Logistic function

Percentage Accurate: 99.8% → 99.9%

Time: 9.1s

Alternatives: 14

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.9% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.8%

   \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
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 82.8%

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

   3. neg-mul-1 82.8%

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

   4. exp-prod 82.8%

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

   5. pow-pow 99.9%

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

   6. div-inv 99.8%

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

4. Applied egg-rr 99.8%

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

5. Taylor expanded in x around inf 99.8%

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

   1. mul-1-neg 99.8%

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

   2. distribute-frac-neg2 99.8%

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

   3. unpow-1 99.8%

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

   4. exp-to-pow 99.8%

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

   5. log1p-undefine 99.8%

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

   6. *-commutative 99.8%

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

   7. neg-mul-1 99.8%

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

7. Simplified 99.8%

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

8. Final simplification 99.8%

   \[\leadsto e^{-\mathsf{log1p}\left(e^{\frac{-x}{s}}\right)} \]
9. 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 82.8%

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

   3. neg-mul-1 82.8%

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

   4. exp-prod 82.8%

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

   5. pow-pow 99.9%

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

   6. div-inv 99.8%

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

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

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

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 99.8%

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

3. Final simplification 99.8%

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

Alternative 4: 55.8% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-x \leq -1.9999999774532045 \cdot 10^{-26}:\\ \;\;\;\;\frac{1}{x \cdot \frac{2}{x}}\\ \mathbf{elif}\;-x \leq 8.500000346269318 \cdot 10^{+20}:\\ \;\;\;\;\frac{1}{\frac{4 + x \cdot \frac{\frac{x}{s}}{s}}{\frac{x}{s} + 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x \cdot \frac{x - s \cdot 2}{x \cdot s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- x) -1.9999999774532045e-26)
   (/ 1.0 (* x (/ 2.0 x)))
   (if (<= (- x) 8.500000346269318e+20)
     (/ 1.0 (/ (+ 4.0 (* x (/ (/ x s) s))) (+ (/ x s) 2.0)))
     (/ -1.0 (* x (/ (- x (* s 2.0)) (* x s)))))))

C

float code(float x, float s) {
	float tmp;
	if (-x <= -1.9999999774532045e-26f) {
		tmp = 1.0f / (x * (2.0f / x));
	} else if (-x <= 8.500000346269318e+20f) {
		tmp = 1.0f / ((4.0f + (x * ((x / s) / s))) / ((x / s) + 2.0f));
	} else {
		tmp = -1.0f / (x * ((x - (s * 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 <= (-1.9999999774532045e-26)) then
        tmp = 1.0e0 / (x * (2.0e0 / x))
    else if (-x <= 8.500000346269318e+20) then
        tmp = 1.0e0 / ((4.0e0 + (x * ((x / s) / s))) / ((x / s) + 2.0e0))
    else
        tmp = (-1.0e0) / (x * ((x - (s * 2.0e0)) / (x * s)))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-x) <= Float32(-1.9999999774532045e-26))
		tmp = Float32(Float32(1.0) / Float32(x * Float32(Float32(2.0) / x)));
	elseif (Float32(-x) <= Float32(8.500000346269318e+20))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(4.0) + Float32(x * Float32(Float32(x / s) / s))) / Float32(Float32(x / s) + Float32(2.0))));
	else
		tmp = Float32(Float32(-1.0) / Float32(x * Float32(Float32(x - Float32(s * Float32(2.0))) / Float32(x * s))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (-x <= single(-1.9999999774532045e-26))
		tmp = single(1.0) / (x * (single(2.0) / x));
	elseif (-x <= single(8.500000346269318e+20))
		tmp = single(1.0) / ((single(4.0) + (x * ((x / s) / s))) / ((x / s) + single(2.0)));
	else
		tmp = single(-1.0) / (x * ((x - (s * single(2.0))) / (x * s)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if (neg.f32 x) < -1.99999998e-26

