Logistic function

Percentage Accurate: 99.8% → 99.9%

Time: 12.2s

Alternatives: 15

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 83.4%

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

   3. neg-mul-1 83.4%

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

   4. exp-prod 83.4%

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

   5. pow-pow 99.7%

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

   6. div-inv 99.7%

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

4. Applied egg-rr 99.7%

   \[\leadsto \frac{1}{1 + \color{blue}{{\left(e^{-1}\right)}^{\left(\frac{x}{s}\right)}}} \]
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 99.7%

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

   3. log1p-expm1-u 99.7%

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

   4. log1p-define 99.8%

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

   5. expm1-log1p-u 99.8%

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

   6. pow-exp 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, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Final simplification 99.8%

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

Alternative 3: 58.0% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{-s}\\ t_1 := 2 + \frac{x}{s}\\ \mathbf{if}\;t\_0 \leq -10:\\ \;\;\;\;\frac{1}{x \cdot \frac{2}{x}}\\ \mathbf{elif}\;t\_0 \leq 1.0000000150474662 \cdot 10^{+29}:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{s} \cdot \frac{x}{s}}{t\_1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x \cdot t\_1}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ x (- s))) (t_1 (+ 2.0 (/ x s))))
   (if (<= t_0 -10.0)
     (/ 1.0 (* x (/ 2.0 x)))
     (if (<= t_0 1.0000000150474662e+29)
       (/ 1.0 (/ (- 4.0 (* (/ x s) (/ x s))) t_1))
       (/ 1.0 (/ (* x t_1) x))))))

C

float code(float x, float s) {
	float t_0 = x / -s;
	float t_1 = 2.0f + (x / s);
	float tmp;
	if (t_0 <= -10.0f) {
		tmp = 1.0f / (x * (2.0f / x));
	} else if (t_0 <= 1.0000000150474662e+29f) {
		tmp = 1.0f / ((4.0f - ((x / s) * (x / s))) / t_1);
	} else {
		tmp = 1.0f / ((x * t_1) / x);
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: t_1
    real(4) :: tmp
    t_0 = x / -s
    t_1 = 2.0e0 + (x / s)
    if (t_0 <= (-10.0e0)) then
        tmp = 1.0e0 / (x * (2.0e0 / x))
    else if (t_0 <= 1.0000000150474662e+29) then
        tmp = 1.0e0 / ((4.0e0 - ((x / s) * (x / s))) / t_1)
    else
        tmp = 1.0e0 / ((x * t_1) / x)
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = Float32(x / Float32(-s))
	t_1 = Float32(Float32(2.0) + Float32(x / s))
	tmp = Float32(0.0)
	if (t_0 <= Float32(-10.0))
		tmp = Float32(Float32(1.0) / Float32(x * Float32(Float32(2.0) / x)));
	elseif (t_0 <= Float32(1.0000000150474662e+29))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(4.0) - Float32(Float32(x / s) * Float32(x / s))) / t_1));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(x * t_1) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = x / -s;
	t_1 = single(2.0) + (x / s);
	tmp = single(0.0);
	if (t_0 <= single(-10.0))
		tmp = single(1.0) / (x * (single(2.0) / x));
	elseif (t_0 <= single(1.0000000150474662e+29))
		tmp = single(1.0) / ((single(4.0) - ((x / s) * (x / s))) / t_1);
	else
		tmp = single(1.0) / ((x * t_1) / x);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if (/.f32 (neg.f32 x) s) < -10

   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 -10 < (/.f32 (neg.f32 x) s) < 1.00000002e29

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0 61.3%

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

      1. mul-1-neg 61.3%

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

      2. unsub-neg 61.3%

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

   5. Simplified 61.3%

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

      1. sub-neg 61.3%

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

      2. neg-mul-1 61.3%

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

      3. rem-log-exp 92.6%

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

      4. pow-exp 92.6%

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

      5. flip-+ 58.2%

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

      6. metadata-eval 58.2%

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

      7. pow-exp 58.2%

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

      8. rem-log-exp 58.2%

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

      9. neg-mul-1 58.2%

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

      10. pow-exp 58.2%

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

      11. rem-log-exp 58.9%

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

      12. neg-mul-1 58.9%

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

      13. distribute-neg-frac2 58.9%

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

      14. distribute-neg-frac2 58.9%

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

      15. pow-exp 58.9%

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

      16. rem-log-exp 73.8%

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

      17. neg-mul-1 73.8%

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

      18. distribute-neg-frac2 73.8%

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

   7. Applied egg-rr 73.8%

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

   if 1.00000002e29 < (/.f32 (neg.f32 x) s)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 86.3%

