HairBSDF, gamma for a refracted ray

Percentage Accurate: 92.4% → 98.1%

Time: 15.1s

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: 92.4% 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: 98.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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(((((eta - sintheta_o) ** (-0.5e0)) * ((eta + sintheta_o) ** (-0.5e0))) * h))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 92.2%

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

   1. clear-num 91.5%

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

   2. associate-/r/ 91.6%

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

   3. pow1/2 91.6%

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

   4. pow-flip 91.6%

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

   5. pow2 91.6%

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

   6. pow2 91.6%

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

   7. pow2 91.6%

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

   8. metadata-eval 91.6%

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

4. Applied egg-rr 91.6%

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

5. Taylor expanded in sinTheta_O around 0 91.5%

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

   1. neg-mul-1 91.5%

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

   2. +-commutative 91.5%

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

   3. unsub-neg 91.5%

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

7. Simplified 91.5%

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

   1. unpow2 91.5%

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

   2. unpow2 91.5%

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

   3. difference-of-squares 91.5%

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

9. Applied egg-rr 91.5%

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

    1. *-commutative 91.5%

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

    2. unpow-prod-down 98.3%

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

    3. +-commutative 98.3%

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

11. Applied egg-rr 98.3%

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

12. Final simplification 98.3%

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

Alternative 2: 97.9% accurate, 2.8× speedup

Math

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

FPCore

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

C

float code(float sinTheta_O, float h, float eta) {
	return asinf((h / (eta + ((sinTheta_O * -0.5f) / (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 + ((sintheta_o * (-0.5e0)) / (eta / sintheta_o)))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 92.2%

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

3. Taylor expanded in sinTheta_O around 0 98.1%

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

   1. unpow2 98.1%

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

   2. *-un-lft-identity 98.1%

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

   3. times-frac 98.3%

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

5. Applied egg-rr 98.3%

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

   1. /-rgt-identity 98.3%

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

   2. associate-*r* 98.3%

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

   3. clear-num 98.3%

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

   4. un-div-inv 98.3%

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

   5. *-commutative 98.3%

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

7. Applied egg-rr 98.3%

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

8. Final simplification 98.3%

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

Alternative 3: 95.1% 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 92.2%

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

3. Taylor expanded in eta around inf 96.2%

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

4. Final simplification 96.2%

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

Reproduce

herbie shell --seed 2024021 
(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)))))))))