1-neg 50.2%

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

      2. unsub-neg 50.2%

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

   5. Simplified 50.2%

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

   6. Taylor expanded in x around inf 50.2%

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

      1. mul-1-neg 50.2%

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

      2. distribute-frac-neg 50.2%

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

   8. Simplified 50.2%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -2:\\ \;\;\;\;1\\ \mathbf{elif}\;\frac{-x}{s} \leq 2:\\ \;\;\;\;0.5 + \frac{x}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{-x}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.0% accurate, 4.9× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if (t_0 <= -2.0f) {
		tmp = 1.0f;
	} else if (t_0 <= 2.0f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / t_0;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: tmp
    t_0 = -x / s
    if (t_0 <= (-2.0e0)) then
        tmp = 1.0e0
    else if (t_0 <= 2.0e0) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 / t_0
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = Float32(Float32(-x) / s)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-2.0))
		tmp = Float32(1.0);
	elseif (t_0 <= Float32(2.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = -x / s;
	tmp = single(0.0);
	if (t_0 <= single(-2.0))
		tmp = single(1.0);
	elseif (t_0 <= single(2.0))
		tmp = single(0.5);
	else
		tmp = single(1.0) / t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

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

   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%

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

      3. add-sqr-sqrt 100.0%

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

      4. sqrt-unprod 99.3%

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

      5. sqr-neg 99.3%

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

      6. sqrt-unprod -0.0%

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

      7. add-sqr-sqrt 6.5%

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

      8. add-sqr-sqrt 6.5%

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

      9. pow2 6.5%

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

      10. metadata-eval 6.5%

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

      11. pow-flip 6.5%

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

      12. add-sqr-sqrt -0.0%

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

      13. sqrt-unprod 99.3%

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

      14. sqr-neg 99.3%

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

      15. sqrt-unprod 100.0%

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

      16. add-sqr-sqrt 100.0%

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

      17. metadata-eval 100.0%

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

      18. metadata-eval 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. add-exp-log 100.0%

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

      2. log-rec 100.0%

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

      3. log1p-udef 100.0%

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

      4. add-exp-log 100.0%

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

      5. sqrt-pow2 100.0%

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

      6. metadata-eval 100.0%

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

      7. log-pow 100.0%

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

      8. add-log-exp 100.0%

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

      9. neg-mul-1 100.0%

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

      10. distribute-neg-frac 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. exp-neg 100.0%

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

      2. log1p-udef 100.0%

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

      3. +-commutative 100.0%

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

      4. add-exp-log 100.0%

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

      5. add-sqr-sqrt 100.0%

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

      6. associate-/r* 100.0%

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

   8. Applied egg-rr 2.3%

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

      1. *-inverses 99.4%

         \[\leadsto \color{blue}{1} \]

   10. Simplified 99.4%

       \[\leadsto \color{blue}{1} \]

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

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 87.2%

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

   if 2 < (/.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 50.2%

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

      1. mul-1-neg 50.2%

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

      2. unsub-neg 50.2%

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

   5. Simplified 50.2%

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

   6. Taylor expanded in x around inf 50.2%

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

      1. mul-1-neg 50.2%

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

      2. distribute-frac-neg 50.2%

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

   8. Simplified 50.2%

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

4. Final simplification 79.1%

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

Alternative 7: 76.4% accurate, 7.2× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= -2.0f) {
		tmp = 1.0f;
	} 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) <= (-2.0e0)) then
        tmp = 1.0e0
    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(-2.0))
		tmp = Float32(1.0);
	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(-2.0))
		tmp = single(1.0);
	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) < -2