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 14.2%

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

      1. mul-1-neg 14.2%

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

      2. unsub-neg 14.2%

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

   5. Simplified 14.2%

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

   6. Taylor expanded in x around inf 14.2%

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

      1. sub-neg 14.2%

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

      2. associate-*r/ 14.2%

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

      3. metadata-eval 14.2%

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

      4. distribute-neg-frac 14.2%

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

      5. metadata-eval 14.2%

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

   8. Simplified 14.2%

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

   9. Taylor expanded in x around 0 34.5%

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

   if -1.99999998e-26 < (neg.f32 x) < 8.50000035e20

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 47.9%

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

      1. mul-1-neg 47.9%

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

      2. unsub-neg 47.9%

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

   5. Simplified 47.9%

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

      1. sub-neg 47.9%

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

      2. neg-mul-1 47.9%

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

      3. rem-log-exp 92.0%

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

      4. pow-exp 92.0%

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

      5. flip-+ 36.7%

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

      6. metadata-eval 36.7%

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

      7. pow-exp 36.7%

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

      8. rem-log-exp 36.7%

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

      9. neg-mul-1 36.7%

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

      10. pow-exp 36.7%

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

      11. rem-log-exp 37.6%

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

      12. neg-mul-1 37.6%

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

      13. distribute-neg-frac2 37.6%

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

      14. distribute-neg-frac2 37.6%

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

      15. pow-exp 37.6%

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

      16. rem-log-exp 60.5%

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

      17. neg-mul-1 60.5%

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

      18. distribute-neg-frac2 60.5%

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

   7. Applied egg-rr 60.5%

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

      1. associate-*l/ 60.5%

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

      2. neg-mul-1 60.5%

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

      3. times-frac 65.2%

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

      4. add-sqr-sqrt -0.0%

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

      5. sqrt-unprod 64.6%

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

      6. sqr-neg 64.6%

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

      7. sqrt-unprod 65.1%

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

      8. add-sqr-sqrt 65.1%

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

   9. Applied egg-rr 65.1%

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

   if 8.50000035e20 < (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 73.8%

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

      1. mul-1-neg 73.8%

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

      2. unsub-neg 73.8%

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

   5. Simplified 73.8%

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

   6. Taylor expanded in x around inf 73.8%

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

      1. sub-neg 73.8%

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

      2. associate-*r/ 73.8%

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

      3. metadata-eval 73.8%

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

      4. distribute-neg-frac 73.8%

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

      5. metadata-eval 73.8%

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

   8. Simplified 73.8%

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

      1. frac-add 73.8%

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

      2. *-commutative 73.8%

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

      3. neg-mul-1 73.8%

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

      4. fma-define 73.8%

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

   10. Applied egg-rr 73.8%

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

       1. fma-undefine 73.8%

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

       2. unsub-neg 73.8%

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

       3. *-commutative 73.8%

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

   12. Simplified 73.8%

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

4. Final simplification 52.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-x \leq -1.9999999774532045 \cdot 10^{-26}:\\ \;\;\;\;\frac{1}{x \cdot \frac{2}{x}}\\ \mathbf{elif}\;-x \leq 8.500000346269318 \cdot 10^{+20}:\\ \;\;\;\;\frac{1}{\frac{4 + x \cdot \frac{\frac{x}{s}}{s}}{\frac{x}{s} + 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x \cdot \frac{x - s \cdot 2}{x \cdot s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 54.4% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-x \leq -1.000000046701102 \cdot 10^{-34}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;-x \leq 8.500000346269318 \cdot 10^{+20}:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{s \cdot \frac{s}{x}}}{\frac{x}{s} + 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x \cdot \frac{x - s \cdot 2}{x \cdot s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- x) -1.000000046701102e-34)
   0.5
   (if (<= (- x) 8.500000346269318e+20)
     (/ 1.0 (/ (- 4.0 (/ x (* s (/ s x)))) (+ (/ x s) 2.0)))
     (/ -1.0 (* x (/ (- x (* s 2.0)) (* x s)))))))

C

float code(float x, float s) {
	float tmp;
	if (-x <= -1.000000046701102e-34f) {
		tmp = 0.5f;
	} else if (-x <= 8.500000346269318e+20f) {
		tmp = 1.0f / ((4.0f - (x / (s * (s / x)))) / ((x / s) + 2.0f));
	} else {
		tmp = -1.0f / (x * ((x - (s * 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 <= (-1.000000046701102e-34)) then
        tmp = 0.5e0
    else if (-x <= 8.500000346269318e+20) then
        tmp = 1.0e0 / ((4.0e0 - (x / (s * (s / x)))) / ((x / s) + 2.0e0))
    else
        tmp = (-1.0e0) / (x * ((x - (s * 2.0e0)) / (x * s)))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-x) <= Float32(-1.000000046701102e-34))
		tmp = Float32(0.5);
	elseif (Float32(-x) <= Float32(8.500000346269318e+20))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(4.0) - Float32(x / Float32(s * Float32(s / x)))) / Float32(Float32(x / s) + Float32(2.0))));
	else
		tmp = Float32(Float32(-1.0) / Float32(x * Float32(Float32(x - Float32(s * Float32(2.0))) / Float32(x * s))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (-x <= single(-1.000000046701102e-34))
		tmp = single(0.5);
	elseif (-x <= single(8.500000346269318e+20))
		tmp = single(1.0) / ((single(4.0) - (x / (s * (s / x)))) / ((x / s) + single(2.0)));
	else
		tmp = single(-1.0) / (x * ((x - (s * single(2.0))) / (x * s)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if (neg.f32 x) < -1.00000005e-34

