Logistic function

Percentage Accurate: 99.8% → 99.8%

Time: 9.1s

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: 99.8% 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. add-sqr-sqrt 50.7%

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

   4. sqrt-unprod 63.9%

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

   5. sqr-neg 63.9%

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

   6. sqrt-unprod 14.8%

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

   7. add-sqr-sqrt 27.8%

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

   8. pow1 27.8%

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

   9. pow1 27.8%

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

   10. add-cbrt-cube 27.8%

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

   11. pow1/3 27.8%

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

   12. pow-flip 27.8%

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

3. Applied egg-rr 99.9%

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

   1. sqr-pow 99.9%

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

   2. *-commutative 99.9%

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

   3. exp-prod 99.8%

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

   4. metadata-eval 99.8%

      \[\leadsto \frac{1}{1 + {\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)}^{\color{blue}{-0.16666666666666666}} \cdot {\left(e^{3 \cdot \frac{x}{s}}\right)}^{\left(\frac{-0.3333333333333333}{2}\right)}} \]

   5. *-commutative 99.8%

      \[\leadsto \frac{1}{1 + {\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)}^{-0.16666666666666666} \cdot {\left(e^{\color{blue}{\frac{x}{s} \cdot 3}}\right)}^{\left(\frac{-0.3333333333333333}{2}\right)}} \]

   6. exp-prod 99.8%

      \[\leadsto \frac{1}{1 + {\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)}^{-0.16666666666666666} \cdot {\color{blue}{\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)}}^{\left(\frac{-0.3333333333333333}{2}\right)}} \]

   7. metadata-eval 99.8%

      \[\leadsto \frac{1}{1 + {\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)}^{-0.16666666666666666} \cdot {\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)}^{\color{blue}{-0.16666666666666666}}} \]

5. Simplified 99.8%

   \[\leadsto \frac{1}{1 + \color{blue}{{\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)}^{-0.16666666666666666} \cdot {\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)}^{-0.16666666666666666}}} \]
6. Step-by-step derivation

   1. pow2 99.8%

      \[\leadsto \frac{1}{1 + \color{blue}{{\left({\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)}^{-0.16666666666666666}\right)}^{2}}} \]

   2. pow-pow 99.8%

      \[\leadsto \frac{1}{1 + {\color{blue}{\left({\left(e^{\frac{x}{s}}\right)}^{\left(3 \cdot -0.16666666666666666\right)}\right)}}^{2}} \]

   3. metadata-eval 99.8%

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

7. Applied egg-rr 99.8%

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

   1. add-exp-log 99.8%

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

   2. log-rec 99.8%

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

   3. log1p-udef 99.9%

      \[\leadsto e^{-\color{blue}{\mathsf{log1p}\left({\left({\left(e^{\frac{x}{s}}\right)}^{-0.5}\right)}^{2}\right)}} \]

   4. add-exp-log 99.9%

      \[\leadsto e^{-\mathsf{log1p}\left(\color{blue}{e^{\log \left({\left({\left(e^{\frac{x}{s}}\right)}^{-0.5}\right)}^{2}\right)}}\right)} \]

   5. pow-pow 100.0%

      \[\leadsto e^{-\mathsf{log1p}\left(e^{\log \color{blue}{\left({\left(e^{\frac{x}{s}}\right)}^{\left(-0.5 \cdot 2\right)}\right)}}\right)} \]

   6. metadata-eval 100.0%

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

   7. log-pow 100.0%

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

   8. add-log-exp 100.0%

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

   9. neg-mul-1 100.0%

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

   10. distribute-neg-frac 100.0%

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

9. Applied egg-rr 100.0%

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

10. Final simplification 100.0%

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

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: 63.2% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-x \leq 9.999999796611898 \cdot 10^{-32}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(2 - \frac{x}{s}\right) + x \cdot \left(0.5 \cdot \frac{x}{s \cdot s}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- x) 9.999999796611898e-32)
   0.5
   (/ 1.0 (+ (- 2.0 (/ x s)) (* x (* 0.5 (/ x (* s s))))))))