   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%

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

      3. add-sqr-sqrt 100.0%

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

      4. sqrt-unprod 99.3%

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

      5. sqr-neg 99.3%

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

      6. sqrt-unprod -0.0%

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

      7. add-sqr-sqrt 6.5%

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

      8. add-sqr-sqrt 6.5%

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

      9. pow2 6.5%

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

      10. metadata-eval 6.5%

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

      11. pow-flip 6.5%

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

      12. add-sqr-sqrt -0.0%

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

      13. sqrt-unprod 99.3%

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

      14. sqr-neg 99.3%

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

      15. sqrt-unprod 100.0%

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

      16. add-sqr-sqrt 100.0%

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

      17. metadata-eval 100.0%

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

      18. metadata-eval 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. add-exp-log 100.0%

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

      2. log-rec 100.0%

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

      3. log1p-udef 100.0%

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

      4. add-exp-log 100.0%

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

      5. sqrt-pow2 100.0%

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

      6. metadata-eval 100.0%

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

      7. log-pow 100.0%

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

      8. add-log-exp 100.0%

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

      9. neg-mul-1 100.0%

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

      10. distribute-neg-frac 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. exp-neg 100.0%

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

      2. log1p-udef 100.0%

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

      3. +-commutative 100.0%

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

      4. add-exp-log 100.0%

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

      5. add-sqr-sqrt 100.0%

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

      6. associate-/r* 100.0%

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

   8. Applied egg-rr 2.3%

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

      1. *-inverses 99.4%

         \[\leadsto \color{blue}{1} \]

   10. Simplified 99.4%

       \[\leadsto \color{blue}{1} \]

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

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 67.7%

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

      1. mul-1-neg 67.7%

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

      2. unsub-neg 67.7%

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

   5. Simplified 67.7%

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

4. Final simplification 80.7%

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

Alternative 8: 69.5% accurate, 9.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0010000000474974513:\\ \;\;\;\;\frac{s}{x}\\ \mathbf{elif}\;x \leq 4.999999841327613 \cdot 10^{-22}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -0.0010000000474974513)
   (/ s x)
   (if (<= x 4.999999841327613e-22) 0.5 1.0)))

C

float code(float x, float s) {
	float tmp;
	if (x <= -0.0010000000474974513f) {
		tmp = s / x;
	} else if (x <= 4.999999841327613e-22f) {
		tmp = 0.5f;
	} else {
		tmp = 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 <= (-0.0010000000474974513e0)) then
        tmp = s / x
    else if (x <= 4.999999841327613e-22) then
        tmp = 0.5e0
    else
        tmp = 1.0e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-0.0010000000474974513))
		tmp = Float32(s / x);
	elseif (x <= Float32(4.999999841327613e-22))
		tmp = Float32(0.5);
	else
		tmp = Float32(1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-0.0010000000474974513))
		tmp = s / x;
	elseif (x <= single(4.999999841327613e-22))
		tmp = single(0.5);
	else
		tmp = single(1.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -0.00100000005

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 61.8%

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

      1. mul-1-neg 61.8%

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

      2. unsub-neg 61.8%

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

   5. Simplified 61.8%

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

   6. Taylor expanded in x around inf 57.7%

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

      1. associate-*r/ 57.7%

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

      2. neg-mul-1 57.7%

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

   8. Simplified 57.7%

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

      1. expm1-log1p-u 57.7%

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

      2. expm1-udef 98.7%

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

      3. add-sqr-sqrt -0.0%

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

      4. sqrt-unprod 98.7%

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

      5. sqr-neg 98.7%

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

      6. sqrt-unprod 98.7%

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

      7. add-sqr-sqrt 98.7%

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

   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 57.7%

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

       2. expm1-log1p 57.7%

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

   12. Simplified 57.7%

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

   if -0.00100000005 < x < 4.9999998e-22

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 62.6%

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

   if 4.9999998e-22 < 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%

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

      3. add-sqr-sqrt 99.9%

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

      4. sqrt-unprod 99.9%

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

      5. sqr-neg 99.9%

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

      6. sqrt-unprod -0.0%

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

      7. add-sqr-sqrt 14.3%

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

      8. add-sqr-sqrt 14.3%

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

      9. pow2 14.3%

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

      10. metadata-eval 14.3%

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

      11. pow-flip 14.3%

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

      12. add-sqr-sqrt -0.0%

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

      13. sqrt-unprod 99.9%

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

      14. sqr-neg 99.9%

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

      15. sqrt-unprod 99.9%

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

      16. add-sqr-sqrt 99.9%

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

      17. metadata-eval 99.9%

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

      18. metadata-eval 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. add-exp-log 99.9%

