Logistic function

Percentage Accurate: 99.8% → 99.9%

Time: 17.7s

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(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 99.7%

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

   2. exp-prod 85.4%

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

   3. neg-mul-1 85.4%

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

   4. exp-prod 85.4%

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

   5. pow-pow 99.7%

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

   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 99.7%

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

   2. log-rec 99.7%

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

   3. log1p-expm1-u 99.7%

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

   4. log1p-define 99.7%

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

   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 99.7%

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

   2. exp-prod 85.4%

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

   3. neg-mul-1 85.4%

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

   4. exp-prod 85.4%

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

   5. pow-pow 99.7%

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

   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: 59.9% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t\_0 \leq -50:\\ \;\;\;\;\frac{1}{x \cdot \frac{2}{x}}\\ \mathbf{elif}\;t\_0 \leq 4.999999840142846 \cdot 10^{+37}:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{s} \cdot \frac{x}{s}}{\frac{x}{s} + 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x \cdot \left(\frac{-1}{s} - \left(\frac{1}{s} + \left(\frac{2}{x} + \frac{1}{s}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)))
   (if (<= t_0 -50.0)
     (/ 1.0 (* x (/ 2.0 x)))
     (if (<= t_0 4.999999840142846e+37)
       (/ 1.0 (/ (- 4.0 (* (/ x s) (/ x s))) (+ (/ x s) 2.0)))
       (/ -1.0 (* x (- (/ -1.0 s) (+ (/ 1.0 s) (+ (/ 2.0 x) (/ 1.0 s))))))))))

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if (t_0 <= -50.0f) {
		tmp = 1.0f / (x * (2.0f / x));
	} else if (t_0 <= 4.999999840142846e+37f) {
		tmp = 1.0f / ((4.0f - ((x / s) * (x / s))) / ((x / s) + 2.0f));
	} else {
		tmp = -1.0f / (x * ((-1.0f / s) - ((1.0f / s) + ((2.0f / x) + (1.0f / s)))));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: tmp
    t_0 = -x / s
    if (t_0 <= (-50.0e0)) then
        tmp = 1.0e0 / (x * (2.0e0 / x))
    else if (t_0 <= 4.999999840142846e+37) then
        tmp = 1.0e0 / ((4.0e0 - ((x / s) * (x / s))) / ((x / s) + 2.0e0))
    else
        tmp = (-1.0e0) / (x * (((-1.0e0) / s) - ((1.0e0 / s) + ((2.0e0 / x) + (1.0e0 / s)))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = Float32(Float32(-x) / s)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-50.0))
		tmp = Float32(Float32(1.0) / Float32(x * Float32(Float32(2.0) / x)));
	elseif (t_0 <= Float32(4.999999840142846e+37))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(4.0) - Float32(Float32(x / s) * Float32(x / s))) / Float32(Float32(x / s) + Float32(2.0))));
	else
		tmp = Float32(Float32(-1.0) / Float32(x * Float32(Float32(Float32(-1.0) / s) - Float32(Float32(Float32(1.0) / s) + Float32(Float32(Float32(2.0) / x) + Float32(Float32(1.0) / s))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = -x / s;
	tmp = single(0.0);
	if (t_0 <= single(-50.0))
		tmp = single(1.0) / (x * (single(2.0) / x));
	elseif (t_0 <= single(4.999999840142846e+37))
		tmp = single(1.0) / ((single(4.0) - ((x / s) * (x / s))) / ((x / s) + single(2.0)));
	else
		tmp = single(-1.0) / (x * ((single(-1.0) / s) - ((single(1.0) / s) + ((single(2.0) / x) + (single(1.0) / s)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if (/.f32 (neg.f32 x) s) < -50

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 5.3%

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

      1. mul-1-neg 5.3%

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

      2. unsub-neg 5.3%

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

   5. Simplified 5.3%

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

   6. Taylor expanded in x around inf 5.3%

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

      1. associate-*r/ 5.3%

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

      2. metadata-eval 5.3%

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

   8. Simplified 5.3%

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

   9. Taylor expanded in x around 0 28.1%

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

   if -50 < (/.f32 (neg.f32 x) s) < 4.99999984e37

   1. Initial program 99.5%

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

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

      1. sub-neg 51.8%

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

      2. neg-mul-1 51.8%

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

      3. rem-log-exp 92.2%

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

      4. pow-exp 92.2%

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

      5. flip-+ 46.6%

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

      6. metadata-eval 46.6%

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

      7. pow-exp 46.6%

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

      8. rem-log-exp 46.6%

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

      9. neg-mul-1 46.6%

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

      10. pow-exp 46.6%

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

      11. rem-log-exp 47.2%

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

      12. neg-mul-1 47.2%

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

      13. distribute-neg-frac2 47.2%

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

      14. distribute-neg-frac2 47.2%

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

      15. pow-exp 47.2%

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

      16. rem-log-exp 76.3%

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

      17. neg-mul-1 76.3%

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

      18. distribute-neg-frac2 76.3%

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

   7. Applied egg-rr 76.3%

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

   if 4.99999984e37 < (/.f32 (neg.f32 x) s)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 97.6%

