HairBSDF, gamma for a refracted ray

Percentage Accurate: 91.8% → 98.0%

Time: 15.3s

Alternatives: 4

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: 91.8% 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: 98.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \sin^{-1} \left(\frac{h}{eta + -0.5 \cdot \left({\left(sinTheta\_O \cdot {eta}^{-0.5}\right)}^{2} \cdot \sqrt{\frac{1}{1 - {sinTheta\_O}^{2}}}\right)}\right) \end{array} \]

FPCore

(FPCore (sinTheta_O h eta)
 :precision binary32
 (asin
  (/
   h
   (+
    eta
    (*
     -0.5
     (*
      (pow (* sinTheta_O (pow eta -0.5)) 2.0)
      (sqrt (/ 1.0 (- 1.0 (pow sinTheta_O 2.0))))))))))

C

float code(float sinTheta_O, float h, float eta) {
	return asinf((h / (eta + (-0.5f * (powf((sinTheta_O * powf(eta, -0.5f)), 2.0f) * sqrtf((1.0f / (1.0f - powf(sinTheta_O, 2.0f)))))))));
}

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 * (eta ** (-0.5e0))) ** 2.0e0) * sqrt((1.0e0 / (1.0e0 - (sintheta_o ** 2.0e0)))))))))
end function

Julia

function code(sinTheta_O, h, eta)
	return asin(Float32(h / Float32(eta + Float32(Float32(-0.5) * Float32((Float32(sinTheta_O * (eta ^ Float32(-0.5))) ^ Float32(2.0)) * sqrt(Float32(Float32(1.0) / Float32(Float32(1.0) - (sinTheta_O ^ Float32(2.0))))))))))
end

MATLAB

function tmp = code(sinTheta_O, h, eta)
	tmp = asin((h / (eta + (single(-0.5) * (((sinTheta_O * (eta ^ single(-0.5))) ^ single(2.0)) * sqrt((single(1.0) / (single(1.0) - (sinTheta_O ^ single(2.0))))))))));
end

Derivation

1. Initial program 90.4%

   \[\sin^{-1} \left(\frac{h}{\sqrt{eta \cdot eta - \frac{sinTheta\_O \cdot sinTheta\_O}{\sqrt{1 - sinTheta\_O \cdot sinTheta\_O}}}}\right) \]
2. Add Preprocessing

3. Taylor expanded in eta around inf 97.1%

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

   1. add-sqr-sqrt 97.1%

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

   2. sqrt-div 97.1%

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

   3. unpow2 97.1%

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

   4. sqrt-prod 43.1%

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

   5. add-sqr-sqrt 95.7%

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

   6. sqrt-div 95.7%

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

   7. unpow2 95.7%

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

   8. sqrt-prod 43.6%

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

   9. add-sqr-sqrt 97.6%

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

5. Applied egg-rr 97.6%

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

   1. unpow2 97.6%

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

7. Simplified 97.6%

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

   1. div-inv 97.6%

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

   2. pow1/2 97.6%

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

   3. pow-flip 97.6%

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

   4. metadata-eval 97.6%

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

9. Applied egg-rr 97.6%

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

10. Final simplification 97.6%

    \[\leadsto \sin^{-1} \left(\frac{h}{eta + -0.5 \cdot \left({\left(sinTheta\_O \cdot {eta}^{-0.5}\right)}^{2} \cdot \sqrt{\frac{1}{1 - {sinTheta\_O}^{2}}}\right)}\right) \]
11. Add Preprocessing

Alternative 2: 98.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sin^{-1} \left(\frac{h}{eta + -0.5 \cdot \left(\frac{sinTheta\_O}{\sqrt{1 - {sinTheta\_O}^{2}}} \cdot \frac{sinTheta\_O}{eta}\right)}\right) \end{array} \]

FPCore

(FPCore (sinTheta_O h eta)
 :precision binary32
 (asin
  (/
   h
   (+
    eta
    (*
     -0.5
     (*
      (/ sinTheta_O (sqrt (- 1.0 (pow sinTheta_O 2.0))))
      (/ sinTheta_O eta)))))))

