Logistic function

Percentage Accurate: 99.8% → 99.9%

Time: 8.5s

Alternatives: 16

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.9% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.8%

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

   1. div-inv 99.8%

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

   2. exp-prod 83.8%

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

   3. neg-mul-1 83.8%

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

   4. exp-prod 83.8%

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

   5. pow-pow 99.8%

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

   6. div-inv 99.8%

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

4. Applied egg-rr 99.8%

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

   1. add-exp-log 99.8%

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

   2. log-rec 99.8%

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

   3. log1p-expm1-u 99.8%

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

   4. log1p-define 99.9%

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

   5. pow-exp 99.9%

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

   6. expm1-log1p-u 99.9%

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

   7. neg-mul-1 99.9%

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

   8. distribute-neg-frac2 99.9%

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

6. Applied egg-rr 99.9%

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

Alternative 2: 99.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(x, s)
	tmp = single(1.0) / (single(1.0) + (exp(single(-1.0)) ^ (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 83.8%

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

   3. neg-mul-1 83.8%

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

   4. exp-prod 83.8%

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

   5. pow-pow 99.8%

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

   6. div-inv 99.8%

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

4. Applied egg-rr 99.8%

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

Alternative 3: 49.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{-s}\\ \mathbf{if}\;t\_0 \leq 2:\\ \;\;\;\;0.5\\ \mathbf{elif}\;t\_0 \leq 999999984306749400:\\ \;\;\;\;\frac{\frac{{s}^{2}}{-s}}{x}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{s} \cdot \frac{x}{s}}{\frac{x}{s} + 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ x (- s))))
   (if (<= t_0 2.0)
     0.5
     (if (<= t_0 999999984306749400.0)
       (/ (/ (pow s 2.0) (- s)) x)
       (if (<= t_0 INFINITY)
         (/ 1.0 (/ (- 4.0 (* (/ x s) (/ x s))) (+ (/ 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 = 0.5f;
	} else if (t_0 <= 999999984306749400.0f) {
		tmp = (powf(s, 2.0f) / -s) / x;
	} else if (t_0 <= ((float) INFINITY)) {
		tmp = 1.0f / ((4.0f - ((x / s) * (x / s))) / ((x / s) + 2.0f));
	} else {
		tmp = 1.0f / (x / s);
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = Float32(x / Float32(-s))
	tmp = Float32(0.0)
	if (t_0 <= Float32(2.0))
		tmp = Float32(0.5);
	elseif (t_0 <= Float32(999999984306749400.0))
		tmp = Float32(Float32((s ^ Float32(2.0)) / Float32(-s)) / x);
	elseif (t_0 <= Float32(Inf))
		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 / 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(0.5);
	elseif (t_0 <= single(999999984306749400.0))
		tmp = ((s ^ single(2.0)) / -s) / x;
	elseif (t_0 <= single(Inf))
		tmp = single(1.0) / ((single(4.0) - ((x / s) * (x / s))) / ((x / s) + single(2.0)));
	else
		tmp = single(1.0) / (x / s);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 4 regimes

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

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 48.9%

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

   if 2 < (/.f32 (neg.f32 x) s) < 9.99999984e17

   1. Initial program 99.2%

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

   3. Taylor expanded in x around 0 8.2%

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

      1. neg-mul-1 8.2%

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

      2. unsub-neg 8.2%

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

   5. Simplified 8.2%

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

   6. Taylor expanded in x around inf 8.1%

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

      1. associate-*r/ 8.1%

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

      2. neg-mul-1 8.1%

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

   8. Simplified 8.1%

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

      1. neg-sub0 8.1%

         \[\leadsto \frac{\color{blue}{0 - s}}{x} \]

      2. flip-- 46.5%

         \[\leadsto \frac{\color{blue}{\frac{0 \cdot 0 - s \cdot s}{0 + s}}}{x} \]

