Logistic function

Percentage Accurate: 99.8% → 99.8%

Time: 11.0s

Alternatives: 13

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, 0.6× speedup

Math

\[\begin{array}{l} \\ \frac{1}{{\left(e^{3}\right)}^{\left(-0.3333333333333333 \cdot \frac{x}{s}\right)} + 1} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (/ 1.0 (+ (pow (exp 3.0) (* -0.3333333333333333 (/ x s))) 1.0)))

C

float code(float x, float s) {
	return 1.0f / (powf(expf(3.0f), (-0.3333333333333333f * (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(3.0e0) ** ((-0.3333333333333333e0) * (x / s))) + 1.0e0)
end function

Julia

function code(x, s)
	return Float32(Float32(1.0) / Float32((exp(Float32(3.0)) ^ Float32(Float32(-0.3333333333333333) * Float32(x / s))) + Float32(1.0)))
end

MATLAB

function tmp = code(x, s)
	tmp = single(1.0) / ((exp(single(3.0)) ^ (single(-0.3333333333333333) * (x / s))) + single(1.0));
end

Derivation

1. Initial program 99.7%

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

   1. lift-exp.f32 N/A

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

   2. *-lft-identity N/A

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

   3. exp-prod N/A

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

   4. lower-pow.f32 N/A

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

   5. exp-1-e N/A

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

   6. lower-E.f32 99.7

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

4. Applied rewrites 99.7%

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

   1. lift-pow.f32 N/A

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

   2. lift-E.f32 N/A

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

   3. add-cbrt-cube N/A

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

   4. pow1/3 N/A

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

   5. metadata-eval N/A

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

   6. log-E N/A

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

   7. lift-E.f32 N/A

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

   8. log-pow N/A

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

   9. pow1/3 N/A

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

   10. lift-E.f32 N/A

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

   11. pow-pow N/A

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

   12. lower-pow.f32 N/A

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

   13. rem-cube-cbrt N/A

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

   14. add-cbrt-cube N/A

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

   15. e-exp-1 N/A

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

   16. pow-exp N/A

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

   17. metadata-eval N/A

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

   18. lower-exp.f32 N/A

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

   19. lower-*.f32 N/A

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

6. Applied rewrites 99.8%

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

7. Taylor expanded in s around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower-/.f32 99.8

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

9. Applied rewrites 99.8%

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

10. Final simplification 99.8%

    \[\leadsto \frac{1}{{\left(e^{3}\right)}^{\left(-0.3333333333333333 \cdot \frac{x}{s}\right)} + 1} \]
11. Add Preprocessing

Alternative 2: 79.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{e^{\frac{-x}{s}} + 1}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\frac{1}{\left(\frac{0.5}{s \cdot s} \cdot x\right) \cdot x + 1}\\ \mathbf{elif}\;t\_0 \leq 0.7496804594993591:\\ \;\;\;\;\frac{1}{\left(\left(\frac{0.5}{s} \cdot \left(\frac{x}{s} \cdot x\right) + 1\right) - \frac{x}{s}\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.5}{s}, x, -1\right), \frac{x}{s}, 1\right), 1, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ 1.0 (+ (exp (/ (- x) s)) 1.0))))
   (if (<= t_0 0.0)
     (/ 1.0 (+ (* (* (/ 0.5 (* s s)) x) x) 1.0))
     (if (<= t_0 0.7496804594993591)
       (/ 1.0 (+ (- (+ (* (/ 0.5 s) (* (/ x s) x)) 1.0) (/ x s)) 1.0))
       (/ 1.0 (fma (fma (fma (/ 0.5 s) x -1.0) (/ x s) 1.0) 1.0 1.0))))))

C

float code(float x, float s) {
	float t_0 = 1.0f / (expf((-x / s)) + 1.0f);
	float tmp;
	if (t_0 <= 0.0f) {
		tmp = 1.0f / ((((0.5f / (s * s)) * x) * x) + 1.0f);
	} else if (t_0 <= 0.7496804594993591f) {
		tmp = 1.0f / (((((0.5f / s) * ((x / s) * x)) + 1.0f) - (x / s)) + 1.0f);
	} else {
		tmp = 1.0f / fmaf(fmaf(fmaf((0.5f / s), x, -1.0f), (x / s), 1.0f), 1.0f, 1.0f);
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = Float32(Float32(1.0) / Float32(exp(Float32(Float32(-x) / s)) + Float32(1.0)))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(Float32(Float32(0.5) / Float32(s * s)) * x) * x) + Float32(1.0)));
	elseif (t_0 <= Float32(0.7496804594993591))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(Float32(Float32(Float32(0.5) / s) * Float32(Float32(x / s) * x)) + Float32(1.0)) - Float32(x / s)) + Float32(1.0)));
	else
		tmp = Float32(Float32(1.0) / fma(fma(fma(Float32(Float32(0.5) / s), x, Float32(-1.0)), Float32(x / s), Float32(1.0)), Float32(1.0), Float32(1.0)));
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s)))) < 0.0

