Logistic function

Percentage Accurate: 99.8% → 99.8%

Time: 12.8s

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: 99.8% 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 99.9%

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

   1. div-inv 99.9%

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

   2. exp-prod 80.8%

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

   3. neg-mul-1 80.8%

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

   4. exp-prod 80.8%

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

   5. pow-pow 99.9%

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

   6. div-inv 100.0%

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

4. Applied egg-rr 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, s)
	return Float32(Float32(1.0) / Float32(Float32(1.0) + (Float32(exp(1)) ^ Float32(Float32(-x) / s))))
end

MATLAB

function tmp = code(x, s)
	tmp = single(1.0) / (single(1.0) + (single(2.71828182845904523536) ^ (-x / s)));
end

Derivation

1. Initial program 99.9%

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

   1. distribute-frac-neg 99.9%

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

   2. exp-neg 99.9%

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

4. Applied egg-rr 99.9%

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

   1. *-un-lft-identity 99.9%

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

   2. exp-prod 99.9%

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

6. Applied egg-rr 99.9%

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

   1. exp-1-e 99.9%

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

8. Simplified 99.9%

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

   1. inv-pow 99.9%

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

   2. pow-pow 100.0%

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

10. Applied egg-rr 100.0%

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

    1. *-commutative 100.0%

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

    2. mul-1-neg 100.0%

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

    3. distribute-neg-frac 100.0%

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

12. Simplified 100.0%

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

13. Final simplification 100.0%

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

Alternative 3: 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

Derivation

1. Initial program 99.9%

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

3. Final simplification 99.9%

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

Alternative 4: 86.1% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq 0.10000000149011612:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{elif}\;t_0 \leq 1.9999999360571385 \cdot 10^{+38}:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{s \cdot \frac{s}{x}}}{\frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)))
   (if (<= t_0 0.10000000149011612)
     (/ 1.0 (+ 1.0 (/ 1.0 (+ 1.0 (/ x s)))))
     (if (<= t_0 1.9999999360571385e+38)
       (/ 1.0 (/ (- 4.0 (/ x (* s (/ s x)))) (/ x s)))
       (/ 1.0 (/ x s))))))

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if (t_0 <= 0.10000000149011612f) {
		tmp = 1.0f / (1.0f + (1.0f / (1.0f + (x / s))));
	} else if (t_0 <= 1.9999999360571385e+38f) {
		tmp = 1.0f / ((4.0f - (x / (s * (s / x)))) / (x / s));
	} 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) :: t_0
    real(4) :: tmp
    t_0 = -x / s
    if (t_0 <= 0.10000000149011612e0) then
        tmp = 1.0e0 / (1.0e0 + (1.0e0 / (1.0e0 + (x / s))))
    else if (t_0 <= 1.9999999360571385e+38) then
        tmp = 1.0e0 / ((4.0e0 - (x / (s * (s / x)))) / (x / s))
    else
        tmp = 1.0e0 / (x / s)
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = Float32(Float32(-x) / s)
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.10000000149011612))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(x / s)))));
	elseif (t_0 <= Float32(1.9999999360571385e+38))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(4.0) - Float32(x / Float32(s * Float32(s / x)))) / Float32(x / s)));
	else
		tmp = Float32(Float32(1.0) / Float32(x / s));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = -x / s;
	tmp = single(0.0);
	if (t_0 <= single(0.10000000149011612))
		tmp = single(1.0) / (single(1.0) + (single(1.0) / (single(1.0) + (x / s))));
	elseif (t_0 <= single(1.9999999360571385e+38))
		tmp = single(1.0) / ((single(4.0) - (x / (s * (s / x)))) / (x / s));
	else
		tmp = single(1.0) / (x / s);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if (/.f32 (neg.f32 x) s) < 0.100000001

