Logistic function

Percentage Accurate: 99.8% → 99.9%

Time: 10.3s

Alternatives: 18

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. distribute-frac-neg 99.8%

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

   2. exp-neg 99.8%

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

4. Applied egg-rr 99.8%

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

6. Applied egg-rr 99.9%

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

7. Final simplification 99.9%

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{-1 + \left(e^{\frac{x}{-s}} + 2\right)} \end{array} \]

FPCore

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

C

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

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)) + 2.0e0))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

   1. distribute-frac-neg 99.8%

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

   2. exp-neg 99.8%

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

4. Applied egg-rr 99.8%

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

   1. inv-pow 99.8%

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

   2. pow-exp 99.8%

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

   3. *-commutative 99.8%

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

   4. pow-exp 99.8%

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

   5. expm1-log1p-u 99.8%

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

   6. expm1-undefine 99.8%

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

   7. pow-exp 99.8%

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

   8. *-commutative 99.8%

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

   9. pow-exp 99.8%

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

   10. inv-pow 99.8%

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

   11. rec-exp 99.8%

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

   12. distribute-neg-frac2 99.8%

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

6. Applied egg-rr 99.8%

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

   1. sub-neg 99.8%

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

   2. metadata-eval 99.8%

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

   3. +-commutative 99.8%

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

   4. log1p-undefine 99.7%

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

   5. rem-exp-log 99.8%

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

   6. associate-+r+ 99.8%

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

   7. metadata-eval 99.8%

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

8. Simplified 99.8%

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

9. Final simplification 99.8%

   \[\leadsto \frac{1}{-1 + \left(e^{\frac{x}{-s}} + 2\right)} \]
10. Add Preprocessing

Alternative 3: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	return 1.0f / (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 + (1.0e0 / exp((x / s))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

   1. distribute-frac-neg 99.8%

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

   2. exp-neg 99.8%

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

4. Applied egg-rr 99.8%

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

5. Final simplification 99.8%

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

Alternative 4: 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 5: 85.3% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{-s}\\ \mathbf{if}\;t\_0 \leq -0.03999999910593033:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{elif}\;t\_0 \leq 4.999999840142846 \cdot 10^{+37}:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{s} \cdot \frac{x}{s}}{2 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{s} \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ x (- s))))
   (if (<= t_0 -0.03999999910593033)
     (/ 1.0 (+ 1.0 (/ 1.0 (+ 1.0 (/ x s)))))
     (if (<= t_0 4.999999840142846e+37)
       (/ 1.0 (/ (- 4.0 (* (/ x s) (/ x s))) (+ 2.0 (/ x s))))
       (/ 1.0 (* (/ x s) 3.0))))))

C

float code(float x, float s) {
	float t_0 = x / -s;
	float tmp;
	if (t_0 <= -0.03999999910593033f) {
		tmp = 1.0f / (1.0f + (1.0f / (1.0f + (x / s))));
	} else if (t_0 <= 4.999999840142846e+37f) {
		tmp = 1.0f / ((4.0f - ((x / s) * (x / s))) / (2.0f + (x / s)));
	} 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) :: t_0
    real(4) :: tmp
    t_0 = x / -s
    if (t_0 <= (-0.03999999910593033e0)) then
        tmp = 1.0e0 / (1.0e0 + (1.0e0 / (1.0e0 + (x / s))))
    else if (t_0 <= 4.999999840142846e+37) then
        tmp = 1.0e0 / ((4.0e0 - ((x / s) * (x / s))) / (2.0e0 + (x / s)))
    else
        tmp = 1.0e0 / ((x / s) * 3.0e0)
    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(-0.03999999910593033))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(x / s)))));
	elseif (t_0 <= Float32(4.999999840142846e+37))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(4.0) - Float32(Float32(x / s) * Float32(x / s))) / Float32(Float32(2.0) + Float32(x / s))));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(x / s) * Float32(3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = x / -s;
	tmp = single(0.0);
	if (t_0 <= single(-0.03999999910593033))
		tmp = single(1.0) / (single(1.0) + (single(1.0) / (single(1.0) + (x / s))));
	elseif (t_0 <= single(4.999999840142846e+37))
		tmp = single(1.0) / ((single(4.0) - ((x / s) * (x / s))) / (single(2.0) + (x / s)));
	else
		tmp = single(1.0) / ((x / s) * single(3.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if (/.f32 (neg.f32 x) s) < -0.0399999991

