?

Average Error: 0.1 → 0.0
Time: 12.4s
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-rr0.1

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

  4. Applied egg-rr0.0

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

  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
Error2.4
Cost776
\[\begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq -5:\\ \;\;\;\;1\\ \mathbf{elif}\;t_0 \leq 1:\\ \;\;\;\;0.5 + \frac{x}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \left(x \cdot \frac{x}{s \cdot \left(-s\right)}\right) \cdot -0.5}\\ \end{array} \]
Alternative 3
Error2.9
Cost772
\[\begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -5:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \left(\left(1 - \frac{x}{s}\right) - x \cdot \frac{\frac{x}{s \cdot -2}}{s}\right)}\\ \end{array} \]
Alternative 4
Error3.0
Cost744
\[\begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq -5:\\ \;\;\;\;1\\ \mathbf{elif}\;t_0 \leq 1:\\ \;\;\;\;0.5 + \frac{x}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + 0.5 \cdot \frac{x \cdot x}{s \cdot s}}\\ \end{array} \]
Alternative 5
Error4.0
Cost616
\[\begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq -5:\\ \;\;\;\;1\\ \mathbf{elif}\;t_0 \leq 1:\\ \;\;\;\;0.5 + \frac{x}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(s \cdot \frac{s}{x \cdot x}\right)\\ \end{array} \]
Alternative 6
Error3.1
Cost616
\[\begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq -5:\\ \;\;\;\;1\\ \mathbf{elif}\;t_0 \leq 1:\\ \;\;\;\;0.5 + \frac{x}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(s \cdot s\right) \cdot 2}{x \cdot x}\\ \end{array} \]
Alternative 7
Error7.1
Cost552
\[\begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq -5:\\ \;\;\;\;1\\ \mathbf{elif}\;t_0 \leq 1:\\ \;\;\;\;0.5 + \frac{x}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_0}\\ \end{array} \]
Alternative 8
Error7.8
Cost520
\[\begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq -5:\\ \;\;\;\;1\\ \mathbf{elif}\;t_0 \leq 1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_0}\\ \end{array} \]
Alternative 9
Error7.3
Cost388
\[\begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -5:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \end{array} \]
Alternative 10
Error10.0
Cost228
\[\begin{array}{l} \mathbf{if}\;x \leq -2.000000026702864 \cdot 10^{-10}:\\ \;\;\;\;s \cdot \frac{-1}{x}\\ \mathbf{elif}\;x \leq 2.000000047484456 \cdot 10^{-32}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
Alternative 11
Error10.0
Cost196
\[\begin{array}{l} \mathbf{if}\;x \leq -2.000000026702864 \cdot 10^{-10}:\\ \;\;\;\;\frac{-s}{x}\\ \mathbf{elif}\;x \leq 2.000000047484456 \cdot 10^{-32}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
Alternative 12
Error10.0
Cost168
\[\begin{array}{l} \mathbf{if}\;x \leq -0.03999999910593033:\\ \;\;\;\;\frac{s}{x}\\ \mathbf{elif}\;x \leq 2.000000047484456 \cdot 10^{-32}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
Alternative 13
Error13.6
Cost100
\[\begin{array}{l} \mathbf{if}\;x \leq 2.000000047484456 \cdot 10^{-32}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
Alternative 14
Error20.6
Cost32
\[0.5 \]

Error

Reproduce?

herbie shell --seed 2023018 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))