   1. Initial program 100.0%

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

   3. Taylor expanded in s around inf

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

   5. Applied rewrites 6.3%

      \[\leadsto \frac{1}{1 + \color{blue}{\mathsf{fma}\left(\frac{x}{s}, \mathsf{fma}\left(\frac{0.5}{s}, x, -1\right), 1\right)}} \]

   6. Taylor expanded in s around 0

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

   8. Applied rewrites 89.0%

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

   if 0.0 < (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s)))) < 0.749680459

   1. Initial program 99.1%

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

   3. Taylor expanded in s around inf

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

   5. Applied rewrites 85.9%

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

   7. Applied rewrites 90.0%

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

   if 0.749680459 < (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s))))

   1. Initial program 100.0%

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

   3. Taylor expanded in s around inf

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

   5. Applied rewrites 28.1%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. *-lft-identity N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 100.0

         \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{x}{s}, \mathsf{fma}\left(\frac{0.5}{s}, x, -1\right), 1\right), 1, 1\right)}} \]

   7. Applied rewrites 100.0%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{e^{\frac{-x}{s}} + 1} \leq 0:\\ \;\;\;\;\frac{1}{\left(\frac{0.5}{s \cdot s} \cdot x\right) \cdot x + 1}\\ \mathbf{elif}\;\frac{1}{e^{\frac{-x}{s}} + 1} \leq 0.7496804594993591:\\ \;\;\;\;\frac{1}{\left(\left(\frac{0.5}{s} \cdot \left(\frac{x}{s} \cdot x\right) + 1\right) - \frac{x}{s}\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.5}{s}, x, -1\right), \frac{x}{s}, 1\right), 1, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{e^{\frac{-x}{s}} + 1}\\ \mathbf{if}\;t\_0 \leq 0.009999999776482582:\\ \;\;\;\;\frac{1}{\left(\frac{0.5}{s \cdot s} \cdot x\right) \cdot x + 1}\\ \mathbf{elif}\;t\_0 \leq 0.7496804594993591:\\ \;\;\;\;0.25 \cdot \frac{x}{s} + 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.5}{s}, x, -1\right), \frac{x}{s}, 1\right), 1, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ 1.0 (+ (exp (/ (- x) s)) 1.0))))
   (if (<= t_0 0.009999999776482582)
     (/ 1.0 (+ (* (* (/ 0.5 (* s s)) x) x) 1.0))
     (if (<= t_0 0.7496804594993591)
       (+ (* 0.25 (/ x s)) 0.5)
       (/ 1.0 (fma (fma (fma (/ 0.5 s) x -1.0) (/ x s) 1.0) 1.0 1.0))))))

C

float code(float x, float s) {
	float t_0 = 1.0f / (expf((-x / s)) + 1.0f);
	float tmp;
	if (t_0 <= 0.009999999776482582f) {
		tmp = 1.0f / ((((0.5f / (s * s)) * x) * x) + 1.0f);
	} else if (t_0 <= 0.7496804594993591f) {
		tmp = (0.25f * (x / s)) + 0.5f;
	} else {
		tmp = 1.0f / fmaf(fmaf(fmaf((0.5f / s), x, -1.0f), (x / s), 1.0f), 1.0f, 1.0f);
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = Float32(Float32(1.0) / Float32(exp(Float32(Float32(-x) / s)) + Float32(1.0)))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.009999999776482582))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(Float32(Float32(0.5) / Float32(s * s)) * x) * x) + Float32(1.0)));
	elseif (t_0 <= Float32(0.7496804594993591))
		tmp = Float32(Float32(Float32(0.25) * Float32(x / s)) + Float32(0.5));
	else
		tmp = Float32(Float32(1.0) / fma(fma(fma(Float32(Float32(0.5) / s), x, Float32(-1.0)), Float32(x / s), Float32(1.0)), Float32(1.0), Float32(1.0)));
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s)))) < 0.00999999978

