Logistic function

Percentage Accurate: 99.8% → 99.8%

Time: 10.0s

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, 0.3× speedup

Math

\[\begin{array}{l} \\ \frac{1}{e^{-0.25 \cdot \frac{x}{s}} \cdot {\left(e^{-0.75} \cdot e^{-0.75}\right)}^{\left(\frac{\frac{x}{s}}{2}\right)} + 1} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (/
  1.0
  (+
   (*
    (exp (* -0.25 (/ x s)))
    (pow (* (exp -0.75) (exp -0.75)) (/ (/ x s) 2.0)))
   1.0)))

C

float code(float x, float s) {
	return 1.0f / ((expf((-0.25f * (x / s))) * powf((expf(-0.75f) * expf(-0.75f)), ((x / s) / 2.0f))) + 1.0f);
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 1.0e0 / ((exp(((-0.25e0) * (x / s))) * ((exp((-0.75e0)) * exp((-0.75e0))) ** ((x / s) / 2.0e0))) + 1.0e0)
end function

Julia

function code(x, s)
	return Float32(Float32(1.0) / Float32(Float32(exp(Float32(Float32(-0.25) * Float32(x / s))) * (Float32(exp(Float32(-0.75)) * exp(Float32(-0.75))) ^ Float32(Float32(x / s) / Float32(2.0)))) + Float32(1.0)))
end

MATLAB

function tmp = code(x, s)
	tmp = single(1.0) / ((exp((single(-0.25) * (x / s))) * ((exp(single(-0.75)) * exp(single(-0.75))) ^ ((x / s) / single(2.0)))) + 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%

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

6. Applied rewrites 99.7%

   \[\leadsto \frac{1}{1 + \color{blue}{{\left({\mathsf{E}\left(\right)}^{1.5}\right)}^{\left(-0.5 \cdot \frac{x}{s}\right)} \cdot {\left(e^{-0.5 \cdot \frac{x}{s}}\right)}^{0.5}}} \]
7. Step-by-step derivation

   1. lift-pow.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. pow-unpow N/A

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

   4. sqr-pow N/A

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

   5. pow-prod-down N/A

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

   6. lower-pow.f32 N/A

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

8. Applied rewrites 99.8%

   \[\leadsto \frac{1}{1 + \color{blue}{{\left(e^{-0.75} \cdot e^{-0.75}\right)}^{\left(\frac{\frac{x}{s}}{2}\right)}} \cdot {\left(e^{-0.5 \cdot \frac{x}{s}}\right)}^{0.5}} \]
9. Step-by-step derivation

   1. lift-pow.f32 N/A

      \[\leadsto \frac{1}{1 + {\left(e^{\frac{-3}{4}} \cdot e^{\frac{-3}{4}}\right)}^{\left(\frac{\frac{x}{s}}{2}\right)} \cdot \color{blue}{{\left(e^{\frac{-1}{2} \cdot \frac{x}{s}}\right)}^{\frac{1}{2}}}} \]

   2. lift-exp.f32 N/A

      \[\leadsto \frac{1}{1 + {\left(e^{\frac{-3}{4}} \cdot e^{\frac{-3}{4}}\right)}^{\left(\frac{\frac{x}{s}}{2}\right)} \cdot {\color{blue}{\left(e^{\frac{-1}{2} \cdot \frac{x}{s}}\right)}}^{\frac{1}{2}}} \]

   3. pow-exp N/A

      \[\leadsto \frac{1}{1 + {\left(e^{\frac{-3}{4}} \cdot e^{\frac{-3}{4}}\right)}^{\left(\frac{\frac{x}{s}}{2}\right)} \cdot \color{blue}{e^{\left(\frac{-1}{2} \cdot \frac{x}{s}\right) \cdot \frac{1}{2}}}} \]

   4. lower-exp.f32 N/A

      \[\leadsto \frac{1}{1 + {\left(e^{\frac{-3}{4}} \cdot e^{\frac{-3}{4}}\right)}^{\left(\frac{\frac{x}{s}}{2}\right)} \cdot \color{blue}{e^{\left(\frac{-1}{2} \cdot \frac{x}{s}\right) \cdot \frac{1}{2}}}} \]

