HairBSDF, gamma for a refracted ray

Average Error: 8.42% → 2.11%

Time: 17.6s

Precision: binary32

Cost: 7328

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{sinTheta_O}{eta} \cdot -0.5\\ \sin^{-1} \left(\frac{\frac{h}{\frac{\mathsf{fma}\left(sinTheta_O, t_0, eta\right)}{\frac{sinTheta_O}{eta} \cdot \left(sinTheta_O \cdot -0.5\right) - eta}}}{sinTheta_O \cdot t_0 - eta}\right) \end{array} \]

FPCore

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

↓

(FPCore (sinTheta_O h eta)
 :precision binary32
 (let* ((t_0 (* (/ sinTheta_O eta) -0.5)))
   (asin
    (/
     (/
      h
      (/
       (fma sinTheta_O t_0 eta)
       (- (* (/ sinTheta_O eta) (* sinTheta_O -0.5)) eta)))
     (- (* sinTheta_O t_0) eta)))))

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) {
	float t_0 = (sinTheta_O / eta) * -0.5f;
	return asinf(((h / (fmaf(sinTheta_O, t_0, eta) / (((sinTheta_O / eta) * (sinTheta_O * -0.5f)) - eta))) / ((sinTheta_O * t_0) - eta)));
}

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)
	t_0 = Float32(Float32(sinTheta_O / eta) * Float32(-0.5))
	return asin(Float32(Float32(h / Float32(fma(sinTheta_O, t_0, eta) / Float32(Float32(Float32(sinTheta_O / eta) * Float32(sinTheta_O * Float32(-0.5))) - eta))) / Float32(Float32(sinTheta_O * t_0) - eta)))
end

Derivation

1. Initial program 8.42

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

2. Simplified 8.42

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

   [Start] 8.42	\[ \sin^{-1} \left(\frac{h}{\sqrt{eta \cdot eta - \frac{sinTheta_O \cdot sinTheta_O}{\sqrt{1 - sinTheta_O \cdot sinTheta_O}}}}\right) \]
   fma-neg [=>] 8.42	\[ \sin^{-1} \left(\frac{h}{\sqrt{\color{blue}{\mathsf{fma}\left(eta, eta, -\frac{sinTheta_O \cdot sinTheta_O}{\sqrt{1 - sinTheta_O \cdot sinTheta_O}}\right)}}}\right) \]
   distribute-neg-frac [=>] 8.42	\[ \sin^{-1} \left(\frac{h}{\sqrt{\mathsf{fma}\left(eta, eta, \color{blue}{\frac{-sinTheta_O \cdot sinTheta_O}{\sqrt{1 - sinTheta_O \cdot sinTheta_O}}}\right)}}\right) \]
   distribute-rgt-neg-in [=>] 8.42	\[ \sin^{-1} \left(\frac{h}{\sqrt{\mathsf{fma}\left(eta, eta, \frac{\color{blue}{sinTheta_O \cdot \left(-sinTheta_O\right)}}{\sqrt{1 - sinTheta_O \cdot sinTheta_O}}\right)}}\right) \]

3. Taylor expanded in sinTheta_O around 0 2.64

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

4. Simplified 2.08

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

   [Start] 2.64	\[ \sin^{-1} \left(\frac{h}{eta + -0.5 \cdot \frac{{sinTheta_O}^{2}}{eta}}\right) \]
   +-commutative [=>] 2.64	\[ \sin^{-1} \left(\frac{h}{\color{blue}{-0.5 \cdot \frac{{sinTheta_O}^{2}}{eta} + eta}}\right) \]
   fma-def [=>] 2.64	\[ \sin^{-1} \left(\frac{h}{\color{blue}{\mathsf{fma}\left(-0.5, \frac{{sinTheta_O}^{2}}{eta}, eta\right)}}\right) \]
   unpow2 [=>] 2.64	\[ \sin^{-1} \left(\frac{h}{\mathsf{fma}\left(-0.5, \frac{\color{blue}{sinTheta_O \cdot sinTheta_O}}{eta}, eta\right)}\right) \]
   associate-/l* [=>] 2.08	\[ \sin^{-1} \left(\frac{h}{\mathsf{fma}\left(-0.5, \color{blue}{\frac{sinTheta_O}{\frac{eta}{sinTheta_O}}}, eta\right)}\right) \]

5. Applied egg-rr 10.85

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

6. Applied egg-rr 2.1

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

7. Applied egg-rr 2.11

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

8. Final simplification 2.11

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

Alternatives

Alternative 1
Error	2.64%
Cost	3552

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

Alternative 2
Error	2.08%
Cost	3552

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

Alternative 3
Error	4.66%
Cost	3296

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

Reproduce

herbie shell --seed 2023121 
(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)))))))))