Disney BSSRDF, sample scattering profile, upper

Average Error: 1.3 → 0.6

Time: 13.3s

Precision: binary32

Cost: 13536

\[\left(0 \leq s \land s \leq 256\right) \land \left(0.25 \leq u \land u \leq 1\right)\]

Math

\[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]

↓

\[s \cdot \left(\left(\mathsf{log1p}\left(2.3703703703703702 \cdot {\left(0.25 - u\right)}^{3}\right) - \mathsf{log1p}\left({\left(0.25 - u\right)}^{2} \cdot 1.7777777777777777 + \left(0.25 - u\right) \cdot -1.3333333333333333\right)\right) \cdot -3\right) \]

FPCore

(FPCore (s u)
 :precision binary32
 (* (* 3.0 s) (log (/ 1.0 (- 1.0 (/ (- u 0.25) 0.75))))))

↓

(FPCore (s u)
 :precision binary32
 (*
  s
  (*
   (-
    (log1p (* 2.3703703703703702 (pow (- 0.25 u) 3.0)))
    (log1p
     (+
      (* (pow (- 0.25 u) 2.0) 1.7777777777777777)
      (* (- 0.25 u) -1.3333333333333333))))
   -3.0)))

C

float code(float s, float u) {
	return (3.0f * s) * logf((1.0f / (1.0f - ((u - 0.25f) / 0.75f))));
}

↓

float code(float s, float u) {
	return s * ((log1pf((2.3703703703703702f * powf((0.25f - u), 3.0f))) - log1pf(((powf((0.25f - u), 2.0f) * 1.7777777777777777f) + ((0.25f - u) * -1.3333333333333333f)))) * -3.0f);
}

Julia

function code(s, u)
	return Float32(Float32(Float32(3.0) * s) * log(Float32(Float32(1.0) / Float32(Float32(1.0) - Float32(Float32(u - Float32(0.25)) / Float32(0.75))))))
end

↓

function code(s, u)
	return Float32(s * Float32(Float32(log1p(Float32(Float32(2.3703703703703702) * (Float32(Float32(0.25) - u) ^ Float32(3.0)))) - log1p(Float32(Float32((Float32(Float32(0.25) - u) ^ Float32(2.0)) * Float32(1.7777777777777777)) + Float32(Float32(Float32(0.25) - u) * Float32(-1.3333333333333333))))) * Float32(-3.0)))
end

Derivation

1. Initial program 1.3

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]

