HairBSDF, gamma for a refracted ray

Percentage Accurate: 91.8% → 97.8%

Time: 14.2s

Alternatives: 9

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: 97.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \sin^{-1} \left(\frac{h}{eta + -0.5 \cdot \left({\left(\mathsf{expm1}\left(\mathsf{log1p}\left(e\right)\right)\right)}^{\log \left(sinTheta_O \cdot \frac{sinTheta_O}{eta}\right)} \cdot \sqrt{\frac{1}{1 - sinTheta_O \cdot sinTheta_O}}\right)}\right) \end{array} \]

FPCore

(FPCore (sinTheta_O h eta)
 :precision binary32
 (asin
  (/
   h
   (+
    eta
    (*
     -0.5
     (*
      (pow (expm1 (log1p E)) (log (* sinTheta_O (/ sinTheta_O eta))))
      (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 * (powf(expm1f(log1pf(((float) M_E))), logf((sinTheta_O * (sinTheta_O / eta)))) * sqrtf((1.0f / (1.0f - (sinTheta_O * sinTheta_O)))))))));
}

Julia

function code(sinTheta_O, h, eta)
	return asin(Float32(h / Float32(eta + Float32(Float32(-0.5) * Float32((expm1(log1p(Float32(exp(1)))) ^ log(Float32(sinTheta_O * Float32(sinTheta_O / eta)))) * sqrt(Float32(Float32(1.0) / Float32(Float32(1.0) - Float32(sinTheta_O * sinTheta_O)))))))))
end

Derivation

1. Initial program 91.1%

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

2. Taylor expanded in eta around inf 96.7%

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

   1. unpow2 96.7%

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

   2. unpow2 96.7%

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

4. Simplified 96.7%

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

   1. add-exp-log 96.7%

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

   2. associate-/l* 97.4%

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

6. Applied egg-rr 97.4%

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

   1. *-un-lft-identity 97.4%

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

   2. exp-prod 97.4%

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

   3. exp-1-e 97.4%

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

   4. associate-/r/ 97.4%

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

8. Applied egg-rr 97.4%

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

   1. expm1-log1p-u 97.4%

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

10. Applied egg-rr 97.4%

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

11. Final simplification 97.4%

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

Alternative 2: 97.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \sin^{-1} \left(\frac{h}{eta + -0.5 \cdot \left(\sqrt{\frac{1}{1 - sinTheta_O \cdot sinTheta_O}} \cdot {\left(\sqrt{e}\right)}^{\left(\log \left(sinTheta_O \cdot \frac{sinTheta_O}{eta}\right) \cdot 2\right)}\right)}\right) \end{array} \]

FPCore

(FPCore (sinTheta_O h eta)
 :precision binary32
 (asin
  (/
   h
   (+
    eta
    (*
     -0.5
     (*
      (sqrt (/ 1.0 (- 1.0 (* sinTheta_O sinTheta_O))))
      (pow (sqrt E) (* (log (* sinTheta_O (/ sinTheta_O eta))) 2.0))))))))

C

float code(float sinTheta_O, float h, float eta) {
	return asinf((h / (eta + (-0.5f * (sqrtf((1.0f / (1.0f - (sinTheta_O * sinTheta_O)))) * powf(sqrtf(((float) M_E)), (logf((sinTheta_O * (sinTheta_O / eta))) * 2.0f)))))));
}

Julia

function code(sinTheta_O, h, eta)
	return asin(Float32(h / Float32(eta + Float32(Float32(-0.5) * Float32(sqrt(Float32(Float32(1.0) / Float32(Float32(1.0) - Float32(sinTheta_O * sinTheta_O)))) * (sqrt(Float32(exp(1))) ^ Float32(log(Float32(sinTheta_O * Float32(sinTheta_O / eta))) * Float32(2.0))))))))
end

MATLAB

function tmp = code(sinTheta_O, h, eta)
	tmp = asin((h / (eta + (single(-0.5) * (sqrt((single(1.0) / (single(1.0) - (sinTheta_O * sinTheta_O)))) * (sqrt(single(2.71828182845904523536)) ^ (log((sinTheta_O * (sinTheta_O / eta))) * single(2.0))))))));
end

