Average Error: 0.3 → 0.3
Time: 1.2min
Precision: binary32
\[\left(2.328306437 \cdot 10^{-10} \leq u \land u \leq 1\right) \land \left(0 \leq s \land s \leq 1.0651631\right)\]
\[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]
↓
\[\begin{array}{l} t_0 := \frac{1}{\frac{u}{1 + e^{-\frac{\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}}\\ t_1 := \sqrt{t_0}\\ \mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(-s, \mathsf{log1p}\left(t_1\right), \left(-s\right) \cdot \log \left(\frac{t_0 + -1}{1 + t_1}\right)\right)\right)\right) \end{array} \]
\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 := \frac{1}{\frac{u}{1 + e^{-\frac{\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}}\\
t_1 := \sqrt{t_0}\\
\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(-s, \mathsf{log1p}\left(t_1\right), \left(-s\right) \cdot \log \left(\frac{t_0 + -1}{1 + t_1}\right)\right)\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
         (/
          1.0
          (+
           (/ u (+ 1.0 (exp (- (/ PI s)))))
           (/ (- 1.0 u) (+ 1.0 (exp (/ PI s)))))))
        (t_1 (sqrt t_0)))
   (log1p
    (expm1
     (fma (- s) (log1p t_1) (* (- s) (log (/ (+ t_0 -1.0) (+ 1.0 t_1)))))))))
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)))) + ((1.0f - u) / (1.0f + expf(((float) M_PI) / s))));
	float t_1 = sqrtf(t_0);
	return log1pf(expm1f(fmaf(-s, log1pf(t_1), (-s * logf((t_0 + -1.0f) / (1.0f + t_1))))));
}

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. Simplified0.4

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

  8. Simplified0.4

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

  9. Applied log1p-expm1-u_binary320.4

    \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\left(-s\right) \cdot \mathsf{log1p}\left(\sqrt{\frac{1}{\frac{u}{1 + e^{-\frac{\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}}}\right) + \left(-s\right) \cdot \log \left(\sqrt{\frac{1}{\frac{u}{1 + e^{-\frac{\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}}} + -1\right)\right)\right)} \]

  10. Simplified0.4

    \[\leadsto \mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\mathsf{fma}\left(-s, \mathsf{log1p}\left(\sqrt{\frac{1}{\frac{u}{1 + e^{-\frac{\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}}}\right), \left(-s\right) \cdot \log \left(\sqrt{\frac{1}{\frac{u}{1 + e^{-\frac{\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}}} + -1\right)\right)\right)}\right) \]

  11. Applied flip-+_binary320.4

    \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(-s, \mathsf{log1p}\left(\sqrt{\frac{1}{\frac{u}{1 + e^{-\frac{\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}}}\right), \left(-s\right) \cdot \log \color{blue}{\left(\frac{\sqrt{\frac{1}{\frac{u}{1 + e^{-\frac{\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}}} \cdot \sqrt{\frac{1}{\frac{u}{1 + e^{-\frac{\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}}} - -1 \cdot -1}{\sqrt{\frac{1}{\frac{u}{1 + e^{-\frac{\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}}} - -1}\right)}\right)\right)\right) \]

  12. Simplified0.3

    \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(-s, \mathsf{log1p}\left(\sqrt{\frac{1}{\frac{u}{1 + e^{-\frac{\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}}}\right), \left(-s\right) \cdot \log \left(\frac{\color{blue}{\frac{1}{\frac{u}{1 + e^{-\frac{\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1}}{\sqrt{\frac{1}{\frac{u}{1 + e^{-\frac{\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}}} - -1}\right)\right)\right)\right) \]

  13. Simplified0.3

    \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(-s, \mathsf{log1p}\left(\sqrt{\frac{1}{\frac{u}{1 + e^{-\frac{\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}}}\right), \left(-s\right) \cdot \log \left(\frac{\frac{1}{\frac{u}{1 + e^{-\frac{\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1}{\color{blue}{\sqrt{\frac{1}{\frac{u}{1 + e^{-\frac{\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}}} + 1}}\right)\right)\right)\right) \]

  14. Final simplification0.3

    \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(-s, \mathsf{log1p}\left(\sqrt{\frac{1}{\frac{u}{1 + e^{-\frac{\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}}}\right), \left(-s\right) \cdot \log \left(\frac{\frac{1}{\frac{u}{1 + e^{-\frac{\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1}{1 + \sqrt{\frac{1}{\frac{u}{1 + e^{-\frac{\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}}}}\right)\right)\right)\right) \]

Reproduce

herbie shell --seed 2021310 
(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))))