   1. Initial program 99.5%

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

   3. Taylor expanded in s around inf

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

   5. Applied rewrites 6.7%

      \[\leadsto \frac{1}{1 + \color{blue}{\mathsf{fma}\left(\frac{x}{s}, \mathsf{fma}\left(\frac{0.5}{s}, x, -1\right), 1\right)}} \]

   6. Taylor expanded in s around 0

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

   8. Applied rewrites 85.0%

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

   if 0.00999999978 < (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s)))) < 0.749680459

   1. Initial program 99.6%

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

   3. Taylor expanded in s around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f32 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{4}, \frac{x}{s}, \frac{1}{2}\right)} \]

      3. lower-/.f32 91.1

         \[\leadsto \mathsf{fma}\left(0.25, \color{blue}{\frac{x}{s}}, 0.5\right) \]

   5. Applied rewrites 89.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.25, \frac{x}{s}, 0.5\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 95.8%

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

   if 0.749680459 < (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s))))

   1. Initial program 100.0%

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

   3. Taylor expanded in s around inf

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

   5. Applied rewrites 28.1%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. *-lft-identity N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 100.0

         \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{x}{s}, \mathsf{fma}\left(\frac{0.5}{s}, x, -1\right), 1\right), 1, 1\right)}} \]

   7. Applied rewrites 100.0%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{e^{\frac{-x}{s}} + 1} \leq 0.009999999776482582:\\ \;\;\;\;\frac{1}{\left(\frac{0.5}{s \cdot s} \cdot x\right) \cdot x + 1}\\ \mathbf{elif}\;\frac{1}{e^{\frac{-x}{s}} + 1} \leq 0.7496804594993591:\\ \;\;\;\;0.25 \cdot \frac{x}{s} + 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.5}{s}, x, -1\right), \frac{x}{s}, 1\right), 1, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{e^{\frac{-x}{s}} + 1}\\ \mathbf{if}\;t\_0 \leq 0.009999999776482582:\\ \;\;\;\;\frac{1}{\left(\frac{0.5}{s \cdot s} \cdot x\right) \cdot x + 1}\\ \mathbf{elif}\;t\_0 \leq 0.7496804594993591:\\ \;\;\;\;0.25 \cdot \frac{x}{s} + 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.5}{s} \cdot x, \frac{x}{s}, 1\right), 1, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ 1.0 (+ (exp (/ (- x) s)) 1.0))))
   (if (<= t_0 0.009999999776482582)
     (/ 1.0 (+ (* (* (/ 0.5 (* s s)) x) x) 1.0))
     (if (<= t_0 0.7496804594993591)
       (+ (* 0.25 (/ x s)) 0.5)
       (/ 1.0 (fma (fma (* (/ 0.5 s) x) (/ x s) 1.0) 1.0 1.0))))))

C

float code(float x, float s) {
	float t_0 = 1.0f / (expf((-x / s)) + 1.0f);
	float tmp;
	if (t_0 <= 0.009999999776482582f) {
		tmp = 1.0f / ((((0.5f / (s * s)) * x) * x) + 1.0f);
	} else if (t_0 <= 0.7496804594993591f) {
		tmp = (0.25f * (x / s)) + 0.5f;
	} else {
		tmp = 1.0f / fmaf(fmaf(((0.5f / s) * x), (x / s), 1.0f), 1.0f, 1.0f);
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = Float32(Float32(1.0) / Float32(exp(Float32(Float32(-x) / s)) + Float32(1.0)))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.009999999776482582))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(Float32(Float32(0.5) / Float32(s * s)) * x) * x) + Float32(1.0)));
	elseif (t_0 <= Float32(0.7496804594993591))
		tmp = Float32(Float32(Float32(0.25) * Float32(x / s)) + Float32(0.5));
	else
		tmp = Float32(Float32(1.0) / fma(fma(Float32(Float32(Float32(0.5) / s) * x), Float32(x / s), Float32(1.0)), Float32(1.0), Float32(1.0)));
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s)))) < 0.00999999978

   1. Initial program 99.5%

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

   3. Taylor expanded in s around inf

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

   5. Applied rewrites 6.7%

      \[\leadsto \frac{1}{1 + \color{blue}{\mathsf{fma}\left(\frac{x}{s}, \mathsf{fma}\left(\frac{0.5}{s}, x, -1\right), 1\right)}} \]

   6. Taylor expanded in s around 0

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

   8. Applied rewrites 85.0%

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

   if 0.00999999978 < (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s)))) < 0.749680459

