HairBSDF, gamma for a refracted ray

Percentage Accurate: 92.0% → 97.8%

Time: 8.7s

Alternatives: 4

Speedup: 1.5×

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: 97.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

MATLAB

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

Derivation

1. Initial program 90.9%

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

3. Taylor expanded in sinTheta_O around 0

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

   1. unpow2 N/A

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

   2. lower-*.f32 87.6

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

5. Applied rewrites 87.6%

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

6. Taylor expanded in sinTheta_O around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

      \[\leadsto \sin^{-1} \left(\frac{h}{\color{blue}{\mathsf{fma}\left(\frac{{sinTheta\_O}^{2}}{eta}, \frac{-1}{2}, eta\right)}}\right) \]

   4. lower-/.f32 N/A

      \[\leadsto \sin^{-1} \left(\frac{h}{\mathsf{fma}\left(\color{blue}{\frac{{sinTheta\_O}^{2}}{eta}}, \frac{-1}{2}, eta\right)}\right) \]

   5. unpow2 N/A

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

   6. lower-*.f32 86.0

      \[\leadsto \sin^{-1} \left(\frac{h}{\mathsf{fma}\left(\frac{\color{blue}{sinTheta\_O \cdot sinTheta\_O}}{eta}, -0.5, eta\right)}\right) \]

8. Applied rewrites 95.4%

   \[\leadsto \sin^{-1} \left(\frac{h}{\color{blue}{\mathsf{fma}\left(\frac{sinTheta\_O \cdot sinTheta\_O}{eta}, -0.5, eta\right)}}\right) \]
9. Step-by-step derivation

10. Applied rewrites 97.9%

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

12. Applied rewrites 98.3%

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

13. Final simplification 98.3%

    \[\leadsto \sin^{-1} \left(\frac{h}{eta - 0.5 \cdot \left(sinTheta\_O \cdot \left(sinTheta\_O \cdot \frac{1}{eta}\right)\right)}\right) \]
14. Add Preprocessing

Alternative 2: 97.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

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 / eta) * sintheta_o) * 0.5e0))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 90.9%

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

3. Taylor expanded in sinTheta_O around 0

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

   1. unpow2 N/A

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

   2. lower-*.f32 87.6

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

5. Applied rewrites 87.6%

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

6. Taylor expanded in sinTheta_O around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

      \[\leadsto \sin^{-1} \left(\frac{h}{\color{blue}{\mathsf{fma}\left(\frac{{sinTheta\_O}^{2}}{eta}, \frac{-1}{2}, eta\right)}}\right) \]

   4. lower-/.f32 N/A

      \[\leadsto \sin^{-1} \left(\frac{h}{\mathsf{fma}\left(\color{blue}{\frac{{sinTheta\_O}^{2}}{eta}}, \frac{-1}{2}, eta\right)}\right) \]

   5. unpow2 N/A

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

   6. lower-*.f32 87.1

      \[\leadsto \sin^{-1} \left(\frac{h}{\mathsf{fma}\left(\frac{\color{blue}{sinTheta\_O \cdot sinTheta\_O}}{eta}, -0.5, eta\right)}\right) \]

8. Applied rewrites 95.4%

   \[\leadsto \sin^{-1} \left(\frac{h}{\color{blue}{\mathsf{fma}\left(\frac{sinTheta\_O \cdot sinTheta\_O}{eta}, -0.5, eta\right)}}\right) \]
9. Step-by-step derivation

10. Applied rewrites 97.9%

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

12. Applied rewrites 98.3%

    \[\leadsto \sin^{-1} \left(\frac{h}{eta - \left(\frac{sinTheta\_O}{eta} \cdot sinTheta\_O\right) \cdot \color{blue}{0.5}}\right) \]
13. Add Preprocessing

Alternative 3: 97.3% accurate, 1.2× 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 90.9%

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

3. Taylor expanded in sinTheta_O around 0

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

   1. unpow2 N/A

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

   2. lower-*.f32 87.6

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

5. Applied rewrites 87.6%

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

6. Taylor expanded in sinTheta_O around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

      \[\leadsto \sin^{-1} \left(\frac{h}{\color{blue}{\mathsf{fma}\left(\frac{{sinTheta\_O}^{2}}{eta}, \frac{-1}{2}, eta\right)}}\right) \]

   4. lower-/.f32 N/A

      \[\leadsto \sin^{-1} \left(\frac{h}{\mathsf{fma}\left(\color{blue}{\frac{{sinTheta\_O}^{2}}{eta}}, \frac{-1}{2}, eta\right)}\right) \]

   5. unpow2 N/A

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

   6. lower-*.f32 74.4

      \[\leadsto \sin^{-1} \left(\frac{h}{\mathsf{fma}\left(\frac{\color{blue}{sinTheta\_O \cdot sinTheta\_O}}{eta}, -0.5, eta\right)}\right) \]

8. Applied rewrites 95.7%

   \[\leadsto \sin^{-1} \left(\frac{h}{\color{blue}{\mathsf{fma}\left(\frac{sinTheta\_O \cdot sinTheta\_O}{eta}, -0.5, eta\right)}}\right) \]
9. Step-by-step derivation

10. Applied rewrites 97.9%

    \[\leadsto \sin^{-1} \left(\frac{h}{eta - \color{blue}{0.5 \cdot \frac{sinTheta\_O \cdot sinTheta\_O}{eta}}}\right) \]
11. Add Preprocessing

Alternative 4: 95.2% accurate, 1.5× 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 90.9%

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

3. Taylor expanded in sinTheta_O around 0

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

   1. lower-/.f32 95.7

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

5. Applied rewrites 95.7%

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

Reproduce

herbie shell --seed 2024305 
(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)))))))))