HairBSDF, gamma for a refracted ray

Average Error: 2.6 → 0.7

Time: 15.5s

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

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

↓

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

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 (+ eta (* -0.5 (/ sinTheta_O (/ 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 / (eta + (-0.5f * (sinTheta_O / (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 / (eta + ((-0.5e0) * (sintheta_o / (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(eta + Float32(Float32(-0.5) * Float32(sinTheta_O / 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 / (eta + (single(-0.5) * (sinTheta_O / (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

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

3. Taylor expanded in sinTheta_O around 0 2.6

   \[\leadsto \sin^{-1} \left(\frac{h}{\sqrt{\color{blue}{-0.5 \cdot {sinTheta_O}^{4} + \left({eta}^{2} + -1 \cdot {sinTheta_O}^{2}\right)}}}\right) \]

4. Simplified 2.6

   \[\leadsto \sin^{-1} \left(\frac{h}{\sqrt{\color{blue}{\mathsf{fma}\left(-0.5, {sinTheta_O}^{4}, eta \cdot eta - sinTheta_O \cdot sinTheta_O\right)}}}\right) \]

5. Applied egg-rr 2.6

   \[\leadsto \sin^{-1} \left(\frac{h}{\sqrt{\mathsf{fma}\left(-0.5, {sinTheta_O}^{4}, \color{blue}{\left(sinTheta_O + eta\right) \cdot \left(eta - sinTheta_O\right)}\right)}}\right) \]

6. Taylor expanded in sinTheta_O around 0 0.9

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

7. Simplified 0.7

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

8. Final simplification 0.7

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

Alternatives

Alternative 1
Error	1.6
Cost	3296

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

Reproduce

herbie shell --seed 2022217 
(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)))))))))