   5. lift-*.f32 N/A

      \[\leadsto \frac{1}{1 + {\left(e^{\frac{-3}{4}} \cdot e^{\frac{-3}{4}}\right)}^{\left(\frac{\frac{x}{s}}{2}\right)} \cdot e^{\color{blue}{\left(\frac{-1}{2} \cdot \frac{x}{s}\right)} \cdot \frac{1}{2}}} \]

   6. *-commutative N/A

      \[\leadsto \frac{1}{1 + {\left(e^{\frac{-3}{4}} \cdot e^{\frac{-3}{4}}\right)}^{\left(\frac{\frac{x}{s}}{2}\right)} \cdot e^{\color{blue}{\left(\frac{x}{s} \cdot \frac{-1}{2}\right)} \cdot \frac{1}{2}}} \]

   7. associate-*l* N/A

      \[\leadsto \frac{1}{1 + {\left(e^{\frac{-3}{4}} \cdot e^{\frac{-3}{4}}\right)}^{\left(\frac{\frac{x}{s}}{2}\right)} \cdot e^{\color{blue}{\frac{x}{s} \cdot \left(\frac{-1}{2} \cdot \frac{1}{2}\right)}}} \]

   8. metadata-eval N/A

      \[\leadsto \frac{1}{1 + {\left(e^{\frac{-3}{4}} \cdot e^{\frac{-3}{4}}\right)}^{\left(\frac{\frac{x}{s}}{2}\right)} \cdot e^{\frac{x}{s} \cdot \color{blue}{\frac{-1}{4}}}} \]

   9. metadata-eval N/A

      \[\leadsto \frac{1}{1 + {\left(e^{\frac{-3}{4}} \cdot e^{\frac{-3}{4}}\right)}^{\left(\frac{\frac{x}{s}}{2}\right)} \cdot e^{\frac{x}{s} \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right)}}} \]

   10. lower-*.f32 N/A

       \[\leadsto \frac{1}{1 + {\left(e^{\frac{-3}{4}} \cdot e^{\frac{-3}{4}}\right)}^{\left(\frac{\frac{x}{s}}{2}\right)} \cdot e^{\color{blue}{\frac{x}{s} \cdot \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)}}} \]

   11. metadata-eval 99.8

       \[\leadsto \frac{1}{1 + {\left(e^{-0.75} \cdot e^{-0.75}\right)}^{\left(\frac{\frac{x}{s}}{2}\right)} \cdot e^{\frac{x}{s} \cdot \color{blue}{-0.25}}} \]

10. Applied rewrites 99.8%

    \[\leadsto \frac{1}{1 + {\left(e^{-0.75} \cdot e^{-0.75}\right)}^{\left(\frac{\frac{x}{s}}{2}\right)} \cdot \color{blue}{e^{\frac{x}{s} \cdot -0.25}}} \]

11. Final simplification 99.8%

    \[\leadsto \frac{1}{e^{-0.25 \cdot \frac{x}{s}} \cdot {\left(e^{-0.75} \cdot e^{-0.75}\right)}^{\left(\frac{\frac{x}{s}}{2}\right)} + 1} \]
12. Add Preprocessing

Alternative 2: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \frac{1}{e^{\left(\frac{x}{s} \cdot -0.75\right) \cdot 0.3333333333333333} \cdot {\left({\mathsf{E}\left(\right)}^{1.5}\right)}^{\left(-0.5 \cdot \frac{x}{s}\right)} + 1} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (/
  1.0
  (+
   (*
    (exp (* (* (/ x s) -0.75) 0.3333333333333333))
    (pow (pow (E) 1.5) (* -0.5 (/ x s))))
   1.0)))

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%

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

6. Applied rewrites 99.7%

   \[\leadsto \frac{1}{1 + \color{blue}{{\left({\mathsf{E}\left(\right)}^{1.5}\right)}^{\left(-0.5 \cdot \frac{x}{s}\right)} \cdot {\left(e^{-0.5 \cdot \frac{x}{s}}\right)}^{0.5}}} \]
7. Step-by-step derivation

