HairBSDF, gamma for a refracted ray

Average Accuracy: 92.2% → 98.4%

Time: 23.0s

Precision: binary32

Cost: 9888

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

Math

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

↓

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

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 (* (sqrt (+ sinTheta_O eta)) (sqrt (- eta sinTheta_O))))))

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 / (sqrtf((sinTheta_O + eta)) * sqrtf((eta - sinTheta_O)))));
}

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 / (sqrt((sintheta_o + eta)) * sqrt((eta - sintheta_o)))))
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(sqrt(Float32(sinTheta_O + eta)) * sqrt(Float32(eta - sinTheta_O)))))
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 / (sqrt((sinTheta_O + eta)) * sqrt((eta - sinTheta_O)))));
end

Derivation

1. Initial program 92.2%

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

2. Applied egg-rr 91.1%

   \[\leadsto \sin^{-1} \left(\frac{h}{\color{blue}{{\left({\left(eta \cdot eta - \frac{sinTheta_O \cdot sinTheta_O}{\sqrt{1 - sinTheta_O \cdot sinTheta_O}}\right)}^{0.25}\right)}^{2}}}\right) \]
   Proof

   [Start] 92.2	\[ \sin^{-1} \left(\frac{h}{\sqrt{eta \cdot eta - \frac{sinTheta_O \cdot sinTheta_O}{\sqrt{1 - sinTheta_O \cdot sinTheta_O}}}}\right) \]
   add-sqr-sqrt [=>] 91.1	\[ \sin^{-1} \left(\frac{h}{\color{blue}{\sqrt{\sqrt{eta \cdot eta - \frac{sinTheta_O \cdot sinTheta_O}{\sqrt{1 - sinTheta_O \cdot sinTheta_O}}}} \cdot \sqrt{\sqrt{eta \cdot eta - \frac{sinTheta_O \cdot sinTheta_O}{\sqrt{1 - sinTheta_O \cdot sinTheta_O}}}}}}\right) \]
   pow2 [=>] 91.1	\[ \sin^{-1} \left(\frac{h}{\color{blue}{{\left(\sqrt{\sqrt{eta \cdot eta - \frac{sinTheta_O \cdot sinTheta_O}{\sqrt{1 - sinTheta_O \cdot sinTheta_O}}}}\right)}^{2}}}\right) \]
   pow1/2 [=>] 91.1	\[ \sin^{-1} \left(\frac{h}{{\left(\sqrt{\color{blue}{{\left(eta \cdot eta - \frac{sinTheta_O \cdot sinTheta_O}{\sqrt{1 - sinTheta_O \cdot sinTheta_O}}\right)}^{0.5}}}\right)}^{2}}\right) \]
   sqrt-pow1 [=>] 91.1	\[ \sin^{-1} \left(\frac{h}{{\color{blue}{\left({\left(eta \cdot eta - \frac{sinTheta_O \cdot sinTheta_O}{\sqrt{1 - sinTheta_O \cdot sinTheta_O}}\right)}^{\left(\frac{0.5}{2}\right)}\right)}}^{2}}\right) \]
   metadata-eval [=>] 91.1	\[ \sin^{-1} \left(\frac{h}{{\left({\left(eta \cdot eta - \frac{sinTheta_O \cdot sinTheta_O}{\sqrt{1 - sinTheta_O \cdot sinTheta_O}}\right)}^{\color{blue}{0.25}}\right)}^{2}}\right) \]

3. Taylor expanded in sinTheta_O around 0 90.9%

   \[\leadsto \sin^{-1} \left(\frac{h}{{\left({\color{blue}{\left({eta}^{2} + -1 \cdot {sinTheta_O}^{2}\right)}}^{0.25}\right)}^{2}}\right) \]

4. Simplified 90.9%

   \[\leadsto \sin^{-1} \left(\frac{h}{{\left({\color{blue}{\left(eta \cdot eta - sinTheta_O \cdot sinTheta_O\right)}}^{0.25}\right)}^{2}}\right) \]
   Proof

   [Start] 90.9	\[ \sin^{-1} \left(\frac{h}{{\left({\left({eta}^{2} + -1 \cdot {sinTheta_O}^{2}\right)}^{0.25}\right)}^{2}}\right) \]
   unpow2 [=>] 90.9	\[ \sin^{-1} \left(\frac{h}{{\left({\left(\color{blue}{eta \cdot eta} + -1 \cdot {sinTheta_O}^{2}\right)}^{0.25}\right)}^{2}}\right) \]
   mul-1-neg [=>] 90.9	\[ \sin^{-1} \left(\frac{h}{{\left({\left(eta \cdot eta + \color{blue}{\left(-{sinTheta_O}^{2}\right)}\right)}^{0.25}\right)}^{2}}\right) \]
   unsub-neg [=>] 90.9	\[ \sin^{-1} \left(\frac{h}{{\left({\color{blue}{\left(eta \cdot eta - {sinTheta_O}^{2}\right)}}^{0.25}\right)}^{2}}\right) \]
   unpow2 [=>] 90.9	\[ \sin^{-1} \left(\frac{h}{{\left({\left(eta \cdot eta - \color{blue}{sinTheta_O \cdot sinTheta_O}\right)}^{0.25}\right)}^{2}}\right) \]

5. Applied egg-rr 98.4%

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

   [Start] 90.9	\[ \sin^{-1} \left(\frac{h}{{\left({\left(eta \cdot eta - sinTheta_O \cdot sinTheta_O\right)}^{0.25}\right)}^{2}}\right) \]
   pow-pow [=>] 92.1	\[ \sin^{-1} \left(\frac{h}{\color{blue}{{\left(eta \cdot eta - sinTheta_O \cdot sinTheta_O\right)}^{\left(0.25 \cdot 2\right)}}}\right) \]
   metadata-eval [=>] 92.1	\[ \sin^{-1} \left(\frac{h}{{\left(eta \cdot eta - sinTheta_O \cdot sinTheta_O\right)}^{\color{blue}{0.5}}}\right) \]
   pow1/2 [<=] 92.1	\[ \sin^{-1} \left(\frac{h}{\color{blue}{\sqrt{eta \cdot eta - sinTheta_O \cdot sinTheta_O}}}\right) \]
   difference-of-squares [=>] 92.1	\[ \sin^{-1} \left(\frac{h}{\sqrt{\color{blue}{\left(eta + sinTheta_O\right) \cdot \left(eta - sinTheta_O\right)}}}\right) \]
   sqrt-prod [=>] 98.4	\[ \sin^{-1} \left(\frac{h}{\color{blue}{\sqrt{eta + sinTheta_O} \cdot \sqrt{eta - sinTheta_O}}}\right) \]

6. Simplified 98.4%

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

   [Start] 98.4	\[ \sin^{-1} \left(\frac{h}{\sqrt{eta + sinTheta_O} \cdot \sqrt{eta - sinTheta_O}}\right) \]
   +-commutative [=>] 98.4	\[ \sin^{-1} \left(\frac{h}{\sqrt{\color{blue}{sinTheta_O + eta}} \cdot \sqrt{eta - sinTheta_O}}\right) \]

7. Final simplification 98.4%

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

Alternatives

Alternative 1
Accuracy	97.5%
Cost	3552

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

Alternative 2
Accuracy	98.0%
Cost	3552

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

Alternative 3
Accuracy	95.6%
Cost	3296

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

Reproduce

herbie shell --seed 2023146 
(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)))))))))