Logistic function

Percentage Accurate: 99.8% → 99.9%

Time: 10.9s

Alternatives: 12

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.9% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.8%

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

   1. div-inv 99.8%

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

   2. exp-prod 83.7%

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

   3. neg-mul-1 83.7%

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

   4. exp-prod 83.7%

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

   5. pow-pow 99.8%

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

   6. div-inv 99.8%

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

4. Applied egg-rr 99.8%

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

   1. add-exp-log 99.8%

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

   2. log-rec 99.8%

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

   3. log1p-expm1-u 99.8%

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

   4. log1p-define 99.8%

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

   5. pow-exp 99.8%

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

   6. expm1-log1p-u 99.8%

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

   7. neg-mul-1 99.8%

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

   8. distribute-neg-frac2 99.8%

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

6. Applied egg-rr 99.8%

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

7. Final simplification 99.8%

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

Alternative 2: 99.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

   1. div-inv 99.8%

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

   2. exp-prod 83.7%

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

   3. neg-mul-1 83.7%

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

   4. exp-prod 83.7%

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

   5. pow-pow 99.8%

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

   6. div-inv 99.8%

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

4. Applied egg-rr 99.8%

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

5. Final simplification 99.8%

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

Alternative 3: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\left(\mathsf{expm1}\left(\frac{x}{-s}\right) + 1\right) + 1} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (/ 1.0 (+ (+ (expm1 (/ x (- s))) 1.0) 1.0)))

C

float code(float x, float s) {
	return 1.0f / ((expm1f((x / -s)) + 1.0f) + 1.0f);
}

Julia

function code(x, s)
	return Float32(Float32(1.0) / Float32(Float32(expm1(Float32(x / Float32(-s))) + Float32(1.0)) + Float32(1.0)))
end

Derivation

1. Initial program 99.8%

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

   1. div-inv 99.8%

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

   2. exp-prod 83.7%

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

   3. neg-mul-1 83.7%

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

   4. exp-prod 83.7%

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

   5. pow-pow 99.8%

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

   6. div-inv 99.8%

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

4. Applied egg-rr 99.8%

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

   1. add-cube-cbrt 99.6%

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

   2. unpow-prod-down 99.7%

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

   3. cbrt-unprod 99.8%

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

   4. prod-exp 99.8%

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

   5. metadata-eval 99.8%

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

6. Applied egg-rr 99.8%

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

   1. expm1-log1p-u 99.7%

      \[\leadsto \frac{1}{1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\sqrt[3]{e^{-2}}\right)}^{\left(\frac{x}{s}\right)} \cdot {\left(\sqrt[3]{e^{-1}}\right)}^{\left(\frac{x}{s}\right)}\right)\right)}} \]

   2. log1p-define 99.7%

      \[\leadsto \frac{1}{1 + \mathsf{expm1}\left(\color{blue}{\log \left(1 + {\left(\sqrt[3]{e^{-2}}\right)}^{\left(\frac{x}{s}\right)} \cdot {\left(\sqrt[3]{e^{-1}}\right)}^{\left(\frac{x}{s}\right)}\right)}\right)} \]

   3. expm1-undefine 99.7%

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

   4. add-exp-log 99.8%

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

   5. pow-prod-down 99.8%

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

   6. metadata-eval 99.8%

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

   7. prod-exp 99.8%

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

   8. cbrt-unprod 99.6%

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

   9. add-cube-cbrt 99.8%

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

8. Applied egg-rr 99.8%

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

   1. associate--l+ 99.8%

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

   2. exp-prod 99.8%

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

   3. *-commutative 99.8%

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

   4. expm1-define 99.8%

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

   5. *-commutative 99.8%

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

   6. mul-1-neg 99.8%

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

   7. distribute-frac-neg2 99.8%

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

10. Simplified 99.8%

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

11. Final simplification 99.8%

    \[\leadsto \frac{1}{\left(\mathsf{expm1}\left(\frac{x}{-s}\right) + 1\right) + 1} \]
12. Add Preprocessing

Alternative 4: 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(x / Float32(-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 99.8%

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

3. Final simplification 99.8%

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

Alternative 5: 54.5% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-x \leq -6.000000019026461 \cdot 10^{-30}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;-x \leq 1999999968613499000:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{\frac{x}{s}}{\frac{s}{x}}}{\frac{x}{s} + 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{-1}{\frac{s}{x - s \cdot 2}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- x) -6.000000019026461e-30)
   0.5
   (if (<= (- x) 1999999968613499000.0)
     (/ 1.0 (/ (- 4.0 (/ (/ x s) (/ s x))) (+ (/ x s) 2.0)))
     (/ 1.0 (/ -1.0 (/ s (- x (* s 2.0))))))))

