Logistic function

Percentage Accurate: 99.8% → 99.8%

Time: 11.4s

Alternatives: 12

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

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(x, s)
	tmp = single(1.0) / (single(1.0) + (exp(single(-2.0)) ^ ((x / single(2.0)) / 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 99.8%

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

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 99.9%

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

   3. *-commutative 99.9%

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

   4. pow-exp 99.8%

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

   5. sqr-pow 99.8%

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

   6. pow-prod-down 99.8%

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

   7. prod-exp 99.9%

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

   8. metadata-eval 99.9%

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

   9. associate-/l/ 99.9%

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

6. Applied egg-rr 99.9%

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

   1. associate-/r* 99.9%

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

8. Simplified 99.9%

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

9. Final simplification 99.9%

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + \left(1 + \mathsf{expm1}\left(\frac{-x}{s}\right)\right)} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (/ 1.0 (+ 1.0 (+ 1.0 (expm1 (/ (- x) s))))))

C

float code(float x, float s) {
	return 1.0f / (1.0f + (1.0f + expm1f((-x / s))));
}

Julia

function code(x, s)
	return Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) + expm1(Float32(Float32(-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 99.8%

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

   2. exp-prod 83.4%

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

   3. neg-mul-1 83.4%

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

   4. exp-prod 83.4%

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

   5. pow-pow 99.8%

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

   6. div-inv 99.8%

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

4. Applied egg-rr 99.8%

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

   1. expm1-log1p-u 99.8%

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

   2. expm1-udef 99.8%

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

   3. log1p-udef 99.9%

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

   4. add-exp-log 99.8%

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

   5. pow-exp 99.9%

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

   6. *-commutative 99.9%

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

   7. pow-exp 99.8%

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

   8. inv-pow 99.8%

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

   9. rec-exp 99.9%

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

   10. distribute-neg-frac 99.9%

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

6. Applied egg-rr 99.9%

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

   1. associate--l+ 99.9%

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

   2. expm1-def 99.9%

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

8. Simplified 99.9%

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

9. Final simplification 99.9%

   \[\leadsto \frac{1}{1 + \left(1 + \mathsf{expm1}\left(\frac{-x}{s}\right)\right)} \]
10. 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: 89.2% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq 0.10000000149011612:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{elif}\;t_0 \leq 9.999999680285692 \cdot 10^{+37}:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{\frac{s \cdot \left(-s\right)}{x}}}{2 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \left(1 + 0.3333333333333333 \cdot \frac{x \cdot -3}{s}\right)}\\ \end{array} \end{array} \]

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 9.999999680285692e+37)
       (/ 1.0 (/ (- 4.0 (/ x (/ (* s (- s)) x))) (+ 2.0 (/ x s))))
       (/ 1.0 (+ 1.0 (+ 1.0 (* 0.3333333333333333 (/ (* x -3.0) 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 <= 9.999999680285692e+37f) {
		tmp = 1.0f / ((4.0f - (x / ((s * -s) / x))) / (2.0f + (x / s)));
	} else {
		tmp = 1.0f / (1.0f + (1.0f + (0.3333333333333333f * ((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) :: 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 <= 9.999999680285692e+37) then
        tmp = 1.0e0 / ((4.0e0 - (x / ((s * -s) / x))) / (2.0e0 + (x / s)))
    else
        tmp = 1.0e0 / (1.0e0 + (1.0e0 + (0.3333333333333333e0 * ((x * (-3.0e0)) / 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(9.999999680285692e+37))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(4.0) - Float32(x / Float32(Float32(s * Float32(-s)) / x))) / Float32(Float32(2.0) + Float32(x / s))));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) + Float32(Float32(0.3333333333333333) * Float32(Float32(x * Float32(-3.0)) / 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(9.999999680285692e+37))
		tmp = single(1.0) / ((single(4.0) - (x / ((s * -s) / x))) / (single(2.0) + (x / s)));
	else
		tmp = single(1.0) / (single(1.0) + (single(1.0) + (single(0.3333333333333333) * ((x * single(-3.0)) / 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 95.9%

