Sample trimmed logistic on [-pi, pi]

Average Error: 0.3 → 0.3

Time: 9.4s

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

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

↓

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

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
      (-
       (/ (+ -1.0 u) (+ 1.0 (exp (/ PI s))))
       (/ 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 / (((-1.0f + u) / (1.0f + expf((((float) M_PI) / s)))) - (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(Float32(Float32(-1.0) + u) / Float32(Float32(1.0) + exp(Float32(Float32(pi) / s)))) - Float32(u / Float32(Float32(1.0) + exp(Float32(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) / (((single(-1.0) + u) / (single(1.0) + exp((single(pi) / s)))) - (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

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

3. Final simplification 0.3

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

Alternatives

Alternative 1
Error	23.9
Cost	16256

\[{\left(\sqrt[3]{s \cdot \left(\log \left(\frac{s}{\pi}\right) - \frac{s}{\pi}\right)}\right)}^{3} \]

Alternative 2
Error	23.9
Cost	16224

\[\mathsf{log1p}\left(\mathsf{expm1}\left(s \cdot \left(\log \left(\frac{s}{\pi}\right) - \frac{s}{\pi}\right)\right)\right) \]

Alternative 3
Error	23.9
Cost	9824

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

Alternative 4
Error	23.9
Cost	9728

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

Alternative 5
Error	24.0
Cost	6624

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

Alternative 6
Error	24.0
Cost	6560

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

Alternative 7
Error	27.9
Cost	3360

\[-\frac{s}{\pi} \cdot s \]

Alternative 8
Error	28.3
Cost	3232

\[-\pi \]

Alternative 9
Error	30.5
Cost	3200

\[\pi \]

Reproduce

herbie shell --seed 2022217 
(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))))