   1. Initial program 99.6%

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

   3. Taylor expanded in s around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f32 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{4}, \frac{x}{s}, \frac{1}{2}\right)} \]

      3. lower-/.f32 91.1

         \[\leadsto \mathsf{fma}\left(0.25, \color{blue}{\frac{x}{s}}, 0.5\right) \]

   5. Applied rewrites 91.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.25, \frac{x}{s}, 0.5\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 95.8%

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

   if 0.749680459 < (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s))))

   1. Initial program 100.0%

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

   3. Taylor expanded in s around inf

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

   5. Applied rewrites 28.1%

      \[\leadsto \frac{1}{1 + \color{blue}{\mathsf{fma}\left(\frac{x}{s}, \mathsf{fma}\left(\frac{0.5}{s}, x, -1\right), 1\right)}} \]

   6. Taylor expanded in s around 0

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

   8. Applied rewrites 29.0%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. *-lft-identity N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 100.0

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

   10. Applied rewrites 100.0%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{e^{\frac{-x}{s}} + 1} \leq 0.009999999776482582:\\ \;\;\;\;\frac{1}{\left(\frac{0.5}{s \cdot s} \cdot x\right) \cdot x + 1}\\ \mathbf{elif}\;\frac{1}{e^{\frac{-x}{s}} + 1} \leq 0.7496804594993591:\\ \;\;\;\;0.25 \cdot \frac{x}{s} + 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.5}{s} \cdot x, \frac{x}{s}, 1\right), 1, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \frac{1}{{\left(e^{2}\right)}^{\left(-0.5 \cdot \frac{x}{s}\right)} + 1} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (/ 1.0 (+ (pow (exp 2.0) (* -0.5 (/ x s))) 1.0)))

C

float code(float x, float s) {
	return 1.0f / (powf(expf(2.0f), (-0.5f * (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(2.0e0) ** ((-0.5e0) * (x / s))) + 1.0e0)
end function

Julia

function code(x, s)
	return Float32(Float32(1.0) / Float32((exp(Float32(2.0)) ^ Float32(Float32(-0.5) * Float32(x / s))) + Float32(1.0)))
end

MATLAB

function tmp = code(x, s)
	tmp = single(1.0) / ((exp(single(2.0)) ^ (single(-0.5) * (x / s))) + single(1.0));
end

Derivation

1. Initial program 99.7%

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

   1. lift-exp.f32 N/A

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

   2. *-lft-identity N/A

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

   3. exp-prod N/A

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

   4. lower-pow.f32 N/A

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

   5. exp-1-e N/A

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

   6. lower-E.f32 99.7

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

4. Applied rewrites 99.7%

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

   1. lift-pow.f32 N/A

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

   2. sqr-pow N/A

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

   3. pow-prod-down N/A

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

   4. lift-E.f32 N/A

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

   5. lift-E.f32 N/A

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

   6. div-inv N/A

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

   7. lift-/.f32 N/A

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

   8. lift-neg.f32 N/A

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

   9. distribute-frac-neg N/A

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

   10. lift-/.f32 N/A

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

   11. neg-mul-1 N/A

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

   12. metadata-eval N/A

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

   13. associate-*r* N/A

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

   14. *-commutative N/A

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

   15. lift-/.f32 N/A

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

   16. associate-*r/ N/A

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

   17. associate-*l/ N/A

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

   18. lift-/.f32 N/A

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

   19. lift-*.f32 N/A

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

   20. lower-pow.f32 N/A

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

6. Applied rewrites 99.7%

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

7. Final simplification 99.7%

   \[\leadsto \frac{1}{{\left(e^{2}\right)}^{\left(-0.5 \cdot \frac{x}{s}\right)} + 1} \]
8. Add Preprocessing

Alternative 6: 99.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. div-inv N/A

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

   4. associate-/r* N/A

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

   5. lower-/.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. metadata-eval N/A

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

   8. lift-neg.f32 N/A

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

   9. frac-2neg N/A

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

   10. lower-/.f32 99.7

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

4. Applied rewrites 99.7%

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

5. Final simplification 99.7%

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

Alternative 7: 99.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. div-inv N/A

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

   4. associate-/r* N/A

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

   5. lower-/.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. metadata-eval N/A