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

   if 0.100000001 < (/.f32 (neg.f32 x) s) < 9.99999968e37

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 11.7%

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

      1. mul-1-neg 11.7%

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

      2. unsub-neg 11.7%

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

   5. Simplified 11.7%

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

      1. sub-neg 11.7%

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

      2. flip-+ 62.9%

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

      3. metadata-eval 62.9%

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

      4. distribute-neg-frac 62.9%

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

      5. distribute-neg-frac 62.9%

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

      6. distribute-neg-frac 62.9%

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

   7. Applied egg-rr 62.9%

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

      1. frac-times 69.8%

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

      2. sqr-neg 69.8%

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

      3. frac-times 62.9%

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

      4. clear-num 62.9%

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

      5. frac-times 66.0%

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

      6. *-un-lft-identity 66.0%

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

   9. Applied egg-rr 66.0%

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

       1. frac-2neg 66.0%

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

       2. associate-*l/ 76.7%

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

       3. add-sqr-sqrt 76.7%

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

       4. sqrt-unprod 69.8%

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

       5. sqr-neg 69.8%

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

       6. sqrt-unprod -0.0%

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

       7. add-sqr-sqrt 76.6%

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

   11. Applied egg-rr 76.6%

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

   if 9.99999968e37 < (/.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. distribute-frac-neg 100.0%

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

      2. exp-neg 100.0%

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

      3. add-sqr-sqrt -0.0%

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

      4. sqrt-unprod 6.3%

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

      5. sqr-neg 6.3%

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

      6. sqrt-unprod 6.3%

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

      7. add-sqr-sqrt 6.3%

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

      8. add-cbrt-cube 6.3%

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

      9. pow1/3 6.3%

         \[\leadsto \frac{1}{1 + \frac{1}{\color{blue}{{\left(\left(e^{\frac{-x}{s}} \cdot e^{\frac{-x}{s}}\right) \cdot e^{\frac{-x}{s}}\right)}^{0.3333333333333333}}}} \]

      10. pow-flip 6.3%

          \[\leadsto \frac{1}{1 + \color{blue}{{\left(\left(e^{\frac{-x}{s}} \cdot e^{\frac{-x}{s}}\right) \cdot e^{\frac{-x}{s}}\right)}^{\left(-0.3333333333333333\right)}}} \]

      11. pow3 6.3%

          \[\leadsto \frac{1}{1 + {\color{blue}{\left({\left(e^{\frac{-x}{s}}\right)}^{3}\right)}}^{\left(-0.3333333333333333\right)}} \]

      12. add-sqr-sqrt 6.3%

          \[\leadsto \frac{1}{1 + {\left({\left(e^{\frac{\color{blue}{\sqrt{-x} \cdot \sqrt{-x}}}{s}}\right)}^{3}\right)}^{\left(-0.3333333333333333\right)}} \]

      13. sqrt-unprod 6.3%

          \[\leadsto \frac{1}{1 + {\left({\left(e^{\frac{\color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}}}{s}}\right)}^{3}\right)}^{\left(-0.3333333333333333\right)}} \]

      14. sqr-neg 6.3%

          \[\leadsto \frac{1}{1 + {\left({\left(e^{\frac{\sqrt{\color{blue}{x \cdot x}}}{s}}\right)}^{3}\right)}^{\left(-0.3333333333333333\right)}} \]

      15. sqrt-unprod -0.0%

          \[\leadsto \frac{1}{1 + {\left({\left(e^{\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{s}}\right)}^{3}\right)}^{\left(-0.3333333333333333\right)}} \]

      16. add-sqr-sqrt 100.0%

          \[\leadsto \frac{1}{1 + {\left({\left(e^{\frac{\color{blue}{x}}{s}}\right)}^{3}\right)}^{\left(-0.3333333333333333\right)}} \]

   4. Applied egg-rr 100.0%

      \[\leadsto \frac{1}{1 + \color{blue}{{\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)}^{-0.3333333333333333}}} \]

