Logistic function

Percentage Accurate: 99.8% → 99.9%

Time: 10.8s

Alternatives: 16

Speedup: 1.0×

Specification

\[0 \leq s \land s \leq 1.0651631\]

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ {\left(e^{2}\right)}^{\left(-0.5 \cdot \mathsf{log1p}\left(e^{-\frac{x}{s}}\right)\right)} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (pow (exp 2.0) (* -0.5 (log1p (exp (- (/ x s)))))))

C

float code(float x, float s) {
	return powf(expf(2.0f), (-0.5f * log1pf(expf(-(x / s)))));
}

Julia

function code(x, s)
	return exp(Float32(2.0)) ^ Float32(Float32(-0.5) * log1p(exp(Float32(-Float32(x / s)))))
end

Derivation

1. Initial program 99.8%

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

   1. lift-/.f32 N/A

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

   2. inv-pow N/A

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

   3. pow-to-exp N/A

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

   4. *-commutative N/A

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

   5. log-pow N/A

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

   6. inv-pow N/A

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

   7. lift-/.f32 N/A

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

   8. lower-exp.f32 N/A

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

   9. lift-/.f32 N/A

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

   10. log-rec N/A

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

   11. lower-neg.f32 N/A

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

   12. lift-+.f32 N/A

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

   13. lower-log1p.f32 99.8

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

4. Applied rewrites 99.8%

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

   1. lift-exp.f32 N/A

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

   2. lift-neg.f32 N/A

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

   3. lift-log1p.f32 N/A

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

   4. lift-+.f32 N/A

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

   5. neg-log N/A

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

   6. inv-pow N/A

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

   7. metadata-eval N/A

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

   8. pow-pow N/A

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

   9. lift-pow.f32 N/A

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

   10. log-pow N/A

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

   11. exp-prod N/A

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

   12. lower-pow.f32 N/A

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

   13. lower-exp.f32 N/A

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

   14. lift-pow.f32 N/A

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

   15. pow-to-exp N/A

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

6. Applied rewrites 99.8%

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

7. Final simplification 99.8%

   \[\leadsto {\left(e^{2}\right)}^{\left(-0.5 \cdot \mathsf{log1p}\left(e^{-\frac{x}{s}}\right)\right)} \]
8. Add Preprocessing

Alternative 2: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ e^{-\mathsf{log1p}\left(e^{-\frac{x}{s}}\right)} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (exp (- (log1p (exp (- (/ x s)))))))

C

float code(float x, float s) {
	return expf(-log1pf(expf(-(x / s))));
}

Julia

function code(x, s)
	return exp(Float32(-log1p(exp(Float32(-Float32(x / s))))))
end

Derivation

1. Initial program 99.8%

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

   1. lift-/.f32 N/A

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

   2. inv-pow N/A

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

   3. pow-to-exp N/A

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

   4. *-commutative N/A

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

   5. log-pow N/A

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

   6. inv-pow N/A

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

   7. lift-/.f32 N/A

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

   8. lower-exp.f32 N/A

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

   9. lift-/.f32 N/A

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

   10. log-rec N/A

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

   11. lower-neg.f32 N/A

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

   12. lift-+.f32 N/A

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

   13. lower-log1p.f32 99.9

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

4. Applied rewrites 99.9%

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

5. Final simplification 99.9%

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

Reproduce

herbie shell --seed 2024230 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))