HairBSDF, gamma for a refracted ray

Percentage Accurate: 92.3% → 98.7%

Time: 16.9s

Alternatives: 4

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: 92.3% 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, 0.4× speedup

Math

\[\begin{array}{l} sinTheta_O_m = \left|sinTheta\_O\right| \\ \begin{array}{l} t_0 := {\left(1 - {sinTheta\_O\_m}^{2}\right)}^{-0.25}\\ \sin^{-1} \left({\left(\mathsf{fma}\left(sinTheta\_O\_m, t\_0, eta\right)\right)}^{-0.5} \cdot \frac{h}{\sqrt{eta - sinTheta\_O\_m \cdot t\_0}}\right) \end{array} \end{array} \]

FPCore

sinTheta_O_m = (fabs.f32 sinTheta_O)
(FPCore (sinTheta_O_m h eta)
 :precision binary32
 (let* ((t_0 (pow (- 1.0 (pow sinTheta_O_m 2.0)) -0.25)))
   (asin
    (*
     (pow (fma sinTheta_O_m t_0 eta) -0.5)
     (/ h (sqrt (- eta (* sinTheta_O_m t_0))))))))

C

sinTheta_O_m = fabs(sinTheta_O);
float code(float sinTheta_O_m, float h, float eta) {
	float t_0 = powf((1.0f - powf(sinTheta_O_m, 2.0f)), -0.25f);
	return asinf((powf(fmaf(sinTheta_O_m, t_0, eta), -0.5f) * (h / sqrtf((eta - (sinTheta_O_m * t_0))))));
}

Julia

sinTheta_O_m = abs(sinTheta_O)
function code(sinTheta_O_m, h, eta)
	t_0 = Float32(Float32(1.0) - (sinTheta_O_m ^ Float32(2.0))) ^ Float32(-0.25)
	return asin(Float32((fma(sinTheta_O_m, t_0, eta) ^ Float32(-0.5)) * Float32(h / sqrt(Float32(eta - Float32(sinTheta_O_m * t_0))))))
end

Derivation

1. Initial program 94.4%

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

   1. add-sqr-sqrt 94.4%

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

   2. difference-of-squares 94.4%

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

   3. sqrt-div 94.4%

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

   4. sqrt-prod 48.9%

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

   5. add-sqr-sqrt 89.6%

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

   6. pow1/2 89.6%

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

   7. sqrt-pow1 89.6%

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

   8. pow2 89.6%

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

   9. metadata-eval 89.6%

      \[\leadsto \sin^{-1} \left(\frac{h}{\sqrt{\left(eta + \frac{sinTheta\_O}{{\left(1 - {sinTheta\_O}^{2}\right)}^{\color{blue}{0.25}}}\right) \cdot \left(eta - \sqrt{\frac{sinTheta\_O \cdot sinTheta\_O}{\sqrt{1 - sinTheta\_O \cdot sinTheta\_O}}}\right)}}\right) \]

   10. sqrt-div 89.6%

       \[\leadsto \sin^{-1} \left(\frac{h}{\sqrt{\left(eta + \frac{sinTheta\_O}{{\left(1 - {sinTheta\_O}^{2}\right)}^{0.25}}\right) \cdot \left(eta - \color{blue}{\frac{\sqrt{sinTheta\_O \cdot sinTheta\_O}}{\sqrt{\sqrt{1 - sinTheta\_O \cdot sinTheta\_O}}}}\right)}}\right) \]

   11. sqrt-prod 49.1%

       \[\leadsto \sin^{-1} \left(\frac{h}{\sqrt{\left(eta + \frac{sinTheta\_O}{{\left(1 - {sinTheta\_O}^{2}\right)}^{0.25}}\right) \cdot \left(eta - \frac{\color{blue}{\sqrt{sinTheta\_O} \cdot \sqrt{sinTheta\_O}}}{\sqrt{\sqrt{1 - sinTheta\_O \cdot sinTheta\_O}}}\right)}}\right) \]

   12. add-sqr-sqrt 94.4%

       \[\leadsto \sin^{-1} \left(\frac{h}{\sqrt{\left(eta + \frac{sinTheta\_O}{{\left(1 - {sinTheta\_O}^{2}\right)}^{0.25}}\right) \cdot \left(eta - \frac{\color{blue}{sinTheta\_O}}{\sqrt{\sqrt{1 - sinTheta\_O \cdot sinTheta\_O}}}\right)}}\right) \]

