Sample trimmed logistic on [-pi, pi]

Average Error: 98.9% → 98.9%

Time: 30.3s

Precision: binary32

Cost: 16736.00

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

Math

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

↓

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

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
 (*
  (- s)
  (log
   (+
    -1.0
    (/
     1.0
     (-
      (/ u (+ 1.0 (exp (/ (- PI) s))))
      (/ (+ u -1.0) (+ 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 / ((u / (1.0f + expf((-((float) M_PI) / s)))) - ((u + -1.0f) / (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(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(u + Float32(-1.0)) / Float32(Float32(1.0) + exp(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)))) - ((u + single(-1.0)) / (single(1.0) + exp((single(pi) / 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. Final simplification 98.9

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

Alternatives

Alternative 1
Error	14.3%
Cost	6880.00

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

Alternative 2
Error	11.7%
Cost	6816.00

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

Alternative 3
Error	11.7%
Cost	6816.00

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

Alternative 4
Error	11.7%
Cost	6784.00

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

Alternative 5
Error	11.7%
Cost	6720.00

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

Alternative 6
Error	11.7%
Cost	3584.00

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

Alternative 7
Error	11.7%
Cost	3392.00

\[\pi \cdot \left(-1 + u \cdot 2\right) \]

Alternative 8
Error	11.5%
Cost	3360.00

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

Alternative 9
Error	11.5%
Cost	3232.00

\[-\pi \]

Alternative 10
Error	4.6%
Cost	3200.00

\[\pi \]

Reproduce

herbie shell --seed 2023093 
(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))))