Logistic function

Average Error: 0.1 → 0.1

Time: 9.5s

Precision: binary32

Cost: 6656

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

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

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

Derivation

1. Initial program 0.1

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

2. Applied egg-rr 0.1

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

3. Applied egg-rr 0.1

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

4. Applied egg-rr 0.1

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

5. Applied egg-rr 0.1

   \[\leadsto \frac{1}{1 + \color{blue}{\left({\left(e^{-1}\right)}^{\left(\frac{x}{s}\right)} + 0\right)}} \]

6. Simplified 0.1

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

   [Start] 0.1	\[ \frac{1}{1 + \left({\left(e^{-1}\right)}^{\left(\frac{x}{s}\right)} + 0\right)} \]
   +-rgt-identity [=>] 0.1	\[ \frac{1}{1 + \color{blue}{{\left(e^{-1}\right)}^{\left(\frac{x}{s}\right)}}} \]

7. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.1
Cost	3456

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

Alternative 2
Error	10.0
Cost	196

\[\begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq 5:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 3
Error	20.8
Cost	32

\[0.5 \]

Reproduce

herbie shell --seed 2022354 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))