Disney BSSRDF, sample scattering profile, upper

Average Error: 1.3 → 0.6

Time: 14.2s

Precision: binary32

Cost: 10112

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

Math

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

↓

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

FPCore

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

↓

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

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 3.0f * (s * (log1pf(((u + -0.25f) * 1.3333333333333333f)) - log1pf((powf((u + -0.25f), 2.0f) * -1.7777777777777777f))));
}

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(Float32(3.0) * Float32(s * Float32(log1p(Float32(Float32(u + Float32(-0.25)) * Float32(1.3333333333333333))) - log1p(Float32((Float32(u + Float32(-0.25)) ^ Float32(2.0)) * Float32(-1.7777777777777777))))))
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 1.1

   \[\leadsto \color{blue}{3 \cdot \left(s \cdot \left(-\log \left(1 - \frac{u + -0.25}{0.75}\right)\right)\right)} \]
   Proof
   (*.f32 3 (*.f32 s (neg.f32 (log.f32 (-.f32 1 (/.f32 (+.f32 u -1/4) 3/4)))))): 0 points increase in error, 0 points decrease in error
   (*.f32 3 (*.f32 s (neg.f32 (log.f32 (-.f32 1 (/.f32 (+.f32 u (Rewrite<= metadata-eval (neg.f32 1/4))) 3/4)))))): 0 points increase in error, 0 points decrease in error
   (*.f32 3 (*.f32 s (neg.f32 (log.f32 (-.f32 1 (/.f32 (Rewrite<= sub-neg_binary32 (-.f32 u 1/4)) 3/4)))))): 0 points increase in error, 0 points decrease in error
   (*.f32 3 (*.f32 s (Rewrite<= log-rec_binary32 (log.f32 (/.f32 1 (-.f32 1 (/.f32 (-.f32 u 1/4) 3/4))))))): 85 points increase in error, 33 points decrease in error
   (Rewrite<= associate-*l*_binary32 (*.f32 (*.f32 3 s) (log.f32 (/.f32 1 (-.f32 1 (/.f32 (-.f32 u 1/4) 3/4)))))): 60 points increase in error, 46 points decrease in error

3. Applied egg-rr 0.6

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

4. Final simplification 0.6

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

Alternatives

Alternative 1
Error	1.2
Cost	3488

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

Alternative 2
Error	0.7
Cost	3488

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

Alternative 3
Error	0.7
Cost	3488

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

Alternative 4
Error	0.7
Cost	3488

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

Alternative 5
Error	0.5
Cost	3488

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

Alternative 6
Error	0.6
Cost	3488

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

Alternative 7
Error	25.9
Cost	3424

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

Alternative 8
Error	23.8
Cost	3424

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

Alternative 9
Error	23.8
Cost	3424

\[s \cdot \left(3 \cdot u + \log 0.421875\right) \]

Alternative 10
Error	29.6
Cost	3296

\[s \cdot \log 0.421875 \]

Reproduce

herbie shell --seed 2022329 
(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))))))