Sample trimmed logistic on [-pi, pi]

Average Error: 0.3 → 0.4

Time: 29.8s

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

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

↓

\[\begin{array}{l} t_0 := \frac{1 - u}{1 + e^{\frac{\pi}{s}}}\\ t_1 := \frac{u}{1 + e^{\frac{-\pi}{s}}}\\ \left(-s\right) \cdot \log \left(\mathsf{fma}\left(\frac{1}{{t_1}^{3} + {t_0}^{3}}, t_1 \cdot t_1 + \left(t_0 \cdot t_0 - t_1 \cdot t_0\right), -1\right)\right) \end{array} \]

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 (/ (- 1.0 u) (+ 1.0 (exp (/ PI s)))))
        (t_1 (/ u (+ 1.0 (exp (/ (- PI) s))))))
   (*
    (- s)
    (log
     (fma
      (/ 1.0 (+ (pow t_1 3.0) (pow t_0 3.0)))
      (+ (* t_1 t_1) (- (* t_0 t_0) (* t_1 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 = (1.0f - u) / (1.0f + expf(((float) M_PI) / s));
	float t_1 = u / (1.0f + expf(-((float) M_PI) / s));
	return -s * logf(fmaf((1.0f / (powf(t_1, 3.0f) + powf(t_0, 3.0f))), ((t_1 * t_1) + ((t_0 * t_0) - (t_1 * 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 flip3-+_binary32 0.4

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{{\left(\frac{u}{1 + e^{\frac{-\pi}{s}}}\right)}^{3} + {\left(\frac{1 - u}{1 + e^{\frac{\pi}{s}}}\right)}^{3}}{\frac{u}{1 + e^{\frac{-\pi}{s}}} \cdot \frac{u}{1 + e^{\frac{-\pi}{s}}} + \left(\frac{1 - u}{1 + e^{\frac{\pi}{s}}} \cdot \frac{1 - u}{1 + e^{\frac{\pi}{s}}} - \frac{u}{1 + e^{\frac{-\pi}{s}}} \cdot \frac{1 - u}{1 + e^{\frac{\pi}{s}}}\right)}}} + -1\right) \]

4. Applied associate-/r/_binary32 0.4

   \[\leadsto \left(-s\right) \cdot \log \left(\color{blue}{\frac{1}{{\left(\frac{u}{1 + e^{\frac{-\pi}{s}}}\right)}^{3} + {\left(\frac{1 - u}{1 + e^{\frac{\pi}{s}}}\right)}^{3}} \cdot \left(\frac{u}{1 + e^{\frac{-\pi}{s}}} \cdot \frac{u}{1 + e^{\frac{-\pi}{s}}} + \left(\frac{1 - u}{1 + e^{\frac{\pi}{s}}} \cdot \frac{1 - u}{1 + e^{\frac{\pi}{s}}} - \frac{u}{1 + e^{\frac{-\pi}{s}}} \cdot \frac{1 - u}{1 + e^{\frac{\pi}{s}}}\right)\right)} + -1\right) \]

5. Applied fma-def_binary32 0.4

   \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\mathsf{fma}\left(\frac{1}{{\left(\frac{u}{1 + e^{\frac{-\pi}{s}}}\right)}^{3} + {\left(\frac{1 - u}{1 + e^{\frac{\pi}{s}}}\right)}^{3}}, \frac{u}{1 + e^{\frac{-\pi}{s}}} \cdot \frac{u}{1 + e^{\frac{-\pi}{s}}} + \left(\frac{1 - u}{1 + e^{\frac{\pi}{s}}} \cdot \frac{1 - u}{1 + e^{\frac{\pi}{s}}} - \frac{u}{1 + e^{\frac{-\pi}{s}}} \cdot \frac{1 - u}{1 + e^{\frac{\pi}{s}}}\right), -1\right)\right)} \]

6. Final simplification 0.4

   \[\leadsto \left(-s\right) \cdot \log \left(\mathsf{fma}\left(\frac{1}{{\left(\frac{u}{1 + e^{\frac{-\pi}{s}}}\right)}^{3} + {\left(\frac{1 - u}{1 + e^{\frac{\pi}{s}}}\right)}^{3}}, \frac{u}{1 + e^{\frac{-\pi}{s}}} \cdot \frac{u}{1 + e^{\frac{-\pi}{s}}} + \left(\frac{1 - u}{1 + e^{\frac{\pi}{s}}} \cdot \frac{1 - u}{1 + e^{\frac{\pi}{s}}} - \frac{u}{1 + e^{\frac{-\pi}{s}}} \cdot \frac{1 - u}{1 + e^{\frac{\pi}{s}}}\right), -1\right)\right) \]

Reproduce

herbie shell --seed 2022020 
(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))))