HairBSDF, gamma for a refracted ray

Average Error: 2.5 → 0.7

Time: 13.7s

Precision: binary32

Cost: 9952

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

Math

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

↓

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

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 (/ h (- eta (exp (log (* sinTheta_O (* (/ sinTheta_O eta) 0.5))))))))

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((h / (eta - expf(logf((sinTheta_O * ((sinTheta_O / eta) * 0.5f)))))));
}

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((h / (eta - exp(log((sintheta_o * ((sintheta_o / eta) * 0.5e0)))))))
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(h / Float32(eta - exp(log(Float32(sinTheta_O * Float32(Float32(sinTheta_O / eta) * Float32(0.5))))))))
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((h / (eta - exp(log((sinTheta_O * ((sinTheta_O / eta) * single(0.5))))))));
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 0.8

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

3. Simplified 0.8

   \[\leadsto \sin^{-1} \left(\frac{h}{\color{blue}{eta + -0.5 \cdot \frac{sinTheta_O \cdot sinTheta_O}{eta}}}\right) \]
   Proof
   (+.f32 eta (*.f32 -1/2 (/.f32 (*.f32 sinTheta_O sinTheta_O) eta))): 0 points increase in error, 0 points decrease in error
   (+.f32 eta (*.f32 -1/2 (/.f32 (Rewrite<= unpow2_binary32 (pow.f32 sinTheta_O 2)) eta))): 0 points increase in error, 0 points decrease in error

4. Applied egg-rr 0.7

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

5. Taylor expanded in h around 0 0.8

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

6. Simplified 0.7

   \[\leadsto \color{blue}{\sin^{-1} \left(\frac{h}{eta - 0.5 \cdot \left(sinTheta_O \cdot \frac{sinTheta_O}{eta}\right)}\right)} \]
   Proof
   (asin.f32 (/.f32 h (-.f32 eta (*.f32 1/2 (*.f32 sinTheta_O (/.f32 sinTheta_O eta)))))): 0 points increase in error, 0 points decrease in error
   (asin.f32 (/.f32 h (-.f32 eta (*.f32 1/2 (Rewrite=> associate-*r/_binary32 (/.f32 (*.f32 sinTheta_O sinTheta_O) eta)))))): 11 points increase in error, 2 points decrease in error
   (asin.f32 (/.f32 h (-.f32 eta (*.f32 1/2 (/.f32 (Rewrite<= unpow2_binary32 (pow.f32 sinTheta_O 2)) eta))))): 0 points increase in error, 0 points decrease in error
   (asin.f32 (/.f32 h (Rewrite=> cancel-sign-sub-inv_binary32 (+.f32 eta (*.f32 (neg.f32 1/2) (/.f32 (pow.f32 sinTheta_O 2) eta)))))): 0 points increase in error, 0 points decrease in error
   (asin.f32 (/.f32 h (+.f32 eta (*.f32 (Rewrite=> metadata-eval -1/2) (/.f32 (pow.f32 sinTheta_O 2) eta))))): 0 points increase in error, 0 points decrease in error
   (asin.f32 (/.f32 h (Rewrite=> +-commutative_binary32 (+.f32 (*.f32 -1/2 (/.f32 (pow.f32 sinTheta_O 2) eta)) eta)))): 0 points increase in error, 0 points decrease in error
   (asin.f32 (/.f32 h (+.f32 (*.f32 -1/2 (/.f32 (pow.f32 sinTheta_O 2) eta)) (Rewrite<= *-lft-identity_binary32 (*.f32 1 eta))))): 0 points increase in error, 0 points decrease in error
   (asin.f32 (/.f32 h (+.f32 (*.f32 -1/2 (/.f32 (pow.f32 sinTheta_O 2) eta)) (*.f32 (Rewrite<= metadata-eval (neg.f32 -1)) eta)))): 0 points increase in error, 0 points decrease in error
   (asin.f32 (/.f32 h (Rewrite<= cancel-sign-sub-inv_binary32 (-.f32 (*.f32 -1/2 (/.f32 (pow.f32 sinTheta_O 2) eta)) (*.f32 -1 eta))))): 0 points increase in error, 0 points decrease in error
   (asin.f32 (/.f32 h (Rewrite=> cancel-sign-sub-inv_binary32 (+.f32 (*.f32 -1/2 (/.f32 (pow.f32 sinTheta_O 2) eta)) (*.f32 (neg.f32 -1) eta))))): 0 points increase in error, 0 points decrease in error
   (asin.f32 (/.f32 h (+.f32 (*.f32 -1/2 (/.f32 (pow.f32 sinTheta_O 2) eta)) (*.f32 (Rewrite=> metadata-eval 1) eta)))): 0 points increase in error, 0 points decrease in error
   (asin.f32 (/.f32 h (+.f32 (*.f32 -1/2 (/.f32 (pow.f32 sinTheta_O 2) eta)) (Rewrite=> *-lft-identity_binary32 eta)))): 0 points increase in error, 0 points decrease in error
   (asin.f32 (/.f32 h (Rewrite<= +-commutative_binary32 (+.f32 eta (*.f32 -1/2 (/.f32 (pow.f32 sinTheta_O 2) eta)))))): 0 points increase in error, 0 points decrease in error
   (asin.f32 (/.f32 h (+.f32 eta (*.f32 (Rewrite<= metadata-eval (neg.f32 1/2)) (/.f32 (pow.f32 sinTheta_O 2) eta))))): 0 points increase in error, 0 points decrease in error
   (asin.f32 (/.f32 h (Rewrite<= cancel-sign-sub-inv_binary32 (-.f32 eta (*.f32 1/2 (/.f32 (pow.f32 sinTheta_O 2) eta)))))): 0 points increase in error, 0 points decrease in error
   (asin.f32 (/.f32 h (Rewrite=> cancel-sign-sub-inv_binary32 (+.f32 eta (*.f32 (neg.f32 1/2) (/.f32 (pow.f32 sinTheta_O 2) eta)))))): 0 points increase in error, 0 points decrease in error
   (asin.f32 (/.f32 h (+.f32 eta (*.f32 (Rewrite=> metadata-eval -1/2) (/.f32 (pow.f32 sinTheta_O 2) eta))))): 0 points increase in error, 0 points decrease in error

7. Applied egg-rr 0.7

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

8. Final simplification 0.7

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

Alternatives

Alternative 1
Error	0.7
Cost	9952

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

Alternative 2
Error	0.8
Cost	3552

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

Alternative 3
Error	0.8
Cost	3552

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

Alternative 4
Error	0.7
Cost	3552

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

Alternative 5
Error	1.4
Cost	3296

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

Reproduce

herbie shell --seed 2022332 
(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)))))))))