Disney BSSRDF, sample scattering profile, lower

Average Error: 12.4 → 0.2

Time: 9.8s

Precision: binary32

Cost: 9984

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

Math

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

↓

\[\mathsf{fma}\left(\mathsf{log1p}\left(4 \cdot u\right), s, s \cdot \left(-\mathsf{log1p}\left(u \cdot \left(u \cdot -16\right)\right)\right)\right) \]

FPCore

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

↓

(FPCore (s u)
 :precision binary32
 (fma (log1p (* 4.0 u)) s (* s (- (log1p (* u (* u -16.0)))))))

C

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

↓

float code(float s, float u) {
	return fmaf(log1pf((4.0f * u)), s, (s * -log1pf((u * (u * -16.0f)))));
}

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 fma(log1p(Float32(Float32(4.0) * u)), s, Float32(s * Float32(-log1p(Float32(u * Float32(u * Float32(-16.0)))))))
end

Derivation

1. Initial program 12.4

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

2. Applied egg-rr 3.9

   \[\leadsto s \cdot \color{blue}{\left(\log \left(\frac{1}{1 - 16 \cdot \left(u \cdot u\right)}\right) + \mathsf{log1p}\left(4 \cdot u\right)\right)} \]

3. Simplified 0.3

   \[\leadsto s \cdot \color{blue}{\left(\mathsf{log1p}\left(4 \cdot u\right) - \mathsf{log1p}\left(\left(-16 \cdot u\right) \cdot u\right)\right)} \]
   Proof
   (-.f32 (log1p.f32 (*.f32 4 u)) (log1p.f32 (*.f32 (*.f32 -16 u) u))): 0 points increase in error, 0 points decrease in error
   (-.f32 (log1p.f32 (*.f32 4 u)) (log1p.f32 (*.f32 (*.f32 (Rewrite<= metadata-eval (neg.f32 16)) u) u))): 0 points increase in error, 0 points decrease in error
   (-.f32 (log1p.f32 (*.f32 4 u)) (log1p.f32 (Rewrite<= associate-*r*_binary32 (*.f32 (neg.f32 16) (*.f32 u u))))): 0 points increase in error, 0 points decrease in error
   (-.f32 (log1p.f32 (*.f32 4 u)) (log1p.f32 (*.f32 (neg.f32 16) (Rewrite<= unpow2_binary32 (pow.f32 u 2))))): 0 points increase in error, 0 points decrease in error
   (-.f32 (log1p.f32 (*.f32 4 u)) (Rewrite<= log1p-def_binary32 (log.f32 (+.f32 1 (*.f32 (neg.f32 16) (pow.f32 u 2)))))): 45 points increase in error, 143 points decrease in error
   (-.f32 (log1p.f32 (*.f32 4 u)) (log.f32 (Rewrite<= cancel-sign-sub-inv_binary32 (-.f32 1 (*.f32 16 (pow.f32 u 2)))))): 0 points increase in error, 0 points decrease in error
   (-.f32 (log1p.f32 (*.f32 4 u)) (log.f32 (-.f32 1 (*.f32 16 (Rewrite=> unpow2_binary32 (*.f32 u u)))))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= unsub-neg_binary32 (+.f32 (log1p.f32 (*.f32 4 u)) (neg.f32 (log.f32 (-.f32 1 (*.f32 16 (*.f32 u u))))))): 0 points increase in error, 0 points decrease in error
   (+.f32 (log1p.f32 (*.f32 4 u)) (Rewrite<= log-rec_binary32 (log.f32 (/.f32 1 (-.f32 1 (*.f32 16 (*.f32 u u))))))): 39 points increase in error, 32 points decrease in error
   (Rewrite<= +-commutative_binary32 (+.f32 (log.f32 (/.f32 1 (-.f32 1 (*.f32 16 (*.f32 u u))))) (log1p.f32 (*.f32 4 u)))): 0 points increase in error, 0 points decrease in error

4. Applied egg-rr 0.2

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{log1p}\left(4 \cdot u\right), s, \left(-\mathsf{log1p}\left(u \cdot \left(u \cdot -16\right)\right)\right) \cdot s\right)} \]

5. Final simplification 0.2

   \[\leadsto \mathsf{fma}\left(\mathsf{log1p}\left(4 \cdot u\right), s, s \cdot \left(-\mathsf{log1p}\left(u \cdot \left(u \cdot -16\right)\right)\right)\right) \]

Alternatives

Alternative 1
Error	0.3
Cost	6816

\[s \cdot \mathsf{log1p}\left(4 \cdot u\right) - s \cdot \mathsf{log1p}\left(u \cdot \left(u \cdot -16\right)\right) \]

Alternative 2
Error	0.2
Cost	3392

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

Alternative 3
Error	4.2
Cost	416

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

Alternative 4
Error	4.2
Cost	352

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

Alternative 5
Error	4.2
Cost	288

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

Alternative 6
Error	4.2
Cost	288

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

Alternative 7
Error	8.4
Cost	160

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

Alternative 8
Error	8.4
Cost	160

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

Reproduce

herbie shell --seed 2022338 
(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))))))