      3. metadata-eval 46.5%

         \[\leadsto \frac{\frac{\color{blue}{0} - s \cdot s}{0 + s}}{x} \]

      4. pow2 46.5%

         \[\leadsto \frac{\frac{0 - \color{blue}{{s}^{2}}}{0 + s}}{x} \]

      5. add-sqr-sqrt 46.5%

         \[\leadsto \frac{\frac{0 - {s}^{2}}{0 + \color{blue}{\sqrt{s} \cdot \sqrt{s}}}}{x} \]

      6. sqrt-unprod 4.8%

         \[\leadsto \frac{\frac{0 - {s}^{2}}{0 + \color{blue}{\sqrt{s \cdot s}}}}{x} \]

      7. sqr-neg 4.8%

         \[\leadsto \frac{\frac{0 - {s}^{2}}{0 + \sqrt{\color{blue}{\left(-s\right) \cdot \left(-s\right)}}}}{x} \]

      8. sqrt-unprod -0.0%

         \[\leadsto \frac{\frac{0 - {s}^{2}}{0 + \color{blue}{\sqrt{-s} \cdot \sqrt{-s}}}}{x} \]

      9. add-sqr-sqrt 45.8%

         \[\leadsto \frac{\frac{0 - {s}^{2}}{0 + \color{blue}{\left(-s\right)}}}{x} \]

      10. sub-neg 45.8%

          \[\leadsto \frac{\frac{0 - {s}^{2}}{\color{blue}{0 - s}}}{x} \]

      11. neg-sub0 45.8%

          \[\leadsto \frac{\frac{0 - {s}^{2}}{\color{blue}{-s}}}{x} \]

      12. add-sqr-sqrt -0.0%

          \[\leadsto \frac{\frac{0 - {s}^{2}}{\color{blue}{\sqrt{-s} \cdot \sqrt{-s}}}}{x} \]

      13. sqrt-unprod 4.8%

          \[\leadsto \frac{\frac{0 - {s}^{2}}{\color{blue}{\sqrt{\left(-s\right) \cdot \left(-s\right)}}}}{x} \]

      14. sqr-neg 4.8%

          \[\leadsto \frac{\frac{0 - {s}^{2}}{\sqrt{\color{blue}{s \cdot s}}}}{x} \]

      15. sqrt-unprod 46.5%

          \[\leadsto \frac{\frac{0 - {s}^{2}}{\color{blue}{\sqrt{s} \cdot \sqrt{s}}}}{x} \]

      16. add-sqr-sqrt 46.5%

          \[\leadsto \frac{\frac{0 - {s}^{2}}{\color{blue}{s}}}{x} \]

   10. Applied egg-rr 46.5%

       \[\leadsto \frac{\color{blue}{\frac{0 - {s}^{2}}{s}}}{x} \]
   11. Step-by-step derivation

