Logistic function

Percentage Accurate: 99.8% → 99.9%

Time: 9.7s

Alternatives: 13

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. 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. div-inv 99.9%

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

   4. add-sqr-sqrt 46.1%

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

   5. sqrt-unprod 57.7%

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

   6. sqr-neg 57.7%

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

   7. sqrt-unprod 13.2%

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

   8. add-sqr-sqrt 26.3%

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

   9. div-inv 26.3%

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

   10. add-cube-cbrt 26.3%

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

   11. pow3 26.3%

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

   12. pow-flip 26.3%

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

3. Applied egg-rr 99.9%

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

   1. add-exp-log 99.9%

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

   2. log-rec 99.8%

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

   3. log1p-udef 99.8%

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

   4. pow1/3 99.8%

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

   5. pow-pow 99.9%

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

   6. metadata-eval 99.9%

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

5. Applied egg-rr 99.9%

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

   1. unpow-1 99.9%

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

   2. rec-exp 99.9%

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

   3. distribute-frac-neg 99.9%

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

7. Simplified 99.9%

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

8. Final simplification 99.9%

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

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. Final simplification 99.9%

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

Alternative 3: 60.7% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: tmp
    t_0 = -x / s
    if (t_0 <= (-1.0e0)) then
        tmp = 0.5e0
    else if (t_0 <= 9.999999680285692e+37) then
        tmp = 1.0e0 / ((4.0e0 - (x / (s * (s / x)))) / ((x / s) + 2.0e0))
    else
        tmp = 1.0e0 / (x / s)
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = Float32(Float32(-x) / s)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-1.0))
		tmp = Float32(0.5);
	elseif (t_0 <= Float32(9.999999680285692e+37))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(4.0) - Float32(x / Float32(s * Float32(s / x)))) / Float32(Float32(x / s) + Float32(2.0))));
	else
		tmp = Float32(Float32(1.0) / Float32(x / s));
	end
	return tmp
end

MATLAB

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 28.2%

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

   if -1 < (/.f32 (neg.f32 x) s) < 9.99999968e37

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 50.6%

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

      1. mul-1-neg 50.6%

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

      2. unsub-neg 50.6%

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

   4. Simplified 50.6%

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

      1. sub-neg 50.6%

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

      2. neg-mul-1 50.6%

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

      3. rem-log-exp 96.9%

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

      4. pow-exp 96.9%

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

      5. flip-+ 45.0%

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

      6. metadata-eval 45.0%

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

      7. pow-exp 45.0%

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

      8. rem-log-exp 45.0%

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

      9. pow-exp 45.0%

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

      10. rem-log-exp 45.7%

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

      11. neg-mul-1 45.7%

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

      12. distribute-neg-frac 45.7%

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

      13. neg-mul-1 45.7%

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

      14. distribute-neg-frac 45.7%

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

      15. pow-exp 45.7%

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

      16. rem-log-exp 76.1%

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

      17. neg-mul-1 76.1%

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

      18. distribute-neg-frac 76.1%

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

   6. Applied egg-rr 76.1%

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

      1. frac-times 78.8%

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

      2. sqr-neg 78.8%

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

      3. frac-times 76.1%

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

      4. clear-num 76.1%

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

      5. frac-times 78.1%

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

      6. *-un-lft-identity 78.1%

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

   8. Applied egg-rr 78.1%

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

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

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 100.0%

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

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

   4. Simplified 100.0%

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

   5. Taylor expanded in x around inf 87.5%

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

      1. associate-*r/ 87.5%

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

      2. neg-mul-1 87.5%

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

   7. Simplified 87.5%

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

      1. clear-num 100.0%

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

      2. inv-pow 100.0%

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

      3. add-sqr-sqrt -0.0%

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

      4. sqrt-unprod 100.0%

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

      5. sqr-neg 100.0%

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

      6. sqrt-unprod 100.0%

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

      7. add-sqr-sqrt 100.0%

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

   9. Applied egg-rr 100.0%

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

       1. unpow-1 100.0%

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

   11. Simplified 100.0%

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

4. Final simplification 64.2%

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

Alternative 4: 62.3% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: tmp
    t_0 = -x / s
    if (t_0 <= (-1.0e0)) then
        tmp = 0.5e0
    else if (t_0 <= 9.999999680285692e+37) then
        tmp = 1.0e0 / ((4.0e0 - (x * ((x / s) / s))) / ((x / s) + 2.0e0))
    else
        tmp = 1.0e0 / (x / s)
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = Float32(Float32(-x) / s)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-1.0))
		tmp = Float32(0.5);
	elseif (t_0 <= Float32(9.999999680285692e+37))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(4.0) - Float32(x * Float32(Float32(x / s) / s))) / Float32(Float32(x / s) + Float32(2.0))));
	else
		tmp = Float32(Float32(1.0) / Float32(x / s));
	end
	return tmp
end

