Logistic function

Percentage Accurate: 99.8% → 99.9%

Time: 8.4s

Alternatives: 14

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.9% 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. div-inv 99.9%

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

   2. exp-prod 88.1%

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

   3. neg-mul-1 88.1%

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

   4. exp-prod 88.1%

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

   5. pow-pow 99.9%

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

   6. div-inv 99.9%

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

4. Applied egg-rr 99.9%

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

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, 0.5× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	return 1.0f / (1.0f + powf(expf(-1.0f), (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((-1.0e0)) ** (x / s)))
end function

Julia

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

MATLAB

function tmp = code(x, s)
	tmp = single(1.0) / (single(1.0) + (exp(single(-1.0)) ^ (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. div-inv 99.9%

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

   2. exp-prod 88.1%

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

   3. neg-mul-1 88.1%

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

   4. exp-prod 88.1%

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

   5. pow-pow 99.9%

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

   6. div-inv 99.9%

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

4. Applied egg-rr 99.9%

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

Alternative 3: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.9%

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

3. Final simplification 99.9%

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

Alternative 4: 79.4% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.999999721059845 \cdot 10^{+27}:\\ \;\;\;\;\frac{1}{2 + \frac{-1}{\frac{s}{x}}}\\ \mathbf{elif}\;x \leq -4.999999955487895 \cdot 10^{-38}:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{s \cdot \frac{s}{x}}}{\frac{x}{s} + 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -4.999999721059845e+27)
   (/ 1.0 (+ 2.0 (/ -1.0 (/ s x))))
   (if (<= x -4.999999955487895e-38)
     (/ 1.0 (/ (- 4.0 (/ x (* s (/ s x)))) (+ (/ x s) 2.0)))
     (/ 1.0 (+ 1.0 (/ 1.0 (+ 1.0 (/ x s))))))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -4.999999721059845e+27f) {
		tmp = 1.0f / (2.0f + (-1.0f / (s / x)));
	} else if (x <= -4.999999955487895e-38f) {
		tmp = 1.0f / ((4.0f - (x / (s * (s / x)))) / ((x / s) + 2.0f));
	} else {
		tmp = 1.0f / (1.0f + (1.0f / (1.0f + (x / s))));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-4.999999721059845e+27)) then
        tmp = 1.0e0 / (2.0e0 + ((-1.0e0) / (s / x)))
    else if (x <= (-4.999999955487895e-38)) then
        tmp = 1.0e0 / ((4.0e0 - (x / (s * (s / x)))) / ((x / s) + 2.0e0))
    else
        tmp = 1.0e0 / (1.0e0 + (1.0e0 / (1.0e0 + (x / s))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-4.999999721059845e+27))
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(Float32(-1.0) / Float32(s / x))));
	elseif (x <= Float32(-4.999999955487895e-38))
		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(Float32(1.0) + Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(x / s)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-4.999999721059845e+27))
		tmp = single(1.0) / (single(2.0) + (single(-1.0) / (s / x)));
	elseif (x <= single(-4.999999955487895e-38))
		tmp = single(1.0) / ((single(4.0) - (x / (s * (s / x)))) / ((x / s) + single(2.0)));
	else
		tmp = single(1.0) / (single(1.0) + (single(1.0) / (single(1.0) + (x / s))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -4.99999972e27