       1. sub0-neg 46.5%

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

   12. Simplified 46.5%

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

   if 9.99999984e17 < (/.f32 (neg.f32 x) s) < +inf.0

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 49.8%

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

      1. neg-mul-1 49.8%

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

      2. unsub-neg 49.8%

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

   5. Simplified 49.8%

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

      1. sub-neg 49.8%

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

      2. neg-mul-1 49.8%

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

      3. rem-log-exp 100.0%

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

      4. pow-exp 100.0%

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

      5. flip-+ -0.0%

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

      6. metadata-eval -0.0%

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

      7. pow-exp -0.0%

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

      8. rem-log-exp -0.0%

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

      9. neg-mul-1 -0.0%

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

      10. pow-exp -0.0%

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

      11. rem-log-exp 0.1%

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

      12. neg-mul-1 0.1%

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

      13. distribute-neg-frac2 0.1%

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

      14. distribute-neg-frac2 0.1%

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

      15. pow-exp 0.1%

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

      16. rem-log-exp 57.1%

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

      17. neg-mul-1 57.1%

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

      18. distribute-neg-frac2 57.1%

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

   7. Applied egg-rr 57.1%

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

   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 35.2%

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

      1. neg-mul-1 35.2%

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

      2. unsub-neg 35.2%

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

   5. Simplified 35.2%

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

   6. Taylor expanded in x around inf 15.2%

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

      1. associate-*r/ 15.2%

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

      2. neg-mul-1 15.2%

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

   8. Simplified 15.2%

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

      1. clear-num 16.1%

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

      2. inv-pow 16.1%

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

      3. add-sqr-sqrt -0.0%

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

      4. sqrt-unprod 27.4%

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

      5. sqr-neg 27.4%

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

      6. sqrt-unprod 17.9%

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

      7. add-sqr-sqrt 17.9%

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

   10. Applied egg-rr 17.9%

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

       1. unpow-1 17.9%

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

   12. Simplified 17.9%

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

4. Final simplification 50.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq 2:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\frac{x}{-s} \leq 999999984306749400:\\ \;\;\;\;\frac{\frac{{s}^{2}}{-s}}{x}\\ \mathbf{elif}\;\frac{x}{-s} \leq \infty:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{s} \cdot \frac{x}{s}}{\frac{x}{s} + 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 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(x / Float32(-s)))))
end

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Final simplification 99.8%

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

Alternative 5: 48.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{-s}\\ \mathbf{if}\;t\_0 \leq 0.20000000298023224:\\ \;\;\;\;0.5\\ \mathbf{elif}\;t\_0 \leq 999999984306749400:\\ \;\;\;\;\frac{1}{x \cdot \frac{s + x \cdot -0.5}{x \cdot \left(s \cdot 0.5\right)}}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{s} \cdot \frac{x}{s}}{\frac{x}{s} + 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ x (- s))))
   (if (<= t_0 0.20000000298023224)
     0.5
     (if (<= t_0 999999984306749400.0)
       (/ 1.0 (* x (/ (+ s (* x -0.5)) (* x (* s 0.5)))))
       (if (<= t_0 INFINITY)
         (/ 1.0 (/ (- 4.0 (* (/ x s) (/ x s))) (+ (/ 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 <= 0.20000000298023224f) {
		tmp = 0.5f;
	} else if (t_0 <= 999999984306749400.0f) {
		tmp = 1.0f / (x * ((s + (x * -0.5f)) / (x * (s * 0.5f))));
	} else if (t_0 <= ((float) INFINITY)) {
		tmp = 1.0f / ((4.0f - ((x / s) * (x / s))) / ((x / s) + 2.0f));
	} else {
		tmp = 1.0f / (x / s);
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = Float32(x / Float32(-s))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.20000000298023224))
		tmp = Float32(0.5);
	elseif (t_0 <= Float32(999999984306749400.0))
		tmp = Float32(Float32(1.0) / Float32(x * Float32(Float32(s + Float32(x * Float32(-0.5))) / Float32(x * Float32(s * Float32(0.5))))));
	elseif (t_0 <= Float32(Inf))
		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 / s));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = x / -s;
	tmp = single(0.0);
	if (t_0 <= single(0.20000000298023224))
		tmp = single(0.5);
	elseif (t_0 <= single(999999984306749400.0))
		tmp = single(1.0) / (x * ((s + (x * single(-0.5))) / (x * (s * single(0.5)))));
	elseif (t_0 <= single(Inf))
		tmp = single(1.0) / ((single(4.0) - ((x / s) * (x / s))) / ((x / s) + single(2.0)));
	else
		tmp = single(1.0) / (x / s);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 4 regimes

