HairBSDF, gamma for a refracted ray

Percentage Accurate: 91.6% → 98.6%

Time: 10.6s

Alternatives: 4

Speedup: 1.5×

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: 91.6% 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: 98.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sinTheta\_O \cdot sinTheta\_O \leq 0:\\ \;\;\;\;\sin^{-1} \left(\frac{h}{eta}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{h}{\sqrt{\left(eta + sinTheta\_O\right) \cdot \left(eta - sinTheta\_O\right)}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (sinTheta_O h eta)
 :precision binary32
 (if (<= (* sinTheta_O sinTheta_O) 0.0)
   (asin (/ h eta))
   (asin (/ h (sqrt (* (+ eta sinTheta_O) (- eta sinTheta_O)))))))

C

float code(float sinTheta_O, float h, float eta) {
	float tmp;
	if ((sinTheta_O * sinTheta_O) <= 0.0f) {
		tmp = asinf((h / eta));
	} else {
		tmp = asinf((h / sqrtf(((eta + sinTheta_O) * (eta - sinTheta_O)))));
	}
	return tmp;
}

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
    real(4) :: tmp
    if ((sintheta_o * sintheta_o) <= 0.0e0) then
        tmp = asin((h / eta))
    else
        tmp = asin((h / sqrt(((eta + sintheta_o) * (eta - sintheta_o)))))
    end if
    code = tmp
end function

Julia

function code(sinTheta_O, h, eta)
	tmp = Float32(0.0)
	if (Float32(sinTheta_O * sinTheta_O) <= Float32(0.0))
		tmp = asin(Float32(h / eta));
	else
		tmp = asin(Float32(h / sqrt(Float32(Float32(eta + sinTheta_O) * Float32(eta - sinTheta_O)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(sinTheta_O, h, eta)
	tmp = single(0.0);
	if ((sinTheta_O * sinTheta_O) <= single(0.0))
		tmp = asin((h / eta));
	else
		tmp = asin((h / sqrt(((eta + sinTheta_O) * (eta - sinTheta_O)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 sinTheta_O sinTheta_O) < 0.0

   1. Initial program 89.4%

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

   3. Taylor expanded in eta around inf

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

      1. lower-/.f32 99.5

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

   5. Applied rewrites 99.5%

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

   if 0.0 < (*.f32 sinTheta_O sinTheta_O)

   1. Initial program 99.3%

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

   3. Taylor expanded in sinTheta_O around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. difference-of-squares N/A

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

      7. lower-*.f32 N/A

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

      8. lower-+.f32 N/A

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

      9. lower--.f32 99.1

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

   5. Applied rewrites 99.1%

      \[\leadsto \sin^{-1} \left(\frac{h}{\sqrt{\color{blue}{\left(eta + sinTheta\_O\right) \cdot \left(eta - sinTheta\_O\right)}}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 97.9% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (sinTheta_O h eta)
 :precision binary32
 (asin (/ h (fma (/ sinTheta_O (* eta -2.0)) sinTheta_O eta))))

C

float code(float sinTheta_O, float h, float eta) {
	return asinf((h / fmaf((sinTheta_O / (eta * -2.0f)), sinTheta_O, eta)));
}

Julia

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

Derivation

1. Initial program 93.9%

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

3. Taylor expanded in sinTheta_O around 0

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

   1. +-commutative N/A

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

   2. associate-*r/ N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. metadata-eval N/A

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

   6. distribute-neg-frac N/A

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

   7. metadata-eval N/A

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

   8. associate-*r/ N/A

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

   9. lower-fma.f32 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f32 N/A

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

   12. associate-*r/ N/A

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

   13. metadata-eval N/A

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

   14. distribute-neg-frac N/A

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

   15. metadata-eval N/A

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

   16. lower-/.f32 97.7

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

5. Applied rewrites 97.7%

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

7. Applied rewrites 98.0%

   \[\leadsto \sin^{-1} \left(\frac{h}{\mathsf{fma}\left(\frac{sinTheta\_O}{eta \cdot -2}, \color{blue}{sinTheta\_O}, eta\right)}\right) \]
8. Add Preprocessing

Alternative 3: 97.3% accurate, 1.2× speedup

Math

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

FPCore

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

C

float code(float sinTheta_O, float h, float eta) {
	return asinf((h / fmaf((sinTheta_O * sinTheta_O), (-0.5f / eta), eta)));
}

Julia

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

Derivation

1. Initial program 93.9%

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

3. Taylor expanded in sinTheta_O around 0

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

   1. +-commutative N/A

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

   2. associate-*r/ N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. metadata-eval N/A

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

   6. distribute-neg-frac N/A

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

   7. metadata-eval N/A

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

   8. associate-*r/ N/A

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

   9. lower-fma.f32 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f32 N/A

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

   12. associate-*r/ N/A

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

   13. metadata-eval N/A

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

   14. distribute-neg-frac N/A

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

   15. metadata-eval N/A

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

   16. lower-/.f32 97.7

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

5. Applied rewrites 97.7%

   \[\leadsto \sin^{-1} \left(\frac{h}{\color{blue}{\mathsf{fma}\left(sinTheta\_O \cdot sinTheta\_O, \frac{-0.5}{eta}, eta\right)}}\right) \]
6. Add Preprocessing

Alternative 4: 95.4% accurate, 1.5× 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 93.9%

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

3. Taylor expanded in eta around inf

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

   1. lower-/.f32 95.0

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

5. Applied rewrites 95.0%

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

Reproduce

herbie shell --seed 2024226 
(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)))))))))