   1. lift-pow.f32 N/A

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

   2. lift-exp.f32 N/A

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

   3. pow-exp N/A

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

   4. *-commutative N/A

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

   5. exp-prod N/A

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

   6. metadata-eval N/A

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

   7. pow-exp N/A

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

   8. e-exp-1 N/A

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

   9. add-cube-cbrt N/A

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

   10. pow3 N/A

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

   11. pow-pow N/A

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

   12. metadata-eval N/A

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

   13. pow-unpow N/A

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

   14. pow-to-exp N/A

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

   15. lower-exp.f32 N/A

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

   16. lower-*.f32 N/A

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

   17. lift-E.f32 N/A

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

   18. pow1/3 N/A

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

   19. log-pow N/A

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

   20. lift-E.f32 N/A

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

   21. log-E N/A

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

   22. metadata-eval N/A

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

8. Applied rewrites 99.7%

   \[\leadsto \frac{1}{1 + {\left({\mathsf{E}\left(\right)}^{1.5}\right)}^{\left(-0.5 \cdot \frac{x}{s}\right)} \cdot \color{blue}{e^{0.3333333333333333 \cdot \left(-0.75 \cdot \frac{x}{s}\right)}}} \]

9. Final simplification 99.7%

   \[\leadsto \frac{1}{e^{\left(\frac{x}{s} \cdot -0.75\right) \cdot 0.3333333333333333} \cdot {\left({\mathsf{E}\left(\right)}^{1.5}\right)}^{\left(-0.5 \cdot \frac{x}{s}\right)} + 1} \]
10. Add Preprocessing

Alternative 3: 92.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{-x}{s}}\\ \mathbf{if}\;t\_0 \leq 9.999999682655225 \cdot 10^{-22}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(1, 1, 1\right)}\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;0.25 \cdot \frac{x}{s} + 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\frac{0.5}{s \cdot s} \cdot x\right) \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ (- x) s))))
   (if (<= t_0 9.999999682655225e-22)
     (/ 1.0 (fma 1.0 1.0 1.0))
     (if (<= t_0 2.0)
       (+ (* 0.25 (/ x s)) 0.5)
       (/ 1.0 (* (* (/ 0.5 (* s s)) x) x))))))

C

float code(float x, float s) {
	float t_0 = expf((-x / s));
	float tmp;
	if (t_0 <= 9.999999682655225e-22f) {
		tmp = 1.0f / fmaf(1.0f, 1.0f, 1.0f);
	} else if (t_0 <= 2.0f) {
		tmp = (0.25f * (x / s)) + 0.5f;
	} else {
		tmp = 1.0f / (((0.5f / (s * s)) * x) * x);
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = exp(Float32(Float32(-x) / s))
	tmp = Float32(0.0)
	if (t_0 <= Float32(9.999999682655225e-22))
		tmp = Float32(Float32(1.0) / fma(Float32(1.0), Float32(1.0), Float32(1.0)));
	elseif (t_0 <= Float32(2.0))
		tmp = Float32(Float32(Float32(0.25) * Float32(x / s)) + Float32(0.5));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(Float32(0.5) / Float32(s * s)) * x) * x));
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if (exp.f32 (/.f32 (neg.f32 x) s)) < 9.9999997e-22

   1. Initial program 100.0%

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

   3. Taylor expanded in s around inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 5.3

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

   5. Applied rewrites 5.3%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. *-lft-identity N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 100.0

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

   7. Applied rewrites 98.9%

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

   8. Taylor expanded in s around inf

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

   10. Applied rewrites 98.9%

       \[\leadsto \frac{1}{\mathsf{fma}\left(1, 1, 1\right)} \]

   if 9.9999997e-22 < (exp.f32 (/.f32 (neg.f32 x) s)) < 2

   1. Initial program 99.7%

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

   3. Taylor expanded in s around inf

      \[\leadsto \color{blue}{\frac{1}{2} + \frac{1}{4} \cdot \frac{x}{s}} \]
   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 89.2

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

   5. Applied rewrites 88.0%

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

   7. Applied rewrites 97.4%

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

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

   1. Initial program 99.5%

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

   3. Taylor expanded in s around inf

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

   5. Applied rewrites 6.8%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\frac{x}{s}, \mathsf{fma}\left(\frac{0.5}{s}, x, -1\right), 2\right)}} \]