   13. pow1/2 94.4%

       \[\leadsto \sin^{-1} \left(\frac{h}{\sqrt{\left(eta + \frac{sinTheta\_O}{{\left(1 - {sinTheta\_O}^{2}\right)}^{0.25}}\right) \cdot \left(eta - \frac{sinTheta\_O}{\sqrt{\color{blue}{{\left(1 - sinTheta\_O \cdot sinTheta\_O\right)}^{0.5}}}}\right)}}\right) \]

   14. sqrt-pow1 94.4%

       \[\leadsto \sin^{-1} \left(\frac{h}{\sqrt{\left(eta + \frac{sinTheta\_O}{{\left(1 - {sinTheta\_O}^{2}\right)}^{0.25}}\right) \cdot \left(eta - \frac{sinTheta\_O}{\color{blue}{{\left(1 - sinTheta\_O \cdot sinTheta\_O\right)}^{\left(\frac{0.5}{2}\right)}}}\right)}}\right) \]

4. Applied egg-rr 94.4%

   \[\leadsto \sin^{-1} \left(\frac{h}{\sqrt{\color{blue}{\left(eta + \frac{sinTheta\_O}{{\left(1 - {sinTheta\_O}^{2}\right)}^{0.25}}\right) \cdot \left(eta - \frac{sinTheta\_O}{{\left(1 - {sinTheta\_O}^{2}\right)}^{0.25}}\right)}}}\right) \]
5. Step-by-step derivation

   1. sqrt-prod 98.3%

      \[\leadsto \sin^{-1} \left(\frac{h}{\color{blue}{\sqrt{eta + \frac{sinTheta\_O}{{\left(1 - {sinTheta\_O}^{2}\right)}^{0.25}}} \cdot \sqrt{eta - \frac{sinTheta\_O}{{\left(1 - {sinTheta\_O}^{2}\right)}^{0.25}}}}}\right) \]

   2. +-commutative 98.3%

      \[\leadsto \sin^{-1} \left(\frac{h}{\sqrt{\color{blue}{\frac{sinTheta\_O}{{\left(1 - {sinTheta\_O}^{2}\right)}^{0.25}} + eta}} \cdot \sqrt{eta - \frac{sinTheta\_O}{{\left(1 - {sinTheta\_O}^{2}\right)}^{0.25}}}}\right) \]

   3. div-inv 98.3%

      \[\leadsto \sin^{-1} \left(\frac{h}{\sqrt{\color{blue}{sinTheta\_O \cdot \frac{1}{{\left(1 - {sinTheta\_O}^{2}\right)}^{0.25}}} + eta} \cdot \sqrt{eta - \frac{sinTheta\_O}{{\left(1 - {sinTheta\_O}^{2}\right)}^{0.25}}}}\right) \]

   4. fma-define 98.3%

      \[\leadsto \sin^{-1} \left(\frac{h}{\sqrt{\color{blue}{\mathsf{fma}\left(sinTheta\_O, \frac{1}{{\left(1 - {sinTheta\_O}^{2}\right)}^{0.25}}, eta\right)}} \cdot \sqrt{eta - \frac{sinTheta\_O}{{\left(1 - {sinTheta\_O}^{2}\right)}^{0.25}}}}\right) \]

   5. pow-flip 98.3%

      \[\leadsto \sin^{-1} \left(\frac{h}{\sqrt{\mathsf{fma}\left(sinTheta\_O, \color{blue}{{\left(1 - {sinTheta\_O}^{2}\right)}^{\left(-0.25\right)}}, eta\right)} \cdot \sqrt{eta - \frac{sinTheta\_O}{{\left(1 - {sinTheta\_O}^{2}\right)}^{0.25}}}}\right) \]

   6. metadata-eval 98.3%

      \[\leadsto \sin^{-1} \left(\frac{h}{\sqrt{\mathsf{fma}\left(sinTheta\_O, {\left(1 - {sinTheta\_O}^{2}\right)}^{\color{blue}{-0.25}}, eta\right)} \cdot \sqrt{eta - \frac{sinTheta\_O}{{\left(1 - {sinTheta\_O}^{2}\right)}^{0.25}}}}\right) \]

   7. div-inv 98.3%

      \[\leadsto \sin^{-1} \left(\frac{h}{\sqrt{\mathsf{fma}\left(sinTheta\_O, {\left(1 - {sinTheta\_O}^{2}\right)}^{-0.25}, eta\right)} \cdot \sqrt{eta - \color{blue}{sinTheta\_O \cdot \frac{1}{{\left(1 - {sinTheta\_O}^{2}\right)}^{0.25}}}}}\right) \]