   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 94.1%

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

      1. +-commutative 94.1%

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

   7. Simplified 94.1%

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

   if -0.0399999991 < (/.f32 (neg.f32 x) s) < 4.99999984e37

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 54.5%

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

      1. mul-1-neg 54.5%

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

      2. unsub-neg 54.5%

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

   5. Simplified 54.5%

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

      1. *-un-lft-identity 54.5%

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

      2. cancel-sign-sub-inv 54.5%

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

      3. metadata-eval 54.5%

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

      4. add-log-exp 96.3%

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

      5. pow-exp 96.3%

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

      6. flip-+ 49.3%

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

      7. metadata-eval 49.3%

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

      8. pow-exp 49.3%

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

      9. add-log-exp 49.3%

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

      10. neg-mul-1 49.3%

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

      11. pow-exp 49.3%

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

      12. add-log-exp 50.0%

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

      13. neg-mul-1 50.0%

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

      14. distribute-neg-frac2 50.0%

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

      15. distribute-neg-frac2 50.0%

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

      16. pow-exp 50.0%

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

   7. Applied egg-rr 75.1%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 98.2%

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

      1. mul-1-neg 98.2%

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

      2. unsub-neg 98.2%

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

   5. Simplified 98.2%

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

   6. Taylor expanded in x around inf 98.2%

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

      1. associate-*r/ 98.2%

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

      2. metadata-eval 98.2%

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

   8. Simplified 98.2%

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

      1. add-sqr-sqrt -0.0%

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

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

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

      3. prod-diff -0.0%

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

      4. associate-/r/ -0.0%

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

      5. clear-num -0.0%

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

      6. distribute-frac-neg2 -0.0%

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

      7. add-sqr-sqrt -0.0%

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

      8. sqrt-unprod -0.0%

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

      9. sqr-neg -0.0%

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

      10. sqrt-unprod -0.0%

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

      11. add-sqr-sqrt -0.0%

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

      12. fma-define -0.0%

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

      13. add-sqr-sqrt 98.2%

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

   10. Applied egg-rr 100.0%

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

       1. +-commutative 100.0%

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

       2. fma-undefine 100.0%

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

       3. *-rgt-identity 100.0%

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

       4. associate-+l+ 100.0%

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

       5. +-commutative 100.0%

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

   12. Simplified 100.0%

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

   13. Taylor expanded in x around inf 100.0%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq -0.03999999910593033:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{elif}\;\frac{x}{-s} \leq 4.999999840142846 \cdot 10^{+37}:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{s} \cdot \frac{x}{s}}{2 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{s} \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.9% accurate, 4.7× speedup

Math

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

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ x (- s)) 20.0)
   (/ 1.0 (+ 1.0 (/ 1.0 (+ 1.0 (/ x s)))))
   (/ 1.0 (* x (/ (+ s (* x 0.5)) (* x (* s 0.5)))))))

C

float code(float x, float s) {
	float tmp;
	if ((x / -s) <= 20.0f) {
		tmp = 1.0f / (1.0f + (1.0f / (1.0f + (x / s))));
	} else {
		tmp = 1.0f / (x * ((s + (x * 0.5f)) / (x * (s * 0.5f))));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((x / -s) <= 20.0e0) then
        tmp = 1.0e0 / (1.0e0 + (1.0e0 / (1.0e0 + (x / s))))
    else
        tmp = 1.0e0 / (x * ((s + (x * 0.5e0)) / (x * (s * 0.5e0))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(x / Float32(-s)) <= Float32(20.0))
		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(x * Float32(Float32(s + Float32(x * Float32(0.5))) / Float32(x * Float32(s * Float32(0.5))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((x / -s) <= single(20.0))
		tmp = single(1.0) / (single(1.0) + (single(1.0) / (single(1.0) + (x / s))));
	else
		tmp = single(1.0) / (x * ((s + (x * single(0.5))) / (x * (s * single(0.5)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.7%