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

      1. mul-1-neg 97.6%

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

      2. unsub-neg 97.6%

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

   5. Simplified 97.6%

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

   6. Taylor expanded in x around inf 97.6%

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

      1. associate-*r/ 97.6%

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

      2. metadata-eval 97.6%

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

   8. Simplified 97.6%

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

      1. add-sqr-sqrt -0.0%

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

      2. *-un-lft-identity -0.0%

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

      3. prod-diff -0.0%

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

      4. associate-/r/ -0.0%

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

      5. clear-num -0.0%

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

      6. fma-define -0.0%

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

      7. add-sqr-sqrt 97.6%

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

      8. distribute-frac-neg2 97.6%

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

      9. add-sqr-sqrt -0.0%

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

      10. sqrt-unprod 100.0%

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

      11. sqr-neg 100.0%

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

      12. sqrt-unprod 97.6%

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

      13. add-sqr-sqrt 97.6%

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

      14. /-rgt-identity 97.6%

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

      15. clear-num 97.6%

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

   10. Applied egg-rr 100.0%

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

       1. +-commutative 100.0%

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

       2. fma-undefine 100.0%

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

       3. *-rgt-identity 100.0%

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

       4. associate-+l+ 100.0%

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

       5. +-commutative 100.0%

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

   12. Simplified 100.0%

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

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -50:\\ \;\;\;\;\frac{1}{x \cdot \frac{2}{x}}\\ \mathbf{elif}\;\frac{-x}{s} \leq 4.999999840142846 \cdot 10^{+37}:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{s} \cdot \frac{x}{s}}{\frac{x}{s} + 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x \cdot \left(\frac{-1}{s} - \left(\frac{1}{s} + \left(\frac{2}{x} + \frac{1}{s}\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 49.1% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -50:\\ \;\;\;\;\frac{1}{x \cdot \frac{2}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{-1}{\frac{s}{s \cdot 2 - x}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) -50.0)
   (/ 1.0 (* x (/ 2.0 x)))
   (/ -1.0 (/ -1.0 (/ s (- (* s 2.0) x))))))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= -50.0f) {
		tmp = 1.0f / (x * (2.0f / x));
	} else {
		tmp = -1.0f / (-1.0f / (s / ((s * 2.0f) - x)));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((-x / s) <= (-50.0e0)) then
        tmp = 1.0e0 / (x * (2.0e0 / x))
    else
        tmp = (-1.0e0) / ((-1.0e0) / (s / ((s * 2.0e0) - x)))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(-x) / s) <= Float32(-50.0))
		tmp = Float32(Float32(1.0) / Float32(x * Float32(Float32(2.0) / x)));
	else
		tmp = Float32(Float32(-1.0) / Float32(Float32(-1.0) / Float32(s / Float32(Float32(s * Float32(2.0)) - x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((-x / s) <= single(-50.0))
		tmp = single(1.0) / (x * (single(2.0) / x));
	else
		tmp = single(-1.0) / (single(-1.0) / (s / ((s * single(2.0)) - x)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < -50

