HairBSDF, gamma for a refracted ray

Average Error: 2.5 → 0.9

Time: 15.9s

Precision: binary32

Cost: 6784

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

Math

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

↓

\[\sin^{-1} \left(\frac{h + h}{eta + \left(eta - \frac{{sinTheta_O}^{2}}{eta}\right)}\right) \]

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 h) (+ eta (- eta (/ (pow sinTheta_O 2.0) 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 + h) / (eta + (eta - (powf(sinTheta_O, 2.0f) / eta)))));
}

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 + h) / (eta + (eta - ((sintheta_o ** 2.0e0) / eta)))))
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(Float32(h + h) / Float32(eta + Float32(eta - Float32((sinTheta_O ^ Float32(2.0)) / eta)))))
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 + h) / (eta + (eta - ((sinTheta_O ^ single(2.0)) / 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 0.9

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

3. Applied egg-rr 0.9

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

4. Simplified 0.9

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

   [Start] 0.9	\[ \sin^{-1} \left(\frac{h}{\left(eta + eta\right) - \frac{{sinTheta_O}^{2}}{eta}} + \frac{h}{\left(eta + eta\right) - \frac{{sinTheta_O}^{2}}{eta}}\right) \]
   rational.json-simplify-45 [=>] 0.9	\[ \sin^{-1} \color{blue}{\left(\frac{h + h}{\left(eta + eta\right) - \frac{{sinTheta_O}^{2}}{eta}}\right)} \]
   rational.json-simplify-5 [=>] 0.9	\[ \sin^{-1} \left(\frac{h + h}{\color{blue}{eta + \left(eta - \frac{{sinTheta_O}^{2}}{eta}\right)}}\right) \]

5. Final simplification 0.9

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

Alternatives

Alternative 1
Error	0.9
Cost	6720

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

Alternative 2
Error	0.9
Cost	6720

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

Alternative 3
Error	1.5
Cost	3296

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

Reproduce

herbie shell --seed 2023073 
(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)))))))))