HairBSDF, gamma for a refracted ray

Average Accuracy: 91.7% → 97.8%

Time: 18.4s

Precision: binary32

Cost: 13088

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

Math

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

↓

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

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 (exp (log (* sinTheta_O (/ sinTheta_O eta)))) 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, expf(logf((sinTheta_O * (sinTheta_O / eta)))), 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), exp(log(Float32(sinTheta_O * Float32(sinTheta_O / eta)))), eta)))
end

Derivation

1. Initial program 91.7%

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

2. Taylor expanded in sinTheta_O around 0 97.2%

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

3. Simplified 97.8%

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

   [Start] 97.2	\[ \sin^{-1} \left(\frac{h}{eta + -0.5 \cdot \frac{{sinTheta_O}^{2}}{eta}}\right) \]
   +-commutative [=>] 97.2	\[ \sin^{-1} \left(\frac{h}{\color{blue}{-0.5 \cdot \frac{{sinTheta_O}^{2}}{eta} + eta}}\right) \]
   fma-def [=>] 97.2	\[ \sin^{-1} \left(\frac{h}{\color{blue}{\mathsf{fma}\left(-0.5, \frac{{sinTheta_O}^{2}}{eta}, eta\right)}}\right) \]
   unpow2 [=>] 97.2	\[ \sin^{-1} \left(\frac{h}{\mathsf{fma}\left(-0.5, \frac{\color{blue}{sinTheta_O \cdot sinTheta_O}}{eta}, eta\right)}\right) \]
   associate-/l* [=>] 97.8	\[ \sin^{-1} \left(\frac{h}{\mathsf{fma}\left(-0.5, \color{blue}{\frac{sinTheta_O}{\frac{eta}{sinTheta_O}}}, eta\right)}\right) \]

4. Applied egg-rr 97.8%

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

5. Final simplification 97.8%

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

Alternatives

Alternative 1
Accuracy	97.2%
Cost	3552

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

Alternative 2
Accuracy	97.8%
Cost	3552

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

Alternative 3
Accuracy	95.3%
Cost	3296

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

Reproduce

herbie shell --seed 2023129 
(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)))))))))