Logistic function

Percentage Accurate: 99.8% → 99.8%

Time: 12.8s

Alternatives: 10

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, 0.4× 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(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. distribute-frac-neg 99.9%

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

   2. exp-neg 99.8%

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

   3. add-sqr-sqrt 46.4%

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

   4. sqrt-unprod 57.4%

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

   5. sqr-neg 57.4%

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

   6. sqrt-unprod 13.3%

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

   7. add-sqr-sqrt 25.9%

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

   8. add-cbrt-cube 25.9%

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

   9. pow1/3 25.9%

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

   10. pow-flip 25.9%

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

4. Applied egg-rr 99.5%

   \[\leadsto \frac{1}{1 + \color{blue}{{\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)}^{-0.3333333333333333}}} \]
5. Step-by-step derivation

   1. metadata-eval 99.5%

      \[\leadsto \frac{1}{1 + {\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)}^{\color{blue}{\left(2 \cdot -0.16666666666666666\right)}}} \]

   2. pow-sqr 99.5%

      \[\leadsto \frac{1}{1 + \color{blue}{{\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)}^{-0.16666666666666666} \cdot {\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)}^{-0.16666666666666666}}} \]

6. Simplified 99.5%

   \[\leadsto \frac{1}{1 + \color{blue}{{\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)}^{-0.16666666666666666} \cdot {\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)}^{-0.16666666666666666}}} \]
7. Step-by-step derivation

   1. add-sqr-sqrt 99.5%

      \[\leadsto \frac{1}{1 + \color{blue}{\sqrt{{\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)}^{-0.16666666666666666} \cdot {\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)}^{-0.16666666666666666}} \cdot \sqrt{{\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)}^{-0.16666666666666666} \cdot {\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)}^{-0.16666666666666666}}}} \]

   2. sqrt-unprod 99.5%

      \[\leadsto \frac{1}{1 + \color{blue}{\sqrt{\left({\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)}^{-0.16666666666666666} \cdot {\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)}^{-0.16666666666666666}\right) \cdot \left({\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)}^{-0.16666666666666666} \cdot {\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)}^{-0.16666666666666666}\right)}}} \]

   3. pow2 99.5%

      \[\leadsto \frac{1}{1 + \sqrt{\color{blue}{{\left({\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)}^{-0.16666666666666666}\right)}^{2}} \cdot \left({\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)}^{-0.16666666666666666} \cdot {\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)}^{-0.16666666666666666}\right)}} \]

   4. pow2 99.5%

      \[\leadsto \frac{1}{1 + \sqrt{{\left({\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)}^{-0.16666666666666666}\right)}^{2} \cdot \color{blue}{{\left({\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)}^{-0.16666666666666666}\right)}^{2}}}} \]

   5. pow-prod-up 99.5%

      \[\leadsto \frac{1}{1 + \sqrt{\color{blue}{{\left({\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)}^{-0.16666666666666666}\right)}^{\left(2 + 2\right)}}}} \]

   6. pow-pow 99.5%

      \[\leadsto \frac{1}{1 + \sqrt{{\color{blue}{\left({\left(e^{\frac{x}{s}}\right)}^{\left(3 \cdot -0.16666666666666666\right)}\right)}}^{\left(2 + 2\right)}}} \]

   7. metadata-eval 99.5%

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

   8. metadata-eval 99.5%

      \[\leadsto \frac{1}{1 + \sqrt{{\left({\left(e^{\frac{x}{s}}\right)}^{-0.5}\right)}^{\color{blue}{4}}}} \]

8. Applied egg-rr 99.5%

   \[\leadsto \frac{1}{1 + \color{blue}{\sqrt{{\left({\left(e^{\frac{x}{s}}\right)}^{-0.5}\right)}^{4}}}} \]
9. Step-by-step derivation

   1. exp-prod 99.5%

      \[\leadsto \frac{1}{1 + \sqrt{{\color{blue}{\left(e^{\frac{x}{s} \cdot -0.5}\right)}}^{4}}} \]