   8. pow-flip 98.3%

      \[\leadsto \sin^{-1} \left(\frac{h}{\sqrt{\mathsf{fma}\left(sinTheta\_O, {\left(1 - {sinTheta\_O}^{2}\right)}^{-0.25}, eta\right)} \cdot \sqrt{eta - sinTheta\_O \cdot \color{blue}{{\left(1 - {sinTheta\_O}^{2}\right)}^{\left(-0.25\right)}}}}\right) \]

   9. metadata-eval 98.3%

      \[\leadsto \sin^{-1} \left(\frac{h}{\sqrt{\mathsf{fma}\left(sinTheta\_O, {\left(1 - {sinTheta\_O}^{2}\right)}^{-0.25}, eta\right)} \cdot \sqrt{eta - sinTheta\_O \cdot {\left(1 - {sinTheta\_O}^{2}\right)}^{\color{blue}{-0.25}}}}\right) \]

6. Applied egg-rr 98.3%

   \[\leadsto \sin^{-1} \left(\frac{h}{\color{blue}{\sqrt{\mathsf{fma}\left(sinTheta\_O, {\left(1 - {sinTheta\_O}^{2}\right)}^{-0.25}, eta\right)} \cdot \sqrt{eta - sinTheta\_O \cdot {\left(1 - {sinTheta\_O}^{2}\right)}^{-0.25}}}}\right) \]
7. Step-by-step derivation

   1. *-un-lft-identity 98.3%

      \[\leadsto \sin^{-1} \left(\frac{\color{blue}{1 \cdot h}}{\sqrt{\mathsf{fma}\left(sinTheta\_O, {\left(1 - {sinTheta\_O}^{2}\right)}^{-0.25}, eta\right)} \cdot \sqrt{eta - sinTheta\_O \cdot {\left(1 - {sinTheta\_O}^{2}\right)}^{-0.25}}}\right) \]

   2. times-frac 98.2%

      \[\leadsto \sin^{-1} \color{blue}{\left(\frac{1}{\sqrt{\mathsf{fma}\left(sinTheta\_O, {\left(1 - {sinTheta\_O}^{2}\right)}^{-0.25}, eta\right)}} \cdot \frac{h}{\sqrt{eta - sinTheta\_O \cdot {\left(1 - {sinTheta\_O}^{2}\right)}^{-0.25}}}\right)} \]

8. Applied egg-rr 98.2%

   \[\leadsto \sin^{-1} \color{blue}{\left(\frac{1}{\sqrt{\mathsf{fma}\left(sinTheta\_O, {\left(1 - {sinTheta\_O}^{2}\right)}^{-0.25}, eta\right)}} \cdot \frac{h}{\sqrt{eta - sinTheta\_O \cdot {\left(1 - {sinTheta\_O}^{2}\right)}^{-0.25}}}\right)} \]
9. Step-by-step derivation

   1. associate-*r/ 98.2%

      \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\frac{1}{\sqrt{\mathsf{fma}\left(sinTheta\_O, {\left(1 - {sinTheta\_O}^{2}\right)}^{-0.25}, eta\right)}} \cdot h}{\sqrt{eta - sinTheta\_O \cdot {\left(1 - {sinTheta\_O}^{2}\right)}^{-0.25}}}\right)} \]

   2. pow1/2 98.2%

      \[\leadsto \sin^{-1} \left(\frac{\frac{1}{\color{blue}{{\left(\mathsf{fma}\left(sinTheta\_O, {\left(1 - {sinTheta\_O}^{2}\right)}^{-0.25}, eta\right)\right)}^{0.5}}} \cdot h}{\sqrt{eta - sinTheta\_O \cdot {\left(1 - {sinTheta\_O}^{2}\right)}^{-0.25}}}\right) \]

   3. pow-flip 98.6%

      \[\leadsto \sin^{-1} \left(\frac{\color{blue}{{\left(\mathsf{fma}\left(sinTheta\_O, {\left(1 - {sinTheta\_O}^{2}\right)}^{-0.25}, eta\right)\right)}^{\left(-0.5\right)}} \cdot h}{\sqrt{eta - sinTheta\_O \cdot {\left(1 - {sinTheta\_O}^{2}\right)}^{-0.25}}}\right) \]