MATLAB

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 28.2%

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

   if -1 < (/.f32 (neg.f32 x) s) < 9.99999968e37

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 50.6%

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

      1. mul-1-neg 50.6%

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

      2. unsub-neg 50.6%

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

   4. Simplified 50.6%

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

      1. sub-neg 50.6%

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

      2. neg-mul-1 50.6%

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

      3. rem-log-exp 96.9%

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

      4. pow-exp 96.9%

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

      5. flip-+ 45.0%

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

      6. metadata-eval 45.0%

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

      7. pow-exp 45.0%

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

      8. rem-log-exp 45.0%

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

      9. pow-exp 45.0%

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

      10. rem-log-exp 45.7%

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

      11. neg-mul-1 45.7%

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

      12. distribute-neg-frac 45.7%

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

      13. neg-mul-1 45.7%

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

      14. distribute-neg-frac 45.7%

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

      15. pow-exp 45.7%

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

      16. rem-log-exp 76.1%

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

      17. neg-mul-1 76.1%

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

      18. distribute-neg-frac 76.1%

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

   6. Applied egg-rr 76.1%

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

      1. frac-times 78.8%

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

      2. sqr-neg 78.8%

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

      3. frac-times 76.1%

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

      4. clear-num 76.1%

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

      5. frac-times 78.1%

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

      6. *-un-lft-identity 78.1%

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

   8. Applied egg-rr 78.1%

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

      1. associate-/r* 76.1%

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

      2. div-inv 76.1%

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

      3. clear-num 76.1%

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

      4. *-un-lft-identity 76.1%

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

      5. times-frac 82.3%

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

   10. Applied egg-rr 82.3%

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

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

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 100.0%

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

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

   4. Simplified 100.0%

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

   5. Taylor expanded in x around inf 87.5%

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

      1. associate-*r/ 87.5%

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

      2. neg-mul-1 87.5%

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

   7. Simplified 87.5%

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

      1. clear-num 100.0%

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

      2. inv-pow 100.0%

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

      3. add-sqr-sqrt -0.0%

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

      4. sqrt-unprod 100.0%

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

      5. sqr-neg 100.0%

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

      6. sqrt-unprod 100.0%

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

      7. add-sqr-sqrt 100.0%

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

   9. Applied egg-rr 100.0%

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

       1. unpow-1 100.0%

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

   11. Simplified 100.0%

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

4. Final simplification 66.4%

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

Alternative 5: 57.7% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: tmp
    t_0 = -x / s
    if (t_0 <= 2.0e0) then
        tmp = 0.5e0
    else if (t_0 <= 9.999999680285692e+37) then
        tmp = 1.0e0 / ((4.0e0 - ((x / s) * (x / s))) / (x / s))
    else
        tmp = 1.0e0 / (x / s)
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = Float32(Float32(-x) / s)
	tmp = Float32(0.0)
	if (t_0 <= Float32(2.0))
		tmp = Float32(0.5);
	elseif (t_0 <= Float32(9.999999680285692e+37))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(4.0) - Float32(Float32(x / s) * Float32(x / s))) / Float32(x / s)));
	else
		tmp = Float32(Float32(1.0) / Float32(x / s));
	end
	return tmp
end

