HairBSDF, gamma for a refracted ray

Percentage Accurate: 92.0% → 98.3%

Time: 13.5s

Alternatives: 3

Speedup: 3.1×

Specification

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

Math

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

FPCore

(FPCore (sinTheta_O h eta)
 :precision binary32
 (asin
  (/
   h
   (sqrt
    (-
     (* eta eta)
     (/
      (* sinTheta_O sinTheta_O)
      (sqrt (- 1.0 (* sinTheta_O 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))))))));
}

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

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

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

Initial Program: 92.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (sinTheta_O h eta)
 :precision binary32
 (asin
  (/
   h
   (sqrt
    (-
     (* eta eta)
     (/
      (* sinTheta_O sinTheta_O)
      (sqrt (- 1.0 (* sinTheta_O 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))))))));
}

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

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

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

Alternative 1: 98.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{sinTheta_O}{\sqrt{\mathsf{hypot}\left(1, sinTheta_O\right)}}\\ \sin^{-1} \left(\frac{h}{\sqrt{eta - t_0} \cdot \sqrt{eta + t_0}}\right) \end{array} \end{array} \]

FPCore

(FPCore (sinTheta_O h eta)
 :precision binary32
 (let* ((t_0 (/ sinTheta_O (sqrt (hypot 1.0 sinTheta_O)))))
   (asin (/ h (* (sqrt (- eta t_0)) (sqrt (+ eta t_0)))))))

C

float code(float sinTheta_O, float h, float eta) {
	float t_0 = sinTheta_O / sqrtf(hypotf(1.0f, sinTheta_O));
	return asinf((h / (sqrtf((eta - t_0)) * sqrtf((eta + t_0)))));
}

Julia

function code(sinTheta_O, h, eta)
	t_0 = Float32(sinTheta_O / sqrt(hypot(Float32(1.0), sinTheta_O)))
	return asin(Float32(h / Float32(sqrt(Float32(eta - t_0)) * sqrt(Float32(eta + t_0)))))
end

MATLAB

function tmp = code(sinTheta_O, h, eta)
	t_0 = sinTheta_O / sqrt(hypot(single(1.0), sinTheta_O));
	tmp = asin((h / (sqrt((eta - t_0)) * sqrt((eta + t_0)))));
end

Derivation

1. Initial program 91.8%

   \[\sin^{-1} \left(\frac{h}{\sqrt{eta \cdot eta - \frac{sinTheta_O \cdot sinTheta_O}{\sqrt{1 - sinTheta_O \cdot sinTheta_O}}}}\right) \]
2. Step-by-step derivation

   1. add-sqr-sqrt 91.8%

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

   2. difference-of-squares 91.8%

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

3. Applied egg-rr 91.6%

   \[\leadsto \sin^{-1} \left(\frac{h}{\sqrt{\color{blue}{\left(eta + \frac{sinTheta_O}{{\left({sinTheta_O}^{2} + 1\right)}^{0.25}}\right) \cdot \left(eta - \frac{sinTheta_O}{{\left({sinTheta_O}^{2} + 1\right)}^{0.25}}\right)}}}\right) \]
4. Step-by-step derivation

   1. unpow2 91.6%

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

   2. fma-def 91.6%

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

   3. unpow2 91.6%

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

   4. fma-def 91.6%

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

5. Simplified 91.6%

   \[\leadsto \sin^{-1} \left(\frac{h}{\sqrt{\color{blue}{\left(eta + \frac{sinTheta_O}{{\left(\mathsf{fma}\left(sinTheta_O, sinTheta_O, 1\right)\right)}^{0.25}}\right) \cdot \left(eta - \frac{sinTheta_O}{{\left(\mathsf{fma}\left(sinTheta_O, sinTheta_O, 1\right)\right)}^{0.25}}\right)}}}\right) \]
6. Step-by-step derivation

   1. *-commutative 91.6%

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

   2. sqrt-prod 98.4%

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

   3. fma-udef 98.4%

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

   4. unpow2 98.4%

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

   5. metadata-eval 98.4%

      \[\leadsto \sin^{-1} \left(\frac{h}{\sqrt{eta - \frac{sinTheta_O}{{\left({sinTheta_O}^{2} + 1\right)}^{\color{blue}{\left(\frac{0.5}{2}\right)}}}} \cdot \sqrt{eta + \frac{sinTheta_O}{{\left(\mathsf{fma}\left(sinTheta_O, sinTheta_O, 1\right)\right)}^{0.25}}}}\right) \]