2. if (/.f32 (neg.f32 x) s) < 0.200000003

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 49.0%

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

   if 0.200000003 < (/.f32 (neg.f32 x) s) < 9.99999984e17

   1. Initial program 99.2%

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

   3. Taylor expanded in x around 0 8.8%

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

      1. neg-mul-1 8.8%

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

      2. unsub-neg 8.8%

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

   5. Simplified 8.8%

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

   6. Taylor expanded in x around inf 8.8%

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

      1. associate-*r/ 8.8%

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

      2. metadata-eval 8.8%

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

   8. Simplified 8.8%

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

      1. sub-neg 8.8%

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

      2. distribute-frac-neg2 8.8%

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

      3. add-sqr-sqrt -0.0%

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

      4. sqrt-unprod 45.0%

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

      5. sqr-neg 45.0%

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

      6. sqrt-unprod 7.7%

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

      7. add-sqr-sqrt 7.7%

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

      8. /-rgt-identity 7.7%

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

      9. clear-num 7.7%

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

      10. clear-num 7.7%

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

      11. frac-2neg 7.7%

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

      12. metadata-eval 7.7%

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

      13. frac-add 39.7%

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

      14. *-un-lft-identity 39.7%

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

      15. add-sqr-sqrt -0.0%

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

      16. sqrt-unprod 39.3%

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

      17. sqr-neg 39.3%

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

      18. sqrt-unprod 39.3%

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

      19. add-sqr-sqrt 39.3%

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

      20. div-inv 39.3%

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

      21. metadata-eval 39.3%

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

      22. div-inv 39.3%

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

      23. metadata-eval 39.3%

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

   10. Applied egg-rr 40.8%

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

       1. associate-*l* 40.8%

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

       2. metadata-eval 40.8%

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

       3. associate-*l* 40.8%

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

   12. Simplified 40.8%

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

   if 9.99999984e17 < (/.f32 (neg.f32 x) s) < +inf.0

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 49.8%

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

      1. neg-mul-1 49.8%

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

      2. unsub-neg 49.8%

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

   5. Simplified 49.8%

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

      1. sub-neg 49.8%

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

      2. neg-mul-1 49.8%

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

      3. rem-log-exp 100.0%

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

      4. pow-exp 100.0%

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

      5. flip-+ -0.0%

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

      6. metadata-eval -0.0%

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

      7. pow-exp -0.0%

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

      8. rem-log-exp -0.0%

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

      9. neg-mul-1 -0.0%

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

      10. pow-exp -0.0%

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

      11. rem-log-exp 0.1%

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

      12. neg-mul-1 0.1%

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

      13. distribute-neg-frac2 0.1%

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

      14. distribute-neg-frac2 0.1%

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

      15. pow-exp 0.1%

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

      16. rem-log-exp 57.1%

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

      17. neg-mul-1 57.1%

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

      18. distribute-neg-frac2 57.1%

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

   7. Applied egg-rr 57.1%

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

   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 35.2%

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

      1. neg-mul-1 35.2%

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

      2. unsub-neg 35.2%

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

   5. Simplified 35.2%

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

   6. Taylor expanded in x around inf 15.2%

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

      1. associate-*r/ 15.2%

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

      2. neg-mul-1 15.2%

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

   8. Simplified 15.2%

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

      1. clear-num 16.1%

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

      2. inv-pow 16.1%

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

      3. add-sqr-sqrt -0.0%

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

      4. sqrt-unprod 27.4%

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

      5. sqr-neg 27.4%

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

      6. sqrt-unprod 17.9%

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

      7. add-sqr-sqrt 17.9%

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

   10. Applied egg-rr 17.9%

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

       1. unpow-1 17.9%

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

   12. Simplified 17.9%

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

4. Final simplification 49.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq 0.20000000298023224:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\frac{x}{-s} \leq 999999984306749400:\\ \;\;\;\;\frac{1}{x \cdot \frac{s + x \cdot -0.5}{x \cdot \left(s \cdot 0.5\right)}}\\ \mathbf{elif}\;\frac{x}{-s} \leq \infty:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{s} \cdot \frac{x}{s}}{\frac{x}{s} + 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 51.5% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq 0.20000000298023224:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \frac{s + x \cdot -0.5}{x \cdot \left(s \cdot 0.5\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ x (- s)) 0.20000000298023224)
   0.5
   (/ 1.0 (* x (/ (+ s (* x -0.5)) (* x (* s 0.5)))))))

