HairBSDF, gamma for a refracted ray

Percentage Accurate: 92.0% → 98.3%

Time: 15.4s

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: 92.0% 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: 98.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \sin^{-1} \left(\frac{h}{\sqrt{eta - sinTheta_O \cdot {\left(\mathsf{fma}\left(sinTheta_O, sinTheta_O, 1\right)\right)}^{-0.25}} \cdot \sqrt{eta + sinTheta_O}}\right) \end{array} \]

FPCore

(FPCore (sinTheta_O h eta)
 :precision binary32
 (asin
  (/
   h
   (*
    (sqrt (- eta (* sinTheta_O (pow (fma sinTheta_O sinTheta_O 1.0) -0.25))))
    (sqrt (+ eta sinTheta_O))))))

C

float code(float sinTheta_O, float h, float eta) {
	return asinf((h / (sqrtf((eta - (sinTheta_O * powf(fmaf(sinTheta_O, sinTheta_O, 1.0f), -0.25f)))) * sqrtf((eta + sinTheta_O)))));
}

Julia

function code(sinTheta_O, h, eta)
	return asin(Float32(h / Float32(sqrt(Float32(eta - Float32(sinTheta_O * (fma(sinTheta_O, sinTheta_O, Float32(1.0)) ^ Float32(-0.25))))) * sqrt(Float32(eta + 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. Step-by-step derivation

   1. add-sqr-sqrt 92.9%

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

   2. difference-of-squares 93.0%

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

3. Applied egg-rr 93.0%

   \[\leadsto \sin^{-1} \left(\frac{h}{\sqrt{\color{blue}{\left(eta + \frac{sinTheta_O}{{\left({sinTheta_O}^{2} + 1\right)}^{0.25}}\right) \cdot \left(eta - \frac{sinTheta_O}{{\left({sinTheta_O}^{2} + 1\right)}^{0.25}}\right)}}}\right) \]
4. Step-by-step derivation

   1. unpow2 93.0%

      \[\leadsto \sin^{-1} \left(\frac{h}{\sqrt{\left(eta + \frac{sinTheta_O}{{\left(\color{blue}{sinTheta_O \cdot sinTheta_O} + 1\right)}^{0.25}}\right) \cdot \left(eta - \frac{sinTheta_O}{{\left({sinTheta_O}^{2} + 1\right)}^{0.25}}\right)}}\right) \]

   2. fma-def 93.0%

      \[\leadsto \sin^{-1} \left(\frac{h}{\sqrt{\left(eta + \frac{sinTheta_O}{{\color{blue}{\left(\mathsf{fma}\left(sinTheta_O, sinTheta_O, 1\right)\right)}}^{0.25}}\right) \cdot \left(eta - \frac{sinTheta_O}{{\left({sinTheta_O}^{2} + 1\right)}^{0.25}}\right)}}\right) \]

   3. unpow2 93.0%

      \[\leadsto \sin^{-1} \left(\frac{h}{\sqrt{\left(eta + \frac{sinTheta_O}{{\left(\mathsf{fma}\left(sinTheta_O, sinTheta_O, 1\right)\right)}^{0.25}}\right) \cdot \left(eta - \frac{sinTheta_O}{{\left(\color{blue}{sinTheta_O \cdot sinTheta_O} + 1\right)}^{0.25}}\right)}}\right) \]

   4. fma-def 93.0%

      \[\leadsto \sin^{-1} \left(\frac{h}{\sqrt{\left(eta + \frac{sinTheta_O}{{\left(\mathsf{fma}\left(sinTheta_O, sinTheta_O, 1\right)\right)}^{0.25}}\right) \cdot \left(eta - \frac{sinTheta_O}{{\color{blue}{\left(\mathsf{fma}\left(sinTheta_O, sinTheta_O, 1\right)\right)}}^{0.25}}\right)}}\right) \]

5. Simplified 93.0%

   \[\leadsto \sin^{-1} \left(\frac{h}{\sqrt{\color{blue}{\left(eta + \frac{sinTheta_O}{{\left(\mathsf{fma}\left(sinTheta_O, sinTheta_O, 1\right)\right)}^{0.25}}\right) \cdot \left(eta - \frac{sinTheta_O}{{\left(\mathsf{fma}\left(sinTheta_O, sinTheta_O, 1\right)\right)}^{0.25}}\right)}}}\right) \]

6. Taylor expanded in sinTheta_O around 0 93.0%

   \[\leadsto \sin^{-1} \left(\frac{h}{\sqrt{\color{blue}{\left(eta + sinTheta_O\right)} \cdot \left(eta - \frac{sinTheta_O}{{\left(\mathsf{fma}\left(sinTheta_O, sinTheta_O, 1\right)\right)}^{0.25}}\right)}}\right) \]
7. Step-by-step derivation

