Average Error: 0.1 → 0.1
Time: 11.0s
Precision: binary32
Cost: 9952
\[0 \leq s \land s \leq 1.0651631\]
\[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
↓
\[\frac{\frac{1}{2 + 2 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cosh \left(\frac{x}{s}\right)\right)\right)}}{s} \]
(FPCore (x s)
 :precision binary32
 (/
  (exp (/ (- (fabs x)) s))
  (* (* s (+ 1.0 (exp (/ (- (fabs x)) s)))) (+ 1.0 (exp (/ (- (fabs x)) s))))))
↓
(FPCore (x s)
 :precision binary32
 (/ (/ 1.0 (+ 2.0 (* 2.0 (expm1 (log1p (cosh (/ x s))))))) s))
float code(float x, float s) {
	return expf((-fabsf(x) / s)) / ((s * (1.0f + expf((-fabsf(x) / s)))) * (1.0f + expf((-fabsf(x) / s))));
}
↓
float code(float x, float s) {
	return (1.0f / (2.0f + (2.0f * expm1f(log1pf(coshf((x / s))))))) / s;
}

function code(x, s)
	return Float32(exp(Float32(Float32(-abs(x)) / s)) / Float32(Float32(s * Float32(Float32(1.0) + exp(Float32(Float32(-abs(x)) / s)))) * Float32(Float32(1.0) + exp(Float32(Float32(-abs(x)) / s)))))
end
↓
function code(x, s)
	return Float32(Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(Float32(2.0) * expm1(log1p(cosh(Float32(x / s))))))) / s)
end

\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)}
↓
\frac{\frac{1}{2 + 2 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cosh \left(\frac{x}{s}\right)\right)\right)}}{s}

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.1

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

  2. Simplified0.1

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

  3. Applied egg-rr0.1

    \[\leadsto \frac{\color{blue}{0 + \frac{1}{2 + 2 \cdot \cosh \left(\frac{x}{s}\right)}}}{s} \]

  4. Applied egg-rr0.1

    \[\leadsto \frac{0 + \frac{1}{2 + 2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cosh \left(\frac{x}{s}\right)\right)\right)}}}{s} \]

  5. Final simplification0.1

    \[\leadsto \frac{\frac{1}{2 + 2 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cosh \left(\frac{x}{s}\right)\right)\right)}}{s} \]

Alternatives

Alternative 1
Error0.1
Cost6848
\[\frac{\frac{1}{e^{\frac{-x}{s}} + \left(2 + e^{\frac{x}{s}}\right)}}{s} \]
Alternative 2
Error0.2
Cost3616
\[\frac{1}{2 + 2 \cdot \cosh \left(\frac{x}{s}\right)} \cdot \frac{1}{s} \]
Alternative 3
Error4.6
Cost3588
\[\begin{array}{l} \mathbf{if}\;x \leq -2.419999923414571 \cdot 10^{-27}:\\ \;\;\;\;\frac{\frac{1}{e^{\frac{-x}{s}} + 3}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{x}{s} \cdot \frac{x}{s} + 4}}{s}\\ \end{array} \]
Alternative 4
Error0.1
Cost3552
\[\frac{\frac{1}{2 + 2 \cdot \cosh \left(\frac{x}{s}\right)}}{s} \]
Alternative 5
Error7.6
Cost480
\[\frac{1}{s} \cdot \frac{1}{\frac{x}{s} \cdot \frac{x}{s} + 4} \]
Alternative 6
Error7.6
Cost416
\[\frac{\frac{1}{\frac{x}{s} \cdot \frac{x}{s} + 4}}{s} \]
Alternative 7
Error23.6
Cost96
\[\frac{0.25}{s} \]

Error

Reproduce

herbie shell --seed 2022284 
(FPCore (x s)
  :name "Logistic distribution"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ (exp (/ (- (fabs x)) s)) (* (* s (+ 1.0 (exp (/ (- (fabs x)) s)))) (+ 1.0 (exp (/ (- (fabs x)) s))))))