Disney BSSRDF, sample scattering profile, upper

Average Error: 1.3 → 0.7

Time: 10.7s

Precision: binary32

Cost: 3488

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

Math

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

↓

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

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 (* (log1p (* (+ u -0.25) -1.3333333333333333)) s)))

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 * (log1pf(((u + -0.25f) * -1.3333333333333333f)) * s);
}

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(log1p(Float32(Float32(u + Float32(-0.25)) * Float32(-1.3333333333333333))) * s))
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. Applied egg-rr 1.4

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

3. Applied egg-rr 1.1

   \[\leadsto \color{blue}{0 + s \cdot \log \left({\left(\mathsf{fma}\left(u + -0.25, -1.3333333333333333, 1\right)\right)}^{-3}\right)} \]

4. Applied egg-rr 0.7

   \[\leadsto 0 + \color{blue}{\left(0 + -3 \cdot \left(\mathsf{log1p}\left(\left(u + -0.25\right) \cdot -1.3333333333333333\right) \cdot s\right)\right)} \]

5. Final simplification 0.7

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

Alternatives

Alternative 1
Error	1.2
Cost	3488

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

Alternative 2
Error	1.0
Cost	3488

\[\mathsf{log1p}\left(u \cdot -1.3333333333333333 + 0.3333333333333333\right) \cdot \left(-3 \cdot s\right) \]

Alternative 3
Error	0.7
Cost	3488

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

Alternative 4
Error	0.7
Cost	3488

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

Alternative 5
Error	23.8
Cost	3424

\[\left(s \cdot 3\right) \cdot \left(u + \log 0.75\right) \]

Alternative 6
Error	29.6
Cost	3360

\[\left(s \cdot 3\right) \cdot \log 0.75 \]

Alternative 7
Error	29.6
Cost	3296

\[s \cdot \log 0.421875 \]

Reproduce

herbie shell --seed 2022228 
(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))))))