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

      1. distribute-frac-neg 99.7%

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

      2. exp-neg 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in x around 0 93.6%

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

      1. +-commutative 93.6%

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

   7. Simplified 93.6%

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

   if 20 < (/.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 45.9%

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

      1. mul-1-neg 45.9%

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

      2. unsub-neg 45.9%

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

   5. Simplified 45.9%

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

   6. Taylor expanded in x around inf 45.9%

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

      1. associate-*r/ 45.9%

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

      2. metadata-eval 45.9%

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

   8. Simplified 45.9%

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

      1. sub-neg 45.9%

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

      2. distribute-frac-neg2 45.9%

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

      3. add-sqr-sqrt -0.0%

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

      4. sqrt-unprod 72.7%

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

      5. sqr-neg 72.7%

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

      6. sqrt-unprod 45.9%

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

      7. add-sqr-sqrt 45.9%

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

      8. /-rgt-identity 45.9%

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

      9. clear-num 45.9%

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

      10. clear-num 45.9%

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

      11. frac-add 53.7%

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

      12. *-un-lft-identity 53.7%

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

      13. div-inv 53.7%

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

      14. metadata-eval 53.7%

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

      15. div-inv 53.7%

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

      16. metadata-eval 53.7%

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

   10. Applied egg-rr 53.7%

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

       1. *-rgt-identity 53.7%

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

       2. associate-*l* 53.7%

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

       3. *-commutative 53.7%

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

   12. Simplified 53.7%

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

4. Final simplification 78.7%

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

Alternative 7: 77.6% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq 0.019999999552965164:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \frac{s \cdot 2 - x}{x \cdot s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ x (- s)) 0.019999999552965164)
   (/ 1.0 (+ 1.0 (/ 1.0 (+ 1.0 (/ x s)))))
   (/ 1.0 (* x (/ (- (* s 2.0) x) (* x s))))))

C

float code(float x, float s) {
	float tmp;
	if ((x / -s) <= 0.019999999552965164f) {
		tmp = 1.0f / (1.0f + (1.0f / (1.0f + (x / s))));
	} else {
		tmp = 1.0f / (x * (((s * 2.0f) - x) / (x * s)));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((x / -s) <= 0.019999999552965164e0) then
        tmp = 1.0e0 / (1.0e0 + (1.0e0 / (1.0e0 + (x / s))))
    else
        tmp = 1.0e0 / (x * (((s * 2.0e0) - x) / (x * s)))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(x / Float32(-s)) <= Float32(0.019999999552965164))
		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(x * Float32(Float32(Float32(s * Float32(2.0)) - x) / Float32(x * s))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((x / -s) <= single(0.019999999552965164))
		tmp = single(1.0) / (single(1.0) + (single(1.0) / (single(1.0) + (x / s))));
	else
		tmp = single(1.0) / (x * (((s * single(2.0)) - x) / (x * s)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 0.0199999996

   1. Initial program 99.8%

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

      1. distribute-frac-neg 99.8%

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

      2. exp-neg 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around 0 95.7%

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

      1. +-commutative 95.7%

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

   7. Simplified 95.7%

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

   if 0.0199999996 < (/.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-sub 51.6%

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

      2. div-inv 57.7%

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

      3. fma-neg 57.7%

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

      4. *-rgt-identity 57.7%

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

   10. Applied egg-rr 57.7%

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

       1. associate-*r/ 51.6%

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

       2. *-rgt-identity 51.6%

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

       3. fma-undefine 51.6%

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

       4. *-commutative 51.6%

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

       5. unsub-neg 51.6%

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

       6. *-commutative 51.6%

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

   12. Simplified 51.6%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq 0.019999999552965164:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \frac{s \cdot 2 - x}{x \cdot s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 49.2% accurate, 6.3× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	float tmp;
	if ((x / -s) <= -2.0f) {
		tmp = 1.0f / (x * (2.0f / x));
	} else {
		tmp = 1.0f / (-1.0f + (3.0f - (x / s)));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((x / -s) <= (-2.0e0)) then
        tmp = 1.0e0 / (x * (2.0e0 / x))
    else
        tmp = 1.0e0 / ((-1.0e0) + (3.0e0 - (x / s)))
    end if
    code = tmp
end function