C

float code(float x, float s) {
	float tmp;
	if (-x <= 9.999999796611898e-32f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / ((2.0f - (x / s)) + (x * (0.5f * (x / (s * 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 <= 9.999999796611898e-32) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 / ((2.0e0 - (x / s)) + (x * (0.5e0 * (x / (s * s)))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-x) <= Float32(9.999999796611898e-32))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(2.0) - Float32(x / s)) + Float32(x * Float32(Float32(0.5) * Float32(x / Float32(s * s))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (-x <= single(9.999999796611898e-32))
		tmp = single(0.5);
	else
		tmp = single(1.0) / ((single(2.0) - (x / s)) + (x * (single(0.5) * (x / (s * s)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (neg.f32 x) < 9.9999998e-32

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 47.1%

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

   if 9.9999998e-32 < (neg.f32 x)

   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. add-sqr-sqrt -0.0%

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

      4. sqrt-unprod 23.1%

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

      5. sqr-neg 23.1%

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

      6. sqrt-unprod 23.1%

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

      7. add-sqr-sqrt 23.1%

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

      8. pow1 23.1%

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

      9. pow1 23.1%

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

      10. add-cbrt-cube 23.1%

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

      11. pow1/3 23.1%

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

      12. pow-flip 23.1%

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

   3. Applied egg-rr 99.8%

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

      1. sqr-pow 99.8%

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

      2. *-commutative 99.8%

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

      3. exp-prod 99.8%

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

      4. metadata-eval 99.8%

         \[\leadsto \frac{1}{1 + {\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)}^{\color{blue}{-0.16666666666666666}} \cdot {\left(e^{3 \cdot \frac{x}{s}}\right)}^{\left(\frac{-0.3333333333333333}{2}\right)}} \]

      5. *-commutative 99.8%

         \[\leadsto \frac{1}{1 + {\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)}^{-0.16666666666666666} \cdot {\left(e^{\color{blue}{\frac{x}{s} \cdot 3}}\right)}^{\left(\frac{-0.3333333333333333}{2}\right)}} \]

      6. exp-prod 99.8%

         \[\leadsto \frac{1}{1 + {\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)}^{-0.16666666666666666} \cdot {\color{blue}{\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)}}^{\left(\frac{-0.3333333333333333}{2}\right)}} \]

      7. metadata-eval 99.8%

         \[\leadsto \frac{1}{1 + {\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)}^{-0.16666666666666666} \cdot {\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)}^{\color{blue}{-0.16666666666666666}}} \]

   5. Simplified 99.8%

      \[\leadsto \frac{1}{1 + \color{blue}{{\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)}^{-0.16666666666666666} \cdot {\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)}^{-0.16666666666666666}}} \]

   6. Taylor expanded in s around inf 59.3%

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

      1. +-commutative 59.3%

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

      2. neg-mul-1 59.3%

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

      3. sub-neg 59.3%

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

      4. mul-1-neg 59.3%

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

      5. unsub-neg 59.3%

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

      6. distribute-rgt-out 83.9%

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

      7. metadata-eval 83.9%

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

      8. unpow2 83.9%

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

      9. times-frac 75.9%

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

      10. unpow2 75.9%

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

      11. associate-*r/ 76.0%

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

   8. Simplified 76.0%

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

      1. *-commutative 76.0%

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

      2. clear-num 76.0%

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

      3. associate-*r/ 75.9%

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

      4. frac-times 83.9%

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

      5. *-un-lft-identity 83.9%

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

      6. div-inv 83.9%

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

      7. metadata-eval 83.9%

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

   10. Applied egg-rr 83.9%

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

       1. sub-neg 83.9%

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

       2. associate-/l* 88.3%

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

       3. distribute-neg-frac 88.3%

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

       4. *-commutative 88.3%

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

       5. associate-*r* 88.3%

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

       6. metadata-eval 88.3%

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

       7. distribute-rgt-neg-in 88.3%

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

       8. distribute-neg-frac 88.3%

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

       9. frac-2neg 88.3%

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

       10. div-inv 88.3%

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

       11. clear-num 88.3%

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

       12. *-un-lft-identity 88.3%

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

       13. *-commutative 88.3%

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

       14. times-frac 88.3%

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

       15. metadata-eval 88.3%

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

   12. Applied egg-rr 88.3%

       \[\leadsto \frac{1}{\color{blue}{\left(2 - \frac{x}{s}\right) + x \cdot \left(0.5 \cdot \frac{x}{s \cdot s}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-x \leq 9.999999796611898 \cdot 10^{-32}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(2 - \frac{x}{s}\right) + x \cdot \left(0.5 \cdot \frac{x}{s \cdot s}\right)}\\ \end{array} \]

