HairBSDF, gamma for a refracted ray

Percentage Accurate: 92.2% → 97.9%

Time: 13.0s

Alternatives: 4

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: 92.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 97.9% accurate, 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 93.3%

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

2. Taylor expanded in eta around inf 97.3%

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

   1. unpow2 97.3%

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

   2. unpow2 97.3%

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

4. Simplified 97.3%

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

   1. *-un-lft-identity 97.3%

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

   2. associate-/l* 98.0%

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

   3. sqrt-div 98.0%

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

   4. metadata-eval 98.0%

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

6. Applied egg-rr 98.0%

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

   1. *-lft-identity 98.0%

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

   2. associate-*r/ 98.0%

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

   3. *-rgt-identity 98.0%

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

   4. associate-/r/ 98.0%

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

8. Simplified 98.0%

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

9. Final simplification 98.0%

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

Alternative 2: 97.4% accurate, 2.8× speedup

Math

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

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(Float32(sinTheta_O * 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 93.3%

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

2. Taylor expanded in sinTheta_O around 0 97.3%

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

   1. unpow2 97.3%

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

4. Simplified 97.3%

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

5. Final simplification 97.3%

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

Alternative 3: 97.9% 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 93.3%

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

2. Taylor expanded in eta around inf 97.3%

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

   1. unpow2 97.3%

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

   2. unpow2 97.3%

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

4. Simplified 97.3%

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

   1. *-un-lft-identity 97.3%

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

   2. associate-/l* 98.0%

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

   3. sqrt-div 98.0%

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

   4. metadata-eval 98.0%

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

6. Applied egg-rr 98.0%

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

   1. *-lft-identity 98.0%

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

   2. associate-*r/ 98.0%

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

   3. *-rgt-identity 98.0%

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

   4. associate-/r/ 98.0%

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

8. Simplified 98.0%

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

9. Taylor expanded in sinTheta_O around 0 97.3%

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

    1. *-commutative 97.3%

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

    2. unpow2 97.3%

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

    3. associate-*r/ 98.0%

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

    4. associate-*l* 98.0%

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

11. Simplified 98.0%

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

12. Final simplification 98.0%

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

Alternative 4: 95.3% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 93.3%

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

2. Taylor expanded in eta around inf 95.5%

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

3. Final simplification 95.5%

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

Reproduce

herbie shell --seed 2023207 
(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)))))))))