Sample trimmed logistic on [-pi, pi]

Average Error: 0.3 → 0.3

Time: 15.6s

Precision: binary32

Cost: 23136

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

Math

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

↓

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

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 (exp (log (/ 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(expf(logf((((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(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(exp(log(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(exp(log((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	\[ \left(-s\right) \cdot \log \color{blue}{\left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} + \left(-1\right)\right)} \]
   sub-neg [=>] 0.3	\[ \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \color{blue}{\left(\frac{1}{1 + e^{\frac{-\pi}{s}}} + \left(-\frac{1}{1 + e^{\frac{\pi}{s}}}\right)\right)} + \frac{1}{1 + e^{\frac{\pi}{s}}}} + \left(-1\right)\right) \]
   distribute-rgt-in [=>] 0.3	\[ \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{\left(\frac{1}{1 + e^{\frac{-\pi}{s}}} \cdot u + \left(-\frac{1}{1 + e^{\frac{\pi}{s}}}\right) \cdot u\right)} + \frac{1}{1 + e^{\frac{\pi}{s}}}} + \left(-1\right)\right) \]
   associate-+l+ [=>] 0.3	\[ \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{1}{1 + e^{\frac{-\pi}{s}}} \cdot u + \left(\left(-\frac{1}{1 + e^{\frac{\pi}{s}}}\right) \cdot u + \frac{1}{1 + e^{\frac{\pi}{s}}}\right)}} + \left(-1\right)\right) \]
   distribute-lft-neg-out [=>] 0.3	\[ \left(-s\right) \cdot \log \left(\frac{1}{\frac{1}{1 + e^{\frac{-\pi}{s}}} \cdot u + \left(\color{blue}{\left(-\frac{1}{1 + e^{\frac{\pi}{s}}} \cdot u\right)} + \frac{1}{1 + e^{\frac{\pi}{s}}}\right)} + \left(-1\right)\right) \]
   neg-sub0 [=>] 0.3	\[ \left(-s\right) \cdot \log \left(\frac{1}{\frac{1}{1 + e^{\frac{-\pi}{s}}} \cdot u + \left(\color{blue}{\left(0 - \frac{1}{1 + e^{\frac{\pi}{s}}} \cdot u\right)} + \frac{1}{1 + e^{\frac{\pi}{s}}}\right)} + \left(-1\right)\right) \]
   associate-+l- [=>] 0.3	\[ \left(-s\right) \cdot \log \left(\frac{1}{\frac{1}{1 + e^{\frac{-\pi}{s}}} \cdot u + \color{blue}{\left(0 - \left(\frac{1}{1 + e^{\frac{\pi}{s}}} \cdot u - \frac{1}{1 + e^{\frac{\pi}{s}}}\right)\right)}} + \left(-1\right)\right) \]
   sub0-neg [=>] 0.3	\[ \left(-s\right) \cdot \log \left(\frac{1}{\frac{1}{1 + e^{\frac{-\pi}{s}}} \cdot u + \color{blue}{\left(-\left(\frac{1}{1 + e^{\frac{\pi}{s}}} \cdot u - \frac{1}{1 + e^{\frac{\pi}{s}}}\right)\right)}} + \left(-1\right)\right) \]
   neg-mul-1 [=>] 0.3	\[ \left(-s\right) \cdot \log \left(\frac{1}{\frac{1}{1 + e^{\frac{-\pi}{s}}} \cdot u + \color{blue}{-1 \cdot \left(\frac{1}{1 + e^{\frac{\pi}{s}}} \cdot u - \frac{1}{1 + e^{\frac{\pi}{s}}}\right)}} + \left(-1\right)\right) \]
   metadata-eval [<=] 0.3	\[ \left(-s\right) \cdot \log \left(\frac{1}{\frac{1}{1 + e^{\frac{-\pi}{s}}} \cdot u + \color{blue}{\left(-1\right)} \cdot \left(\frac{1}{1 + e^{\frac{\pi}{s}}} \cdot u - \frac{1}{1 + e^{\frac{\pi}{s}}}\right)} + \left(-1\right)\right) \]
   cancel-sign-sub-inv [<=] 0.3	\[ \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{1}{1 + e^{\frac{-\pi}{s}}} \cdot u - 1 \cdot \left(\frac{1}{1 + e^{\frac{\pi}{s}}} \cdot u - \frac{1}{1 + e^{\frac{\pi}{s}}}\right)}} + \left(-1\right)\right) \]
   associate-*l/ [=>] 0.3	\[ \left(-s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{1 \cdot u}{1 + e^{\frac{-\pi}{s}}}} - 1 \cdot \left(\frac{1}{1 + e^{\frac{\pi}{s}}} \cdot u - \frac{1}{1 + e^{\frac{\pi}{s}}}\right)} + \left(-1\right)\right) \]
   *-lft-identity [=>] 0.3	\[ \left(-s\right) \cdot \log \left(\frac{1}{\frac{\color{blue}{u}}{1 + e^{\frac{-\pi}{s}}} - 1 \cdot \left(\frac{1}{1 + e^{\frac{\pi}{s}}} \cdot u - \frac{1}{1 + e^{\frac{\pi}{s}}}\right)} + \left(-1\right)\right) \]
   *-lft-identity [=>] 0.3	\[ \left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{-\pi}{s}}} - \color{blue}{\left(\frac{1}{1 + e^{\frac{\pi}{s}}} \cdot u - \frac{1}{1 + e^{\frac{\pi}{s}}}\right)}} + \left(-1\right)\right) \]
   associate-*l/ [=>] 0.3	\[ \left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{-\pi}{s}}} - \left(\color{blue}{\frac{1 \cdot u}{1 + e^{\frac{\pi}{s}}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right)} + \left(-1\right)\right) \]
   *-lft-identity [=>] 0.3	\[ \left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{-\pi}{s}}} - \left(\frac{\color{blue}{u}}{1 + e^{\frac{\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right)} + \left(-1\right)\right) \]
   div-sub [<=] 0.3	\[ \left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{-\pi}{s}}} - \color{blue}{\frac{u - 1}{1 + e^{\frac{\pi}{s}}}}} + \left(-1\right)\right) \]
   sub-neg [=>] 0.3	\[ \left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{-\pi}{s}}} - \frac{\color{blue}{u + \left(-1\right)}}{1 + e^{\frac{\pi}{s}}}} + \left(-1\right)\right) \]
   metadata-eval [=>] 0.3	\[ \left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{-\pi}{s}}} - \frac{u + \color{blue}{-1}}{1 + e^{\frac{\pi}{s}}}} + \left(-1\right)\right) \]
   metadata-eval [=>] 0.3	\[ \left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{-\pi}{s}}} - \frac{u + -1}{1 + e^{\frac{\pi}{s}}}} + \color{blue}{-1}\right) \]