   1. +-commutative 93.0%

      \[\leadsto \sin^{-1} \left(\frac{h}{\sqrt{\color{blue}{\left(sinTheta_O + eta\right)} \cdot \left(eta - \frac{sinTheta_O}{{\left(\mathsf{fma}\left(sinTheta_O, sinTheta_O, 1\right)\right)}^{0.25}}\right)}}\right) \]

8. Simplified 93.0%

   \[\leadsto \sin^{-1} \left(\frac{h}{\sqrt{\color{blue}{\left(sinTheta_O + eta\right)} \cdot \left(eta - \frac{sinTheta_O}{{\left(\mathsf{fma}\left(sinTheta_O, sinTheta_O, 1\right)\right)}^{0.25}}\right)}}\right) \]
9. Step-by-step derivation

   1. *-commutative 93.0%

      \[\leadsto \sin^{-1} \left(\frac{h}{\sqrt{\color{blue}{\left(eta - \frac{sinTheta_O}{{\left(\mathsf{fma}\left(sinTheta_O, sinTheta_O, 1\right)\right)}^{0.25}}\right) \cdot \left(sinTheta_O + eta\right)}}}\right) \]

   2. sqrt-prod 98.6%

      \[\leadsto \sin^{-1} \left(\frac{h}{\color{blue}{\sqrt{eta - \frac{sinTheta_O}{{\left(\mathsf{fma}\left(sinTheta_O, sinTheta_O, 1\right)\right)}^{0.25}}} \cdot \sqrt{sinTheta_O + eta}}}\right) \]

   3. div-inv 98.6%

      \[\leadsto \sin^{-1} \left(\frac{h}{\sqrt{eta - \color{blue}{sinTheta_O \cdot \frac{1}{{\left(\mathsf{fma}\left(sinTheta_O, sinTheta_O, 1\right)\right)}^{0.25}}}} \cdot \sqrt{sinTheta_O + eta}}\right) \]

   4. pow-flip 98.6%

      \[\leadsto \sin^{-1} \left(\frac{h}{\sqrt{eta - sinTheta_O \cdot \color{blue}{{\left(\mathsf{fma}\left(sinTheta_O, sinTheta_O, 1\right)\right)}^{\left(-0.25\right)}}} \cdot \sqrt{sinTheta_O + eta}}\right) \]

   5. metadata-eval 98.6%

      \[\leadsto \sin^{-1} \left(\frac{h}{\sqrt{eta - sinTheta_O \cdot {\left(\mathsf{fma}\left(sinTheta_O, sinTheta_O, 1\right)\right)}^{\color{blue}{-0.25}}} \cdot \sqrt{sinTheta_O + eta}}\right) \]

   6. +-commutative 98.6%

      \[\leadsto \sin^{-1} \left(\frac{h}{\sqrt{eta - sinTheta_O \cdot {\left(\mathsf{fma}\left(sinTheta_O, sinTheta_O, 1\right)\right)}^{-0.25}} \cdot \sqrt{\color{blue}{eta + sinTheta_O}}}\right) \]

10. Applied egg-rr 98.6%

    \[\leadsto \sin^{-1} \left(\frac{h}{\color{blue}{\sqrt{eta - sinTheta_O \cdot {\left(\mathsf{fma}\left(sinTheta_O, sinTheta_O, 1\right)\right)}^{-0.25}} \cdot \sqrt{eta + sinTheta_O}}}\right) \]

11. Final simplification 98.6%

    \[\leadsto \sin^{-1} \left(\frac{h}{\sqrt{eta - sinTheta_O \cdot {\left(\mathsf{fma}\left(sinTheta_O, sinTheta_O, 1\right)\right)}^{-0.25}} \cdot \sqrt{eta + sinTheta_O}}\right) \]

Alternative 2: 98.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 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. Step-by-step derivation

   1. add-sqr-sqrt 92.9%

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

   2. difference-of-squares 93.0%

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

3. Applied egg-rr 93.0%

   \[\leadsto \sin^{-1} \left(\frac{h}{\sqrt{\color{blue}{\left(eta + \frac{sinTheta_O}{{\left({sinTheta_O}^{2} + 1\right)}^{0.25}}\right) \cdot \left(eta - \frac{sinTheta_O}{{\left({sinTheta_O}^{2} + 1\right)}^{0.25}}\right)}}}\right) \]
4. Step-by-step derivation

   1. unpow2 93.0%

      \[\leadsto \sin^{-1} \left(\frac{h}{\sqrt{\left(eta + \frac{sinTheta_O}{{\left(\color{blue}{sinTheta_O \cdot sinTheta_O} + 1\right)}^{0.25}}\right) \cdot \left(eta - \frac{sinTheta_O}{{\left({sinTheta_O}^{2} + 1\right)}^{0.25}}\right)}}\right) \]

