HairBSDF, gamma for a refracted ray

Percentage Accurate: 91.9% → 98.4%

Time: 36.8s

Alternatives: 5

Speedup: 3.1×

Specification

\[\left(\left(-1 \leq sinTheta_O \land sinTheta_O \leq 1\right) \land \left(-1 \leq h \land h \leq 1\right)\right) \land \left(0 \leq eta \land eta \leq 10\right)\]

Math

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

FPCore

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

C

float code(float sinTheta_O, float h, float eta) {
	return asinf((h / sqrtf(((eta * eta) - ((sinTheta_O * sinTheta_O) / sqrtf((1.0f - (sinTheta_O * sinTheta_O))))))));
}

Fortran

real(4) function code(sintheta_o, h, eta)
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: h
    real(4), intent (in) :: eta
    code = asin((h / sqrt(((eta * eta) - ((sintheta_o * sintheta_o) / sqrt((1.0e0 - (sintheta_o * sintheta_o))))))))
end function

Julia

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

MATLAB

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

Initial Program: 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: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float sinTheta_O, float h, float eta) {
	return asinf((h / (sqrtf((eta + sinTheta_O)) * sqrtf((eta - sinTheta_O)))));
}

Fortran

real(4) function code(sintheta_o, h, eta)
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: h
    real(4), intent (in) :: eta
    code = asin((h / (sqrt((eta + sintheta_o)) * sqrt((eta - sintheta_o)))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 96.5%

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

2. Taylor expanded in sinTheta_O around 0 96.0%

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

   1. mul-1-neg 96.0%

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

   2. unpow2 96.0%

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

   3. distribute-rgt-neg-out 96.0%

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

   4. unpow2 96.0%

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

   5. +-commutative 96.0%

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

   6. distribute-rgt-neg-out 96.0%

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

   7. unpow2 96.0%

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

   8. unsub-neg 96.0%

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

   9. unpow2 96.0%

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

4. Simplified 96.0%

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

   1. pow1/2 96.0%

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

   2. difference-of-squares 96.0%

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

   3. unpow-prod-down 98.1%

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

6. Applied egg-rr 98.1%

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

   1. unpow1/2 98.1%

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

   2. unpow1/2 98.1%

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

8. Simplified 98.1%

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

9. Final simplification 98.1%

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

Alternative 2: 97.8% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

MATLAB

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

Derivation

1. Initial program 96.5%

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

2. Taylor expanded in eta around inf 97.7%

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

   1. unpow2 97.7%

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

   2. unpow2 97.7%

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

4. Simplified 97.7%

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

   1. div-inv 97.1%

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

   2. associate-*l/ 97.1%

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

   3. sqrt-div 97.1%

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

   4. metadata-eval 97.1%

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

   5. div-inv 97.1%

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

   6. associate-/l* 97.1%

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

6. Applied egg-rr 97.1%

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

   1. associate-*r/ 97.7%

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

   2. *-rgt-identity 97.7%

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

   3. associate-/l/ 97.9%

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

8. Simplified 97.9%

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

9. Final simplification 97.9%

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

Alternative 3: 97.8% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

MATLAB

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

Derivation

1. Initial program 96.5%

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

2. Taylor expanded in eta around inf 97.7%

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

   1. unpow2 97.7%

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

   2. unpow2 97.7%

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

4. Simplified 97.7%

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

   1. div-inv 97.1%

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

   2. associate-*l/ 97.1%

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

   3. sqrt-div 97.1%

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

   4. metadata-eval 97.1%

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

   5. div-inv 97.1%

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

   6. associate-/l* 97.1%

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

6. Applied egg-rr 97.1%

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

   1. associate-*r/ 97.7%

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

   2. *-rgt-identity 97.7%

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

   3. associate-/l/ 97.9%

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

8. Simplified 97.9%

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

9. Taylor expanded in sinTheta_O around 0 97.9%

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

10. Final simplification 97.9%

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

Alternative 4: 97.8% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(sintheta_o, h, eta)
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: h
    real(4), intent (in) :: eta
    code = asin((h / (eta + ((-0.5e0) * (sintheta_o * (sintheta_o / eta))))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 96.5%

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

2. Taylor expanded in sinTheta_O around 0 97.7%

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

   1. associate-*r/ 97.7%

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

   2. unpow2 97.7%

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

4. Simplified 97.7%

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

5. Taylor expanded in sinTheta_O around 0 97.7%

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

   1. unpow2 97.7%

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

   2. associate-*l/ 97.9%

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

   3. *-commutative 97.9%

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

7. Simplified 97.9%

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

8. Final simplification 97.9%

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

Alternative 5: 95.2% accurate, 3.1× speedup

Math

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

FPCore

(FPCore (sinTheta_O h eta) :precision binary32 (asin (/ h eta)))

C

float code(float sinTheta_O, float h, float eta) {
	return asinf((h / eta));
}

Fortran

real(4) function code(sintheta_o, h, eta)
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: h
    real(4), intent (in) :: eta
    code = asin((h / eta))
end function

Julia

function code(sinTheta_O, h, eta)
	return asin(Float32(h / eta))
end

MATLAB

function tmp = code(sinTheta_O, h, eta)
	tmp = asin((h / eta));
end

Derivation

1. Initial program 96.5%

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

2. Taylor expanded in eta around inf 94.7%

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

3. Final simplification 94.7%

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

Reproduce

herbie shell --seed 2023299 
(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)))))))))