HairBSDF, gamma for a refracted ray

Percentage Accurate: 92.4% → 98.4%

Time: 14.5s

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: 92.4% 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

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 93.3%

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

   1. associate-/l* 93.3%

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

3. Simplified 93.3%

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

4. Taylor expanded in sinTheta_O around 0 92.9%

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

   1. unpow2 92.9%

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

6. Simplified 92.9%

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

   1. pow1/2 92.9%

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

   2. difference-of-squares 92.9%

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

   3. unpow-prod-down 98.1%

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

8. Applied egg-rr 98.1%

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

   1. unpow1/2 98.1%

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

   2. unpow1/2 98.1%

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

10. Simplified 98.1%

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

11. Final simplification 98.1%

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

Alternative 2: 97.8% accurate, 1.5× speedup

Math

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

FPCore

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

C

float code(float sinTheta_O, float h, float eta) {
	return asinf((h / (eta + (-0.5f * ((sinTheta_O * (sinTheta_O / eta)) / 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 / (eta + ((-0.5e0) * ((sintheta_o * (sintheta_o / eta)) / sqrt((1.0e0 - (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(sinTheta_O * Float32(sinTheta_O / eta)) / sqrt(Float32(Float32(1.0) - Float32(sinTheta_O * sinTheta_O))))))))
end

MATLAB

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

Derivation

1. Initial program 93.3%

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

   1. associate-/l* 93.3%

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

3. Simplified 93.3%

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

4. Taylor expanded in eta around inf 96.6%

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

   1. unpow2 96.6%

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

   2. unpow2 96.6%

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

6. Simplified 96.6%

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

   1. expm1-log1p-u 96.6%

      \[\leadsto \sin^{-1} \left(\frac{h}{eta + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-0.5 \cdot \left(\frac{sinTheta_O \cdot sinTheta_O}{eta} \cdot \sqrt{\frac{1}{1 - sinTheta_O \cdot sinTheta_O}}\right)\right)\right)}}\right) \]

   2. expm1-udef 95.0%

      \[\leadsto \sin^{-1} \left(\frac{h}{eta + \color{blue}{\left(e^{\mathsf{log1p}\left(-0.5 \cdot \left(\frac{sinTheta_O \cdot sinTheta_O}{eta} \cdot \sqrt{\frac{1}{1 - sinTheta_O \cdot sinTheta_O}}\right)\right)} - 1\right)}}\right) \]

   3. associate-/l* 95.0%

      \[\leadsto \sin^{-1} \left(\frac{h}{eta + \left(e^{\mathsf{log1p}\left(-0.5 \cdot \left(\color{blue}{\frac{sinTheta_O}{\frac{eta}{sinTheta_O}}} \cdot \sqrt{\frac{1}{1 - sinTheta_O \cdot sinTheta_O}}\right)\right)} - 1\right)}\right) \]

   4. sqrt-div 95.0%

      \[\leadsto \sin^{-1} \left(\frac{h}{eta + \left(e^{\mathsf{log1p}\left(-0.5 \cdot \left(\frac{sinTheta_O}{\frac{eta}{sinTheta_O}} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{1 - sinTheta_O \cdot sinTheta_O}}}\right)\right)} - 1\right)}\right) \]

   5. metadata-eval 95.0%

      \[\leadsto \sin^{-1} \left(\frac{h}{eta + \left(e^{\mathsf{log1p}\left(-0.5 \cdot \left(\frac{sinTheta_O}{\frac{eta}{sinTheta_O}} \cdot \frac{\color{blue}{1}}{\sqrt{1 - sinTheta_O \cdot sinTheta_O}}\right)\right)} - 1\right)}\right) \]

8. Applied egg-rr 95.0%

   \[\leadsto \sin^{-1} \left(\frac{h}{eta + \color{blue}{\left(e^{\mathsf{log1p}\left(-0.5 \cdot \left(\frac{sinTheta_O}{\frac{eta}{sinTheta_O}} \cdot \frac{1}{\sqrt{1 - sinTheta_O \cdot sinTheta_O}}\right)\right)} - 1\right)}}\right) \]
9. Step-by-step derivation

   1. expm1-def 97.1%

      \[\leadsto \sin^{-1} \left(\frac{h}{eta + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-0.5 \cdot \left(\frac{sinTheta_O}{\frac{eta}{sinTheta_O}} \cdot \frac{1}{\sqrt{1 - sinTheta_O \cdot sinTheta_O}}\right)\right)\right)}}\right) \]

   2. expm1-log1p 97.1%

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

   3. associate-*r/ 97.1%

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

   4. *-rgt-identity 97.1%

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

   5. associate-/r/ 97.1%

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

10. Simplified 97.1%

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

11. Final simplification 97.1%

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

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 93.3%

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

   1. associate-/l* 93.3%

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

3. Simplified 93.3%

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

4. Taylor expanded in sinTheta_O around 0 96.6%

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

   1. unpow2 96.6%

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

6. Simplified 96.6%

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

7. Final simplification 96.6%

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

Alternative 4: 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 93.3%

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

   1. associate-/l* 93.3%

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

3. Simplified 93.3%

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

4. Taylor expanded in eta around inf 96.6%

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

   1. unpow2 96.6%

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

   2. unpow2 96.6%

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

6. Simplified 96.6%

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

7. Taylor expanded in sinTheta_O around 0 96.6%

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

   1. *-commutative 96.6%

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

   2. unpow2 96.6%

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

9. Simplified 96.6%

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

    1. expm1-log1p-u 96.6%

       \[\leadsto \sin^{-1} \left(\frac{h}{eta + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{sinTheta_O \cdot sinTheta_O}{eta}\right)\right)} \cdot -0.5}\right) \]

    2. expm1-udef 94.9%

       \[\leadsto \sin^{-1} \left(\frac{h}{eta + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{sinTheta_O \cdot sinTheta_O}{eta}\right)} - 1\right)} \cdot -0.5}\right) \]

    3. associate-/l* 94.9%

       \[\leadsto \sin^{-1} \left(\frac{h}{eta + \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{sinTheta_O}{\frac{eta}{sinTheta_O}}}\right)} - 1\right) \cdot -0.5}\right) \]

11. Applied egg-rr 94.9%

    \[\leadsto \sin^{-1} \left(\frac{h}{eta + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{sinTheta_O}{\frac{eta}{sinTheta_O}}\right)} - 1\right)} \cdot -0.5}\right) \]
12. Step-by-step derivation

    1. expm1-def 97.0%

       \[\leadsto \sin^{-1} \left(\frac{h}{eta + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{sinTheta_O}{\frac{eta}{sinTheta_O}}\right)\right)} \cdot -0.5}\right) \]

    2. expm1-log1p 97.0%

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

    3. associate-/r/ 97.0%

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

13. Simplified 97.0%

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

14. Final simplification 97.0%

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

Alternative 5: 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 93.3%

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

   1. associate-/l* 93.3%

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

3. Simplified 93.3%

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

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 2023252 
(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)))))))))