Derivation

1. Initial program 91.1%

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

2. Taylor expanded in eta around inf 96.7%

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

   1. unpow2 96.7%

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

   2. unpow2 96.7%

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

4. Simplified 96.7%

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

   1. add-exp-log 96.7%

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

   2. associate-/l* 97.4%

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

6. Applied egg-rr 97.4%

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

   1. *-un-lft-identity 97.4%

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

   2. exp-prod 97.4%

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

   3. exp-1-e 97.4%

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

   4. associate-/r/ 97.4%

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

8. Applied egg-rr 97.4%

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

   1. add-sqr-sqrt 97.4%

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

   2. unpow-prod-down 97.4%

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

   3. associate-*l/ 96.7%

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

   4. associate-*l/ 96.7%

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

10. Applied egg-rr 96.7%

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

    1. pow-sqr 96.7%

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

    2. associate-*r/ 97.4%

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

12. Simplified 97.4%

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

13. Final simplification 97.4%

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

Alternative 3: 49.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \sin^{-1} \left(\frac{h}{eta + -0.5 \cdot \left(\sqrt{\frac{1}{1 - sinTheta_O \cdot sinTheta_O}} \cdot {e}^{\left(2 \cdot \log sinTheta_O - \log eta\right)}\right)}\right) \end{array} \]

FPCore

(FPCore (sinTheta_O h eta)
 :precision binary32
 (asin
  (/
   h
   (+
    eta
    (*
     -0.5
     (*
      (sqrt (/ 1.0 (- 1.0 (* sinTheta_O sinTheta_O))))
      (pow E (- (* 2.0 (log sinTheta_O)) (log eta)))))))))

C

float code(float sinTheta_O, float h, float eta) {
	return asinf((h / (eta + (-0.5f * (sqrtf((1.0f / (1.0f - (sinTheta_O * sinTheta_O)))) * powf(((float) M_E), ((2.0f * logf(sinTheta_O)) - logf(eta))))))));
}

Julia

function code(sinTheta_O, h, eta)
	return asin(Float32(h / Float32(eta + Float32(Float32(-0.5) * Float32(sqrt(Float32(Float32(1.0) / Float32(Float32(1.0) - Float32(sinTheta_O * sinTheta_O)))) * (Float32(exp(1)) ^ Float32(Float32(Float32(2.0) * log(sinTheta_O)) - log(eta))))))))
end

MATLAB

function tmp = code(sinTheta_O, h, eta)
	tmp = asin((h / (eta + (single(-0.5) * (sqrt((single(1.0) / (single(1.0) - (sinTheta_O * sinTheta_O)))) * (single(2.71828182845904523536) ^ ((single(2.0) * log(sinTheta_O)) - log(eta))))))));
end

Derivation

1. Initial program 91.1%

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

2. Taylor expanded in eta around inf 96.7%

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

   1. unpow2 96.7%

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

   2. unpow2 96.7%

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

4. Simplified 96.7%

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

   1. add-exp-log 96.7%

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

   2. associate-/l* 97.4%

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

6. Applied egg-rr 97.4%

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

   1. *-un-lft-identity 97.4%

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

   2. exp-prod 97.4%

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

   3. exp-1-e 97.4%

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

   4. associate-/r/ 97.4%

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

8. Applied egg-rr 97.4%

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

   1. associate-*l/ 96.7%

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

   2. log-div 96.7%

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

   3. pow2 96.7%

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

   4. log-pow 45.3%

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

10. Applied egg-rr 45.3%

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

11. Final simplification 45.3%

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

Alternative 4: 49.0% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 91.1%

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

2. Taylor expanded in eta around inf 96.7%

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

   1. unpow2 96.7%

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

   2. unpow2 96.7%

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

4. Simplified 96.7%

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

   1. add-exp-log 96.7%

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

   2. associate-/l* 97.4%

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

6. Applied egg-rr 97.4%

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

   1. log-div 45.3%

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

8. Applied egg-rr 45.3%

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

9. Final simplification 45.3%

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

Alternative 5: 97.8% accurate, 0.6× speedup

Math

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

FPCore

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

C

float code(float sinTheta_O, float h, float eta) {
	return asinf((h / (eta + (-0.5f * (sqrtf((1.0f / (1.0f - (sinTheta_O * sinTheta_O)))) * powf(((float) M_E), logf((sinTheta_O * (sinTheta_O / eta)))))))));
}

Julia

function code(sinTheta_O, h, eta)
	return asin(Float32(h / Float32(eta + Float32(Float32(-0.5) * Float32(sqrt(Float32(Float32(1.0) / Float32(Float32(1.0) - Float32(sinTheta_O * sinTheta_O)))) * (Float32(exp(1)) ^ log(Float32(sinTheta_O * Float32(sinTheta_O / eta)))))))))
end

MATLAB

function tmp = code(sinTheta_O, h, eta)
	tmp = asin((h / (eta + (single(-0.5) * (sqrt((single(1.0) / (single(1.0) - (sinTheta_O * sinTheta_O)))) * (single(2.71828182845904523536) ^ log((sinTheta_O * (sinTheta_O / eta)))))))));
end

