Average Error: 0.3 → 0.4
Time: 8.9s
Precision: binary32
\[\left(2.328306437 \cdot 10^{-10} \leq u \land u \leq 1\right) \land \left(0 \leq s \land s \leq 1.0651631\right)\]
\[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]
↓
\[\begin{array}{l} t_0 := \frac{-1 + u}{1 + e^{\frac{\pi}{s}}} - \frac{u}{1 + e^{\frac{-\pi}{s}}}\\ t_1 := \frac{-1}{t_0}\\ \left(-s\right) \cdot \log \left(e^{\log \left(\frac{-1 + \frac{-1}{{t_0}^{3}}}{{t_1}^{2} + \left(1 + t_1\right)}\right)}\right) \end{array} \]
(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))))
          (/ u (+ 1.0 (exp (/ (- PI) s))))))
        (t_1 (/ -1.0 t_0)))
   (*
    (- s)
    (log
     (exp
      (log
       (/ (+ -1.0 (/ -1.0 (pow t_0 3.0))) (+ (pow t_1 2.0) (+ 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)))) - (u / (1.0f + expf((-((float) M_PI) / s))));
	float t_1 = -1.0f / t_0;
	return -s * logf(expf(logf(((-1.0f + (-1.0f / powf(t_0, 3.0f))) / (powf(t_1, 2.0f) + (1.0f + t_1))))));
}

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(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)))))
	t_1 = Float32(Float32(-1.0) / t_0)
	return Float32(Float32(-s) * log(exp(log(Float32(Float32(Float32(-1.0) + Float32(Float32(-1.0) / (t_0 ^ Float32(3.0)))) / Float32((t_1 ^ Float32(2.0)) + Float32(Float32(1.0) + t_1)))))))
end

function tmp = code(u, s)
	tmp = -s * log(((single(1.0) / ((u * ((single(1.0) / (single(1.0) + exp((-single(pi) / s)))) - (single(1.0) / (single(1.0) + exp((single(pi) / s)))))) + (single(1.0) / (single(1.0) + exp((single(pi) / s)))))) - single(1.0)));
end
↓
function tmp = code(u, s)
	t_0 = ((single(-1.0) + u) / (single(1.0) + exp((single(pi) / s)))) - (u / (single(1.0) + exp((-single(pi) / s))));
	t_1 = single(-1.0) / t_0;
	tmp = -s * log(exp(log(((single(-1.0) + (single(-1.0) / (t_0 ^ single(3.0)))) / ((t_1 ^ single(2.0)) + (single(1.0) + t_1))))));
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 + u}{1 + e^{\frac{\pi}{s}}} - \frac{u}{1 + e^{\frac{-\pi}{s}}}\\
t_1 := \frac{-1}{t_0}\\
\left(-s\right) \cdot \log \left(e^{\log \left(\frac{-1 + \frac{-1}{{t_0}^{3}}}{{t_1}^{2} + \left(1 + t_1\right)}\right)}\right)
\end{array}

Error

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-rr0.4

    \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\frac{{\left(\frac{-1}{\frac{-1 + u}{1 + e^{\frac{\pi}{s}}} - \frac{u}{1 + e^{-\frac{\pi}{s}}}}\right)}^{3} + -1}{{\left(\frac{-1}{\frac{-1 + u}{1 + e^{\frac{\pi}{s}}} - \frac{u}{1 + e^{-\frac{\pi}{s}}}}\right)}^{2} + \left(1 - \frac{-1}{\frac{-1 + u}{1 + e^{\frac{\pi}{s}}} - \frac{u}{1 + e^{-\frac{\pi}{s}}}} \cdot -1\right)}\right)} \]

  4. Applied egg-rr0.4

    \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(e^{\log \left(\frac{-1 + \frac{-1}{{\left(\frac{-1 + u}{1 + e^{\frac{\pi}{s}}} - \frac{u}{1 + e^{\frac{-\pi}{s}}}\right)}^{3}}}{{\left(\frac{-1}{\frac{-1 + u}{1 + e^{\frac{\pi}{s}}} - \frac{u}{1 + e^{\frac{-\pi}{s}}}}\right)}^{2} + \left(1 - \frac{1}{\frac{-1 + u}{1 + e^{\frac{\pi}{s}}} - \frac{u}{1 + e^{\frac{-\pi}{s}}}}\right)}\right)}\right)} \]

  5. Final simplification0.4

    \[\leadsto \left(-s\right) \cdot \log \left(e^{\log \left(\frac{-1 + \frac{-1}{{\left(\frac{-1 + u}{1 + e^{\frac{\pi}{s}}} - \frac{u}{1 + e^{\frac{-\pi}{s}}}\right)}^{3}}}{{\left(\frac{-1}{\frac{-1 + u}{1 + e^{\frac{\pi}{s}}} - \frac{u}{1 + e^{\frac{-\pi}{s}}}}\right)}^{2} + \left(1 + \frac{-1}{\frac{-1 + u}{1 + e^{\frac{\pi}{s}}} - \frac{u}{1 + e^{\frac{-\pi}{s}}}}\right)}\right)}\right) \]

Reproduce

herbie shell --seed 2022181 
(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))))