HairBSDF, gamma for a refracted ray

Percentage Accurate: 91.7% → 98.5%

Time: 11.9s

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: 91.7% 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: 98.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} sinTheta_O_m = \left|sinTheta\_O\right| \\ \begin{array}{l} \mathbf{if}\;sinTheta\_O\_m \leq 4.22000007260429 \cdot 10^{-25}:\\ \;\;\;\;\sin^{-1} \left(\frac{h}{eta}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{h}{\sqrt{\left(eta - sinTheta\_O\_m\right) \cdot \left(eta + sinTheta\_O\_m\right)}}\right)\\ \end{array} \end{array} \]

FPCore

sinTheta_O_m = (fabs.f32 sinTheta_O)
(FPCore (sinTheta_O_m h eta)
 :precision binary32
 (if (<= sinTheta_O_m 4.22000007260429e-25)
   (asin (/ h eta))
   (asin (/ h (sqrt (* (- eta sinTheta_O_m) (+ eta sinTheta_O_m)))))))

C

sinTheta_O_m = fabs(sinTheta_O);
float code(float sinTheta_O_m, float h, float eta) {
	float tmp;
	if (sinTheta_O_m <= 4.22000007260429e-25f) {
		tmp = asinf((h / eta));
	} else {
		tmp = asinf((h / sqrtf(((eta - sinTheta_O_m) * (eta + sinTheta_O_m)))));
	}
	return tmp;
}

Fortran

sinTheta_O_m = abs(sintheta_o)
real(4) function code(sintheta_o_m, h, eta)
    real(4), intent (in) :: sintheta_o_m
    real(4), intent (in) :: h
    real(4), intent (in) :: eta
    real(4) :: tmp
    if (sintheta_o_m <= 4.22000007260429e-25) then
        tmp = asin((h / eta))
    else
        tmp = asin((h / sqrt(((eta - sintheta_o_m) * (eta + sintheta_o_m)))))
    end if
    code = tmp
end function

Julia

sinTheta_O_m = abs(sinTheta_O)
function code(sinTheta_O_m, h, eta)
	tmp = Float32(0.0)
	if (sinTheta_O_m <= Float32(4.22000007260429e-25))
		tmp = asin(Float32(h / eta));
	else
		tmp = asin(Float32(h / sqrt(Float32(Float32(eta - sinTheta_O_m) * Float32(eta + sinTheta_O_m)))));
	end
	return tmp
end

MATLAB

sinTheta_O_m = abs(sinTheta_O);
function tmp_2 = code(sinTheta_O_m, h, eta)
	tmp = single(0.0);
	if (sinTheta_O_m <= single(4.22000007260429e-25))
		tmp = asin((h / eta));
	else
		tmp = asin((h / sqrt(((eta - sinTheta_O_m) * (eta + sinTheta_O_m)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sinTheta_O < 4.22000007e-25

   1. Initial program 91.1%

      \[\sin^{-1} \left(\frac{h}{\sqrt{eta \cdot eta - \frac{sinTheta\_O \cdot sinTheta\_O}{\sqrt{1 - sinTheta\_O \cdot sinTheta\_O}}}}\right) \]
   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 94.4

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

   5. Simplified 94.4%

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

   if 4.22000007e-25 < sinTheta_O

   1. Initial program 98.5%

      \[\sin^{-1} \left(\frac{h}{\sqrt{eta \cdot eta - \frac{sinTheta\_O \cdot sinTheta\_O}{\sqrt{1 - sinTheta\_O \cdot sinTheta\_O}}}}\right) \]
   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 97.0

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

   5. Simplified 97.0%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sinTheta\_O \leq 4.22000007260429 \cdot 10^{-25}:\\ \;\;\;\;\sin^{-1} \left(\frac{h}{eta}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{h}{\sqrt{\left(eta - sinTheta\_O\right) \cdot \left(eta + sinTheta\_O\right)}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.4% accurate, 1.1× speedup

Math

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

FPCore

sinTheta_O_m = (fabs.f32 sinTheta_O)
(FPCore (sinTheta_O_m h eta)
 :precision binary32
 (asin (/ h (* (sqrt (- eta sinTheta_O_m)) (sqrt (+ eta sinTheta_O_m))))))

C

sinTheta_O_m = fabs(sinTheta_O);
float code(float sinTheta_O_m, float h, float eta) {
	return asinf((h / (sqrtf((eta - sinTheta_O_m)) * sqrtf((eta + sinTheta_O_m)))));
}

Fortran

sinTheta_O_m = abs(sintheta_o)
real(4) function code(sintheta_o_m, h, eta)
    real(4), intent (in) :: sintheta_o_m
    real(4), intent (in) :: h
    real(4), intent (in) :: eta
    code = asin((h / (sqrt((eta - sintheta_o_m)) * sqrt((eta + sintheta_o_m)))))
end function