C

float code(float x, float s) {
	float tmp;
	if ((x / -s) <= 0.20000000298023224f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / (x * ((s + (x * -0.5f)) / (x * (s * 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 / -s) <= 0.20000000298023224e0) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 / (x * ((s + (x * (-0.5e0))) / (x * (s * 0.5e0))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(x / Float32(-s)) <= Float32(0.20000000298023224))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / Float32(x * Float32(Float32(s + Float32(x * Float32(-0.5))) / Float32(x * Float32(s * Float32(0.5))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((x / -s) <= single(0.20000000298023224))
		tmp = single(0.5);
	else
		tmp = single(1.0) / (x * ((s + (x * single(-0.5))) / (x * (s * single(0.5)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 0.200000003

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 49.0%

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

   if 0.200000003 < (/.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 34.3%

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

      1. neg-mul-1 34.3%

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

      2. unsub-neg 34.3%

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

   5. Simplified 34.3%

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

   6. Taylor expanded in x around inf 35.2%

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

      1. associate-*r/ 35.2%

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

      2. metadata-eval 35.2%

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

   8. Simplified 35.2%

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

      1. sub-neg 35.2%

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

      2. distribute-frac-neg2 35.2%

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

      3. add-sqr-sqrt -0.0%

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

      4. sqrt-unprod 61.7%

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

      5. sqr-neg 61.7%

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

      6. sqrt-unprod 34.8%

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

      7. add-sqr-sqrt 34.8%

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

      8. /-rgt-identity 34.8%

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

      9. clear-num 34.8%

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

      10. clear-num 34.8%

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

      11. frac-2neg 34.8%

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

      12. metadata-eval 34.8%

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

      13. frac-add 47.8%

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

      14. *-un-lft-identity 47.8%

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

      15. add-sqr-sqrt -0.0%

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

      16. sqrt-unprod 47.7%

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

      17. sqr-neg 47.7%

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

      18. sqrt-unprod 47.7%

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

      19. add-sqr-sqrt 47.7%

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

      20. div-inv 47.7%

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

      21. metadata-eval 47.7%

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

      22. div-inv 47.7%

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

      23. metadata-eval 47.7%

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

   10. Applied egg-rr 48.2%

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

       1. associate-*l* 48.2%

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

       2. metadata-eval 48.2%

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

       3. associate-*l* 48.2%

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

   12. Simplified 48.2%

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

4. Final simplification 48.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq 0.20000000298023224:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \frac{s + x \cdot -0.5}{x \cdot \left(s \cdot 0.5\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 51.8% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-x \leq 3.499999888929329 \cdot 10^{-21}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(s \cdot 2 - x\right)} \cdot \left(x \cdot s\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- x) 3.499999888929329e-21)
   0.5
   (* (/ 1.0 (* x (- (* s 2.0) x))) (* x s))))

C

float code(float x, float s) {
	float tmp;
	if (-x <= 3.499999888929329e-21f) {
		tmp = 0.5f;
	} else {
		tmp = (1.0f / (x * ((s * 2.0f) - x))) * (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 <= 3.499999888929329e-21) then
        tmp = 0.5e0
    else
        tmp = (1.0e0 / (x * ((s * 2.0e0) - x))) * (x * s)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-x) <= Float32(3.499999888929329e-21))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(Float32(1.0) / Float32(x * Float32(Float32(s * Float32(2.0)) - x))) * Float32(x * s));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (-x <= single(3.499999888929329e-21))
		tmp = single(0.5);
	else
		tmp = (single(1.0) / (x * ((s * single(2.0)) - x))) * (x * s);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (neg.f32 x) < 3.49999989e-21

