?

Average Error: 0.1 → 0.1
Time: 10.8s
Precision: binary32
Cost: 29440

?

\[0 \leq s \land s \leq 1.0651631\]
\[\frac{1}{1 + e^{\frac{-x}{s}}} \]
↓
\[\begin{array}{l} t_0 := {\left(\sqrt{e}\right)}^{\left(\frac{x}{s}\right)}\\ \frac{1}{1 + \frac{\frac{1}{{\left(\sqrt[3]{t_0 \cdot t_0}\right)}^{2}}}{e^{\frac{x}{s} \cdot 0.3333333333333333}}} \end{array} \]
(FPCore (x s) :precision binary32 (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))
↓
(FPCore (x s)
 :precision binary32
 (let* ((t_0 (pow (sqrt E) (/ x s))))
   (/
    1.0
    (+
     1.0
     (/
      (/ 1.0 (pow (cbrt (* t_0 t_0)) 2.0))
      (exp (* (/ x s) 0.3333333333333333)))))))
float code(float x, float s) {
	return 1.0f / (1.0f + expf((-x / s)));
}
↓
float code(float x, float s) {
	float t_0 = powf(sqrtf(((float) M_E)), (x / s));
	return 1.0f / (1.0f + ((1.0f / powf(cbrtf((t_0 * t_0)), 2.0f)) / expf(((x / s) * 0.3333333333333333f))));
}

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 = sqrt(Float32(exp(1))) ^ Float32(x / s)
	return Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(Float32(1.0) / (cbrt(Float32(t_0 * t_0)) ^ Float32(2.0))) / exp(Float32(Float32(x / s) * Float32(0.3333333333333333))))))
end

\frac{1}{1 + e^{\frac{-x}{s}}}
↓
\begin{array}{l}
t_0 := {\left(\sqrt{e}\right)}^{\left(\frac{x}{s}\right)}\\
\frac{1}{1 + \frac{\frac{1}{{\left(\sqrt[3]{t_0 \cdot t_0}\right)}^{2}}}{e^{\frac{x}{s} \cdot 0.3333333333333333}}}
\end{array}

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

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

  3. Applied egg-rr0.1

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

  4. Applied egg-rr0.1

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

  5. Applied egg-rr0.1

    \[\leadsto \frac{1}{1 + \frac{\frac{1}{{\left(\sqrt[3]{{\left(\sqrt{e}\right)}^{\left(\frac{x}{s}\right)} \cdot {\left(\sqrt{e}\right)}^{\left(\frac{x}{s}\right)}}\right)}^{2}}}{\color{blue}{e^{\frac{x}{s} \cdot 0.3333333333333333}}}} \]

  6. Final simplification0.1

    \[\leadsto \frac{1}{1 + \frac{\frac{1}{{\left(\sqrt[3]{{\left(\sqrt{e}\right)}^{\left(\frac{x}{s}\right)} \cdot {\left(\sqrt{e}\right)}^{\left(\frac{x}{s}\right)}}\right)}^{2}}}{e^{\frac{x}{s} \cdot 0.3333333333333333}}} \]

Alternatives

Alternative 1
Error0.1
Cost22848
\[\begin{array}{l} t_0 := \sqrt[3]{{e}^{\left(\frac{x}{s}\right)}}\\ \frac{1}{1 + \frac{\frac{1}{{t_0}^{2}}}{t_0}} \end{array} \]
Alternative 2
Error0.1
Cost19744
\[\frac{1}{1 + \frac{\frac{1}{{\left(\sqrt[3]{{e}^{\left(\frac{x}{s}\right)}}\right)}^{2}}}{\frac{1}{\sqrt[3]{e^{\frac{-x}{s}}}}}} \]
Alternative 3
Error0.1
Cost19648
\[\frac{1}{1 + \frac{\frac{1}{{\left(\sqrt[3]{{e}^{\left(\frac{x}{s}\right)}}\right)}^{2}}}{\sqrt[3]{e^{\frac{x}{s}}}}} \]
Alternative 4
Error0.1
Cost16512
\[\frac{1}{1 + \frac{\frac{1}{{\left(\sqrt{e^{\frac{x}{\frac{s}{0.6666666666666666}}}}\right)}^{2}}}{\sqrt[3]{e^{\frac{x}{s}}}}} \]
Alternative 5
Error0.1
Cost9856
\[\frac{1}{1 + \mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{-x}{s}}\right)\right)} \]
Alternative 6
Error0.1
Cost3456
\[\frac{1}{1 + e^{\frac{-x}{s}}} \]
Alternative 7
Error0.7
Cost552
\[\begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq -5:\\ \;\;\;\;1\\ \mathbf{elif}\;t_0 \leq 1:\\ \;\;\;\;0.5 + \frac{x}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
Alternative 8
Error0.7
Cost552
\[\begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq -5:\\ \;\;\;\;1\\ \mathbf{elif}\;t_0 \leq 1:\\ \;\;\;\;0.5 + \frac{0.25}{\frac{s}{x}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
Alternative 9
Error1.4
Cost360
\[\begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq -1:\\ \;\;\;\;1\\ \mathbf{elif}\;t_0 \leq 20:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
Alternative 10
Error13.9
Cost100
\[\begin{array}{l} \mathbf{if}\;x \leq 5.000000015855384 \cdot 10^{-31}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
Alternative 11
Error20.8
Cost32
\[0.5 \]

Error

Reproduce?

herbie shell --seed 2023066 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))