Average Error: 0.3 → 0.3
Time: 19.5s
Precision: binary32
Cost: 16800
\[\left(2.328306437 \cdot 10^{-10} \leq u \land u \leq 1\right) \land \left(0 \leq s \land s \leq 1.0651631\right)\]
\[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]
↓
\[s \cdot \left(-\log \left(\frac{1}{\frac{u}{1 + e^{\frac{-\pi}{s}}} + \left(1 - u\right) \cdot \frac{1}{1 + e^{\frac{\pi}{s}}}} + -1\right)\right) \]
(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
 (*
  s
  (-
   (log
    (+
     (/
      1.0
      (+
       (/ u (+ 1.0 (exp (/ (- PI) s))))
       (* (- 1.0 u) (/ 1.0 (+ 1.0 (exp (/ PI s)))))))
     -1.0)))))
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) {
	return s * -logf(((1.0f / ((u / (1.0f + expf((-((float) M_PI) / s)))) + ((1.0f - u) * (1.0f / (1.0f + expf((((float) M_PI) / s))))))) + -1.0f));
}

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)
	return Float32(s * Float32(-log(Float32(Float32(Float32(1.0) / Float32(Float32(u / Float32(Float32(1.0) + exp(Float32(Float32(-Float32(pi)) / s)))) + Float32(Float32(Float32(1.0) - u) * Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(pi) / s))))))) + Float32(-1.0)))))
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)
	tmp = s * -log(((single(1.0) / ((u / (single(1.0) + exp((-single(pi) / s)))) + ((single(1.0) - u) * (single(1.0) / (single(1.0) + exp((single(pi) / s))))))) + single(-1.0)));
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)
↓
s \cdot \left(-\log \left(\frac{1}{\frac{u}{1 + e^{\frac{-\pi}{s}}} + \left(1 - u\right) \cdot \frac{1}{1 + e^{\frac{\pi}{s}}}} + -1\right)\right)

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

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

    [Start]0.3

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

    sub-neg [=>]0.3

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

    sub-neg [=>]0.3

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

    distribute-rgt-in [=>]0.3

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

    associate-+l+ [=>]0.3

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

    associate-*l/ [=>]0.3

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

    *-lft-identity [=>]0.3

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

    *-commutative [=>]0.3

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

    neg-mul-1 [=>]0.3

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

    metadata-eval [<=]0.3

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

    associate-*r* [=>]0.3

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

    distribute-lft1-in [=>]0.3

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

    *-commutative [=>]0.3

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

    metadata-eval [=>]0.3

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

    mul-1-neg [=>]0.3

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

    metadata-eval [=>]0.3

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

  3. Final simplification0.3

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

Alternatives

Alternative 1
Error0.3
Cost16736
\[s \cdot \left(-\log \left(-1 + \frac{1}{\frac{u}{1 + e^{\frac{-\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}}\right)\right) \]
Alternative 2
Error7.7
Cost16672
\[s \cdot \left(-\log \left(\frac{-1}{u \cdot \left(\frac{1}{1 + e^{\frac{\pi}{s}}} + \frac{-1}{1 + e^{\frac{-\pi}{s}}}\right)}\right)\right) \]
Alternative 3
Error8.2
Cost13536
\[s \cdot \left(-\log \left(\frac{-1}{u \cdot \left(\frac{1}{1 + \left(1 + \frac{\pi}{s}\right)} - \frac{1}{1 + e^{\frac{-\pi}{s}}}\right)}\right)\right) \]
Alternative 4
Error24.0
Cost10112
\[s \cdot \left(-\log \left(1 + \frac{4}{\frac{s}{\left(u \cdot \pi\right) \cdot -0.5 + \pi \cdot 0.25}}\right)\right) \]
Alternative 5
Error20.1
Cost10080
\[\left(-s\right) \cdot \log \left(\frac{-1}{u \cdot \left(\frac{1}{1 + e^{\frac{\pi}{s}}} + -0.5\right)}\right) \]
Alternative 6
Error24.7
Cost9856
\[s \cdot \left(\log \left(2 \cdot \frac{s}{\pi}\right) - \log u\right) \]
Alternative 7
Error24.7
Cost6656
\[s \cdot \log \left(s \cdot \frac{2}{u \cdot \pi}\right) \]
Alternative 8
Error28.3
Cost3712
\[4 \cdot \left(\pi \cdot \frac{0.0625 + u \cdot \left(u \cdot -0.25\right)}{-0.25 + u \cdot -0.5}\right) \]
Alternative 9
Error28.3
Cost3584
\[4 \cdot \frac{1}{\frac{1}{\pi \cdot \left(-0.25 + u \cdot 0.5\right)}} \]
Alternative 10
Error28.3
Cost3456
\[4 \cdot \left(\pi \cdot \left(-0.25 + u \cdot 0.5\right)\right) \]
Alternative 11
Error28.3
Cost3392
\[4 \cdot \left(\pi \cdot \left(u + -0.25\right)\right) \]
Alternative 12
Error28.3
Cost3232
\[-\pi \]

Error

Reproduce

herbie shell --seed 2022364 
(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))))