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 86.6%

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

      1. neg-mul-1 86.6%

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

      2. unsub-neg 86.6%

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

   5. Simplified 86.6%

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

      1. clear-num 86.6%

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

      2. inv-pow 86.6%

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

   7. Applied egg-rr 86.6%

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

      1. unpow-1 86.6%

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

   9. Simplified 86.6%

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

   if -4.99999972e27 < x < -4.99999996e-38

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 43.5%

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

      1. neg-mul-1 43.5%

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

      2. unsub-neg 43.5%

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

   5. Simplified 43.5%

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

      1. *-un-lft-identity 43.5%

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

      2. cancel-sign-sub-inv 43.5%

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

      3. metadata-eval 43.5%

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

      4. add-log-exp 95.1%

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

      5. pow-exp 95.1%

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

      6. flip-+ 25.1%

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

      7. metadata-eval 25.1%

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

      8. pow-exp 25.1%

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

      9. add-log-exp 25.1%

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

      10. neg-mul-1 25.1%

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

      11. pow-exp 25.1%

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

      12. add-log-exp 25.9%

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

      13. neg-mul-1 25.9%

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

      14. distribute-neg-frac2 25.9%

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

      15. distribute-neg-frac2 25.9%

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

      16. pow-exp 25.9%

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

   7. Applied egg-rr 58.2%

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

      1. clear-num 58.2%

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

      2. frac-times 62.8%

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

      3. *-un-lft-identity 62.8%

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

      4. add-sqr-sqrt -0.0%

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

      5. sqrt-unprod 59.8%

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

      6. sqr-neg 59.8%

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

      7. sqrt-unprod 62.7%

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

      8. add-sqr-sqrt 62.7%

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

      9. add-sqr-sqrt -0.0%

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

      10. sqrt-unprod 59.8%

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

      11. sqr-neg 59.8%

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

      12. sqrt-unprod 62.8%

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

      13. add-sqr-sqrt 62.8%

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

   9. Applied egg-rr 62.8%

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

   if -4.99999996e-38 < x

   1. Initial program 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. exp-neg 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0 97.6%

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

      1. +-commutative 97.6%

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

   7. Simplified 97.6%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.999999721059845 \cdot 10^{+27}:\\ \;\;\;\;\frac{1}{2 + \frac{-1}{\frac{s}{x}}}\\ \mathbf{elif}\;x \leq -4.999999955487895 \cdot 10^{-38}:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{s \cdot \frac{s}{x}}}{\frac{x}{s} + 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.7% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.999999721059845 \cdot 10^{+27}:\\ \;\;\;\;\frac{1}{2 + \frac{-1}{\frac{s}{x}}}\\ \mathbf{elif}\;x \leq -5.00000006675716 \cdot 10^{-11}:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{s} \cdot \frac{x}{s}}{\frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -4.999999721059845e+27)
   (/ 1.0 (+ 2.0 (/ -1.0 (/ s x))))
   (if (<= x -5.00000006675716e-11)
     (/ 1.0 (/ (- 4.0 (* (/ x s) (/ x s))) (/ x s)))
     (/ 1.0 (+ 1.0 (/ 1.0 (+ 1.0 (/ x s))))))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -4.999999721059845e+27f) {
		tmp = 1.0f / (2.0f + (-1.0f / (s / x)));
	} else if (x <= -5.00000006675716e-11f) {
		tmp = 1.0f / ((4.0f - ((x / s) * (x / s))) / (x / s));
	} else {
		tmp = 1.0f / (1.0f + (1.0f / (1.0f + (x / s))));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-4.999999721059845e+27)) then
        tmp = 1.0e0 / (2.0e0 + ((-1.0e0) / (s / x)))
    else if (x <= (-5.00000006675716e-11)) then
        tmp = 1.0e0 / ((4.0e0 - ((x / s) * (x / s))) / (x / s))
    else
        tmp = 1.0e0 / (1.0e0 + (1.0e0 / (1.0e0 + (x / s))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-4.999999721059845e+27))
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(Float32(-1.0) / Float32(s / x))));
	elseif (x <= Float32(-5.00000006675716e-11))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(4.0) - Float32(Float32(x / s) * Float32(x / s))) / Float32(x / s)));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(x / s)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-4.999999721059845e+27))
		tmp = single(1.0) / (single(2.0) + (single(-1.0) / (s / x)));
	elseif (x <= single(-5.00000006675716e-11))
		tmp = single(1.0) / ((single(4.0) - ((x / s) * (x / s))) / (x / s));
	else
		tmp = single(1.0) / (single(1.0) + (single(1.0) / (single(1.0) + (x / s))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -4.99999972e27

