Average Error: 0.1 → 0.1
Time: 10.4s
Precision: binary32
Cost: 10048
\[0 \leq s \land s \leq 1.0651631\]
\[\frac{1}{1 + e^{\frac{-x}{s}}} \]
↓
\[\frac{1}{1 + e^{\frac{-0.6666666666666666}{\frac{s}{x}}} \cdot \sqrt[3]{e^{\frac{-x}{s}}}} \]
(FPCore (x s) :precision binary32 (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))
↓
(FPCore (x s)
 :precision binary32
 (/
  1.0
  (+ 1.0 (* (exp (/ -0.6666666666666666 (/ s x))) (cbrt (exp (/ (- x) s)))))))
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 + (expf((-0.6666666666666666f / (s / x))) * cbrtf(expf((-x / s)))));
}

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) + Float32(exp(Float32(Float32(-0.6666666666666666) / Float32(s / x))) * cbrt(exp(Float32(Float32(-x) / s))))))
end

\frac{1}{1 + e^{\frac{-x}{s}}}
↓
\frac{1}{1 + e^{\frac{-0.6666666666666666}{\frac{s}{x}}} \cdot \sqrt[3]{e^{\frac{-x}{s}}}}

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.1

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

  2. Applied egg-rr0.1

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

  3. Applied egg-rr0.1

    \[\leadsto \frac{1}{1 + \frac{1}{\color{blue}{{e}^{\left(2 \cdot \left(0.3333333333333333 \cdot \frac{x}{s}\right)\right)} \cdot {e}^{\left(0.3333333333333333 \cdot \frac{x}{s}\right)}}}} \]

  4. Applied egg-rr0.1

    \[\leadsto \frac{1}{1 + \color{blue}{e^{-\frac{0.6666666666666666}{\frac{s}{x}}} \cdot \sqrt[3]{e^{-\frac{x}{s}}}}} \]

  5. Final simplification0.1

    \[\leadsto \frac{1}{1 + e^{\frac{-0.6666666666666666}{\frac{s}{x}}} \cdot \sqrt[3]{e^{\frac{-x}{s}}}} \]

Alternatives

Alternative 1
Error0.1
Cost10048
\[\frac{1}{1 + \sqrt[3]{e^{\frac{-x}{s}}} \cdot e^{\frac{x}{s} \cdot -0.6666666666666666}} \]
Alternative 2
Error0.1
Cost6656
\[\frac{1}{1 + {e}^{\left(\frac{-x}{s}\right)}} \]
Alternative 3
Error0.1
Cost3456
\[\frac{1}{1 + e^{\frac{-x}{s}}} \]
Alternative 4
Error1.5
Cost552
\[\begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq -20:\\ \;\;\;\;1 - \frac{s}{x}\\ \mathbf{elif}\;t_0 \leq 50:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
Alternative 5
Error1.2
Cost552
\[\begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq -20:\\ \;\;\;\;1 - \frac{s}{x}\\ \mathbf{elif}\;t_0 \leq 2:\\ \;\;\;\;0.5 + \frac{x}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
Alternative 6
Error1.4
Cost516
\[\begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 50:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
Alternative 7
Error1.9
Cost360
\[\begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq -20:\\ \;\;\;\;1 - \frac{s}{x}\\ \mathbf{elif}\;t_0 \leq 50:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
Alternative 8
Error11.7
Cost100
\[\begin{array}{l} \mathbf{if}\;x \leq -3.999999935100636 \cdot 10^{-17}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
Alternative 9
Error21.0
Cost32
\[0.5 \]

Error

Reproduce

herbie shell --seed 2022300 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))