   4. metadata-eval 98.6%

      \[\leadsto \sin^{-1} \left(\frac{{\left(\mathsf{fma}\left(sinTheta\_O, {\left(1 - {sinTheta\_O}^{2}\right)}^{-0.25}, eta\right)\right)}^{\color{blue}{-0.5}} \cdot h}{\sqrt{eta - sinTheta\_O \cdot {\left(1 - {sinTheta\_O}^{2}\right)}^{-0.25}}}\right) \]

10. Applied egg-rr 98.6%

    \[\leadsto \sin^{-1} \color{blue}{\left(\frac{{\left(\mathsf{fma}\left(sinTheta\_O, {\left(1 - {sinTheta\_O}^{2}\right)}^{-0.25}, eta\right)\right)}^{-0.5} \cdot h}{\sqrt{eta - sinTheta\_O \cdot {\left(1 - {sinTheta\_O}^{2}\right)}^{-0.25}}}\right)} \]
11. Step-by-step derivation

    1. associate-/l* 98.7%

       \[\leadsto \sin^{-1} \color{blue}{\left({\left(\mathsf{fma}\left(sinTheta\_O, {\left(1 - {sinTheta\_O}^{2}\right)}^{-0.25}, eta\right)\right)}^{-0.5} \cdot \frac{h}{\sqrt{eta - sinTheta\_O \cdot {\left(1 - {sinTheta\_O}^{2}\right)}^{-0.25}}}\right)} \]

12. Simplified 98.7%

    \[\leadsto \sin^{-1} \color{blue}{\left({\left(\mathsf{fma}\left(sinTheta\_O, {\left(1 - {sinTheta\_O}^{2}\right)}^{-0.25}, eta\right)\right)}^{-0.5} \cdot \frac{h}{\sqrt{eta - sinTheta\_O \cdot {\left(1 - {sinTheta\_O}^{2}\right)}^{-0.25}}}\right)} \]
13. Add Preprocessing

Alternative 2: 98.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} sinTheta_O_m = \left|sinTheta\_O\right| \\ \begin{array}{l} t_0 := {\left(1 - {sinTheta\_O\_m}^{2}\right)}^{-0.25}\\ \sin^{-1} \left(\frac{h}{\sqrt{eta - sinTheta\_O\_m \cdot t\_0} \cdot \sqrt{\mathsf{fma}\left(sinTheta\_O\_m, t\_0, eta\right)}}\right) \end{array} \end{array} \]

FPCore

sinTheta_O_m = (fabs.f32 sinTheta_O)
(FPCore (sinTheta_O_m h eta)
 :precision binary32
 (let* ((t_0 (pow (- 1.0 (pow sinTheta_O_m 2.0)) -0.25)))
   (asin
    (/
     h
     (*
      (sqrt (- eta (* sinTheta_O_m t_0)))
      (sqrt (fma sinTheta_O_m t_0 eta)))))))

C

sinTheta_O_m = fabs(sinTheta_O);
float code(float sinTheta_O_m, float h, float eta) {
	float t_0 = powf((1.0f - powf(sinTheta_O_m, 2.0f)), -0.25f);
	return asinf((h / (sqrtf((eta - (sinTheta_O_m * t_0))) * sqrtf(fmaf(sinTheta_O_m, t_0, eta)))));
}

Julia

sinTheta_O_m = abs(sinTheta_O)
function code(sinTheta_O_m, h, eta)
	t_0 = Float32(Float32(1.0) - (sinTheta_O_m ^ Float32(2.0))) ^ Float32(-0.25)
	return asin(Float32(h / Float32(sqrt(Float32(eta - Float32(sinTheta_O_m * t_0))) * sqrt(fma(sinTheta_O_m, t_0, eta)))))
end

Derivation

1. Initial program 94.4%

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

   1. add-sqr-sqrt 94.4%

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

   2. difference-of-squares 94.4%

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

   3. sqrt-div 94.4%

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

   4. sqrt-prod 48.9%

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

   5. add-sqr-sqrt 89.6%

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

   6. pow1/2 89.6%

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

   7. sqrt-pow1 89.6%

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

   8. pow2 89.6%

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

   9. metadata-eval 89.6%

      \[\leadsto \sin^{-1} \left(\frac{h}{\sqrt{\left(eta + \frac{sinTheta\_O}{{\left(1 - {sinTheta\_O}^{2}\right)}^{\color{blue}{0.25}}}\right) \cdot \left(eta - \sqrt{\frac{sinTheta\_O \cdot sinTheta\_O}{\sqrt{1 - sinTheta\_O \cdot sinTheta\_O}}}\right)}}\right) \]

