HairBSDF, gamma for a refracted ray

Average Error: 8.63% → 2.22%

Time: 17.4s

Precision: binary32

Cost: 3552

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

Math

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

↓

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

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 (+ eta (* sinTheta_O (* (/ sinTheta_O eta) -0.5))))))

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 / (eta + (sinTheta_O * ((sinTheta_O / eta) * -0.5f)))));
}

Fortran

real(4) function code(sintheta_o, h, eta)
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: h
    real(4), intent (in) :: eta
    code = asin((h / sqrt(((eta * eta) - ((sintheta_o * sintheta_o) / sqrt((1.0e0 - (sintheta_o * sintheta_o))))))))
end function

↓

real(4) function code(sintheta_o, h, eta)
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: h
    real(4), intent (in) :: eta
    code = asin((h / (eta + (sintheta_o * ((sintheta_o / eta) * (-0.5e0))))))
end function

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 / Float32(eta + Float32(sinTheta_O * Float32(Float32(sinTheta_O / eta) * Float32(-0.5))))))
end

MATLAB

function tmp = code(sinTheta_O, h, eta)
	tmp = asin((h / sqrt(((eta * eta) - ((sinTheta_O * sinTheta_O) / sqrt((single(1.0) - (sinTheta_O * sinTheta_O))))))));
end

↓

function tmp = code(sinTheta_O, h, eta)
	tmp = asin((h / (eta + (sinTheta_O * ((sinTheta_O / eta) * single(-0.5))))));
end

Derivation

1. Initial program 8.63

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

2. Simplified 8.62

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

   [Start] 8.63	\[ \sin^{-1} \left(\frac{h}{\sqrt{eta \cdot eta - \frac{sinTheta_O \cdot sinTheta_O}{\sqrt{1 - sinTheta_O \cdot sinTheta_O}}}}\right) \]
   associate-*r/ [<=] 8.62	\[ \sin^{-1} \left(\frac{h}{\sqrt{eta \cdot eta - \color{blue}{sinTheta_O \cdot \frac{sinTheta_O}{\sqrt{1 - sinTheta_O \cdot sinTheta_O}}}}}\right) \]

3. Taylor expanded in eta around inf 2.81

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

4. Simplified 2.81

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

   [Start] 2.81	\[ \sin^{-1} \left(\frac{h}{eta + -0.5 \cdot \left(\frac{{sinTheta_O}^{2}}{eta} \cdot \sqrt{\frac{1}{1 - {sinTheta_O}^{2}}}\right)}\right) \]
   unpow2 [=>] 2.81	\[ \sin^{-1} \left(\frac{h}{eta + -0.5 \cdot \left(\frac{\color{blue}{sinTheta_O \cdot sinTheta_O}}{eta} \cdot \sqrt{\frac{1}{1 - {sinTheta_O}^{2}}}\right)}\right) \]
   unpow2 [=>] 2.81	\[ \sin^{-1} \left(\frac{h}{eta + -0.5 \cdot \left(\frac{sinTheta_O \cdot sinTheta_O}{eta} \cdot \sqrt{\frac{1}{1 - \color{blue}{sinTheta_O \cdot sinTheta_O}}}\right)}\right) \]

5. Taylor expanded in sinTheta_O around 0 2.85

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

6. Simplified 2.22

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

   [Start] 2.85	\[ \sin^{-1} \left(\frac{h}{eta + -0.5 \cdot \frac{{sinTheta_O}^{2}}{eta}}\right) \]
   *-commutative [=>] 2.85	\[ \sin^{-1} \left(\frac{h}{eta + \color{blue}{\frac{{sinTheta_O}^{2}}{eta} \cdot -0.5}}\right) \]
   unpow2 [=>] 2.85	\[ \sin^{-1} \left(\frac{h}{eta + \frac{\color{blue}{sinTheta_O \cdot sinTheta_O}}{eta} \cdot -0.5}\right) \]
   associate-*r/ [<=] 2.22	\[ \sin^{-1} \left(\frac{h}{eta + \color{blue}{\left(sinTheta_O \cdot \frac{sinTheta_O}{eta}\right)} \cdot -0.5}\right) \]
   associate-*l* [=>] 2.22	\[ \sin^{-1} \left(\frac{h}{eta + \color{blue}{sinTheta_O \cdot \left(\frac{sinTheta_O}{eta} \cdot -0.5\right)}}\right) \]

7. Final simplification 2.22

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

Alternatives

Alternative 1
Error	2.85%
Cost	3552

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

Alternative 2
Error	4.89%
Cost	3296

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

Reproduce

herbie shell --seed 2023102 
(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)))))))))