   5. Taylor expanded in s around -inf 100.0%

      \[\leadsto \frac{1}{1 + \color{blue}{\left(1 + 0.3333333333333333 \cdot \frac{-2 \cdot x + -1 \cdot x}{s}\right)}} \]
   6. Step-by-step derivation

      1. distribute-rgt-out 100.0%

         \[\leadsto \frac{1}{1 + \left(1 + 0.3333333333333333 \cdot \frac{\color{blue}{x \cdot \left(-2 + -1\right)}}{s}\right)} \]

      2. metadata-eval 100.0%

         \[\leadsto \frac{1}{1 + \left(1 + 0.3333333333333333 \cdot \frac{x \cdot \color{blue}{-3}}{s}\right)} \]

   7. Simplified 100.0%

      \[\leadsto \frac{1}{1 + \color{blue}{\left(1 + 0.3333333333333333 \cdot \frac{x \cdot -3}{s}\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 0.10000000149011612:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{elif}\;\frac{-x}{s} \leq 9.999999680285692 \cdot 10^{+37}:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{\frac{s \cdot \left(-s\right)}{x}}}{2 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \left(1 + 0.3333333333333333 \cdot \frac{x \cdot -3}{s}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.3% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;t_0 \leq 9.999999680285692 \cdot 10^{+37}:\\ \;\;\;\;\frac{1}{\frac{4 - x \cdot \frac{\frac{x}{s}}{s}}{2 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \left(1 + 0.3333333333333333 \cdot \frac{x \cdot -3}{s}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)))
   (if (<= t_0 -1.0)
     1.0
     (if (<= t_0 9.999999680285692e+37)
       (/ 1.0 (/ (- 4.0 (* x (/ (/ x s) s))) (+ 2.0 (/ x s))))
       (/ 1.0 (+ 1.0 (+ 1.0 (* 0.3333333333333333 (/ (* x -3.0) s)))))))))

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if (t_0 <= -1.0f) {
		tmp = 1.0f;
	} else if (t_0 <= 9.999999680285692e+37f) {
		tmp = 1.0f / ((4.0f - (x * ((x / s) / s))) / (2.0f + (x / s)));
	} else {
		tmp = 1.0f / (1.0f + (1.0f + (0.3333333333333333f * ((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) :: t_0
    real(4) :: tmp
    t_0 = -x / s
    if (t_0 <= (-1.0e0)) then
        tmp = 1.0e0
    else if (t_0 <= 9.999999680285692e+37) then
        tmp = 1.0e0 / ((4.0e0 - (x * ((x / s) / s))) / (2.0e0 + (x / s)))
    else
        tmp = 1.0e0 / (1.0e0 + (1.0e0 + (0.3333333333333333e0 * ((x * (-3.0e0)) / 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(-1.0))
		tmp = Float32(1.0);
	elseif (t_0 <= Float32(9.999999680285692e+37))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(4.0) - Float32(x * Float32(Float32(x / s) / s))) / Float32(Float32(2.0) + Float32(x / s))));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) + Float32(Float32(0.3333333333333333) * Float32(Float32(x * Float32(-3.0)) / s)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = -x / s;
	tmp = single(0.0);
	if (t_0 <= single(-1.0))
		tmp = single(1.0);
	elseif (t_0 <= single(9.999999680285692e+37))
		tmp = single(1.0) / ((single(4.0) - (x * ((x / s) / s))) / (single(2.0) + (x / s)));
	else
		tmp = single(1.0) / (single(1.0) + (single(1.0) + (single(0.3333333333333333) * ((x * single(-3.0)) / s))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 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 96.6%

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

   6. Taylor expanded in x around inf 96.5%

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

      1. mul-1-neg 96.5%

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

      2. unsub-neg 96.5%

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

   8. Simplified 96.5%

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

   9. Taylor expanded in s around 0 98.8%

      \[\leadsto \color{blue}{1} \]