   2. *-commutative 99.5%

      \[\leadsto \frac{1}{1 + \sqrt{{\left(e^{\color{blue}{-0.5 \cdot \frac{x}{s}}}\right)}^{4}}} \]

   3. associate-*r/ 99.5%

      \[\leadsto \frac{1}{1 + \sqrt{{\left(e^{\color{blue}{\frac{-0.5 \cdot x}{s}}}\right)}^{4}}} \]

10. Simplified 99.5%

    \[\leadsto \frac{1}{1 + \color{blue}{\sqrt{{\left(e^{\frac{-0.5 \cdot x}{s}}\right)}^{4}}}} \]

11. Taylor expanded in x around inf 99.8%

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

    1. add-exp-log 99.8%

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

    2. log-rec 99.8%

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

    3. log1p-udef 99.8%

       \[\leadsto e^{-\color{blue}{\mathsf{log1p}\left({\left(e^{-0.5 \cdot \frac{x}{s}}\right)}^{2}\right)}} \]

    4. unpow2 99.8%

       \[\leadsto e^{-\mathsf{log1p}\left(\color{blue}{e^{-0.5 \cdot \frac{x}{s}} \cdot e^{-0.5 \cdot \frac{x}{s}}}\right)} \]

    5. *-commutative 99.8%

       \[\leadsto e^{-\mathsf{log1p}\left(e^{\color{blue}{\frac{x}{s} \cdot -0.5}} \cdot e^{-0.5 \cdot \frac{x}{s}}\right)} \]

    6. pow-exp 99.8%

       \[\leadsto e^{-\mathsf{log1p}\left(\color{blue}{{\left(e^{\frac{x}{s}}\right)}^{-0.5}} \cdot e^{-0.5 \cdot \frac{x}{s}}\right)} \]

    7. *-commutative 99.8%

       \[\leadsto e^{-\mathsf{log1p}\left({\left(e^{\frac{x}{s}}\right)}^{-0.5} \cdot e^{\color{blue}{\frac{x}{s} \cdot -0.5}}\right)} \]

    8. pow-exp 99.8%

       \[\leadsto e^{-\mathsf{log1p}\left({\left(e^{\frac{x}{s}}\right)}^{-0.5} \cdot \color{blue}{{\left(e^{\frac{x}{s}}\right)}^{-0.5}}\right)} \]

    9. pow-prod-up 99.8%

       \[\leadsto e^{-\mathsf{log1p}\left(\color{blue}{{\left(e^{\frac{x}{s}}\right)}^{\left(-0.5 + -0.5\right)}}\right)} \]

    10. metadata-eval 99.8%

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

13. Applied egg-rr 99.8%

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

    1. *-rgt-identity 99.8%

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

    2. *-rgt-identity 99.8%

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

    3. unpow-1 99.8%

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

    4. rec-exp 99.9%

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

    5. distribute-neg-frac 99.9%

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

15. Simplified 99.9%

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

16. Final simplification 99.9%

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

function code(x, s)
	return Float32(Float32(1.0) / Float32(exp(Float32(Float32(-x) / 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 99.9%

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

3. Final simplification 99.9%

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

Alternative 3: 63.1% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq 100:\\ \;\;\;\;0.5\\ \mathbf{elif}\;t_0 \leq 1.9999999360571385 \cdot 10^{+38}:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{\frac{s \cdot \left(-s\right)}{x}}}{2 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)))
   (if (<= t_0 100.0)
     0.5
     (if (<= t_0 1.9999999360571385e+38)
       (/ 1.0 (/ (- 4.0 (/ x (/ (* s (- s)) x))) (+ 2.0 (/ x s))))
       (/ 1.0 (/ x s))))))

