Sample trimmed logistic on [-pi, pi]

Average Error: 0.3 → 0.3

Time: 14.0s

Precision: binary32

Cost: 16800

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

Math

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

↓

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

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
      (+
       (/ u (+ 1.0 (exp (/ (- PI) s))))
       (/ (- 1.0 u) (+ 1.0 (exp (* PI (/ 1.0 s)))))))
     -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) {
	return s * -logf(((1.0f / ((u / (1.0f + expf((-((float) M_PI) / s)))) + ((1.0f - u) / (1.0f + expf((((float) M_PI) * (1.0f / s))))))) + -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)
	return Float32(s * Float32(-log(Float32(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(Float32(pi) * Float32(Float32(1.0) / s))))))) + 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)
	tmp = s * -log(((single(1.0) / ((u / (single(1.0) + exp((-single(pi) / s)))) + ((single(1.0) - u) / (single(1.0) + exp((single(pi) * (single(1.0) / s))))))) + 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

   \[\leadsto \color{blue}{\left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{-\pi}{s}}} + \frac{1 - u}{1 + e^{\frac{\pi}{s}}}} + -1\right)} \]
   Proof
   (*.f32 (neg.f32 s) (log.f32 (+.f32 (/.f32 1 (+.f32 (/.f32 u (+.f32 1 (exp.f32 (/.f32 (neg.f32 (PI.f32)) s)))) (/.f32 (-.f32 1 u) (+.f32 1 (exp.f32 (/.f32 (PI.f32) s)))))) -1))): 0 points increase in error, 0 points decrease in error
   (*.f32 (neg.f32 s) (log.f32 (+.f32 (/.f32 1 (+.f32 (/.f32 (Rewrite<= *-rgt-identity_binary32 (*.f32 u 1)) (+.f32 1 (exp.f32 (/.f32 (neg.f32 (PI.f32)) s)))) (/.f32 (-.f32 1 u) (+.f32 1 (exp.f32 (/.f32 (PI.f32) s)))))) -1))): 0 points increase in error, 0 points decrease in error
   (*.f32 (neg.f32 s) (log.f32 (+.f32 (/.f32 1 (+.f32 (Rewrite<= associate-*r/_binary32 (*.f32 u (/.f32 1 (+.f32 1 (exp.f32 (/.f32 (neg.f32 (PI.f32)) s)))))) (/.f32 (-.f32 1 u) (+.f32 1 (exp.f32 (/.f32 (PI.f32) s)))))) -1))): 0 points increase in error, 0 points decrease in error
   (*.f32 (neg.f32 s) (log.f32 (+.f32 (/.f32 1 (+.f32 (*.f32 u (/.f32 1 (+.f32 1 (exp.f32 (/.f32 (neg.f32 (PI.f32)) s))))) (Rewrite=> div-sub_binary32 (-.f32 (/.f32 1 (+.f32 1 (exp.f32 (/.f32 (PI.f32) s)))) (/.f32 u (+.f32 1 (exp.f32 (/.f32 (PI.f32) s)))))))) -1))): 0 points increase in error, 0 points decrease in error
   (*.f32 (neg.f32 s) (log.f32 (+.f32 (/.f32 1 (+.f32 (*.f32 u (/.f32 1 (+.f32 1 (exp.f32 (/.f32 (neg.f32 (PI.f32)) s))))) (-.f32 (/.f32 1 (+.f32 1 (exp.f32 (/.f32 (PI.f32) s)))) (/.f32 (Rewrite<= *-lft-identity_binary32 (*.f32 1 u)) (+.f32 1 (exp.f32 (/.f32 (PI.f32) s))))))) -1))): 0 points increase in error, 0 points decrease in error
   (*.f32 (neg.f32 s) (log.f32 (+.f32 (/.f32 1 (+.f32 (*.f32 u (/.f32 1 (+.f32 1 (exp.f32 (/.f32 (neg.f32 (PI.f32)) s))))) (-.f32 (/.f32 1 (+.f32 1 (exp.f32 (/.f32 (PI.f32) s)))) (Rewrite<= associate-*l/_binary32 (*.f32 (/.f32 1 (+.f32 1 (exp.f32 (/.f32 (PI.f32) s)))) u))))) -1))): 0 points increase in error, 0 points decrease in error
   (*.f32 (neg.f32 s) (log.f32 (+.f32 (/.f32 1 (+.f32 (*.f32 u (/.f32 1 (+.f32 1 (exp.f32 (/.f32 (neg.f32 (PI.f32)) s))))) (Rewrite=> cancel-sign-sub-inv_binary32 (+.f32 (/.f32 1 (+.f32 1 (exp.f32 (/.f32 (PI.f32) s)))) (*.f32 (neg.f32 (/.f32 1 (+.f32 1 (exp.f32 (/.f32 (PI.f32) s))))) u))))) -1))): 0 points increase in error, 0 points decrease in error
   (*.f32 (neg.f32 s) (log.f32 (+.f32 (/.f32 1 (+.f32 (*.f32 u (/.f32 1 (+.f32 1 (exp.f32 (/.f32 (neg.f32 (PI.f32)) s))))) (Rewrite<= +-commutative_binary32 (+.f32 (*.f32 (neg.f32 (/.f32 1 (+.f32 1 (exp.f32 (/.f32 (PI.f32) s))))) u) (/.f32 1 (+.f32 1 (exp.f32 (/.f32 (PI.f32) s)))))))) -1))): 0 points increase in error, 0 points decrease in error
   (*.f32 (neg.f32 s) (log.f32 (+.f32 (/.f32 1 (+.f32 (*.f32 u (/.f32 1 (+.f32 1 (exp.f32 (/.f32 (neg.f32 (PI.f32)) s))))) (+.f32 (Rewrite=> *-commutative_binary32 (*.f32 u (neg.f32 (/.f32 1 (+.f32 1 (exp.f32 (/.f32 (PI.f32) s))))))) (/.f32 1 (+.f32 1 (exp.f32 (/.f32 (PI.f32) s))))))) -1))): 0 points increase in error, 0 points decrease in error
   (*.f32 (neg.f32 s) (log.f32 (+.f32 (/.f32 1 (Rewrite<= associate-+l+_binary32 (+.f32 (+.f32 (*.f32 u (/.f32 1 (+.f32 1 (exp.f32 (/.f32 (neg.f32 (PI.f32)) s))))) (*.f32 u (neg.f32 (/.f32 1 (+.f32 1 (exp.f32 (/.f32 (PI.f32) s))))))) (/.f32 1 (+.f32 1 (exp.f32 (/.f32 (PI.f32) s))))))) -1))): 0 points increase in error, 0 points decrease in error
   (*.f32 (neg.f32 s) (log.f32 (+.f32 (/.f32 1 (+.f32 (Rewrite<= distribute-lft-in_binary32 (*.f32 u (+.f32 (/.f32 1 (+.f32 1 (exp.f32 (/.f32 (neg.f32 (PI.f32)) s)))) (neg.f32 (/.f32 1 (+.f32 1 (exp.f32 (/.f32 (PI.f32) s)))))))) (/.f32 1 (+.f32 1 (exp.f32 (/.f32 (PI.f32) s)))))) -1))): 0 points increase in error, 0 points decrease in error
   (*.f32 (neg.f32 s) (log.f32 (+.f32 (/.f32 1 (+.f32 (*.f32 u (Rewrite<= sub-neg_binary32 (-.f32 (/.f32 1 (+.f32 1 (exp.f32 (/.f32 (neg.f32 (PI.f32)) s)))) (/.f32 1 (+.f32 1 (exp.f32 (/.f32 (PI.f32) s))))))) (/.f32 1 (+.f32 1 (exp.f32 (/.f32 (PI.f32) s)))))) -1))): 0 points increase in error, 0 points decrease in error
   (*.f32 (neg.f32 s) (log.f32 (+.f32 (/.f32 1 (+.f32 (*.f32 u (-.f32 (/.f32 1 (+.f32 1 (exp.f32 (/.f32 (neg.f32 (PI.f32)) s)))) (/.f32 1 (+.f32 1 (exp.f32 (/.f32 (PI.f32) s)))))) (/.f32 1 (+.f32 1 (exp.f32 (/.f32 (PI.f32) s)))))) (Rewrite<= metadata-eval (neg.f32 1))))): 0 points increase in error, 0 points decrease in error
   (*.f32 (neg.f32 s) (log.f32 (Rewrite<= sub-neg_binary32 (-.f32 (/.f32 1 (+.f32 (*.f32 u (-.f32 (/.f32 1 (+.f32 1 (exp.f32 (/.f32 (neg.f32 (PI.f32)) s)))) (/.f32 1 (+.f32 1 (exp.f32 (/.f32 (PI.f32) s)))))) (/.f32 1 (+.f32 1 (exp.f32 (/.f32 (PI.f32) s)))))) 1)))): 0 points increase in error, 0 points decrease in error