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 5.1%

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

      1. mul-1-neg 5.1%

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

      2. unsub-neg 5.1%

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

   5. Simplified 5.1%

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

   6. Taylor expanded in x around inf 5.1%

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

      1. associate-*r/ 5.1%

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

      2. metadata-eval 5.1%

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

   8. Simplified 5.1%

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

   9. Taylor expanded in x around 0 28.1%

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

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

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 64.8%

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

      1. mul-1-neg 64.8%

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

      2. unsub-neg 64.8%

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

   5. Simplified 64.8%

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

      1. expm1-log1p-u 64.8%

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

   7. Applied egg-rr 64.8%

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

      1. expm1-undefine 64.8%

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

      2. sub-neg 64.8%

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

      3. log1p-undefine 64.8%

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

      4. rem-exp-log 64.8%

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

      5. associate-+r- 64.8%

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

      6. metadata-eval 64.8%

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

      7. metadata-eval 64.8%

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

   9. Simplified 64.8%

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

4. Final simplification 51.4%

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

Alternative 9: 49.9% accurate, 6.3× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	float tmp;
	if (-x <= 4.999999999099794e-24f) {
		tmp = 0.5f;
	} else {
		tmp = s * (x / (x * ((s * 2.0f) - x)));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (-x <= 4.999999999099794e-24) then
        tmp = 0.5e0
    else
        tmp = s * (x / (x * ((s * 2.0e0) - x)))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-x) <= Float32(4.999999999099794e-24))
		tmp = Float32(0.5);
	else
		tmp = Float32(s * Float32(x / Float32(x * Float32(Float32(s * Float32(2.0)) - x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (-x <= single(4.999999999099794e-24))
		tmp = single(0.5);
	else
		tmp = s * (x / (x * ((s * single(2.0)) - x)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (neg.f32 x) < 5e-24

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 50.5%

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

   if 5e-24 < (neg.f32 x)

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 50.3%

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

      1. mul-1-neg 50.3%

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

      2. unsub-neg 50.3%

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

   5. Simplified 50.3%

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

   6. Taylor expanded in x around inf 50.3%

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

      1. associate-*r/ 50.3%

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

      2. metadata-eval 50.3%

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

   8. Simplified 50.3%

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

      1. associate-/r* 46.0%

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

      2. frac-sub 49.4%

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

      3. associate-/r/ 53.0%

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

      4. fma-neg 53.0%

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

      5. *-rgt-identity 53.0%

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

   10. Applied egg-rr 53.0%

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

       1. associate-*r* 51.4%

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

       2. *-commutative 51.4%

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

       3. associate-/l/ 52.3%

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

       4. associate-*l/ 53.3%

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

       5. *-commutative 53.3%

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

       6. *-rgt-identity 53.3%

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

       7. *-commutative 53.3%

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

       8. fma-undefine 53.3%

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

       9. *-commutative 53.3%

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

       10. unsub-neg 53.3%

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

       11. *-commutative 53.3%

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

   12. Simplified 53.3%

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

4. Final simplification 51.6%

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

Alternative 10: 76.1% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-x \leq 2.00000006274879 \cdot 10^{-22}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;s \cdot \frac{x}{x \cdot \left(s \cdot 2 - x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- x) 2.00000006274879e-22)
   (/ 1.0 (+ 1.0 (/ 1.0 (+ 1.0 (/ x s)))))
   (* s (/ x (* x (- (* s 2.0) x))))))