Alternative 4: 58.5% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 2000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(s \cdot \frac{s}{x \cdot x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) 2000.0) 0.5 (* 2.0 (* s (/ s (* x x))))))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= 2000.0f) {
		tmp = 0.5f;
	} else {
		tmp = 2.0f * (s * (s / (x * x)));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((-x / s) <= 2000.0e0) then
        tmp = 0.5e0
    else
        tmp = 2.0e0 * (s * (s / (x * x)))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(-x) / s) <= Float32(2000.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(2.0) * Float32(s * Float32(s / Float32(x * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((-x / s) <= single(2000.0))
		tmp = single(0.5);
	else
		tmp = single(2.0) * (s * (s / (x * x)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 2e3

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 51.7%

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

   if 2e3 < (/.f32 (neg.f32 x) s)

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 85.0%

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

      1. mul-1-neg 85.0%

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

      2. unsub-neg 85.0%

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

      3. unpow2 85.0%

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

      4. unpow2 85.0%

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

      5. times-frac 73.8%

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

   4. Simplified 73.8%

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

   5. Taylor expanded in x around inf 83.2%

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

      1. unpow2 83.2%

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

      2. unpow2 83.2%

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

   7. Simplified 83.2%

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

      1. associate-/l* 69.4%

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

      2. associate-/r/ 69.4%

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

   9. Applied egg-rr 69.4%

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

4. Final simplification 57.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 2000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(s \cdot \frac{s}{x \cdot x}\right)\\ \end{array} \]

Alternative 5: 61.0% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 2000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{s \cdot s}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) 2000.0) 0.5 (* 2.0 (/ (* s s) (* x x)))))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= 2000.0f) {
		tmp = 0.5f;
	} else {
		tmp = 2.0f * ((s * s) / (x * x));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((-x / s) <= 2000.0e0) then
        tmp = 0.5e0
    else
        tmp = 2.0e0 * ((s * s) / (x * x))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(-x) / s) <= Float32(2000.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(2.0) * Float32(Float32(s * s) / Float32(x * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((-x / s) <= single(2000.0))
		tmp = single(0.5);
	else
		tmp = single(2.0) * ((s * s) / (x * x));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 2e3

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 51.7%

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

   if 2e3 < (/.f32 (neg.f32 x) s)

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 85.0%

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

      1. mul-1-neg 85.0%

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

      2. unsub-neg 85.0%

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

      3. unpow2 85.0%

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

      4. unpow2 85.0%

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

      5. times-frac 73.8%

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

   4. Simplified 73.8%

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

   5. Taylor expanded in x around inf 83.2%

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

      1. unpow2 83.2%

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

      2. unpow2 83.2%

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

   7. Simplified 83.2%

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

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 2000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{s \cdot s}{x \cdot x}\\ \end{array} \]

Alternative 6: 62.5% accurate, 7.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((-x / s) <= single(2.0))
		tmp = single(0.5);
	else
		tmp = ((s * s) / x) * (single(2.0) / x);
	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 52.3%

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

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

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 83.0%

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

      1. mul-1-neg 83.0%

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

      2. unsub-neg 83.0%

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

      3. unpow2 83.0%

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

      4. unpow2 83.0%

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

      5. times-frac 72.3%

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

   4. Simplified 72.3%

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

   5. Taylor expanded in x around inf 81.3%

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

      1. unpow2 81.3%

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

      2. unpow2 81.3%

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

   7. Simplified 81.3%

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

      1. associate-*r/ 81.3%

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

      2. associate-/r* 87.0%

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

      3. *-commutative 87.0%

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

   9. Applied egg-rr 87.0%

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

       1. associate-/l/ 81.3%

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

       2. times-frac 87.0%

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

   11. Applied egg-rr 87.0%

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

4. Final simplification 64.2%

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

Alternative 7: 62.5% accurate, 7.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((-x / s) <= single(2.0))
		tmp = single(0.5);
	else
		tmp = ((single(2.0) * (s * s)) / x) / x;
	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 52.3%