   10. sqrt-div 89.6%

       \[\leadsto \sin^{-1} \left(\frac{h}{\sqrt{\left(eta + \frac{sinTheta\_O}{{\left(1 - {sinTheta\_O}^{2}\right)}^{0.25}}\right) \cdot \left(eta - \color{blue}{\frac{\sqrt{sinTheta\_O \cdot sinTheta\_O}}{\sqrt{\sqrt{1 - sinTheta\_O \cdot sinTheta\_O}}}}\right)}}\right) \]

   11. sqrt-prod 49.1%

       \[\leadsto \sin^{-1} \left(\frac{h}{\sqrt{\left(eta + \frac{sinTheta\_O}{{\left(1 - {sinTheta\_O}^{2}\right)}^{0.25}}\right) \cdot \left(eta - \frac{\color{blue}{\sqrt{sinTheta\_O} \cdot \sqrt{sinTheta\_O}}}{\sqrt{\sqrt{1 - sinTheta\_O \cdot sinTheta\_O}}}\right)}}\right) \]

   12. add-sqr-sqrt 94.4%

       \[\leadsto \sin^{-1} \left(\frac{h}{\sqrt{\left(eta + \frac{sinTheta\_O}{{\left(1 - {sinTheta\_O}^{2}\right)}^{0.25}}\right) \cdot \left(eta - \frac{\color{blue}{sinTheta\_O}}{\sqrt{\sqrt{1 - sinTheta\_O \cdot sinTheta\_O}}}\right)}}\right) \]

   13. pow1/2 94.4%

       \[\leadsto \sin^{-1} \left(\frac{h}{\sqrt{\left(eta + \frac{sinTheta\_O}{{\left(1 - {sinTheta\_O}^{2}\right)}^{0.25}}\right) \cdot \left(eta - \frac{sinTheta\_O}{\sqrt{\color{blue}{{\left(1 - sinTheta\_O \cdot sinTheta\_O\right)}^{0.5}}}}\right)}}\right) \]

   14. sqrt-pow1 94.4%

       \[\leadsto \sin^{-1} \left(\frac{h}{\sqrt{\left(eta + \frac{sinTheta\_O}{{\left(1 - {sinTheta\_O}^{2}\right)}^{0.25}}\right) \cdot \left(eta - \frac{sinTheta\_O}{\color{blue}{{\left(1 - sinTheta\_O \cdot sinTheta\_O\right)}^{\left(\frac{0.5}{2}\right)}}}\right)}}\right) \]

4. Applied egg-rr 94.4%

   \[\leadsto \sin^{-1} \left(\frac{h}{\sqrt{\color{blue}{\left(eta + \frac{sinTheta\_O}{{\left(1 - {sinTheta\_O}^{2}\right)}^{0.25}}\right) \cdot \left(eta - \frac{sinTheta\_O}{{\left(1 - {sinTheta\_O}^{2}\right)}^{0.25}}\right)}}}\right) \]
5. Step-by-step derivation

   1. sqrt-prod 98.3%

      \[\leadsto \sin^{-1} \left(\frac{h}{\color{blue}{\sqrt{eta + \frac{sinTheta\_O}{{\left(1 - {sinTheta\_O}^{2}\right)}^{0.25}}} \cdot \sqrt{eta - \frac{sinTheta\_O}{{\left(1 - {sinTheta\_O}^{2}\right)}^{0.25}}}}}\right) \]

   2. +-commutative 98.3%

      \[\leadsto \sin^{-1} \left(\frac{h}{\sqrt{\color{blue}{\frac{sinTheta\_O}{{\left(1 - {sinTheta\_O}^{2}\right)}^{0.25}} + eta}} \cdot \sqrt{eta - \frac{sinTheta\_O}{{\left(1 - {sinTheta\_O}^{2}\right)}^{0.25}}}}\right) \]

   3. div-inv 98.3%

      \[\leadsto \sin^{-1} \left(\frac{h}{\sqrt{\color{blue}{sinTheta\_O \cdot \frac{1}{{\left(1 - {sinTheta\_O}^{2}\right)}^{0.25}}} + eta} \cdot \sqrt{eta - \frac{sinTheta\_O}{{\left(1 - {sinTheta\_O}^{2}\right)}^{0.25}}}}\right) \]

