Logistic function

Percentage Accurate: 99.8% → 99.9%

Time: 9.0s

Alternatives: 8

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. Step-by-step derivation

   1. add-exp-log 99.8%

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

   2. log-rec 99.8%

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

   3. log1p-expm1-u 99.8%

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

   4. log1p-define 99.8%

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

   5. pow-exp 99.8%

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

   6. expm1-log1p-u 99.8%

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

   7. neg-mul-1 99.8%

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

   8. distribute-neg-frac2 99.8%

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

6. Applied egg-rr 99.8%

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

7. Final simplification 99.8%

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

Alternative 2: 99.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

   1. div-inv 99.8%

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

   2. exp-prod 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: 48.6% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t\_0 \leq -2:\\ \;\;\;\;\frac{1}{x \cdot \frac{2}{x}}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{s \cdot \frac{s}{x}}}{\frac{x}{s} + 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

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

C

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

Julia

function code(x, s)
	t_0 = Float32(Float32(-x) / s)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-2.0))
		tmp = Float32(Float32(1.0) / Float32(x * Float32(Float32(2.0) / x)));
	elseif (t_0 <= Float32(Inf))
		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 / s));
	end
	return tmp
end

MATLAB

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 5.1%

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

      1. mul-1-neg 5.1%

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

      2. unsub-neg 5.1%

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

   5. Simplified 5.1%

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

   6. Taylor expanded in x around inf 5.1%

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

      1. associate-*r/ 5.1%

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

      2. metadata-eval 5.1%

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

   8. Simplified 5.1%

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

   9. Taylor expanded in x around 0 28.1%

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

   if -2 < (/.f32 (neg.f32 x) s) < +inf.0

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 57.2%

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

      1. mul-1-neg 57.2%

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

      2. unsub-neg 57.2%

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

   5. Simplified 57.2%

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

      1. sub-neg 57.2%

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

      2. neg-mul-1 57.2%

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

      3. rem-log-exp 95.6%

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

      4. pow-exp 95.6%

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

      5. flip-+ 35.1%

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

      6. metadata-eval 35.1%

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

      7. pow-exp 35.1%

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

      8. rem-log-exp 35.1%

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

      9. neg-mul-1 35.1%

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

      10. pow-exp 35.1%

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

      11. rem-log-exp 35.8%

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

      12. neg-mul-1 35.8%

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

      13. distribute-neg-frac2 35.8%

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

      14. distribute-neg-frac2 35.8%

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

      15. pow-exp 35.8%

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

      16. rem-log-exp 59.1%

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

      17. neg-mul-1 59.1%

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

      18. distribute-neg-frac2 59.1%

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

   7. Applied egg-rr 59.1%

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

      1. distribute-frac-neg2 59.1%

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

      2. distribute-frac-neg2 59.1%

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

      3. sqr-neg 59.1%

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

      4. clear-num 59.1%

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

      5. frac-times 60.3%

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

      6. *-un-lft-identity 60.3%

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

   9. Applied egg-rr 60.3%

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

   if +inf.0 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 36.0%

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

      1. mul-1-neg 36.0%

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

      2. unsub-neg 36.0%

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

   5. Simplified 36.0%

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

      1. sub-neg 36.0%

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

      2. neg-mul-1 36.0%

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

      3. rem-log-exp 59.3%

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

      4. pow-exp 59.3%

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

      5. flip-+ 20.9%

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

      6. metadata-eval 20.9%

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

      7. pow-exp 20.9%

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

      8. rem-log-exp 20.9%

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

      9. neg-mul-1 20.9%

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

      10. pow-exp 20.9%

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

      11. rem-log-exp 21.5%

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

      12. neg-mul-1 21.5%

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

      13. distribute-neg-frac2 21.5%

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

      14. distribute-neg-frac2 21.5%

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

      15. pow-exp 21.5%

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

      16. rem-log-exp 36.4%

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

      17. neg-mul-1 36.4%

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

      18. distribute-neg-frac2 36.4%

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

   7. Applied egg-rr 36.4%

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

      1. distribute-frac-neg2 36.4%

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

      2. distribute-frac-neg2 36.4%

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

      3. sqr-neg 36.4%

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

      4. clear-num 36.4%

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

      5. frac-2neg 36.4%

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

      6. frac-times 37.1%

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

      7. *-un-lft-identity 37.1%

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

      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 39.1%

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

      10. sqr-neg 39.1%

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

      11. sqrt-unprod 38.3%

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

      12. add-sqr-sqrt 38.3%

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

   9. Applied egg-rr 38.3%

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

   10. Taylor expanded in x around inf 18.6%

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

4. Final simplification 47.2%

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

Alternative 5: 45.9% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t\_0 \leq 5:\\ \;\;\;\;0.5\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{-1}{\frac{\frac{x}{s} \cdot \frac{x}{s} - 4}{\frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)))
   (if (<= t_0 5.0)
     0.5
     (if (<= t_0 INFINITY)
       (/ -1.0 (/ (- (* (/ x s) (/ x s)) 4.0) (/ x s)))
       (/ 1.0 (/ x s))))))