   if -1 < (/.f32 (neg.f32 x) s) < 9.99999968e37

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 53.5%

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

      1. mul-1-neg 53.5%

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

      2. unsub-neg 53.5%

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

   5. Simplified 53.5%

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

      1. sub-neg 53.5%

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

      2. flip-+ 79.0%

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

      3. metadata-eval 79.0%

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

      4. distribute-neg-frac 79.0%

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

      5. distribute-neg-frac 79.0%

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

      6. distribute-neg-frac 79.0%

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

   7. Applied egg-rr 79.0%

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

      1. frac-times 71.0%

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

      2. sqr-neg 71.0%

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

      3. frac-times 79.0%

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

      4. clear-num 79.0%

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

      5. frac-times 80.5%

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

      6. *-un-lft-identity 80.5%

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

   9. Applied egg-rr 80.5%

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

       1. clear-num 80.5%

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

       2. associate-/r/ 82.9%

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

       3. associate-/r* 82.9%

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

       4. clear-num 82.9%

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

   11. Applied egg-rr 82.9%

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

   if 9.99999968e37 < (/.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. distribute-frac-neg 100.0%

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

      2. exp-neg 100.0%

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

      3. add-sqr-sqrt -0.0%

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

      4. sqrt-unprod 6.3%

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

      5. sqr-neg 6.3%

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

      6. sqrt-unprod 6.3%

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

      7. add-sqr-sqrt 6.3%

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

      8. add-cbrt-cube 6.3%

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

      9. pow1/3 6.3%

         \[\leadsto \frac{1}{1 + \frac{1}{\color{blue}{{\left(\left(e^{\frac{-x}{s}} \cdot e^{\frac{-x}{s}}\right) \cdot e^{\frac{-x}{s}}\right)}^{0.3333333333333333}}}} \]

      10. pow-flip 6.3%

          \[\leadsto \frac{1}{1 + \color{blue}{{\left(\left(e^{\frac{-x}{s}} \cdot e^{\frac{-x}{s}}\right) \cdot e^{\frac{-x}{s}}\right)}^{\left(-0.3333333333333333\right)}}} \]

      11. pow3 6.3%

          \[\leadsto \frac{1}{1 + {\color{blue}{\left({\left(e^{\frac{-x}{s}}\right)}^{3}\right)}}^{\left(-0.3333333333333333\right)}} \]

      12. add-sqr-sqrt 6.3%

          \[\leadsto \frac{1}{1 + {\left({\left(e^{\frac{\color{blue}{\sqrt{-x} \cdot \sqrt{-x}}}{s}}\right)}^{3}\right)}^{\left(-0.3333333333333333\right)}} \]

      13. sqrt-unprod 6.3%

          \[\leadsto \frac{1}{1 + {\left({\left(e^{\frac{\color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}}}{s}}\right)}^{3}\right)}^{\left(-0.3333333333333333\right)}} \]

      14. sqr-neg 6.3%

          \[\leadsto \frac{1}{1 + {\left({\left(e^{\frac{\sqrt{\color{blue}{x \cdot x}}}{s}}\right)}^{3}\right)}^{\left(-0.3333333333333333\right)}} \]

      15. sqrt-unprod -0.0%

          \[\leadsto \frac{1}{1 + {\left({\left(e^{\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{s}}\right)}^{3}\right)}^{\left(-0.3333333333333333\right)}} \]

      16. add-sqr-sqrt 100.0%

          \[\leadsto \frac{1}{1 + {\left({\left(e^{\frac{\color{blue}{x}}{s}}\right)}^{3}\right)}^{\left(-0.3333333333333333\right)}} \]

   4. Applied egg-rr 100.0%

      \[\leadsto \frac{1}{1 + \color{blue}{{\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)}^{-0.3333333333333333}}} \]

   5. Taylor expanded in s around -inf 100.0%

      \[\leadsto \frac{1}{1 + \color{blue}{\left(1 + 0.3333333333333333 \cdot \frac{-2 \cdot x + -1 \cdot x}{s}\right)}} \]
   6. Step-by-step derivation

      1. distribute-rgt-out 100.0%

         \[\leadsto \frac{1}{1 + \left(1 + 0.3333333333333333 \cdot \frac{\color{blue}{x \cdot \left(-2 + -1\right)}}{s}\right)} \]