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if (t_0 <= 100.0f) {
		tmp = 0.5f;
	} else if (t_0 <= 1.9999999360571385e+38f) {
		tmp = 1.0f / ((4.0f - (x / ((s * -s) / x))) / (2.0f + (x / s)));
	} else {
		tmp = 1.0f / (x / s);
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: tmp
    t_0 = -x / s
    if (t_0 <= 100.0e0) then
        tmp = 0.5e0
    else if (t_0 <= 1.9999999360571385e+38) then
        tmp = 1.0e0 / ((4.0e0 - (x / ((s * -s) / x))) / (2.0e0 + (x / s)))
    else
        tmp = 1.0e0 / (x / s)
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = Float32(Float32(-x) / s)
	tmp = Float32(0.0)
	if (t_0 <= Float32(100.0))
		tmp = Float32(0.5);
	elseif (t_0 <= Float32(1.9999999360571385e+38))
		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(x / s));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = -x / s;
	tmp = single(0.0);
	if (t_0 <= single(100.0))
		tmp = single(0.5);
	elseif (t_0 <= single(1.9999999360571385e+38))
		tmp = single(1.0) / ((single(4.0) - (x / ((s * -s) / x))) / (single(2.0) + (x / s)));
	else
		tmp = single(1.0) / (x / s);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 52.0%

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

   if 100 < (/.f32 (neg.f32 x) s) < 1.99999994e38

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 10.5%

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

      1. mul-1-neg 10.5%

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

      2. unsub-neg 10.5%

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

   5. Simplified 10.5%

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

      1. sub-neg 10.5%

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

      2. flip-+ 48.7%

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

      3. metadata-eval 48.7%

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

      4. distribute-neg-frac 48.7%

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

      5. distribute-neg-frac 48.7%

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

      6. distribute-neg-frac 48.7%

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

   7. Applied egg-rr 48.7%

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

      1. clear-num 48.7%

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

      2. frac-times 52.6%

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

      3. *-un-lft-identity 52.6%

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

      4. add-sqr-sqrt 52.6%

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

      5. sqrt-unprod 52.5%

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

      6. sqr-neg 52.5%

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

      7. sqrt-unprod -0.0%

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

      8. add-sqr-sqrt 52.6%

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

      9. add-sqr-sqrt 52.6%

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

      10. sqrt-unprod 52.5%

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

      11. sqr-neg 52.5%

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

      12. sqrt-unprod -0.0%

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

      13. add-sqr-sqrt 52.6%

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

   9. Applied egg-rr 52.6%

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

       1. frac-2neg 52.6%

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

       2. associate-*l/ 73.7%

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

       3. add-sqr-sqrt 73.7%

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

       4. sqrt-unprod 67.9%

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

       5. sqr-neg 67.9%

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

       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 73.7%

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

   11. Applied egg-rr 73.7%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. distribute-frac-neg 100.0%

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

   8. Simplified 100.0%

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

      1. inv-pow 100.0%

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

      2. add-sqr-sqrt 100.0%

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

      3. sqrt-unprod 100.0%

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

      4. sqr-neg 100.0%

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

      5. sqrt-unprod -0.0%

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

      6. add-sqr-sqrt 100.0%

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

   10. Applied egg-rr 100.0%

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

       1. unpow-1 100.0%

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

   12. Simplified 100.0%

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

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 100:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\frac{-x}{s} \leq 1.9999999360571385 \cdot 10^{+38}:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{\frac{s \cdot \left(-s\right)}{x}}}{2 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 63.5% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq -1:\\ \;\;\;\;0.5\\ \mathbf{elif}\;t_0 \leq 1.9999999360571385 \cdot 10^{+38}:\\ \;\;\;\;\frac{1}{\frac{4 - x \cdot \frac{\frac{x}{s}}{s}}{2 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)))
   (if (<= t_0 -1.0)
     0.5
     (if (<= t_0 1.9999999360571385e+38)
       (/ 1.0 (/ (- 4.0 (* x (/ (/ x s) s))) (+ 2.0 (/ x s))))
       (/ 1.0 (/ x s))))))

