HairBSDF, gamma for a refracted ray

Percentage Accurate: 91.7% → 97.9%

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: 91.7% 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: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 90.1%

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

3. Taylor expanded in sinTheta_O around 0 96.2%

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

   1. add-sqr-sqrt 96.2%

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

   2. sqrt-div 96.2%

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

   3. unpow2 96.2%

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

   4. sqrt-prod 45.2%

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

   5. add-sqr-sqrt 94.5%

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

   6. sqrt-div 94.5%

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

   7. unpow2 94.5%

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

   8. sqrt-prod 45.5%

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

   9. add-sqr-sqrt 97.0%

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

5. Applied egg-rr 97.0%

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

   1. unpow2 97.0%

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

7. Simplified 97.0%

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

   1. unpow2 97.0%

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

   2. clear-num 97.0%

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

   3. frac-times 97.0%

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

   4. metadata-eval 97.0%

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

   5. div-inv 97.0%

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

   6. /-rgt-identity 97.0%

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

9. Applied egg-rr 97.0%

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

    1. associate-*r/ 97.0%

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

    2. rem-square-sqrt 97.0%

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

11. Simplified 97.0%

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

    1. add-exp-log 97.0%

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

    2. div-inv 97.0%

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

    3. clear-num 97.0%

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

13. Applied egg-rr 97.0%

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

Alternative 2: 97.9% 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 90.1%

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

3. Taylor expanded in sinTheta_O around 0 96.2%

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

   1. add-sqr-sqrt 96.2%

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

   2. sqrt-div 96.2%

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

   3. unpow2 96.2%

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

   4. sqrt-prod 45.2%

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

   5. add-sqr-sqrt 94.5%

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

   6. sqrt-div 94.5%

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

   7. unpow2 94.5%

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

   8. sqrt-prod 45.5%

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

   9. add-sqr-sqrt 97.0%

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

5. Applied egg-rr 97.0%

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

   1. unpow2 97.0%

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

7. Simplified 97.0%

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

Alternative 3: 97.9% 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 90.1%

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

3. Taylor expanded in sinTheta_O around 0 96.2%

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

   1. add-sqr-sqrt 96.2%

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

   2. sqrt-div 96.2%

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

   3. unpow2 96.2%

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

   4. sqrt-prod 45.2%

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

   5. add-sqr-sqrt 94.5%

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

   6. sqrt-div 94.5%

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

   7. unpow2 94.5%

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

   8. sqrt-prod 45.5%

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

   9. add-sqr-sqrt 97.0%

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

5. Applied egg-rr 97.0%

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

   1. unpow2 97.0%

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

7. Simplified 97.0%

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

   1. unpow2 97.0%

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

   2. clear-num 97.0%

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

   3. frac-times 97.0%

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

   4. metadata-eval 97.0%

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

   5. div-inv 97.0%

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

   6. /-rgt-identity 97.0%

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

9. Applied egg-rr 97.0%

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

    1. associate-*r/ 97.0%

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

    2. rem-square-sqrt 97.0%

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

11. Simplified 97.0%

    \[\leadsto \sin^{-1} \left(\frac{h}{eta + -0.5 \cdot \color{blue}{\frac{sinTheta\_O}{\frac{eta}{sinTheta\_O}}}}\right) \]
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 90.1%

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

3. Taylor expanded in eta around inf 93.6%

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

Reproduce

herbie shell --seed 2024132 
(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)))))))))