   2. fma-def 93.0%

      \[\leadsto \sin^{-1} \left(\frac{h}{\sqrt{\left(eta + \frac{sinTheta_O}{{\color{blue}{\left(\mathsf{fma}\left(sinTheta_O, sinTheta_O, 1\right)\right)}}^{0.25}}\right) \cdot \left(eta - \frac{sinTheta_O}{{\left({sinTheta_O}^{2} + 1\right)}^{0.25}}\right)}}\right) \]

   3. unpow2 93.0%

      \[\leadsto \sin^{-1} \left(\frac{h}{\sqrt{\left(eta + \frac{sinTheta_O}{{\left(\mathsf{fma}\left(sinTheta_O, sinTheta_O, 1\right)\right)}^{0.25}}\right) \cdot \left(eta - \frac{sinTheta_O}{{\left(\color{blue}{sinTheta_O \cdot sinTheta_O} + 1\right)}^{0.25}}\right)}}\right) \]

   4. fma-def 93.0%

      \[\leadsto \sin^{-1} \left(\frac{h}{\sqrt{\left(eta + \frac{sinTheta_O}{{\left(\mathsf{fma}\left(sinTheta_O, sinTheta_O, 1\right)\right)}^{0.25}}\right) \cdot \left(eta - \frac{sinTheta_O}{{\color{blue}{\left(\mathsf{fma}\left(sinTheta_O, sinTheta_O, 1\right)\right)}}^{0.25}}\right)}}\right) \]

5. Simplified 93.0%

   \[\leadsto \sin^{-1} \left(\frac{h}{\sqrt{\color{blue}{\left(eta + \frac{sinTheta_O}{{\left(\mathsf{fma}\left(sinTheta_O, sinTheta_O, 1\right)\right)}^{0.25}}\right) \cdot \left(eta - \frac{sinTheta_O}{{\left(\mathsf{fma}\left(sinTheta_O, sinTheta_O, 1\right)\right)}^{0.25}}\right)}}}\right) \]

6. Taylor expanded in sinTheta_O around 0 93.0%

   \[\leadsto \sin^{-1} \left(\frac{h}{\sqrt{\color{blue}{\left(eta + sinTheta_O\right)} \cdot \left(eta - \frac{sinTheta_O}{{\left(\mathsf{fma}\left(sinTheta_O, sinTheta_O, 1\right)\right)}^{0.25}}\right)}}\right) \]
7. Step-by-step derivation

   1. +-commutative 93.0%

      \[\leadsto \sin^{-1} \left(\frac{h}{\sqrt{\color{blue}{\left(sinTheta_O + eta\right)} \cdot \left(eta - \frac{sinTheta_O}{{\left(\mathsf{fma}\left(sinTheta_O, sinTheta_O, 1\right)\right)}^{0.25}}\right)}}\right) \]

8. Simplified 93.0%

   \[\leadsto \sin^{-1} \left(\frac{h}{\sqrt{\color{blue}{\left(sinTheta_O + eta\right)} \cdot \left(eta - \frac{sinTheta_O}{{\left(\mathsf{fma}\left(sinTheta_O, sinTheta_O, 1\right)\right)}^{0.25}}\right)}}\right) \]

9. Taylor expanded in sinTheta_O around 0 93.0%

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

    1. neg-mul-1 93.0%

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

    2. unsub-neg 93.0%

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

11. Simplified 93.0%

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

    1. *-un-lft-identity 93.0%

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

    2. sqrt-prod 98.6%

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

    3. +-commutative 98.6%

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

    4. times-frac 98.1%

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

13. Applied egg-rr 98.1%

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

    1. associate-*l/ 98.3%

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

    2. *-lft-identity 98.3%

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

    3. +-commutative 98.3%

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

15. Simplified 98.3%

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

16. Final simplification 98.3%

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

Alternative 3: 98.0% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(sinTheta_O, h, eta)
	tmp = asin((h / (eta + (single(-0.5) * (sinTheta_O / (eta / 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 sinTheta_O around 0 97.7%

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

   1. unpow2 97.7%

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

   2. *-un-lft-identity 97.7%

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

   3. times-frac 98.1%

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

4. Applied egg-rr 98.1%

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

   1. clear-num 98.1%

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

   2. frac-times 98.1%

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

   3. metadata-eval 98.1%

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

   4. div-inv 98.1%

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

   5. /-rgt-identity 98.1%

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

   6. metadata-eval 98.1%

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

   7. times-frac 98.1%

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

   8. *-un-lft-identity 98.1%

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

   9. *-un-lft-identity 98.1%

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

6. Applied egg-rr 98.1%

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

7. Final simplification 98.1%

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

Alternative 4: 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 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 sinTheta_O around 0 97.7%

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

   1. unpow2 97.7%

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

   2. *-un-lft-identity 97.7%

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

   3. times-frac 98.1%

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

4. Applied egg-rr 98.1%

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

5. Final simplification 98.1%

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

Alternative 5: 95.6% 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 96.1%

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

3. Final simplification 96.1%

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

Reproduce

herbie shell --seed 2023315 
(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)))))))))