?

Average Error: 0.18% → 0.22%
Time: 13.5s
Precision: binary32
Cost: 26144

?

\[0 \leq s \land s \leq 1.0651631\]
\[\frac{1}{1 + e^{\frac{-x}{s}}} \]
↓
\[\begin{array}{l} t_0 := e^{\frac{x}{s}}\\ \frac{1}{1 + \frac{\frac{1}{\sqrt[3]{\sqrt{t_0}} \cdot \sqrt{{e}^{\left(\frac{x}{s}\right)}}}}{\sqrt[3]{t_0}}} \end{array} \]
(FPCore (x s) :precision binary32 (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))
↓
(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ x s))))
   (/
    1.0
    (+
     1.0
     (/ (/ 1.0 (* (cbrt (sqrt t_0)) (sqrt (pow E (/ x s))))) (cbrt t_0))))))
float code(float x, float s) {
	return 1.0f / (1.0f + expf((-x / s)));
}
↓
float code(float x, float s) {
	float t_0 = expf((x / s));
	return 1.0f / (1.0f + ((1.0f / (cbrtf(sqrtf(t_0)) * sqrtf(powf(((float) M_E), (x / s))))) / cbrtf(t_0)));
}

function code(x, s)
	return Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(-x) / s))))
end
↓
function code(x, s)
	t_0 = exp(Float32(x / s))
	return Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(Float32(1.0) / Float32(cbrt(sqrt(t_0)) * sqrt((Float32(exp(1)) ^ Float32(x / s))))) / cbrt(t_0))))
end

\frac{1}{1 + e^{\frac{-x}{s}}}
↓
\begin{array}{l}
t_0 := e^{\frac{x}{s}}\\
\frac{1}{1 + \frac{\frac{1}{\sqrt[3]{\sqrt{t_0}} \cdot \sqrt{{e}^{\left(\frac{x}{s}\right)}}}}{\sqrt[3]{t_0}}}
\end{array}

Error?

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation?

  1. Initial program 0.18

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

  2. Applied egg-rr0.24

    \[\leadsto \frac{1}{1 + \color{blue}{\frac{\frac{1}{{\left(\sqrt[3]{e^{\frac{x}{s}}}\right)}^{2}}}{\sqrt[3]{e^{\frac{x}{s}}}}}} \]

  3. Applied egg-rr0.22

    \[\leadsto \frac{1}{1 + \frac{\frac{1}{\color{blue}{\sqrt{e^{\frac{x}{s}}} \cdot {\left(e^{\frac{x}{s}}\right)}^{0.16666666666666666}}}}{\sqrt[3]{e^{\frac{x}{s}}}}} \]

  4. Simplified0.22

    \[\leadsto \frac{1}{1 + \frac{\frac{1}{\color{blue}{{\left(e^{\frac{x}{s}}\right)}^{0.16666666666666666} \cdot \sqrt{e^{\frac{x}{s}}}}}}{\sqrt[3]{e^{\frac{x}{s}}}}} \]
    Proof

    [Start]0.22

    \[ \frac{1}{1 + \frac{\frac{1}{\sqrt{e^{\frac{x}{s}}} \cdot {\left(e^{\frac{x}{s}}\right)}^{0.16666666666666666}}}{\sqrt[3]{e^{\frac{x}{s}}}}} \]

    *-commutative [=>]0.22

    \[ \frac{1}{1 + \frac{\frac{1}{\color{blue}{{\left(e^{\frac{x}{s}}\right)}^{0.16666666666666666} \cdot \sqrt{e^{\frac{x}{s}}}}}}{\sqrt[3]{e^{\frac{x}{s}}}}} \]

  5. Applied egg-rr0.22

    \[\leadsto \frac{1}{1 + \frac{\frac{1}{{\left(e^{\frac{x}{s}}\right)}^{0.16666666666666666} \cdot \sqrt{\color{blue}{{e}^{\left(\frac{x}{s}\right)}}}}}{\sqrt[3]{e^{\frac{x}{s}}}}} \]

