?

Average Error: 99.0% → 99.0%
Time: 18.1s
Precision: binary32
Cost: 23168.00

?

\[\left(2.328306437 \cdot 10^{-10} \leq u \land u \leq 1\right) \land \left(0 \leq s \land s \leq 1.0651631\right)\]
\[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]
↓
\[\left(-s\right) \cdot \log \left(-1 + \frac{1}{\frac{u}{1 + e^{\frac{-\pi}{s}}} + \frac{1 - u}{1 + e^{{\left(\sqrt{\frac{\pi}{s}}\right)}^{2}}}}\right) \]
(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 (pow (sqrt (/ PI s)) 2.0))))))))))
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 + (1.0f / ((u / (1.0f + expf((-((float) M_PI) / s)))) + ((1.0f - u) / (1.0f + expf(powf(sqrtf((((float) M_PI) / s)), 2.0f))))))));
}

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(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((sqrt(Float32(Float32(pi) / s)) ^ Float32(2.0))))))))))
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) + (single(1.0) / ((u / (single(1.0) + exp((-single(pi) / s)))) + ((single(1.0) - u) / (single(1.0) + exp((sqrt((single(pi) / s)) ^ single(2.0)))))))));
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)
↓
\left(-s\right) \cdot \log \left(-1 + \frac{1}{\frac{u}{1 + e^{\frac{-\pi}{s}}} + \frac{1 - u}{1 + e^{{\left(\sqrt{\frac{\pi}{s}}\right)}^{2}}}}\right)

Error?

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation?

  1. Initial program 99.0

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

  2. Simplified99.0

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

    [Start]99.0

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

    sub-neg [=>]99.0

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

  3. Applied egg-rr99.0

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

    [Start]99.0

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

    add-sqr-sqrt [=>]99.0

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

    pow2 [=>]99.0

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

  4. Final simplification99.0

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

Alternatives

Alternative 1
Error99.0%
Cost16864.00
\[s \cdot \left(-\log \left(-1 + \frac{1}{\frac{u}{1 + e^{\frac{-\pi}{s}}} + \frac{1 - u}{1 + e^{-1 + \left(1 + \frac{\pi}{s}\right)}}}\right)\right) \]
Alternative 2
Error99.0%
Cost16736.00
\[s \cdot \left(-\log \left(-1 + \frac{1}{\frac{u}{1 + e^{\frac{-\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}}\right)\right) \]
Alternative 3
Error37.8%
Cost10208.00
\[\left(-s\right) \cdot \log \left(-1 + \frac{1}{u \cdot 0.5 + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}}\right) \]
Alternative 4
Error37.2%
Cost10144.00
\[s \cdot \left(-\log \left(-1 + \frac{1}{u \cdot \left(0.5 + \frac{-1}{1 + e^{\frac{\pi}{s}}}\right)}\right)\right) \]
Alternative 5
Error36.3%
Cost7072.00
\[\left(-s\right) \cdot \log \left(-1 + \frac{1}{\frac{u}{2} + \frac{1 - u}{1 + \left(1 + \frac{\pi}{s}\right)}}\right) \]
Alternative 6
Error11.6%
Cost6656.00
\[4 \cdot \frac{\pi}{\frac{1}{\mathsf{fma}\left(u, 0.5, -0.25\right)}} \]
Alternative 7
Error11.6%
Cost3712.00
\[4 \cdot \frac{\pi \cdot \left(0.0625 + -0.25 \cdot \left(u \cdot u\right)\right)}{-0.25 + u \cdot -0.5} \]
Alternative 8
Error11.6%
Cost3584.00
\[4 \cdot \frac{1}{\frac{1}{\pi \cdot \left(u \cdot 0.5 + -0.25\right)}} \]
Alternative 9
Error11.6%
Cost3456.00
\[4 \cdot \left(\pi \cdot \left(u \cdot 0.5 + -0.25\right)\right) \]
Alternative 10
Error11.4%
Cost3232.00
\[-\pi \]
Alternative 11
Error4.6%
Cost3200.00
\[\pi \]

Error

Reproduce?

herbie shell --seed 2023121 
(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))))