HairBSDF, gamma for a refracted ray

Average Error: 2.6 → 0.6

Time: 17.6s

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

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

↓

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

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

Derivation

1. Initial program 2.6

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

2. Simplified 2.6

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

   [Start] 2.6	\[ \sin^{-1} \left(\frac{h}{\sqrt{eta \cdot eta - \frac{sinTheta_O \cdot sinTheta_O}{\sqrt{1 - sinTheta_O \cdot sinTheta_O}}}}\right) \]
   fma-neg [=>] 2.6	\[ \sin^{-1} \left(\frac{h}{\sqrt{\color{blue}{\mathsf{fma}\left(eta, eta, -\frac{sinTheta_O \cdot sinTheta_O}{\sqrt{1 - sinTheta_O \cdot sinTheta_O}}\right)}}}\right) \]
   distribute-neg-frac [=>] 2.6	\[ \sin^{-1} \left(\frac{h}{\sqrt{\mathsf{fma}\left(eta, eta, \color{blue}{\frac{-sinTheta_O \cdot sinTheta_O}{\sqrt{1 - sinTheta_O \cdot sinTheta_O}}}\right)}}\right) \]
   distribute-rgt-neg-in [=>] 2.6	\[ \sin^{-1} \left(\frac{h}{\sqrt{\mathsf{fma}\left(eta, eta, \frac{\color{blue}{sinTheta_O \cdot \left(-sinTheta_O\right)}}{\sqrt{1 - sinTheta_O \cdot sinTheta_O}}\right)}}\right) \]

3. Taylor expanded in sinTheta_O around 0 2.7

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

4. Simplified 2.7

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

   [Start] 2.7	\[ \sin^{-1} \left(\frac{h}{\sqrt{\mathsf{fma}\left(eta, eta, -1 \cdot {sinTheta_O}^{2}\right)}}\right) \]
   mul-1-neg [=>] 2.7	\[ \sin^{-1} \left(\frac{h}{\sqrt{\mathsf{fma}\left(eta, eta, \color{blue}{-{sinTheta_O}^{2}}\right)}}\right) \]
   unpow2 [=>] 2.7	\[ \sin^{-1} \left(\frac{h}{\sqrt{\mathsf{fma}\left(eta, eta, -\color{blue}{sinTheta_O \cdot sinTheta_O}\right)}}\right) \]
   distribute-rgt-neg-in [=>] 2.7	\[ \sin^{-1} \left(\frac{h}{\sqrt{\mathsf{fma}\left(eta, eta, \color{blue}{sinTheta_O \cdot \left(-sinTheta_O\right)}\right)}}\right) \]

5. Taylor expanded in h around 0 4.1

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

6. Simplified 4.1

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

   [Start] 4.1	\[ \sin^{-1} \left(\sqrt{\frac{1}{{eta}^{2} + -1 \cdot {sinTheta_O}^{2}}} \cdot h\right) \]
   *-commutative [=>] 4.1	\[ \sin^{-1} \color{blue}{\left(h \cdot \sqrt{\frac{1}{{eta}^{2} + -1 \cdot {sinTheta_O}^{2}}}\right)} \]
   mul-1-neg [=>] 4.1	\[ \sin^{-1} \left(h \cdot \sqrt{\frac{1}{{eta}^{2} + \color{blue}{\left(-{sinTheta_O}^{2}\right)}}}\right) \]
   unsub-neg [=>] 4.1	\[ \sin^{-1} \left(h \cdot \sqrt{\frac{1}{\color{blue}{{eta}^{2} - {sinTheta_O}^{2}}}}\right) \]
   unpow2 [=>] 4.1	\[ \sin^{-1} \left(h \cdot \sqrt{\frac{1}{\color{blue}{eta \cdot eta} - {sinTheta_O}^{2}}}\right) \]
   unpow2 [=>] 4.1	\[ \sin^{-1} \left(h \cdot \sqrt{\frac{1}{eta \cdot eta - \color{blue}{sinTheta_O \cdot sinTheta_O}}}\right) \]

7. Applied egg-rr 0.6

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

8. Final simplification 0.6

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

Alternatives

Alternative 1
Error	0.9
Cost	3552

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

Alternative 2
Error	0.7
Cost	3552

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

Alternative 3
Error	0.7
Cost	3552

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

Alternative 4
Error	1.5
Cost	3296

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

Reproduce

herbie shell --seed 2023056 
(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)))))))))