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 5.3%

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

      1. mul-1-neg 5.3%

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

      2. unsub-neg 5.3%

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

   5. Simplified 5.3%

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

   6. Taylor expanded in x around inf 5.3%

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

      1. associate-*r/ 5.3%

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

      2. metadata-eval 5.3%

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

   8. Simplified 5.3%

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

   9. Taylor expanded in x around 0 28.1%

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

   if -50 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 60.1%

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

      1. mul-1-neg 60.1%

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

      2. unsub-neg 60.1%

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

   5. Simplified 60.1%

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

   6. Taylor expanded in x around inf 59.5%

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

      1. associate-*r/ 59.5%

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

      2. metadata-eval 59.5%

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

   8. Simplified 59.5%

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

      1. frac-sub 47.1%

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

      2. associate-*r/ 50.4%

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

      3. *-commutative 50.4%

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

      4. *-rgt-identity 50.4%

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

   10. Applied egg-rr 50.4%

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

       1. *-commutative 50.4%

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

       2. associate-/l* 47.0%

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

       3. *-commutative 47.0%

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

   12. Simplified 47.0%

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

       1. associate-*r/ 50.4%

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

       2. *-commutative 50.4%

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

       3. *-commutative 50.4%

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

       4. times-frac 60.1%

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

       5. *-inverses 60.1%

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

       6. metadata-eval 60.1%

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

       7. div-inv 60.1%

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

       8. clear-num 60.1%

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

       9. clear-num 60.1%

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

   14. Applied egg-rr 60.1%

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

4. Final simplification 47.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -50:\\ \;\;\;\;\frac{1}{x \cdot \frac{2}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{-1}{\frac{s}{s \cdot 2 - x}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 50.4% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-x \leq 4.9999998413276127 \cdot 10^{-20}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot s\right) \cdot \frac{\frac{1}{x}}{s \cdot 2 - x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- x) 4.9999998413276127e-20)
   0.5
   (* (* x s) (/ (/ 1.0 x) (- (* s 2.0) x)))))

C

float code(float x, float s) {
	float tmp;
	if (-x <= 4.9999998413276127e-20f) {
		tmp = 0.5f;
	} else {
		tmp = (x * s) * ((1.0f / x) / ((s * 2.0f) - x));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (-x <= 4.9999998413276127e-20) then
        tmp = 0.5e0
    else
        tmp = (x * s) * ((1.0e0 / x) / ((s * 2.0e0) - x))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-x) <= Float32(4.9999998413276127e-20))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(x * s) * Float32(Float32(Float32(1.0) / x) / Float32(Float32(s * Float32(2.0)) - x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (-x <= single(4.9999998413276127e-20))
		tmp = single(0.5);
	else
		tmp = (x * s) * ((single(1.0) / x) / ((s * single(2.0)) - x));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (neg.f32 x) < 4.99999984e-20

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 48.4%

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

   if 4.99999984e-20 < (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 42.0%

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

      1. mul-1-neg 42.0%

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

      2. unsub-neg 42.0%

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

   5. Simplified 42.0%

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

   6. Taylor expanded in x around inf 42.0%

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

      1. associate-*r/ 42.0%

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

      2. metadata-eval 42.0%

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

   8. Simplified 42.0%

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

      1. associate-/r* 37.8%

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

      2. frac-sub 39.8%

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

      3. associate-/r/ 47.1%

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

      4. *-commutative 47.1%

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

      5. *-rgt-identity 47.1%

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

   10. Applied egg-rr 47.1%

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

4. Final simplification 47.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-x \leq 4.9999998413276127 \cdot 10^{-20}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot s\right) \cdot \frac{\frac{1}{x}}{s \cdot 2 - x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 52.9% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-x \leq 4.999999999099794 \cdot 10^{-24}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{x \cdot \left(x - s \cdot 2\right)}{x \cdot s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- x) 4.999999999099794e-24)
   0.5
   (/ -1.0 (/ (* x (- x (* s 2.0))) (* x s)))))

C

float code(float x, float s) {
	float tmp;
	if (-x <= 4.999999999099794e-24f) {
		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 <= 4.999999999099794e-24) 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(4.999999999099794e-24))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(-1.0) / Float32(Float32(x * 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(4.999999999099794e-24))
		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) < 5e-24

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 48.0%

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

   if 5e-24 < (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 42.7%

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

      1. mul-1-neg 42.7%

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

      2. unsub-neg 42.7%

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

   5. Simplified 42.7%

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

   6. Taylor expanded in x around inf 42.7%

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

      1. associate-*r/ 42.7%

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

      2. metadata-eval 42.7%

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

   8. Simplified 42.7%

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

      1. *-commutative 42.7%

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

      2. frac-sub 44.7%

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

      3. associate-*l/ 53.9%

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

      4. *-commutative 53.9%

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

      5. *-rgt-identity 53.9%

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

   10. Applied egg-rr 53.9%

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

4. Final simplification 50.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-x \leq 4.999999999099794 \cdot 10^{-24}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{x \cdot \left(x - s \cdot 2\right)}{x \cdot s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 49.1% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -50:\\ \;\;\;\;\frac{1}{x \cdot \frac{2}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{x - s \cdot 2}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) -50.0)
   (/ 1.0 (* x (/ 2.0 x)))
   (/ -1.0 (/ (- x (* s 2.0)) s))))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= -50.0f) {
		tmp = 1.0f / (x * (2.0f / x));
	} 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 / s) <= (-50.0e0)) then
        tmp = 1.0e0 / (x * (2.0e0 / x))
    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(Float32(-x) / s) <= Float32(-50.0))
		tmp = Float32(Float32(1.0) / Float32(x * Float32(Float32(2.0) / x)));
	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 / s) <= single(-50.0))
		tmp = single(1.0) / (x * (single(2.0) / x));
	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 (/.f32 (neg.f32 x) s) < -50