   1. Initial program 99.9%

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

      1. distribute-frac-neg 99.9%

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

      2. exp-neg 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0 94.6%

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

   if 0.100000001 < (/.f32 (neg.f32 x) s) < 1.99999994e38

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 11.2%

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

      1. mul-1-neg 11.2%

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

      2. unsub-neg 11.2%

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

   5. Simplified 11.2%

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

      1. *-un-lft-identity 11.2%

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

      2. cancel-sign-sub-inv 11.2%

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

      3. metadata-eval 11.2%

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

      4. add-log-exp 97.9%

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

      5. pow-exp 97.9%

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

      6. flip-+ 0.8%

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

      7. metadata-eval 0.8%

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

      8. pow-exp 0.8%

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

      9. add-log-exp 0.8%

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

      10. neg-mul-1 0.8%

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

      11. pow-exp 0.8%

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

      12. add-log-exp 2.2%

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

      13. neg-mul-1 2.2%

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

      14. distribute-neg-frac 2.2%

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

      15. distribute-neg-frac 2.2%

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

      16. pow-exp 2.2%

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

   7. Applied egg-rr 56.4%

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

   8. Taylor expanded in x around inf 56.0%

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

      1. distribute-frac-neg 56.0%

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

      2. distribute-frac-neg 56.0%

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

      3. sqr-neg 56.0%

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

      4. clear-num 56.0%

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

      5. frac-times 62.5%

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

      6. *-un-lft-identity 62.5%

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

   10. Applied egg-rr 62.5%

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

   if 1.99999994e38 < (/.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 0 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf 86.6%

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

      1. associate-*r/ 86.6%

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

      2. neg-mul-1 86.6%

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

   8. Simplified 86.6%

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

      1. remove-double-neg 86.6%

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

      2. frac-2neg 86.6%

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

      3. clear-num 100.0%

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

      4. inv-pow 100.0%

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

      5. add-sqr-sqrt 100.0%

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

      6. sqrt-unprod 100.0%

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

      7. sqr-neg 100.0%

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

      8. sqrt-unprod -0.0%

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

      9. add-sqr-sqrt 100.0%

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

   10. Applied egg-rr 100.0%

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

       1. unpow-1 100.0%

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

   12. Simplified 100.0%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 0.10000000149011612:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{elif}\;\frac{-x}{s} \leq 1.9999999360571385 \cdot 10^{+38}:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{s \cdot \frac{s}{x}}}{\frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 90.4% accurate, 3.7× speedup

Math

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

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) 5.0)
   (/ 1.0 (+ 1.0 (/ 1.0 (+ 1.0 (/ x s)))))
   (/ 1.0 (+ 2.0 (- (* 0.5 (* x (* x (* (/ 1.0 s) (/ 1.0 s))))) (/ x s))))))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= 5.0f) {
		tmp = 1.0f / (1.0f + (1.0f / (1.0f + (x / s))));
	} else {
		tmp = 1.0f / (2.0f + ((0.5f * (x * (x * ((1.0f / s) * (1.0f / s))))) - (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) <= 5.0e0) then
        tmp = 1.0e0 / (1.0e0 + (1.0e0 / (1.0e0 + (x / s))))
    else
        tmp = 1.0e0 / (2.0e0 + ((0.5e0 * (x * (x * ((1.0e0 / s) * (1.0e0 / s))))) - (x / s)))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(-x) / s) <= Float32(5.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(x / s)))));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(Float32(Float32(0.5) * Float32(x * Float32(x * Float32(Float32(Float32(1.0) / s) * Float32(Float32(1.0) / s))))) - Float32(x / s))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((-x / s) <= single(5.0))
		tmp = single(1.0) / (single(1.0) + (single(1.0) / (single(1.0) + (x / s))));
	else
		tmp = single(1.0) / (single(2.0) + ((single(0.5) * (x * (x * ((single(1.0) / s) * (single(1.0) / s))))) - (x / s)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

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

      1. distribute-frac-neg 99.9%

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

      2. exp-neg 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0 93.6%

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

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

   1. Initial program 100.0%

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

      1. div-inv 100.0%

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

      2. exp-prod 75.6%

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

      3. neg-mul-1 75.6%

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

      4. exp-prod 75.6%

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

      5. pow-pow 100.0%

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

      6. div-inv 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0 82.2%

      \[\leadsto \frac{1}{\color{blue}{2 + \left(-1 \cdot \frac{x}{s} + 0.5 \cdot \frac{{x}^{2}}{{s}^{2}}\right)}} \]
   6. Step-by-step derivation

