Sample trimmed logistic on [-pi, pi]

Average Accuracy: 99.0% → 99.0%

Time: 16.5s

Precision: binary32

Cost: 19936

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

Math

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

↓

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

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
 (*
  (log
   (+
    (/
     1.0
     (+
      (/ u (+ 1.0 (exp (/ (- PI) s))))
      (/ (- 1.0 u) (+ 1.0 (pow E (/ PI s))))))
    -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 logf(((1.0f / ((u / (1.0f + expf((-((float) M_PI) / s)))) + ((1.0f - u) / (1.0f + powf(((float) M_E), (((float) M_PI) / s)))))) + -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(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(exp(1)) ^ Float32(Float32(pi) / s)))))) + Float32(-1.0))) * Float32(-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 = log(((single(1.0) / ((u / (single(1.0) + exp((-single(pi) / s)))) + ((single(1.0) - u) / (single(1.0) + (single(2.71828182845904523536) ^ (single(pi) / s)))))) + single(-1.0))) * -s;
end

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. Simplified 99.0%

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

   [Start] 99.0	\[ \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]
   distribute-lft-neg-out [=>] 99.0	\[ \color{blue}{-s \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right)} \]
   distribute-rgt-neg-in [=>] 99.0	\[ \color{blue}{s \cdot \left(-\log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right)\right)} \]
   sub-neg [=>] 99.0	\[ s \cdot \left(-\log \color{blue}{\left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} + \left(-1\right)\right)}\right) \]

3. Applied egg-rr 99.0%

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

   [Start] 99.0	\[ s \cdot \left(-\log \left(\frac{1}{\frac{u}{1 + e^{\frac{-\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right)\right) \]
   *-un-lft-identity [=>] 99.0	\[ s \cdot \left(-\log \left(\frac{1}{\frac{u}{1 + e^{\frac{-\pi}{s}}} + \frac{1 - u}{1 + e^{\color{blue}{1 \cdot \frac{\pi}{s}}}}} + -1\right)\right) \]
   exp-prod [=>] 99.0	\[ s \cdot \left(-\log \left(\frac{1}{\frac{u}{1 + e^{\frac{-\pi}{s}}} + \frac{1 - u}{1 + \color{blue}{{\left(e^{1}\right)}^{\left(\frac{\pi}{s}\right)}}}} + -1\right)\right) \]
   exp-1-e [=>] 99.0	\[ s \cdot \left(-\log \left(\frac{1}{\frac{u}{1 + e^{\frac{-\pi}{s}}} + \frac{1 - u}{1 + {\color{blue}{e}}^{\left(\frac{\pi}{s}\right)}}} + -1\right)\right) \]

4. Final simplification 99.0%

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

Alternatives

Alternative 1
Accuracy	99.0%
Cost	16736

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

Alternative 2
Accuracy	98.6%
Cost	16672

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

Alternative 3
Accuracy	85.9%
Cost	13600

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

Alternative 4
Accuracy	85.8%
Cost	13536

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

Alternative 5
Accuracy	37.6%
Cost	10144

\[s \cdot \left(-\log \left(-1 + \frac{1}{\frac{1}{1 + e^{\frac{\pi}{s}}} + u \cdot 0.5}\right)\right) \]

Alternative 6
Accuracy	36.2%
Cost	7008

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

Alternative 7
Accuracy	16.2%
Cost	3648

\[4 \cdot \left(s \cdot \left(u \cdot \left(\frac{1}{2 - \frac{\pi}{s}} - 0.5\right)\right)\right) \]

Alternative 8
Accuracy	16.2%
Cost	3616

\[s \cdot \left(u \cdot \left(\left(-2\right) - \frac{-4}{2 - \frac{\pi}{s}}\right)\right) \]

Alternative 9
Accuracy	16.2%
Cost	3520

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

Alternative 10
Accuracy	16.2%
Cost	160

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

Alternative 11
Accuracy	16.2%
Cost	160

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

Reproduce

herbie shell --seed 2023151 
(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))))