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if (t_0 <= 5.0f) {
		tmp = 0.5f;
	} else if (t_0 <= ((float) INFINITY)) {
		tmp = -1.0f / ((((x / s) * (x / s)) - 4.0f) / (x / s));
	} else {
		tmp = 1.0f / (x / s);
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = Float32(Float32(-x) / s)
	tmp = Float32(0.0)
	if (t_0 <= Float32(5.0))
		tmp = Float32(0.5);
	elseif (t_0 <= Float32(Inf))
		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 / s));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = -x / s;
	tmp = single(0.0);
	if (t_0 <= single(5.0))
		tmp = single(0.5);
	elseif (t_0 <= single(Inf))
		tmp = single(-1.0) / ((((x / s) * (x / s)) - single(4.0)) / (x / s));
	else
		tmp = single(1.0) / (x / s);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if (/.f32 (neg.f32 x) s) < 5

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 48.7%

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

   if 5 < (/.f32 (neg.f32 x) s) < +inf.0

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 36.2%

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

      1. mul-1-neg 36.2%

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

      2. unsub-neg 36.2%

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

   5. Simplified 36.2%

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

      1. sub-neg 36.2%

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

      2. neg-mul-1 36.2%

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

      3. rem-log-exp 99.1%

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

      4. pow-exp 99.1%

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

      5. flip-+ 0.1%

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

      6. metadata-eval 0.1%

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

      7. pow-exp 0.1%

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

      8. rem-log-exp 0.1%

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

      9. neg-mul-1 0.1%

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

      10. pow-exp 0.1%

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

      11. rem-log-exp 1.2%

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

      12. neg-mul-1 1.2%

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

      13. distribute-neg-frac2 1.2%

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

      14. distribute-neg-frac2 1.2%

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

      15. pow-exp 1.2%

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

      16. rem-log-exp 39.3%

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

      17. neg-mul-1 39.3%

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

      18. distribute-neg-frac2 39.3%

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

   7. Applied egg-rr 39.3%

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

   8. Taylor expanded in x around inf 39.3%

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

   if +inf.0 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 36.0%

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

      1. mul-1-neg 36.0%

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

      2. unsub-neg 36.0%

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

   5. Simplified 36.0%

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

      1. sub-neg 36.0%

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

      2. neg-mul-1 36.0%

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

      3. rem-log-exp 59.3%

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

      4. pow-exp 59.3%

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

      5. flip-+ 20.9%

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

      6. metadata-eval 20.9%

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

      7. pow-exp 20.9%

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

      8. rem-log-exp 20.9%

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

      9. neg-mul-1 20.9%

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

      10. pow-exp 20.9%

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

      11. rem-log-exp 21.5%

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

      12. neg-mul-1 21.5%

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

      13. distribute-neg-frac2 21.5%

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

      14. distribute-neg-frac2 21.5%

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

      15. pow-exp 21.5%

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

      16. rem-log-exp 36.4%

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

      17. neg-mul-1 36.4%

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

      18. distribute-neg-frac2 36.4%

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

   7. Applied egg-rr 36.4%

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

      1. distribute-frac-neg2 36.4%

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

      2. distribute-frac-neg2 36.4%

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

      3. sqr-neg 36.4%

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

      4. clear-num 36.4%

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

      5. frac-2neg 36.4%

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

      6. frac-times 37.1%

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

      7. *-un-lft-identity 37.1%

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

      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 39.1%

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

      10. sqr-neg 39.1%

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

      11. sqrt-unprod 38.3%

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

      12. add-sqr-sqrt 38.3%

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

   9. Applied egg-rr 38.3%

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

   10. Taylor expanded in x around inf 18.6%

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

4. Final simplification 45.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 5:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\frac{-x}{s} \leq \infty:\\ \;\;\;\;\frac{-1}{\frac{\frac{x}{s} \cdot \frac{x}{s} - 4}{\frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 63.5% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -0.5:\\ \;\;\;\;\frac{1}{x \cdot \frac{2}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\frac{x}{s} \cdot \frac{x}{s} - 4}{\frac{1}{0.5 + \frac{x \cdot -0.25}{s}}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) -0.5)
   (/ 1.0 (* x (/ 2.0 x)))
   (/ -1.0 (/ (- (* (/ x s) (/ x s)) 4.0) (/ 1.0 (+ 0.5 (/ (* x -0.25) s)))))))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= -0.5f) {
		tmp = 1.0f / (x * (2.0f / x));
	} else {
		tmp = -1.0f / ((((x / s) * (x / s)) - 4.0f) / (1.0f / (0.5f + ((x * -0.25f) / 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 = 1.0e0 / (x * (2.0e0 / x))
    else
        tmp = (-1.0e0) / ((((x / s) * (x / s)) - 4.0e0) / (1.0e0 / (0.5e0 + ((x * (-0.25e0)) / 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(Float32(1.0) / Float32(x * Float32(Float32(2.0) / x)));
	else
		tmp = Float32(Float32(-1.0) / Float32(Float32(Float32(Float32(x / s) * Float32(x / s)) - Float32(4.0)) / Float32(Float32(1.0) / Float32(Float32(0.5) + Float32(Float32(x * Float32(-0.25)) / s)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((-x / s) <= single(-0.5))
		tmp = single(1.0) / (x * (single(2.0) / x));
	else
		tmp = single(-1.0) / ((((x / s) * (x / s)) - single(4.0)) / (single(1.0) / (single(0.5) + ((x * single(-0.25)) / 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 100.0%