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

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

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 83.0%

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

      1. mul-1-neg 83.0%

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

      2. unsub-neg 83.0%

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

      3. unpow2 83.0%

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

      4. unpow2 83.0%

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

      5. times-frac 72.3%

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

   4. Simplified 72.3%

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

   5. Taylor expanded in x around inf 81.3%

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

      1. unpow2 81.3%

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

      2. unpow2 81.3%

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

   7. Simplified 81.3%

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

      1. associate-*r/ 81.3%

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

      2. associate-/r* 87.0%

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

      3. *-commutative 87.0%

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

   9. Applied egg-rr 87.0%

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

4. Final simplification 64.2%

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

Alternative 8: 49.7% 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 28.1%

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

   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 63.9%

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

      1. mul-1-neg 63.9%

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

      2. unsub-neg 63.9%

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

   4. Simplified 63.9%

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

4. Final simplification 50.2%

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

Alternative 9: 48.1% 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 52.3%

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

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

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 42.1%

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

      1. mul-1-neg 42.1%

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

      2. unsub-neg 42.1%

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

   4. Simplified 42.1%

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

   5. Taylor expanded in x around inf 42.1%

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

      1. neg-mul-1 42.1%

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

      2. distribute-neg-frac 42.1%

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

   7. Simplified 42.1%

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

4. Final simplification 48.8%

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

Alternative 10: 46.4% accurate, 15.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if x < -1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 59.6%

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

      1. mul-1-neg 59.6%

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

      2. unsub-neg 59.6%

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

   4. Simplified 59.6%

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

   5. Taylor expanded in x around inf 59.6%

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

      1. neg-mul-1 59.6%

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

      2. distribute-neg-frac 59.6%

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

   7. Simplified 59.6%

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

      1. associate-/r/ 52.4%

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

      2. add-sqr-sqrt 52.4%

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

      3. sqrt-unprod 63.4%

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

      4. sqr-neg 63.4%

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

      5. sqrt-prod -0.0%

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

      6. add-sqr-sqrt 52.4%

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

   9. Applied egg-rr 52.4%

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

   if -1 < x

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 45.3%

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

4. Final simplification 46.9%

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

Alternative 11: 47.8% accurate, 15.2× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	float tmp;
	if (x <= -1.0f) {
		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 <= (-1.0e0)) 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(-1.0))
		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(-1.0))
		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 < -1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 59.6%

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

      1. mul-1-neg 59.6%

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

      2. unsub-neg 59.6%

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

   4. Simplified 59.6%

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

   5. Taylor expanded in x around inf 59.6%

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

      1. neg-mul-1 59.6%

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

      2. distribute-neg-frac 59.6%

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

   7. Simplified 59.6%

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

      1. clear-num 59.6%

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

      2. inv-pow 59.6%

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

      3. /-rgt-identity 59.6%

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

      4. add-sqr-sqrt 59.6%

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

      5. sqrt-unprod 66.9%

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

      6. sqr-neg 66.9%

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

      7. sqrt-prod -0.0%

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

      8. add-sqr-sqrt 59.6%

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

   9. Applied egg-rr 59.6%

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

       1. unpow-1 59.6%

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

   11. Simplified 59.6%

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

   if -1 < x

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 45.3%

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

4. Final simplification 48.6%

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

Alternative 12: 46.4% accurate, 21.1× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	float tmp;
	if (x <= -1.0f) {
		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 <= (-1.0e0)) 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(-1.0))
		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(-1.0))
		tmp = s / x;
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 59.6%

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

      1. mul-1-neg 59.6%

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

      2. unsub-neg 59.6%

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

   4. Simplified 59.6%

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

   5. Taylor expanded in x around inf 59.6%

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

      1. neg-mul-1 59.6%

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

      2. distribute-neg-frac 59.6%

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

   7. Simplified 59.6%

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

      1. expm1-log1p-u 59.6%

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

      2. expm1-udef 98.4%

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

      3. clear-num 98.4%

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

      4. add-sqr-sqrt 98.4%

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

      5. add-sqr-sqrt 98.4%

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

      6. sqrt-prod 98.4%

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

      7. sqrt-unprod 98.4%

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

      8. sqr-neg 98.4%

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

      9. sqrt-div 98.4%

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

      10. times-frac 98.4%

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

      11. sqrt-prod 34.5%

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

      12. add-sqr-sqrt 98.4%

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

   9. Applied egg-rr 98.4%

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

       1. expm1-def 52.4%

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

       2. expm1-log1p 52.4%

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

   11. Simplified 52.4%

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

   if -1 < x

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 45.3%

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

4. Final simplification 46.9%

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

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 36.5%

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

3. Final simplification 36.5%

   \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023230 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))