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if (t_0 <= -1.0f) {
		tmp = 0.5f;
	} else if (t_0 <= 1.9999999360571385e+38f) {
		tmp = 1.0f / ((4.0f - (x * ((x / s) / s))) / (2.0f + (x / s)));
	} else {
		tmp = 1.0f / (x / s);
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: tmp
    t_0 = -x / s
    if (t_0 <= (-1.0e0)) then
        tmp = 0.5e0
    else if (t_0 <= 1.9999999360571385e+38) then
        tmp = 1.0e0 / ((4.0e0 - (x * ((x / s) / s))) / (2.0e0 + (x / s)))
    else
        tmp = 1.0e0 / (x / s)
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = Float32(Float32(-x) / s)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-1.0))
		tmp = Float32(0.5);
	elseif (t_0 <= Float32(1.9999999360571385e+38))
		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(x / s));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = -x / s;
	tmp = single(0.0);
	if (t_0 <= single(-1.0))
		tmp = single(0.5);
	elseif (t_0 <= single(1.9999999360571385e+38))
		tmp = single(1.0) / ((single(4.0) - (x * ((x / s) / s))) / (single(2.0) + (x / s)));
	else
		tmp = single(1.0) / (x / s);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 28.2%

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

   if -1 < (/.f32 (neg.f32 x) s) < 1.99999994e38

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 48.4%

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

      1. mul-1-neg 48.4%

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

      2. unsub-neg 48.4%

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

   5. Simplified 48.4%

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

      1. sub-neg 48.4%

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

      2. flip-+ 68.4%

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

      3. metadata-eval 68.4%

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

      4. distribute-neg-frac 68.4%

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

      5. distribute-neg-frac 68.4%

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

      6. distribute-neg-frac 68.4%

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

   7. Applied egg-rr 68.4%

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

      1. clear-num 68.4%

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

      2. frac-times 70.5%

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

      3. *-un-lft-identity 70.5%

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

      4. add-sqr-sqrt 49.3%

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

      5. sqrt-unprod 70.5%

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

      6. sqr-neg 70.5%

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

      7. sqrt-unprod 21.2%

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

      8. add-sqr-sqrt 70.2%

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

      9. add-sqr-sqrt 49.0%

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

      10. sqrt-unprod 70.4%

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

      11. sqr-neg 70.4%

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

      12. sqrt-unprod 21.2%

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

      13. add-sqr-sqrt 70.5%

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

   9. Applied egg-rr 70.5%

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

       1. clear-num 70.5%

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

       2. associate-/r/ 77.4%

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

       3. associate-/r* 77.4%

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

       4. clear-num 77.4%

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

   11. Applied egg-rr 77.4%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. distribute-frac-neg 100.0%

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

   8. Simplified 100.0%

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

      1. inv-pow 100.0%

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

      2. add-sqr-sqrt 100.0%

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

      3. sqrt-unprod 100.0%

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

      4. sqr-neg 100.0%

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

      5. sqrt-unprod -0.0%

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

      6. add-sqr-sqrt 100.0%

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

   10. Applied egg-rr 100.0%

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

       1. unpow-1 100.0%

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

   12. Simplified 100.0%

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

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -1:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\frac{-x}{s} \leq 1.9999999360571385 \cdot 10^{+38}:\\ \;\;\;\;\frac{1}{\frac{4 - x \cdot \frac{\frac{x}{s}}{s}}{2 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 49.7% accurate, 8.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 28.2%

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

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

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 58.9%

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

      1. mul-1-neg 58.9%

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

      2. unsub-neg 58.9%

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

   5. Simplified 58.9%

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

4. Final simplification 48.2%

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

Alternative 6: 48.0% accurate, 9.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp_2 = code(x, s)
	t_0 = -x / s;
	tmp = single(0.0);
	if (t_0 <= single(2.0))
		tmp = single(0.5);
	else
		tmp = single(1.0) / t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 52.8%

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

   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 39.1%

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

      1. mul-1-neg 39.1%

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

      2. unsub-neg 39.1%

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

   5. Simplified 39.1%

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

   6. Taylor expanded in x around inf 39.1%

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

      1. mul-1-neg 39.1%

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

      2. distribute-frac-neg 39.1%

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

   8. Simplified 39.1%

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

4. Final simplification 47.1%

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

Alternative 7: 47.3% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.000000013351432 \cdot 10^{-10}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -1.000000013351432e-10) (/ 1.0 (/ x s)) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -1.000000013351432e-10f) {
		tmp = 1.0f / (x / s);
	} else {
		tmp = 0.5f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-1.000000013351432e-10)) then
        tmp = 1.0e0 / (x / s)
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-1.000000013351432e-10))
		tmp = Float32(Float32(1.0) / Float32(x / s));
	else
		tmp = Float32(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-1.000000013351432e-10))
		tmp = single(1.0) / (x / s);
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -1.00000001e-10