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 5.3%

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

      1. mul-1-neg 5.3%

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

      2. unsub-neg 5.3%

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

   5. Simplified 5.3%

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

   6. Taylor expanded in x around inf 5.3%

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

      1. associate-*r/ 5.3%

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

      2. metadata-eval 5.3%

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

   8. Simplified 5.3%

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

   9. Taylor expanded in x around 0 28.1%

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

   if -50 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 60.1%

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

      1. mul-1-neg 60.1%

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

      2. unsub-neg 60.1%

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

   5. Simplified 60.1%

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

   6. Taylor expanded in s around 0 60.1%

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

      1. *-commutative 60.1%

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

   8. Simplified 60.1%

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

4. Final simplification 47.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -50:\\ \;\;\;\;\frac{1}{x \cdot \frac{2}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{x - s \cdot 2}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 49.1% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq -50:\\ \;\;\;\;\frac{1}{x \cdot \frac{2}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ x (- s)) -50.0) (/ 1.0 (* x (/ 2.0 x))) (/ 1.0 (- 2.0 (/ x s)))))

C

float code(float x, float s) {
	float tmp;
	if ((x / -s) <= -50.0f) {
		tmp = 1.0f / (x * (2.0f / x));
	} else {
		tmp = 1.0f / (2.0f - (x / s));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((x / -s) <= (-50.0e0)) then
        tmp = 1.0e0 / (x * (2.0e0 / x))
    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(-s)) <= Float32(-50.0))
		tmp = Float32(Float32(1.0) / Float32(x * Float32(Float32(2.0) / x)));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) - Float32(x / s)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((x / -s) <= single(-50.0))
		tmp = single(1.0) / (x * (single(2.0) / x));
	else
		tmp = single(1.0) / (single(2.0) - (x / s));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < -50

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 5.3%

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

      1. mul-1-neg 5.3%

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

      2. unsub-neg 5.3%

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

   5. Simplified 5.3%

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

   6. Taylor expanded in x around inf 5.3%

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

      1. associate-*r/ 5.3%

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

      2. metadata-eval 5.3%

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

   8. Simplified 5.3%

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

   9. Taylor expanded in x around 0 28.1%

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

   if -50 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 60.1%

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

      1. mul-1-neg 60.1%

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

      2. unsub-neg 60.1%

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

   5. Simplified 60.1%

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

4. Final simplification 47.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq -50:\\ \;\;\;\;\frac{1}{x \cdot \frac{2}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 47.8% accurate, 8.3× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= 0.5f) {
		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 / s) <= 0.5e0) 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(Float32(-x) / s) <= Float32(0.5))
		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 / s) <= single(0.5))
		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 (/.f32 (neg.f32 x) s) < 0.5

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 50.1%

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

   if 0.5 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0 38.1%

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

      1. mul-1-neg 38.1%

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

      2. unsub-neg 38.1%

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

   5. Simplified 38.1%

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

   6. Taylor expanded in x around inf 38.0%

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

      1. mul-1-neg 38.0%

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

      2. distribute-frac-neg2 38.0%

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

   8. Simplified 38.0%

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

4. Final simplification 45.7%

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

Alternative 11: 46.2% accurate, 12.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.00019999999494757503:\\ \;\;\;\;\frac{s}{-x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -0.00019999999494757503) (/ s (- x)) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -0.00019999999494757503f) {
		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 <= (-0.00019999999494757503e0)) 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(-0.00019999999494757503))
		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(-0.00019999999494757503))
		tmp = s / -x;
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -1.99999995e-4

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 48.4%

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

      1. mul-1-neg 48.4%

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

      2. unsub-neg 48.4%

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

   5. Simplified 48.4%

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

   6. Taylor expanded in x around inf 42.7%

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

      1. associate-*r/ 42.7%

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

      2. neg-mul-1 42.7%

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

   8. Simplified 42.7%

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

   if -1.99999995e-4 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 44.6%

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

4. Final simplification 44.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00019999999494757503:\\ \;\;\;\;\frac{s}{-x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 34.7% 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 34.6%

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

4. Final simplification 34.6%

   \[\leadsto 0.5 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024076 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))