C

float code(float x, float s) {
	float tmp;
	if (-x <= -6.000000019026461e-30f) {
		tmp = 0.5f;
	} else if (-x <= 1999999968613499000.0f) {
		tmp = 1.0f / ((4.0f - ((x / s) / (s / x))) / ((x / s) + 2.0f));
	} else {
		tmp = 1.0f / (-1.0f / (s / (x - (s * 2.0f))));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (-x <= (-6.000000019026461e-30)) then
        tmp = 0.5e0
    else if (-x <= 1999999968613499000.0e0) then
        tmp = 1.0e0 / ((4.0e0 - ((x / s) / (s / x))) / ((x / s) + 2.0e0))
    else
        tmp = 1.0e0 / ((-1.0e0) / (s / (x - (s * 2.0e0))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-x) <= Float32(-6.000000019026461e-30))
		tmp = Float32(0.5);
	elseif (Float32(-x) <= Float32(1999999968613499000.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(4.0) - Float32(Float32(x / s) / Float32(s / x))) / Float32(Float32(x / s) + Float32(2.0))));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(-1.0) / Float32(s / Float32(x - Float32(s * Float32(2.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (-x <= single(-6.000000019026461e-30))
		tmp = single(0.5);
	elseif (-x <= single(1999999968613499000.0))
		tmp = single(1.0) / ((single(4.0) - ((x / s) / (s / x))) / ((x / s) + single(2.0)));
	else
		tmp = single(1.0) / (single(-1.0) / (s / (x - (s * single(2.0)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if (neg.f32 x) < -6e-30

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 38.1%

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

   if -6e-30 < (neg.f32 x) < 1.99999997e18

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0 43.3%

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

      1. mul-1-neg 43.3%

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

      2. unsub-neg 43.3%

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

   5. Simplified 43.3%

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

      1. sub-neg 43.3%

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

      2. neg-mul-1 43.3%

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

      3. rem-log-exp 94.3%

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

      4. pow-exp 94.3%

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

      5. flip-+ 31.9%

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

      6. metadata-eval 31.9%

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

      7. pow-exp 31.9%

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

      8. rem-log-exp 31.9%

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

      9. neg-mul-1 31.9%

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

      10. pow-exp 31.9%

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

      11. rem-log-exp 33.1%

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

      12. neg-mul-1 33.1%

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

      13. distribute-neg-frac2 33.1%

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

      14. distribute-neg-frac2 33.1%

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

      15. pow-exp 33.1%

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

      16. rem-log-exp 52.8%

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

      17. neg-mul-1 52.8%

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

      18. distribute-neg-frac2 52.8%

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

   7. Applied egg-rr 52.8%

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

      1. associate-*l/ 52.8%

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

      2. add-sqr-sqrt -0.0%

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

      3. sqrt-unprod 57.8%

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

      4. sqr-neg 57.8%

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

      5. sqrt-unprod 52.3%

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

      6. add-sqr-sqrt 52.3%

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

      7. remove-double-neg 52.3%

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

      8. distribute-neg-frac2 52.3%

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

      9. associate-*l/ 52.3%

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

      10. distribute-lft-neg-in 52.3%

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

      11. clear-num 52.3%

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

      12. un-div-inv 52.3%

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

      13. distribute-frac-neg2 52.3%

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

      14. remove-double-neg 52.3%

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

      15. add-sqr-sqrt -0.0%

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

      16. sqrt-unprod 58.1%

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

      17. sqr-neg 58.1%

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

      18. sqrt-unprod 52.8%

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

      19. add-sqr-sqrt 52.8%

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

   9. Applied egg-rr 52.8%

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

   10. Taylor expanded in x around 0 52.8%

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

   if 1.99999997e18 < (neg.f32 x)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 82.6%