      1. mul-1-neg 82.2%

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

      2. distribute-frac-neg 82.2%

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

      3. +-commutative 82.2%

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

      4. distribute-frac-neg 82.2%

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

      5. unsub-neg 82.2%

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

      6. unpow2 82.2%

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

      7. unpow2 82.2%

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

      8. times-frac 70.4%

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

      9. unpow2 70.4%

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

   7. Simplified 70.4%

      \[\leadsto \frac{1}{\color{blue}{2 + \left(0.5 \cdot {\left(\frac{x}{s}\right)}^{2} - \frac{x}{s}\right)}} \]
   8. Step-by-step derivation

      1. unpow2 70.4%

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

      2. sqr-neg 70.4%

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

      3. distribute-frac-neg 70.4%

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

      4. distribute-frac-neg 70.4%

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

      5. associate-*l/ 70.4%

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

      6. *-un-lft-identity 70.4%

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

      7. times-frac 79.6%

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

      8. add-sqr-sqrt 79.6%

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

      9. sqrt-unprod 78.5%

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

      10. sqr-neg 78.5%

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

      11. sqrt-unprod -0.0%

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

      12. add-sqr-sqrt 51.4%

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

      13. add-sqr-sqrt 51.4%

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

      14. sqrt-unprod 50.5%

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

      15. sqr-neg 50.5%

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

      16. sqrt-unprod -0.0%

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

      17. add-sqr-sqrt 79.6%

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

   9. Applied egg-rr 79.6%

      \[\leadsto \frac{1}{2 + \left(0.5 \cdot \color{blue}{\left(\frac{x}{1} \cdot \frac{\frac{x}{s}}{s}\right)} - \frac{x}{s}\right)} \]
   10. Step-by-step derivation

       1. div-inv 79.6%

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

       2. div-inv 79.6%

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

       3. associate-*l* 88.9%

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

   11. Applied egg-rr 88.9%

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

4. Final simplification 91.8%

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

Alternative 6: 90.4% accurate, 4.0× speedup

Math

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

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) 5.0)
   (/ 1.0 (+ 1.0 (/ 1.0 (+ 1.0 (/ x s)))))
   (/ 1.0 (+ 2.0 (- (* 0.5 (* x (* x (/ (/ 1.0 s) s)))) (/ x s))))))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= 5.0f) {
		tmp = 1.0f / (1.0f + (1.0f / (1.0f + (x / s))));
	} else {
		tmp = 1.0f / (2.0f + ((0.5f * (x * (x * ((1.0f / s) / s)))) - (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) <= 5.0e0) then
        tmp = 1.0e0 / (1.0e0 + (1.0e0 / (1.0e0 + (x / s))))
    else
        tmp = 1.0e0 / (2.0e0 + ((0.5e0 * (x * (x * ((1.0e0 / s) / s)))) - (x / s)))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(-x) / s) <= Float32(5.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(x / s)))));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(Float32(Float32(0.5) * Float32(x * Float32(x * Float32(Float32(Float32(1.0) / s) / s)))) - Float32(x / s))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((-x / s) <= single(5.0))
		tmp = single(1.0) / (single(1.0) + (single(1.0) / (single(1.0) + (x / s))));
	else
		tmp = single(1.0) / (single(2.0) + ((single(0.5) * (x * (x * ((single(1.0) / s) / s)))) - (x / s)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

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

      1. distribute-frac-neg 99.9%

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

      2. exp-neg 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0 93.6%

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

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

   1. Initial program 100.0%

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

      1. div-inv 100.0%

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

      2. exp-prod 75.6%

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

      3. neg-mul-1 75.6%

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

      4. exp-prod 75.6%

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

      5. pow-pow 100.0%

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

      6. div-inv 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0 82.2%

      \[\leadsto \frac{1}{\color{blue}{2 + \left(-1 \cdot \frac{x}{s} + 0.5 \cdot \frac{{x}^{2}}{{s}^{2}}\right)}} \]
   6. Step-by-step derivation

      1. mul-1-neg 82.2%

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

      2. distribute-frac-neg 82.2%

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

      3. +-commutative 82.2%

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

      4. distribute-frac-neg 82.2%

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

      5. unsub-neg 82.2%

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

      6. unpow2 82.2%

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

      7. unpow2 82.2%

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

      8. times-frac 70.4%

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

      9. unpow2 70.4%

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

   7. Simplified 70.4%

      \[\leadsto \frac{1}{\color{blue}{2 + \left(0.5 \cdot {\left(\frac{x}{s}\right)}^{2} - \frac{x}{s}\right)}} \]
   8. Step-by-step derivation