MATLAB

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 53.6%

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

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

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 10.9%

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

      1. mul-1-neg 10.9%

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

      2. unsub-neg 10.9%

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

   4. Simplified 10.9%

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

      1. sub-neg 10.9%

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

      2. neg-mul-1 10.9%

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

      3. rem-log-exp 97.7%

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

      4. pow-exp 97.7%

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

      5. flip-+ 0.5%

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

      6. metadata-eval 0.5%

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

      7. pow-exp 0.5%

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

      8. rem-log-exp 0.5%

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

      9. pow-exp 0.5%

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

      10. rem-log-exp 1.8%

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

      11. neg-mul-1 1.8%

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

      12. distribute-neg-frac 1.8%

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

      13. neg-mul-1 1.8%

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

      14. distribute-neg-frac 1.8%

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

      15. pow-exp 1.8%

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

      16. rem-log-exp 58.7%

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

      17. neg-mul-1 58.7%

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

      18. distribute-neg-frac 58.7%

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

   6. Applied egg-rr 58.7%

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

   7. Taylor expanded in x around inf 58.7%

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

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

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 100.0%

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

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

   4. Simplified 100.0%

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

   5. Taylor expanded in x around inf 87.5%

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

      1. associate-*r/ 87.5%

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

      2. neg-mul-1 87.5%

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

   7. Simplified 87.5%

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

      1. clear-num 100.0%

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

      2. inv-pow 100.0%

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

      3. add-sqr-sqrt -0.0%

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

      4. sqrt-unprod 100.0%

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

      5. sqr-neg 100.0%

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

      6. sqrt-unprod 100.0%

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

      7. add-sqr-sqrt 100.0%

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

   9. Applied egg-rr 100.0%

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

       1. unpow-1 100.0%

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

   11. Simplified 100.0%

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

4. Final simplification 61.5%

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

Alternative 6: 49.2% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + x \cdot \left(\frac{-0.3333333333333333}{s} + \frac{-0.6666666666666666}{s}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) -1.0)
   0.5
   (/
    1.0
    (+ 2.0 (* x (+ (/ -0.3333333333333333 s) (/ -0.6666666666666666 s)))))))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= -1.0f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / (2.0f + (x * ((-0.3333333333333333f / s) + (-0.6666666666666666f / s))));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((-x / s) <= (-1.0e0)) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 / (2.0e0 + (x * (((-0.3333333333333333e0) / s) + ((-0.6666666666666666e0) / s))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(-x) / s) <= Float32(-1.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(x * Float32(Float32(Float32(-0.3333333333333333) / s) + Float32(Float32(-0.6666666666666666) / s)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((-x / s) <= single(-1.0))
		tmp = single(0.5);
	else
		tmp = single(1.0) / (single(2.0) + (x * ((single(-0.3333333333333333) / s) + (single(-0.6666666666666666) / s))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 28.2%

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

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

   1. Initial program 99.8%

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

      1. distribute-frac-neg 99.8%

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

      2. exp-neg 99.8%

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

      3. div-inv 99.8%

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

      4. add-sqr-sqrt 18.3%

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

      5. sqrt-unprod 37.7%

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

      6. sqr-neg 37.7%

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

      7. sqrt-unprod 20.0%

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

      8. add-sqr-sqrt 36.5%

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

      9. div-inv 36.5%

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

      10. add-cube-cbrt 36.5%

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

      11. pow3 36.5%

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

      12. pow-flip 36.5%

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

   3. Applied egg-rr 99.8%

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

   4. Taylor expanded in s around inf 61.1%

      \[\leadsto \frac{1}{\color{blue}{2 + \left(-0.6666666666666666 \cdot \frac{x}{s} + -0.3333333333333333 \cdot \frac{x}{s}\right)}} \]
   5. Step-by-step derivation

      1. +-commutative 61.1%

         \[\leadsto \frac{1}{2 + \color{blue}{\left(-0.3333333333333333 \cdot \frac{x}{s} + -0.6666666666666666 \cdot \frac{x}{s}\right)}} \]

      2. *-un-lft-identity 61.1%

         \[\leadsto \frac{1}{2 + \left(\color{blue}{1 \cdot \left(-0.3333333333333333 \cdot \frac{x}{s}\right)} + -0.6666666666666666 \cdot \frac{x}{s}\right)} \]

      3. fma-def 61.1%

         \[\leadsto \frac{1}{2 + \color{blue}{\mathsf{fma}\left(1, -0.3333333333333333 \cdot \frac{x}{s}, -0.6666666666666666 \cdot \frac{x}{s}\right)}} \]

      4. clear-num 61.1%

         \[\leadsto \frac{1}{2 + \mathsf{fma}\left(1, -0.3333333333333333 \cdot \color{blue}{\frac{1}{\frac{s}{x}}}, -0.6666666666666666 \cdot \frac{x}{s}\right)} \]

      5. un-div-inv 61.1%

         \[\leadsto \frac{1}{2 + \mathsf{fma}\left(1, \color{blue}{\frac{-0.3333333333333333}{\frac{s}{x}}}, -0.6666666666666666 \cdot \frac{x}{s}\right)} \]

      6. clear-num 61.1%

         \[\leadsto \frac{1}{2 + \mathsf{fma}\left(1, \frac{-0.3333333333333333}{\frac{s}{x}}, -0.6666666666666666 \cdot \color{blue}{\frac{1}{\frac{s}{x}}}\right)} \]

      7. un-div-inv 61.1%

         \[\leadsto \frac{1}{2 + \mathsf{fma}\left(1, \frac{-0.3333333333333333}{\frac{s}{x}}, \color{blue}{\frac{-0.6666666666666666}{\frac{s}{x}}}\right)} \]

