HairBSDF, gamma for a refracted ray

Percentage Accurate: 92.0% → 98.5%

Time: 1.3min

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: 92.0% 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: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 93.5%

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

3. Taylor expanded in h around 0 89.8%

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

4. Taylor expanded in sinTheta_O around 0 89.8%

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

   1. neg-mul-1 89.8%

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

   2. +-commutative 89.8%

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

   3. unsub-neg 89.8%

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

6. Simplified 89.8%

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

   1. unpow2 89.8%

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

   2. unpow2 89.8%

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

   3. difference-of-squares 89.8%

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

8. Applied egg-rr 89.8%

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

   1. associate-/r* 89.9%

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

   2. sqrt-div 98.8%

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

   3. +-commutative 98.8%

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

10. Applied egg-rr 98.8%

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

11. Final simplification 98.8%

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

Alternative 2: 97.8% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 93.5%

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

3. Taylor expanded in sinTheta_O around 0 97.0%

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

   1. add-sqr-sqrt 97.0%

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

   2. div-inv 97.0%

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

   3. sqrt-prod 97.0%

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

   4. sqrt-pow1 96.3%

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

   5. metadata-eval 96.3%

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

   6. pow1 96.3%

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

   7. inv-pow 96.3%

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

   8. sqrt-pow1 96.3%

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

   9. metadata-eval 96.3%

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

   10. div-inv 96.3%

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

   11. sqrt-prod 96.3%

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

   12. sqrt-pow1 97.8%

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

   13. metadata-eval 97.8%

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

   14. pow1 97.8%

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

   15. inv-pow 97.8%

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

   16. sqrt-pow1 97.8%

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

   17. metadata-eval 97.8%

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

5. Applied egg-rr 97.8%

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

   1. unpow2 97.8%

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

7. Simplified 97.8%

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

   1. unpow-prod-down 97.0%

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

   2. pow-pow 97.0%

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

   3. metadata-eval 97.0%

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

   4. inv-pow 97.0%

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

   5. unpow2 97.0%

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

   6. associate-*l* 97.8%

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

9. Applied egg-rr 97.8%

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

10. Final simplification 97.8%

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

Alternative 3: 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 93.5%

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

3. Taylor expanded in sinTheta_O around 0 97.0%

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

   1. add-sqr-sqrt 97.0%

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

   2. div-inv 97.0%

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

   3. sqrt-prod 97.0%

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

   4. sqrt-pow1 96.3%

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

   5. metadata-eval 96.3%

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

   6. pow1 96.3%

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

   7. inv-pow 96.3%

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

   8. sqrt-pow1 96.3%

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

   9. metadata-eval 96.3%

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

   10. div-inv 96.3%

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

   11. sqrt-prod 96.3%

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

   12. sqrt-pow1 97.8%

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

   13. metadata-eval 97.8%

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

   14. pow1 97.8%

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

   15. inv-pow 97.8%

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

   16. sqrt-pow1 97.8%

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

   17. metadata-eval 97.8%

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

5. Applied egg-rr 97.8%

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

   1. unpow2 97.8%

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

7. Simplified 97.8%

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

   1. unpow-prod-down 97.0%

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

   2. pow-pow 97.0%

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

   3. metadata-eval 97.0%

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

   4. inv-pow 97.0%

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

   5. div-inv 97.0%

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

   6. unpow2 97.0%

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

   7. associate-/l* 97.8%

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

9. Applied egg-rr 97.8%

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

10. Final simplification 97.8%

    \[\leadsto \sin^{-1} \left(\frac{h}{eta + -0.5 \cdot \left(sinTheta\_O \cdot \frac{sinTheta\_O}{eta}\right)}\right) \]
11. 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 93.5%

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

3. Taylor expanded in eta around inf 95.1%

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

4. Final simplification 95.1%

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

Reproduce

herbie shell --seed 2024115 
(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)))))))))