C

float code(float sinTheta_O, float h, float eta) {
	return asinf((h / (eta + (-0.5f * ((sinTheta_O / sqrtf((1.0f - powf(sinTheta_O, 2.0f)))) * (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 / sqrt((1.0e0 - (sintheta_o ** 2.0e0)))) * (sintheta_o / eta))))))
end function

Julia

function code(sinTheta_O, h, eta)
	return asin(Float32(h / Float32(eta + Float32(Float32(-0.5) * Float32(Float32(sinTheta_O / sqrt(Float32(Float32(1.0) - (sinTheta_O ^ Float32(2.0))))) * Float32(sinTheta_O / eta))))))
end

MATLAB

function tmp = code(sinTheta_O, h, eta)
	tmp = asin((h / (eta + (single(-0.5) * ((sinTheta_O / sqrt((single(1.0) - (sinTheta_O ^ single(2.0))))) * (sinTheta_O / eta))))));
end

Derivation

1. Initial program 90.4%

   \[\sin^{-1} \left(\frac{h}{\sqrt{eta \cdot eta - \frac{sinTheta\_O \cdot sinTheta\_O}{\sqrt{1 - sinTheta\_O \cdot sinTheta\_O}}}}\right) \]
2. Add Preprocessing

3. Taylor expanded in eta around inf 97.1%

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

   1. pow1 97.1%

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

   2. associate-*l/ 97.1%

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

   3. sqrt-div 97.1%

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

   4. metadata-eval 97.1%

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

   5. div-inv 97.1%

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

5. Applied egg-rr 97.1%

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

   1. unpow1 97.1%

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

   2. associate-/l/ 97.1%

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

7. Simplified 97.1%

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

   1. unpow2 97.1%

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

   2. *-commutative 97.1%

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

   3. times-frac 97.6%

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

9. Applied egg-rr 97.6%

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

10. Final simplification 97.6%

    \[\leadsto \sin^{-1} \left(\frac{h}{eta + -0.5 \cdot \left(\frac{sinTheta\_O}{\sqrt{1 - {sinTheta\_O}^{2}}} \cdot \frac{sinTheta\_O}{eta}\right)}\right) \]
11. Add Preprocessing

Alternative 3: 97.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \sin^{-1} \left(\frac{h}{eta + -0.5 \cdot \frac{{sinTheta\_O}^{2}}{eta}}\right) \end{array} \]

FPCore

(FPCore (sinTheta_O h eta)
 :precision binary32
 (asin (/ h (+ eta (* -0.5 (/ (pow sinTheta_O 2.0) eta))))))

C

float code(float sinTheta_O, float h, float eta) {
	return asinf((h / (eta + (-0.5f * (powf(sinTheta_O, 2.0f) / 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 ** 2.0e0) / eta)))))
end function

Julia

function code(sinTheta_O, h, eta)
	return asin(Float32(h / Float32(eta + Float32(Float32(-0.5) * Float32((sinTheta_O ^ Float32(2.0)) / eta)))))
end

MATLAB

function tmp = code(sinTheta_O, h, eta)
	tmp = asin((h / (eta + (single(-0.5) * ((sinTheta_O ^ single(2.0)) / eta)))));
end

Derivation

1. Initial program 90.4%

   \[\sin^{-1} \left(\frac{h}{\sqrt{eta \cdot eta - \frac{sinTheta\_O \cdot sinTheta\_O}{\sqrt{1 - sinTheta\_O \cdot sinTheta\_O}}}}\right) \]
2. Add Preprocessing

3. Taylor expanded in sinTheta_O around 0 96.9%

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

4. Final simplification 96.9%

   \[\leadsto \sin^{-1} \left(\frac{h}{eta + -0.5 \cdot \frac{{sinTheta\_O}^{2}}{eta}}\right) \]
5. Add Preprocessing

Alternative 4: 95.4% 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 90.4%

   \[\sin^{-1} \left(\frac{h}{\sqrt{eta \cdot eta - \frac{sinTheta\_O \cdot sinTheta\_O}{\sqrt{1 - sinTheta\_O \cdot sinTheta\_O}}}}\right) \]
2. Add Preprocessing

3. Taylor expanded in eta around inf 94.6%

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

4. Final simplification 94.6%

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

Reproduce

herbie shell --seed 2024044 
(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)))))))))