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

      1. mul-1-neg 86.3%

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

      2. unsub-neg 86.3%

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

   5. Simplified 86.3%

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

   6. Taylor expanded in x around inf 86.3%

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

      1. associate-*r/ 86.3%

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

      2. metadata-eval 86.3%

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

   8. Simplified 86.3%

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

   9. Taylor expanded in x around 0 86.3%

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

       1. mul-1-neg 86.3%

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

       2. unsub-neg 86.3%

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

   11. Simplified 86.3%

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

       1. associate-*r/ 100.0%

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

       2. sub-neg 100.0%

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

       3. distribute-frac-neg2 100.0%

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

       4. add-sqr-sqrt -0.0%

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

       5. sqrt-unprod 100.0%

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

       6. sqr-neg 100.0%

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

       7. sqrt-unprod 100.0%

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

       8. add-sqr-sqrt 100.0%

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

   13. Applied egg-rr 100.0%

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

4. Final simplification 60.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq -10:\\ \;\;\;\;\frac{1}{x \cdot \frac{2}{x}}\\ \mathbf{elif}\;\frac{x}{-s} \leq 1.0000000150474662 \cdot 10^{+29}:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{s} \cdot \frac{x}{s}}{2 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x \cdot \left(2 + \frac{x}{s}\right)}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 55.6% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{-s}\\ \mathbf{if}\;t\_0 \leq 10:\\ \;\;\;\;0.5\\ \mathbf{elif}\;t\_0 \leq 1.9999999556392617 \cdot 10^{+23}:\\ \;\;\;\;\frac{1}{x \cdot \frac{s + x \cdot 0.5}{x \cdot \left(s \cdot 0.5\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x \cdot \left(2 + \frac{x}{s}\right)}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ x (- s))))
   (if (<= t_0 10.0)
     0.5
     (if (<= t_0 1.9999999556392617e+23)
       (/ 1.0 (* x (/ (+ s (* x 0.5)) (* x (* s 0.5)))))
       (/ 1.0 (/ (* x (+ 2.0 (/ x s))) x))))))

C

float code(float x, float s) {
	float t_0 = x / -s;
	float tmp;
	if (t_0 <= 10.0f) {
		tmp = 0.5f;
	} else if (t_0 <= 1.9999999556392617e+23f) {
		tmp = 1.0f / (x * ((s + (x * 0.5f)) / (x * (s * 0.5f))));
	} else {
		tmp = 1.0f / ((x * (2.0f + (x / s))) / x);
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: tmp
    t_0 = x / -s
    if (t_0 <= 10.0e0) then
        tmp = 0.5e0
    else if (t_0 <= 1.9999999556392617e+23) then
        tmp = 1.0e0 / (x * ((s + (x * 0.5e0)) / (x * (s * 0.5e0))))
    else
        tmp = 1.0e0 / ((x * (2.0e0 + (x / s))) / x)
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = Float32(x / Float32(-s))
	tmp = Float32(0.0)
	if (t_0 <= Float32(10.0))
		tmp = Float32(0.5);
	elseif (t_0 <= Float32(1.9999999556392617e+23))
		tmp = Float32(Float32(1.0) / Float32(x * Float32(Float32(s + Float32(x * Float32(0.5))) / Float32(x * Float32(s * Float32(0.5))))));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(x * Float32(Float32(2.0) + Float32(x / s))) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = x / -s;
	tmp = single(0.0);
	if (t_0 <= single(10.0))
		tmp = single(0.5);
	elseif (t_0 <= single(1.9999999556392617e+23))
		tmp = single(1.0) / (x * ((s + (x * single(0.5))) / (x * (s * single(0.5)))));
	else
		tmp = single(1.0) / ((x * (single(2.0) + (x / s))) / x);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if (/.f32 (neg.f32 x) s) < 10