      1. unpow2 70.4%

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

      2. sqr-neg 70.4%

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

      3. distribute-frac-neg 70.4%

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

      4. distribute-frac-neg 70.4%

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

      5. associate-*l/ 70.4%

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

      6. *-un-lft-identity 70.4%

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

      7. times-frac 79.6%

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

      8. add-sqr-sqrt 79.6%

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

      9. sqrt-unprod 78.5%

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

      10. sqr-neg 78.5%

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

      11. sqrt-unprod -0.0%

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

      12. add-sqr-sqrt 51.4%

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

      13. add-sqr-sqrt 51.4%

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

      14. sqrt-unprod 50.5%

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

      15. sqr-neg 50.5%

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

      16. sqrt-unprod -0.0%

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

      17. add-sqr-sqrt 79.6%

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

   9. Applied egg-rr 79.6%

      \[\leadsto \frac{1}{2 + \left(0.5 \cdot \color{blue}{\left(\frac{x}{1} \cdot \frac{\frac{x}{s}}{s}\right)} - \frac{x}{s}\right)} \]
   10. Step-by-step derivation

       1. div-inv 79.6%

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

       2. *-un-lft-identity 79.6%

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

       3. times-frac 88.9%

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

       4. /-rgt-identity 88.9%

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

   11. Applied egg-rr 88.9%

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

4. Final simplification 91.8%

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

Alternative 7: 86.2% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < -1

   1. Initial program 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. exp-neg 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0 93.9%

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

   if -1 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.9%

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

      1. div-inv 99.9%

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

      2. exp-prod 80.7%

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

      3. neg-mul-1 80.7%

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

      4. exp-prod 80.7%

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

      5. pow-pow 99.9%

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

      6. div-inv 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0 81.9%

      \[\leadsto \frac{1}{\color{blue}{2 + \left(-1 \cdot \frac{x}{s} + 0.5 \cdot \frac{{x}^{2}}{{s}^{2}}\right)}} \]
   6. Step-by-step derivation

      1. mul-1-neg 81.9%

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

      2. distribute-frac-neg 81.9%

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

      3. +-commutative 81.9%

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

      4. distribute-frac-neg 81.9%

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

      5. unsub-neg 81.9%

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

      6. unpow2 81.9%

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

      7. unpow2 81.9%

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

      8. times-frac 80.2%

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

      9. unpow2 80.2%

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

   7. Simplified 80.2%

      \[\leadsto \frac{1}{\color{blue}{2 + \left(0.5 \cdot {\left(\frac{x}{s}\right)}^{2} - \frac{x}{s}\right)}} \]
   8. Step-by-step derivation

      1. unpow2 80.2%

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

   9. Applied egg-rr 80.2%

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

4. Final simplification 85.5%

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

Alternative 8: 87.3% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < -1

   1. Initial program 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. exp-neg 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0 93.9%

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

   if -1 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.9%

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

      1. div-inv 99.9%

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

      2. exp-prod 80.7%

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

      3. neg-mul-1 80.7%

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

      4. exp-prod 80.7%

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

      5. pow-pow 99.9%

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

      6. div-inv 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0 81.9%

      \[\leadsto \frac{1}{\color{blue}{2 + \left(-1 \cdot \frac{x}{s} + 0.5 \cdot \frac{{x}^{2}}{{s}^{2}}\right)}} \]
   6. Step-by-step derivation

      1. mul-1-neg 81.9%

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

      2. distribute-frac-neg 81.9%

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

      3. +-commutative 81.9%

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

      4. distribute-frac-neg 81.9%

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

      5. unsub-neg 81.9%

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

      6. unpow2 81.9%

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

      7. unpow2 81.9%

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

      8. times-frac 80.2%

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

      9. unpow2 80.2%

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

   7. Simplified 80.2%

      \[\leadsto \frac{1}{\color{blue}{2 + \left(0.5 \cdot {\left(\frac{x}{s}\right)}^{2} - \frac{x}{s}\right)}} \]
   8. Step-by-step derivation