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 39.2%

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

   if -1.00000005e-34 < (neg.f32 x) < 8.50000035e20

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 44.3%

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

      1. mul-1-neg 44.3%

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

      2. unsub-neg 44.3%

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

   5. Simplified 44.3%

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

      1. sub-neg 44.3%

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

      2. neg-mul-1 44.3%

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

      3. rem-log-exp 94.4%

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

      4. pow-exp 94.4%

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

      5. flip-+ 31.7%

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

      6. metadata-eval 31.7%

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

      7. pow-exp 31.7%

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

      8. rem-log-exp 31.7%

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

      9. neg-mul-1 31.7%

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

      10. pow-exp 31.7%

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

      11. rem-log-exp 32.7%

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

      12. neg-mul-1 32.7%

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

      13. distribute-neg-frac2 32.7%

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

      14. distribute-neg-frac2 32.7%

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

      15. pow-exp 32.7%

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

      16. rem-log-exp 58.6%

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

      17. neg-mul-1 58.6%

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

      18. distribute-neg-frac2 58.6%

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

   7. Applied egg-rr 58.6%

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

      1. distribute-frac-neg2 58.6%

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

      2. distribute-frac-neg2 58.6%

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

      3. sqr-neg 58.6%

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

      4. clear-num 58.6%

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

      5. frac-times 60.4%

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

      6. *-un-lft-identity 60.4%

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

   9. Applied egg-rr 60.4%

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

   if 8.50000035e20 < (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 73.8%

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

      1. mul-1-neg 73.8%

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

      2. unsub-neg 73.8%

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

   5. Simplified 73.8%

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

   6. Taylor expanded in x around inf 73.8%

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

      1. sub-neg 73.8%

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

      2. associate-*r/ 73.8%

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

      3. metadata-eval 73.8%

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

      4. distribute-neg-frac 73.8%

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

      5. metadata-eval 73.8%

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

   8. Simplified 73.8%

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

      1. frac-add 73.8%

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

      2. *-commutative 73.8%

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

      3. neg-mul-1 73.8%

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

      4. fma-define 73.8%

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

   10. Applied egg-rr 73.8%

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

       1. fma-undefine 73.8%

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

       2. unsub-neg 73.8%

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

       3. *-commutative 73.8%

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

   12. Simplified 73.8%

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

4. Final simplification 51.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-x \leq -1.000000046701102 \cdot 10^{-34}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;-x \leq 8.500000346269318 \cdot 10^{+20}:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{s \cdot \frac{s}{x}}}{\frac{x}{s} + 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x \cdot \frac{x - s \cdot 2}{x \cdot s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 50.5% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-x \leq 4.999999873689376 \cdot 10^{-5}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;-x \leq 8.500000346269318 \cdot 10^{+20}:\\ \;\;\;\;\frac{-1}{\frac{\frac{x}{s} \cdot \frac{x}{s} - 4}{\frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x \cdot \frac{x - s \cdot 2}{x \cdot s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- x) 4.999999873689376e-5)
   0.5
   (if (<= (- x) 8.500000346269318e+20)
     (/ -1.0 (/ (- (* (/ x s) (/ x s)) 4.0) (/ x s)))
     (/ -1.0 (* x (/ (- x (* s 2.0)) (* x s)))))))

