Average Error: 0.3 → 0.3
Time: 1.4min
Precision: binary32
\[2.328306437 \cdot 10^{-10} \leq u \land u \leq 1 \land 0 \leq s \land s \leq 1.0651631\]
\[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]
↓
\[\begin{array}{l} t_0 := 1 + e^{\frac{\pi}{s}}\\ t_1 := \left(\frac{u}{1 + e^{-\frac{\pi}{s}}} + \frac{1}{t_0}\right) - \frac{u}{t_0}\\ t_2 := {t_1}^{2}\\ t_3 := \frac{1}{t_2}\\ \left(-s\right) \cdot \log \left(\frac{\frac{t_3}{t_2} + -1}{\left(1 + \frac{1}{t_1}\right) \cdot \left(1 + t_3\right)}\right) \end{array} \]
\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 := 1 + e^{\frac{\pi}{s}}\\
t_1 := \left(\frac{u}{1 + e^{-\frac{\pi}{s}}} + \frac{1}{t_0}\right) - \frac{u}{t_0}\\
t_2 := {t_1}^{2}\\
t_3 := \frac{1}{t_2}\\
\left(-s\right) \cdot \log \left(\frac{\frac{t_3}{t_2} + -1}{\left(1 + \frac{1}{t_1}\right) \cdot \left(1 + t_3\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 (exp (/ PI s))))
        (t_1 (- (+ (/ u (+ 1.0 (exp (- (/ PI s))))) (/ 1.0 t_0)) (/ u t_0)))
        (t_2 (pow t_1 2.0))
        (t_3 (/ 1.0 t_2)))
   (*
    (- s)
    (log (/ (+ (/ t_3 t_2) -1.0) (* (+ 1.0 (/ 1.0 t_1)) (+ 1.0 t_3)))))))
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 + expf(((float) M_PI) / s);
	float t_1 = ((u / (1.0f + expf(-(((float) M_PI) / s)))) + (1.0f / t_0)) - (u / t_0);
	float t_2 = powf(t_1, 2.0f);
	float t_3 = 1.0f / t_2;
	return -s * logf(((t_3 / t_2) + -1.0f) / ((1.0f + (1.0f / t_1)) * (1.0f + t_3)));
}

Error

Bits error versus u

Bits error versus s

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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. Using strategy rm

  4. Applied flip-+_binary320.3

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

  5. Simplified0.3

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

  6. Simplified0.3

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

  7. Taylor expanded around 0 0.3

    \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\frac{\frac{1}{{\left(\left(\frac{u}{e^{-\frac{\pi}{s}} + 1} + \frac{1}{1 + e^{\frac{\pi}{s}}}\right) - \frac{u}{1 + e^{\frac{\pi}{s}}}\right)}^{2}} - 1}{1 + \frac{1}{\left(\frac{u}{e^{-\frac{\pi}{s}} + 1} + \frac{1}{e^{\frac{\pi}{s}} + 1}\right) - \frac{u}{e^{\frac{\pi}{s}} + 1}}}\right)} \]

  8. Simplified0.3

    \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\frac{\frac{1}{{\left(\left(\frac{u}{e^{-\frac{\pi}{s}} + 1} + \frac{1}{1 + e^{\frac{\pi}{s}}}\right) - \frac{u}{1 + e^{\frac{\pi}{s}}}\right)}^{2}} + -1}{1 + \frac{1}{\left(\frac{u}{e^{-\frac{\pi}{s}} + 1} + \frac{1}{e^{\frac{\pi}{s}} + 1}\right) - \frac{u}{e^{\frac{\pi}{s}} + 1}}}\right)} \]
  9. Using strategy rm

  10. Applied flip-+_binary320.4

    \[\leadsto \left(-s\right) \cdot \log \left(\frac{\color{blue}{\frac{\frac{1}{{\left(\left(\frac{u}{e^{-\frac{\pi}{s}} + 1} + \frac{1}{1 + e^{\frac{\pi}{s}}}\right) - \frac{u}{1 + e^{\frac{\pi}{s}}}\right)}^{2}} \cdot \frac{1}{{\left(\left(\frac{u}{e^{-\frac{\pi}{s}} + 1} + \frac{1}{1 + e^{\frac{\pi}{s}}}\right) - \frac{u}{1 + e^{\frac{\pi}{s}}}\right)}^{2}} - -1 \cdot -1}{\frac{1}{{\left(\left(\frac{u}{e^{-\frac{\pi}{s}} + 1} + \frac{1}{1 + e^{\frac{\pi}{s}}}\right) - \frac{u}{1 + e^{\frac{\pi}{s}}}\right)}^{2}} - -1}}}{1 + \frac{1}{\left(\frac{u}{e^{-\frac{\pi}{s}} + 1} + \frac{1}{e^{\frac{\pi}{s}} + 1}\right) - \frac{u}{e^{\frac{\pi}{s}} + 1}}}\right) \]

