HairBSDF, gamma for a refracted ray

Percentage Accurate: 91.9% → 97.8%

Time: 13.0s

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: 91.9% 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: 97.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \sin^{-1} \left(\frac{h}{eta + -0.5 \cdot {\left(\frac{\frac{sinTheta\_O}{{eta}^{0.25}}}{{eta}^{0.25}}\right)}^{2}}\right) \end{array} \]

FPCore

(FPCore (sinTheta_O h eta)
 :precision binary32
 (asin
  (/
   h
   (+
    eta
    (* -0.5 (pow (/ (/ sinTheta_O (pow eta 0.25)) (pow eta 0.25)) 2.0))))))

C

float code(float sinTheta_O, float h, float eta) {
	return asinf((h / (eta + (-0.5f * powf(((sinTheta_O / powf(eta, 0.25f)) / powf(eta, 0.25f)), 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 / (eta ** 0.25e0)) / (eta ** 0.25e0)) ** 2.0e0)))))
end function

Julia

function code(sinTheta_O, h, eta)
	return asin(Float32(h / Float32(eta + Float32(Float32(-0.5) * (Float32(Float32(sinTheta_O / (eta ^ Float32(0.25))) / (eta ^ Float32(0.25))) ^ Float32(2.0))))))
end

MATLAB

function tmp = code(sinTheta_O, h, eta)
	tmp = asin((h / (eta + (single(-0.5) * (((sinTheta_O / (eta ^ single(0.25))) / (eta ^ single(0.25))) ^ single(2.0))))));
end

Derivation

1. Initial program 94.5%

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

3. Taylor expanded in sinTheta_O around 0 98.3%

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

   1. pow2 98.3%

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

   2. add-sqr-sqrt 98.3%

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

   3. sqrt-div 98.3%

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

   4. sqrt-prod 52.6%

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

   5. add-sqr-sqrt 97.7%

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

   6. sqrt-div 97.7%

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

   7. sqrt-prod 52.9%

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

   8. add-sqr-sqrt 98.7%

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

5. Applied egg-rr 98.7%

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

   1. unpow2 98.7%

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

7. Simplified 98.7%

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

   1. *-un-lft-identity 98.7%

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

   2. add-sqr-sqrt 98.7%

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

   3. times-frac 98.7%

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

   4. pow1/2 98.7%

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

   5. sqrt-pow1 98.7%

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

   6. metadata-eval 98.7%

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

   7. pow1/2 98.7%

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

   8. sqrt-pow1 98.7%

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

   9. metadata-eval 98.7%

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

9. Applied egg-rr 98.7%

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

    1. associate-*l/ 98.7%

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

    2. *-lft-identity 98.7%

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

11. Simplified 98.7%

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

Alternative 2: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sin^{-1} \left(\frac{h}{eta + -0.5 \cdot \frac{sinTheta\_O}{{\left(\sqrt[3]{\frac{eta}{sinTheta\_O}}\right)}^{3}}}\right) \end{array} \]

FPCore

(FPCore (sinTheta_O h eta)
 :precision binary32
 (asin
  (/ h (+ eta (* -0.5 (/ sinTheta_O (pow (cbrt (/ eta sinTheta_O)) 3.0)))))))

C

float code(float sinTheta_O, float h, float eta) {
	return asinf((h / (eta + (-0.5f * (sinTheta_O / powf(cbrtf((eta / sinTheta_O)), 3.0f))))));
}

Julia

function code(sinTheta_O, h, eta)
	return asin(Float32(h / Float32(eta + Float32(Float32(-0.5) * Float32(sinTheta_O / (cbrt(Float32(eta / sinTheta_O)) ^ Float32(3.0)))))))
end

Derivation

1. Initial program 94.5%

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

3. Taylor expanded in sinTheta_O around 0 98.3%

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

   1. pow2 98.3%

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

   2. add-sqr-sqrt 98.3%

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

   3. sqrt-div 98.3%

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

   4. sqrt-prod 52.6%

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

   5. add-sqr-sqrt 97.7%

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

   6. sqrt-div 97.7%

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

   7. sqrt-prod 52.9%

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

   8. add-sqr-sqrt 98.7%

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

5. Applied egg-rr 98.7%

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

   1. unpow2 98.7%

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

7. Simplified 98.7%

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

   1. unpow2 98.7%

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

   2. clear-num 98.7%

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

   3. frac-times 98.7%

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

   4. metadata-eval 98.7%

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

   5. div-inv 98.7%

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

   6. /-rgt-identity 98.7%

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

9. Applied egg-rr 98.7%

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

    1. associate-*r/ 98.7%

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

    2. rem-square-sqrt 98.7%

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

11. Simplified 98.7%

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

    1. add-cube-cbrt 98.7%

       \[\leadsto \sin^{-1} \left(\frac{h}{eta + -0.5 \cdot \frac{sinTheta\_O}{\color{blue}{\left(\sqrt[3]{\frac{eta}{sinTheta\_O}} \cdot \sqrt[3]{\frac{eta}{sinTheta\_O}}\right) \cdot \sqrt[3]{\frac{eta}{sinTheta\_O}}}}}\right) \]

    2. pow3 98.7%

       \[\leadsto \sin^{-1} \left(\frac{h}{eta + -0.5 \cdot \frac{sinTheta\_O}{\color{blue}{{\left(\sqrt[3]{\frac{eta}{sinTheta\_O}}\right)}^{3}}}}\right) \]

13. Applied egg-rr 98.7%

    \[\leadsto \sin^{-1} \left(\frac{h}{eta + -0.5 \cdot \frac{sinTheta\_O}{\color{blue}{{\left(\sqrt[3]{\frac{eta}{sinTheta\_O}}\right)}^{3}}}}\right) \]
14. Add Preprocessing

Alternative 3: 97.8% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 94.5%

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

3. Taylor expanded in sinTheta_O around 0 98.3%

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

   1. pow2 98.3%

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

   2. add-sqr-sqrt 98.3%

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

   3. sqrt-div 98.3%

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

   4. sqrt-prod 52.6%

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

   5. add-sqr-sqrt 97.7%

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

   6. sqrt-div 97.7%

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

   7. sqrt-prod 52.9%

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

   8. add-sqr-sqrt 98.7%

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

5. Applied egg-rr 98.7%

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

   1. unpow2 98.7%

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

7. Simplified 98.7%

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

   1. unpow2 98.7%

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

   2. clear-num 98.7%

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

   3. frac-times 98.7%

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

   4. metadata-eval 98.7%

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

   5. div-inv 98.7%

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

   6. /-rgt-identity 98.7%

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

9. Applied egg-rr 98.7%

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

    1. associate-*r/ 98.7%

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

    2. rem-square-sqrt 98.7%

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

11. Simplified 98.7%

    \[\leadsto \sin^{-1} \left(\frac{h}{eta + -0.5 \cdot \color{blue}{\frac{sinTheta\_O}{\frac{eta}{sinTheta\_O}}}}\right) \]
12. 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 94.5%

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

3. Taylor expanded in eta around inf 97.0%

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

Reproduce

herbie shell --seed 2024141 
(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)))))))))