C

float code(float x, float s) {
	float tmp;
	if (-x <= 2.00000006274879e-22f) {
		tmp = 1.0f / (1.0f + (1.0f / (1.0f + (x / s))));
	} else {
		tmp = s * (x / (x * ((s * 2.0f) - x)));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (-x <= 2.00000006274879e-22) then
        tmp = 1.0e0 / (1.0e0 + (1.0e0 / (1.0e0 + (x / s))))
    else
        tmp = s * (x / (x * ((s * 2.0e0) - x)))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-x) <= Float32(2.00000006274879e-22))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(x / s)))));
	else
		tmp = Float32(s * Float32(x / Float32(x * Float32(Float32(s * Float32(2.0)) - x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (-x <= single(2.00000006274879e-22))
		tmp = single(1.0) / (single(1.0) + (single(1.0) / (single(1.0) + (x / s))));
	else
		tmp = s * (x / (x * ((s * single(2.0)) - x)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (neg.f32 x) < 2.00000006e-22

   1. Initial program 99.8%

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

      1. distribute-frac-neg 99.8%

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

      2. exp-neg 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around 0 93.6%

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

      1. +-commutative 93.6%

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

   7. Simplified 93.6%

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

   if 2.00000006e-22 < (neg.f32 x)

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 49.8%

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

      1. mul-1-neg 49.8%

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

      2. unsub-neg 49.8%

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

   5. Simplified 49.8%

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

   6. Taylor expanded in x around inf 49.8%

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

      1. associate-*r/ 49.8%

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

      2. metadata-eval 49.8%

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

   8. Simplified 49.8%

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

      1. associate-/r* 45.5%

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

      2. frac-sub 48.9%

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

      3. associate-/r/ 52.6%

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

      4. fma-neg 52.6%

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

      5. *-rgt-identity 52.6%

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

   10. Applied egg-rr 52.6%

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

       1. associate-*r* 51.0%

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

       2. *-commutative 51.0%

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

       3. associate-/l/ 51.9%

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

       4. associate-*l/ 52.9%

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

       5. *-commutative 52.9%

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

       6. *-rgt-identity 52.9%

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

       7. *-commutative 52.9%

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

       8. fma-undefine 52.9%

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

       9. *-commutative 52.9%

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

       10. unsub-neg 52.9%

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

       11. *-commutative 52.9%

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

   12. Simplified 52.9%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-x \leq 2.00000006274879 \cdot 10^{-22}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;s \cdot \frac{x}{x \cdot \left(s \cdot 2 - x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 47.9% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq 20:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{s} \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ x (- s)) 20.0) 0.5 (/ 1.0 (* (/ x s) 3.0))))

C

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

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 52.1%

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

   if 20 < (/.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 45.9%

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

      1. mul-1-neg 45.9%

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

      2. unsub-neg 45.9%

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

   5. Simplified 45.9%

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

   6. Taylor expanded in x around inf 45.9%

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

      1. associate-*r/ 45.9%

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

      2. metadata-eval 45.9%

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

   8. Simplified 45.9%

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

      1. add-sqr-sqrt -0.0%

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

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

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

      3. prod-diff -0.0%

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

      4. associate-/r/ -0.0%

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

      5. clear-num -0.0%

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

      6. distribute-frac-neg2 -0.0%

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

      7. add-sqr-sqrt -0.0%

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

      8. sqrt-unprod -0.0%

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

      9. sqr-neg -0.0%

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

      10. sqrt-unprod -0.0%

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

      11. add-sqr-sqrt -0.0%

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

      12. fma-define -0.0%

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

      13. add-sqr-sqrt 45.9%

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

   10. Applied egg-rr 46.9%

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

       1. +-commutative 46.9%

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

       2. fma-undefine 46.9%

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

       3. *-rgt-identity 46.9%

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

       4. associate-+l+ 46.9%

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

       5. +-commutative 46.9%

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

   12. Simplified 46.9%

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

   13. Taylor expanded in x around inf 46.9%

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

4. Final simplification 50.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq 20:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{s} \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 48.1% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq 20:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \frac{3}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ x (- s)) 20.0) 0.5 (/ 1.0 (* x (/ 3.0 s)))))