Julia

sinTheta_O_m = abs(sinTheta_O)
function code(sinTheta_O_m, h, eta)
	return asin(Float32(h / Float32(sqrt(Float32(eta - sinTheta_O_m)) * sqrt(Float32(eta + sinTheta_O_m)))))
end

MATLAB

sinTheta_O_m = abs(sinTheta_O);
function tmp = code(sinTheta_O_m, h, eta)
	tmp = asin((h / (sqrt((eta - sinTheta_O_m)) * sqrt((eta + sinTheta_O_m)))));
end

Derivation

1. Initial program 92.9%

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

3. Taylor expanded in sinTheta_O around 0

   \[\leadsto \sin^{-1} \left(\frac{h}{\sqrt{\color{blue}{-1 \cdot {sinTheta\_O}^{2} + {eta}^{2}}}}\right) \]
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 92.4

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

5. Simplified 92.4%

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

   1. lift-+.f32 N/A

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

   2. lift--.f32 N/A

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

   3. sqrt-prod N/A

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

   4. pow1/2 N/A

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

   5. pow1/2 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. pow1/2 N/A

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

   9. lower-sqrt.f32 N/A

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

   10. pow1/2 N/A

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

   11. lower-sqrt.f32 97.9

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

   12. lift-+.f32 N/A

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

   13. +-commutative N/A

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

   14. lower-+.f32 97.9

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

7. Applied egg-rr 97.9%

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

8. Final simplification 97.9%

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

Alternative 3: 97.8% accurate, 1.2× speedup

Math

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

FPCore

sinTheta_O_m = (fabs.f32 sinTheta_O)
(FPCore (sinTheta_O_m h eta)
 :precision binary32
 (asin (/ h (fma (* sinTheta_O_m (/ -0.5 eta)) sinTheta_O_m eta))))

C

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

Julia

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

Derivation

1. Initial program 92.9%

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

3. Taylor expanded in sinTheta_O around 0

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

   1. +-commutative N/A

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

   2. associate-*r/ N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. metadata-eval N/A

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

   6. distribute-neg-frac N/A

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

   7. metadata-eval N/A

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

   8. associate-*r/ N/A

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

   9. lower-fma.f32 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f32 N/A

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

   12. associate-*r/ N/A

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

   13. metadata-eval N/A

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

   14. distribute-neg-frac N/A

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

   15. metadata-eval N/A

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

   16. lower-/.f32 95.4

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

5. Simplified 95.4%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f32 N/A

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

   7. lower-*.f32 96.3

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

7. Applied egg-rr 96.3%

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

Alternative 4: 97.2% accurate, 1.2× speedup

Math

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

FPCore

sinTheta_O_m = (fabs.f32 sinTheta_O)
(FPCore (sinTheta_O_m h eta)
 :precision binary32
 (asin (/ h (fma (* sinTheta_O_m sinTheta_O_m) (/ -0.5 eta) eta))))

C

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

Julia

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

Derivation

1. Initial program 92.9%

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

3. Taylor expanded in sinTheta_O around 0

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

   1. +-commutative N/A

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

   2. associate-*r/ N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. metadata-eval N/A

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

   6. distribute-neg-frac N/A

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

   7. metadata-eval N/A

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

   8. associate-*r/ N/A

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

   9. lower-fma.f32 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f32 N/A

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

   12. associate-*r/ N/A

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

   13. metadata-eval N/A

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

   14. distribute-neg-frac N/A

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

   15. metadata-eval N/A

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

   16. lower-/.f32 95.4

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

5. Simplified 95.4%

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

Alternative 5: 95.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} sinTheta_O_m = \left|sinTheta\_O\right| \\ \sin^{-1} \left(\frac{h}{eta}\right) \end{array} \]

FPCore

sinTheta_O_m = (fabs.f32 sinTheta_O)
(FPCore (sinTheta_O_m h eta) :precision binary32 (asin (/ h eta)))

C

sinTheta_O_m = fabs(sinTheta_O);
float code(float sinTheta_O_m, float h, float eta) {
	return asinf((h / eta));
}

Fortran

sinTheta_O_m = abs(sintheta_o)
real(4) function code(sintheta_o_m, h, eta)
    real(4), intent (in) :: sintheta_o_m
    real(4), intent (in) :: h
    real(4), intent (in) :: eta
    code = asin((h / eta))
end function

Julia

sinTheta_O_m = abs(sinTheta_O)
function code(sinTheta_O_m, h, eta)
	return asin(Float32(h / eta))
end

MATLAB

sinTheta_O_m = abs(sinTheta_O);
function tmp = code(sinTheta_O_m, h, eta)
	tmp = asin((h / eta));
end

Derivation

1. Initial program 92.9%

   \[\sin^{-1} \left(\frac{h}{\sqrt{eta \cdot eta - \frac{sinTheta\_O \cdot sinTheta\_O}{\sqrt{1 - sinTheta\_O \cdot sinTheta\_O}}}}\right) \]
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 92.5

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

5. Simplified 92.5%

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

Reproduce

herbie shell --seed 2024208 
(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)))))))))