Average Error: 0.1 → 0.1
Time: 5.2s
Precision: binary32
\[0 \leq s \land s \leq 1.0651631\]
\[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
↓
\[\frac{1}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(s, \left(e^{\mathsf{log1p}\left(e^{\frac{x}{s}}\right)} - 1\right) + 2, \frac{s}{{e}^{\left(\frac{x}{s}\right)}}\right)\right)\right)} \]
\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{1}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(s, \left(e^{\mathsf{log1p}\left(e^{\frac{x}{s}}\right)} - 1\right) + 2, \frac{s}{{e}^{\left(\frac{x}{s}\right)}}\right)\right)\right)}

(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
  (expm1
   (log1p
    (fma
     s
     (+ (- (exp (log1p (exp (/ x s)))) 1.0) 2.0)
     (/ s (pow E (/ x 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 / expm1f(log1pf(fmaf(s, ((expf(log1pf(expf((x / s)))) - 1.0f) + 2.0f), (s / powf(((float) M_E), (x / s))))));
}

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. Simplified0.2

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

  3. Taylor expanded in s 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

    \[\leadsto \frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(s, e^{\frac{x}{s}} + 2, \frac{s}{e^{\frac{x}{s}}}\right)\right)\right)}} \]

  6. Applied egg-rr0.1

    \[\leadsto \frac{1}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(s, e^{\frac{x}{s}} + 2, \frac{s}{\color{blue}{{\left(e^{1}\right)}^{\left(\frac{x}{s}\right)}}}\right)\right)\right)} \]

  7. Applied egg-rr0.1

    \[\leadsto \frac{1}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(s, \color{blue}{\left(e^{\mathsf{log1p}\left(e^{\frac{x}{s}}\right)} - 1\right)} + 2, \frac{s}{{\left(e^{1}\right)}^{\left(\frac{x}{s}\right)}}\right)\right)\right)} \]

  8. Final simplification0.1

    \[\leadsto \frac{1}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(s, \left(e^{\mathsf{log1p}\left(e^{\frac{x}{s}}\right)} - 1\right) + 2, \frac{s}{{e}^{\left(\frac{x}{s}\right)}}\right)\right)\right)} \]

Reproduce

herbie shell --seed 2022130 
(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))))))