HairBSDF, gamma for a refracted ray

Percentage Accurate: 92.0% → 98.0%

Time: 10.7s

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: 92.0% 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: 98.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sin^{-1} \left(\frac{h}{eta \cdot \left(1 + \left(-0.5 \cdot \left(\frac{sinTheta\_O}{eta} \cdot \frac{sinTheta\_O}{eta}\right)\right) \cdot \sqrt{\frac{1}{1 - {sinTheta\_O}^{2}}}\right)}\right) \end{array} \]

FPCore

(FPCore (sinTheta_O h eta)
 :precision binary32
 (asin
  (/
   h
   (*
    eta
    (+
     1.0
     (*
      (* -0.5 (* (/ sinTheta_O eta) (/ sinTheta_O eta)))
      (sqrt (/ 1.0 (- 1.0 (pow sinTheta_O 2.0))))))))))

C

float code(float sinTheta_O, float h, float eta) {
	return asinf((h / (eta * (1.0f + ((-0.5f * ((sinTheta_O / eta) * (sinTheta_O / eta))) * sqrtf((1.0f / (1.0f - powf(sinTheta_O, 2.0f)))))))));
}

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 * (1.0e0 + (((-0.5e0) * ((sintheta_o / eta) * (sintheta_o / eta))) * sqrt((1.0e0 / (1.0e0 - (sintheta_o ** 2.0e0)))))))))
end function

Julia

function code(sinTheta_O, h, eta)
	return asin(Float32(h / Float32(eta * Float32(Float32(1.0) + Float32(Float32(Float32(-0.5) * Float32(Float32(sinTheta_O / eta) * Float32(sinTheta_O / eta))) * sqrt(Float32(Float32(1.0) / Float32(Float32(1.0) - (sinTheta_O ^ Float32(2.0))))))))))
end

MATLAB

function tmp = code(sinTheta_O, h, eta)
	tmp = asin((h / (eta * (single(1.0) + ((single(-0.5) * ((sinTheta_O / eta) * (sinTheta_O / eta))) * sqrt((single(1.0) / (single(1.0) - (sinTheta_O ^ single(2.0))))))))));
end

Derivation

1. Initial program 94.5%

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

3. Taylor expanded in eta around inf 94.2%

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

   1. associate-*r* 94.2%

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

5. Simplified 94.2%

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

   1. pow2 94.2%

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

   2. pow2 94.2%

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

   3. times-frac 98.5%

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

7. Applied egg-rr 98.5%

   \[\leadsto \sin^{-1} \left(\frac{h}{eta \cdot \left(1 + \left(-0.5 \cdot \color{blue}{\left(\frac{sinTheta\_O}{eta} \cdot \frac{sinTheta\_O}{eta}\right)}\right) \cdot \sqrt{\frac{1}{1 - {sinTheta\_O}^{2}}}\right)}\right) \]
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 94.5%

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

3. Taylor expanded in sinTheta_O around 0 98.2%

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

   1. pow2 98.2%

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

   2. associate-/l* 98.5%

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

5. Applied egg-rr 98.5%

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

Alternative 3: 95.4% 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 94.5%

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

3. Taylor expanded in eta around inf 96.2%

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

Reproduce

herbie shell --seed 2024111 
(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)))))))))