HairBSDF, gamma for a refracted ray

Percentage Accurate: 92.1% → 97.3%

Time: 10.6s

Alternatives: 4

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: 92.1% 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: 97.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\sin \tan^{-1} sinTheta\_O \cdot sinTheta\_O}\\ \sin^{-1} \left(\frac{h}{\sqrt{eta + e^{\log t\_0}} \cdot \sqrt{eta - t\_0}}\right) \end{array} \end{array} \]

FPCore

(FPCore (sinTheta_O h eta)
 :precision binary32
 (let* ((t_0 (sqrt (* (sin (atan sinTheta_O)) sinTheta_O))))
   (asin (/ h (* (sqrt (+ eta (exp (log t_0)))) (sqrt (- eta t_0)))))))

C

float code(float sinTheta_O, float h, float eta) {
	float t_0 = sqrtf((sinf(atanf(sinTheta_O)) * sinTheta_O));
	return asinf((h / (sqrtf((eta + expf(logf(t_0)))) * sqrtf((eta - t_0)))));
}

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) :: t_0
    t_0 = sqrt((sin(atan(sintheta_o)) * sintheta_o))
    code = asin((h / (sqrt((eta + exp(log(t_0)))) * sqrt((eta - t_0)))))
end function

Julia

function code(sinTheta_O, h, eta)
	t_0 = sqrt(Float32(sin(atan(sinTheta_O)) * sinTheta_O))
	return asin(Float32(h / Float32(sqrt(Float32(eta + exp(log(t_0)))) * sqrt(Float32(eta - t_0)))))
end

MATLAB

function tmp = code(sinTheta_O, h, eta)
	t_0 = sqrt((sin(atan(sinTheta_O)) * sinTheta_O));
	tmp = asin((h / (sqrt((eta + exp(log(t_0)))) * sqrt((eta - t_0)))));
end

Derivation

1. Initial program 90.6%

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

4. Applied rewrites 52.2%

   \[\leadsto \sin^{-1} \left(\frac{h}{\sqrt{\color{blue}{\frac{{eta}^{6} + {\left(\sin \tan^{-1} sinTheta\_O \cdot \left(-sinTheta\_O\right)\right)}^{3}}{{eta}^{4} + \left({\left(\sin \tan^{-1} sinTheta\_O \cdot sinTheta\_O\right)}^{2} - \left(eta \cdot eta\right) \cdot \left(\sin \tan^{-1} sinTheta\_O \cdot \left(-sinTheta\_O\right)\right)\right)}}}}\right) \]

6. Applied rewrites 97.4%

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

   1. lift-sqrt.f32 N/A

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

   2. pow1/2 N/A

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

   3. metadata-eval N/A

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

   4. pow-to-exp N/A

      \[\leadsto \sin^{-1} \left(\frac{h}{\sqrt{eta + \color{blue}{e^{\log \left(\sin \tan^{-1} sinTheta\_O \cdot sinTheta\_O\right) \cdot \left(\frac{1}{4} \cdot 2\right)}}} \cdot \sqrt{eta - \sqrt{\sin \tan^{-1} sinTheta\_O \cdot sinTheta\_O}}}\right) \]

   5. lower-exp.f32 N/A

      \[\leadsto \sin^{-1} \left(\frac{h}{\sqrt{eta + \color{blue}{e^{\log \left(\sin \tan^{-1} sinTheta\_O \cdot sinTheta\_O\right) \cdot \left(\frac{1}{4} \cdot 2\right)}}} \cdot \sqrt{eta - \sqrt{\sin \tan^{-1} sinTheta\_O \cdot sinTheta\_O}}}\right) \]

   6. rem-log-exp N/A

      \[\leadsto \sin^{-1} \left(\frac{h}{\sqrt{eta + e^{\color{blue}{\log \left(e^{\log \left(\sin \tan^{-1} sinTheta\_O \cdot sinTheta\_O\right) \cdot \left(\frac{1}{4} \cdot 2\right)}\right)}}} \cdot \sqrt{eta - \sqrt{\sin \tan^{-1} sinTheta\_O \cdot sinTheta\_O}}}\right) \]

