Average Error: 0.3 → 0.4
Time: 28.4s
Precision: binary32
\[\left(2.328306437 \cdot 10^{-10} \leq u \land u \leq 1\right) \land \left(0 \leq s \land s \leq 1.0651631\right)\]
\[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]
↓
\[\begin{array}{l} t_0 := \frac{-1}{\frac{-1 + u}{1 + e^{\frac{\pi}{s}}} - \frac{u}{1 + e^{\frac{-\pi}{s}}}}\\ s \cdot \left(-\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\frac{-1 + {t_0}^{2}}{1 + t_0}\right)\right)\right)\right) \end{array} \]
(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
          (-
           (/ (+ -1.0 u) (+ 1.0 (exp (/ PI s))))
           (/ u (+ 1.0 (exp (/ (- PI) s))))))))
   (* s (- (log1p (expm1 (log (/ (+ -1.0 (pow t_0 2.0)) (+ 1.0 t_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 = -1.0f / (((-1.0f + u) / (1.0f + expf((((float) M_PI) / s)))) - (u / (1.0f + expf((-((float) M_PI) / s)))));
	return s * -log1pf(expm1f(logf(((-1.0f + powf(t_0, 2.0f)) / (1.0f + t_0)))));
}

function code(u, s)
	return Float32(Float32(-s) * log(Float32(Float32(Float32(1.0) / Float32(Float32(u * Float32(Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(-Float32(pi)) / s)))) - Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(pi) / s)))))) + Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(pi) / s)))))) - Float32(1.0))))
end
↓
function code(u, s)
	t_0 = Float32(Float32(-1.0) / Float32(Float32(Float32(Float32(-1.0) + u) / Float32(Float32(1.0) + exp(Float32(Float32(pi) / s)))) - Float32(u / Float32(Float32(1.0) + exp(Float32(Float32(-Float32(pi)) / s))))))
	return Float32(s * Float32(-log1p(expm1(log(Float32(Float32(Float32(-1.0) + (t_0 ^ Float32(2.0))) / Float32(Float32(1.0) + t_0)))))))
end

\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{-1 + u}{1 + e^{\frac{\pi}{s}}} - \frac{u}{1 + e^{\frac{-\pi}{s}}}}\\
s \cdot \left(-\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\frac{-1 + {t_0}^{2}}{1 + t_0}\right)\right)\right)\right)
\end{array}

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

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

  3. Applied egg-rr0.5

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

  4. Applied egg-rr0.4

    \[\leadsto \left(-s\right) \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\frac{{\left(\frac{-1}{\frac{-1 + u}{1 + e^{\frac{\pi}{s}}} - \frac{u}{1 + e^{\frac{-\pi}{s}}}}\right)}^{2} + -1}{\frac{-1}{\frac{-1 + u}{1 + e^{\frac{\pi}{s}}} - \frac{u}{1 + e^{\frac{-\pi}{s}}}} + 1}\right)\right)\right)} \]

  5. Final simplification0.4

    \[\leadsto s \cdot \left(-\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\frac{-1 + {\left(\frac{-1}{\frac{-1 + u}{1 + e^{\frac{\pi}{s}}} - \frac{u}{1 + e^{\frac{-\pi}{s}}}}\right)}^{2}}{1 + \frac{-1}{\frac{-1 + u}{1 + e^{\frac{\pi}{s}}} - \frac{u}{1 + e^{\frac{-\pi}{s}}}}}\right)\right)\right)\right) \]

Reproduce

herbie shell --seed 2022156 
(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))))