      2. metadata-eval 100.0%

         \[\leadsto \frac{1}{1 + \left(1 + 0.3333333333333333 \cdot \frac{x \cdot \color{blue}{-3}}{s}\right)} \]

   7. Simplified 100.0%

      \[\leadsto \frac{1}{1 + \color{blue}{\left(1 + 0.3333333333333333 \cdot \frac{x \cdot -3}{s}\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;\frac{-x}{s} \leq 9.999999680285692 \cdot 10^{+37}:\\ \;\;\;\;\frac{1}{\frac{4 - x \cdot \frac{\frac{x}{s}}{s}}{2 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \left(1 + 0.3333333333333333 \cdot \frac{x \cdot -3}{s}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.1% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq -2:\\ \;\;\;\;1\\ \mathbf{elif}\;t_0 \leq 2:\\ \;\;\;\;0.5 + \frac{x}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)))
   (if (<= t_0 -2.0)
     1.0
     (if (<= t_0 2.0) (+ 0.5 (* (/ x s) 0.25)) (/ 1.0 t_0)))))

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if (t_0 <= -2.0f) {
		tmp = 1.0f;
	} else if (t_0 <= 2.0f) {
		tmp = 0.5f + ((x / s) * 0.25f);
	} 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 = 1.0e0
    else if (t_0 <= 2.0e0) then
        tmp = 0.5e0 + ((x / s) * 0.25e0)
    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(1.0);
	elseif (t_0 <= Float32(2.0))
		tmp = Float32(Float32(0.5) + Float32(Float32(x / s) * Float32(0.25)));
	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(1.0);
	elseif (t_0 <= single(2.0))
		tmp = single(0.5) + ((x / s) * single(0.25));
	else
		tmp = single(1.0) / t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing
   3. 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 97.2%

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

   6. Taylor expanded in x around inf 97.1%

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

      1. mul-1-neg 97.1%

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

      2. unsub-neg 97.1%

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

   8. Simplified 97.1%

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

   9. Taylor expanded in s around 0 99.4%

      \[\leadsto \color{blue}{1} \]

   if -2 < (/.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 96.0%

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

      1. *-commutative 96.0%

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

   5. Simplified 96.0%

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

   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 41.3%

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

      1. mul-1-neg 41.3%

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

      2. unsub-neg 41.3%

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

   5. Simplified 41.3%

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

   6. Taylor expanded in x around inf 41.3%

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

      1. mul-1-neg 41.3%

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

      2. distribute-frac-neg 41.3%

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

   8. Simplified 41.3%

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

4. Final simplification 78.2%

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

Alternative 7: 73.9% accurate, 4.9× speedup

Math

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

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)))
   (if (<= t_0 -1.0) 1.0 (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 <= -1.0f) {
		tmp = 1.0f;
	} else 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 <= (-1.0e0)) then
        tmp = 1.0e0
    else 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(-1.0))
		tmp = Float32(1.0);
	elseif (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(-1.0))
		tmp = single(1.0);
	elseif (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 3 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 96.6%

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

   6. Taylor expanded in x around inf 96.5%

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

      1. mul-1-neg 96.5%

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

      2. unsub-neg 96.5%

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

   8. Simplified 96.5%

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

   9. Taylor expanded in s around 0 98.8%

      \[\leadsto \color{blue}{1} \]

   if -1 < (/.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 86.4%

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

   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 41.3%

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

      1. mul-1-neg 41.3%

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

      2. unsub-neg 41.3%

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

   5. Simplified 41.3%

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

   6. Taylor expanded in x around inf 41.3%

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

      1. mul-1-neg 41.3%

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

      2. distribute-frac-neg 41.3%

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

   8. Simplified 41.3%

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

4. Final simplification 75.6%

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

Alternative 8: 75.6% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -1:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \left(1 + 0.3333333333333333 \cdot \frac{x \cdot -3}{s}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) -1.0)
   1.0
   (/ 1.0 (+ 1.0 (+ 1.0 (* 0.3333333333333333 (/ (* x -3.0) s)))))))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= -1.0f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f / (1.0f + (1.0f + (0.3333333333333333f * ((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) <= (-1.0e0)) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 / (1.0e0 + (1.0e0 + (0.3333333333333333e0 * ((x * (-3.0e0)) / 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(1.0);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) + Float32(Float32(0.3333333333333333) * Float32(Float32(x * Float32(-3.0)) / 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);
	else
		tmp = single(1.0) / (single(1.0) + (single(1.0) + (single(0.3333333333333333) * ((x * single(-3.0)) / 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 96.6%