   6. sqrt-pow1 98.4%

      \[\leadsto \sin^{-1} \left(\frac{h}{\sqrt{eta - \frac{sinTheta_O}{\color{blue}{\sqrt{{\left({sinTheta_O}^{2} + 1\right)}^{0.5}}}}} \cdot \sqrt{eta + \frac{sinTheta_O}{{\left(\mathsf{fma}\left(sinTheta_O, sinTheta_O, 1\right)\right)}^{0.25}}}}\right) \]

   7. pow1/2 98.4%

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

   8. +-commutative 98.4%

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

   9. unpow2 98.4%

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

   10. hypot-1-def 98.4%

       \[\leadsto \sin^{-1} \left(\frac{h}{\sqrt{eta - \frac{sinTheta_O}{\sqrt{\color{blue}{\mathsf{hypot}\left(1, sinTheta_O\right)}}}} \cdot \sqrt{eta + \frac{sinTheta_O}{{\left(\mathsf{fma}\left(sinTheta_O, sinTheta_O, 1\right)\right)}^{0.25}}}}\right) \]

   11. fma-udef 98.4%

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

   12. unpow2 98.4%

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

   13. metadata-eval 98.4%

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

7. Applied egg-rr 98.4%

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

8. Final simplification 98.4%

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

Alternative 2: 97.8% accurate, 2.8× speedup

Math

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

FPCore

(FPCore (sinTheta_O h eta)
 :precision binary32
 (asin (/ h (+ eta (* -0.5 (* sinTheta_O (/ sinTheta_O eta)))))))

C

float code(float sinTheta_O, float h, float eta) {
	return asinf((h / (eta + (-0.5f * (sinTheta_O * (sinTheta_O / 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 / (eta + ((-0.5e0) * (sintheta_o * (sintheta_o / eta))))))
end function

Julia

function code(sinTheta_O, h, eta)
	return asin(Float32(h / Float32(eta + Float32(Float32(-0.5) * Float32(sinTheta_O * Float32(sinTheta_O / eta))))))
end

MATLAB

function tmp = code(sinTheta_O, h, eta)
	tmp = asin((h / (eta + (single(-0.5) * (sinTheta_O * (sinTheta_O / eta))))));
end

Derivation

1. Initial program 91.8%

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

2. Taylor expanded in sinTheta_O around 0 98.1%

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

   1. add-sqr-sqrt 98.1%

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

   2. sqrt-div 98.1%

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

   3. unpow2 98.1%

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

   4. sqrt-prod 47.4%

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

   5. add-sqr-sqrt 97.1%

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

   6. sqrt-div 97.1%

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

   7. unpow2 97.1%

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

   8. sqrt-prod 47.5%

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

   9. add-sqr-sqrt 98.3%

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

4. Applied egg-rr 98.3%

   \[\leadsto \sin^{-1} \left(\frac{h}{eta + -0.5 \cdot \color{blue}{\left(\frac{sinTheta_O}{\sqrt{eta}} \cdot \frac{sinTheta_O}{\sqrt{eta}}\right)}}\right) \]
5. Step-by-step derivation

   1. unpow2 98.3%

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

6. Simplified 98.3%

   \[\leadsto \sin^{-1} \left(\frac{h}{eta + -0.5 \cdot \color{blue}{{\left(\frac{sinTheta_O}{\sqrt{eta}}\right)}^{2}}}\right) \]
7. Step-by-step derivation

   1. unpow2 98.3%

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

   2. frac-times 98.1%

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

   3. add-sqr-sqrt 98.1%

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

   4. *-un-lft-identity 98.1%

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

   5. times-frac 98.3%

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

8. Applied egg-rr 98.3%

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

9. Final simplification 98.3%

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

Alternative 3: 95.0% accurate, 3.1× speedup

Math

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

FPCore

(FPCore (sinTheta_O h eta) :precision binary32 (asin (/ h eta)))

C

float code(float sinTheta_O, float h, float eta) {
	return asinf((h / 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 / eta))
end function

Julia

function code(sinTheta_O, h, eta)
	return asin(Float32(h / eta))
end

MATLAB

function tmp = code(sinTheta_O, h, eta)
	tmp = asin((h / eta));
end

Derivation

1. Initial program 91.8%

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

2. Taylor expanded in eta around inf 96.1%

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

3. Final simplification 96.1%

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

Reproduce

herbie shell --seed 2023307 
(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)))))))))