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

   3. Taylor expanded in x around 0 5.4%

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

      1. mul-1-neg 5.4%

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

      2. unsub-neg 5.4%

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

   5. Simplified 5.4%

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

   6. Taylor expanded in x around inf 5.4%

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

      1. associate-*r/ 5.4%

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

      2. metadata-eval 5.4%

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

   8. Simplified 5.4%

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

   9. Taylor expanded in x around 0 28.2%

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

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

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 57.3%

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

      1. mul-1-neg 57.3%

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

      2. unsub-neg 57.3%

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

   5. Simplified 57.3%

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

      1. sub-neg 57.3%

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

      2. neg-mul-1 57.3%

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

      3. rem-log-exp 96.0%

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

      4. pow-exp 96.0%

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

      5. flip-+ 35.1%

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

      6. metadata-eval 35.1%

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

      7. pow-exp 35.1%

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

      8. rem-log-exp 35.1%

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

      9. neg-mul-1 35.1%

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

      10. pow-exp 35.1%

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

      11. rem-log-exp 35.7%

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

      12. neg-mul-1 35.7%

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

      13. distribute-neg-frac2 35.7%

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

      14. distribute-neg-frac2 35.7%

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

      15. pow-exp 35.7%

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

      16. rem-log-exp 59.2%

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

      17. neg-mul-1 59.2%

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

      18. distribute-neg-frac2 59.2%

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

   7. Applied egg-rr 59.2%

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

      1. div-inv 59.2%

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

      2. cancel-sign-sub-inv 59.2%

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

      3. div-inv 59.2%

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

      4. frac-2neg 59.2%

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

      5. add-sqr-sqrt 59.2%

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

      6. sqrt-unprod 47.6%

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

      7. sqr-neg 47.6%

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

      8. sqrt-unprod -0.0%

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

      9. add-sqr-sqrt 56.5%

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

      10. flip-+ 34.0%

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

      11. metadata-eval 34.0%

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

      12. clear-num 34.0%

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

      13. clear-num 34.0%

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

      14. metadata-eval 34.0%

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

      15. flip-+ 56.5%

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

   9. Applied egg-rr 59.2%

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

   10. Taylor expanded in x around 0 83.5%

       \[\leadsto \frac{1}{\frac{4 - \frac{x}{-s} \cdot \frac{x}{-s}}{\frac{1}{\color{blue}{0.5 + -0.25 \cdot \frac{x}{s}}}}} \]
   11. Step-by-step derivation

       1. associate-*r/ 83.5%

          \[\leadsto \frac{1}{\frac{4 - \frac{x}{-s} \cdot \frac{x}{-s}}{\frac{1}{0.5 + \color{blue}{\frac{-0.25 \cdot x}{s}}}}} \]

       2. *-commutative 83.5%

          \[\leadsto \frac{1}{\frac{4 - \frac{x}{-s} \cdot \frac{x}{-s}}{\frac{1}{0.5 + \frac{\color{blue}{x \cdot -0.25}}{s}}}} \]

   12. Simplified 83.5%

       \[\leadsto \frac{1}{\frac{4 - \frac{x}{-s} \cdot \frac{x}{-s}}{\frac{1}{\color{blue}{0.5 + \frac{x \cdot -0.25}{s}}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -0.5:\\ \;\;\;\;\frac{1}{x \cdot \frac{2}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\frac{x}{s} \cdot \frac{x}{s} - 4}{\frac{1}{0.5 + \frac{x \cdot -0.25}{s}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 40.1% accurate, 15.4× speedup

Math

\[\begin{array}{l} \\ \frac{1}{2 - \frac{x}{s}} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (/ 1.0 (- 2.0 (/ x s))))

C

float code(float x, float s) {
	return 1.0f / (2.0f - (x / s));
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 1.0e0 / (2.0e0 - (x / s))
end function

Julia

function code(x, s)
	return Float32(Float32(1.0) / Float32(Float32(2.0) - Float32(x / s)))
end

MATLAB

function tmp = code(x, s)
	tmp = single(1.0) / (single(2.0) - (x / s));
end

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in x around 0 36.0%

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

   1. mul-1-neg 36.0%

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

   2. unsub-neg 36.0%

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

5. Simplified 36.0%

   \[\leadsto \frac{1}{\color{blue}{2 - \frac{x}{s}}} \]
6. Add Preprocessing

Alternative 8: 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)))))