HairBSDF, gamma for a refracted ray

Percentage Accurate: 91.8% → 98.4%

Time: 13.6s

Alternatives: 6

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: 91.8% 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 92.0%

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

2. Taylor expanded in sinTheta_O around 0 91.6%

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

   1. +-commutative 91.6%

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

   2. mul-1-neg 91.6%

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

   3. unsub-neg 91.6%

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

   4. unpow2 91.6%

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

   5. unpow2 91.6%

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

4. Simplified 91.6%

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

5. Taylor expanded in eta around 0 91.6%

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

   1. +-commutative 91.6%

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

   2. neg-mul-1 91.6%

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

   3. sub-neg 91.6%

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

   4. unpow2 91.6%

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

   5. unpow2 91.6%

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

   6. difference-of-squares 91.6%

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

   7. +-commutative 91.6%

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

7. Simplified 91.6%

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

   1. sqrt-prod 98.4%

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

9. Applied egg-rr 98.4%

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

10. Final simplification 98.4%

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

Alternative 2: 97.9% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (sinTheta_O h eta)
 :precision binary32
 (asin
  (/
   h
   (+
    eta
    (*
     (* -0.5 (/ sinTheta_O eta))
     (/ sinTheta_O (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 / eta)) * (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 / (eta + (((-0.5e0) * (sintheta_o / eta)) * (sintheta_o / sqrt((1.0e0 - (sintheta_o * sintheta_o))))))))
end function

Julia

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

Derivation

1. Initial program 92.0%

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

2. Taylor expanded in eta around inf 97.4%

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

   1. associate-*r* 97.4%

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

   2. unpow2 97.4%

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

   3. unpow2 97.4%

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

4. Simplified 97.4%

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

   1. *-un-lft-identity 97.4%

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

   2. associate-/l* 97.9%

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

6. Applied egg-rr 97.9%

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

   1. *-lft-identity 97.9%

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

   2. associate-/r/ 97.9%

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

8. Simplified 97.9%

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

   1. pow1 97.9%

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

   2. associate-*r* 97.9%

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

   3. sqrt-div 97.9%

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

   4. metadata-eval 97.9%

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

10. Applied egg-rr 97.9%

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

    1. unpow1 97.9%

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

    2. associate-*l* 97.9%

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

    3. associate-*r/ 97.9%

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

    4. *-rgt-identity 97.9%

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

12. Simplified 97.9%

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

13. Final simplification 97.9%

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

Alternative 3: 97.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

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 + ((sintheta_o ** 3.0e0) * 0.5e0))))))
end function

Julia

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

MATLAB

function tmp = code(sinTheta_O, h, eta)
	tmp = asin((h / (eta + ((single(-0.5) * (sinTheta_O / eta)) * (sinTheta_O + ((sinTheta_O ^ single(3.0)) * single(0.5)))))));
end

Derivation

1. Initial program 92.0%

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

2. Taylor expanded in eta around inf 97.4%

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

   1. associate-*r* 97.4%

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

   2. unpow2 97.4%

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

   3. unpow2 97.4%

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

4. Simplified 97.4%

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

   1. *-un-lft-identity 97.4%

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

   2. associate-/l* 97.9%

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

6. Applied egg-rr 97.9%

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

   1. *-lft-identity 97.9%

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

   2. associate-/r/ 97.9%

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

8. Simplified 97.9%

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

   1. pow1 97.9%

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

   2. associate-*r* 97.9%

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

   3. sqrt-div 97.9%

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

   4. metadata-eval 97.9%

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

10. Applied egg-rr 97.9%

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

    1. unpow1 97.9%

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

    2. associate-*l* 97.9%

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

    3. associate-*r/ 97.9%

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

    4. *-rgt-identity 97.9%

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

12. Simplified 97.9%

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

13. Taylor expanded in sinTheta_O around 0 97.9%

    \[\leadsto \sin^{-1} \left(\frac{h}{eta + \left(-0.5 \cdot \frac{sinTheta_O}{eta}\right) \cdot \color{blue}{\left(sinTheta_O + 0.5 \cdot {sinTheta_O}^{3}\right)}}\right) \]
14. Step-by-step derivation

    1. *-commutative 97.9%

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

15. Simplified 97.9%

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

16. Final simplification 97.9%

    \[\leadsto \sin^{-1} \left(\frac{h}{eta + \left(-0.5 \cdot \frac{sinTheta_O}{eta}\right) \cdot \left(sinTheta_O + {sinTheta_O}^{3} \cdot 0.5\right)}\right) \]

Alternative 4: 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 92.0%

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

2. Taylor expanded in sinTheta_O around 0 97.3%

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

   1. unpow2 97.3%

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

4. Simplified 97.3%

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

5. Final simplification 97.3%

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

Alternative 5: 97.8% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 92.0%

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

2. Taylor expanded in eta around inf 97.4%

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

   1. associate-*r* 97.4%

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

   2. unpow2 97.4%

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

   3. unpow2 97.4%

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

4. Simplified 97.4%

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

   1. *-un-lft-identity 97.4%

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

   2. associate-/l* 97.9%

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

6. Applied egg-rr 97.9%

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

   1. *-lft-identity 97.9%

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

   2. associate-/r/ 97.9%

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

8. Simplified 97.9%

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

9. Taylor expanded in sinTheta_O around 0 97.3%

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

    1. unpow2 97.3%

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

    2. associate-*r/ 97.8%

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

    3. *-commutative 97.8%

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

    4. associate-*l* 97.8%

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

    5. *-commutative 97.8%

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

    6. associate-*r/ 97.8%

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

11. Simplified 97.8%

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

12. Final simplification 97.8%

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

Alternative 6: 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 92.0%

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

2. Taylor expanded in eta around inf 94.5%

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

3. Final simplification 94.5%

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

Reproduce

herbie shell --seed 2023292 
(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)))))))))