HairBSDF, gamma for a refracted ray

Percentage Accurate: 92.3% → 98.0%

Time: 16.1s

Alternatives: 6

Speedup: 3.1×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 92.3% 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: 98.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \sin^{-1} \left(\frac{h}{eta + -0.5 \cdot \left(\frac{sinTheta_O}{{\left(\sqrt[3]{eta}\right)}^{2}} \cdot \left(sinTheta_O \cdot {\left(e^{0.3333333333333333}\right)}^{\left(-\log eta\right)}\right)\right)}\right) \end{array} \]

FPCore

(FPCore (sinTheta_O h eta)
 :precision binary32
 (asin
  (/
   h
   (+
    eta
    (*
     -0.5
     (*
      (/ sinTheta_O (pow (cbrt eta) 2.0))
      (* sinTheta_O (pow (exp 0.3333333333333333) (- (log eta))))))))))

C

float code(float sinTheta_O, float h, float eta) {
	return asinf((h / (eta + (-0.5f * ((sinTheta_O / powf(cbrtf(eta), 2.0f)) * (sinTheta_O * powf(expf(0.3333333333333333f), -logf(eta))))))));
}

Julia

function code(sinTheta_O, h, eta)
	return asin(Float32(h / Float32(eta + Float32(Float32(-0.5) * Float32(Float32(sinTheta_O / (cbrt(eta) ^ Float32(2.0))) * Float32(sinTheta_O * (exp(Float32(0.3333333333333333)) ^ Float32(-log(eta)))))))))
end

Derivation

1. Initial program 91.5%

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

3. Taylor expanded in sinTheta_O around 0 97.3%

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

   1. unpow2 97.3%

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

   2. add-cbrt-cube 77.3%

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

   3. unpow2 77.3%

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

   4. cbrt-prod 91.0%

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

   5. times-frac 91.6%

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

   6. unpow2 91.6%

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

   7. cbrt-prod 98.1%

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

   8. pow2 98.1%

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

5. Applied egg-rr 98.1%

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

6. Taylor expanded in sinTheta_O around 0 98.1%

   \[\leadsto \sin^{-1} \left(\frac{h}{eta + -0.5 \cdot \left(\frac{sinTheta_O}{{\left(\sqrt[3]{eta}\right)}^{2}} \cdot \color{blue}{\left({\left(\frac{1}{eta}\right)}^{0.3333333333333333} \cdot sinTheta_O\right)}\right)}\right) \]
7. Step-by-step derivation

   1. add-exp-log 98.1%

      \[\leadsto \sin^{-1} \left(\frac{h}{eta + -0.5 \cdot \left(\frac{sinTheta_O}{{\left(\sqrt[3]{eta}\right)}^{2}} \cdot \left(\color{blue}{e^{\log \left({\left(\frac{1}{eta}\right)}^{0.3333333333333333}\right)}} \cdot sinTheta_O\right)\right)}\right) \]

   2. log-pow 98.1%

      \[\leadsto \sin^{-1} \left(\frac{h}{eta + -0.5 \cdot \left(\frac{sinTheta_O}{{\left(\sqrt[3]{eta}\right)}^{2}} \cdot \left(e^{\color{blue}{0.3333333333333333 \cdot \log \left(\frac{1}{eta}\right)}} \cdot sinTheta_O\right)\right)}\right) \]

   3. log-rec 98.1%

      \[\leadsto \sin^{-1} \left(\frac{h}{eta + -0.5 \cdot \left(\frac{sinTheta_O}{{\left(\sqrt[3]{eta}\right)}^{2}} \cdot \left(e^{0.3333333333333333 \cdot \color{blue}{\left(-\log eta\right)}} \cdot sinTheta_O\right)\right)}\right) \]

8. Applied egg-rr 98.1%

   \[\leadsto \sin^{-1} \left(\frac{h}{eta + -0.5 \cdot \left(\frac{sinTheta_O}{{\left(\sqrt[3]{eta}\right)}^{2}} \cdot \left(\color{blue}{e^{0.3333333333333333 \cdot \left(-\log eta\right)}} \cdot sinTheta_O\right)\right)}\right) \]
9. Step-by-step derivation

   1. exp-prod 98.1%

      \[\leadsto \sin^{-1} \left(\frac{h}{eta + -0.5 \cdot \left(\frac{sinTheta_O}{{\left(\sqrt[3]{eta}\right)}^{2}} \cdot \left(\color{blue}{{\left(e^{0.3333333333333333}\right)}^{\left(-\log eta\right)}} \cdot sinTheta_O\right)\right)}\right) \]

10. Simplified 98.1%

    \[\leadsto \sin^{-1} \left(\frac{h}{eta + -0.5 \cdot \left(\frac{sinTheta_O}{{\left(\sqrt[3]{eta}\right)}^{2}} \cdot \left(\color{blue}{{\left(e^{0.3333333333333333}\right)}^{\left(-\log eta\right)}} \cdot sinTheta_O\right)\right)}\right) \]

11. Final simplification 98.1%

    \[\leadsto \sin^{-1} \left(\frac{h}{eta + -0.5 \cdot \left(\frac{sinTheta_O}{{\left(\sqrt[3]{eta}\right)}^{2}} \cdot \left(sinTheta_O \cdot {\left(e^{0.3333333333333333}\right)}^{\left(-\log eta\right)}\right)\right)}\right) \]
12. Add Preprocessing

Alternative 2: 98.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \sin^{-1} \left(\frac{h}{eta + -0.5 \cdot \left(\frac{sinTheta_O}{{\left(\sqrt[3]{eta}\right)}^{2}} \cdot \left(sinTheta_O \cdot {\left(\frac{1}{eta}\right)}^{0.3333333333333333}\right)\right)}\right) \end{array} \]