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

      1. mul-1-neg 82.6%

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

      2. unsub-neg 82.6%

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

   5. Simplified 82.6%

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

   6. Taylor expanded in s around 0 82.6%

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

      1. *-commutative 82.6%

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

   8. Simplified 82.6%

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

      1. clear-num 82.6%

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

      2. inv-pow 82.6%

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

   10. Applied egg-rr 82.6%

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

       1. unpow-1 82.6%

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

   12. Simplified 82.6%

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

4. Final simplification 48.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-x \leq -6.000000019026461 \cdot 10^{-30}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;-x \leq 1999999968613499000:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{\frac{x}{s}}{\frac{s}{x}}}{\frac{x}{s} + 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{-1}{\frac{s}{x - s \cdot 2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 57.3% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-x \leq -6.000000019026461 \cdot 10^{-30}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;-x \leq 1999999968613499000:\\ \;\;\;\;\frac{1}{\frac{4 - x \cdot \frac{\frac{x}{s}}{s}}{\frac{x}{s} + 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{-1}{\frac{s}{x - s \cdot 2}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- x) -6.000000019026461e-30)
   0.5
   (if (<= (- x) 1999999968613499000.0)
     (/ 1.0 (/ (- 4.0 (* x (/ (/ x s) s))) (+ (/ x s) 2.0)))
     (/ 1.0 (/ -1.0 (/ s (- x (* s 2.0))))))))

C

float code(float x, float s) {
	float tmp;
	if (-x <= -6.000000019026461e-30f) {
		tmp = 0.5f;
	} else if (-x <= 1999999968613499000.0f) {
		tmp = 1.0f / ((4.0f - (x * ((x / s) / s))) / ((x / s) + 2.0f));
	} else {
		tmp = 1.0f / (-1.0f / (s / (x - (s * 2.0f))));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (-x <= (-6.000000019026461e-30)) then
        tmp = 0.5e0
    else if (-x <= 1999999968613499000.0e0) then
        tmp = 1.0e0 / ((4.0e0 - (x * ((x / s) / s))) / ((x / s) + 2.0e0))
    else
        tmp = 1.0e0 / ((-1.0e0) / (s / (x - (s * 2.0e0))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-x) <= Float32(-6.000000019026461e-30))
		tmp = Float32(0.5);
	elseif (Float32(-x) <= Float32(1999999968613499000.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(4.0) - Float32(x * Float32(Float32(x / s) / s))) / Float32(Float32(x / s) + Float32(2.0))));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(-1.0) / Float32(s / Float32(x - Float32(s * Float32(2.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (-x <= single(-6.000000019026461e-30))
		tmp = single(0.5);
	elseif (-x <= single(1999999968613499000.0))
		tmp = single(1.0) / ((single(4.0) - (x * ((x / s) / s))) / ((x / s) + single(2.0)));
	else
		tmp = single(1.0) / (single(-1.0) / (s / (x - (s * single(2.0)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if (neg.f32 x) < -6e-30

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 38.1%

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

   if -6e-30 < (neg.f32 x) < 1.99999997e18

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0 43.3%

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

      1. mul-1-neg 43.3%

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

      2. unsub-neg 43.3%

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

   5. Simplified 43.3%

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

      1. sub-neg 43.3%

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

      2. neg-mul-1 43.3%

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

      3. rem-log-exp 94.3%

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

      4. pow-exp 94.3%

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

      5. flip-+ 31.9%

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

      6. metadata-eval 31.9%

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

      7. pow-exp 31.9%

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

      8. rem-log-exp 31.9%

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

      9. neg-mul-1 31.9%

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

      10. pow-exp 31.9%

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

      11. rem-log-exp 33.1%

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

      12. neg-mul-1 33.1%

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

      13. distribute-neg-frac2 33.1%

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

      14. distribute-neg-frac2 33.1%

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

      15. pow-exp 33.1%

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

      16. rem-log-exp 52.8%

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

      17. neg-mul-1 52.8%

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

      18. distribute-neg-frac2 52.8%

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

   7. Applied egg-rr 52.8%

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

      1. associate-*l/ 52.8%

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

      2. add-sqr-sqrt -0.0%

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

      3. sqrt-unprod 57.8%

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

      4. sqr-neg 57.8%

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

      5. sqrt-unprod 52.3%

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

      6. add-sqr-sqrt 52.3%

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

      7. remove-double-neg 52.3%

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

      8. distribute-neg-frac2 52.3%

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

      9. associate-*l/ 52.3%

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

      10. distribute-lft-neg-in 52.3%

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

      11. clear-num 52.3%

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

      12. un-div-inv 52.3%

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

      13. distribute-frac-neg2 52.3%

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

      14. remove-double-neg 52.3%

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

      15. add-sqr-sqrt -0.0%

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

      16. sqrt-unprod 58.1%

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

      17. sqr-neg 58.1%

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

      18. sqrt-unprod 52.8%

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

      19. add-sqr-sqrt 52.8%

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

   9. Applied egg-rr 52.8%

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

       1. associate-/r/ 60.0%

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

   11. Applied egg-rr 60.0%

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

   if 1.99999997e18 < (neg.f32 x)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 82.6%