  6. Applied egg-rr0.22

    \[\leadsto \frac{1}{1 + \frac{\frac{1}{\color{blue}{\left({\left(\sqrt{e^{\frac{x}{s}}}\right)}^{0.16666666666666666} \cdot {\left(\sqrt{e^{\frac{x}{s}}}\right)}^{0.16666666666666666}\right)} \cdot \sqrt{{e}^{\left(\frac{x}{s}\right)}}}}{\sqrt[3]{e^{\frac{x}{s}}}}} \]

  7. Simplified0.22

    \[\leadsto \frac{1}{1 + \frac{\frac{1}{\color{blue}{\sqrt[3]{\sqrt{e^{\frac{x}{s}}}}} \cdot \sqrt{{e}^{\left(\frac{x}{s}\right)}}}}{\sqrt[3]{e^{\frac{x}{s}}}}} \]
    Proof

    [Start]0.22

    \[ \frac{1}{1 + \frac{\frac{1}{\left({\left(\sqrt{e^{\frac{x}{s}}}\right)}^{0.16666666666666666} \cdot {\left(\sqrt{e^{\frac{x}{s}}}\right)}^{0.16666666666666666}\right) \cdot \sqrt{{e}^{\left(\frac{x}{s}\right)}}}}{\sqrt[3]{e^{\frac{x}{s}}}}} \]

    pow-sqr [=>]0.22

    \[ \frac{1}{1 + \frac{\frac{1}{\color{blue}{{\left(\sqrt{e^{\frac{x}{s}}}\right)}^{\left(2 \cdot 0.16666666666666666\right)}} \cdot \sqrt{{e}^{\left(\frac{x}{s}\right)}}}}{\sqrt[3]{e^{\frac{x}{s}}}}} \]

    metadata-eval [=>]0.22

    \[ \frac{1}{1 + \frac{\frac{1}{{\left(\sqrt{e^{\frac{x}{s}}}\right)}^{\color{blue}{0.3333333333333333}} \cdot \sqrt{{e}^{\left(\frac{x}{s}\right)}}}}{\sqrt[3]{e^{\frac{x}{s}}}}} \]

    unpow1/3 [=>]0.22

    \[ \frac{1}{1 + \frac{\frac{1}{\color{blue}{\sqrt[3]{\sqrt{e^{\frac{x}{s}}}}} \cdot \sqrt{{e}^{\left(\frac{x}{s}\right)}}}}{\sqrt[3]{e^{\frac{x}{s}}}}} \]

  8. Final simplification0.22

    \[\leadsto \frac{1}{1 + \frac{\frac{1}{\sqrt[3]{\sqrt{e^{\frac{x}{s}}}} \cdot \sqrt{{e}^{\left(\frac{x}{s}\right)}}}}{\sqrt[3]{e^{\frac{x}{s}}}}} \]

Alternatives

Alternative 1
Error0.11%
Cost9760
\[e^{-\mathsf{log1p}\left(e^{\frac{-x}{s}}\right)} \]
Alternative 2
Error0.2%
Cost3488
\[\frac{1}{1 + \frac{1}{e^{\frac{x}{s}}}} \]
Alternative 3
Error0.18%
Cost3456
\[\frac{1}{1 + e^{\frac{-x}{s}}} \]
Alternative 4
Error7.35%
Cost516
\[\begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 50:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{s}{x} + -1\right)\\ \end{array} \]
Alternative 5
Error34.44%
Cost388
\[\begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 50:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{s}{x} + -1\right)\\ \end{array} \]
Alternative 6
Error51.86%
Cost356
\[\begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq 0.5:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_0}\\ \end{array} \]
Alternative 7
Error53.37%
Cost196
\[\begin{array}{l} \mathbf{if}\;x \leq -0.004999999888241291:\\ \;\;\;\;\frac{-s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
Alternative 8
Error53.43%
Cost164
\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
Alternative 9
Error64.74%
Cost32
\[0.5 \]

Error

Reproduce?

herbie shell --seed 2023089 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))