HairBSDF, gamma for a refracted ray

Percentage Accurate: 91.7% → 97.8%

Time: 28.1s

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: 91.7% 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: 97.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 92.6%

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

   1. sqr-neg 92.6%

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

   2. sqr-neg 92.6%

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

   3. sqr-neg 92.6%

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

   4. sqr-neg 92.6%

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

3. Simplified 92.6%

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

5. Taylor expanded in sinTheta_O around 0 96.5%

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

   1. add-sqr-sqrt 96.5%

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

   2. sqrt-div 96.5%

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

   3. sqrt-pow1 95.1%

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

   4. metadata-eval 95.1%

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

   5. pow1 95.1%

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

   6. sqrt-div 95.1%

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

   7. sqrt-pow1 97.4%

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

   8. metadata-eval 97.4%

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

   9. pow1 97.4%

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

7. Applied egg-rr 97.4%

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

   1. unpow2 97.4%

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

9. Simplified 97.4%

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

Alternative 2: 97.3% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 92.6%

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

   1. sqr-neg 92.6%

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

   2. sqr-neg 92.6%

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

   3. sqr-neg 92.6%

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

   4. sqr-neg 92.6%

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

3. Simplified 92.6%

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

5. Taylor expanded in sinTheta_O around 0 88.4%

   \[\leadsto \sin^{-1} \left(\frac{h}{\color{blue}{eta + {sinTheta\_O}^{2} \cdot \left(-0.5 \cdot \frac{{sinTheta\_O}^{2} \cdot \left(0.5 + 0.25 \cdot \frac{1}{{eta}^{2}}\right)}{eta} - 0.5 \cdot \frac{1}{eta}\right)}}\right) \]
6. Step-by-step derivation

   1. unpow2 88.4%

      \[\leadsto \sin^{-1} \left(\frac{h}{eta + {sinTheta\_O}^{2} \cdot \left(-0.5 \cdot \frac{\color{blue}{\left(sinTheta\_O \cdot sinTheta\_O\right)} \cdot \left(0.5 + 0.25 \cdot \frac{1}{{eta}^{2}}\right)}{eta} - 0.5 \cdot \frac{1}{eta}\right)}\right) \]

7. Applied egg-rr 88.4%

   \[\leadsto \sin^{-1} \left(\frac{h}{eta + {sinTheta\_O}^{2} \cdot \left(-0.5 \cdot \frac{\color{blue}{\left(sinTheta\_O \cdot sinTheta\_O\right)} \cdot \left(0.5 + 0.25 \cdot \frac{1}{{eta}^{2}}\right)}{eta} - 0.5 \cdot \frac{1}{eta}\right)}\right) \]

8. Taylor expanded in eta around inf 96.5%

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

   1. unpow2 88.4%

      \[\leadsto \sin^{-1} \left(\frac{h}{eta + {sinTheta\_O}^{2} \cdot \left(-0.5 \cdot \frac{\color{blue}{\left(sinTheta\_O \cdot sinTheta\_O\right)} \cdot \left(0.5 + 0.25 \cdot \frac{1}{{eta}^{2}}\right)}{eta} - 0.5 \cdot \frac{1}{eta}\right)}\right) \]

10. Applied egg-rr 96.5%

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

11. Final simplification 96.5%

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

Alternative 3: 97.3% accurate, 2.8× speedup

Math

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

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(Float32(sinTheta_O * 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 92.6%

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

   1. sqr-neg 92.6%

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

   2. sqr-neg 92.6%

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

   3. sqr-neg 92.6%

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

   4. sqr-neg 92.6%

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

3. Simplified 92.6%

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

5. Taylor expanded in sinTheta_O around 0 96.5%

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

   1. unpow2 88.4%

      \[\leadsto \sin^{-1} \left(\frac{h}{eta + {sinTheta\_O}^{2} \cdot \left(-0.5 \cdot \frac{\color{blue}{\left(sinTheta\_O \cdot sinTheta\_O\right)} \cdot \left(0.5 + 0.25 \cdot \frac{1}{{eta}^{2}}\right)}{eta} - 0.5 \cdot \frac{1}{eta}\right)}\right) \]

7. Applied egg-rr 96.5%

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

Alternative 4: 95.2% 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 92.6%

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

   1. sqr-neg 92.6%

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

   2. sqr-neg 92.6%

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

   3. sqr-neg 92.6%

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

   4. sqr-neg 92.6%

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

3. Simplified 92.6%

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

5. Taylor expanded in eta around inf 94.2%

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

Reproduce

herbie shell --seed 2024143 
(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)))))))))