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

      1. mul-1-neg 82.6%

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

      2. unsub-neg 82.6%

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

   5. Simplified 82.6%

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

   6. Taylor expanded in s around 0 82.6%

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

      1. *-commutative 82.6%

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

   8. Simplified 82.6%

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

      1. clear-num 82.6%

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

      2. inv-pow 82.6%

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

   10. Applied egg-rr 82.6%

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

       1. unpow-1 82.6%

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

   12. Simplified 82.6%

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

4. Final simplification 51.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-x \leq -6.000000019026461 \cdot 10^{-30}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;-x \leq 1999999968613499000:\\ \;\;\;\;\frac{1}{\frac{4 - x \cdot \frac{\frac{x}{s}}{s}}{\frac{x}{s} + 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{-1}{\frac{s}{x - s \cdot 2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 49.7% accurate, 8.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-x \leq -6.000000019026461 \cdot 10^{-30}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- x) -6.000000019026461e-30) 0.5 (/ 1.0 (- 2.0 (/ x s)))))

C

float code(float x, float s) {
	float tmp;
	if (-x <= -6.000000019026461e-30f) {
		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 <= (-6.000000019026461e-30)) 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(-x) <= Float32(-6.000000019026461e-30))
		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 <= single(-6.000000019026461e-30))
		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 (neg.f32 x) < -6e-30

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 38.1%

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

   if -6e-30 < (neg.f32 x)

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 51.8%

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

      1. mul-1-neg 51.8%

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

      2. unsub-neg 51.8%

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

   5. Simplified 51.8%

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

4. Final simplification 45.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-x \leq -6.000000019026461 \cdot 10^{-30}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 48.3% accurate, 9.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-x \leq 4.999999858590343 \cdot 10^{-10}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- x) 4.999999858590343e-10) 0.5 (/ 1.0 (/ x s))))

C

float code(float x, float s) {
	float tmp;
	if (-x <= 4.999999858590343e-10f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / (x / s);
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (-x <= 4.999999858590343e-10) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 / (x / s)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-x) <= Float32(4.999999858590343e-10))
		tmp = Float32(0.5);
	else
		tmp = 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(4.999999858590343e-10))
		tmp = single(0.5);
	else
		tmp = single(1.0) / (x / s);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (neg.f32 x) < 4.99999986e-10

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 45.0%

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

   if 4.99999986e-10 < (neg.f32 x)

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 42.8%

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

      1. mul-1-neg 42.8%

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

      2. unsub-neg 42.8%

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

   5. Simplified 42.8%

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

      1. sub-neg 42.8%

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

      2. neg-mul-1 42.8%

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

      3. rem-log-exp 97.9%

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

      4. pow-exp 97.9%

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

      5. flip-+ 0.4%

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

      6. metadata-eval 0.4%

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

      7. pow-exp 0.4%

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

      8. rem-log-exp 0.4%

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

      9. neg-mul-1 0.4%

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

      10. pow-exp 0.4%

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

      11. rem-log-exp 1.4%

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

      12. neg-mul-1 1.4%

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

      13. distribute-neg-frac2 1.4%

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

      14. distribute-neg-frac2 1.4%

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

      15. pow-exp 1.4%

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

      16. rem-log-exp 32.1%

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

      17. neg-mul-1 32.1%

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

      18. distribute-neg-frac2 32.1%

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

   7. Applied egg-rr 32.1%

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

      1. sub-neg 32.1%

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

      2. distribute-frac-neg2 32.1%

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

      3. remove-double-neg 32.1%

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

      4. add-sqr-sqrt 32.1%

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

      5. sqrt-unprod 21.9%

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

      6. sqr-neg 21.9%

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

      7. sqrt-unprod -0.0%

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

      8. add-sqr-sqrt 31.8%

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

      9. flip-+ 2.7%

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

      10. metadata-eval 2.7%

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

      11. clear-num 2.7%

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

      12. clear-num 2.7%

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

      13. metadata-eval 2.7%

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

      14. flip-+ 31.8%

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

   9. Applied egg-rr 31.8%

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

   10. Taylor expanded in x around inf 42.5%

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

4. Final simplification 44.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-x \leq 4.999999858590343 \cdot 10^{-10}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 48.4% accurate, 9.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-x \leq 4.999999858590343 \cdot 10^{-10}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- x) 4.999999858590343e-10) 0.5 (/ -1.0 (/ x s))))

