Sample trimmed logistic on [-pi, pi]

Average Error: 0.3 → 0.3

Time: 17.6s

Precision: binary32

Cost: 29664

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

Math

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

↓

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

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 (* (pow (cbrt PI) 2.0) (* (cbrt 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((powf(cbrtf(((float) M_PI)), 2.0f) * (cbrtf(((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(Float32(-s) * 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((cbrt(Float32(pi)) ^ Float32(2.0)) * Float32(cbrt(Float32(pi)) * Float32(Float32(1.0) / s)))))))) + Float32(-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

   \[\leadsto \left(-s\right) \cdot \log \left(\frac{1}{\frac{u}{1 + e^{\frac{-\pi}{s}}} + \frac{1 - u}{1 + e^{\color{blue}{{\left(\sqrt[3]{\pi}\right)}^{2} \cdot \left(\sqrt[3]{\pi} \cdot \frac{1}{s}\right)}}}} + -1\right) \]

4. Final simplification 0.3

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

Alternatives

Alternative 1
Error	0.3
Cost	19968

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

Alternative 2
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 3
Error	19.9
Cost	13408

\[\left(-s\right) \cdot \mathsf{log1p}\left(-1 + \left(-1 + \frac{1}{\mathsf{fma}\left(u, 0.5, \frac{1 - u}{1 + e^{\frac{\pi}{s}}}\right)}\right)\right) \]

Alternative 4
Error	19.9
Cost	13408

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

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	3456

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

Alternative 7
Error	28.4
Cost	3232

\[-\pi \]

Alternative 8
Error	28.7
Cost	32

\[0 \]

Reproduce

herbie shell --seed 2022308 
(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))))