      1. unpow2 80.2%

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

      2. sqr-neg 80.2%

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

      3. distribute-frac-neg 80.2%

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

      4. distribute-frac-neg 80.2%

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

      5. associate-*l/ 80.2%

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

      6. *-un-lft-identity 80.2%

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

      7. times-frac 85.8%

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

      8. add-sqr-sqrt 62.4%

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

      9. sqrt-unprod 84.4%

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

      10. sqr-neg 84.4%

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

      11. sqrt-unprod 22.8%

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

      12. add-sqr-sqrt 67.7%

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

      13. add-sqr-sqrt 44.9%

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

      14. sqrt-unprod 67.4%

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

      15. sqr-neg 67.4%

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

      16. sqrt-unprod 23.4%

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

      17. add-sqr-sqrt 85.8%

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

   9. Applied egg-rr 85.8%

      \[\leadsto \frac{1}{2 + \left(0.5 \cdot \color{blue}{\left(\frac{x}{1} \cdot \frac{\frac{x}{s}}{s}\right)} - \frac{x}{s}\right)} \]
   10. Step-by-step derivation

       1. /-rgt-identity 85.8%

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

       2. associate-*r/ 80.2%

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

       3. associate-/l* 83.0%

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

       4. div-inv 83.0%

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

       5. clear-num 83.0%

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

   11. Applied egg-rr 83.0%

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

4. Final simplification 87.2%

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

Alternative 9: 88.8% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < -1

   1. Initial program 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. exp-neg 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0 93.9%

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

   if -1 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.9%

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

      1. div-inv 99.9%

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

      2. exp-prod 80.7%

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

      3. neg-mul-1 80.7%

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

      4. exp-prod 80.7%

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

      5. pow-pow 99.9%

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

      6. div-inv 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0 81.9%

      \[\leadsto \frac{1}{\color{blue}{2 + \left(-1 \cdot \frac{x}{s} + 0.5 \cdot \frac{{x}^{2}}{{s}^{2}}\right)}} \]
   6. Step-by-step derivation

      1. mul-1-neg 81.9%

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

      2. distribute-frac-neg 81.9%

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

      3. +-commutative 81.9%

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

      4. distribute-frac-neg 81.9%

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

      5. unsub-neg 81.9%

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

      6. unpow2 81.9%

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

      7. unpow2 81.9%

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

      8. times-frac 80.2%

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

      9. unpow2 80.2%

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

   7. Simplified 80.2%

      \[\leadsto \frac{1}{\color{blue}{2 + \left(0.5 \cdot {\left(\frac{x}{s}\right)}^{2} - \frac{x}{s}\right)}} \]
   8. Step-by-step derivation

      1. unpow2 80.2%

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

      2. sqr-neg 80.2%

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

      3. distribute-frac-neg 80.2%

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

      4. distribute-frac-neg 80.2%

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

      5. associate-*l/ 80.2%

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

      6. *-un-lft-identity 80.2%

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

      7. times-frac 85.8%

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

      8. add-sqr-sqrt 62.4%

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

      9. sqrt-unprod 84.4%

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

      10. sqr-neg 84.4%

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

      11. sqrt-unprod 22.8%

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

      12. add-sqr-sqrt 67.7%

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

      13. add-sqr-sqrt 44.9%

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

      14. sqrt-unprod 67.4%

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

      15. sqr-neg 67.4%

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

      16. sqrt-unprod 23.4%

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

      17. add-sqr-sqrt 85.8%

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

   9. Applied egg-rr 85.8%

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

4. Final simplification 88.9%

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

Alternative 10: 74.3% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -0.0010000000474974513:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) -0.0010000000474974513)
   (/ 1.0 (+ 1.0 (/ 1.0 (+ 1.0 (/ x s)))))
   (/ 1.0 (- 2.0 (/ x s)))))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= -0.0010000000474974513f) {
		tmp = 1.0f / (1.0f + (1.0f / (1.0f + (x / s))));
	} 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) <= (-0.0010000000474974513e0)) then
        tmp = 1.0e0 / (1.0e0 + (1.0e0 / (1.0e0 + (x / s))))
    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(Float32(-x) / s) <= Float32(-0.0010000000474974513))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(x / s)))));
	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(-0.0010000000474974513))
		tmp = single(1.0) / (single(1.0) + (single(1.0) / (single(1.0) + (x / s))));
	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) < -0.00100000005

