Sample trimmed logistic on [-pi, pi]

Average Error: 0.3 → 0.3

Time: 16.1s

Precision: binary32

Cost: 23296

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

Math

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

↓

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

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 (+ 1.0 (exp (/ PI s)))))
   (*
    s
    (-
     (log
      (+
       (/ 1.0 (- (- (/ u (+ 1.0 (exp (/ (- PI) s)))) (/ -1.0 t_0)) (/ 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 = 1.0f + expf((((float) M_PI) / s));
	return s * -logf(((1.0f / (((u / (1.0f + expf((-((float) M_PI) / s)))) - (-1.0f / t_0)) - (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(Float32(1.0) + exp(Float32(Float32(pi) / s)))
	return Float32(s * Float32(-log(Float32(Float32(Float32(1.0) / Float32(Float32(Float32(u / Float32(Float32(1.0) + exp(Float32(Float32(-Float32(pi)) / s)))) - Float32(Float32(-1.0) / t_0)) - 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 = single(1.0) + exp((single(pi) / s));
	tmp = s * -log(((single(1.0) / (((u / (single(1.0) + exp((-single(pi) / s)))) - (single(-1.0) / t_0)) - (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. 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. Taylor expanded in s around 0 0.3

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

4. Simplified 0.3

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

   [Start] 0.3	\[ -1 \cdot \left(s \cdot \log \left(\frac{1}{\left(\frac{1}{e^{\frac{\pi}{s}} + 1} + \frac{u}{1 + e^{-1 \cdot \frac{\pi}{s}}}\right) - \frac{u}{e^{\frac{\pi}{s}} + 1}} - 1\right)\right) \]
   mul-1-neg [=>] 0.3	\[ \color{blue}{-s \cdot \log \left(\frac{1}{\left(\frac{1}{e^{\frac{\pi}{s}} + 1} + \frac{u}{1 + e^{-1 \cdot \frac{\pi}{s}}}\right) - \frac{u}{e^{\frac{\pi}{s}} + 1}} - 1\right)} \]
   sub-neg [=>] 0.3	\[ -s \cdot \log \color{blue}{\left(\frac{1}{\left(\frac{1}{e^{\frac{\pi}{s}} + 1} + \frac{u}{1 + e^{-1 \cdot \frac{\pi}{s}}}\right) - \frac{u}{e^{\frac{\pi}{s}} + 1}} + \left(-1\right)\right)} \]
   +-commutative [=>] 0.3	\[ -s \cdot \log \left(\frac{1}{\color{blue}{\left(\frac{u}{1 + e^{-1 \cdot \frac{\pi}{s}}} + \frac{1}{e^{\frac{\pi}{s}} + 1}\right)} - \frac{u}{e^{\frac{\pi}{s}} + 1}} + \left(-1\right)\right) \]
   mul-1-neg [=>] 0.3	\[ -s \cdot \log \left(\frac{1}{\left(\frac{u}{1 + e^{\color{blue}{-\frac{\pi}{s}}}} + \frac{1}{e^{\frac{\pi}{s}} + 1}\right) - \frac{u}{e^{\frac{\pi}{s}} + 1}} + \left(-1\right)\right) \]
   distribute-neg-frac [=>] 0.3	\[ -s \cdot \log \left(\frac{1}{\left(\frac{u}{1 + e^{\color{blue}{\frac{-\pi}{s}}}} + \frac{1}{e^{\frac{\pi}{s}} + 1}\right) - \frac{u}{e^{\frac{\pi}{s}} + 1}} + \left(-1\right)\right) \]
   +-commutative [<=] 0.3	\[ -s \cdot \log \left(\frac{1}{\left(\frac{u}{1 + e^{\frac{-\pi}{s}}} + \frac{1}{\color{blue}{1 + e^{\frac{\pi}{s}}}}\right) - \frac{u}{e^{\frac{\pi}{s}} + 1}} + \left(-1\right)\right) \]
   +-commutative [<=] 0.3	\[ -s \cdot \log \left(\frac{1}{\left(\frac{u}{1 + e^{\frac{-\pi}{s}}} + \frac{1}{1 + e^{\frac{\pi}{s}}}\right) - \frac{u}{\color{blue}{1 + e^{\frac{\pi}{s}}}}} + \left(-1\right)\right) \]
   metadata-eval [=>] 0.3	\[ -s \cdot \log \left(\frac{1}{\left(\frac{u}{1 + e^{\frac{-\pi}{s}}} + \frac{1}{1 + e^{\frac{\pi}{s}}}\right) - \frac{u}{1 + e^{\frac{\pi}{s}}}} + \color{blue}{-1}\right) \]

5. Final simplification 0.3

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

Alternatives

Alternative 1
Error	0.3
Cost	16736

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

Alternative 2
Error	0.4
Cost	16672

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

Alternative 3
Error	28.3
Cost	16640

\[4 \cdot \left(\frac{\pi \cdot \left(0.0625 + -0.25 \cdot \left(u \cdot u\right)\right)}{\log \left(e^{\mathsf{fma}\left(-0.25, u \cdot u, 0.0625\right)}\right)} \cdot \mathsf{fma}\left(u, 0.5, -0.25\right)\right) \]

Alternative 4
Error	28.3
Cost	9984

\[4 \cdot \left(\left(\pi \cdot 0.5 + 0.5 \cdot \left(u \cdot \pi\right)\right) + \pi \cdot -0.75\right) \]

Alternative 5
Error	28.3
Cost	9952

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

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	3584

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

Alternative 9
Error	28.3
Cost	3456

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

Alternative 10
Error	28.4
Cost	3232

\[-\pi \]

Reproduce

herbie shell --seed 2022356 
(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))))