  11. Applied associate-/l/_binary320.4

    \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\frac{\frac{1}{{\left(\left(\frac{u}{e^{-\frac{\pi}{s}} + 1} + \frac{1}{1 + e^{\frac{\pi}{s}}}\right) - \frac{u}{1 + e^{\frac{\pi}{s}}}\right)}^{2}} \cdot \frac{1}{{\left(\left(\frac{u}{e^{-\frac{\pi}{s}} + 1} + \frac{1}{1 + e^{\frac{\pi}{s}}}\right) - \frac{u}{1 + e^{\frac{\pi}{s}}}\right)}^{2}} - -1 \cdot -1}{\left(1 + \frac{1}{\left(\frac{u}{e^{-\frac{\pi}{s}} + 1} + \frac{1}{e^{\frac{\pi}{s}} + 1}\right) - \frac{u}{e^{\frac{\pi}{s}} + 1}}\right) \cdot \left(\frac{1}{{\left(\left(\frac{u}{e^{-\frac{\pi}{s}} + 1} + \frac{1}{1 + e^{\frac{\pi}{s}}}\right) - \frac{u}{1 + e^{\frac{\pi}{s}}}\right)}^{2}} - -1\right)}\right)} \]

  12. Simplified0.4

    \[\leadsto \left(-s\right) \cdot \log \left(\frac{\frac{1}{{\left(\left(\frac{u}{e^{-\frac{\pi}{s}} + 1} + \frac{1}{1 + e^{\frac{\pi}{s}}}\right) - \frac{u}{1 + e^{\frac{\pi}{s}}}\right)}^{2}} \cdot \frac{1}{{\left(\left(\frac{u}{e^{-\frac{\pi}{s}} + 1} + \frac{1}{1 + e^{\frac{\pi}{s}}}\right) - \frac{u}{1 + e^{\frac{\pi}{s}}}\right)}^{2}} - -1 \cdot -1}{\color{blue}{\left(1 + \frac{1}{\left(\frac{u}{e^{-\frac{\pi}{s}} + 1} + \frac{1}{e^{\frac{\pi}{s}} + 1}\right) - \frac{u}{e^{\frac{\pi}{s}} + 1}}\right) \cdot \left(\frac{1}{{\left(\left(\frac{u}{e^{-\frac{\pi}{s}} + 1} + \frac{1}{1 + e^{\frac{\pi}{s}}}\right) - \frac{u}{1 + e^{\frac{\pi}{s}}}\right)}^{2}} + 1\right)}}\right) \]
  13. Using strategy rm

  14. Applied un-div-inv_binary320.3

    \[\leadsto \left(-s\right) \cdot \log \left(\frac{\color{blue}{\frac{\frac{1}{{\left(\left(\frac{u}{e^{-\frac{\pi}{s}} + 1} + \frac{1}{1 + e^{\frac{\pi}{s}}}\right) - \frac{u}{1 + e^{\frac{\pi}{s}}}\right)}^{2}}}{{\left(\left(\frac{u}{e^{-\frac{\pi}{s}} + 1} + \frac{1}{1 + e^{\frac{\pi}{s}}}\right) - \frac{u}{1 + e^{\frac{\pi}{s}}}\right)}^{2}}} - -1 \cdot -1}{\left(1 + \frac{1}{\left(\frac{u}{e^{-\frac{\pi}{s}} + 1} + \frac{1}{e^{\frac{\pi}{s}} + 1}\right) - \frac{u}{e^{\frac{\pi}{s}} + 1}}\right) \cdot \left(\frac{1}{{\left(\left(\frac{u}{e^{-\frac{\pi}{s}} + 1} + \frac{1}{1 + e^{\frac{\pi}{s}}}\right) - \frac{u}{1 + e^{\frac{\pi}{s}}}\right)}^{2}} + 1\right)}\right) \]

  15. Final simplification0.3

    \[\leadsto \left(-s\right) \cdot \log \left(\frac{\frac{\frac{1}{{\left(\left(\frac{u}{1 + e^{-\frac{\pi}{s}}} + \frac{1}{1 + e^{\frac{\pi}{s}}}\right) - \frac{u}{1 + e^{\frac{\pi}{s}}}\right)}^{2}}}{{\left(\left(\frac{u}{1 + e^{-\frac{\pi}{s}}} + \frac{1}{1 + e^{\frac{\pi}{s}}}\right) - \frac{u}{1 + e^{\frac{\pi}{s}}}\right)}^{2}} + -1}{\left(1 + \frac{1}{\left(\frac{u}{1 + e^{-\frac{\pi}{s}}} + \frac{1}{1 + e^{\frac{\pi}{s}}}\right) - \frac{u}{1 + e^{\frac{\pi}{s}}}}\right) \cdot \left(1 + \frac{1}{{\left(\left(\frac{u}{1 + e^{-\frac{\pi}{s}}} + \frac{1}{1 + e^{\frac{\pi}{s}}}\right) - \frac{u}{1 + e^{\frac{\pi}{s}}}\right)}^{2}}\right)}\right) \]

Reproduce

herbie shell --seed 2021204 
(FPCore (u s)
  :name "Sample trimmed logistic on [-pi, pi]"
  :precision binary32
  :pre (and (<= 2.328306437e-10 u 1.0) (<= 0.0 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))))