Sample trimmed logistic on [-pi, pi]

Average Accuracy: 98.9% → 98.9%

Time: 17.4s

Precision: binary32

Cost: 16800

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

Math

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

↓

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

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 (/ 1.0 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) * (1.0f / 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) * Float32(Float32(1.0) / 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) * (single(1.0) / s)))))))));
end

Derivation

1. Initial program 98.9%

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

2. Simplified 98.9%

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

   [Start] 98.9	\[ \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]
   sub-neg [=>] 98.9	\[ \left(-s\right) \cdot \log \color{blue}{\left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} + \left(-1\right)\right)} \]

3. Applied egg-rr 98.9%

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

   [Start] 98.9	\[ \left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{-\pi}{s}}} - \frac{u + -1}{1 + e^{\frac{\pi}{s}}}} + -1\right) \]
   div-inv [=>] 98.9	\[ \left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{-\pi}{s}}} - \frac{u + -1}{1 + e^{\color{blue}{\pi \cdot \frac{1}{s}}}}} + -1\right) \]
   *-commutative [=>] 98.9	\[ \left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{-\pi}{s}}} - \frac{u + -1}{1 + e^{\color{blue}{\frac{1}{s} \cdot \pi}}}} + -1\right) \]

4. Final simplification 98.9%

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

Alternatives

Alternative 1
Accuracy	98.9%
Cost	16736

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

Alternative 2
Accuracy	25.2%
Cost	10048

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

Alternative 3
Accuracy	25.2%
Cost	6848

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

Alternative 4
Accuracy	25.0%
Cost	6560

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

Alternative 5
Accuracy	25.0%
Cost	6560

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

Alternative 6
Accuracy	14.4%
Cost	3716

\[\begin{array}{l} \mathbf{if}\;s \leq 4.999999918875795 \cdot 10^{-18}:\\ \;\;\;\;{u}^{3} \cdot \left(s \cdot 2.6666666666666665\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi \cdot \left(1 + \left(u \cdot u\right) \cdot -4\right)}{-1 + u \cdot -2}\\ \end{array} \]

Alternative 7
Accuracy	14.4%
Cost	3460

\[\begin{array}{l} \mathbf{if}\;s \leq 1.4999999523982838 \cdot 10^{-20}:\\ \;\;\;\;s \cdot 0\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \left(-1 + u \cdot 2\right)\\ \end{array} \]

Alternative 8
Accuracy	14.4%
Cost	3460

\[\begin{array}{l} \mathbf{if}\;s \leq 4.999999918875795 \cdot 10^{-18}:\\ \;\;\;\;{u}^{3} \cdot \left(s \cdot 2.6666666666666665\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \left(-1 + u \cdot 2\right)\\ \end{array} \]

Alternative 9
Accuracy	14.1%
Cost	3300

\[\begin{array}{l} \mathbf{if}\;s \leq 4.999999918875795 \cdot 10^{-18}:\\ \;\;\;\;s \cdot 0\\ \mathbf{else}:\\ \;\;\;\;-\pi\\ \end{array} \]

Alternative 10
Accuracy	11.4%
Cost	3232

\[-\pi \]

Reproduce

herbie shell --seed 2023135 
(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))))