   6. Taylor expanded in s around 0

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

   8. Applied rewrites 77.4%

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

Alternative 4: 75.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{e^{\frac{-x}{s}} + 1} \leq 0.5666321516036987:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(1, 1, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ 1.0 (+ (exp (/ (- x) s)) 1.0)) 0.5666321516036987)
   (/ 1.0 (- 2.0 (/ x s)))
   (/ 1.0 (fma 1.0 1.0 1.0))))

C

float code(float x, float s) {
	float tmp;
	if ((1.0f / (expf((-x / s)) + 1.0f)) <= 0.5666321516036987f) {
		tmp = 1.0f / (2.0f - (x / s));
	} else {
		tmp = 1.0f / fmaf(1.0f, 1.0f, 1.0f);
	}
	return tmp;
}

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) / Float32(exp(Float32(Float32(-x) / s)) + Float32(1.0))) <= Float32(0.5666321516036987))
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) - Float32(x / s)));
	else
		tmp = Float32(Float32(1.0) / fma(Float32(1.0), Float32(1.0), Float32(1.0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s)))) < 0.566632152

   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 67.2

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

   5. Applied rewrites 67.2%

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

   if 0.566632152 < (/.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

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 5.3

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

   5. Applied rewrites 5.3%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. *-lft-identity N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 100.0

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

   7. Applied rewrites 98.9%

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

   8. Taylor expanded in s around inf

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

   10. Applied rewrites 98.9%

       \[\leadsto \frac{1}{\mathsf{fma}\left(1, 1, 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{e^{\frac{-x}{s}} + 1} \leq 0.5666321516036987:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(1, 1, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 62.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{e^{\frac{-x}{s}} + 1} \leq 0.5666321516036987:\\ \;\;\;\;0.25 \cdot \frac{x}{s} + 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(1, 1, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ 1.0 (+ (exp (/ (- x) s)) 1.0)) 0.5666321516036987)
   (+ (* 0.25 (/ x s)) 0.5)
   (/ 1.0 (fma 1.0 1.0 1.0))))

C

float code(float x, float s) {
	float tmp;
	if ((1.0f / (expf((-x / s)) + 1.0f)) <= 0.5666321516036987f) {
		tmp = (0.25f * (x / s)) + 0.5f;
	} else {
		tmp = 1.0f / fmaf(1.0f, 1.0f, 1.0f);
	}
	return tmp;
}

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) / Float32(exp(Float32(Float32(-x) / s)) + Float32(1.0))) <= Float32(0.5666321516036987))
		tmp = Float32(Float32(Float32(0.25) * Float32(x / s)) + Float32(0.5));
	else
		tmp = Float32(Float32(1.0) / fma(Float32(1.0), Float32(1.0), Float32(1.0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s)))) < 0.566632152

   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 44.7

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

   5. Applied rewrites 44.2%

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

   7. Applied rewrites 47.0%

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

   if 0.566632152 < (/.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

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 5.3

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

   5. Applied rewrites 5.3%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. *-lft-identity N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 100.0

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

   7. Applied rewrites 98.9%

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

   8. Taylor expanded in s around inf

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

   10. Applied rewrites 98.9%

       \[\leadsto \frac{1}{\mathsf{fma}\left(1, 1, 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{e^{\frac{-x}{s}} + 1} \leq 0.5666321516036987:\\ \;\;\;\;0.25 \cdot \frac{x}{s} + 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(1, 1, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 60.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{e^{\frac{-x}{s}} + 1} \leq 0.5666321516036987:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(1, 1, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ 1.0 (+ (exp (/ (- x) s)) 1.0)) 0.5666321516036987)
   0.5
   (/ 1.0 (fma 1.0 1.0 1.0))))

C

float code(float x, float s) {
	float tmp;
	if ((1.0f / (expf((-x / s)) + 1.0f)) <= 0.5666321516036987f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / fmaf(1.0f, 1.0f, 1.0f);
	}
	return tmp;
}