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 53.7%

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

   if 10 < (/.f32 (neg.f32 x) s) < 1.99999996e23

   1. Initial program 99.2%

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

   3. Taylor expanded in x around 0 8.1%

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

      1. mul-1-neg 8.1%

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

      2. unsub-neg 8.1%

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

   5. Simplified 8.1%

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

   6. Taylor expanded in x around inf 8.1%

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

      1. associate-*r/ 8.1%

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

      2. metadata-eval 8.1%

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

   8. Simplified 8.1%

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

      1. frac-sub 34.2%

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

      2. sub-neg 34.2%

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

      3. *-rgt-identity 34.2%

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

      4. add-sqr-sqrt 34.2%

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

      5. sqrt-unprod 34.2%

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

      6. sqr-neg 34.2%

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

      7. sqrt-unprod -0.0%

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

      8. add-sqr-sqrt 34.1%

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

      9. *-rgt-identity 34.1%

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

      10. frac-add 8.0%

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

      11. clear-num 8.0%

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

      12. frac-add 34.2%

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

      13. *-un-lft-identity 34.2%

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

      14. div-inv 34.2%

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

      15. metadata-eval 34.2%

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

      16. div-inv 34.2%

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

      17. metadata-eval 34.2%

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

   10. Applied egg-rr 34.2%

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

       1. *-rgt-identity 34.2%

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

       2. associate-*l* 34.2%

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

   12. Simplified 34.2%

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

   if 1.99999996e23 < (/.f32 (neg.f32 x) s)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 68.3%

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

      1. mul-1-neg 68.3%

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

      2. unsub-neg 68.3%

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

   5. Simplified 68.3%

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

   6. Taylor expanded in x around inf 68.3%

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

      1. associate-*r/ 68.3%

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

      2. metadata-eval 68.3%

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

   8. Simplified 68.3%

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

   9. Taylor expanded in x around 0 68.3%

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

       1. mul-1-neg 68.3%

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

       2. unsub-neg 68.3%

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

   11. Simplified 68.3%

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

       1. associate-*r/ 85.8%

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

       2. sub-neg 85.8%

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

       3. distribute-frac-neg2 85.8%

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

       4. add-sqr-sqrt -0.0%

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

       5. sqrt-unprod 94.7%

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

       6. sqr-neg 94.7%

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

       7. sqrt-unprod 85.8%

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

       8. add-sqr-sqrt 85.8%

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

   13. Applied egg-rr 85.8%

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

4. Final simplification 57.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq 10:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\frac{x}{-s} \leq 1.9999999556392617 \cdot 10^{+23}:\\ \;\;\;\;\frac{1}{x \cdot \frac{s + x \cdot 0.5}{x \cdot \left(s \cdot 0.5\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x \cdot \left(2 + \frac{x}{s}\right)}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 55.5% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{-s}\\ \mathbf{if}\;t\_0 \leq 10:\\ \;\;\;\;0.5\\ \mathbf{elif}\;t\_0 \leq 4.999999889098154 \cdot 10^{+21}:\\ \;\;\;\;\frac{1}{x \cdot \frac{x + s \cdot -2}{x \cdot s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x \cdot \left(2 + \frac{x}{s}\right)}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ x (- s))))
   (if (<= t_0 10.0)
     0.5
     (if (<= t_0 4.999999889098154e+21)
       (/ 1.0 (* x (/ (+ x (* s -2.0)) (* x s))))
       (/ 1.0 (/ (* x (+ 2.0 (/ x s))) x))))))

C

float code(float x, float s) {
	float t_0 = x / -s;
	float tmp;
	if (t_0 <= 10.0f) {
		tmp = 0.5f;
	} else if (t_0 <= 4.999999889098154e+21f) {
		tmp = 1.0f / (x * ((x + (s * -2.0f)) / (x * s)));
	} else {
		tmp = 1.0f / ((x * (2.0f + (x / s))) / x);
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: tmp
    t_0 = x / -s
    if (t_0 <= 10.0e0) then
        tmp = 0.5e0
    else if (t_0 <= 4.999999889098154e+21) then
        tmp = 1.0e0 / (x * ((x + (s * (-2.0e0))) / (x * s)))
    else
        tmp = 1.0e0 / ((x * (2.0e0 + (x / s))) / x)
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = Float32(x / Float32(-s))
	tmp = Float32(0.0)
	if (t_0 <= Float32(10.0))
		tmp = Float32(0.5);
	elseif (t_0 <= Float32(4.999999889098154e+21))
		tmp = Float32(Float32(1.0) / Float32(x * Float32(Float32(x + Float32(s * Float32(-2.0))) / Float32(x * s))));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(x * Float32(Float32(2.0) + Float32(x / s))) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = x / -s;
	tmp = single(0.0);
	if (t_0 <= single(10.0))
		tmp = single(0.5);
	elseif (t_0 <= single(4.999999889098154e+21))
		tmp = single(1.0) / (x * ((x + (s * single(-2.0))) / (x * s)));
	else
		tmp = single(1.0) / ((x * (single(2.0) + (x / s))) / x);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if (/.f32 (neg.f32 x) s) < 10