FPCore

(FPCore (sinTheta_O h eta)
 :precision binary32
 (asin
  (/
   h
   (+
    eta
    (*
     -0.5
     (*
      (/ sinTheta_O (pow (cbrt eta) 2.0))
      (* sinTheta_O (pow (/ 1.0 eta) 0.3333333333333333))))))))

C

float code(float sinTheta_O, float h, float eta) {
	return asinf((h / (eta + (-0.5f * ((sinTheta_O / powf(cbrtf(eta), 2.0f)) * (sinTheta_O * powf((1.0f / eta), 0.3333333333333333f)))))));
}

Julia

function code(sinTheta_O, h, eta)
	return asin(Float32(h / Float32(eta + Float32(Float32(-0.5) * Float32(Float32(sinTheta_O / (cbrt(eta) ^ Float32(2.0))) * Float32(sinTheta_O * (Float32(Float32(1.0) / eta) ^ Float32(0.3333333333333333))))))))
end

Derivation

1. Initial program 91.5%

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

3. Taylor expanded in sinTheta_O around 0 97.3%

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

   1. unpow2 97.3%

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

   2. add-cbrt-cube 77.3%

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

   3. unpow2 77.3%

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

   4. cbrt-prod 91.0%

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

   5. times-frac 91.6%

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

   6. unpow2 91.6%

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

   7. cbrt-prod 98.1%

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

   8. pow2 98.1%

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

5. Applied egg-rr 98.1%

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

6. Taylor expanded in sinTheta_O around 0 98.1%

   \[\leadsto \sin^{-1} \left(\frac{h}{eta + -0.5 \cdot \left(\frac{sinTheta_O}{{\left(\sqrt[3]{eta}\right)}^{2}} \cdot \color{blue}{\left({\left(\frac{1}{eta}\right)}^{0.3333333333333333} \cdot sinTheta_O\right)}\right)}\right) \]

7. Final simplification 98.1%

   \[\leadsto \sin^{-1} \left(\frac{h}{eta + -0.5 \cdot \left(\frac{sinTheta_O}{{\left(\sqrt[3]{eta}\right)}^{2}} \cdot \left(sinTheta_O \cdot {\left(\frac{1}{eta}\right)}^{0.3333333333333333}\right)\right)}\right) \]
8. Add Preprocessing

Alternative 3: 98.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \sin^{-1} \left(\frac{h}{eta + -0.5 \cdot \left(\frac{sinTheta_O}{{\left(\sqrt[3]{eta}\right)}^{2}} \cdot \frac{sinTheta_O}{\sqrt[3]{eta}}\right)}\right) \end{array} \]

FPCore

(FPCore (sinTheta_O h eta)
 :precision binary32
 (asin
  (/
   h
   (+
    eta
    (*
     -0.5
     (* (/ sinTheta_O (pow (cbrt eta) 2.0)) (/ sinTheta_O (cbrt eta))))))))

C

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

Julia

function code(sinTheta_O, h, eta)
	return asin(Float32(h / Float32(eta + Float32(Float32(-0.5) * Float32(Float32(sinTheta_O / (cbrt(eta) ^ Float32(2.0))) * Float32(sinTheta_O / cbrt(eta)))))))
end

Derivation

1. Initial program 91.5%

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

3. Taylor expanded in sinTheta_O around 0 97.3%

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

   1. unpow2 97.3%

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

   2. add-cbrt-cube 77.3%

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

   3. unpow2 77.3%

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

   4. cbrt-prod 91.0%

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

   5. times-frac 91.6%

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

   6. unpow2 91.6%

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

   7. cbrt-prod 98.1%

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

   8. pow2 98.1%

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

5. Applied egg-rr 98.1%

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

6. Final simplification 98.1%

   \[\leadsto \sin^{-1} \left(\frac{h}{eta + -0.5 \cdot \left(\frac{sinTheta_O}{{\left(\sqrt[3]{eta}\right)}^{2}} \cdot \frac{sinTheta_O}{\sqrt[3]{eta}}\right)}\right) \]
7. Add Preprocessing

Alternative 4: 98.0% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

MATLAB

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

Derivation

1. Initial program 91.5%

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

3. Taylor expanded in sinTheta_O around 0 97.3%

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

   1. unpow2 97.3%

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

   2. add-cbrt-cube 77.3%

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

   3. unpow2 77.3%

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

   4. cbrt-prod 91.0%

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

   5. times-frac 91.6%

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

   6. unpow2 91.6%

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

   7. cbrt-prod 98.1%

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

   8. pow2 98.1%

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

5. Applied egg-rr 98.1%

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

   1. *-commutative 98.1%

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

   2. clear-num 98.1%

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

   3. clear-num 98.1%

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

   4. frac-times 98.1%

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

   5. metadata-eval 98.1%

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

7. Applied egg-rr 98.1%

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

   1. associate-*l/ 98.1%

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

   2. *-commutative 98.1%

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

   3. associate-*l/ 98.1%

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

   4. unpow2 98.1%

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

   5. rem-3cbrt-lft 98.1%

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

9. Simplified 98.1%

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

10. Final simplification 98.1%

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

Alternative 5: 98.0% 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 91.5%

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

3. Taylor expanded in sinTheta_O around 0 97.3%

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

   1. unpow2 97.3%

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

   2. *-un-lft-identity 97.3%

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

   3. times-frac 98.1%

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

5. Applied egg-rr 98.1%

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

6. Final simplification 98.1%

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

Alternative 6: 95.5% 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 91.5%

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

3. Taylor expanded in eta around inf 95.6%

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

4. Final simplification 95.6%

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

Reproduce

herbie shell --seed 2024019 
(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)))))))))