C

float code(float x, float s) {
	float tmp;
	if ((x / -s) <= 20.0f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / (x * (3.0f / s));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((x / -s) <= 20.0e0) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 / (x * (3.0e0 / s))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(x / Float32(-s)) <= Float32(20.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / Float32(x * Float32(Float32(3.0) / s)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((x / -s) <= single(20.0))
		tmp = single(0.5);
	else
		tmp = single(1.0) / (x * (single(3.0) / s));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 52.1%

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

   if 20 < (/.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 45.9%

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

      1. mul-1-neg 45.9%

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

      2. unsub-neg 45.9%

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

   5. Simplified 45.9%

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

   6. Taylor expanded in x around inf 45.9%

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

      1. associate-*r/ 45.9%

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

      2. metadata-eval 45.9%

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

   8. Simplified 45.9%

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

      1. add-sqr-sqrt -0.0%

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

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

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

      3. prod-diff -0.0%

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

      4. associate-/r/ -0.0%

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

      5. clear-num -0.0%

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

      6. distribute-frac-neg2 -0.0%

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

      7. add-sqr-sqrt -0.0%

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

      8. sqrt-unprod -0.0%

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

      9. sqr-neg -0.0%

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

      10. sqrt-unprod -0.0%

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

      11. add-sqr-sqrt -0.0%

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

      12. fma-define -0.0%

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

      13. add-sqr-sqrt 45.9%

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

   10. Applied egg-rr 46.9%

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

       1. +-commutative 46.9%

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

       2. fma-undefine 46.9%

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

       3. *-rgt-identity 46.9%

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

       4. associate-+l+ 46.9%

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

       5. +-commutative 46.9%

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

   12. Simplified 46.9%

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

   13. Taylor expanded in s around 0 46.9%

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

4. Final simplification 50.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq 20:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \frac{3}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 49.2% accurate, 7.2× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	float tmp;
	if ((x / -s) <= -2.0f) {
		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) <= (-2.0e0)) 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(-2.0))
		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(-2.0))
		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) < -2

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 5.1%

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

      1. mul-1-neg 5.1%

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

      2. unsub-neg 5.1%

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

   5. Simplified 5.1%

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

   6. Taylor expanded in x around inf 5.1%

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

      1. associate-*r/ 5.1%

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

      2. metadata-eval 5.1%

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

   8. Simplified 5.1%

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

   9. Taylor expanded in x around 0 28.1%

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

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

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 64.8%

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

      1. mul-1-neg 64.8%

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

      2. unsub-neg 64.8%

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

   5. Simplified 64.8%

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

4. Final simplification 51.3%

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

Alternative 14: 47.6% accurate, 8.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 52.6%

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

   if 2 < (/.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 45.4%

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

      1. mul-1-neg 45.4%

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

      2. unsub-neg 45.4%

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

   5. Simplified 45.4%

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

   6. Taylor expanded in x around inf 45.3%

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

      1. mul-1-neg 45.3%

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

      2. distribute-frac-neg2 45.3%

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

   8. Simplified 45.3%

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

4. Final simplification 49.8%

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

Alternative 15: 46.3% accurate, 10.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.999999873689376 \cdot 10^{-5}:\\ \;\;\;\;s \cdot \frac{0.3333333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -4.999999873689376e-5) (* s (/ 0.3333333333333333 x)) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -4.999999873689376e-5f) {
		tmp = s * (0.3333333333333333f / 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 * (0.3333333333333333e0 / 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(Float32(0.3333333333333333) / 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 * (single(0.3333333333333333) / 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 100.0%