   6. Applied egg-rr 61.1%

      \[\leadsto \frac{1}{2 + \color{blue}{\mathsf{fma}\left(1, \frac{-0.3333333333333333}{\frac{s}{x}}, \frac{-0.6666666666666666}{\frac{s}{x}}\right)}} \]
   7. Step-by-step derivation

      1. fma-udef 61.1%

         \[\leadsto \frac{1}{2 + \color{blue}{\left(1 \cdot \frac{-0.3333333333333333}{\frac{s}{x}} + \frac{-0.6666666666666666}{\frac{s}{x}}\right)}} \]

      2. *-lft-identity 61.1%

         \[\leadsto \frac{1}{2 + \left(\color{blue}{\frac{-0.3333333333333333}{\frac{s}{x}}} + \frac{-0.6666666666666666}{\frac{s}{x}}\right)} \]

      3. associate-/r/ 61.1%

         \[\leadsto \frac{1}{2 + \left(\color{blue}{\frac{-0.3333333333333333}{s} \cdot x} + \frac{-0.6666666666666666}{\frac{s}{x}}\right)} \]

      4. associate-/r/ 61.1%

         \[\leadsto \frac{1}{2 + \left(\frac{-0.3333333333333333}{s} \cdot x + \color{blue}{\frac{-0.6666666666666666}{s} \cdot x}\right)} \]

      5. distribute-rgt-out 61.1%

         \[\leadsto \frac{1}{2 + \color{blue}{x \cdot \left(\frac{-0.3333333333333333}{s} + \frac{-0.6666666666666666}{s}\right)}} \]

   8. Simplified 61.1%

      \[\leadsto \frac{1}{2 + \color{blue}{x \cdot \left(\frac{-0.3333333333333333}{s} + \frac{-0.6666666666666666}{s}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + x \cdot \left(\frac{-0.3333333333333333}{s} + \frac{-0.6666666666666666}{s}\right)}\\ \end{array} \]

Alternative 7: 49.2% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + \frac{x \cdot -0.6666666666666666 + x \cdot -0.3333333333333333}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) -1.0)
   0.5
   (/
    1.0
    (+ 2.0 (/ (+ (* x -0.6666666666666666) (* x -0.3333333333333333)) s)))))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= -1.0f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / (2.0f + (((x * -0.6666666666666666f) + (x * -0.3333333333333333f)) / s));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((-x / s) <= (-1.0e0)) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 / (2.0e0 + (((x * (-0.6666666666666666e0)) + (x * (-0.3333333333333333e0))) / s))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(-x) / s) <= Float32(-1.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(Float32(Float32(x * Float32(-0.6666666666666666)) + Float32(x * Float32(-0.3333333333333333))) / s)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((-x / s) <= single(-1.0))
		tmp = single(0.5);
	else
		tmp = single(1.0) / (single(2.0) + (((x * single(-0.6666666666666666)) + (x * single(-0.3333333333333333))) / s));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 28.2%

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

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

   1. Initial program 99.8%

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

      1. distribute-frac-neg 99.8%

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

      2. exp-neg 99.8%

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

      3. div-inv 99.8%

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

      4. add-sqr-sqrt 18.3%

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

      5. sqrt-unprod 37.7%

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

      6. sqr-neg 37.7%

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

      7. sqrt-unprod 20.0%

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

      8. add-sqr-sqrt 36.5%

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

      9. div-inv 36.5%

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

      10. add-cube-cbrt 36.5%

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

      11. pow3 36.5%

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

      12. pow-flip 36.5%

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

   3. Applied egg-rr 99.8%

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

   4. Taylor expanded in s around inf 61.1%

      \[\leadsto \frac{1}{\color{blue}{2 + \left(-0.6666666666666666 \cdot \frac{x}{s} + -0.3333333333333333 \cdot \frac{x}{s}\right)}} \]

   5. Taylor expanded in s around 0 61.1%

      \[\leadsto \frac{1}{2 + \color{blue}{\frac{-0.6666666666666666 \cdot x + -0.3333333333333333 \cdot x}{s}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + \frac{x \cdot -0.6666666666666666 + x \cdot -0.3333333333333333}{s}}\\ \end{array} \]

Alternative 8: 49.2% accurate, 8.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 28.2%

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

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

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 61.1%

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

      1. mul-1-neg 61.1%

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

      2. unsub-neg 61.1%

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

   4. Simplified 61.1%

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

4. Final simplification 49.9%

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

Alternative 9: 47.6% accurate, 9.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 53.6%

