?

Average Error: 0.3 → 0.3
Time: 14.7s
Precision: binary32
Cost: 16800

?

\[\left(2.328306437 \cdot 10^{-10} \leq u \land u \leq 1\right) \land \left(0 \leq s \land s \leq 1.0651631\right)\]
\[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]
↓
\[\left(-s\right) \cdot \log \left(-1 + \frac{1}{\frac{u}{1 + e^{\frac{-\pi}{s}}} + \frac{1 - u}{1 + e^{\pi \cdot \frac{1}{s}}}}\right) \]
(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
    (/
     1.0
     (+
      (/ u (+ 1.0 (exp (/ (- PI) s))))
      (/ (- 1.0 u) (+ 1.0 (exp (* PI (/ 1.0 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) {
	return -s * logf((-1.0f + (1.0f / ((u / (1.0f + expf((-((float) M_PI) / s)))) + ((1.0f - u) / (1.0f + expf((((float) M_PI) * (1.0f / s)))))))));
}

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

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

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

  3. Applied egg-rr0.3

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

  4. Final simplification0.3

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

Alternatives

Alternative 1
Error0.3
Cost16736
\[\left(-s\right) \cdot \log \left(-1 + \frac{1}{\frac{u}{1 + e^{\frac{-\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}}\right) \]
Alternative 2
Error20.0
Cost16640
\[\left(-s\right) \cdot \log \left(-1 + \frac{1}{{\left(\sqrt[3]{u \cdot 0.5 + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}}\right)}^{3}}\right) \]
Alternative 3
Error20.0
Cost16640
\[s \cdot \left(-\log \left(-1 + \frac{1}{\frac{u}{2} + \frac{1 - u}{1 + e^{{\left(\sqrt[3]{\frac{\pi}{s}}\right)}^{3}}}}\right)\right) \]
Alternative 4
Error20.0
Cost16576
\[\log \left(\sqrt{-1 + \frac{1}{\mathsf{fma}\left(0.5, u, \frac{1 - u}{1 + e^{\frac{\pi}{s}}}\right)}}\right) \cdot \left(s \cdot -2\right) \]
Alternative 5
Error20.0
Cost10272
\[s \cdot \left(-\log \left(-1 + \frac{1}{\frac{u}{2} + \frac{1 - u}{1 + e^{\pi \cdot \frac{1}{s}}}}\right)\right) \]
Alternative 6
Error20.0
Cost10208
\[s \cdot \left(-\log \left(-1 + \frac{1}{u \cdot 0.5 + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}}\right)\right) \]
Alternative 7
Error20.1
Cost10144
\[\left(-s\right) \cdot \mathsf{log1p}\left(-2 + \frac{1}{u \cdot \left(0.5 + \frac{-1}{1 + e^{\frac{\pi}{s}}}\right)}\right) \]
Alternative 8
Error20.4
Cost7072
\[\left(-s\right) \cdot \log \left(-1 + \frac{1}{\frac{u}{2} + \frac{1 - u}{1 + \left(1 + \frac{\pi}{s}\right)}}\right) \]
Alternative 9
Error27.4
Cost6724
\[\begin{array}{l} \mathbf{if}\;s \leq 2.59999998825566 \cdot 10^{-14}:\\ \;\;\;\;s \cdot \left(-\mathsf{log1p}\left(-2 + \frac{2}{\left(1 + u\right) - u}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \frac{\pi}{\frac{1}{\mathsf{fma}\left(u, 0.5, -0.25\right)}}\\ \end{array} \]
Alternative 10
Error27.4
Cost3652
\[\begin{array}{l} \mathbf{if}\;s \leq 2.59999998825566 \cdot 10^{-14}:\\ \;\;\;\;s \cdot \left(-\mathsf{log1p}\left(-2 + \frac{2}{\left(1 + u\right) - u}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(\pi \cdot \left(u \cdot 0.5 + -0.25\right)\right)\\ \end{array} \]
Alternative 11
Error28.3
Cost3456
\[4 \cdot \left(\pi \cdot \left(u \cdot 0.5 + -0.25\right)\right) \]
Alternative 12
Error28.3
Cost3232
\[-\pi \]
Alternative 13
Error30.5
Cost3200
\[\pi \]

Error

Reproduce?

herbie shell --seed 2023045 
(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))))