2. Simplified 0.5

   \[\leadsto \color{blue}{\left(3 \cdot s\right) \cdot \left(-\mathsf{log1p}\left(\frac{-\left(u + -0.25\right)}{0.75}\right)\right)} \]
   Proof
   (*.f32 (*.f32 3 s) (neg.f32 (log1p.f32 (/.f32 (neg.f32 (+.f32 u -1/4)) 3/4)))): 0 points increase in error, 0 points decrease in error
   (*.f32 (*.f32 3 s) (neg.f32 (log1p.f32 (/.f32 (neg.f32 (+.f32 u (Rewrite<= metadata-eval (neg.f32 1/4)))) 3/4)))): 0 points increase in error, 0 points decrease in error
   (*.f32 (*.f32 3 s) (neg.f32 (log1p.f32 (/.f32 (neg.f32 (Rewrite<= sub-neg_binary32 (-.f32 u 1/4))) 3/4)))): 0 points increase in error, 0 points decrease in error
   (*.f32 (*.f32 3 s) (neg.f32 (log1p.f32 (/.f32 (neg.f32 (Rewrite<= --rgt-identity_binary32 (-.f32 (-.f32 u 1/4) 0))) 3/4)))): 0 points increase in error, 0 points decrease in error
   (*.f32 (*.f32 3 s) (neg.f32 (log1p.f32 (/.f32 (neg.f32 (-.f32 (-.f32 u 1/4) (Rewrite<= metadata-eval (log.f32 1)))) 3/4)))): 0 points increase in error, 0 points decrease in error
   (*.f32 (*.f32 3 s) (neg.f32 (log1p.f32 (/.f32 (Rewrite<= sub0-neg_binary32 (-.f32 0 (-.f32 (-.f32 u 1/4) (log.f32 1)))) 3/4)))): 0 points increase in error, 0 points decrease in error
   (*.f32 (*.f32 3 s) (neg.f32 (log1p.f32 (/.f32 (-.f32 (Rewrite<= metadata-eval (log.f32 1)) (-.f32 (-.f32 u 1/4) (log.f32 1))) 3/4)))): 0 points increase in error, 0 points decrease in error
   (*.f32 (*.f32 3 s) (neg.f32 (log1p.f32 (Rewrite=> div-sub_binary32 (-.f32 (/.f32 (log.f32 1) 3/4) (/.f32 (-.f32 (-.f32 u 1/4) (log.f32 1)) 3/4)))))): 0 points increase in error, 0 points decrease in error
   (*.f32 (*.f32 3 s) (neg.f32 (log1p.f32 (-.f32 (/.f32 (Rewrite=> metadata-eval 0) 3/4) (/.f32 (-.f32 (-.f32 u 1/4) (log.f32 1)) 3/4))))): 0 points increase in error, 0 points decrease in error
   (*.f32 (*.f32 3 s) (neg.f32 (log1p.f32 (-.f32 (Rewrite=> metadata-eval 0) (/.f32 (-.f32 (-.f32 u 1/4) (log.f32 1)) 3/4))))): 0 points increase in error, 0 points decrease in error
   (*.f32 (*.f32 3 s) (neg.f32 (log1p.f32 (-.f32 0 (/.f32 (-.f32 (-.f32 u 1/4) (Rewrite=> metadata-eval 0)) 3/4))))): 0 points increase in error, 0 points decrease in error
   (*.f32 (*.f32 3 s) (neg.f32 (log1p.f32 (-.f32 0 (/.f32 (Rewrite=> --rgt-identity_binary32 (-.f32 u 1/4)) 3/4))))): 0 points increase in error, 0 points decrease in error
   (*.f32 (*.f32 3 s) (neg.f32 (log1p.f32 (Rewrite<= neg-sub0_binary32 (neg.f32 (/.f32 (-.f32 u 1/4) 3/4)))))): 0 points increase in error, 0 points decrease in error
   (*.f32 (*.f32 3 s) (neg.f32 (Rewrite<= log1p-def_binary32 (log.f32 (+.f32 1 (neg.f32 (/.f32 (-.f32 u 1/4) 3/4))))))): 81 points increase in error, 4 points decrease in error
   (*.f32 (*.f32 3 s) (neg.f32 (log.f32 (Rewrite<= sub-neg_binary32 (-.f32 1 (/.f32 (-.f32 u 1/4) 3/4)))))): 0 points increase in error, 0 points decrease in error
   (*.f32 (*.f32 3 s) (Rewrite<= log-rec_binary32 (log.f32 (/.f32 1 (-.f32 1 (/.f32 (-.f32 u 1/4) 3/4)))))): 88 points increase in error, 27 points decrease in error

3. Taylor expanded in s around 0 1.2

   \[\leadsto \color{blue}{-3 \cdot \left(s \cdot \log \left(1 + 1.3333333333333333 \cdot \left(0.25 - u\right)\right)\right)} \]

4. Simplified 0.7

   \[\leadsto \color{blue}{s \cdot \left(\mathsf{log1p}\left(1.3333333333333333 \cdot \left(0.25 - u\right)\right) \cdot -3\right)} \]
   Proof
   (*.f32 s (*.f32 (log1p.f32 (*.f32 4/3 (-.f32 1/4 u))) -3)): 0 points increase in error, 0 points decrease in error
   (*.f32 s (*.f32 (Rewrite<= log1p-def_binary32 (log.f32 (+.f32 1 (*.f32 4/3 (-.f32 1/4 u))))) -3)): 72 points increase in error, 3 points decrease in error
   (Rewrite<= associate-*l*_binary32 (*.f32 (*.f32 s (log.f32 (+.f32 1 (*.f32 4/3 (-.f32 1/4 u))))) -3)): 46 points increase in error, 47 points decrease in error
   (Rewrite<= *-commutative_binary32 (*.f32 -3 (*.f32 s (log.f32 (+.f32 1 (*.f32 4/3 (-.f32 1/4 u))))))): 0 points increase in error, 0 points decrease in error

5. Applied egg-rr 1.3

   \[\leadsto s \cdot \left(\color{blue}{\left(\mathsf{log1p}\left({\left(1.3333333333333333 \cdot \left(0.25 - u\right)\right)}^{3}\right) - \log \left(1 + \left({\left(0.25 - u\right)}^{2} \cdot 1.7777777777777777 - 1.3333333333333333 \cdot \left(0.25 - u\right)\right)\right)\right)} \cdot -3\right) \]