   1. Initial program 99.9%

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

      1. distribute-frac-neg 99.9%

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

      2. exp-neg 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0 92.6%

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

   if -0.00100000005 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 58.2%

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

      1. mul-1-neg 58.2%

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

      2. unsub-neg 58.2%

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

   5. Simplified 58.2%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -0.0010000000474974513:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 48.8% accurate, 7.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < -1

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 28.2%

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

   if -1 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 58.5%

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

      1. mul-1-neg 58.5%

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

      2. unsub-neg 58.5%

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

   5. Simplified 58.5%

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

4. Final simplification 46.9%

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

Alternative 12: 47.3% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq 2:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s))) (if (<= t_0 2.0) 0.5 (/ 1.0 t_0))))

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if (t_0 <= 2.0f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / t_0;
	}
	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 <= 2.0e0) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 / t_0
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = Float32(Float32(-x) / s)
	tmp = Float32(0.0)
	if (t_0 <= Float32(2.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / t_0);
	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);
	else
		tmp = single(1.0) / t_0;
	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 51.7%

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

   if 2 < (/.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 35.7%

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

      1. mul-1-neg 35.7%

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

      2. unsub-neg 35.7%

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

   5. Simplified 35.7%

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

   6. Taylor expanded in x around inf 35.7%

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

      1. mul-1-neg 35.7%

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

      2. distribute-frac-neg 35.7%

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

   8. Simplified 35.7%

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

4. Final simplification 45.6%

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

Alternative 13: 47.0% accurate, 10.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.99999978589949 \cdot 10^{-8}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -5.99999978589949e-8) (/ 1.0 (/ x s)) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -5.99999978589949e-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 <= (-5.99999978589949e-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(-5.99999978589949e-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(-5.99999978589949e-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 < -5.99999979e-8

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 45.5%

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

      1. mul-1-neg 45.5%

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

      2. unsub-neg 45.5%

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

   5. Simplified 45.5%

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

   6. Taylor expanded in x around inf 40.4%

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

      1. associate-*r/ 40.4%

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

      2. neg-mul-1 40.4%

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

   8. Simplified 40.4%

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

      1. remove-double-neg 40.4%

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

      2. frac-2neg 40.4%

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

      3. clear-num 45.5%

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

      4. inv-pow 45.5%

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

      5. add-sqr-sqrt 45.5%

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

      6. sqrt-unprod 58.5%

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

      7. sqr-neg 58.5%

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

      8. sqrt-unprod -0.0%

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

      9. add-sqr-sqrt 45.5%

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

   10. Applied egg-rr 45.5%

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

       1. unpow-1 45.5%

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

   12. Simplified 45.5%

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

   if -5.99999979e-8 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 45.4%

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

4. Final simplification 45.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.99999978589949 \cdot 10^{-8}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 45.7% accurate, 13.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.99999978589949 \cdot 10^{-8}:\\ \;\;\;\;\frac{s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (if (<= x -5.99999978589949e-8) (/ s x) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -5.99999978589949e-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 <= (-5.99999978589949e-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(-5.99999978589949e-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(-5.99999978589949e-8))
		tmp = s / x;
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -5.99999979e-8

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 45.5%

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

      1. mul-1-neg 45.5%

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

      2. unsub-neg 45.5%

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

   5. Simplified 45.5%

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

   6. Taylor expanded in x around inf 40.4%

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

      1. associate-*r/ 40.4%

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

      2. neg-mul-1 40.4%

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

   8. Simplified 40.4%

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

      1. expm1-log1p-u 40.4%

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

      2. expm1-udef 96.1%

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

      3. add-sqr-sqrt -0.0%

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

      4. sqrt-unprod 96.1%

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

      5. sqr-neg 96.1%

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

      6. sqrt-unprod 96.1%

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

      7. add-sqr-sqrt 96.1%

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

   10. Applied egg-rr 96.1%

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

       1. expm1-def 40.4%

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

       2. expm1-log1p 40.4%

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

   12. Simplified 40.4%

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

   if -5.99999979e-8 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 45.4%

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

4. Final simplification 44.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.99999978589949 \cdot 10^{-8}:\\ \;\;\;\;\frac{s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 34.8% accurate, 108.0× speedup

Math

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

FPCore

(FPCore (x s) :precision binary32 0.5)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around 0 34.6%

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

4. Final simplification 34.6%

   \[\leadsto 0.5 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024026 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))