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 53.7%

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

   if 10 < (/.f32 (neg.f32 x) s) < 4.99999989e21

   1. Initial program 99.1%

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

   3. Taylor expanded in x around 0 7.9%

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

      1. mul-1-neg 7.9%

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

      2. unsub-neg 7.9%

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

   5. Simplified 7.9%

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

   6. Taylor expanded in x around inf 7.9%

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

      1. associate-*r/ 7.9%

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

      2. metadata-eval 7.9%

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

   8. Simplified 7.9%

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

      1. frac-sub 36.6%

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

      2. sub-neg 36.6%

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

      3. *-rgt-identity 36.6%

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

      4. add-sqr-sqrt 36.6%

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

      5. sqrt-unprod 36.6%

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

      6. sqr-neg 36.6%

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

      7. sqrt-unprod -0.0%

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

      8. add-sqr-sqrt 36.6%

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

      9. *-rgt-identity 36.6%

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

      10. frac-add 7.8%

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

      11. frac-2neg 7.8%

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

      12. frac-2neg 7.8%

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

      13. metadata-eval 7.8%

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

      14. frac-add 36.6%

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

   10. Applied egg-rr 36.6%

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

   if 4.99999989e21 < (/.f32 (neg.f32 x) s)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 65.0%

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

      1. mul-1-neg 65.0%

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

      2. unsub-neg 65.0%

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

   5. Simplified 65.0%

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

   6. Taylor expanded in x around inf 65.0%

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

      1. associate-*r/ 65.0%

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

      2. metadata-eval 65.0%

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

   8. Simplified 65.0%

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

   9. Taylor expanded in x around 0 65.0%

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

       1. mul-1-neg 65.0%

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

       2. unsub-neg 65.0%

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

   11. Simplified 65.0%

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

       1. associate-*r/ 81.6%

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

       2. sub-neg 81.6%

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

       3. distribute-frac-neg2 81.6%

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

       4. add-sqr-sqrt -0.0%

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

       5. sqrt-unprod 91.6%

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

       6. sqr-neg 91.6%

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

       7. sqrt-unprod 81.6%

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

       8. add-sqr-sqrt 81.6%

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

   13. Applied egg-rr 81.6%

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

4. Final simplification 57.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq 10:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\frac{x}{-s} \leq 4.999999889098154 \cdot 10^{+21}:\\ \;\;\;\;\frac{1}{x \cdot \frac{x + s \cdot -2}{x \cdot s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x \cdot \left(2 + \frac{x}{s}\right)}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 52.5% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq 100:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x \cdot \left(2 + \frac{x}{s}\right)}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ x (- s)) 100.0) 0.5 (/ 1.0 (/ (* x (+ 2.0 (/ x s))) x))))

C

float code(float x, float s) {
	float tmp;
	if ((x / -s) <= 100.0f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / ((x * (2.0f + (x / s))) / x);
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((x / -s) <= 100.0e0) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 / ((x * (2.0e0 + (x / s))) / x)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(x / Float32(-s)) <= Float32(100.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(x * Float32(Float32(2.0) + Float32(x / s))) / x));
	end
	return tmp
end

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 100

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 53.2%

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

   if 100 < (/.f32 (neg.f32 x) s)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 45.7%