6. Simplified 0.6

   \[\leadsto s \cdot \left(\color{blue}{\left(\mathsf{log1p}\left(2.3703703703703702 \cdot {\left(0.25 - u\right)}^{3}\right) - \mathsf{log1p}\left({\left(0.25 - u\right)}^{2} \cdot 1.7777777777777777 + -1.3333333333333333 \cdot \left(0.25 - u\right)\right)\right)} \cdot -3\right) \]
   Proof
   (-.f32 (log1p.f32 (*.f32 64/27 (pow.f32 (-.f32 1/4 u) 3))) (log1p.f32 (+.f32 (*.f32 (pow.f32 (-.f32 1/4 u) 2) 16/9) (*.f32 -4/3 (-.f32 1/4 u))))): 0 points increase in error, 0 points decrease in error
   (-.f32 (log1p.f32 (*.f32 (Rewrite<= metadata-eval (pow.f32 4/3 3)) (pow.f32 (-.f32 1/4 u) 3))) (log1p.f32 (+.f32 (*.f32 (pow.f32 (-.f32 1/4 u) 2) 16/9) (*.f32 -4/3 (-.f32 1/4 u))))): 78 points increase in error, 0 points decrease in error
   (-.f32 (log1p.f32 (Rewrite<= cube-prod_binary32 (pow.f32 (*.f32 4/3 (-.f32 1/4 u)) 3))) (log1p.f32 (+.f32 (*.f32 (pow.f32 (-.f32 1/4 u) 2) 16/9) (*.f32 -4/3 (-.f32 1/4 u))))): 10 points increase in error, 41 points decrease in error
   (-.f32 (log1p.f32 (pow.f32 (*.f32 4/3 (-.f32 1/4 u)) 3)) (log1p.f32 (+.f32 (*.f32 (pow.f32 (-.f32 1/4 u) 2) 16/9) (*.f32 (Rewrite<= metadata-eval (neg.f32 4/3)) (-.f32 1/4 u))))): 0 points increase in error, 0 points decrease in error
   (-.f32 (log1p.f32 (pow.f32 (*.f32 4/3 (-.f32 1/4 u)) 3)) (log1p.f32 (Rewrite<= cancel-sign-sub-inv_binary32 (-.f32 (*.f32 (pow.f32 (-.f32 1/4 u) 2) 16/9) (*.f32 4/3 (-.f32 1/4 u)))))): 0 points increase in error, 0 points decrease in error
   (-.f32 (log1p.f32 (pow.f32 (*.f32 4/3 (-.f32 1/4 u)) 3)) (Rewrite<= log1p-def_binary32 (log.f32 (+.f32 1 (-.f32 (*.f32 (pow.f32 (-.f32 1/4 u) 2) 16/9) (*.f32 4/3 (-.f32 1/4 u))))))): 68 points increase in error, 63 points decrease in error

7. Final simplification 0.6

   \[\leadsto s \cdot \left(\left(\mathsf{log1p}\left(2.3703703703703702 \cdot {\left(0.25 - u\right)}^{3}\right) - \mathsf{log1p}\left({\left(0.25 - u\right)}^{2} \cdot 1.7777777777777777 + \left(0.25 - u\right) \cdot -1.3333333333333333\right)\right) \cdot -3\right) \]

Alternatives

Alternative 1
Error	1.3
Cost	3488

\[-3 \cdot \left(s \cdot \log \left(1.3333333333333333 + u \cdot -1.3333333333333333\right)\right) \]

Alternative 2
Error	0.7
Cost	3488

\[-3 \cdot \left(s \cdot \mathsf{log1p}\left(\left(0.25 - u\right) \cdot 1.3333333333333333\right)\right) \]

Alternative 3
Error	0.7
Cost	3488

\[\left(s \cdot -3\right) \cdot \mathsf{log1p}\left(\left(0.25 - u\right) \cdot 1.3333333333333333\right) \]

Alternative 4
Error	0.5
Cost	3488

\[s \cdot \left(-3 \cdot \mathsf{log1p}\left(\frac{0.25 - u}{0.75}\right)\right) \]

Alternative 5
Error	0.5
Cost	3488

\[\mathsf{log1p}\left(\frac{0.25 - u}{0.75}\right) \cdot \left(s \cdot -3\right) \]

Alternative 6
Error	26.0
Cost	3424

\[-3 \cdot \left(s \cdot \mathsf{log1p}\left(u \cdot -1.3333333333333333\right)\right) \]

Alternative 7
Error	23.8
Cost	3424

\[-3 \cdot \left(s \cdot \left(\log 1.3333333333333333 - u\right)\right) \]

Alternative 8
Error	29.6
Cost	3360

\[-3 \cdot \left(s \cdot \mathsf{log1p}\left(0.3333333333333333\right)\right) \]

Alternative 9
Error	29.6
Cost	3296

\[s \cdot \mathsf{log1p}\left(-0.578125\right) \]

Reproduce

herbie shell --seed 2022330 
(FPCore (s u)
  :name "Disney BSSRDF, sample scattering profile, upper"
  :precision binary32
  :pre (and (and (<= 0.0 s) (<= s 256.0)) (and (<= 0.25 u) (<= u 1.0)))
  (* (* 3.0 s) (log (/ 1.0 (- 1.0 (/ (- u 0.25) 0.75))))))