HairBSDF, gamma for a refracted ray

Percentage Accurate: 92.4% → 97.9%

Time: 12.5s

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: 92.4% 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, 0.8× speedup

Math

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

FPCore

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

C

float code(float sinTheta_O, float h, float eta) {
	return asinf((h / (eta + (-0.5f * (expf(logf((sinTheta_O / (eta / sinTheta_O)))) * sqrtf((1.0f / (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) * (exp(log((sintheta_o / (eta / sintheta_o)))) * sqrt((1.0e0 / (1.0e0 - (sintheta_o * sintheta_o)))))))))
end function

Julia

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

Derivation

1. Initial program 92.9%

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

2. Taylor expanded in eta around inf 97.8%

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

   1. unpow2 97.8%

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

   2. unpow2 97.8%

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

4. Simplified 97.8%

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

   1. add-exp-log 97.8%

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

   2. associate-/l* 98.3%

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

6. Applied egg-rr 98.3%

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

7. Final simplification 98.3%

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

Alternative 2: 97.9% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 92.9%

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

2. Taylor expanded in eta around inf 97.8%

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

   1. unpow2 97.8%

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

   2. unpow2 97.8%

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

4. Simplified 97.8%

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

   1. expm1-log1p-u 97.8%

      \[\leadsto \sin^{-1} \left(\frac{h}{eta + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-0.5 \cdot \left(\frac{sinTheta_O \cdot sinTheta_O}{eta} \cdot \sqrt{\frac{1}{1 - sinTheta_O \cdot sinTheta_O}}\right)\right)\right)}}\right) \]

   2. expm1-udef 96.2%

      \[\leadsto \sin^{-1} \left(\frac{h}{eta + \color{blue}{\left(e^{\mathsf{log1p}\left(-0.5 \cdot \left(\frac{sinTheta_O \cdot sinTheta_O}{eta} \cdot \sqrt{\frac{1}{1 - sinTheta_O \cdot sinTheta_O}}\right)\right)} - 1\right)}}\right) \]

   3. associate-/l* 96.2%

      \[\leadsto \sin^{-1} \left(\frac{h}{eta + \left(e^{\mathsf{log1p}\left(-0.5 \cdot \left(\color{blue}{\frac{sinTheta_O}{\frac{eta}{sinTheta_O}}} \cdot \sqrt{\frac{1}{1 - sinTheta_O \cdot sinTheta_O}}\right)\right)} - 1\right)}\right) \]

   4. pow1/2 96.2%

      \[\leadsto \sin^{-1} \left(\frac{h}{eta + \left(e^{\mathsf{log1p}\left(-0.5 \cdot \left(\frac{sinTheta_O}{\frac{eta}{sinTheta_O}} \cdot \color{blue}{{\left(\frac{1}{1 - sinTheta_O \cdot sinTheta_O}\right)}^{0.5}}\right)\right)} - 1\right)}\right) \]

   5. inv-pow 96.2%

      \[\leadsto \sin^{-1} \left(\frac{h}{eta + \left(e^{\mathsf{log1p}\left(-0.5 \cdot \left(\frac{sinTheta_O}{\frac{eta}{sinTheta_O}} \cdot {\color{blue}{\left({\left(1 - sinTheta_O \cdot sinTheta_O\right)}^{-1}\right)}}^{0.5}\right)\right)} - 1\right)}\right) \]

   6. pow-pow 96.2%

      \[\leadsto \sin^{-1} \left(\frac{h}{eta + \left(e^{\mathsf{log1p}\left(-0.5 \cdot \left(\frac{sinTheta_O}{\frac{eta}{sinTheta_O}} \cdot \color{blue}{{\left(1 - sinTheta_O \cdot sinTheta_O\right)}^{\left(-1 \cdot 0.5\right)}}\right)\right)} - 1\right)}\right) \]

   7. metadata-eval 96.2%

      \[\leadsto \sin^{-1} \left(\frac{h}{eta + \left(e^{\mathsf{log1p}\left(-0.5 \cdot \left(\frac{sinTheta_O}{\frac{eta}{sinTheta_O}} \cdot {\left(1 - sinTheta_O \cdot sinTheta_O\right)}^{\color{blue}{-0.5}}\right)\right)} - 1\right)}\right) \]

6. Applied egg-rr 96.2%

   \[\leadsto \sin^{-1} \left(\frac{h}{eta + \color{blue}{\left(e^{\mathsf{log1p}\left(-0.5 \cdot \left(\frac{sinTheta_O}{\frac{eta}{sinTheta_O}} \cdot {\left(1 - sinTheta_O \cdot sinTheta_O\right)}^{-0.5}\right)\right)} - 1\right)}}\right) \]
7. Step-by-step derivation

   1. expm1-def 98.3%

      \[\leadsto \sin^{-1} \left(\frac{h}{eta + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-0.5 \cdot \left(\frac{sinTheta_O}{\frac{eta}{sinTheta_O}} \cdot {\left(1 - sinTheta_O \cdot sinTheta_O\right)}^{-0.5}\right)\right)\right)}}\right) \]