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

      1. mul-1-neg 45.7%

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

      2. unsub-neg 45.7%

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

   5. Simplified 45.7%

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

   6. Taylor expanded in x around inf 45.7%

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

      1. associate-*r/ 45.7%

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

      2. metadata-eval 45.7%

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

   8. Simplified 45.7%

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

   9. Taylor expanded in x around 0 45.7%

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

       1. mul-1-neg 45.7%

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

       2. unsub-neg 45.7%

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

   11. Simplified 45.7%

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

       1. associate-*r/ 56.7%

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

       2. sub-neg 56.7%

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

       3. distribute-frac-neg2 56.7%

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

       4. add-sqr-sqrt -0.0%

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

       5. sqrt-unprod 78.3%

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

       6. sqr-neg 78.3%

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

       7. sqrt-unprod 56.7%

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

       8. add-sqr-sqrt 56.7%

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

   13. Applied egg-rr 56.7%

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

4. Final simplification 54.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq 100:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x \cdot \left(2 + \frac{x}{s}\right)}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 48.7% accurate, 6.3× speedup

Math

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

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ x (- s)) -10.0)
   (/ 1.0 (* x (/ 2.0 x)))
   (/ -1.0 (/ (- x (* s 2.0)) s))))

C

float code(float x, float s) {
	float tmp;
	if ((x / -s) <= -10.0f) {
		tmp = 1.0f / (x * (2.0f / x));
	} else {
		tmp = -1.0f / ((x - (s * 2.0f)) / s);
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((x / -s) <= (-10.0e0)) then
        tmp = 1.0e0 / (x * (2.0e0 / x))
    else
        tmp = (-1.0e0) / ((x - (s * 2.0e0)) / s)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(x / Float32(-s)) <= Float32(-10.0))
		tmp = Float32(Float32(1.0) / Float32(x * Float32(Float32(2.0) / x)));
	else
		tmp = Float32(Float32(-1.0) / Float32(Float32(x - Float32(s * Float32(2.0))) / s));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((x / -s) <= single(-10.0))
		tmp = single(1.0) / (x * (single(2.0) / x));
	else
		tmp = single(-1.0) / ((x - (s * single(2.0))) / s);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < -10

   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 -10 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 67.3%

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

      1. mul-1-neg 67.3%

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

      2. unsub-neg 67.3%

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

   5. Simplified 67.3%

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

   6. Taylor expanded in s around 0 67.3%

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

4. Final simplification 52.6%

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

Alternative 8: 47.5% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq 60:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{s} \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ x (- s)) 60.0) 0.5 (/ 1.0 (* (/ x s) 3.0))))

C

float code(float x, float s) {
	float tmp;
	if ((x / -s) <= 60.0f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / ((x / s) * 3.0f);
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((x / -s) <= 60.0e0) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 / ((x / s) * 3.0e0)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(x / Float32(-s)) <= Float32(60.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(x / s) * Float32(3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((x / -s) <= single(60.0))
		tmp = single(0.5);
	else
		tmp = single(1.0) / ((x / s) * single(3.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 60