C

float code(float x, float s) {
	float tmp;
	if (-x <= 4.999999873689376e-5f) {
		tmp = 0.5f;
	} else if (-x <= 8.500000346269318e+20f) {
		tmp = -1.0f / ((((x / s) * (x / s)) - 4.0f) / (x / s));
	} else {
		tmp = -1.0f / (x * ((x - (s * 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 <= 4.999999873689376e-5) then
        tmp = 0.5e0
    else if (-x <= 8.500000346269318e+20) then
        tmp = (-1.0e0) / ((((x / s) * (x / s)) - 4.0e0) / (x / s))
    else
        tmp = (-1.0e0) / (x * ((x - (s * 2.0e0)) / (x * s)))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-x) <= Float32(4.999999873689376e-5))
		tmp = Float32(0.5);
	elseif (Float32(-x) <= Float32(8.500000346269318e+20))
		tmp = Float32(Float32(-1.0) / Float32(Float32(Float32(Float32(x / s) * Float32(x / s)) - Float32(4.0)) / Float32(x / s)));
	else
		tmp = Float32(Float32(-1.0) / Float32(x * Float32(Float32(x - Float32(s * Float32(2.0))) / Float32(x * s))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (-x <= single(4.999999873689376e-5))
		tmp = single(0.5);
	elseif (-x <= single(8.500000346269318e+20))
		tmp = single(-1.0) / ((((x / s) * (x / s)) - single(4.0)) / (x / s));
	else
		tmp = single(-1.0) / (x * ((x - (s * single(2.0))) / (x * s)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if (neg.f32 x) < 4.99999987e-5

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 44.5%

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

   if 4.99999987e-5 < (neg.f32 x) < 8.50000035e20

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 24.8%

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

      1. mul-1-neg 24.8%

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

      2. unsub-neg 24.8%

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

   5. Simplified 24.8%

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

      1. sub-neg 24.8%

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

      2. neg-mul-1 24.8%

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

      3. rem-log-exp 98.1%

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

      4. pow-exp 98.1%

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

      5. flip-+ 0.3%

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

      6. metadata-eval 0.3%

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

      7. pow-exp 0.3%

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

      8. rem-log-exp 0.3%

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

      9. neg-mul-1 0.3%

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

      10. pow-exp 0.3%

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

      11. rem-log-exp 1.3%

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

      12. neg-mul-1 1.3%

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

      13. distribute-neg-frac2 1.3%

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

      14. distribute-neg-frac2 1.3%

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

      15. pow-exp 1.3%

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

      16. rem-log-exp 53.9%

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

      17. neg-mul-1 53.9%

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

      18. distribute-neg-frac2 53.9%

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

   7. Applied egg-rr 53.9%

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

   8. Taylor expanded in x around inf 53.9%

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

   if 8.50000035e20 < (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 73.8%

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

      1. mul-1-neg 73.8%

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

      2. unsub-neg 73.8%

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

   5. Simplified 73.8%

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

   6. Taylor expanded in x around inf 73.8%

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

      1. sub-neg 73.8%

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

      2. associate-*r/ 73.8%

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

      3. metadata-eval 73.8%

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

      4. distribute-neg-frac 73.8%

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

      5. metadata-eval 73.8%

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

   8. Simplified 73.8%

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

      1. frac-add 73.8%

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

      2. *-commutative 73.8%

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

      3. neg-mul-1 73.8%

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

      4. fma-define 73.8%

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

   10. Applied egg-rr 73.8%

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

       1. fma-undefine 73.8%

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

       2. unsub-neg 73.8%

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

       3. *-commutative 73.8%

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

   12. Simplified 73.8%

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

4. Final simplification 49.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-x \leq 4.999999873689376 \cdot 10^{-5}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;-x \leq 8.500000346269318 \cdot 10^{+20}:\\ \;\;\;\;\frac{-1}{\frac{\frac{x}{s} \cdot \frac{x}{s} - 4}{\frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x \cdot \frac{x - s \cdot 2}{x \cdot s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 49.8% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-x \leq 4.999999898305949 \cdot 10^{-32}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x \cdot \frac{x - s \cdot 2}{x \cdot s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- x) 4.999999898305949e-32)
   0.5
   (/ -1.0 (* x (/ (- x (* s 2.0)) (* x s))))))

C

float code(float x, float s) {
	float tmp;
	if (-x <= 4.999999898305949e-32f) {
		tmp = 0.5f;
	} else {
		tmp = -1.0f / (x * ((x - (s * 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 <= 4.999999898305949e-32) then
        tmp = 0.5e0
    else
        tmp = (-1.0e0) / (x * ((x - (s * 2.0e0)) / (x * s)))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-x) <= Float32(4.999999898305949e-32))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(-1.0) / Float32(x * Float32(Float32(x - Float32(s * Float32(2.0))) / Float32(x * s))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (-x <= single(4.999999898305949e-32))
		tmp = single(0.5);
	else
		tmp = single(-1.0) / (x * ((x - (s * single(2.0))) / (x * s)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (neg.f32 x) < 4.9999999e-32

