Average Error: 0.1 → 0.1
Time: 12.4s
Precision: binary32
\[0 \leq s \land s \leq 1.0651631\]
\[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
↓
\[\frac{{\left(\frac{1}{\sqrt[3]{2 + 2 \cdot e^{\log \cosh \left(\frac{x}{s}\right)}}}\right)}^{3}}{s} \]
(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
 (/ (pow (/ 1.0 (cbrt (+ 2.0 (* 2.0 (exp (log (cosh (/ x s)))))))) 3.0) 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 powf((1.0f / cbrtf((2.0f + (2.0f * expf(logf(coshf((x / s)))))))), 3.0f) / 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) / cbrt(Float32(Float32(2.0) + Float32(Float32(2.0) * exp(log(cosh(Float32(x / s)))))))) ^ Float32(3.0)) / 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{{\left(\frac{1}{\sqrt[3]{2 + 2 \cdot e^{\log \cosh \left(\frac{x}{s}\right)}}}\right)}^{3}}{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

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

  4. Applied egg-rr0.1

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

  5. Final simplification0.1

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

Reproduce

herbie shell --seed 2022210 
(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))))))