Sample trimmed logistic on [-pi, pi]

Average Error: 0.3 → 0.3

Time: 23.1s

Precision: binary32

Cost: 23328

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

Math

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

↓

\[\begin{array}{l} t_0 := e^{\frac{\pi}{s}} + 1\\ -1 \cdot \left(s \cdot \log \left(\frac{1}{\left(\frac{1}{t_0} + \frac{u}{1 + e^{-\frac{\pi}{s}}}\right) - \frac{u}{t_0}} - 1\right)\right) \end{array} \]

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
 (let* ((t_0 (+ (exp (/ PI s)) 1.0)))
   (*
    -1.0
    (*
     s
     (log
      (-
       (/ 1.0 (- (+ (/ 1.0 t_0) (/ u (+ 1.0 (exp (- (/ PI s)))))) (/ u t_0)))
       1.0))))))

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) {
	float t_0 = expf((((float) M_PI) / s)) + 1.0f;
	return -1.0f * (s * logf(((1.0f / (((1.0f / t_0) + (u / (1.0f + expf(-(((float) M_PI) / s))))) - (u / t_0))) - 1.0f)));
}

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)
	t_0 = Float32(exp(Float32(Float32(pi) / s)) + Float32(1.0))
	return Float32(Float32(-1.0) * Float32(s * log(Float32(Float32(Float32(1.0) / Float32(Float32(Float32(Float32(1.0) / t_0) + Float32(u / Float32(Float32(1.0) + exp(Float32(-Float32(Float32(pi) / s)))))) - Float32(u / t_0))) - Float32(1.0)))))
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)
	t_0 = exp((single(pi) / s)) + single(1.0);
	tmp = single(-1.0) * (s * log(((single(1.0) / (((single(1.0) / t_0) + (u / (single(1.0) + exp(-(single(pi) / s))))) - (u / t_0))) - single(1.0))));
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. Applied egg-rr 0.3

   \[\leadsto \color{blue}{s \cdot \left(-\log \left(\frac{1}{\frac{1}{1 + e^{\frac{\pi}{s}}} \cdot \left(\left(-u\right) + 1\right) - u \cdot \left(-\frac{1}{1 + e^{\frac{-\pi}{s}}}\right)} + -1\right)\right) + 0} \]

3. Simplified 0.3

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

   [Start] 0.3	\[ s \cdot \left(-\log \left(\frac{1}{\frac{1}{1 + e^{\frac{\pi}{s}}} \cdot \left(\left(-u\right) + 1\right) - u \cdot \left(-\frac{1}{1 + e^{\frac{-\pi}{s}}}\right)} + -1\right)\right) + 0 \]
   rational_best_oopsla_all_46_json_45_simplify-85 [=>] 0.3	\[ \color{blue}{s \cdot \left(-\log \left(\frac{1}{\frac{1}{1 + e^{\frac{\pi}{s}}} \cdot \left(\left(-u\right) + 1\right) - u \cdot \left(-\frac{1}{1 + e^{\frac{-\pi}{s}}}\right)} + -1\right)\right)} \]
   rational_best_oopsla_all_46_json_45_simplify-35 [=>] 0.3	\[ s \cdot \left(-\log \color{blue}{\left(-1 + \frac{1}{\frac{1}{1 + e^{\frac{\pi}{s}}} \cdot \left(\left(-u\right) + 1\right) - u \cdot \left(-\frac{1}{1 + e^{\frac{-\pi}{s}}}\right)}\right)}\right) \]

4. Applied egg-rr 0.3

   \[\leadsto \color{blue}{\log \left(-1 + \frac{1}{\frac{1}{1 + e^{\frac{\pi}{s}}} \cdot \left(1 - u\right) + u \cdot \frac{1}{1 + e^{\frac{-\pi}{s}}}}\right) \cdot \left(-s\right) + 0} \]

5. Simplified 0.3

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

   [Start] 0.3	\[ \log \left(-1 + \frac{1}{\frac{1}{1 + e^{\frac{\pi}{s}}} \cdot \left(1 - u\right) + u \cdot \frac{1}{1 + e^{\frac{-\pi}{s}}}}\right) \cdot \left(-s\right) + 0 \]
   rational_best_oopsla_all_46_json_45_simplify-85 [=>] 0.3	\[ \color{blue}{\log \left(-1 + \frac{1}{\frac{1}{1 + e^{\frac{\pi}{s}}} \cdot \left(1 - u\right) + u \cdot \frac{1}{1 + e^{\frac{-\pi}{s}}}}\right) \cdot \left(-s\right)} \]

6. Taylor expanded in s around 0 0.3

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

7. Applied egg-rr 0.3

   \[\leadsto -1 \cdot \left(s \cdot \log \left(\frac{1}{\left(\frac{1}{e^{\frac{\pi}{s}} + 1} + \frac{u}{1 + \color{blue}{\left(e^{-\frac{\pi}{s}} + 0\right)}}\right) - \frac{u}{e^{\frac{\pi}{s}} + 1}} - 1\right)\right) \]

8. Simplified 0.3

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

   [Start] 0.3	\[ -1 \cdot \left(s \cdot \log \left(\frac{1}{\left(\frac{1}{e^{\frac{\pi}{s}} + 1} + \frac{u}{1 + \left(e^{-\frac{\pi}{s}} + 0\right)}\right) - \frac{u}{e^{\frac{\pi}{s}} + 1}} - 1\right)\right) \]
   rational_best_oopsla_all_46_json_45_simplify-85 [=>] 0.3	\[ -1 \cdot \left(s \cdot \log \left(\frac{1}{\left(\frac{1}{e^{\frac{\pi}{s}} + 1} + \frac{u}{1 + \color{blue}{e^{-\frac{\pi}{s}}}}\right) - \frac{u}{e^{\frac{\pi}{s}} + 1}} - 1\right)\right) \]

9. Final simplification 0.3

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

Alternatives

Alternative 1
Error	0.3
Cost	16864

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

Alternative 2
Error	0.9
Cost	10048

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

Alternative 3
Error	23.9
Cost	9760

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

Alternative 4
Error	24.0
Cost	6624

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

Alternative 5
Error	28.2
Cost	3456

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

Alternative 6
Error	28.3
Cost	3232

\[-\pi \]

Reproduce

herbie shell --seed 2023090 
(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))))