Average Error: 0.1 → 0.0
Time: 13.9s
Precision: binary32
Cost: 9760
\[0 \leq s \land s \leq 1.0651631\]
\[\frac{1}{1 + e^{\frac{-x}{s}}} \]
↓
\[e^{-\mathsf{log1p}\left(e^{\frac{-x}{s}}\right)} \]
(FPCore (x s) :precision binary32 (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))
↓
(FPCore (x s) :precision binary32 (exp (- (log1p (exp (/ (- x) s))))))
float code(float x, float s) {
	return 1.0f / (1.0f + expf((-x / s)));
}
↓
float code(float x, float s) {
	return expf(-log1pf(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 exp(Float32(-log1p(exp(Float32(Float32(-x) / s)))))
end

\frac{1}{1 + e^{\frac{-x}{s}}}
↓
e^{-\mathsf{log1p}\left(e^{\frac{-x}{s}}\right)}

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

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

  3. Applied egg-rr0.0

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

  4. Simplified0.0

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

    [Start]0.0

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

    *-rgt-identity [<=]0.0

    \[ e^{\color{blue}{\left(-\mathsf{log1p}\left(e^{-\frac{x}{s}}\right)\right) \cdot 1}} \]

    *-rgt-identity [=>]0.0

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

    distribute-neg-frac [=>]0.0

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

  5. Final simplification0.0

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

Alternatives

Alternative 1
Error0.1
Cost3456
\[\frac{1}{e^{\frac{-x}{s}} + 1} \]
Alternative 2
Error3.8
Cost772
\[\begin{array}{l} \mathbf{if}\;-x \leq -5.000000156871975 \cdot 10^{-23}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \left(\frac{x}{s} + 0.5 \cdot \frac{x \cdot x}{s \cdot s}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + \left(x \cdot \frac{\frac{x \cdot 0.5}{s}}{s} - \frac{x}{s}\right)}\\ \end{array} \]
Alternative 3
Error3.5
Cost708
\[\begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -1:\\ \;\;\;\;\frac{1}{-1 + \left(2 + \frac{-1}{-1 - \frac{x}{s}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + \left(x \cdot \frac{\frac{x \cdot 0.5}{s}}{s} - \frac{x}{s}\right)}\\ \end{array} \]
Alternative 4
Error3.6
Cost680
\[\begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq -4:\\ \;\;\;\;\frac{1}{1 + \frac{s}{x}}\\ \mathbf{elif}\;t_0 \leq 4:\\ \;\;\;\;0.5 + \frac{x}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{0.5 \cdot \frac{x \cdot x}{s \cdot s}}\\ \end{array} \]
Alternative 5
Error3.7
Cost616
\[\begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq -4:\\ \;\;\;\;\frac{1}{1 + \frac{s}{x}}\\ \mathbf{elif}\;t_0 \leq 4:\\ \;\;\;\;0.5 + \frac{x}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{s \cdot s}{x \cdot x}\\ \end{array} \]
Alternative 6
Error3.8
Cost580
\[\begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 20:\\ \;\;\;\;\frac{1}{-1 + \left(2 + \frac{-1}{-1 - \frac{x}{s}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{0.5 \cdot \frac{x \cdot x}{s \cdot s}}\\ \end{array} \]
Alternative 7
Error7.9
Cost552
\[\begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq -4:\\ \;\;\;\;1 - \frac{s}{x}\\ \mathbf{elif}\;t_0 \leq 0.5:\\ \;\;\;\;0.5 + \frac{x}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_0}\\ \end{array} \]
Alternative 8
Error7.9
Cost552
\[\begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq -4:\\ \;\;\;\;\frac{1}{1 + \frac{s}{x}}\\ \mathbf{elif}\;t_0 \leq 0.5:\\ \;\;\;\;0.5 + \frac{x}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_0}\\ \end{array} \]
Alternative 9
Error3.8
Cost548
\[\begin{array}{l} \mathbf{if}\;x \leq -5.000000156871975 \cdot 10^{-23}:\\ \;\;\;\;\frac{1}{2 + \left(\frac{x \cdot x}{s \cdot s} - \frac{x}{s}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-1 + \left(2 + \frac{-1}{-1 - \frac{x}{s}}\right)}\\ \end{array} \]
Alternative 10
Error8.6
Cost520
\[\begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq -2:\\ \;\;\;\;1 - \frac{s}{x}\\ \mathbf{elif}\;t_0 \leq 0.5:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_0}\\ \end{array} \]
Alternative 11
Error3.8
Cost516
\[\begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 20:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{0.5 \cdot \frac{x \cdot x}{s \cdot s}}\\ \end{array} \]
Alternative 12
Error8.1
Cost388
\[\begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -1:\\ \;\;\;\;\frac{1}{1 + \frac{s}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \end{array} \]
Alternative 13
Error10.3
Cost296
\[\begin{array}{l} \mathbf{if}\;x \leq -4.999999987376214 \cdot 10^{-7}:\\ \;\;\;\;s \cdot \frac{-1}{x}\\ \mathbf{elif}\;x \leq 1.99999996490334 \cdot 10^{-13}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{s}{x}\\ \end{array} \]
Alternative 14
Error17.2
Cost228
\[\begin{array}{l} \mathbf{if}\;x \leq -4.999999987376214 \cdot 10^{-7}:\\ \;\;\;\;s \cdot \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
Alternative 15
Error17.2
Cost196
\[\begin{array}{l} \mathbf{if}\;x \leq -4.999999987376214 \cdot 10^{-7}:\\ \;\;\;\;\frac{-s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
Alternative 16
Error17.2
Cost164
\[\begin{array}{l} \mathbf{if}\;x \leq -0.0002500000118743628:\\ \;\;\;\;\frac{s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
Alternative 17
Error20.9
Cost32
\[0.5 \]

Error

Reproduce

herbie shell --seed 2022356 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))