   7. pow-to-exp N/A

      \[\leadsto \sin^{-1} \left(\frac{h}{\sqrt{eta + e^{\log \color{blue}{\left({\left(\sin \tan^{-1} sinTheta\_O \cdot sinTheta\_O\right)}^{\left(\frac{1}{4} \cdot 2\right)}\right)}}} \cdot \sqrt{eta - \sqrt{\sin \tan^{-1} sinTheta\_O \cdot sinTheta\_O}}}\right) \]

   8. metadata-eval N/A

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

   9. pow1/2 N/A

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

   10. lift-sqrt.f32 N/A

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

   11. lower-log.f32 97.4

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

8. Applied rewrites 97.4%

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

Alternative 2: 97.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\sin \tan^{-1} sinTheta\_O \cdot sinTheta\_O}\\ \sin^{-1} \left(\frac{h}{\sqrt{eta + t\_0} \cdot \sqrt{eta - t\_0}}\right) \end{array} \end{array} \]

FPCore

(FPCore (sinTheta_O h eta)
 :precision binary32
 (let* ((t_0 (sqrt (* (sin (atan sinTheta_O)) sinTheta_O))))
   (asin (/ h (* (sqrt (+ eta t_0)) (sqrt (- eta t_0)))))))

C

float code(float sinTheta_O, float h, float eta) {
	float t_0 = sqrtf((sinf(atanf(sinTheta_O)) * sinTheta_O));
	return asinf((h / (sqrtf((eta + t_0)) * sqrtf((eta - t_0)))));
}

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) :: t_0
    t_0 = sqrt((sin(atan(sintheta_o)) * sintheta_o))
    code = asin((h / (sqrt((eta + t_0)) * sqrt((eta - t_0)))))
end function

Julia

function code(sinTheta_O, h, eta)
	t_0 = sqrt(Float32(sin(atan(sinTheta_O)) * sinTheta_O))
	return asin(Float32(h / Float32(sqrt(Float32(eta + t_0)) * sqrt(Float32(eta - t_0)))))
end

MATLAB

function tmp = code(sinTheta_O, h, eta)
	t_0 = sqrt((sin(atan(sinTheta_O)) * sinTheta_O));
	tmp = asin((h / (sqrt((eta + t_0)) * sqrt((eta - t_0)))));
end

Derivation

1. Initial program 90.6%

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

4. Applied rewrites 52.2%

   \[\leadsto \sin^{-1} \left(\frac{h}{\sqrt{\color{blue}{\frac{{eta}^{6} + {\left(\sin \tan^{-1} sinTheta\_O \cdot \left(-sinTheta\_O\right)\right)}^{3}}{{eta}^{4} + \left({\left(\sin \tan^{-1} sinTheta\_O \cdot sinTheta\_O\right)}^{2} - \left(eta \cdot eta\right) \cdot \left(\sin \tan^{-1} sinTheta\_O \cdot \left(-sinTheta\_O\right)\right)\right)}}}}\right) \]

6. Applied rewrites 97.4%

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

Alternative 3: 96.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(sinTheta_O, h, eta)
	tmp = asin(((((single(0.5) / eta) * (((sinTheta_O * sinTheta_O) * h) / eta)) + h) / eta));
end

Derivation

1. Initial program 90.6%

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

3. Taylor expanded in h around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

5. Applied rewrites 86.2%

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

6. Taylor expanded in sinTheta_O around 0

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

8. Applied rewrites 94.8%

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

9. Taylor expanded in eta around inf

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

    1. lower-/.f32 N/A

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

11. Applied rewrites 17.6%

    \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\mathsf{fma}\left(0.5 \cdot \left(\frac{sinTheta\_O \cdot sinTheta\_O}{eta} \cdot \frac{h}{eta}\right), \sqrt{\frac{1}{1 - sinTheta\_O \cdot sinTheta\_O}}, \left(h \cdot \frac{\frac{{sinTheta\_O}^{4}}{1 - sinTheta\_O \cdot sinTheta\_O} \cdot -0.75}{{eta}^{4}}\right) \cdot -0.5\right) + h}{eta}\right)} \]

12. Taylor expanded in sinTheta_O around 0

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

14. Applied rewrites 96.1%

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

Alternative 4: 95.0% 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 90.6%

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

3. Taylor expanded in sinTheta_O around 0

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

   1. lower-/.f32 95.3

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

5. Applied rewrites 95.3%

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

Reproduce

herbie shell --seed 2024340 
(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)))))))))