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 45.2%

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

   if 3.49999989e-21 < (neg.f32 x)

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 41.6%

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

      1. neg-mul-1 41.6%

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

      2. unsub-neg 41.6%

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

   5. Simplified 41.6%

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

   6. Taylor expanded in x around inf 42.4%

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

      1. associate-*r/ 42.4%

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

      2. metadata-eval 42.4%

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

   8. Simplified 42.4%

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

      1. associate-/r* 40.1%

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

      2. frac-sub 46.0%

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

      3. associate-/r/ 49.1%

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

      4. *-rgt-identity 49.1%

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

      5. *-commutative 49.1%

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

      6. fmm-def 49.1%

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

   10. Applied egg-rr 49.1%

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

       1. associate-/l/ 50.1%

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

       2. *-commutative 50.1%

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

       3. fmm-undef 50.1%

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

       4. *-commutative 50.1%

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

   12. Simplified 50.1%

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

4. Final simplification 46.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-x \leq 3.499999888929329 \cdot 10^{-21}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(s \cdot 2 - x\right)} \cdot \left(x \cdot s\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 49.8% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq -20:\\ \;\;\;\;\frac{1}{x \cdot \frac{2}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + x \cdot \frac{-1}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ x (- s)) -20.0)
   (/ 1.0 (* x (/ 2.0 x)))
   (/ 1.0 (+ 2.0 (* x (/ -1.0 s))))))

C

float code(float x, float s) {
	float tmp;
	if ((x / -s) <= -20.0f) {
		tmp = 1.0f / (x * (2.0f / x));
	} else {
		tmp = 1.0f / (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) :: tmp
    if ((x / -s) <= (-20.0e0)) then
        tmp = 1.0e0 / (x * (2.0e0 / x))
    else
        tmp = 1.0e0 / (2.0e0 + (x * ((-1.0e0) / s)))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(x / Float32(-s)) <= Float32(-20.0))
		tmp = Float32(Float32(1.0) / Float32(x * Float32(Float32(2.0) / x)));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(x * Float32(Float32(-1.0) / s))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((x / -s) <= single(-20.0))
		tmp = single(1.0) / (x * (single(2.0) / x));
	else
		tmp = single(1.0) / (single(2.0) + (x * (single(-1.0) / s)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < -20

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 5.2%

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

      1. neg-mul-1 5.2%

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

      2. unsub-neg 5.2%

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

   5. Simplified 5.2%

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

   6. Taylor expanded in x around inf 5.2%

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

      1. associate-*r/ 5.2%

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

      2. metadata-eval 5.2%

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

   8. Simplified 5.2%

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

   9. Taylor expanded in x around 0 28.1%

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

   if -20 < (/.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 57.0%

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

      1. neg-mul-1 57.0%

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

      2. unsub-neg 57.0%

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

   5. Simplified 57.0%

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

      1. clear-num 57.0%

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

      2. associate-/r/ 57.6%

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

   7. Applied egg-rr 57.6%

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

4. Final simplification 45.1%

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

Alternative 9: 49.7% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq -20:\\ \;\;\;\;\frac{1}{x \cdot \frac{2}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-1 + \left(3 - \frac{x}{s}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ x (- s)) -20.0)
   (/ 1.0 (* x (/ 2.0 x)))
   (/ 1.0 (+ -1.0 (- 3.0 (/ x s))))))

C

float code(float x, float s) {
	float tmp;
	if ((x / -s) <= -20.0f) {
		tmp = 1.0f / (x * (2.0f / x));
	} else {
		tmp = 1.0f / (-1.0f + (3.0f - (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) <= (-20.0e0)) then
        tmp = 1.0e0 / (x * (2.0e0 / x))
    else
        tmp = 1.0e0 / ((-1.0e0) + (3.0e0 - (x / s)))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(x / Float32(-s)) <= Float32(-20.0))
		tmp = Float32(Float32(1.0) / Float32(x * Float32(Float32(2.0) / x)));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(-1.0) + Float32(Float32(3.0) - Float32(x / s))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((x / -s) <= single(-20.0))
		tmp = single(1.0) / (x * (single(2.0) / x));
	else
		tmp = single(1.0) / (single(-1.0) + (single(3.0) - (x / s)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < -20

