Disney BSSRDF, sample scattering profile, lower

Average Error: 12.4 → 0.2

Time: 7.5s

Precision: binary32

Cost: 3392

\[\left(0 \leq s \land s \leq 256\right) \land \left(2.328306437 \cdot 10^{-10} \leq u \land u \leq 0.25\right)\]

Math

\[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]

↓

\[\mathsf{log1p}\left(u \cdot -4\right) \cdot \left(-s\right) \]

FPCore

(FPCore (s u) :precision binary32 (* s (log (/ 1.0 (- 1.0 (* 4.0 u))))))

↓

(FPCore (s u) :precision binary32 (* (log1p (* u -4.0)) (- s)))

C

float code(float s, float u) {
	return s * logf((1.0f / (1.0f - (4.0f * u))));
}

↓

float code(float s, float u) {
	return log1pf((u * -4.0f)) * -s;
}

Julia

function code(s, u)
	return Float32(s * log(Float32(Float32(1.0) / Float32(Float32(1.0) - Float32(Float32(4.0) * u)))))
end

↓

function code(s, u)
	return Float32(log1p(Float32(u * Float32(-4.0))) * Float32(-s))
end

Derivation

1. Initial program 12.4

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]

2. Simplified 0.2

   \[\leadsto \color{blue}{\mathsf{log1p}\left(u \cdot -4\right) \cdot \left(-s\right)} \]
   Proof
   (*.f32 (log1p.f32 (*.f32 u -4)) (neg.f32 s)): 0 points increase in error, 0 points decrease in error
   (*.f32 (log1p.f32 (*.f32 u (Rewrite<= metadata-eval (neg.f32 4)))) (neg.f32 s)): 0 points increase in error, 0 points decrease in error
   (*.f32 (log1p.f32 (Rewrite<= distribute-rgt-neg-in_binary32 (neg.f32 (*.f32 u 4)))) (neg.f32 s)): 0 points increase in error, 0 points decrease in error
   (*.f32 (log1p.f32 (neg.f32 (Rewrite<= *-commutative_binary32 (*.f32 4 u)))) (neg.f32 s)): 0 points increase in error, 0 points decrease in error
   (*.f32 (Rewrite<= log1p-def_binary32 (log.f32 (+.f32 1 (neg.f32 (*.f32 4 u))))) (neg.f32 s)): 233 points increase in error, 0 points decrease in error
   (*.f32 (log.f32 (Rewrite<= sub-neg_binary32 (-.f32 1 (*.f32 4 u)))) (neg.f32 s)): 0 points increase in error, 0 points decrease in error
   (Rewrite<= distribute-rgt-neg-in_binary32 (neg.f32 (*.f32 (log.f32 (-.f32 1 (*.f32 4 u))) s))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= distribute-lft-neg-out_binary32 (*.f32 (neg.f32 (log.f32 (-.f32 1 (*.f32 4 u)))) s)): 0 points increase in error, 0 points decrease in error
   (*.f32 (Rewrite<= log-rec_binary32 (log.f32 (/.f32 1 (-.f32 1 (*.f32 4 u))))) s): 133 points increase in error, 50 points decrease in error
   (Rewrite<= *-commutative_binary32 (*.f32 s (log.f32 (/.f32 1 (-.f32 1 (*.f32 4 u)))))): 0 points increase in error, 0 points decrease in error

3. Final simplification 0.2

   \[\leadsto \mathsf{log1p}\left(u \cdot -4\right) \cdot \left(-s\right) \]

Alternatives

Alternative 1
Error	3.6
Cost	320

\[s \cdot \frac{-u}{u \cdot 0.5 + -0.25} \]

Alternative 2
Error	4.3
Cost	288

\[s \cdot \left(u \cdot \left(4 + u \cdot 8\right)\right) \]

Alternative 3
Error	4.3
Cost	288

\[u \cdot \left(s \cdot \left(4 + u \cdot 8\right)\right) \]

Alternative 4
Error	8.4
Cost	160

\[4 \cdot \left(u \cdot s\right) \]

Alternative 5
Error	8.4
Cost	160

\[u \cdot \left(s \cdot 4\right) \]

Reproduce

herbie shell --seed 2022325 
(FPCore (s u)
  :name "Disney BSSRDF, sample scattering profile, lower"
  :precision binary32
  :pre (and (and (<= 0.0 s) (<= s 256.0)) (and (<= 2.328306437e-10 u) (<= u 0.25)))
  (* s (log (/ 1.0 (- 1.0 (* 4.0 u))))))