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 53.5%

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

   if 60 < (/.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 45.3%

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

      1. mul-1-neg 45.3%

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

      2. unsub-neg 45.3%

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

   5. Simplified 45.3%

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

   6. Taylor expanded in x around inf 45.3%

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

      1. associate-*r/ 45.3%

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

      2. metadata-eval 45.3%

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

   8. Simplified 45.3%

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

      1. frac-2neg 45.3%

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

      2. div-inv 45.3%

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

      3. add-sqr-sqrt 45.3%

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

      4. sqrt-unprod 48.7%

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

      5. sqr-neg 48.7%

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

      6. sqrt-unprod -0.0%

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

      7. add-sqr-sqrt 45.3%

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

      8. *-un-lft-identity 45.3%

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

      9. prod-diff 45.3%

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

      10. metadata-eval 45.3%

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

      11. associate-/r/ 45.3%

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

      12. clear-num 45.3%

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

      13. distribute-frac-neg2 45.3%

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

      14. add-sqr-sqrt -0.0%

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

      15. sqrt-unprod 69.1%

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

      16. sqr-neg 69.1%

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

      17. sqrt-unprod 45.2%

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

      18. add-sqr-sqrt 45.2%

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

   10. Applied egg-rr 48.7%

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

       1. +-commutative 48.7%

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

       2. fma-undefine 48.7%

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

       3. *-rgt-identity 48.7%

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

       4. associate-+l+ 48.7%

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

       5. fma-undefine 48.7%

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

       6. *-rgt-identity 48.7%

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

       7. metadata-eval 48.7%

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

       8. distribute-rgt-neg-in 48.7%

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

       9. unsub-neg 48.7%

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

       10. metadata-eval 48.7%

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

       11. distribute-lft-neg-in 48.7%

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

       12. associate-*r/ 48.7%

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

       13. metadata-eval 48.7%

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

       14. distribute-neg-frac 48.7%

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

       15. metadata-eval 48.7%

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

       16. associate-*l/ 48.7%

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

       17. metadata-eval 48.7%

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

   12. Simplified 48.7%

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

   13. Taylor expanded in x around inf 48.7%

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

4. Final simplification 52.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq 60:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{s} \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 48.7% accurate, 7.2× speedup

Math

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

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ x (- s)) -10.0) (/ 1.0 (* x (/ 2.0 x))) (/ 1.0 (- 2.0 (/ x s)))))

C

float code(float x, float s) {
	float tmp;
	if ((x / -s) <= -10.0f) {
		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) <= (-10.0e0)) 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(-10.0))
		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(-10.0))
		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) < -10

   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 -10 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 67.3%

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

      1. mul-1-neg 67.3%

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

      2. unsub-neg 67.3%

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

   5. Simplified 67.3%

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

4. Final simplification 52.6%

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

Alternative 10: 47.3% accurate, 8.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq 0.20000000298023224:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

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

C

float code(float x, float s) {
	float tmp;
	if ((x / -s) <= 0.20000000298023224f) {
		tmp = 0.5f;
	} else {
		tmp = -1.0f / (x / s);
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((x / -s) <= 0.20000000298023224e0) 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(0.20000000298023224))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(-1.0) / Float32(x / s));
	end
	return tmp
end

MATLAB

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 54.1%

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

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

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 44.2%

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

      1. mul-1-neg 44.2%

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

      2. unsub-neg 44.2%

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

   5. Simplified 44.2%

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

   6. Taylor expanded in x around inf 44.2%

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

      1. mul-1-neg 44.2%

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

      2. distribute-neg-frac 44.2%

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

   8. Simplified 44.2%

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

4. Final simplification 50.9%

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

Alternative 11: 46.0% accurate, 10.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;0.3333333333333333 \cdot \frac{s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -1.0) (* 0.3333333333333333 (/ s x)) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -1.0f) {
		tmp = 0.3333333333333333f * (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 <= (-1.0e0)) then
        tmp = 0.3333333333333333e0 * (s / x)
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-1.0))
		tmp = Float32(Float32(0.3333333333333333) * 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(-1.0))
		tmp = single(0.3333333333333333) * (s / x);
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -1

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 61.7%

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

      1. mul-1-neg 61.7%

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

      2. unsub-neg 61.7%

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

   5. Simplified 61.7%

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

   6. Taylor expanded in x around inf 61.7%

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

      1. associate-*r/ 61.7%

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

      2. metadata-eval 61.7%

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

   8. Simplified 61.7%

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

      1. frac-2neg 61.7%

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

      2. div-inv 61.7%

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

      3. add-sqr-sqrt 61.7%

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

      4. sqrt-unprod 61.7%

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

      5. sqr-neg 61.7%

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

      6. sqrt-unprod -0.0%

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

      7. add-sqr-sqrt 61.7%

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

      8. *-un-lft-identity 61.7%

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

      9. prod-diff 61.7%

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

      10. metadata-eval 61.7%

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

      11. associate-/r/ 61.7%

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

      12. clear-num 61.7%

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

      13. distribute-frac-neg2 61.7%

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

      14. add-sqr-sqrt -0.0%

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

      15. sqrt-unprod 65.3%

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

      16. sqr-neg 65.3%

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

      17. sqrt-unprod 61.7%

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

      18. add-sqr-sqrt 61.7%

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

   10. Applied egg-rr 66.7%

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

       1. +-commutative 66.7%

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

       2. fma-undefine 66.7%

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

       3. *-rgt-identity 66.7%

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

       4. associate-+l+ 66.7%

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

       5. fma-undefine 66.7%

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

       6. *-rgt-identity 66.7%

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

       7. metadata-eval 66.7%

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

       8. distribute-rgt-neg-in 66.7%

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

       9. unsub-neg 66.7%

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

       10. metadata-eval 66.7%

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

       11. distribute-lft-neg-in 66.7%

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

       12. associate-*r/ 66.7%

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

       13. metadata-eval 66.7%

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

       14. distribute-neg-frac 66.7%

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

       15. metadata-eval 66.7%

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

       16. associate-*l/ 66.7%

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

       17. metadata-eval 66.7%

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

   12. Simplified 66.7%

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

   13. Taylor expanded in x around inf 55.1%

       \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{s}{x}} \]