3. Applied egg-rr 0.3

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

4. Final simplification 0.3

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

Alternatives

Alternative 1
Error	0.3
Cost	16736

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

Alternative 2
Error	28.3
Cost	13248

\[4 \cdot \log \left(-1 + \frac{1}{\frac{1}{\mathsf{expm1}\left(\pi \cdot \mathsf{fma}\left(u, 0.5, -0.25\right)\right) + 2}}\right) \]

Alternative 3
Error	28.3
Cost	13120

\[4 \cdot \log \left(-1 + \left(\mathsf{expm1}\left(\pi \cdot \mathsf{fma}\left(u, 0.5, -0.25\right)\right) + 2\right)\right) \]

Alternative 4
Error	28.3
Cost	9984

\[4 \cdot \mathsf{log1p}\left(\left(2 + \mathsf{expm1}\left(\pi \cdot \left(-0.25 + u \cdot 0.5\right)\right)\right) + -2\right) \]

Alternative 5
Error	28.3
Cost	9920

\[4 \cdot \log \left(1 + \mathsf{expm1}\left(\pi \cdot \left(-0.25 + u \cdot 0.5\right)\right)\right) \]

Alternative 6
Error	28.3
Cost	9856

\[4 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\pi \cdot \left(-0.25 + u \cdot 0.5\right)\right)\right) \]

Alternative 7
Error	28.3
Cost	6656

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

Alternative 8
Error	28.3
Cost	3712

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

Alternative 9
Error	28.3
Cost	3584

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

Alternative 10
Error	28.3
Cost	3456

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

Alternative 11
Error	28.4
Cost	3232

\[-\pi \]

Alternative 12
Error	30.5
Cost	3200

\[\pi \]

Reproduce

herbie shell --seed 2022354 
(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))))