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

      2. log-rec 99.9%

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

      3. log1p-udef 100.0%

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

      4. add-exp-log 100.0%

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

      5. sqrt-pow2 99.9%

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

      6. metadata-eval 99.9%

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

      7. log-pow 99.9%

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

      8. add-log-exp 99.9%

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

      9. neg-mul-1 99.9%

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

      10. distribute-neg-frac 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. exp-neg 99.9%

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

      2. log1p-udef 99.9%

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

      3. +-commutative 99.9%

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

      4. add-exp-log 99.9%

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

      5. add-sqr-sqrt 99.7%

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

      6. associate-/r* 99.9%

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

   8. Applied egg-rr 4.8%

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

      1. *-inverses 92.6%

         \[\leadsto \color{blue}{1} \]

   10. Simplified 92.6%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0010000000474974513:\\ \;\;\;\;\frac{s}{x}\\ \mathbf{elif}\;x \leq 4.999999841327613 \cdot 10^{-22}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 69.5% accurate, 9.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0010000000474974513:\\ \;\;\;\;s \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq 4.999999841327613 \cdot 10^{-22}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -0.0010000000474974513)
   (* s (/ 1.0 x))
   (if (<= x 4.999999841327613e-22) 0.5 1.0)))

C

float code(float x, float s) {
	float tmp;
	if (x <= -0.0010000000474974513f) {
		tmp = s * (1.0f / x);
	} else if (x <= 4.999999841327613e-22f) {
		tmp = 0.5f;
	} else {
		tmp = 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 <= (-0.0010000000474974513e0)) then
        tmp = s * (1.0e0 / x)
    else if (x <= 4.999999841327613e-22) then
        tmp = 0.5e0
    else
        tmp = 1.0e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-0.0010000000474974513))
		tmp = Float32(s * Float32(Float32(1.0) / x));
	elseif (x <= Float32(4.999999841327613e-22))
		tmp = Float32(0.5);
	else
		tmp = Float32(1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-0.0010000000474974513))
		tmp = s * (single(1.0) / x);
	elseif (x <= single(4.999999841327613e-22))
		tmp = single(0.5);
	else
		tmp = single(1.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -0.00100000005

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 61.8%

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

      1. mul-1-neg 61.8%

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

      2. unsub-neg 61.8%

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

   5. Simplified 61.8%

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

   6. Taylor expanded in x around inf 57.7%

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

      1. associate-*r/ 57.7%

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

      2. neg-mul-1 57.7%

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

   8. Simplified 57.7%

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

      1. div-inv 57.7%

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

      2. *-commutative 57.7%

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

      3. add-sqr-sqrt -0.0%

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

      4. sqrt-unprod 69.4%

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

      5. sqr-neg 69.4%

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

      6. sqrt-unprod 57.7%

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

      7. add-sqr-sqrt 57.7%

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

   10. Applied egg-rr 57.7%

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

   if -0.00100000005 < x < 4.9999998e-22

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 62.6%

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

   if 4.9999998e-22 < 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%

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

      3. add-sqr-sqrt 99.9%

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

      4. sqrt-unprod 99.9%

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

      5. sqr-neg 99.9%

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

      6. sqrt-unprod -0.0%

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

      7. add-sqr-sqrt 14.3%

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

      8. add-sqr-sqrt 14.3%

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

      9. pow2 14.3%

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

      10. metadata-eval 14.3%

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

      11. pow-flip 14.3%

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

      12. add-sqr-sqrt -0.0%

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

      13. sqrt-unprod 99.9%

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

      14. sqr-neg 99.9%

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

      15. sqrt-unprod 99.9%

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

      16. add-sqr-sqrt 99.9%

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

      17. metadata-eval 99.9%

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

      18. metadata-eval 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. add-exp-log 99.9%

