HairBSDF, gamma for a refracted ray

Percentage Accurate: 91.9% → 98.4%

Time: 12.5s

Alternatives: 5

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: 91.9% 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: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float sinTheta_O, float h, float eta) {
	return asinf((h / (sqrtf((sinTheta_O + eta)) * sqrtf((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((sintheta_o + eta)) * sqrt((eta - sintheta_o)))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 92.8%

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

   1. associate-/l* 92.8%

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

3. Simplified 92.8%

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

4. Taylor expanded in sinTheta_O around 0 92.7%

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

   1. unpow2 92.7%

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

6. Simplified 92.7%

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

   1. expm1-log1p-u 92.5%

      \[\leadsto \sin^{-1} \left(\frac{h}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{eta \cdot eta - sinTheta_O \cdot sinTheta_O}\right)\right)}}\right) \]

   2. expm1-udef 37.3%

      \[\leadsto \sin^{-1} \left(\frac{h}{\color{blue}{e^{\mathsf{log1p}\left(\sqrt{eta \cdot eta - sinTheta_O \cdot sinTheta_O}\right)} - 1}}\right) \]

   3. difference-of-squares 37.3%

      \[\leadsto \sin^{-1} \left(\frac{h}{e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\left(eta + sinTheta_O\right) \cdot \left(eta - sinTheta_O\right)}}\right)} - 1}\right) \]

8. Applied egg-rr 37.3%

   \[\leadsto \sin^{-1} \left(\frac{h}{\color{blue}{e^{\mathsf{log1p}\left(\sqrt{\left(eta + sinTheta_O\right) \cdot \left(eta - sinTheta_O\right)}\right)} - 1}}\right) \]
9. Step-by-step derivation

   1. expm1-def 92.5%

      \[\leadsto \sin^{-1} \left(\frac{h}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\left(eta + sinTheta_O\right) \cdot \left(eta - sinTheta_O\right)}\right)\right)}}\right) \]

   2. expm1-log1p 92.7%

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

   3. +-commutative 92.7%

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

10. Simplified 92.7%

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

    1. sqrt-prod 98.6%

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

12. Applied egg-rr 98.6%

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

13. Final simplification 98.6%

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

Alternative 2: 98.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \sin^{-1} \left(\frac{h}{eta + -0.5 \cdot \left(\frac{0.5 \cdot {sinTheta_O}^{4}}{eta} + sinTheta_O \cdot \frac{sinTheta_O}{eta}\right)}\right) \end{array} \]

FPCore

(FPCore (sinTheta_O h eta)
 :precision binary32
 (asin
  (/
   h
   (+
    eta
    (*
     -0.5
     (+
      (/ (* 0.5 (pow sinTheta_O 4.0)) eta)
      (* sinTheta_O (/ sinTheta_O eta))))))))

C

float code(float sinTheta_O, float h, float eta) {
	return asinf((h / (eta + (-0.5f * (((0.5f * powf(sinTheta_O, 4.0f)) / eta) + (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) * (((0.5e0 * (sintheta_o ** 4.0e0)) / eta) + (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(Float32(Float32(0.5) * (sinTheta_O ^ Float32(4.0))) / eta) + Float32(sinTheta_O * Float32(sinTheta_O / eta)))))))
end

MATLAB

function tmp = code(sinTheta_O, h, eta)
	tmp = asin((h / (eta + (single(-0.5) * (((single(0.5) * (sinTheta_O ^ single(4.0))) / eta) + (sinTheta_O * (sinTheta_O / eta)))))));
end

Derivation

1. Initial program 92.8%

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

   1. associate-/l* 92.8%

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

3. Simplified 92.8%

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

4. Taylor expanded in eta around inf 97.0%

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

   1. unpow2 97.0%

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

   2. unpow2 97.0%

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

6. Simplified 97.0%

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

7. Taylor expanded in sinTheta_O around 0 97.0%

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

   1. +-commutative 97.0%

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

   2. associate-*r/ 97.0%

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

   3. unpow2 97.0%

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

   4. associate-*r/ 97.4%

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

9. Simplified 97.4%

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

10. Final simplification 97.4%

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

Alternative 3: 97.9% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 92.8%

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

   1. associate-/l* 92.8%

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

3. Simplified 92.8%

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

4. Taylor expanded in eta around inf 97.0%

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

   1. unpow2 97.0%

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

   2. unpow2 97.0%

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

6. Simplified 97.0%

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

7. Taylor expanded in sinTheta_O around 0 96.9%

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

   1. unpow2 96.9%

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

   2. associate-*l/ 97.3%

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

   3. *-commutative 97.3%

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

9. Simplified 97.3%

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

    1. *-commutative 97.3%

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

    2. associate-/r/ 97.3%

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

    3. clear-num 97.3%

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

11. Applied egg-rr 97.3%

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

12. Final simplification 97.3%

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

Alternative 4: 98.0% 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 92.8%

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

   1. associate-/l* 92.8%

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

3. Simplified 92.8%

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

4. Taylor expanded in eta around inf 97.0%

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

   1. unpow2 97.0%

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

   2. unpow2 97.0%

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

6. Simplified 97.0%

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

7. Taylor expanded in sinTheta_O around 0 96.9%

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

   1. unpow2 96.9%

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

   2. associate-*l/ 97.3%

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

   3. *-commutative 97.3%

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

9. Simplified 97.3%

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

10. Final simplification 97.3%

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

Alternative 5: 95.5% 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 92.8%

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

   1. associate-/l* 92.8%

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

3. Simplified 92.8%

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

4. Taylor expanded in eta around inf 94.4%

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

5. Final simplification 94.4%

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

Reproduce

herbie shell --seed 2023258 
(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)))))))))