HairBSDF, gamma for a refracted ray

Percentage Accurate: 91.7% → 98.4%

Time: 12.3s

Alternatives: 3

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: 91.7% 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

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

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(h / Float32(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 90.6%

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

3. Taylor expanded in sinTheta_O around 0 90.4%

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

   1. /-rgt-identity 90.4%

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

   2. difference-of-squares 90.4%

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

5. Applied egg-rr 90.4%

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

   1. *-commutative 90.4%

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

   2. sqrt-prod 98.4%

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

7. Applied egg-rr 98.4%

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

Alternative 2: 97.9% 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 90.6%

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

3. Taylor expanded in sinTheta_O around 0 97.4%

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

   1. add-sqr-sqrt 97.4%

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

   2. sqrt-div 97.4%

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

   3. sqrt-pow1 96.4%

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

   4. metadata-eval 96.4%

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

   5. pow1 96.4%

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

   6. sqrt-div 96.4%

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

   7. sqrt-pow1 98.0%

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

   8. metadata-eval 98.0%

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

   9. pow1 98.0%

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

5. Applied egg-rr 98.0%

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

   1. unpow2 98.0%

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

7. Simplified 98.0%

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

   1. unpow2 98.0%

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

   2. clear-num 98.0%

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

   3. clear-num 98.0%

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

   4. frac-times 98.0%

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

   5. metadata-eval 98.0%

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

9. Applied egg-rr 98.0%

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

    1. associate-*r/ 98.0%

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

    2. associate-*l/ 98.0%

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

    3. rem-square-sqrt 98.0%

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

11. Simplified 98.0%

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

    1. metadata-eval 98.0%

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

    2. associate-/l/ 97.4%

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

    3. add-sqr-sqrt 97.4%

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

    4. times-frac 98.0%

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

    5. frac-times 98.0%

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

    6. clear-num 98.0%

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

    7. clear-num 98.0%

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

13. Applied egg-rr 98.0%

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

    1. associate-*l/ 97.7%

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

    2. associate-*r/ 98.0%

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

    3. associate-/l/ 98.0%

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

    4. rem-square-sqrt 98.0%

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

15. Simplified 98.0%

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

Alternative 3: 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 90.6%

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

3. Taylor expanded in eta around inf 95.8%

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

Reproduce

herbie shell --seed 2024139 
(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)))))))))