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

   6. Taylor expanded in x around inf 96.5%

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

      1. mul-1-neg 96.5%

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

      2. unsub-neg 96.5%

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

   8. Simplified 96.5%

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

   9. Taylor expanded in s around 0 98.8%

      \[\leadsto \color{blue}{1} \]

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

   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 99.7%

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

      3. add-sqr-sqrt 25.1%

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

      4. sqrt-unprod 40.8%

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

      5. sqr-neg 40.8%

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

      6. sqrt-unprod 17.3%

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

      7. add-sqr-sqrt 38.1%

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

      8. add-cbrt-cube 38.1%

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

      9. pow1/3 38.1%

         \[\leadsto \frac{1}{1 + \frac{1}{\color{blue}{{\left(\left(e^{\frac{-x}{s}} \cdot e^{\frac{-x}{s}}\right) \cdot e^{\frac{-x}{s}}\right)}^{0.3333333333333333}}}} \]

      10. pow-flip 38.1%

          \[\leadsto \frac{1}{1 + \color{blue}{{\left(\left(e^{\frac{-x}{s}} \cdot e^{\frac{-x}{s}}\right) \cdot e^{\frac{-x}{s}}\right)}^{\left(-0.3333333333333333\right)}}} \]

      11. pow3 38.1%

          \[\leadsto \frac{1}{1 + {\color{blue}{\left({\left(e^{\frac{-x}{s}}\right)}^{3}\right)}}^{\left(-0.3333333333333333\right)}} \]

      12. add-sqr-sqrt 17.3%

          \[\leadsto \frac{1}{1 + {\left({\left(e^{\frac{\color{blue}{\sqrt{-x} \cdot \sqrt{-x}}}{s}}\right)}^{3}\right)}^{\left(-0.3333333333333333\right)}} \]

      13. sqrt-unprod 40.8%

          \[\leadsto \frac{1}{1 + {\left({\left(e^{\frac{\color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}}}{s}}\right)}^{3}\right)}^{\left(-0.3333333333333333\right)}} \]

      14. sqr-neg 40.8%

          \[\leadsto \frac{1}{1 + {\left({\left(e^{\frac{\sqrt{\color{blue}{x \cdot x}}}{s}}\right)}^{3}\right)}^{\left(-0.3333333333333333\right)}} \]

      15. sqrt-unprod 25.1%

          \[\leadsto \frac{1}{1 + {\left({\left(e^{\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{s}}\right)}^{3}\right)}^{\left(-0.3333333333333333\right)}} \]

      16. add-sqr-sqrt 98.7%

          \[\leadsto \frac{1}{1 + {\left({\left(e^{\frac{\color{blue}{x}}{s}}\right)}^{3}\right)}^{\left(-0.3333333333333333\right)}} \]

   4. Applied egg-rr 98.7%

      \[\leadsto \frac{1}{1 + \color{blue}{{\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)}^{-0.3333333333333333}}} \]

   5. Taylor expanded in s around -inf 63.4%

      \[\leadsto \frac{1}{1 + \color{blue}{\left(1 + 0.3333333333333333 \cdot \frac{-2 \cdot x + -1 \cdot x}{s}\right)}} \]
   6. Step-by-step derivation

      1. distribute-rgt-out 63.4%

         \[\leadsto \frac{1}{1 + \left(1 + 0.3333333333333333 \cdot \frac{\color{blue}{x \cdot \left(-2 + -1\right)}}{s}\right)} \]

      2. metadata-eval 63.4%

         \[\leadsto \frac{1}{1 + \left(1 + 0.3333333333333333 \cdot \frac{x \cdot \color{blue}{-3}}{s}\right)} \]