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 44.1%

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

   if 4.9999999e-32 < (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 45.8%

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

      1. mul-1-neg 45.8%

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

      2. unsub-neg 45.8%

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

   5. Simplified 45.8%

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

   6. Taylor expanded in x around inf 45.7%

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

      1. sub-neg 45.7%

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

      2. associate-*r/ 45.7%

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

      3. metadata-eval 45.7%

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

      4. distribute-neg-frac 45.7%

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

      5. metadata-eval 45.7%

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

   8. Simplified 45.7%

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

      1. frac-add 47.6%

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

      2. *-commutative 47.6%

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

      3. neg-mul-1 47.6%

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

      4. fma-define 47.6%

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

   10. Applied egg-rr 47.6%

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

       1. fma-undefine 47.6%

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

       2. unsub-neg 47.6%

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

       3. *-commutative 47.6%

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

   12. Simplified 47.6%

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

4. Final simplification 45.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-x \leq 4.999999898305949 \cdot 10^{-32}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x \cdot \frac{x - s \cdot 2}{x \cdot s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 48.3% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-x \leq -1.000000046701102 \cdot 10^{-34}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{x - s \cdot 2}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- x) -1.000000046701102e-34) 0.5 (/ -1.0 (/ (- x (* s 2.0)) s))))

C

float code(float x, float s) {
	float tmp;
	if (-x <= -1.000000046701102e-34f) {
		tmp = 0.5f;
	} else {
		tmp = -1.0f / ((x - (s * 2.0f)) / 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 <= (-1.000000046701102e-34)) then
        tmp = 0.5e0
    else
        tmp = (-1.0e0) / ((x - (s * 2.0e0)) / s)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-x) <= Float32(-1.000000046701102e-34))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(-1.0) / Float32(Float32(x - Float32(s * Float32(2.0))) / s));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (-x <= single(-1.000000046701102e-34))
		tmp = single(0.5);
	else
		tmp = single(-1.0) / ((x - (s * single(2.0))) / s);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (neg.f32 x) < -1.00000005e-34

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 39.2%

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

   if -1.00000005e-34 < (neg.f32 x)

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 50.7%

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

      1. mul-1-neg 50.7%

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

      2. unsub-neg 50.7%

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

   5. Simplified 50.7%

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

   6. Taylor expanded in s around 0 50.7%

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

      1. *-commutative 50.7%

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

   8. Simplified 50.7%

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

4. Final simplification 45.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-x \leq -1.000000046701102 \cdot 10^{-34}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{x - s \cdot 2}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 48.3% accurate, 8.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-x \leq -1.000000046701102 \cdot 10^{-34}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- x) -1.000000046701102e-34) 0.5 (/ 1.0 (- 2.0 (/ x s)))))

C

float code(float x, float s) {
	float tmp;
	if (-x <= -1.000000046701102e-34f) {
		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 <= (-1.000000046701102e-34)) 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(-1.000000046701102e-34))
		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(-1.000000046701102e-34))
		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) < -1.00000005e-34

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 39.2%

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

   if -1.00000005e-34 < (neg.f32 x)

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 50.7%

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

      1. mul-1-neg 50.7%

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

      2. unsub-neg 50.7%

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

   5. Simplified 50.7%

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

Alternative 10: 47.2% accurate, 9.8× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	float tmp;
	if (-x <= 4.999999873689376e-5f) {
		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 <= 4.999999873689376e-5) 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(4.999999873689376e-5))
		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(4.999999873689376e-5))
		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) < 4.99999987e-5

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 44.5%

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

   if 4.99999987e-5 < (neg.f32 x)

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 43.1%

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

      1. mul-1-neg 43.1%

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

      2. unsub-neg 43.1%

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

   5. Simplified 43.1%

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

   6. Taylor expanded in x around inf 43.1%

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

      1. mul-1-neg 43.1%

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

      2. distribute-frac-neg2 43.1%

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

   8. Simplified 43.1%

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

4. Final simplification 44.1%

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

Alternative 11: 47.2% accurate, 9.8× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	float tmp;
	if (-x <= 4.999999873689376e-5f) {
		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 <= 4.999999873689376e-5) 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(4.999999873689376e-5))
		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(4.999999873689376e-5))
		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) < 4.99999987e-5

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 44.5%

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

   if 4.99999987e-5 < (neg.f32 x)