3. Applied egg-rr 0.3

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

4. Final simplification 0.3

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

Alternatives

Alternative 1
Error	0.3
Cost	16736

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

Alternative 2
Error	19.9
Cost	16640

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

Alternative 3
Error	19.9
Cost	13408

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

Alternative 4
Error	19.9
Cost	10272

\[s \cdot \left(-\log \left(-1 + \frac{1}{\frac{u}{2} + \frac{1 - u}{1 + \left(1 + \mathsf{expm1}\left(\frac{\pi}{s}\right)\right)}}\right)\right) \]

Alternative 5
Error	19.9
Cost	10208

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

Alternative 6
Error	20.1
Cost	9888

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

Alternative 7
Error	20.1
Cost	9856

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

Alternative 8
Error	20.1
Cost	7072

\[\left(-s\right) \cdot \log \left(\frac{-1 + \frac{8}{{u}^{3}}}{1 + \frac{2}{u} \cdot \left(1 + \frac{2}{u}\right)}\right) \]

Alternative 9
Error	20.1
Cost	3456

\[s \cdot \left(-\log \left(-1 + \frac{2}{u}\right)\right) \]

Alternative 10
Error	20.1
Cost	3392

\[\left(-s\right) \cdot \log \left(\frac{2}{u}\right) \]

Alternative 11
Error	28.4
Cost	3360

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

Alternative 12
Error	28.4
Cost	3232

\[-\pi \]

Alternative 13
Error	30.5
Cost	3200

\[\pi \]

Reproduce

herbie shell --seed 2022331 
(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))))