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

   8. lift-neg.f32 N/A

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

   9. frac-2neg N/A

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

   10. lower-/.f32 99.7

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

4. Applied rewrites 99.7%

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

   1. lift-/.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. frac-2neg N/A

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

   4. metadata-eval N/A

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

   5. associate-/l/ N/A

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

   6. lower-/.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lower-neg.f32 99.7

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

6. Applied rewrites 99.7%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. frac-2neg N/A

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

   5. metadata-eval N/A

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

   6. lift-neg.f32 N/A

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

   7. un-div-inv N/A

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

   8. lift-neg.f32 N/A

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

   9. lift-neg.f32 N/A

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

   10. frac-2neg N/A

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

   11. div-inv N/A

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

   12. lower-*.f32 N/A

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

   13. lower-/.f32 99.7

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

8. Applied rewrites 99.7%

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

9. Final simplification 99.7%

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

Alternative 8: 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 99.7%

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

3. Final simplification 99.7%

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

Alternative 9: 64.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 40:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\frac{0.5}{s \cdot s} \cdot x\right) \cdot x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) 40.0) 0.5 (/ 1.0 (+ (* (* (/ 0.5 (* s s)) x) x) 1.0))))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= 40.0f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / ((((0.5f / (s * s)) * x) * x) + 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 / s) <= 40.0e0) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 / ((((0.5e0 / (s * s)) * x) * x) + 1.0e0)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(-x) / s) <= Float32(40.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(Float32(Float32(0.5) / Float32(s * s)) * x) * x) + Float32(1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((-x / s) <= single(40.0))
		tmp = single(0.5);
	else
		tmp = single(1.0) / ((((single(0.5) / (s * s)) * x) * x) + single(1.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 40

   1. Initial program 99.7%

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

   3. Taylor expanded in s around inf

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

   5. Applied rewrites 55.7%

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

   if 40 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.8%

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

   3. Taylor expanded in s around inf

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

   5. Applied rewrites 6.3%

      \[\leadsto \frac{1}{1 + \color{blue}{\mathsf{fma}\left(\frac{x}{s}, \mathsf{fma}\left(\frac{0.5}{s}, x, -1\right), 1\right)}} \]

   6. Taylor expanded in s around 0

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

   8. Applied rewrites 88.1%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 40:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\frac{0.5}{s \cdot s} \cdot x\right) \cdot x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 50.0% accurate, 2.3× speedup

Math

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

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) -1.0)
   (/ 1.0 (+ (fma (* (/ 1.0 s) x) -1.0 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 / (fmaf(((1.0f / s) * x), -1.0f, 1.0f) + 1.0f);
	} else {
		tmp = 1.0f / (2.0f - (x / s));
	}
	return tmp;
}

Julia

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in s around inf

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

   5. Applied rewrites 28.2%

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

   7. Applied rewrites 29.0%

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

   9. Applied rewrites 29.0%

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

   10. Taylor expanded in s around inf

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

   12. Applied rewrites 29.0%

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

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

   1. Initial program 99.6%

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

   3. Taylor expanded in s around inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 63.2

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

   5. Applied rewrites 63.2%

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

4. Final simplification 51.6%

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

Alternative 11: 47.5% accurate, 2.5× speedup

Math

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

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) -1.0)
   (/ 1.0 (+ (fma (/ x s) -1.0 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 / (fmaf((x / s), -1.0f, 1.0f) + 1.0f);
	} else {
		tmp = 1.0f / (2.0f - (x / s));
	}
	return tmp;
}

Julia

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in s around inf

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

   5. Applied rewrites 28.2%

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

   7. Applied rewrites 29.0%

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

   9. Applied rewrites 28.2%

      \[\leadsto \frac{1}{1 + \mathsf{fma}\left(\frac{x}{s}, \mathsf{fma}\left(x, \color{blue}{\frac{0.5}{s}}, -1\right), 1\right)} \]

   10. Taylor expanded in s around inf

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

   12. Applied rewrites 29.0%

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

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

   1. Initial program 99.6%

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

   3. Taylor expanded in s around inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 63.2

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

   5. Applied rewrites 63.2%

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

4. Final simplification 51.6%

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

Alternative 12: 50.0% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in s around inf

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

   5. Applied rewrites 28.2%

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

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

   1. Initial program 99.6%

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

   3. Taylor expanded in s around inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 63.2

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

   5. Applied rewrites 63.2%

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

Alternative 13: 35.5% accurate, 128.0× 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 99.7%

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

3. Taylor expanded in s around inf

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

5. Applied rewrites 36.8%

   \[\leadsto \color{blue}{0.5} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024288 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))