Sample trimmed logistic on [-pi, pi]

Average Error: 0.3 → 0.3

Time: 45.1s

Precision: binary32

\[\left(2.328306437 \cdot 10^{-10} \leq u \land u \leq 1\right) \land \left(0 \leq s \land s \leq 1.0651631\right)\]

Math

\[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]

↓

\[\begin{array}{l} t_0 := \sqrt{\frac{1}{\frac{u}{1 + e^{-\frac{\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}}}\\ -\mathsf{fma}\left(s, \mathsf{log1p}\left(t_0\right), s \cdot \log \left(t_0 + -1\right)\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary32
 (*
  (- s)
  (log
   (-
    (/
     1.0
     (+
      (*
       u
       (- (/ 1.0 (+ 1.0 (exp (/ (- PI) s)))) (/ 1.0 (+ 1.0 (exp (/ PI s))))))
      (/ 1.0 (+ 1.0 (exp (/ PI s))))))
    1.0))))

↓

(FPCore (u s)
 :precision binary32
 (let* ((t_0
         (sqrt
          (/
           1.0
           (+
            (/ u (+ 1.0 (exp (- (/ PI s)))))
            (/ (- 1.0 u) (+ 1.0 (exp (/ PI s)))))))))
   (- (fma s (log1p t_0) (* s (log (+ t_0 -1.0)))))))

C

float code(float u, float s) {
	return -s * logf((1.0f / ((u * ((1.0f / (1.0f + expf(-((float) M_PI) / s))) - (1.0f / (1.0f + expf(((float) M_PI) / s))))) + (1.0f / (1.0f + expf(((float) M_PI) / s))))) - 1.0f);
}

↓

float code(float u, float s) {
	float t_0 = sqrtf(1.0f / ((u / (1.0f + expf(-(((float) M_PI) / s)))) + ((1.0f - u) / (1.0f + expf(((float) M_PI) / s)))));
	return -fmaf(s, log1pf(t_0), (s * logf(t_0 + -1.0f)));
}

Error

Bits error versus u

Bits error versus s

Derivation

1. Initial program 0.3

   \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]

2. Simplified 0.3

   \[\leadsto \color{blue}{\left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{-\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right)} \]

3. Applied add-sqr-sqrt_binary32 0.4

   \[\leadsto \left(-s\right) \cdot \log \left(\color{blue}{\sqrt{\frac{1}{\frac{u}{1 + e^{\frac{-\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}}} \cdot \sqrt{\frac{1}{\frac{u}{1 + e^{\frac{-\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}}}} + -1\right) \]

4. Applied difference-of-sqr--1_binary32 0.4

   \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\left(\sqrt{\frac{1}{\frac{u}{1 + e^{\frac{-\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}}} + 1\right) \cdot \left(\sqrt{\frac{1}{\frac{u}{1 + e^{\frac{-\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}}} - 1\right)\right)} \]

5. Applied log-prod_binary32 0.5

   \[\leadsto \left(-s\right) \cdot \color{blue}{\left(\log \left(\sqrt{\frac{1}{\frac{u}{1 + e^{\frac{-\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}}} + 1\right) + \log \left(\sqrt{\frac{1}{\frac{u}{1 + e^{\frac{-\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}}} - 1\right)\right)} \]

6. Applied distribute-rgt-in_binary32 0.4

   \[\leadsto \color{blue}{\log \left(\sqrt{\frac{1}{\frac{u}{1 + e^{\frac{-\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}}} + 1\right) \cdot \left(-s\right) + \log \left(\sqrt{\frac{1}{\frac{u}{1 + e^{\frac{-\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}}} - 1\right) \cdot \left(-s\right)} \]

7. Simplified 0.4

   \[\leadsto \color{blue}{\left(-s\right) \cdot \mathsf{log1p}\left(\sqrt{\frac{1}{\frac{u}{1 + e^{-\frac{\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}}}\right)} + \log \left(\sqrt{\frac{1}{\frac{u}{1 + e^{\frac{-\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}}} - 1\right) \cdot \left(-s\right) \]

8. Simplified 0.4

   \[\leadsto \left(-s\right) \cdot \mathsf{log1p}\left(\sqrt{\frac{1}{\frac{u}{1 + e^{-\frac{\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}}}\right) + \color{blue}{\left(-s\right) \cdot \log \left(\sqrt{\frac{1}{\frac{u}{1 + e^{-\frac{\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}}} + -1\right)} \]

9. Applied distribute-lft-neg-out_binary32 0.4

   \[\leadsto \left(-s\right) \cdot \mathsf{log1p}\left(\sqrt{\frac{1}{\frac{u}{1 + e^{-\frac{\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}}}\right) + \color{blue}{\left(-s \cdot \log \left(\sqrt{\frac{1}{\frac{u}{1 + e^{-\frac{\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}}} + -1\right)\right)} \]

10. Applied distribute-lft-neg-out_binary32 0.4

    \[\leadsto \color{blue}{\left(-s \cdot \mathsf{log1p}\left(\sqrt{\frac{1}{\frac{u}{1 + e^{-\frac{\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}}}\right)\right)} + \left(-s \cdot \log \left(\sqrt{\frac{1}{\frac{u}{1 + e^{-\frac{\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}}} + -1\right)\right) \]

11. Applied distribute-neg-out_binary32 0.4

    \[\leadsto \color{blue}{-\left(s \cdot \mathsf{log1p}\left(\sqrt{\frac{1}{\frac{u}{1 + e^{-\frac{\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}}}\right) + s \cdot \log \left(\sqrt{\frac{1}{\frac{u}{1 + e^{-\frac{\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}}} + -1\right)\right)} \]

12. Simplified 0.3

    \[\leadsto -\color{blue}{\mathsf{fma}\left(s, \mathsf{log1p}\left(\sqrt{\frac{1}{\frac{u}{1 + e^{-\frac{\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}}}\right), s \cdot \log \left(\sqrt{\frac{1}{\frac{u}{1 + e^{-\frac{\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}}} + -1\right)\right)} \]

13. Final simplification 0.3

    \[\leadsto -\mathsf{fma}\left(s, \mathsf{log1p}\left(\sqrt{\frac{1}{\frac{u}{1 + e^{-\frac{\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}}}\right), s \cdot \log \left(\sqrt{\frac{1}{\frac{u}{1 + e^{-\frac{\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}}} + -1\right)\right) \]

Reproduce

herbie shell --seed 2022077 
(FPCore (u s)
  :name "Sample trimmed logistic on [-pi, pi]"
  :precision binary32
  :pre (and (and (<= 2.328306437e-10 u) (<= u 1.0)) (and (<= 0.0 s) (<= s 1.0651631)))
  (* (- s) (log (- (/ 1.0 (+ (* u (- (/ 1.0 (+ 1.0 (exp (/ (- PI) s)))) (/ 1.0 (+ 1.0 (exp (/ PI s)))))) (/ 1.0 (+ 1.0 (exp (/ PI s)))))) 1.0))))