Average Error: 0.3 → 0.4
Time: 1.9min
Precision: binary32
\[\left(2.328306437 \cdot 10^{-10} \leq u \land u \leq 1\right) \land \left(0 \leq s \land s \leq 1.0651631\right)\]
\[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]
↓
\[\begin{array}{l} t_0 := \frac{u}{1 + e^{\frac{-\pi}{s}}}\\ \left(-s\right) \cdot \left(\log \left(\sqrt{\frac{1}{t_0 + \frac{1 - u}{1 + e^{\frac{\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}}{\frac{s}{\sqrt[3]{\pi}}}}}} + -1}\right) + \log \left(\sqrt{-1 + \frac{1}{t_0 + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}}}\right)\right) \end{array} \]
\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{u}{1 + e^{\frac{-\pi}{s}}}\\
\left(-s\right) \cdot \left(\log \left(\sqrt{\frac{1}{t_0 + \frac{1 - u}{1 + e^{\frac{\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}}{\frac{s}{\sqrt[3]{\pi}}}}}} + -1}\right) + \log \left(\sqrt{-1 + \frac{1}{t_0 + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}}}\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 (/ u (+ 1.0 (exp (/ (- PI) s))))))
   (*
    (- s)
    (+
     (log
      (sqrt
       (+
        (/
         1.0
         (+
          t_0
          (/
           (- 1.0 u)
           (+ 1.0 (exp (/ (* (cbrt PI) (cbrt PI)) (/ s (cbrt PI))))))))
        -1.0)))
     (log
      (sqrt (+ -1.0 (/ 1.0 (+ t_0 (/ (- 1.0 u) (+ 1.0 (exp (/ PI s)))))))))))))
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 = u / (1.0f + expf((-((float) M_PI) / s)));
	return -s * (logf(sqrtf(((1.0f / (t_0 + ((1.0f - u) / (1.0f + expf(((cbrtf(((float) M_PI)) * cbrtf(((float) M_PI))) / (s / cbrtf(((float) M_PI))))))))) + -1.0f))) + logf(sqrtf((-1.0f + (1.0f / (t_0 + ((1.0f - u) / (1.0f + expf((((float) M_PI) / s))))))))));
}

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. Applied add-sqr-sqrt_binary320.4

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

  4. Applied log-prod_binary320.4

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

  5. Applied add-cube-cbrt_binary320.4

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

  6. Applied associate-/l*_binary320.4

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

  7. Final simplification0.4

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

Reproduce

herbie shell --seed 2022125 
(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))))