HairBSDF, gamma for a refracted ray

Average Error: 2.5 → 0.6

Time: 18.6s

Precision: binary32

Cost: 9952

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

Math

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

↓

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

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 (* (pow (+ sinTheta_O eta) -0.5) (pow (- eta sinTheta_O) -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 * (powf((sinTheta_O + eta), -0.5f) * powf((eta - sinTheta_O), -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 * (((sintheta_o + eta) ** (-0.5e0)) * ((eta - sintheta_o) ** (-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((Float32(sinTheta_O + eta) ^ Float32(-0.5)) * (Float32(eta - sinTheta_O) ^ 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 * (((sinTheta_O + eta) ^ single(-0.5)) * ((eta - sinTheta_O) ^ single(-0.5)))));
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. Applied egg-rr 2.7

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

3. Simplified 2.7

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

   [Start] 2.7	\[ \sin^{-1} \left(h \cdot {\left(eta \cdot eta - sinTheta_O \cdot \frac{sinTheta_O}{\sqrt{1 - sinTheta_O \cdot sinTheta_O}}\right)}^{-0.5}\right) \]
   fma-neg [=>] 2.7	\[ \sin^{-1} \left(h \cdot {\color{blue}{\left(\mathsf{fma}\left(eta, eta, -sinTheta_O \cdot \frac{sinTheta_O}{\sqrt{1 - sinTheta_O \cdot sinTheta_O}}\right)\right)}}^{-0.5}\right) \]
   distribute-lft-neg-in [=>] 2.7	\[ \sin^{-1} \left(h \cdot {\left(\mathsf{fma}\left(eta, eta, \color{blue}{\left(-sinTheta_O\right) \cdot \frac{sinTheta_O}{\sqrt{1 - sinTheta_O \cdot sinTheta_O}}}\right)\right)}^{-0.5}\right) \]
   *-commutative [=>] 2.7	\[ \sin^{-1} \left(h \cdot {\left(\mathsf{fma}\left(eta, eta, \color{blue}{\frac{sinTheta_O}{\sqrt{1 - sinTheta_O \cdot sinTheta_O}} \cdot \left(-sinTheta_O\right)}\right)\right)}^{-0.5}\right) \]

4. Taylor expanded in sinTheta_O around 0 2.8

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

5. Taylor expanded in eta around 0 2.8

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

6. Simplified 2.8

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

   [Start] 2.8	\[ \sin^{-1} \left(h \cdot {\left({eta}^{2} + -1 \cdot {sinTheta_O}^{2}\right)}^{-0.5}\right) \]
   unpow2 [=>] 2.8	\[ \sin^{-1} \left(h \cdot {\left(\color{blue}{eta \cdot eta} + -1 \cdot {sinTheta_O}^{2}\right)}^{-0.5}\right) \]
   mul-1-neg [=>] 2.8	\[ \sin^{-1} \left(h \cdot {\left(eta \cdot eta + \color{blue}{\left(-{sinTheta_O}^{2}\right)}\right)}^{-0.5}\right) \]
   unpow2 [=>] 2.8	\[ \sin^{-1} \left(h \cdot {\left(eta \cdot eta + \left(-\color{blue}{sinTheta_O \cdot sinTheta_O}\right)\right)}^{-0.5}\right) \]
   sub-neg [<=] 2.8	\[ \sin^{-1} \left(h \cdot {\color{blue}{\left(eta \cdot eta - sinTheta_O \cdot sinTheta_O\right)}}^{-0.5}\right) \]

7. Applied egg-rr 0.6

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

8. Simplified 0.6

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

   [Start] 0.6	\[ \sin^{-1} \left(h \cdot \left({\left(eta + sinTheta_O\right)}^{-0.5} \cdot {\left(eta - sinTheta_O\right)}^{-0.5}\right)\right) \]
   +-commutative [=>] 0.6	\[ \sin^{-1} \left(h \cdot \left({\color{blue}{\left(sinTheta_O + eta\right)}}^{-0.5} \cdot {\left(eta - sinTheta_O\right)}^{-0.5}\right)\right) \]

9. Final simplification 0.6

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

Alternatives

Alternative 1
Error	0.7
Cost	3616

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

Alternative 2
Error	0.8
Cost	3552

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

Alternative 3
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 4
Error	0.7
Cost	3552

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

Alternative 5
Error	1.5
Cost	3296

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

Reproduce

herbie shell --seed 2023047 
(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)))))))))