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 86.6%

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

      1. neg-mul-1 86.6%

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

      2. unsub-neg 86.6%

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

   5. Simplified 86.6%

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

      1. clear-num 86.6%

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

      2. inv-pow 86.6%

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

   7. Applied egg-rr 86.6%

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

      1. unpow-1 86.6%

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

   9. Simplified 86.6%

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

   if -4.99999972e27 < x < -5.00000007e-11

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 28.3%

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

      1. neg-mul-1 28.3%

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

      2. unsub-neg 28.3%

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

   5. Simplified 28.3%

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

      1. *-un-lft-identity 28.3%

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

      2. cancel-sign-sub-inv 28.3%

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

      3. metadata-eval 28.3%

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

      4. add-log-exp 100.0%

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

      5. pow-exp 100.0%

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

      6. flip-+ -0.0%

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

      7. metadata-eval -0.0%

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

      8. pow-exp -0.0%

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

      9. add-log-exp -0.0%

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

      10. neg-mul-1 -0.0%

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

      11. pow-exp -0.0%

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

      12. add-log-exp 1.0%

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

      13. neg-mul-1 1.0%

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

      14. distribute-neg-frac2 1.0%

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

      15. distribute-neg-frac2 1.0%

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

      16. pow-exp 1.0%

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

   7. Applied egg-rr 51.7%

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

   8. Taylor expanded in x around inf 51.7%

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

   if -5.00000007e-11 < x

   1. Initial program 99.8%

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

      1. distribute-frac-neg 99.8%

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

      2. exp-neg 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around 0 91.1%

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

      1. +-commutative 91.1%

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

   7. Simplified 91.1%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.999999721059845 \cdot 10^{+27}:\\ \;\;\;\;\frac{1}{2 + \frac{-1}{\frac{s}{x}}}\\ \mathbf{elif}\;x \leq -5.00000006675716 \cdot 10^{-11}:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{s} \cdot \frac{x}{s}}{\frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 69.8% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.00000006675716 \cdot 10^{-11}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \mathbf{elif}\;x \leq 1.000000031374395 \cdot 10^{-22}:\\ \;\;\;\;0.5 + \frac{x \cdot 0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{s}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -5.00000006675716e-11)
   (/ 1.0 (/ x s))
   (if (<= x 1.000000031374395e-22)
     (+ 0.5 (/ (* x 0.25) s))
     (/ 1.0 (+ 1.0 (/ s x))))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -5.00000006675716e-11f) {
		tmp = 1.0f / (x / s);
	} else if (x <= 1.000000031374395e-22f) {
		tmp = 0.5f + ((x * 0.25f) / s);
	} else {
		tmp = 1.0f / (1.0f + (s / x));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-5.00000006675716e-11)) then
        tmp = 1.0e0 / (x / s)
    else if (x <= 1.000000031374395e-22) then
        tmp = 0.5e0 + ((x * 0.25e0) / s)
    else
        tmp = 1.0e0 / (1.0e0 + (s / x))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-5.00000006675716e-11))
		tmp = Float32(Float32(1.0) / Float32(x / s));
	elseif (x <= Float32(1.000000031374395e-22))
		tmp = Float32(Float32(0.5) + Float32(Float32(x * Float32(0.25)) / s));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(s / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-5.00000006675716e-11))
		tmp = single(1.0) / (x / s);
	elseif (x <= single(1.000000031374395e-22))
		tmp = single(0.5) + ((x * single(0.25)) / s);
	else
		tmp = single(1.0) / (single(1.0) + (s / x));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -5.00000007e-11

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 46.6%

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

      1. neg-mul-1 46.6%

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

      2. unsub-neg 46.6%

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

   5. Simplified 46.6%

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

   6. Taylor expanded in x around inf 40.0%

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

      1. associate-*r/ 40.0%

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

      2. neg-mul-1 40.0%

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

   8. Simplified 40.0%

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

      1. add-sqr-sqrt -0.0%

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

      2. sqrt-unprod 54.5%

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

      3. sqr-neg 54.5%

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

      4. sqrt-unprod 40.0%

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

      5. add-sqr-sqrt 40.0%

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

      6. clear-num 46.6%

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

      7. inv-pow 46.6%

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

   10. Applied egg-rr 46.6%

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

       1. unpow-1 46.6%

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

   12. Simplified 46.6%

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

   if -5.00000007e-11 < x < 1.00000003e-22

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0 81.0%

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

      1. associate-*r/ 81.0%

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

   5. Simplified 81.0%

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

   if 1.00000003e-22 < x

   1. Initial program 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. exp-neg 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0 98.5%

