HairBSDF, gamma for a refracted ray

Percentage Accurate: 91.6% → 98.7%

Time: 14.0s

Alternatives: 5

Speedup: 1.5×

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: 91.6% 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: 98.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sinTheta\_O \cdot sinTheta\_O \leq 0:\\ \;\;\;\;\sin^{-1} \left(\frac{h}{eta}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{h}{\sqrt{\mathsf{fma}\left(eta, eta, sinTheta\_O \cdot \left(-sinTheta\_O\right)\right)}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (sinTheta_O h eta)
 :precision binary32
 (if (<= (* sinTheta_O sinTheta_O) 0.0)
   (asin (/ h eta))
   (asin (/ h (sqrt (fma eta eta (* sinTheta_O (- sinTheta_O))))))))

C

float code(float sinTheta_O, float h, float eta) {
	float tmp;
	if ((sinTheta_O * sinTheta_O) <= 0.0f) {
		tmp = asinf((h / eta));
	} else {
		tmp = asinf((h / sqrtf(fmaf(eta, eta, (sinTheta_O * -sinTheta_O)))));
	}
	return tmp;
}

Julia

function code(sinTheta_O, h, eta)
	tmp = Float32(0.0)
	if (Float32(sinTheta_O * sinTheta_O) <= Float32(0.0))
		tmp = asin(Float32(h / eta));
	else
		tmp = asin(Float32(h / sqrt(fma(eta, eta, Float32(sinTheta_O * Float32(-sinTheta_O))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 sinTheta_O sinTheta_O) < 0.0

   1. Initial program 90.8%

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

   3. Taylor expanded in eta around inf

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

      1. lower-/.f32 98.7

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

   5. Applied rewrites 98.7%

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

   if 0.0 < (*.f32 sinTheta_O sinTheta_O)

   1. Initial program 99.5%

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

   3. Taylor expanded in sinTheta_O around 0

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f32 N/A

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

      11. lower-*.f32 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f32 99.5

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in sinTheta_O around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f32 N/A

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

      4. mul-1-neg N/A

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

      5. unpow2 N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. mul-1-neg N/A

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

      8. lower-*.f32 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f32 99.5

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

   8. Applied rewrites 99.5%

      \[\leadsto \sin^{-1} \left(\frac{h}{\sqrt{\color{blue}{\mathsf{fma}\left(eta, eta, sinTheta\_O \cdot \left(-sinTheta\_O\right)\right)}}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 96.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 94.9%

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

3. Taylor expanded in sinTheta_O around 0

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

   1. associate-*r/ N/A

      \[\leadsto \sin^{-1} \left(\color{blue}{\frac{\frac{1}{2} \cdot \left(h \cdot {sinTheta\_O}^{2}\right)}{{eta}^{3}}} + \frac{h}{eta}\right) \]

   2. associate-*r* N/A

      \[\leadsto \sin^{-1} \left(\frac{\color{blue}{\left(\frac{1}{2} \cdot h\right) \cdot {sinTheta\_O}^{2}}}{{eta}^{3}} + \frac{h}{eta}\right) \]

   3. associate-*l/ N/A

      \[\leadsto \sin^{-1} \left(\color{blue}{\frac{\frac{1}{2} \cdot h}{{eta}^{3}} \cdot {sinTheta\_O}^{2}} + \frac{h}{eta}\right) \]

   4. associate-*r/ N/A

      \[\leadsto \sin^{-1} \left(\color{blue}{\left(\frac{1}{2} \cdot \frac{h}{{eta}^{3}}\right)} \cdot {sinTheta\_O}^{2} + \frac{h}{eta}\right) \]

   5. *-commutative N/A

      \[\leadsto \sin^{-1} \left(\color{blue}{{sinTheta\_O}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{{eta}^{3}}\right)} + \frac{h}{eta}\right) \]

   6. unpow2 N/A

      \[\leadsto \sin^{-1} \left(\color{blue}{\left(sinTheta\_O \cdot sinTheta\_O\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{{eta}^{3}}\right) + \frac{h}{eta}\right) \]

   7. associate-*l* N/A

      \[\leadsto \sin^{-1} \left(\color{blue}{sinTheta\_O \cdot \left(sinTheta\_O \cdot \left(\frac{1}{2} \cdot \frac{h}{{eta}^{3}}\right)\right)} + \frac{h}{eta}\right) \]

   8. lower-fma.f32 N/A

      \[\leadsto \sin^{-1} \color{blue}{\left(\mathsf{fma}\left(sinTheta\_O, sinTheta\_O \cdot \left(\frac{1}{2} \cdot \frac{h}{{eta}^{3}}\right), \frac{h}{eta}\right)\right)} \]

   9. associate-*r/ N/A

      \[\leadsto \sin^{-1} \left(\mathsf{fma}\left(sinTheta\_O, sinTheta\_O \cdot \color{blue}{\frac{\frac{1}{2} \cdot h}{{eta}^{3}}}, \frac{h}{eta}\right)\right) \]

