Sample trimmed logistic on [-pi, pi]

Average Error: 0.3 → 0.3

Time: 15.7s

Precision: binary32

Cost: 16736

\[\left(2.328306437 \cdot 10^{-10} \leq u \land u \leq 1\right) \land \left(0 \leq s \land s \leq 1.0651631\right)\]

Math

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

↓

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

FPCore

(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 s)))))))))))

C

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) / s))))))));
}

Julia

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(-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) / s))))))))))
end

MATLAB

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) / s))))))));
end

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. Simplified 0.3

   \[\leadsto \color{blue}{\left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{-\pi}{s}}} - \frac{u + -1}{1 + e^{\frac{\pi}{s}}}} + -1\right)} \]
   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. Final simplification 0.3

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

Alternatives

Alternative 1
Error	23.9
Cost	13216

\[s \cdot \left(u \cdot 2 + \left(\log s - \log \pi\right)\right) - \frac{s}{\frac{\pi}{s}} \]

Alternative 2
Error	23.9
Cost	13120

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

Alternative 3
Error	23.9
Cost	10048

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

Alternative 4
Error	23.9
Cost	6656

\[s \cdot \left(u \cdot 2 - \mathsf{log1p}\left(\frac{\pi}{s}\right)\right) \]

Alternative 5
Error	23.9
Cost	6624

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

Alternative 6
Error	23.9
Cost	6560

\[s \cdot \left(-\mathsf{log1p}\left(\frac{\pi}{s}\right)\right) \]

Alternative 7
Error	28.3
Cost	3712

\[4 \cdot \left(\pi \cdot \frac{0.0625 + u \cdot \left(u \cdot -0.25\right)}{-0.25 - u \cdot 0.5}\right) \]

Alternative 8
Error	28.3
Cost	3712

\[4 \cdot \frac{\pi}{\frac{-0.25 - u \cdot 0.5}{0.0625 + \left(u \cdot u\right) \cdot -0.25}} \]

Alternative 9
Error	28.3
Cost	3712

\[4 \cdot \frac{\pi \cdot \left(0.25 \cdot \left(u \cdot u\right) + -0.0625\right)}{0.25 + u \cdot 0.5} \]

Alternative 10
Error	28.3
Cost	3584

\[4 \cdot \frac{1}{\frac{1}{\pi \cdot \left(u \cdot 0.5 + -0.25\right)}} \]

Alternative 11
Error	28.3
Cost	3456

\[4 \cdot \left(\pi \cdot \left(u \cdot 0.5 + -0.25\right)\right) \]

Alternative 12
Error	28.3
Cost	3392

\[s \cdot \left(u \cdot 2\right) - \pi \]

Alternative 13
Error	28.3
Cost	3232

\[-\pi \]

Alternative 14
Error	29.2
Cost	160

\[2 \cdot \left(s \cdot u\right) \]

Alternative 15
Error	29.2
Cost	160

\[s \cdot \left(u \cdot 2\right) \]

Reproduce

herbie shell --seed 2023034 
(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))))