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

      1. +-commutative 98.5%

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

   7. Simplified 98.5%

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

   8. Taylor expanded in x around inf 93.9%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.00000006675716 \cdot 10^{-11}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \mathbf{elif}\;x \leq 1.000000031374395 \cdot 10^{-22}:\\ \;\;\;\;0.5 + \frac{x \cdot 0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{s}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.3% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.00000006675716 \cdot 10^{-11}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \mathbf{elif}\;x \leq 1.000000031374395 \cdot 10^{-22}:\\ \;\;\;\;0.5 + \frac{x \cdot 0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{s}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -5.00000006675716e-11)
   (/ 1.0 (/ x s))
   (if (<= x 1.000000031374395e-22) (+ 0.5 (/ (* x 0.25) s)) (- 1.0 (/ s x)))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -5.00000006675716e-11f) {
		tmp = 1.0f / (x / s);
	} else if (x <= 1.000000031374395e-22f) {
		tmp = 0.5f + ((x * 0.25f) / s);
	} else {
		tmp = 1.0f - (s / x);
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-5.00000006675716e-11)) then
        tmp = 1.0e0 / (x / s)
    else if (x <= 1.000000031374395e-22) then
        tmp = 0.5e0 + ((x * 0.25e0) / s)
    else
        tmp = 1.0e0 - (s / x)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-5.00000006675716e-11))
		tmp = Float32(Float32(1.0) / Float32(x / s));
	elseif (x <= Float32(1.000000031374395e-22))
		tmp = Float32(Float32(0.5) + Float32(Float32(x * Float32(0.25)) / s));
	else
		tmp = Float32(Float32(1.0) - Float32(s / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-5.00000006675716e-11))
		tmp = single(1.0) / (x / s);
	elseif (x <= single(1.000000031374395e-22))
		tmp = single(0.5) + ((x * single(0.25)) / s);
	else
		tmp = single(1.0) - (s / x);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -5.00000007e-11

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 46.6%

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

      1. neg-mul-1 46.6%

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

      2. unsub-neg 46.6%

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

   5. Simplified 46.6%

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

   6. Taylor expanded in x around inf 40.0%

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

      1. associate-*r/ 40.0%

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

      2. neg-mul-1 40.0%

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

   8. Simplified 40.0%

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

      1. add-sqr-sqrt -0.0%

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

      2. sqrt-unprod 54.5%

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

      3. sqr-neg 54.5%

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

      4. sqrt-unprod 40.0%

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

      5. add-sqr-sqrt 40.0%

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

      6. clear-num 46.6%

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

      7. inv-pow 46.6%

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

   10. Applied egg-rr 46.6%

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

       1. unpow-1 46.6%

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

   12. Simplified 46.6%

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

   if -5.00000007e-11 < x < 1.00000003e-22

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0 81.0%

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

      1. associate-*r/ 81.0%

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

   5. Simplified 81.0%

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

   if 1.00000003e-22 < x

   1. Initial program 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. exp-neg 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0 98.5%

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

      1. +-commutative 98.5%

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

   7. Simplified 98.5%

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

   8. Taylor expanded in x around inf 93.3%

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

      1. neg-mul-1 93.3%

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

      2. unsub-neg 93.3%

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

   10. Simplified 93.3%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.00000006675716 \cdot 10^{-11}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \mathbf{elif}\;x \leq 1.000000031374395 \cdot 10^{-22}:\\ \;\;\;\;0.5 + \frac{x \cdot 0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{s}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.6% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.999999955487895 \cdot 10^{-38}:\\ \;\;\;\;\frac{1}{2 + \frac{-1}{\frac{s}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -4.999999955487895e-38)
   (/ 1.0 (+ 2.0 (/ -1.0 (/ s x))))
   (/ 1.0 (+ 1.0 (/ 1.0 (+ 1.0 (/ x s)))))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -4.999999955487895e-38f) {
		tmp = 1.0f / (2.0f + (-1.0f / (s / x)));
	} else {
		tmp = 1.0f / (1.0f + (1.0f / (1.0f + (x / s))));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-4.999999955487895e-38)) then
        tmp = 1.0e0 / (2.0e0 + ((-1.0e0) / (s / x)))
    else
        tmp = 1.0e0 / (1.0e0 + (1.0e0 / (1.0e0 + (x / s))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-4.999999955487895e-38))
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(Float32(-1.0) / Float32(s / x))));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(x / s)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-4.999999955487895e-38))
		tmp = single(1.0) / (single(2.0) + (single(-1.0) / (s / x)));
	else
		tmp = single(1.0) / (single(1.0) + (single(1.0) / (single(1.0) + (x / s))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -4.99999996e-38

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 53.2%

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

      1. neg-mul-1 53.2%

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

      2. unsub-neg 53.2%

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

   5. Simplified 53.2%

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

      1. clear-num 53.2%

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

      2. inv-pow 53.2%

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

   7. Applied egg-rr 53.2%

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

      1. unpow-1 53.2%

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

   9. Simplified 53.2%

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

   if -4.99999996e-38 < x

   1. Initial program 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. exp-neg 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0 97.6%

