HairBSDF, gamma for a refracted ray

Percentage Accurate: 91.7% → 97.8%

Time: 12.0s

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: 97.8% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 93.4%

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

2. Taylor expanded in sinTheta_O around 0 98.0%

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

   1. unpow2 98.0%

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

   2. associate-*r/ 98.0%

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

4. Simplified 98.0%

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

   1. div-inv 97.3%

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

   2. associate-/l* 97.3%

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

6. Applied egg-rr 97.3%

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

7. Taylor expanded in h around 0 98.0%

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

   1. +-commutative 98.0%

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

   2. unpow2 98.0%

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

   3. associate-*r/ 98.0%

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

   4. associate-*l/ 98.0%

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

   5. associate-*r* 98.6%

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

   6. associate-/r/ 98.6%

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

   7. *-commutative 98.6%

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

   8. fma-def 98.6%

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

   9. associate-/r/ 98.6%

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

   10. *-commutative 98.6%

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

9. Simplified 98.6%

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

10. Final simplification 98.6%

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

Alternative 2: 97.3% accurate, 2.8× speedup

Math

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

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(Float32(-0.5) * 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 93.4%

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

2. Taylor expanded in sinTheta_O around 0 98.0%

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

   1. unpow2 98.0%

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

   2. associate-*r/ 98.0%

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

4. Simplified 98.0%

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

5. Final simplification 98.0%

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

Alternative 3: 95.2% 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 93.4%

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

2. Taylor expanded in eta around inf 96.3%

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

3. Final simplification 96.3%

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

Reproduce

herbie shell --seed 2023176 
(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)))))))))