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

      2. log-rec 99.9%

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

      3. log1p-udef 100.0%

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

      4. add-exp-log 100.0%

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

      5. sqrt-pow2 99.9%

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

      6. metadata-eval 99.9%

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

      7. log-pow 99.9%

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

      8. add-log-exp 99.9%

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

      9. neg-mul-1 99.9%

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

      10. distribute-neg-frac 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. exp-neg 99.9%

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

      2. log1p-udef 99.9%

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

      3. +-commutative 99.9%

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

      4. add-exp-log 99.9%

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

      5. add-sqr-sqrt 99.7%

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

      6. associate-/r* 99.9%

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

   8. Applied egg-rr 4.8%

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

      1. *-inverses 92.6%

         \[\leadsto \color{blue}{1} \]

   10. Simplified 92.6%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0010000000474974513:\\ \;\;\;\;s \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq 4.999999841327613 \cdot 10^{-22}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 70.8% accurate, 9.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0010000000474974513:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \mathbf{elif}\;x \leq 4.999999841327613 \cdot 10^{-22}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -0.0010000000474974513)
   (/ 1.0 (/ x s))
   (if (<= x 4.999999841327613e-22) 0.5 1.0)))

C

float code(float x, float s) {
	float tmp;
	if (x <= -0.0010000000474974513f) {
		tmp = 1.0f / (x / s);
	} else if (x <= 4.999999841327613e-22f) {
		tmp = 0.5f;
	} else {
		tmp = 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 <= (-0.0010000000474974513e0)) then
        tmp = 1.0e0 / (x / s)
    else if (x <= 4.999999841327613e-22) then
        tmp = 0.5e0
    else
        tmp = 1.0e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-0.0010000000474974513))
		tmp = Float32(Float32(1.0) / Float32(x / s));
	elseif (x <= Float32(4.999999841327613e-22))
		tmp = Float32(0.5);
	else
		tmp = Float32(1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-0.0010000000474974513))
		tmp = single(1.0) / (x / s);
	elseif (x <= single(4.999999841327613e-22))
		tmp = single(0.5);
	else
		tmp = single(1.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -0.00100000005

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 61.8%

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

      1. mul-1-neg 61.8%

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

      2. unsub-neg 61.8%

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

   5. Simplified 61.8%

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

   6. Taylor expanded in x around inf 57.7%

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

      1. associate-*r/ 57.7%

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

      2. neg-mul-1 57.7%

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

   8. Simplified 57.7%

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

      1. remove-double-neg 57.7%

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

      2. frac-2neg 57.7%

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

      3. clear-num 61.8%

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

      4. inv-pow 61.8%

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

      5. add-sqr-sqrt 61.8%

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

      6. sqrt-unprod 70.2%

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

      7. sqr-neg 70.2%

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

      8. sqrt-unprod -0.0%

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

      9. add-sqr-sqrt 61.8%

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

   10. Applied egg-rr 61.8%

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

       1. unpow-1 61.8%

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

   12. Simplified 61.8%

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

   if -0.00100000005 < x < 4.9999998e-22

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 62.6%

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

   if 4.9999998e-22 < 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%

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

      3. add-sqr-sqrt 99.9%

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

      4. sqrt-unprod 99.9%

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

      5. sqr-neg 99.9%

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

      6. sqrt-unprod -0.0%

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

      7. add-sqr-sqrt 14.3%

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

      8. add-sqr-sqrt 14.3%

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

      9. pow2 14.3%

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

      10. metadata-eval 14.3%

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

      11. pow-flip 14.3%

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

      12. add-sqr-sqrt -0.0%

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

      13. sqrt-unprod 99.9%

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

      14. sqr-neg 99.9%

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

      15. sqrt-unprod 99.9%

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

      16. add-sqr-sqrt 99.9%

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

      17. metadata-eval 99.9%

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

      18. metadata-eval 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. add-exp-log 99.9%