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 5.2%

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

      1. neg-mul-1 5.2%

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

      2. unsub-neg 5.2%

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

   5. Simplified 5.2%

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

   6. Taylor expanded in x around inf 5.2%

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

      1. associate-*r/ 5.2%

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

      2. metadata-eval 5.2%

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

   8. Simplified 5.2%

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

   9. Taylor expanded in x around 0 28.1%

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

   if -20 < (/.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 57.0%

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

      1. neg-mul-1 57.0%

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

      2. unsub-neg 57.0%

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

   5. Simplified 57.0%

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

      1. expm1-log1p-u 57.0%

         \[\leadsto \frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 - \frac{x}{s}\right)\right)}} \]

   7. Applied egg-rr 57.0%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 - \frac{x}{s}\right)\right)}} \]
   8. Step-by-step derivation

      1. expm1-undefine 57.0%

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

      2. sub-neg 57.0%

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

      3. log1p-undefine 56.9%

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

      4. rem-exp-log 57.0%

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

      5. associate-+r- 57.1%

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

      6. metadata-eval 57.1%

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

      7. metadata-eval 57.1%

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

   9. Simplified 57.1%

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

4. Final simplification 44.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq -20:\\ \;\;\;\;\frac{1}{x \cdot \frac{2}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-1 + \left(3 - \frac{x}{s}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 49.7% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq -20:\\ \;\;\;\;\frac{1}{x \cdot \frac{2}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ x (- s)) -20.0) (/ 1.0 (* x (/ 2.0 x))) (/ 1.0 (- 2.0 (/ x s)))))

C

float code(float x, float s) {
	float tmp;
	if ((x / -s) <= -20.0f) {
		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) <= (-20.0e0)) 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(-20.0))
		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(-20.0))
		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) < -20

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 5.2%

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

      1. neg-mul-1 5.2%

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

      2. unsub-neg 5.2%

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

   5. Simplified 5.2%

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

   6. Taylor expanded in x around inf 5.2%

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

      1. associate-*r/ 5.2%

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

      2. metadata-eval 5.2%

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

   8. Simplified 5.2%

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

   9. Taylor expanded in x around 0 28.1%

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

   if -20 < (/.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 57.0%

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

      1. neg-mul-1 57.0%

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

      2. unsub-neg 57.0%

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

   5. Simplified 57.0%

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

4. Final simplification 44.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq -20:\\ \;\;\;\;\frac{1}{x \cdot \frac{2}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 48.4% accurate, 7.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 48.9%

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

   if 2 < (/.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 34.4%

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

      1. neg-mul-1 34.4%

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

      2. unsub-neg 34.4%

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

   5. Simplified 34.4%

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

   6. Taylor expanded in x around inf 35.3%

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

      1. associate-*r/ 35.3%

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

      2. metadata-eval 35.3%

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

   8. Simplified 35.3%

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

   9. Taylor expanded in x around inf 35.2%

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

4. Final simplification 44.0%

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

Alternative 12: 48.3% accurate, 8.3× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(x / Float32(-s)) <= Float32(2.0))
		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(2.0))
		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) < 2

