Average Error: 0.1 → 0.1
Time: 3.7s
Precision: binary32
\[0 \leq s \land s \leq 1.0651631\]
\[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
↓
\[\begin{array}{l} t_0 := e^{\frac{x}{s}}\\ {\left(\mathsf{fma}\left(s, t_0 + 2, \frac{s}{t_0}\right)\right)}^{-1} \end{array} \]
(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
 (let* ((t_0 (exp (/ x s)))) (pow (fma s (+ t_0 2.0) (/ s t_0)) -1.0)))
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) {
	float t_0 = expf((x / s));
	return powf(fmaf(s, (t_0 + 2.0f), (s / t_0)), -1.0f);
}

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)
	t_0 = exp(Float32(x / s))
	return fma(s, Float32(t_0 + Float32(2.0)), Float32(s / t_0)) ^ Float32(-1.0)
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)}
↓
\begin{array}{l}
t_0 := e^{\frac{x}{s}}\\
{\left(\mathsf{fma}\left(s, t_0 + 2, \frac{s}{t_0}\right)\right)}^{-1}
\end{array}

Error

Bits error versus x

Bits error versus s

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. Taylor expanded in x around 0 0.1

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

  4. Simplified0.1

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

  5. Applied egg-rr0.1

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

  6. Applied egg-rr0.1

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

  7. Applied egg-rr0.1

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

  8. Final simplification0.1

    \[\leadsto {\left(\mathsf{fma}\left(s, e^{\frac{x}{s}} + 2, \frac{s}{e^{\frac{x}{s}}}\right)\right)}^{-1} \]

Reproduce

herbie shell --seed 2022153 
(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))))))