   4. fma-define 98.3%

      \[\leadsto \sin^{-1} \left(\frac{h}{\sqrt{\color{blue}{\mathsf{fma}\left(sinTheta\_O, \frac{1}{{\left(1 - {sinTheta\_O}^{2}\right)}^{0.25}}, eta\right)}} \cdot \sqrt{eta - \frac{sinTheta\_O}{{\left(1 - {sinTheta\_O}^{2}\right)}^{0.25}}}}\right) \]

   5. pow-flip 98.3%

      \[\leadsto \sin^{-1} \left(\frac{h}{\sqrt{\mathsf{fma}\left(sinTheta\_O, \color{blue}{{\left(1 - {sinTheta\_O}^{2}\right)}^{\left(-0.25\right)}}, eta\right)} \cdot \sqrt{eta - \frac{sinTheta\_O}{{\left(1 - {sinTheta\_O}^{2}\right)}^{0.25}}}}\right) \]

   6. metadata-eval 98.3%

      \[\leadsto \sin^{-1} \left(\frac{h}{\sqrt{\mathsf{fma}\left(sinTheta\_O, {\left(1 - {sinTheta\_O}^{2}\right)}^{\color{blue}{-0.25}}, eta\right)} \cdot \sqrt{eta - \frac{sinTheta\_O}{{\left(1 - {sinTheta\_O}^{2}\right)}^{0.25}}}}\right) \]

   7. div-inv 98.3%

      \[\leadsto \sin^{-1} \left(\frac{h}{\sqrt{\mathsf{fma}\left(sinTheta\_O, {\left(1 - {sinTheta\_O}^{2}\right)}^{-0.25}, eta\right)} \cdot \sqrt{eta - \color{blue}{sinTheta\_O \cdot \frac{1}{{\left(1 - {sinTheta\_O}^{2}\right)}^{0.25}}}}}\right) \]

   8. pow-flip 98.3%

      \[\leadsto \sin^{-1} \left(\frac{h}{\sqrt{\mathsf{fma}\left(sinTheta\_O, {\left(1 - {sinTheta\_O}^{2}\right)}^{-0.25}, eta\right)} \cdot \sqrt{eta - sinTheta\_O \cdot \color{blue}{{\left(1 - {sinTheta\_O}^{2}\right)}^{\left(-0.25\right)}}}}\right) \]

   9. metadata-eval 98.3%

      \[\leadsto \sin^{-1} \left(\frac{h}{\sqrt{\mathsf{fma}\left(sinTheta\_O, {\left(1 - {sinTheta\_O}^{2}\right)}^{-0.25}, eta\right)} \cdot \sqrt{eta - sinTheta\_O \cdot {\left(1 - {sinTheta\_O}^{2}\right)}^{\color{blue}{-0.25}}}}\right) \]

6. Applied egg-rr 98.3%

   \[\leadsto \sin^{-1} \left(\frac{h}{\color{blue}{\sqrt{\mathsf{fma}\left(sinTheta\_O, {\left(1 - {sinTheta\_O}^{2}\right)}^{-0.25}, eta\right)} \cdot \sqrt{eta - sinTheta\_O \cdot {\left(1 - {sinTheta\_O}^{2}\right)}^{-0.25}}}}\right) \]

7. Final simplification 98.3%

   \[\leadsto \sin^{-1} \left(\frac{h}{\sqrt{eta - sinTheta\_O \cdot {\left(1 - {sinTheta\_O}^{2}\right)}^{-0.25}} \cdot \sqrt{\mathsf{fma}\left(sinTheta\_O, {\left(1 - {sinTheta\_O}^{2}\right)}^{-0.25}, eta\right)}}\right) \]
8. Add Preprocessing

Alternative 3: 97.5% accurate, 1.5× speedup

Math

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

FPCore

sinTheta_O_m = (fabs.f32 sinTheta_O)
(FPCore (sinTheta_O_m h eta)
 :precision binary32
 (asin (/ h (+ eta (* -0.5 (/ (pow sinTheta_O_m 2.0) eta))))))

C

sinTheta_O_m = fabs(sinTheta_O);
float code(float sinTheta_O_m, float h, float eta) {
	return asinf((h / (eta + (-0.5f * (powf(sinTheta_O_m, 2.0f) / 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 + ((-0.5e0) * ((sintheta_o_m ** 2.0e0) / eta)))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 94.4%

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

3. Taylor expanded in sinTheta_O around 0 96.8%

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

Alternative 4: 95.6% accurate, 3.1× 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 94.4%

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

3. Taylor expanded in eta around inf 93.9%

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

Reproduce

herbie shell --seed 2024087 
(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)))))))))