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) / Float32(exp(Float32(Float32(-x) / s)) + Float32(1.0))) <= Float32(0.5666321516036987))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / fma(Float32(1.0), Float32(1.0), Float32(1.0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s)))) < 0.566632152

   1. Initial program 99.6%

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

   3. Taylor expanded in s around inf

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

   5. Applied rewrites 44.7%

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

   if 0.566632152 < (/.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

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 5.3

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

   5. Applied rewrites 5.3%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. *-lft-identity N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 100.0

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

   7. Applied rewrites 98.9%

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

   8. Taylor expanded in s around inf

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

   10. Applied rewrites 98.9%

       \[\leadsto \frac{1}{\mathsf{fma}\left(1, 1, 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{e^{\frac{-x}{s}} + 1} \leq 0.5666321516036987:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(1, 1, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{\frac{-x}{s}} \leq 9.999999682655225 \cdot 10^{-22}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(1, 1, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(1 - \frac{x}{s}\right) + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (exp (/ (- x) s)) 9.999999682655225e-22)
   (/ 1.0 (fma 1.0 1.0 1.0))
   (/ 1.0 (+ (- 1.0 (/ x s)) 1.0))))

C

float code(float x, float s) {
	float tmp;
	if (expf((-x / s)) <= 9.999999682655225e-22f) {
		tmp = 1.0f / fmaf(1.0f, 1.0f, 1.0f);
	} else {
		tmp = 1.0f / ((1.0f - (x / s)) + 1.0f);
	}
	return tmp;
}

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (exp(Float32(Float32(-x) / s)) <= Float32(9.999999682655225e-22))
		tmp = Float32(Float32(1.0) / fma(Float32(1.0), Float32(1.0), Float32(1.0)));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(1.0) - Float32(x / s)) + Float32(1.0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (exp.f32 (/.f32 (neg.f32 x) s)) < 9.9999997e-22

   1. Initial program 100.0%

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

   3. Taylor expanded in s around inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 5.3

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

   5. Applied rewrites 5.3%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. *-lft-identity N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 100.0

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

   7. Applied rewrites 98.9%

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

   8. Taylor expanded in s around inf

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

   10. Applied rewrites 98.9%

       \[\leadsto \frac{1}{\mathsf{fma}\left(1, 1, 1\right)} \]

   if 9.9999997e-22 < (exp.f32 (/.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

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 67.3

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

   5. Applied rewrites 67.3%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\frac{-x}{s}} \leq 9.999999682655225 \cdot 10^{-22}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(1, 1, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(1 - \frac{x}{s}\right) + 1}\\ \end{array} \]
5. 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: 89.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -40:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(1, 1, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(0.5 \cdot \frac{\frac{x}{s}}{s}\right) \cdot x + 2\right) - \frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) -40.0)
   (/ 1.0 (fma 1.0 1.0 1.0))
   (/ 1.0 (- (+ (* (* 0.5 (/ (/ x s) s)) x) 2.0) (/ x s)))))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= -40.0f) {
		tmp = 1.0f / fmaf(1.0f, 1.0f, 1.0f);
	} else {
		tmp = 1.0f / ((((0.5f * ((x / s) / s)) * x) + 2.0f) - (x / s));
	}
	return tmp;
}

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(-x) / s) <= Float32(-40.0))
		tmp = Float32(Float32(1.0) / fma(Float32(1.0), Float32(1.0), Float32(1.0)));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(Float32(Float32(0.5) * Float32(Float32(x / s) / s)) * x) + Float32(2.0)) - Float32(x / s)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in s around inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 5.3

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

   5. Applied rewrites 5.3%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. *-lft-identity N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 100.0

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

   7. Applied rewrites 98.9%

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

   8. Taylor expanded in s around inf

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

   10. Applied rewrites 98.9%

       \[\leadsto \frac{1}{\mathsf{fma}\left(1, 1, 1\right)} \]

   if -40 < (/.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

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

   5. Applied rewrites 44.7%

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

   7. Applied rewrites 85.3%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -40:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(1, 1, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(0.5 \cdot \frac{\frac{x}{s}}{s}\right) \cdot x + 2\right) - \frac{x}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 35.4% 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 38.8%

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

Reproduce

herbie shell --seed 2024267 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))