HairBSDF, gamma for a refracted ray

Percentage Accurate: 91.8% → 97.2%

Time: 16.3s

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: 91.8% 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: 97.2% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (sinTheta_O h eta)
 :precision binary32
 (asin (/ h (fma (pow sinTheta_O 2.0) (/ -0.5 eta) eta))))

C

float code(float sinTheta_O, float h, float eta) {
	return asinf((h / fmaf(powf(sinTheta_O, 2.0f), (-0.5f / eta), eta)));
}

Julia

function code(sinTheta_O, h, eta)
	return asin(Float32(h / fma((sinTheta_O ^ Float32(2.0)), Float32(Float32(-0.5) / eta), 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 96.2%

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

   1. associate-*r/ 96.2%

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

   2. *-commutative 96.2%

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

5. Simplified 96.2%

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

6. Taylor expanded in h around 0 96.2%

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

   1. +-commutative 96.2%

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

   2. associate-*r/ 96.2%

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

   3. *-commutative 96.2%

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

   4. associate-*r/ 96.2%

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

   5. fma-undefine 96.2%

      \[\leadsto \sin^{-1} \left(\frac{h}{\color{blue}{\mathsf{fma}\left({sinTheta\_O}^{2}, \frac{-0.5}{eta}, eta\right)}}\right) \]

8. Simplified 96.2%

   \[\leadsto \color{blue}{\sin^{-1} \left(\frac{h}{\mathsf{fma}\left({sinTheta\_O}^{2}, \frac{-0.5}{eta}, eta\right)}\right)} \]

9. Final simplification 96.2%

   \[\leadsto \sin^{-1} \left(\frac{h}{\mathsf{fma}\left({sinTheta\_O}^{2}, \frac{-0.5}{eta}, eta\right)}\right) \]
10. Add Preprocessing

Alternative 2: 97.2% accurate, 1.5× speedup

Math

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

FPCore

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

C

float code(float sinTheta_O, float h, float eta) {
	return asinf((h / (eta + (powf(sinTheta_O, 2.0f) * (-0.5f / 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 + ((sintheta_o ** 2.0e0) * ((-0.5e0) / eta)))))
end function

Julia

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

MATLAB

function tmp = code(sinTheta_O, h, eta)
	tmp = asin((h / (eta + ((sinTheta_O ^ single(2.0)) * (single(-0.5) / 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 96.2%

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

   1. associate-*r/ 96.2%

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

   2. *-commutative 96.2%

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

5. Simplified 96.2%

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

6. Taylor expanded in sinTheta_O around 0 96.2%

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

   1. associate-*r/ 96.2%

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

   2. *-commutative 96.2%

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

   3. associate-*r/ 96.2%

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

8. Simplified 96.2%

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

9. Final simplification 96.2%

   \[\leadsto \sin^{-1} \left(\frac{h}{eta + {sinTheta\_O}^{2} \cdot \frac{-0.5}{eta}}\right) \]
10. 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 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 eta around inf 94.4%

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

4. Final simplification 94.4%

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

Reproduce

herbie shell --seed 2024079 
(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)))))))))