HairBSDF, gamma for a refracted ray

Percentage Accurate: 92.1% → 98.6%

Time: 16.0s

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: 92.1% 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: 98.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 - {sinTheta_O}^{2}\right)}^{-0.25}\\ \sin^{-1} \left(\frac{h}{\sqrt{\mathsf{fma}\left(sinTheta_O, t_0, eta\right)} \cdot \sqrt{eta - sinTheta_O \cdot t_0}}\right) \end{array} \end{array} \]

FPCore

(FPCore (sinTheta_O h eta)
 :precision binary32
 (let* ((t_0 (pow (- 1.0 (pow sinTheta_O 2.0)) -0.25)))
   (asin
    (/
     h
     (* (sqrt (fma sinTheta_O t_0 eta)) (sqrt (- eta (* sinTheta_O t_0))))))))

C

float code(float sinTheta_O, float h, float eta) {
	float t_0 = powf((1.0f - powf(sinTheta_O, 2.0f)), -0.25f);
	return asinf((h / (sqrtf(fmaf(sinTheta_O, t_0, eta)) * sqrtf((eta - (sinTheta_O * t_0))))));
}

Julia

function code(sinTheta_O, h, eta)
	t_0 = Float32(Float32(1.0) - (sinTheta_O ^ Float32(2.0))) ^ Float32(-0.25)
	return asin(Float32(h / Float32(sqrt(fma(sinTheta_O, t_0, eta)) * sqrt(Float32(eta - Float32(sinTheta_O * t_0))))))
end

Derivation

1. Initial program 89.5%

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

   1. add-sqr-sqrt 89.5%

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

   2. difference-of-squares 89.5%

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

   3. sqrt-div 89.5%

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

   4. sqrt-prod 41.5%

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

   5. add-sqr-sqrt 85.7%

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

   6. pow1/2 85.7%

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

   7. sqrt-pow1 85.7%

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

   8. pow2 85.7%

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

   9. metadata-eval 85.7%

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

   10. sqrt-div 85.7%

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

   11. sqrt-prod 41.6%

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

   12. add-sqr-sqrt 89.5%

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

   13. pow1/2 89.5%

       \[\leadsto \sin^{-1} \left(\frac{h}{\sqrt{\left(eta + \frac{sinTheta_O}{{\left(1 - {sinTheta_O}^{2}\right)}^{0.25}}\right) \cdot \left(eta - \frac{sinTheta_O}{\sqrt{\color{blue}{{\left(1 - sinTheta_O \cdot sinTheta_O\right)}^{0.5}}}}\right)}}\right) \]

   14. sqrt-pow1 89.5%

       \[\leadsto \sin^{-1} \left(\frac{h}{\sqrt{\left(eta + \frac{sinTheta_O}{{\left(1 - {sinTheta_O}^{2}\right)}^{0.25}}\right) \cdot \left(eta - \frac{sinTheta_O}{\color{blue}{{\left(1 - sinTheta_O \cdot sinTheta_O\right)}^{\left(\frac{0.5}{2}\right)}}}\right)}}\right) \]

3. Applied egg-rr 89.5%

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

   1. sqrt-prod 98.6%

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

   2. +-commutative 98.6%

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

   3. div-inv 98.6%

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

   4. fma-def 98.6%

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

   5. pow-flip 98.6%

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

   6. metadata-eval 98.6%

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

   7. div-inv 98.6%

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

   8. pow-flip 98.6%

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

   9. metadata-eval 98.6%

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

5. Applied egg-rr 98.6%

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

6. Final simplification 98.6%

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

Alternative 2: 50.2% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

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 / sqrt(eta)))) ** 2.0e0)))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 89.5%

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

2. Taylor expanded in sinTheta_O around 0 97.5%

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

   1. add-sqr-sqrt 97.5%

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

   2. sqrt-div 97.5%

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

   3. unpow2 97.5%

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

   4. sqrt-prod 47.0%

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

   5. add-sqr-sqrt 96.6%

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

   6. sqrt-div 96.6%

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

   7. unpow2 96.6%

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

   8. sqrt-prod 47.3%

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

   9. add-sqr-sqrt 98.2%

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

4. Applied egg-rr 98.2%

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

   1. unpow2 98.2%

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

6. Simplified 98.2%

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

   1. add-exp-log 47.3%

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

8. Applied egg-rr 47.3%

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

9. Final simplification 47.3%

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

Alternative 3: 97.6% 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 89.5%

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

2. Taylor expanded in sinTheta_O around 0 97.5%

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

   1. add-sqr-sqrt 97.5%

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

   2. sqrt-div 97.5%

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

   3. unpow2 97.5%

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

   4. sqrt-prod 47.0%

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

   5. add-sqr-sqrt 96.6%

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

   6. sqrt-div 96.6%

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

   7. unpow2 96.6%

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

   8. sqrt-prod 47.3%

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

   9. add-sqr-sqrt 98.2%

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

4. Applied egg-rr 98.2%

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

   1. unpow2 98.2%

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

6. Simplified 98.2%

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

   1. unpow2 98.2%

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

   2. clear-num 98.2%

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

   3. clear-num 98.2%

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

   4. frac-times 98.2%

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

   5. metadata-eval 98.2%

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

8. Applied egg-rr 98.2%

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

   1. associate-*r/ 98.2%

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

   2. associate-*l/ 98.2%

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

   3. rem-square-sqrt 98.2%

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

10. Simplified 98.2%

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

11. Final simplification 98.2%

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

Alternative 4: 97.7% 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 89.5%

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

2. Taylor expanded in sinTheta_O around 0 97.5%

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

   1. add-sqr-sqrt 97.5%

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

   2. sqrt-div 97.5%

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

   3. unpow2 97.5%

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

   4. sqrt-prod 47.0%

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

   5. add-sqr-sqrt 96.6%

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

   6. sqrt-div 96.6%

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

   7. unpow2 96.6%

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

   8. sqrt-prod 47.3%

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

   9. add-sqr-sqrt 98.2%

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

4. Applied egg-rr 98.2%

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

   1. unpow2 98.2%

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

6. Simplified 98.2%

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

   1. unpow2 98.2%

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

   2. clear-num 98.2%

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

   3. frac-times 98.2%

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

   4. *-un-lft-identity 98.2%

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

8. Applied egg-rr 98.2%

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

   1. associate-*l/ 98.2%

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

   2. rem-square-sqrt 98.2%

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

10. Simplified 98.2%

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

11. Final simplification 98.2%

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

Alternative 5: 95.2% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 89.5%

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

2. Taylor expanded in eta around inf 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 2023311 
(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)))))))))