HairBSDF, gamma for a refracted ray

Average Error: 92.2% → 97.8%

Time: 24.3s

Precision: binary32

Cost: 23104.00

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

Math

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

↓

\[\begin{array}{l} t_0 := sinTheta_O \cdot \sqrt[3]{sinTheta_O}\\ \sin^{-1} \left(\frac{h}{\mathsf{fma}\left(-0.5, \frac{t_0}{\sqrt[3]{eta \cdot \left(\frac{eta}{t_0} \cdot \sqrt[3]{eta \cdot {\left(\frac{eta}{sinTheta_O}\right)}^{2}}\right)}}, eta\right)}\right) \end{array} \]

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 (cbrt sinTheta_O))))
   (asin
    (/
     h
     (fma
      -0.5
      (/
       t_0
       (cbrt
        (* eta (* (/ eta t_0) (cbrt (* eta (pow (/ eta sinTheta_O) 2.0)))))))
      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 * cbrtf(sinTheta_O);
	return asinf((h / fmaf(-0.5f, (t_0 / cbrtf((eta * ((eta / t_0) * cbrtf((eta * powf((eta / sinTheta_O), 2.0f))))))), 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(sinTheta_O * cbrt(sinTheta_O))
	return asin(Float32(h / fma(Float32(-0.5), Float32(t_0 / cbrt(Float32(eta * Float32(Float32(eta / t_0) * cbrt(Float32(eta * (Float32(eta / sinTheta_O) ^ Float32(2.0)))))))), eta)))
end

Derivation

1. Initial program 92.2

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

2. Taylor expanded in sinTheta_O around 0 97.4

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

3. Simplified 97.8

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

   [Start] 97.4	\[ \sin^{-1} \left(\frac{h}{eta + -0.5 \cdot \frac{{sinTheta_O}^{2}}{eta}}\right) \]
   +-commutative [=>] 97.4	\[ \sin^{-1} \left(\frac{h}{\color{blue}{-0.5 \cdot \frac{{sinTheta_O}^{2}}{eta} + eta}}\right) \]
   fma-def [=>] 97.4	\[ \sin^{-1} \left(\frac{h}{\color{blue}{\mathsf{fma}\left(-0.5, \frac{{sinTheta_O}^{2}}{eta}, eta\right)}}\right) \]
   unpow2 [=>] 97.4	\[ \sin^{-1} \left(\frac{h}{\mathsf{fma}\left(-0.5, \frac{\color{blue}{sinTheta_O \cdot sinTheta_O}}{eta}, eta\right)}\right) \]
   associate-/l* [=>] 97.8	\[ \sin^{-1} \left(\frac{h}{\mathsf{fma}\left(-0.5, \color{blue}{\frac{sinTheta_O}{\frac{eta}{sinTheta_O}}}, eta\right)}\right) \]

4. Applied egg-rr 97.8

   \[\leadsto \sin^{-1} \left(\frac{h}{\mathsf{fma}\left(-0.5, \color{blue}{\frac{\frac{sinTheta_O}{{\left(\sqrt[3]{\frac{eta}{sinTheta_O}}\right)}^{2}}}{\sqrt[3]{eta}} \cdot \sqrt[3]{sinTheta_O}}, eta\right)}\right) \]

5. Simplified 97.8

   \[\leadsto \sin^{-1} \left(\frac{h}{\mathsf{fma}\left(-0.5, \color{blue}{\frac{sinTheta_O \cdot \sqrt[3]{sinTheta_O}}{\sqrt[3]{eta} \cdot {\left(\sqrt[3]{\frac{eta}{sinTheta_O}}\right)}^{2}}}, eta\right)}\right) \]
   Proof

   [Start] 97.8	\[ \sin^{-1} \left(\frac{h}{\mathsf{fma}\left(-0.5, \frac{\frac{sinTheta_O}{{\left(\sqrt[3]{\frac{eta}{sinTheta_O}}\right)}^{2}}}{\sqrt[3]{eta}} \cdot \sqrt[3]{sinTheta_O}, eta\right)}\right) \]
   associate-/l/ [=>] 97.8	\[ \sin^{-1} \left(\frac{h}{\mathsf{fma}\left(-0.5, \color{blue}{\frac{sinTheta_O}{\sqrt[3]{eta} \cdot {\left(\sqrt[3]{\frac{eta}{sinTheta_O}}\right)}^{2}}} \cdot \sqrt[3]{sinTheta_O}, eta\right)}\right) \]
   associate-*l/ [=>] 97.8	\[ \sin^{-1} \left(\frac{h}{\mathsf{fma}\left(-0.5, \color{blue}{\frac{sinTheta_O \cdot \sqrt[3]{sinTheta_O}}{\sqrt[3]{eta} \cdot {\left(\sqrt[3]{\frac{eta}{sinTheta_O}}\right)}^{2}}}, eta\right)}\right) \]

6. Applied egg-rr 97.8

   \[\leadsto \sin^{-1} \left(\frac{h}{\mathsf{fma}\left(-0.5, \frac{sinTheta_O \cdot \sqrt[3]{sinTheta_O}}{\color{blue}{\sqrt[3]{{\left(\frac{eta}{sinTheta_O}\right)}^{2} \cdot eta}}}, eta\right)}\right) \]

7. Applied egg-rr 97.8

   \[\leadsto \sin^{-1} \left(\frac{h}{\mathsf{fma}\left(-0.5, \frac{sinTheta_O \cdot \sqrt[3]{sinTheta_O}}{\sqrt[3]{\color{blue}{\left(\frac{eta}{\frac{sinTheta_O \cdot \sqrt[3]{sinTheta_O}}{1}} \cdot \sqrt[3]{eta \cdot {\left(\frac{eta}{sinTheta_O}\right)}^{2}}\right)} \cdot eta}}, eta\right)}\right) \]

8. Final simplification 97.8

   \[\leadsto \sin^{-1} \left(\frac{h}{\mathsf{fma}\left(-0.5, \frac{sinTheta_O \cdot \sqrt[3]{sinTheta_O}}{\sqrt[3]{eta \cdot \left(\frac{eta}{sinTheta_O \cdot \sqrt[3]{sinTheta_O}} \cdot \sqrt[3]{eta \cdot {\left(\frac{eta}{sinTheta_O}\right)}^{2}}\right)}}, eta\right)}\right) \]

Alternatives

Alternative 1
Error	97.8%
Cost	13344.00

\[\sin^{-1} \left(\frac{h}{\mathsf{fma}\left(-0.5, \frac{sinTheta_O \cdot \sqrt[3]{sinTheta_O}}{\sqrt[3]{\frac{eta \cdot \frac{eta}{sinTheta_O}}{\frac{sinTheta_O}{eta}}}}, eta\right)}\right) \]

Alternative 2
Error	97.4%
Cost	3552.00

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

Alternative 3
Error	97.8%
Cost	3552.00

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

Alternative 4
Error	95.2%
Cost	3296.00

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

Reproduce

herbie shell --seed 2023093 
(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)))))))))