Derivation

1. Initial program 91.1%

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

2. Taylor expanded in eta around inf 96.7%

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

   1. unpow2 96.7%

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

   2. unpow2 96.7%

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

4. Simplified 96.7%

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

   1. add-exp-log 96.7%

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

   2. associate-/l* 97.4%

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

6. Applied egg-rr 97.4%

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

   1. *-un-lft-identity 97.4%

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

   2. exp-prod 97.4%

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

   3. exp-1-e 97.4%

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

   4. associate-/r/ 97.4%

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

8. Applied egg-rr 97.4%

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

9. Final simplification 97.4%

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

Alternative 6: 97.8% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (sinTheta_O h eta)
 :precision binary32
 (asin
  (/
   h
   (+
    eta
    (*
     -0.5
     (exp
      (log
       (/
        sinTheta_O
        (* (/ eta sinTheta_O) (sqrt (- 1.0 (* 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 - (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 - (sintheta_o * sintheta_o)))))))))))
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(Float32(eta / 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) * exp(log((sinTheta_O / ((eta / sinTheta_O) * sqrt((single(1.0) - (sinTheta_O * sinTheta_O)))))))))));
end

Derivation

1. Initial program 91.1%

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

2. Taylor expanded in eta around inf 96.7%

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

   1. unpow2 96.7%

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

   2. unpow2 96.7%

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

4. Simplified 96.7%

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

   1. expm1-log1p-u 96.7%

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

   2. expm1-udef 24.3%

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

   3. associate-/l* 24.4%

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

   4. sqrt-div 24.4%

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

   5. metadata-eval 24.4%

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

6. Applied egg-rr 24.4%

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

   1. expm1-def 97.4%

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

   2. expm1-log1p 97.4%

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

   3. associate-*r/ 97.4%

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

   4. *-rgt-identity 97.4%

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

   5. associate-/r/ 97.4%

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

8. Simplified 97.4%

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

   1. associate-/r/ 97.4%

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

   2. un-div-inv 97.4%

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

   3. add-exp-log 97.4%

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

   4. associate-/r/ 97.4%

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

   5. *-commutative 97.4%

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

   6. associate-/r/ 97.4%

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

   7. frac-times 97.4%

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

   8. *-un-lft-identity 97.4%

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

10. Applied egg-rr 97.4%

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

11. Final simplification 97.4%

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

Alternative 7: 97.8% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (sinTheta_O h eta)
 :precision binary32
 (asin
  (/
   h
   (+
    eta
    (*
     -0.5
     (/
      (* sinTheta_O (/ sinTheta_O eta))
      (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 * (sinTheta_O / eta)) / 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 * (sintheta_o / eta)) / sqrt((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(Float32(sinTheta_O * Float32(sinTheta_O / eta)) / 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 * (sinTheta_O / eta)) / sqrt((single(1.0) - (sinTheta_O * sinTheta_O))))))));
end

Derivation

1. Initial program 91.1%

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

2. Taylor expanded in eta around inf 96.7%

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

   1. unpow2 96.7%

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

   2. unpow2 96.7%

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

4. Simplified 96.7%

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

   1. expm1-log1p-u 96.7%

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

   2. expm1-udef 24.3%

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

   3. associate-/l* 24.4%

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

   4. sqrt-div 24.4%

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

   5. metadata-eval 24.4%

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

6. Applied egg-rr 24.4%

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

   1. expm1-def 97.4%

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

   2. expm1-log1p 97.4%

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

   3. associate-*r/ 97.4%

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

   4. *-rgt-identity 97.4%

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

   5. associate-/r/ 97.4%

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

8. Simplified 97.4%

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

9. Final simplification 97.4%

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

Alternative 8: 97.8% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 91.1%

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

2. Taylor expanded in eta around inf 96.7%

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

   1. unpow2 96.7%

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

   2. unpow2 96.7%

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

4. Simplified 96.7%

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

   1. expm1-log1p-u 96.7%

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

   2. expm1-udef 24.3%

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

   3. associate-/l* 24.4%

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

   4. sqrt-div 24.4%

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

   5. metadata-eval 24.4%

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

6. Applied egg-rr 24.4%

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

   1. expm1-def 97.4%

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

   2. expm1-log1p 97.4%

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

   3. associate-*r/ 97.4%

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

   4. *-rgt-identity 97.4%

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

   5. associate-/r/ 97.4%

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

8. Simplified 97.4%

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

9. Taylor expanded in sinTheta_O around 0 96.7%

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

    1. *-commutative 96.7%

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

    2. unpow2 96.7%

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

    3. associate-*r/ 97.4%

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

    4. associate-*l* 97.4%

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

11. Simplified 97.4%

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

12. Final simplification 97.4%

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

Alternative 9: 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 91.1%

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

2. Taylor expanded in eta around inf 94.8%

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

3. Final simplification 94.8%

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

Reproduce

herbie shell --seed 2023240 
(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)))))))))