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

      1. +-commutative 97.6%

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

   7. Simplified 97.6%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.999999955487895 \cdot 10^{-38}:\\ \;\;\;\;\frac{1}{2 + \frac{-1}{\frac{s}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 68.2% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.00000006675716 \cdot 10^{-11}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \mathbf{elif}\;x \leq 1.000000031374395 \cdot 10^{-22}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{s}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -5.00000006675716e-11)
   (/ 1.0 (/ x s))
   (if (<= x 1.000000031374395e-22) 0.5 (- 1.0 (/ s x)))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -5.00000006675716e-11f) {
		tmp = 1.0f / (x / s);
	} else if (x <= 1.000000031374395e-22f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f - (s / x);
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-5.00000006675716e-11)) then
        tmp = 1.0e0 / (x / s)
    else if (x <= 1.000000031374395e-22) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 - (s / x)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-5.00000006675716e-11))
		tmp = Float32(Float32(1.0) / Float32(x / s));
	elseif (x <= Float32(1.000000031374395e-22))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) - Float32(s / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-5.00000006675716e-11))
		tmp = single(1.0) / (x / s);
	elseif (x <= single(1.000000031374395e-22))
		tmp = single(0.5);
	else
		tmp = single(1.0) - (s / x);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -5.00000007e-11

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 46.6%

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

      1. neg-mul-1 46.6%

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

      2. unsub-neg 46.6%

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

   5. Simplified 46.6%

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

   6. Taylor expanded in x around inf 40.0%

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

      1. associate-*r/ 40.0%

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

      2. neg-mul-1 40.0%

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

   8. Simplified 40.0%

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

      1. add-sqr-sqrt -0.0%

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

      2. sqrt-unprod 54.5%

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

      3. sqr-neg 54.5%

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

      4. sqrt-unprod 40.0%

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

      5. add-sqr-sqrt 40.0%

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

      6. clear-num 46.6%

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

      7. inv-pow 46.6%

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

   10. Applied egg-rr 46.6%

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

       1. unpow-1 46.6%

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

   12. Simplified 46.6%

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

   if -5.00000007e-11 < x < 1.00000003e-22

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0 76.9%

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

   if 1.00000003e-22 < x

   1. Initial program 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. exp-neg 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0 98.5%

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

      1. +-commutative 98.5%

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

   7. Simplified 98.5%

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

   8. Taylor expanded in x around inf 93.3%

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

      1. neg-mul-1 93.3%

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

      2. unsub-neg 93.3%

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

   10. Simplified 93.3%

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

Alternative 10: 66.9% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.00000006675716 \cdot 10^{-11}:\\ \;\;\;\;\frac{s}{x}\\ \mathbf{elif}\;x \leq 1.000000031374395 \cdot 10^{-22}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{s}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -5.00000006675716e-11)
   (/ s x)
   (if (<= x 1.000000031374395e-22) 0.5 (- 1.0 (/ s x)))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -5.00000006675716e-11f) {
		tmp = s / x;
	} else if (x <= 1.000000031374395e-22f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f - (s / x);
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-5.00000006675716e-11)) then
        tmp = s / x
    else if (x <= 1.000000031374395e-22) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 - (s / x)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-5.00000006675716e-11))
		tmp = Float32(s / x);
	elseif (x <= Float32(1.000000031374395e-22))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) - Float32(s / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-5.00000006675716e-11))
		tmp = s / x;
	elseif (x <= single(1.000000031374395e-22))
		tmp = single(0.5);
	else
		tmp = single(1.0) - (s / x);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -5.00000007e-11

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 46.6%

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

      1. neg-mul-1 46.6%

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

      2. unsub-neg 46.6%

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

   5. Simplified 46.6%

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

   6. Taylor expanded in x around inf 40.0%

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

      1. associate-*r/ 40.0%

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

      2. neg-mul-1 40.0%

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

   8. Simplified 40.0%

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

      1. neg-sub0 40.0%

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

      2. sub-neg 40.0%

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

      3. add-sqr-sqrt -0.0%

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

      4. sqrt-unprod 54.5%

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

      5. sqr-neg 54.5%

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

      6. sqrt-unprod 40.0%

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

      7. add-sqr-sqrt 40.0%

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

   10. Applied egg-rr 40.0%

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

       1. +-lft-identity 40.0%

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

   12. Simplified 40.0%

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

   if -5.00000007e-11 < x < 1.00000003e-22

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0 76.9%

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

   if 1.00000003e-22 < x

   1. Initial program 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. exp-neg 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0 98.5%