   10. associate-*r/ N/A

       \[\leadsto \sin^{-1} \left(\mathsf{fma}\left(sinTheta\_O, \color{blue}{\frac{sinTheta\_O \cdot \left(\frac{1}{2} \cdot h\right)}{{eta}^{3}}}, \frac{h}{eta}\right)\right) \]

   11. lower-/.f32 N/A

       \[\leadsto \sin^{-1} \left(\mathsf{fma}\left(sinTheta\_O, \color{blue}{\frac{sinTheta\_O \cdot \left(\frac{1}{2} \cdot h\right)}{{eta}^{3}}}, \frac{h}{eta}\right)\right) \]

   12. lower-*.f32 N/A

       \[\leadsto \sin^{-1} \left(\mathsf{fma}\left(sinTheta\_O, \frac{\color{blue}{sinTheta\_O \cdot \left(\frac{1}{2} \cdot h\right)}}{{eta}^{3}}, \frac{h}{eta}\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \sin^{-1} \left(\mathsf{fma}\left(sinTheta\_O, \frac{sinTheta\_O \cdot \color{blue}{\left(h \cdot \frac{1}{2}\right)}}{{eta}^{3}}, \frac{h}{eta}\right)\right) \]

   14. lower-*.f32 N/A

       \[\leadsto \sin^{-1} \left(\mathsf{fma}\left(sinTheta\_O, \frac{sinTheta\_O \cdot \color{blue}{\left(h \cdot \frac{1}{2}\right)}}{{eta}^{3}}, \frac{h}{eta}\right)\right) \]

   15. cube-mult N/A

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

   16. unpow2 N/A

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

   17. lower-*.f32 N/A

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

   18. unpow2 N/A

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

   19. lower-*.f32 N/A

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

   20. lower-/.f32 79.8

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

5. Applied rewrites 79.8%

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

7. Applied rewrites 92.2%

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

9. Applied rewrites 96.3%

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

10. Taylor expanded in h around 0

    \[\leadsto \sin^{-1} \left(\frac{\mathsf{fma}\left(\frac{sinTheta\_O}{eta}, \frac{1}{2} \cdot sinTheta\_O, eta\right)}{eta \cdot \frac{eta}{h}}\right) \]
11. Step-by-step derivation

12. Applied rewrites 96.3%

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

13. Final simplification 96.3%

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

Alternative 3: 96.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 94.9%

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

3. Taylor expanded in sinTheta_O around 0

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

   1. associate-*r/ N/A

      \[\leadsto \sin^{-1} \left(\color{blue}{\frac{\frac{1}{2} \cdot \left(h \cdot {sinTheta\_O}^{2}\right)}{{eta}^{3}}} + \frac{h}{eta}\right) \]

   2. associate-*r* N/A

      \[\leadsto \sin^{-1} \left(\frac{\color{blue}{\left(\frac{1}{2} \cdot h\right) \cdot {sinTheta\_O}^{2}}}{{eta}^{3}} + \frac{h}{eta}\right) \]

   3. associate-*l/ N/A

      \[\leadsto \sin^{-1} \left(\color{blue}{\frac{\frac{1}{2} \cdot h}{{eta}^{3}} \cdot {sinTheta\_O}^{2}} + \frac{h}{eta}\right) \]

   4. associate-*r/ N/A

      \[\leadsto \sin^{-1} \left(\color{blue}{\left(\frac{1}{2} \cdot \frac{h}{{eta}^{3}}\right)} \cdot {sinTheta\_O}^{2} + \frac{h}{eta}\right) \]

   5. *-commutative N/A

      \[\leadsto \sin^{-1} \left(\color{blue}{{sinTheta\_O}^{2} \cdot \left(\frac{1}{2} \cdot \frac{h}{{eta}^{3}}\right)} + \frac{h}{eta}\right) \]

   6. unpow2 N/A

      \[\leadsto \sin^{-1} \left(\color{blue}{\left(sinTheta\_O \cdot sinTheta\_O\right)} \cdot \left(\frac{1}{2} \cdot \frac{h}{{eta}^{3}}\right) + \frac{h}{eta}\right) \]

   7. associate-*l* N/A

      \[\leadsto \sin^{-1} \left(\color{blue}{sinTheta\_O \cdot \left(sinTheta\_O \cdot \left(\frac{1}{2} \cdot \frac{h}{{eta}^{3}}\right)\right)} + \frac{h}{eta}\right) \]

   8. lower-fma.f32 N/A

      \[\leadsto \sin^{-1} \color{blue}{\left(\mathsf{fma}\left(sinTheta\_O, sinTheta\_O \cdot \left(\frac{1}{2} \cdot \frac{h}{{eta}^{3}}\right), \frac{h}{eta}\right)\right)} \]

   9. associate-*r/ N/A

      \[\leadsto \sin^{-1} \left(\mathsf{fma}\left(sinTheta\_O, sinTheta\_O \cdot \color{blue}{\frac{\frac{1}{2} \cdot h}{{eta}^{3}}}, \frac{h}{eta}\right)\right) \]

