Sample trimmed logistic on [-pi, pi]

Average Error: 0.3 → 0.3

Time: 18.7s

Precision: binary32

Cost: 29664

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

Math

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

↓

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

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 (* (pow (cbrt PI) 2.0) (* (cbrt 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((powf(cbrtf(((float) M_PI)), 2.0f) * (cbrtf(((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((cbrt(Float32(pi)) ^ Float32(2.0)) * Float32(cbrt(Float32(pi)) * Float32(Float32(1.0) / 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. Applied egg-rr 0.3

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

4. Final simplification 0.3

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

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	24.0
Cost	13536

\[\begin{array}{l} t_0 := \pi \cdot \left(0.25 + u \cdot -0.5\right)\\ s \cdot \left(\log s + \left(\frac{s}{t_0} \cdot -0.25 - \log \left(4 \cdot t_0\right)\right)\right) \end{array} \]

Alternative 3
Error	24.0
Cost	10112

\[s \cdot \left(1 + \left(\log s + \left(-1 - \log \left(4 \cdot \left(\pi \cdot \left(0.25 + u \cdot -0.5\right)\right)\right)\right)\right)\right) \]

Alternative 4
Error	24.0
Cost	9984

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

Alternative 5
Error	24.0
Cost	6560

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

Alternative 6
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 7
Error	28.3
Cost	3584

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

Alternative 8
Error	28.3
Cost	3456

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

Alternative 9
Error	28.4
Cost	3232

\[-\pi \]

Alternative 10
Error	28.7
Cost	96

\[s \cdot 0 \]

Reproduce

herbie shell --seed 2023046 
(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))))