Average Error: 0.3 → 0.4
Time: 1.1min
Precision: binary32
\[\left(2.328306437 \cdot 10^{-10} \leq u \land u \leq 1\right) \land \left(0 \leq s \land s \leq 1.0651631\right)\]
\[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]
↓
\[\begin{array}{l} t_0 := \sqrt{\frac{1}{\frac{u}{1 + e^{\frac{-\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}}}\\ \mathsf{fma}\left(\log \left(1 + t_0\right), -s, \left(-s\right) \cdot \log \left(t_0 - 1\right)\right) \end{array} \]
\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(\log \left(1 + t_0\right), -s, \left(-s\right) \cdot \log \left(t_0 - 1\right)\right)
\end{array}

(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 (log (+ 1.0 t_0)) (- s) (* (- s) (log (- t_0 1.0))))))
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(logf(1.0f + t_0), -s, (-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. Simplified0.3

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

  3. Applied add-sqr-sqrt_binary320.4

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

  4. Applied difference-of-sqr--1_binary320.4

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

  5. Applied log-prod_binary320.5

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

  6. Applied distribute-rgt-in_binary320.4

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

  7. Applied fma-def_binary320.4

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

  8. Final simplification0.4

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

Reproduce

herbie shell --seed 2022087 
(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))))