?

Average Error: 0.1 → 0.1
Time: 9.6s
Precision: binary32
Cost: 10144

?

\[0 \leq s \land s \leq 1.0651631\]
\[\frac{1}{1 + e^{\frac{-x}{s}}} \]
↓
\[\begin{array}{l} t_0 := \frac{x}{s} \cdot 0.5\\ \frac{1}{1 + \frac{\frac{1}{{e}^{t_0}}}{e^{t_0}}} \end{array} \]
(FPCore (x s) :precision binary32 (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))
↓
(FPCore (x s)
 :precision binary32
 (let* ((t_0 (* (/ x s) 0.5)))
   (/ 1.0 (+ 1.0 (/ (/ 1.0 (pow E t_0)) (exp 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 = (x / s) * 0.5f;
	return 1.0f / (1.0f + ((1.0f / powf(((float) M_E), t_0)) / expf(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 = Float32(Float32(x / s) * Float32(0.5))
	return Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(Float32(1.0) / (Float32(exp(1)) ^ t_0)) / exp(t_0))))
end

function tmp = code(x, s)
	tmp = single(1.0) / (single(1.0) + exp((-x / s)));
end
↓
function tmp = code(x, s)
	t_0 = (x / s) * single(0.5);
	tmp = single(1.0) / (single(1.0) + ((single(1.0) / (single(2.71828182845904523536) ^ t_0)) / exp(t_0)));
end

\frac{1}{1 + e^{\frac{-x}{s}}}
↓
\begin{array}{l}
t_0 := \frac{x}{s} \cdot 0.5\\
\frac{1}{1 + \frac{\frac{1}{{e}^{t_0}}}{e^{t_0}}}
\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

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

  3. Applied egg-rr0.1

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

  4. Applied egg-rr0.1

    \[\leadsto \frac{1}{1 + \frac{\frac{1}{\color{blue}{{e}^{\left(\frac{x}{s} \cdot 0.5\right)}}}}{\sqrt{e^{\frac{x}{s}}}}} \]

  5. Applied egg-rr0.1

    \[\leadsto \frac{1}{1 + \frac{\frac{1}{{e}^{\left(\frac{x}{s} \cdot 0.5\right)}}}{\color{blue}{e^{\frac{x}{s} \cdot 0.5}}}} \]

  6. Final simplification0.1

    \[\leadsto \frac{1}{1 + \frac{\frac{1}{{e}^{\left(\frac{x}{s} \cdot 0.5\right)}}}{e^{\frac{x}{s} \cdot 0.5}}} \]

Alternatives

Alternative 1
Error0.1
Cost3456
\[\frac{1}{1 + e^{\frac{-x}{s}}} \]
Alternative 2
Error10.0
Cost196
\[\begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 20:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
Alternative 3
Error20.6
Cost32
\[0.5 \]

Error

Reproduce?

herbie shell --seed 2023187 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))