HairBSDF, gamma for a refracted ray

Percentage Accurate: 91.8% → 98.4%

Time: 20.7s

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: 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: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float sinTheta_O, float h, float eta) {
	return asinf((h / (sqrtf((sinTheta_O + eta)) * 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((sintheta_o + eta)) * sqrt((eta - sintheta_o)))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 89.5%

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

3. Simplified 89.5%

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

4. Taylor expanded in sinTheta_O around 0 89.2%

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

   1. +-commutative 89.2%

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

   2. mul-1-neg 89.2%

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

   3. unsub-neg 89.2%

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

   4. unpow2 89.2%

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

   5. unpow2 89.2%

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

6. Simplified 89.2%

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

7. Taylor expanded in eta around 0 89.2%

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

   1. neg-mul-1 89.2%

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

   2. +-commutative 89.2%

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

   3. sub-neg 89.2%

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

   4. unpow2 89.2%

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

   5. unpow2 89.2%

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

   6. difference-of-squares 89.3%

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

   7. +-commutative 89.3%

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

9. Simplified 89.3%

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

    1. sqrt-prod 98.1%

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

    2. +-commutative 98.1%

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

11. Applied egg-rr 98.1%

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

    1. +-commutative 98.1%

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

13. Simplified 98.1%

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

14. Final simplification 98.1%

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

Alternative 2: 97.5% 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 89.5%

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

3. Simplified 89.5%

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

4. Taylor expanded in sinTheta_O around 0 96.8%

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

   1. unpow2 96.8%

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

6. Simplified 96.8%

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

7. Final simplification 96.8%

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

Alternative 3: 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 89.5%

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

3. Simplified 89.5%

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

4. Taylor expanded in sinTheta_O around 0 89.2%

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

   1. +-commutative 89.2%

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

   2. mul-1-neg 89.2%

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

   3. unsub-neg 89.2%

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

   4. unpow2 89.2%

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

   5. unpow2 89.2%

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

6. Simplified 89.2%

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

7. Taylor expanded in eta around 0 89.2%

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

   1. neg-mul-1 89.2%

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

   2. +-commutative 89.2%

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

   3. sub-neg 89.2%

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

   4. unpow2 89.2%

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

   5. unpow2 89.2%

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

   6. difference-of-squares 89.3%

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

   7. +-commutative 89.3%

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

9. Simplified 89.3%

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

10. Taylor expanded in eta around inf 96.8%

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

12. Simplified 97.8%

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

13. Final simplification 97.8%

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

Alternative 4: 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 89.5%

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

3. Simplified 89.5%

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

4. Taylor expanded in eta around inf 95.3%

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

5. Final simplification 95.3%

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

Reproduce

herbie shell --seed 2023275 
(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)))))))))