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 48.9%

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

   if 2 < (/.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 34.4%

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

      1. neg-mul-1 34.4%

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

      2. unsub-neg 34.4%

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

   5. Simplified 34.4%

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

   6. Taylor expanded in x around inf 34.4%

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

      1. associate-*r/ 34.4%

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

      2. neg-mul-1 34.4%

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

   8. Simplified 34.4%

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

4. Final simplification 43.7%

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

Alternative 13: 47.9% accurate, 10.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.800000054037355 \cdot 10^{-8}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -2.800000054037355e-8) (/ 1.0 (/ x s)) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -2.800000054037355e-8f) {
		tmp = 1.0f / (x / s);
	} else {
		tmp = 0.5f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-2.800000054037355e-8)) then
        tmp = 1.0e0 / (x / s)
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-2.800000054037355e-8))
		tmp = Float32(Float32(1.0) / Float32(x / s));
	else
		tmp = Float32(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-2.800000054037355e-8))
		tmp = single(1.0) / (x / s);
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -2.80000005e-8

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 42.8%

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

      1. neg-mul-1 42.8%

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

      2. unsub-neg 42.8%

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

   5. Simplified 42.8%

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

   6. Taylor expanded in x around inf 39.5%

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

      1. associate-*r/ 39.5%

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

      2. neg-mul-1 39.5%

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

   8. Simplified 39.5%

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

      1. clear-num 42.8%

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

      2. inv-pow 42.8%

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

      3. add-sqr-sqrt -0.0%

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

      4. sqrt-unprod 59.4%

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

      5. sqr-neg 59.4%

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

      6. sqrt-unprod 42.5%

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

      7. add-sqr-sqrt 42.5%

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

   10. Applied egg-rr 42.5%

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

       1. unpow-1 42.5%

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

   12. Simplified 42.5%

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

   if -2.80000005e-8 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 43.9%

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

Alternative 14: 46.8% accurate, 12.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.999999987845058 \cdot 10^{-8}:\\ \;\;\;\;\frac{s}{-x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -1.999999987845058e-8) (/ s (- x)) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -1.999999987845058e-8f) {
		tmp = s / -x;
	} else {
		tmp = 0.5f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-1.999999987845058e-8)) then
        tmp = s / -x
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-1.999999987845058e-8))
		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(-1.999999987845058e-8))
		tmp = s / -x;
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -1.99999999e-8

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 42.3%

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

      1. neg-mul-1 42.3%

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

      2. unsub-neg 42.3%

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

   5. Simplified 42.3%

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

   6. Taylor expanded in x around inf 39.1%

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

      1. associate-*r/ 39.1%

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

      2. neg-mul-1 39.1%

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

   8. Simplified 39.1%

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

   if -1.99999999e-8 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 44.1%

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

4. Final simplification 42.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.999999987845058 \cdot 10^{-8}:\\ \;\;\;\;\frac{s}{-x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 46.7% accurate, 13.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.800000054037355 \cdot 10^{-8}:\\ \;\;\;\;\frac{s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -2.800000054037355e-8) (/ s x) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -2.800000054037355e-8f) {
		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 <= (-2.800000054037355e-8)) 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(-2.800000054037355e-8))
		tmp = Float32(s / x);
	else
		tmp = Float32(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-2.800000054037355e-8))
		tmp = s / x;
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -2.80000005e-8

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 42.8%

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

      1. neg-mul-1 42.8%

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

      2. unsub-neg 42.8%

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

   5. Simplified 42.8%

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

   6. Taylor expanded in x around inf 39.5%

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

      1. associate-*r/ 39.5%

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

      2. neg-mul-1 39.5%

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

   8. Simplified 39.5%

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

      1. add-sqr-sqrt -0.0%

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

      2. sqrt-unprod 58.4%

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

      3. sqr-neg 58.4%

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

      4. sqrt-unprod 39.2%

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

      5. add-sqr-sqrt 39.2%

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

      6. div-inv 39.2%

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

   10. Applied egg-rr 39.2%

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

       1. associate-*r/ 39.2%

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

       2. *-rgt-identity 39.2%

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

   12. Simplified 39.2%

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

   if -2.80000005e-8 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 43.9%

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

Alternative 16: 34.9% 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 33.7%

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

Reproduce

herbie shell --seed 2024181 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))