HairBSDF, gamma for a refracted ray

Average Error: 2.5 → 0.5

Time: 17.5s

Precision: binary32

Cost: 9920

\[\left(\left(-1 \leq sinTheta_O \land sinTheta_O \leq 1\right) \land \left(-1 \leq h \land h \leq 1\right)\right) \land \left(0 \leq eta \land eta \leq 10\right)\]

Math

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

↓

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

FPCore

(FPCore (sinTheta_O h eta)
 :precision binary32
 (asin
  (/
   h
   (sqrt
    (-
     (* eta eta)
     (/
      (* sinTheta_O sinTheta_O)
      (sqrt (- 1.0 (* sinTheta_O sinTheta_O)))))))))

↓

(FPCore (sinTheta_O h eta)
 :precision binary32
 (asin (* (pow (- eta sinTheta_O) -0.5) (/ h (sqrt (+ eta 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))))))));
}

↓

float code(float sinTheta_O, float h, float eta) {
	return asinf((powf((eta - sinTheta_O), -0.5f) * (h / sqrtf((eta + 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

↓

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((((eta - sintheta_o) ** (-0.5e0)) * (h / sqrt((eta + 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

↓

function code(sinTheta_O, h, eta)
	return asin(Float32((Float32(eta - sinTheta_O) ^ Float32(-0.5)) * Float32(h / sqrt(Float32(eta + 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

↓

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

Derivation

1. Initial program 2.5

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

2. Taylor expanded in sinTheta_O around 0 2.6

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

3. Simplified 2.6

   \[\leadsto \sin^{-1} \left(\frac{h}{\sqrt{\color{blue}{eta \cdot eta - sinTheta_O \cdot sinTheta_O}}}\right) \]
   Proof
   (-.f32 (*.f32 eta eta) (*.f32 sinTheta_O sinTheta_O)): 0 points increase in error, 0 points decrease in error
   (Rewrite<= unsub-neg_binary32 (+.f32 (*.f32 eta eta) (neg.f32 (*.f32 sinTheta_O sinTheta_O)))): 0 points increase in error, 0 points decrease in error
   (+.f32 (Rewrite<= unpow2_binary32 (pow.f32 eta 2)) (neg.f32 (*.f32 sinTheta_O sinTheta_O))): 0 points increase in error, 0 points decrease in error
   (+.f32 (pow.f32 eta 2) (Rewrite<= mul-1-neg_binary32 (*.f32 -1 (*.f32 sinTheta_O sinTheta_O)))): 0 points increase in error, 0 points decrease in error
   (+.f32 (pow.f32 eta 2) (*.f32 -1 (Rewrite<= unpow2_binary32 (pow.f32 sinTheta_O 2)))): 0 points increase in error, 0 points decrease in error

4. Applied egg-rr 0.5

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

5. Simplified 0.5

   \[\leadsto \sin^{-1} \left(\frac{h}{\color{blue}{\sqrt{sinTheta_O + eta} \cdot \sqrt{eta - sinTheta_O}}}\right) \]
   Proof
   (*.f32 (sqrt.f32 (+.f32 sinTheta_O eta)) (sqrt.f32 (-.f32 eta sinTheta_O))): 0 points increase in error, 0 points decrease in error
   (*.f32 (sqrt.f32 (Rewrite<= +-commutative_binary32 (+.f32 eta sinTheta_O))) (sqrt.f32 (-.f32 eta sinTheta_O))): 0 points increase in error, 0 points decrease in error

6. Applied egg-rr 0.6

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

7. Simplified 0.6

   \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\frac{h}{\sqrt{eta + sinTheta_O}}}{\sqrt{eta - sinTheta_O}}\right)} \]
   Proof
   (/.f32 (/.f32 h (sqrt.f32 (+.f32 eta sinTheta_O))) (sqrt.f32 (-.f32 eta sinTheta_O))): 0 points increase in error, 0 points decrease in error
   (/.f32 (/.f32 h (sqrt.f32 (Rewrite<= +-commutative_binary32 (+.f32 sinTheta_O eta)))) (sqrt.f32 (-.f32 eta sinTheta_O))): 0 points increase in error, 0 points decrease in error
   (/.f32 (Rewrite<= *-rgt-identity_binary32 (*.f32 (/.f32 h (sqrt.f32 (+.f32 sinTheta_O eta))) 1)) (sqrt.f32 (-.f32 eta sinTheta_O))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-*r/_binary32 (*.f32 (/.f32 h (sqrt.f32 (+.f32 sinTheta_O eta))) (/.f32 1 (sqrt.f32 (-.f32 eta sinTheta_O))))): 36 points increase in error, 32 points decrease in error

8. Applied egg-rr 0.5

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

9. Simplified 0.5

   \[\leadsto \sin^{-1} \color{blue}{\left({\left(eta - sinTheta_O\right)}^{-0.5} \cdot \frac{h}{\sqrt{eta + sinTheta_O}}\right)} \]
   Proof
   (*.f32 (pow.f32 (-.f32 eta sinTheta_O) -1/2) (/.f32 h (sqrt.f32 (+.f32 eta sinTheta_O)))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= *-commutative_binary32 (*.f32 (/.f32 h (sqrt.f32 (+.f32 eta sinTheta_O))) (pow.f32 (-.f32 eta sinTheta_O) -1/2))): 0 points increase in error, 0 points decrease in error

10. Final simplification 0.5

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

Alternatives

Alternative 1
Error	0.5
Cost	9888

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

Alternative 2
Error	0.7
Cost	3552

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

Alternative 3
Error	1.6
Cost	3296

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

Reproduce

herbie shell --seed 2022329 
(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)))))))))