C

float code(float x, float s) {
	float tmp;
	if (-x <= 4.999999858590343e-10f) {
		tmp = 0.5f;
	} else {
		tmp = -1.0f / (x / s);
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (-x <= 4.999999858590343e-10) then
        tmp = 0.5e0
    else
        tmp = (-1.0e0) / (x / s)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-x) <= Float32(4.999999858590343e-10))
		tmp = Float32(0.5);
	else
		tmp = 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(4.999999858590343e-10))
		tmp = single(0.5);
	else
		tmp = single(-1.0) / (x / s);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (neg.f32 x) < 4.99999986e-10

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 45.0%

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

   if 4.99999986e-10 < (neg.f32 x)

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 42.8%

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

      1. mul-1-neg 42.8%

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

      2. unsub-neg 42.8%

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

   5. Simplified 42.8%

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

   6. Taylor expanded in x around inf 42.8%

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

      1. mul-1-neg 42.8%

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

      2. distribute-frac-neg2 42.8%

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

   8. Simplified 42.8%

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

4. Final simplification 44.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-x \leq 4.999999858590343 \cdot 10^{-10}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{x}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 47.2% accurate, 12.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.999999717180685 \cdot 10^{-10}:\\ \;\;\;\;\frac{s}{-x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -9.999999717180685e-10) (/ s (- x)) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -9.999999717180685e-10f) {
		tmp = s / -x;
	} else {
		tmp = 0.5f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-9.999999717180685e-10)) then
        tmp = s / -x
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-9.999999717180685e-10))
		tmp = Float32(s / Float32(-x));
	else
		tmp = Float32(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-9.999999717180685e-10))
		tmp = s / -x;
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -9.99999972e-10

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 42.8%

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

      1. mul-1-neg 42.8%

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

      2. unsub-neg 42.8%

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

   5. Simplified 42.8%

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

   6. Taylor expanded in x around inf 39.1%

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

      1. associate-*r/ 39.1%

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

      2. neg-mul-1 39.1%

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

   8. Simplified 39.1%

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

   if -9.99999972e-10 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 45.0%

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

4. Final simplification 43.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.999999717180685 \cdot 10^{-10}:\\ \;\;\;\;\frac{s}{-x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 47.0% accurate, 13.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.999999717180685 \cdot 10^{-10}:\\ \;\;\;\;\frac{s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -9.999999717180685e-10) (/ s x) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -9.999999717180685e-10f) {
		tmp = s / x;
	} else {
		tmp = 0.5f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-9.999999717180685e-10)) then
        tmp = s / x
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-9.999999717180685e-10))
		tmp = Float32(s / x);
	else
		tmp = Float32(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-9.999999717180685e-10))
		tmp = s / x;
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -9.99999972e-10

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 42.8%

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

      1. mul-1-neg 42.8%

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

      2. unsub-neg 42.8%

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

   5. Simplified 42.8%

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

   6. Taylor expanded in x around inf 42.8%

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

      1. mul-1-neg 42.8%

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

      2. distribute-frac-neg2 42.8%

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

   8. Simplified 42.8%

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

      1. clear-num 39.1%

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

      2. div-inv 39.1%

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

      3. add-sqr-sqrt -0.0%

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

      4. sqrt-unprod 51.3%

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

      5. sqr-neg 51.3%

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

      6. sqrt-unprod 38.8%

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

      7. add-sqr-sqrt 38.8%

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

   10. Applied egg-rr 38.8%

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

       1. associate-*r/ 38.8%

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

       2. *-rgt-identity 38.8%

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

   12. Simplified 38.8%

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

   if -9.99999972e-10 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 45.0%

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

4. Final simplification 43.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.999999717180685 \cdot 10^{-10}:\\ \;\;\;\;\frac{s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 35.5% accurate, 108.0× speedup

Math

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

FPCore

(FPCore (x s) :precision binary32 0.5)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in x around 0 33.1%

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

4. Final simplification 33.1%

   \[\leadsto 0.5 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024059 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))