HairBSDF, gamma for a refracted ray

Average Error: 2.5 → 0.7

Time: 16.5s

Precision: binary32

Cost: 6688

\[\left(\left(-1 \leq sinTheta_O \land sinTheta_O \leq 1\right) \land \left(-1 \leq h \land h \leq 1\right)\right) \land \left(0 \leq eta \land eta \leq 10\right)\]

Math

\[\sin^{-1} \left(\frac{h}{\sqrt{eta \cdot eta - \frac{sinTheta_O \cdot sinTheta_O}{\sqrt{1 - sinTheta_O \cdot sinTheta_O}}}}\right) \]

↓

\[\sin^{-1} \left(\frac{h}{\mathsf{fma}\left(-0.5, \frac{sinTheta_O}{\frac{eta}{sinTheta_O}}, eta\right)}\right) \]

FPCore

(FPCore (sinTheta_O h eta)
 :precision binary32
 (asin
  (/
   h
   (sqrt
    (-
     (* eta eta)
     (/
      (* sinTheta_O sinTheta_O)
      (sqrt (- 1.0 (* sinTheta_O sinTheta_O)))))))))

↓

(FPCore (sinTheta_O h eta)
 :precision binary32
 (asin (/ h (fma -0.5 (/ sinTheta_O (/ eta sinTheta_O)) eta))))

C

float code(float sinTheta_O, float h, float eta) {
	return asinf((h / sqrtf(((eta * eta) - ((sinTheta_O * sinTheta_O) / sqrtf((1.0f - (sinTheta_O * sinTheta_O))))))));
}

↓

float code(float sinTheta_O, float h, float eta) {
	return asinf((h / fmaf(-0.5f, (sinTheta_O / (eta / sinTheta_O)), eta)));
}

Julia

function code(sinTheta_O, h, eta)
	return asin(Float32(h / sqrt(Float32(Float32(eta * eta) - Float32(Float32(sinTheta_O * sinTheta_O) / sqrt(Float32(Float32(1.0) - Float32(sinTheta_O * sinTheta_O))))))))
end

↓

function code(sinTheta_O, h, eta)
	return asin(Float32(h / fma(Float32(-0.5), Float32(sinTheta_O / Float32(eta / sinTheta_O)), eta)))
end

Derivation

1. Initial program 2.5

   \[\sin^{-1} \left(\frac{h}{\sqrt{eta \cdot eta - \frac{sinTheta_O \cdot sinTheta_O}{\sqrt{1 - sinTheta_O \cdot sinTheta_O}}}}\right) \]

2. Taylor expanded in sinTheta_O around 0 0.8

   \[\leadsto \sin^{-1} \left(\frac{h}{\color{blue}{eta + -0.5 \cdot \frac{{sinTheta_O}^{2}}{eta}}}\right) \]

3. Simplified 0.7

   \[\leadsto \sin^{-1} \left(\frac{h}{\color{blue}{\mathsf{fma}\left(-0.5, \frac{sinTheta_O}{\frac{eta}{sinTheta_O}}, eta\right)}}\right) \]
   Proof
   (fma.f32 -1/2 (/.f32 sinTheta_O (/.f32 eta sinTheta_O)) eta): 0 points increase in error, 0 points decrease in error
   (fma.f32 -1/2 (Rewrite<= associate-/l*_binary32 (/.f32 (*.f32 sinTheta_O sinTheta_O) eta)) eta): 3 points increase in error, 6 points decrease in error
   (fma.f32 -1/2 (/.f32 (Rewrite<= unpow2_binary32 (pow.f32 sinTheta_O 2)) eta) eta): 0 points increase in error, 0 points decrease in error
   (Rewrite<= fma-def_binary32 (+.f32 (*.f32 -1/2 (/.f32 (pow.f32 sinTheta_O 2) eta)) eta)): 0 points increase in error, 0 points decrease in error
   (Rewrite<= +-commutative_binary32 (+.f32 eta (*.f32 -1/2 (/.f32 (pow.f32 sinTheta_O 2) eta)))): 0 points increase in error, 0 points decrease in error

4. Final simplification 0.7

   \[\leadsto \sin^{-1} \left(\frac{h}{\mathsf{fma}\left(-0.5, \frac{sinTheta_O}{\frac{eta}{sinTheta_O}}, eta\right)}\right) \]

Alternatives

Alternative 1
Error	1.5
Cost	3296

\[\sin^{-1} \left(\frac{h}{eta}\right) \]

Reproduce

herbie shell --seed 2022297 
(FPCore (sinTheta_O h eta)
  :name "HairBSDF, gamma for a refracted ray"
  :precision binary32
  :pre (and (and (and (<= -1.0 sinTheta_O) (<= sinTheta_O 1.0)) (and (<= -1.0 h) (<= h 1.0))) (and (<= 0.0 eta) (<= eta 10.0)))
  (asin (/ h (sqrt (- (* eta eta) (/ (* sinTheta_O sinTheta_O) (sqrt (- 1.0 (* sinTheta_O sinTheta_O)))))))))