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 43.1%

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

      1. mul-1-neg 43.1%

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

      2. unsub-neg 43.1%

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

   5. Simplified 43.1%

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

      1. sub-neg 43.1%

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

      2. neg-mul-1 43.1%

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

      3. rem-log-exp 98.8%

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

      4. pow-exp 98.8%

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

      5. flip-+ 0.2%

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

      6. metadata-eval 0.2%

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

      7. pow-exp 0.2%

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

      8. rem-log-exp 0.2%

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

      9. neg-mul-1 0.2%

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

      10. pow-exp 0.2%

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

      11. rem-log-exp 0.8%

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

      12. neg-mul-1 0.8%

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

      13. distribute-neg-frac2 0.8%

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

      14. distribute-neg-frac2 0.8%

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

      15. pow-exp 0.8%

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

      16. rem-log-exp 45.7%

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

      17. neg-mul-1 45.7%

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

      18. distribute-neg-frac2 45.7%

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

   7. Applied egg-rr 45.7%

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

      1. distribute-frac-neg2 45.7%

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

      2. distribute-frac-neg2 45.7%

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

      3. sqr-neg 45.7%

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

      4. clear-num 45.7%

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

      5. frac-2neg 45.7%

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

      6. frac-times 45.7%

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

      7. *-un-lft-identity 45.7%

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

      8. add-sqr-sqrt -0.0%

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

      9. sqrt-unprod 45.6%

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

      10. sqr-neg 45.6%

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

      11. sqrt-unprod 45.6%

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

      12. add-sqr-sqrt 45.6%

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

   9. Applied egg-rr 45.6%

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

   10. Taylor expanded in x around inf 43.0%

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

Alternative 12: 46.2% accurate, 10.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if (neg.f32 x) < 4.99999987e-5

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 44.5%

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

   if 4.99999987e-5 < (neg.f32 x)

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 43.1%

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

      1. mul-1-neg 43.1%

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

      2. unsub-neg 43.1%

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

   5. Simplified 43.1%

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

   6. Taylor expanded in x around inf 39.2%

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

      1. mul-1-neg 39.2%

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

      2. distribute-neg-frac2 39.2%

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

   8. Simplified 39.2%

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

Alternative 13: 46.1% accurate, 13.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0005000000237487257:\\ \;\;\;\;\frac{s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -0.0005000000237487257) (/ s x) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -0.0005000000237487257f) {
		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 <= (-0.0005000000237487257e0)) 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(-0.0005000000237487257))
		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(-0.0005000000237487257))
		tmp = s / x;
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -5.00000024e-4

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 43.1%

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

      1. mul-1-neg 43.1%

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

      2. unsub-neg 43.1%

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

   5. Simplified 43.1%

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

      1. sub-neg 43.1%

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

      2. neg-mul-1 43.1%

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

      3. rem-log-exp 98.8%

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

      4. pow-exp 98.8%

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

      5. flip-+ 0.2%

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

      6. metadata-eval 0.2%

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

      7. pow-exp 0.2%

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

      8. rem-log-exp 0.2%

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

      9. neg-mul-1 0.2%

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

      10. pow-exp 0.2%

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

      11. rem-log-exp 0.8%

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

      12. neg-mul-1 0.8%

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

      13. distribute-neg-frac2 0.8%

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

      14. distribute-neg-frac2 0.8%

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

      15. pow-exp 0.8%

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

      16. rem-log-exp 45.7%

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

      17. neg-mul-1 45.7%

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

      18. distribute-neg-frac2 45.7%

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

   7. Applied egg-rr 45.7%

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

      1. distribute-frac-neg2 45.7%

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

      2. distribute-frac-neg2 45.7%

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

      3. sqr-neg 45.7%

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

      4. clear-num 45.7%

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

      5. frac-2neg 45.7%

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

      6. frac-times 45.7%

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

      7. *-un-lft-identity 45.7%

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

      8. add-sqr-sqrt -0.0%

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

      9. sqrt-unprod 45.6%

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

      10. sqr-neg 45.6%

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

      11. sqrt-unprod 45.6%

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

      12. add-sqr-sqrt 45.6%

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

   9. Applied egg-rr 45.6%

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

   10. Taylor expanded in x around inf 39.1%

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

   if -5.00000024e-4 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 44.5%

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

Alternative 14: 34.8% 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 33.3%

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

Reproduce

herbie shell --seed 2024096 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))