   2. expm1-log1p 98.3%

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

   3. associate-*r* 98.3%

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

   4. associate-/r/ 98.3%

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

8. Simplified 98.3%

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

   1. add-exp-log 98.3%

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

   2. *-commutative 98.3%

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

10. Applied egg-rr 98.3%

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

11. Final simplification 98.3%

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

Alternative 3: 97.9% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 92.9%

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

2. Taylor expanded in eta around inf 97.8%

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

   1. unpow2 97.8%

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

   2. unpow2 97.8%

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

4. Simplified 97.8%

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

   1. expm1-log1p-u 97.8%

      \[\leadsto \sin^{-1} \left(\frac{h}{eta + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-0.5 \cdot \left(\frac{sinTheta_O \cdot sinTheta_O}{eta} \cdot \sqrt{\frac{1}{1 - sinTheta_O \cdot sinTheta_O}}\right)\right)\right)}}\right) \]

   2. expm1-udef 96.2%

      \[\leadsto \sin^{-1} \left(\frac{h}{eta + \color{blue}{\left(e^{\mathsf{log1p}\left(-0.5 \cdot \left(\frac{sinTheta_O \cdot sinTheta_O}{eta} \cdot \sqrt{\frac{1}{1 - sinTheta_O \cdot sinTheta_O}}\right)\right)} - 1\right)}}\right) \]

   3. associate-/l* 96.2%

      \[\leadsto \sin^{-1} \left(\frac{h}{eta + \left(e^{\mathsf{log1p}\left(-0.5 \cdot \left(\color{blue}{\frac{sinTheta_O}{\frac{eta}{sinTheta_O}}} \cdot \sqrt{\frac{1}{1 - sinTheta_O \cdot sinTheta_O}}\right)\right)} - 1\right)}\right) \]

   4. pow1/2 96.2%

      \[\leadsto \sin^{-1} \left(\frac{h}{eta + \left(e^{\mathsf{log1p}\left(-0.5 \cdot \left(\frac{sinTheta_O}{\frac{eta}{sinTheta_O}} \cdot \color{blue}{{\left(\frac{1}{1 - sinTheta_O \cdot sinTheta_O}\right)}^{0.5}}\right)\right)} - 1\right)}\right) \]

   5. inv-pow 96.2%

      \[\leadsto \sin^{-1} \left(\frac{h}{eta + \left(e^{\mathsf{log1p}\left(-0.5 \cdot \left(\frac{sinTheta_O}{\frac{eta}{sinTheta_O}} \cdot {\color{blue}{\left({\left(1 - sinTheta_O \cdot sinTheta_O\right)}^{-1}\right)}}^{0.5}\right)\right)} - 1\right)}\right) \]

   6. pow-pow 96.2%

      \[\leadsto \sin^{-1} \left(\frac{h}{eta + \left(e^{\mathsf{log1p}\left(-0.5 \cdot \left(\frac{sinTheta_O}{\frac{eta}{sinTheta_O}} \cdot \color{blue}{{\left(1 - sinTheta_O \cdot sinTheta_O\right)}^{\left(-1 \cdot 0.5\right)}}\right)\right)} - 1\right)}\right) \]

   7. metadata-eval 96.2%

      \[\leadsto \sin^{-1} \left(\frac{h}{eta + \left(e^{\mathsf{log1p}\left(-0.5 \cdot \left(\frac{sinTheta_O}{\frac{eta}{sinTheta_O}} \cdot {\left(1 - sinTheta_O \cdot sinTheta_O\right)}^{\color{blue}{-0.5}}\right)\right)} - 1\right)}\right) \]

6. Applied egg-rr 96.2%

   \[\leadsto \sin^{-1} \left(\frac{h}{eta + \color{blue}{\left(e^{\mathsf{log1p}\left(-0.5 \cdot \left(\frac{sinTheta_O}{\frac{eta}{sinTheta_O}} \cdot {\left(1 - sinTheta_O \cdot sinTheta_O\right)}^{-0.5}\right)\right)} - 1\right)}}\right) \]
7. Step-by-step derivation

   1. expm1-def 98.3%

      \[\leadsto \sin^{-1} \left(\frac{h}{eta + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-0.5 \cdot \left(\frac{sinTheta_O}{\frac{eta}{sinTheta_O}} \cdot {\left(1 - sinTheta_O \cdot sinTheta_O\right)}^{-0.5}\right)\right)\right)}}\right) \]

   2. expm1-log1p 98.3%

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

   3. associate-*r* 98.3%

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

   4. associate-/r/ 98.3%

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

8. Simplified 98.3%

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

9. Final simplification 98.3%

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

Alternative 4: 97.9% 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 92.9%

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

2. Taylor expanded in eta around inf 97.8%

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

   1. unpow2 97.8%

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

   2. unpow2 97.8%

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

4. Simplified 97.8%

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

5. 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/ 98.2%

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

7. Simplified 98.2%

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

8. Final simplification 98.2%

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

Alternative 5: 95.3% 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 92.9%

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

2. Taylor expanded in eta around inf 95.6%

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

3. Final simplification 95.6%

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

Reproduce

herbie shell --seed 2023182 
(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)))))))))