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

      1. +-commutative 98.5%

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

   7. Simplified 98.5%

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

   8. Taylor expanded in x around inf 93.3%

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

      1. neg-mul-1 93.3%

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

      2. unsub-neg 93.3%

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

   10. Simplified 93.3%

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

Alternative 11: 70.1% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.000000031374395 \cdot 10^{-22}:\\ \;\;\;\;\frac{1}{2 + \frac{-1}{\frac{s}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{s}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 1.000000031374395e-22)
   (/ 1.0 (+ 2.0 (/ -1.0 (/ s x))))
   (/ 1.0 (+ 1.0 (/ s x)))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 1.000000031374395e-22f) {
		tmp = 1.0f / (2.0f + (-1.0f / (s / x)));
	} else {
		tmp = 1.0f / (1.0f + (s / x));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 1.000000031374395e-22) then
        tmp = 1.0e0 / (2.0e0 + ((-1.0e0) / (s / x)))
    else
        tmp = 1.0e0 / (1.0e0 + (s / x))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(1.000000031374395e-22))
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(Float32(-1.0) / Float32(s / x))));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(s / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(1.000000031374395e-22))
		tmp = single(1.0) / (single(2.0) + (single(-1.0) / (s / x)));
	else
		tmp = single(1.0) / (single(1.0) + (s / x));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 1.00000003e-22

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 60.3%

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

      1. neg-mul-1 60.3%

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

      2. unsub-neg 60.3%

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

   5. Simplified 60.3%

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

      1. clear-num 60.3%

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

      2. inv-pow 60.3%

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

   7. Applied egg-rr 60.3%

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

      1. unpow-1 60.3%

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

   9. Simplified 60.3%

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

   if 1.00000003e-22 < x

   1. Initial program 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. exp-neg 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0 98.5%

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

      1. +-commutative 98.5%

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

   7. Simplified 98.5%

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

   8. Taylor expanded in x around inf 93.9%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.000000031374395 \cdot 10^{-22}:\\ \;\;\;\;\frac{1}{2 + \frac{-1}{\frac{s}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{s}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 70.1% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.000000031374395 \cdot 10^{-22}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{s}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 1.000000031374395e-22)
   (/ 1.0 (- 2.0 (/ x s)))
   (/ 1.0 (+ 1.0 (/ s x)))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 1.000000031374395e-22f) {
		tmp = 1.0f / (2.0f - (x / s));
	} else {
		tmp = 1.0f / (1.0f + (s / x));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 1.000000031374395e-22) then
        tmp = 1.0e0 / (2.0e0 - (x / s))
    else
        tmp = 1.0e0 / (1.0e0 + (s / x))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(1.000000031374395e-22))
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) - Float32(x / s)));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(s / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(1.000000031374395e-22))
		tmp = single(1.0) / (single(2.0) - (x / s));
	else
		tmp = single(1.0) / (single(1.0) + (s / x));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 1.00000003e-22

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 60.3%

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

      1. neg-mul-1 60.3%

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

      2. unsub-neg 60.3%

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

   5. Simplified 60.3%

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

   if 1.00000003e-22 < x

   1. Initial program 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. exp-neg 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0 98.5%

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

      1. +-commutative 98.5%

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

   7. Simplified 98.5%

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

   8. Taylor expanded in x around inf 93.9%

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

Alternative 13: 45.7% accurate, 13.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.00000006675716 \cdot 10^{-11}:\\ \;\;\;\;\frac{s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -5.00000006675716e-11) (/ s x) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -5.00000006675716e-11f) {
		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 <= (-5.00000006675716e-11)) 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(-5.00000006675716e-11))
		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(-5.00000006675716e-11))
		tmp = s / x;
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -5.00000007e-11

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 46.6%

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

      1. neg-mul-1 46.6%

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

      2. unsub-neg 46.6%

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

   5. Simplified 46.6%

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

   6. Taylor expanded in x around inf 40.0%

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

      1. associate-*r/ 40.0%

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

      2. neg-mul-1 40.0%

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

   8. Simplified 40.0%

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

      1. neg-sub0 40.0%

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

      2. sub-neg 40.0%

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

      3. add-sqr-sqrt -0.0%

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

      4. sqrt-unprod 54.5%

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

      5. sqr-neg 54.5%

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

      6. sqrt-unprod 40.0%

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

      7. add-sqr-sqrt 40.0%

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

   10. Applied egg-rr 40.0%

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

       1. +-lft-identity 40.0%

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

   12. Simplified 40.0%

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

   if -5.00000007e-11 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 50.9%

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

Alternative 14: 34.2% 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. Add Preprocessing

3. Taylor expanded in x around 0 34.9%

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

Reproduce

herbie shell --seed 2024185 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))