   10. associate-*r/ N/A

       \[\leadsto \sin^{-1} \left(\mathsf{fma}\left(sinTheta\_O, \color{blue}{\frac{sinTheta\_O \cdot \left(\frac{1}{2} \cdot h\right)}{{eta}^{3}}}, \frac{h}{eta}\right)\right) \]

   11. lower-/.f32 N/A

       \[\leadsto \sin^{-1} \left(\mathsf{fma}\left(sinTheta\_O, \color{blue}{\frac{sinTheta\_O \cdot \left(\frac{1}{2} \cdot h\right)}{{eta}^{3}}}, \frac{h}{eta}\right)\right) \]

   12. lower-*.f32 N/A

       \[\leadsto \sin^{-1} \left(\mathsf{fma}\left(sinTheta\_O, \frac{\color{blue}{sinTheta\_O \cdot \left(\frac{1}{2} \cdot h\right)}}{{eta}^{3}}, \frac{h}{eta}\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \sin^{-1} \left(\mathsf{fma}\left(sinTheta\_O, \frac{sinTheta\_O \cdot \color{blue}{\left(h \cdot \frac{1}{2}\right)}}{{eta}^{3}}, \frac{h}{eta}\right)\right) \]

   14. lower-*.f32 N/A

       \[\leadsto \sin^{-1} \left(\mathsf{fma}\left(sinTheta\_O, \frac{sinTheta\_O \cdot \color{blue}{\left(h \cdot \frac{1}{2}\right)}}{{eta}^{3}}, \frac{h}{eta}\right)\right) \]

   15. cube-mult N/A

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

   16. unpow2 N/A

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

   17. lower-*.f32 N/A

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

   18. unpow2 N/A

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

   19. lower-*.f32 N/A

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

   20. lower-/.f32 79.8

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

5. Applied rewrites 79.8%

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

7. Applied rewrites 92.2%

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

8. Taylor expanded in sinTheta_O around 0

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

10. Applied rewrites 96.0%

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

Alternative 4: 98.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sinTheta\_O \cdot sinTheta\_O \leq 0:\\ \;\;\;\;\sin^{-1} \left(\frac{h}{eta}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{h}{\sqrt{\left(eta + sinTheta\_O\right) \cdot \left(eta - sinTheta\_O\right)}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (sinTheta_O h eta)
 :precision binary32
 (if (<= (* sinTheta_O sinTheta_O) 0.0)
   (asin (/ h eta))
   (asin (/ h (sqrt (* (+ eta sinTheta_O) (- eta sinTheta_O)))))))

C

float code(float sinTheta_O, float h, float eta) {
	float tmp;
	if ((sinTheta_O * sinTheta_O) <= 0.0f) {
		tmp = asinf((h / eta));
	} else {
		tmp = asinf((h / sqrtf(((eta + sinTheta_O) * (eta - sinTheta_O)))));
	}
	return tmp;
}

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
    real(4) :: tmp
    if ((sintheta_o * sintheta_o) <= 0.0e0) then
        tmp = asin((h / eta))
    else
        tmp = asin((h / sqrt(((eta + sintheta_o) * (eta - sintheta_o)))))
    end if
    code = tmp
end function

Julia

function code(sinTheta_O, h, eta)
	tmp = Float32(0.0)
	if (Float32(sinTheta_O * sinTheta_O) <= Float32(0.0))
		tmp = asin(Float32(h / eta));
	else
		tmp = asin(Float32(h / sqrt(Float32(Float32(eta + sinTheta_O) * Float32(eta - sinTheta_O)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(sinTheta_O, h, eta)
	tmp = single(0.0);
	if ((sinTheta_O * sinTheta_O) <= single(0.0))
		tmp = asin((h / eta));
	else
		tmp = asin((h / sqrt(((eta + sinTheta_O) * (eta - sinTheta_O)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 sinTheta_O sinTheta_O) < 0.0

   1. Initial program 90.8%

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

   3. Taylor expanded in eta around inf

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

      1. lower-/.f32 98.7

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

   5. Applied rewrites 98.7%

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

   if 0.0 < (*.f32 sinTheta_O sinTheta_O)

   1. Initial program 99.5%

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

   3. Taylor expanded in sinTheta_O around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. difference-of-squares N/A

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

      7. lower-*.f32 N/A

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

      8. lower-+.f32 N/A

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

      9. lower--.f32 99.5

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

   5. Applied rewrites 99.5%

      \[\leadsto \sin^{-1} \left(\frac{h}{\sqrt{\color{blue}{\left(eta + sinTheta\_O\right) \cdot \left(eta - sinTheta\_O\right)}}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 95.5% accurate, 1.5× 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 94.9%

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

3. Taylor expanded in eta around inf

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

   1. lower-/.f32 94.5

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

5. Applied rewrites 94.5%

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

Reproduce

herbie shell --seed 2024214 
(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)))))))))