   if -1 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 47.6%

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

4. Final simplification 49.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;0.3333333333333333 \cdot \frac{s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 47.0% accurate, 10.8× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	float tmp;
	if (x <= -1.0f) {
		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 <= (-1.0e0)) 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(-1.0))
		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(-1.0))
		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 < -1

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 61.7%

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

      1. mul-1-neg 61.7%

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

      2. unsub-neg 61.7%

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

   5. Simplified 61.7%

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

   6. Taylor expanded in x around inf 61.7%

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

      1. associate-*r/ 61.7%

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

      2. metadata-eval 61.7%

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

   8. Simplified 61.7%

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

      1. frac-sub 61.7%

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

      2. sub-neg 61.7%

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

      3. *-rgt-identity 61.7%

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

      4. add-sqr-sqrt 61.7%

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

      5. sqrt-unprod 68.9%

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

      6. sqr-neg 68.9%

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

      7. sqrt-unprod -0.0%

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

      8. add-sqr-sqrt 61.7%

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

      9. *-rgt-identity 61.7%

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

      10. frac-add 61.7%

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

      11. frac-2neg 61.7%

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

      12. frac-2neg 61.7%

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

      13. metadata-eval 61.7%

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

      14. frac-add 61.7%

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

   10. Applied egg-rr 61.7%

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

   11. Taylor expanded in x around inf 61.7%

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

   if -1 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 47.6%

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

4. Final simplification 50.6%

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

Alternative 13: 45.8% accurate, 12.0× speedup

Math

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

FPCore

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

C

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

Derivation

1. Split input into 2 regimes

2. if x < -4.99999987e-5

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 59.2%

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

      1. mul-1-neg 59.2%

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

      2. unsub-neg 59.2%

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

   5. Simplified 59.2%

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

   6. Taylor expanded in x around inf 51.4%

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

      1. associate-*r/ 51.4%

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

      2. neg-mul-1 51.4%

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

   8. Simplified 51.4%

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

   if -4.99999987e-5 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 48.1%

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

4. Final simplification 48.9%

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

Alternative 14: 45.7% accurate, 13.4× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	float tmp;
	if (x <= -1.0f) {
		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 <= (-1.0e0)) 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(-1.0))
		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(-1.0))
		tmp = s / x;
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -1

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 61.7%

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

      1. mul-1-neg 61.7%

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

      2. unsub-neg 61.7%

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

   5. Simplified 61.7%

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

   6. Taylor expanded in x around inf 61.7%

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

      1. associate-*r/ 61.7%

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

      2. metadata-eval 61.7%

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

   8. Simplified 61.7%

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

      1. frac-sub 61.7%

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

      2. sub-neg 61.7%

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

      3. *-rgt-identity 61.7%

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

      4. add-sqr-sqrt 61.7%

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

      5. sqrt-unprod 68.9%

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

      6. sqr-neg 68.9%

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

      7. sqrt-unprod -0.0%

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

      8. add-sqr-sqrt 61.7%

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

      9. *-rgt-identity 61.7%

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

      10. frac-add 61.7%

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

      11. frac-2neg 61.7%

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

      12. frac-2neg 61.7%

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

      13. metadata-eval 61.7%

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

      14. frac-add 61.7%

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

   10. Applied egg-rr 61.7%

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

   11. Taylor expanded in x around inf 53.4%

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

   if -1 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 47.6%

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

4. Final simplification 48.9%

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

Alternative 15: 34.7% accurate, 108.0× speedup

Math

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

FPCore

(FPCore (x s) :precision binary32 0.5)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in x around 0 38.5%

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

4. Final simplification 38.5%

   \[\leadsto 0.5 \]
5. Add Preprocessing

Reproduce

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