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 46.9%

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

      1. mul-1-neg 46.9%

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

      2. unsub-neg 46.9%

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

   5. Simplified 46.9%

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

   6. Taylor expanded in x around inf 46.9%

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

      1. mul-1-neg 46.9%

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

      2. distribute-frac-neg 46.9%

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

   8. Simplified 46.9%

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

      1. inv-pow 46.9%

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

      2. add-sqr-sqrt 46.9%

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

      3. sqrt-unprod 52.8%

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

      4. sqr-neg 52.8%

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

      5. sqrt-unprod -0.0%

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

      6. add-sqr-sqrt 46.9%

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

   10. Applied egg-rr 46.9%

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

       1. unpow-1 46.9%

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

   12. Simplified 46.9%

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

   if -1.00000001e-10 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 46.9%

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

4. Final simplification 46.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.000000013351432 \cdot 10^{-10}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 46.2% accurate, 17.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.000000013351432 \cdot 10^{-10}:\\ \;\;\;\;\frac{-s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -1.000000013351432e-10) (/ (- s) x) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -1.000000013351432e-10f) {
		tmp = -s / x;
	} else {
		tmp = 0.5f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-1.000000013351432e-10)) then
        tmp = -s / x
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-1.000000013351432e-10))
		tmp = Float32(Float32(-s) / x);
	else
		tmp = Float32(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-1.000000013351432e-10))
		tmp = -s / x;
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -1.00000001e-10

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 46.9%

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

      1. mul-1-neg 46.9%

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

      2. unsub-neg 46.9%

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

   5. Simplified 46.9%

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

   6. Taylor expanded in x around inf 42.7%

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

      1. associate-*r/ 42.7%

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

      2. neg-mul-1 42.7%

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

   8. Simplified 42.7%

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

   if -1.00000001e-10 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 46.9%

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

4. Final simplification 45.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.000000013351432 \cdot 10^{-10}:\\ \;\;\;\;\frac{-s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 46.0% accurate, 21.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.000000013351432 \cdot 10^{-10}:\\ \;\;\;\;\frac{s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -1.000000013351432e-10) (/ s x) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -1.000000013351432e-10f) {
		tmp = s / x;
	} else {
		tmp = 0.5f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-1.000000013351432e-10)) then
        tmp = s / x
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-1.000000013351432e-10))
		tmp = Float32(s / x);
	else
		tmp = Float32(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-1.000000013351432e-10))
		tmp = s / x;
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -1.00000001e-10

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 46.9%

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

      1. mul-1-neg 46.9%

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

      2. unsub-neg 46.9%

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

   5. Simplified 46.9%

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

   6. Taylor expanded in x around inf 46.9%

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

      1. mul-1-neg 46.9%

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

      2. distribute-frac-neg 46.9%

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

   8. Simplified 46.9%

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

      1. expm1-log1p-u 46.9%

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

      2. expm1-udef 94.5%

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

      3. clear-num 94.5%

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

      4. add-sqr-sqrt 94.5%

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

      5. sqrt-unprod 94.5%

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

      6. sqr-neg 94.5%

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

      7. sqrt-unprod -0.0%

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

      8. add-sqr-sqrt 93.4%

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

   10. Applied egg-rr 93.4%

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

       1. expm1-def 42.7%

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

       2. expm1-log1p 42.7%

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

   12. Simplified 42.7%

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

   if -1.00000001e-10 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 46.9%

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

4. Final simplification 45.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.000000013351432 \cdot 10^{-10}:\\ \;\;\;\;\frac{s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 35.1% 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 33.5%

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

4. Final simplification 33.5%

   \[\leadsto 0.5 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024010 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))