   7. Simplified 63.4%

      \[\leadsto \frac{1}{1 + \color{blue}{\left(1 + 0.3333333333333333 \cdot \frac{x \cdot -3}{s}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -1:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \left(1 + 0.3333333333333333 \cdot \frac{x \cdot -3}{s}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 75.4% accurate, 7.2× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= -1.0f) {
		tmp = 1.0f;
	} 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 = 1.0e0
    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(1.0);
	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(1.0);
	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. 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 96.6%

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

   6. Taylor expanded in x around inf 96.5%

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

      1. mul-1-neg 96.5%

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

      2. unsub-neg 96.5%

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

   8. Simplified 96.5%

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

   9. Taylor expanded in s around 0 98.8%

      \[\leadsto \color{blue}{1} \]

   if -1 < (/.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 62.5%

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

      1. mul-1-neg 62.5%

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

      2. unsub-neg 62.5%

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

   5. Simplified 62.5%

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

4. Final simplification 77.4%

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

Alternative 10: 68.5% accurate, 9.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.999999987845058 \cdot 10^{-8}:\\ \;\;\;\;-\frac{s}{x}\\ \mathbf{elif}\;x \leq 9.999999998199587 \cdot 10^{-24}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -1.999999987845058e-8)
   (- (/ s x))
   (if (<= x 9.999999998199587e-24) 0.5 1.0)))

C

float code(float x, float s) {
	float tmp;
	if (x <= -1.999999987845058e-8f) {
		tmp = -(s / x);
	} else if (x <= 9.999999998199587e-24f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-1.999999987845058e-8)) then
        tmp = -(s / x)
    else if (x <= 9.999999998199587e-24) then
        tmp = 0.5e0
    else
        tmp = 1.0e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-1.999999987845058e-8))
		tmp = Float32(-Float32(s / x));
	elseif (x <= Float32(9.999999998199587e-24))
		tmp = Float32(0.5);
	else
		tmp = Float32(1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-1.999999987845058e-8))
		tmp = -(s / x);
	elseif (x <= single(9.999999998199587e-24))
		tmp = single(0.5);
	else
		tmp = single(1.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -1.99999999e-8

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 48.6%

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

      1. mul-1-neg 48.6%

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

      2. unsub-neg 48.6%

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

   5. Simplified 48.6%

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

   6. Taylor expanded in x around inf 46.1%

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

      1. associate-*r/ 46.1%

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

      2. neg-mul-1 46.1%

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

   8. Simplified 46.1%

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

   if -1.99999999e-8 < x < 1e-23

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 67.3%

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

   if 1e-23 < x

   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 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 97.4%

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

   6. Taylor expanded in x around inf 90.0%

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

      1. mul-1-neg 90.0%

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

      2. unsub-neg 90.0%

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

   8. Simplified 90.0%

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

   9. Taylor expanded in s around 0 93.5%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.999999987845058 \cdot 10^{-8}:\\ \;\;\;\;-\frac{s}{x}\\ \mathbf{elif}\;x \leq 9.999999998199587 \cdot 10^{-24}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 57.4% accurate, 17.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 9.999999998199587 \cdot 10^{-24}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (if (<= x 9.999999998199587e-24) 0.5 1.0))

C

float code(float x, float s) {
	float tmp;
	if (x <= 9.999999998199587e-24f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 9.999999998199587e-24) then
        tmp = 0.5e0
    else
        tmp = 1.0e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(9.999999998199587e-24))
		tmp = Float32(0.5);
	else
		tmp = Float32(1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(9.999999998199587e-24))
		tmp = single(0.5);
	else
		tmp = single(1.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 1e-23

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 36.4%

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

   if 1e-23 < x

   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 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 97.4%

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

   6. Taylor expanded in x around inf 90.0%

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

      1. mul-1-neg 90.0%

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

      2. unsub-neg 90.0%

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

   8. Simplified 90.0%

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

   9. Taylor expanded in s around 0 93.5%

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

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9.999999998199587 \cdot 10^{-24}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 35.2% 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 34.4%

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

4. Final simplification 34.4%

   \[\leadsto 0.5 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024019 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))