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

   3. Taylor expanded in x around 0 56.5%

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

      1. mul-1-neg 56.5%

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

      2. unsub-neg 56.5%

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

   5. Simplified 56.5%

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

   6. Taylor expanded in x around inf 56.5%

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

      1. associate-*r/ 56.5%

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

      2. metadata-eval 56.5%

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

   8. Simplified 56.5%

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

      1. add-sqr-sqrt -0.0%

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

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

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

      3. prod-diff -0.0%

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

      4. associate-/r/ -0.0%

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

      5. clear-num -0.0%

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

      6. distribute-frac-neg2 -0.0%

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

      7. add-sqr-sqrt -0.0%

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

      8. sqrt-unprod -0.0%

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

      9. sqr-neg -0.0%

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

      10. sqrt-unprod -0.0%

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

      11. add-sqr-sqrt -0.0%

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

      12. fma-define -0.0%

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

      13. add-sqr-sqrt 56.5%

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

   10. Applied egg-rr 57.7%

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

       1. +-commutative 57.7%

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

       2. fma-undefine 57.7%

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

       3. *-rgt-identity 57.7%

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

       4. associate-+l+ 57.7%

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

       5. +-commutative 57.7%

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

   12. Simplified 57.7%

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

   13. Taylor expanded in x around inf 51.6%

       \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{s}{x}} \]
   14. Step-by-step derivation

       1. associate-*r/ 51.6%

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

       2. *-commutative 51.6%

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

       3. associate-*r/ 51.6%

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

   15. Simplified 51.6%

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

   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 46.8%

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

4. Final simplification 48.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.999999873689376 \cdot 10^{-5}:\\ \;\;\;\;s \cdot \frac{0.3333333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 46.3% accurate, 10.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.999999873689376 \cdot 10^{-5}:\\ \;\;\;\;\frac{s}{x} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -4.999999873689376e-5) (* (/ s x) 0.3333333333333333) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -4.999999873689376e-5f) {
		tmp = (s / x) * 0.3333333333333333f;
	} 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) * 0.3333333333333333e0
    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(Float32(s / x) * Float32(0.3333333333333333));
	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) * single(0.3333333333333333);
	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 100.0%

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

   3. Taylor expanded in x around 0 56.5%

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

      1. mul-1-neg 56.5%

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

      2. unsub-neg 56.5%

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

   5. Simplified 56.5%

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

   6. Taylor expanded in x around inf 56.5%

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

      1. associate-*r/ 56.5%

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

      2. metadata-eval 56.5%

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

   8. Simplified 56.5%

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

      1. add-sqr-sqrt -0.0%

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

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

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

      3. prod-diff -0.0%

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

      4. associate-/r/ -0.0%

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

      5. clear-num -0.0%

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

      6. distribute-frac-neg2 -0.0%

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

      7. add-sqr-sqrt -0.0%

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

      8. sqrt-unprod -0.0%

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

      9. sqr-neg -0.0%

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

      10. sqrt-unprod -0.0%

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

      11. add-sqr-sqrt -0.0%

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

      12. fma-define -0.0%

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

      13. add-sqr-sqrt 56.5%

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

   10. Applied egg-rr 57.7%

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

       1. +-commutative 57.7%

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

       2. fma-undefine 57.7%

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

       3. *-rgt-identity 57.7%

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

       4. associate-+l+ 57.7%

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

       5. +-commutative 57.7%

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

   12. Simplified 57.7%

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

   13. Taylor expanded in x around inf 51.6%

       \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{s}{x}} \]
   14. Step-by-step derivation

       1. *-commutative 51.6%

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

   15. Simplified 51.6%

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

   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 46.8%

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

4. Final simplification 48.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.999999873689376 \cdot 10^{-5}:\\ \;\;\;\;\frac{s}{x} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 46.1% 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 100.0%

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

   3. Taylor expanded in x around 0 56.5%

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

      1. mul-1-neg 56.5%

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

      2. unsub-neg 56.5%

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

   5. Simplified 56.5%

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

   6. Taylor expanded in x around inf 50.4%

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

      1. associate-*r/ 50.4%

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

      2. neg-mul-1 50.4%

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

   8. Simplified 50.4%

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

   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 46.8%

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

4. Final simplification 47.9%

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

Alternative 18: 34.8% accurate, 108.0× speedup

Math

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

FPCore

(FPCore (x s) :precision binary32 0.5)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in x around 0 35.1%

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

4. Final simplification 35.1%

   \[\leadsto 0.5 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024080 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))