Sample trimmed logistic on [-pi, pi]

Average Error: 0.3 → 0.3

Time: 10.0s

Precision: binary32

Cost: 16736

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

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 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
   (*.f32 (neg.f32 s) (log.f32 (+.f32 (/.f32 1 (-.f32 (/.f32 u (+.f32 1 (exp.f32 (/.f32 (neg.f32 (PI.f32)) s)))) (/.f32 (+.f32 u -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 (Rewrite<= *-lft-identity_binary32 (*.f32 1 u)) (+.f32 1 (exp.f32 (/.f32 (neg.f32 (PI.f32)) s)))) (/.f32 (+.f32 u -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<= associate-*l/_binary32 (*.f32 (/.f32 1 (+.f32 1 (exp.f32 (/.f32 (neg.f32 (PI.f32)) s)))) u)) (/.f32 (+.f32 u -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 (/.f32 1 (+.f32 1 (exp.f32 (/.f32 (neg.f32 (PI.f32)) s)))) u) (/.f32 (+.f32 u (Rewrite<= metadata-eval (neg.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 (/.f32 1 (+.f32 1 (exp.f32 (/.f32 (neg.f32 (PI.f32)) s)))) u) (/.f32 (Rewrite<= sub-neg_binary32 (-.f32 u 1)) (+.f32 1 (exp.f32 (/.f32 (PI.f32) s)))))) -1))): 22 points increase in error, 0 points decrease in error
   (*.f32 (neg.f32 s) (log.f32 (+.f32 (/.f32 1 (-.f32 (*.f32 (/.f32 1 (+.f32 1 (exp.f32 (/.f32 (neg.f32 (PI.f32)) s)))) u) (Rewrite=> div-sub_binary32 (-.f32 (/.f32 u (+.f32 1 (exp.f32 (/.f32 (PI.f32) s)))) (/.f32 1 (+.f32 1 (exp.f32 (/.f32 (PI.f32) s)))))))) -1))): 0 points increase in error, 22 points decrease in error
   (*.f32 (neg.f32 s) (log.f32 (+.f32 (/.f32 1 (-.f32 (*.f32 (/.f32 1 (+.f32 1 (exp.f32 (/.f32 (neg.f32 (PI.f32)) s)))) u) (-.f32 (/.f32 (Rewrite<= *-rgt-identity_binary32 (*.f32 u 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 (/.f32 1 (+.f32 1 (exp.f32 (/.f32 (neg.f32 (PI.f32)) s)))) u) (-.f32 (Rewrite<= associate-*r/_binary32 (*.f32 u (/.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 (/.f32 1 (+.f32 1 (exp.f32 (/.f32 (neg.f32 (PI.f32)) s)))) u) (-.f32 (Rewrite<= *-commutative_binary32 (*.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 (/.f32 1 (+.f32 1 (exp.f32 (/.f32 (neg.f32 (PI.f32)) s)))) u) (Rewrite<= *-lft-identity_binary32 (*.f32 1 (-.f32 (*.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 (Rewrite=> cancel-sign-sub-inv_binary32 (+.f32 (*.f32 (/.f32 1 (+.f32 1 (exp.f32 (/.f32 (neg.f32 (PI.f32)) s)))) u) (*.f32 (neg.f32 1) (-.f32 (*.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 (/.f32 1 (+.f32 1 (exp.f32 (/.f32 (neg.f32 (PI.f32)) s)))) u) (*.f32 (Rewrite=> metadata-eval -1) (-.f32 (*.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 (/.f32 1 (+.f32 1 (exp.f32 (/.f32 (neg.f32 (PI.f32)) s)))) u) (Rewrite<= neg-mul-1_binary32 (neg.f32 (-.f32 (*.f32 (/.f32 1 (+.f32 1 (exp.f32 (/.f32 (PI.f32) s)))) u) (/.f32 1 (+.f32 1 (exp.f32 (/.f32 (PI.f32) s))))))))) -1))): 22 points increase in error, 0 points decrease in error
   (*.f32 (neg.f32 s) (log.f32 (+.f32 (/.f32 1 (+.f32 (*.f32 (/.f32 1 (+.f32 1 (exp.f32 (/.f32 (neg.f32 (PI.f32)) s)))) u) (Rewrite<= sub0-neg_binary32 (-.f32 0 (-.f32 (*.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, 22 points decrease in error
   (*.f32 (neg.f32 s) (log.f32 (+.f32 (/.f32 1 (+.f32 (*.f32 (/.f32 1 (+.f32 1 (exp.f32 (/.f32 (neg.f32 (PI.f32)) s)))) u) (Rewrite<= associate-+l-_binary32 (+.f32 (-.f32 0 (*.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 (/.f32 1 (+.f32 1 (exp.f32 (/.f32 (neg.f32 (PI.f32)) s)))) u) (+.f32 (Rewrite<= neg-sub0_binary32 (neg.f32 (*.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 (/.f32 1 (+.f32 1 (exp.f32 (/.f32 (neg.f32 (PI.f32)) s)))) u) (+.f32 (Rewrite<= distribute-lft-neg-out_binary32 (*.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))): 22 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 (/.f32 1 (+.f32 1 (exp.f32 (/.f32 (neg.f32 (PI.f32)) s)))) u) (*.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))): 22 points increase in error, 0 points decrease in error
   (*.f32 (neg.f32 s) (log.f32 (+.f32 (/.f32 1 (+.f32 (Rewrite<= distribute-rgt-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, 22 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. Final simplification 0.3

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

Alternatives

Alternative 1
Error	24.0
Cost	9920

\[\left(-s\right) \cdot \left(\mathsf{log1p}\left(-1 + \frac{\pi}{s}\right) + \frac{s}{\pi}\right) \]

Alternative 2
Error	24.0
Cost	9856

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

Alternative 3
Error	24.0
Cost	6752

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

Alternative 4
Error	24.0
Cost	6560

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

Alternative 5
Error	24.0
Cost	6560

\[s \cdot \left(-\mathsf{log1p}\left(\frac{\pi}{s}\right)\right) \]

Alternative 6
Error	23.9
Cost	6528

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

Alternative 7
Error	27.9
Cost	3360

\[s \cdot \frac{-s}{\pi} \]

Alternative 8
Error	28.4
Cost	3232

\[-\pi \]

Alternative 9
Error	30.5
Cost	3200

\[\pi \]

Reproduce

herbie shell --seed 2022343 
(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))))