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

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

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 40.9%

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

      1. mul-1-neg 40.9%

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

      2. unsub-neg 40.9%

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

   4. Simplified 40.9%

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

   5. Taylor expanded in x around inf 40.9%

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

      1. mul-1-neg 40.9%

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

      2. distribute-frac-neg 40.9%

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

   7. Simplified 40.9%

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

4. Final simplification 48.3%

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

Alternative 10: 47.3% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.999999873689376 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -4.999999873689376e-5) (/ 1.0 (/ x s)) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -4.999999873689376e-5f) {
		tmp = 1.0f / (x / s);
	} else {
		tmp = 0.5f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-4.999999873689376e-5)) then
        tmp = 1.0e0 / (x / s)
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-4.999999873689376e-5))
		tmp = Float32(Float32(1.0) / Float32(x / s));
	else
		tmp = Float32(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-4.999999873689376e-5))
		tmp = single(1.0) / (x / s);
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -4.99999987e-5

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 51.2%

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

      1. mul-1-neg 51.2%

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

      2. unsub-neg 51.2%

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

   4. Simplified 51.2%

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

   5. Taylor expanded in x around inf 45.7%

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

      1. associate-*r/ 45.7%

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

      2. neg-mul-1 45.7%

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

   7. Simplified 45.7%

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

      1. clear-num 51.2%

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

      2. inv-pow 51.2%

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

      3. add-sqr-sqrt -0.0%

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

      4. sqrt-unprod 67.1%

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

      5. sqr-neg 67.1%

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

      6. sqrt-unprod 51.1%

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

      7. add-sqr-sqrt 51.1%

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

   9. Applied egg-rr 51.1%

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

       1. unpow-1 51.1%

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

   11. Simplified 51.1%

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

   if -4.99999987e-5 < x

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 46.6%

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

4. Final simplification 48.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.999999873689376 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 11: 46.2% accurate, 17.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.9999999949504854 \cdot 10^{-6}:\\ \;\;\;\;\frac{-s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -1.9999999949504854e-6) (/ (- s) x) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -1.9999999949504854e-6f) {
		tmp = -s / x;
	} else {
		tmp = 0.5f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-1.9999999949504854e-6)) then
        tmp = -s / x
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-1.9999999949504854e-6))
		tmp = Float32(Float32(-s) / x);
	else
		tmp = Float32(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-1.9999999949504854e-6))
		tmp = -s / x;
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -1.99999999e-6

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 50.9%

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

      1. mul-1-neg 50.9%

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

      2. unsub-neg 50.9%

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

   4. Simplified 50.9%

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

   5. Taylor expanded in x around inf 45.4%

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

      1. associate-*r/ 45.4%

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

      2. neg-mul-1 45.4%

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

   7. Simplified 45.4%

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

   if -1.99999999e-6 < x

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 46.8%

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

4. Final simplification 46.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.9999999949504854 \cdot 10^{-6}:\\ \;\;\;\;\frac{-s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 12: 46.1% accurate, 21.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.999999873689376 \cdot 10^{-5}:\\ \;\;\;\;\frac{s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -4.999999873689376e-5) (/ s x) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -4.999999873689376e-5f) {
		tmp = s / x;
	} else {
		tmp = 0.5f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-4.999999873689376e-5)) then
        tmp = s / x
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-4.999999873689376e-5))
		tmp = Float32(s / x);
	else
		tmp = Float32(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-4.999999873689376e-5))
		tmp = s / x;
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -4.99999987e-5

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 51.2%

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

      1. mul-1-neg 51.2%

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

      2. unsub-neg 51.2%

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

   4. Simplified 51.2%

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

   5. Taylor expanded in x around inf 45.7%

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

      1. associate-*r/ 45.7%

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

      2. neg-mul-1 45.7%

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

   7. Simplified 45.7%

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

      1. expm1-log1p-u 45.7%

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

      2. expm1-udef 97.7%

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

      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 97.7%

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

      5. sqr-neg 97.7%

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

      6. sqrt-unprod 97.7%

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

      7. add-sqr-sqrt 97.7%

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

   9. Applied egg-rr 97.7%

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

       1. expm1-def 45.6%

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

       2. expm1-log1p 45.6%

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

   11. Simplified 45.6%

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

   if -4.99999987e-5 < x

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 46.6%

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

4. Final simplification 46.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.999999873689376 \cdot 10^{-5}:\\ \;\;\;\;\frac{s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 13: 35.2% accurate, 108.0× speedup

Math

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

FPCore

(FPCore (x s) :precision binary32 0.5)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.9%

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

2. Taylor expanded in x around 0 33.9%

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

3. Final simplification 33.9%

   \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023311 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))