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

      2. log-rec 99.9%

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

      3. log1p-udef 100.0%

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

      4. add-exp-log 100.0%

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

      5. sqrt-pow2 99.9%

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

      6. metadata-eval 99.9%

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

      7. log-pow 99.9%

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

      8. add-log-exp 99.9%

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

      9. neg-mul-1 99.9%

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

      10. distribute-neg-frac 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. exp-neg 99.9%

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

      2. log1p-udef 99.9%

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

      3. +-commutative 99.9%

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

      4. add-exp-log 99.9%

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

      5. add-sqr-sqrt 99.7%

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

      6. associate-/r* 99.9%

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

   8. Applied egg-rr 4.8%

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

      1. *-inverses 92.6%

         \[\leadsto \color{blue}{1} \]

   10. Simplified 92.6%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0010000000474974513:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \mathbf{elif}\;x \leq 4.999999841327613 \cdot 10^{-22}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 58.1% accurate, 17.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.999999841327613 \cdot 10^{-22}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (if (<= x 4.999999841327613e-22) 0.5 1.0))

C

float code(float x, float s) {
	float tmp;
	if (x <= 4.999999841327613e-22f) {
		tmp = 0.5f;
	} else {
		tmp = 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 <= 4.999999841327613e-22) then
        tmp = 0.5e0
    else
        tmp = 1.0e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(4.999999841327613e-22))
		tmp = Float32(0.5);
	else
		tmp = Float32(1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(4.999999841327613e-22))
		tmp = single(0.5);
	else
		tmp = single(1.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 4.9999998e-22

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 34.4%

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

   if 4.9999998e-22 < 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%

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

      3. add-sqr-sqrt 99.9%

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

      4. sqrt-unprod 99.9%

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

      5. sqr-neg 99.9%

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

      6. sqrt-unprod -0.0%

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

      7. add-sqr-sqrt 14.3%

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

      8. add-sqr-sqrt 14.3%

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

      9. pow2 14.3%

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

      10. metadata-eval 14.3%

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

      11. pow-flip 14.3%

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

      12. add-sqr-sqrt -0.0%

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

      13. sqrt-unprod 99.9%

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

      14. sqr-neg 99.9%

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

      15. sqrt-unprod 99.9%

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

      16. add-sqr-sqrt 99.9%

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

      17. metadata-eval 99.9%

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

      18. metadata-eval 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. add-exp-log 99.9%

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

      2. log-rec 99.9%

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

      3. log1p-udef 100.0%

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

      4. add-exp-log 100.0%

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

      5. sqrt-pow2 99.9%

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

      6. metadata-eval 99.9%

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

      7. log-pow 99.9%

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

      8. add-log-exp 99.9%

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

      9. neg-mul-1 99.9%

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

      10. distribute-neg-frac 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. exp-neg 99.9%

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

      2. log1p-udef 99.9%

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

      3. +-commutative 99.9%

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

      4. add-exp-log 99.9%

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

      5. add-sqr-sqrt 99.7%

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

      6. associate-/r* 99.9%

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

   8. Applied egg-rr 4.8%

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

      1. *-inverses 92.6%

         \[\leadsto \color{blue}{1} \]

   10. Simplified 92.6%

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

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.999999841327613 \cdot 10^{-22}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 35.3% accurate, 108.0× speedup

Math

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

FPCore

(FPCore (x s) :precision binary32 0.5)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around 0 34.3%

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

4. Final simplification 34.3%

   \[\leadsto 0.5 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024020 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))