HairBSDF, gamma for a refracted ray

Average Error: 2.6 → 0.4

Time: 21.7s

Precision: binary32

Cost: 19904

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

Math

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

↓

\[\begin{array}{l} t_0 := {\left(1 - sinTheta_O \cdot sinTheta_O\right)}^{-0.25}\\ \sin^{-1} \left({\left(\mathsf{fma}\left(sinTheta_O, t_0, eta\right)\right)}^{-0.5} \cdot \frac{h}{\sqrt{eta - sinTheta_O \cdot t_0}}\right) \end{array} \]

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 (pow (- 1.0 (* sinTheta_O sinTheta_O)) -0.25)))
   (asin
    (*
     (pow (fma sinTheta_O t_0 eta) -0.5)
     (/ h (sqrt (- eta (* sinTheta_O t_0))))))))

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 = powf((1.0f - (sinTheta_O * sinTheta_O)), -0.25f);
	return asinf((powf(fmaf(sinTheta_O, t_0, eta), -0.5f) * (h / sqrtf((eta - (sinTheta_O * t_0))))));
}

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(1.0) - Float32(sinTheta_O * sinTheta_O)) ^ Float32(-0.25)
	return asin(Float32((fma(sinTheta_O, t_0, eta) ^ Float32(-0.5)) * Float32(h / sqrt(Float32(eta - Float32(sinTheta_O * t_0))))))
end

Derivation

1. Initial program 2.6

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

2. Applied egg-rr 2.6

   \[\leadsto \sin^{-1} \left(\frac{h}{\sqrt{\color{blue}{\left(eta + \frac{sinTheta_O}{{\left(1 - sinTheta_O \cdot sinTheta_O\right)}^{0.25}}\right) \cdot \left(eta - \frac{sinTheta_O}{{\left(1 - sinTheta_O \cdot sinTheta_O\right)}^{0.25}}\right)}}}\right) \]

3. Applied egg-rr 0.6

   \[\leadsto \sin^{-1} \color{blue}{\left(\frac{1}{\sqrt{\mathsf{fma}\left(sinTheta_O, {\left(1 - sinTheta_O \cdot sinTheta_O\right)}^{-0.25}, eta\right)}} \cdot \frac{h}{\sqrt{eta - sinTheta_O \cdot {\left(1 - sinTheta_O \cdot sinTheta_O\right)}^{-0.25}}}\right)} \]

4. Applied egg-rr 0.4

   \[\leadsto \sin^{-1} \left(\color{blue}{{\left(\mathsf{fma}\left(sinTheta_O, {\left(1 - sinTheta_O \cdot sinTheta_O\right)}^{-0.25}, eta\right)\right)}^{-0.5}} \cdot \frac{h}{\sqrt{eta - sinTheta_O \cdot {\left(1 - sinTheta_O \cdot sinTheta_O\right)}^{-0.25}}}\right) \]

5. Final simplification 0.4

   \[\leadsto \sin^{-1} \left({\left(\mathsf{fma}\left(sinTheta_O, {\left(1 - sinTheta_O \cdot sinTheta_O\right)}^{-0.25}, eta\right)\right)}^{-0.5} \cdot \frac{h}{\sqrt{eta - sinTheta_O \cdot {\left(1 - sinTheta_O \cdot sinTheta_O\right)}^{-0.25}}}\right) \]

Alternatives

Alternative 1
Error	0.5
Cost	19872

\[\begin{array}{l} t_0 := {\left(1 - sinTheta_O \cdot sinTheta_O\right)}^{-0.25}\\ \sin^{-1} \left(\frac{h}{\sqrt{eta - sinTheta_O \cdot t_0} \cdot \sqrt{\mathsf{fma}\left(sinTheta_O, t_0, eta\right)}}\right) \end{array} \]

Alternative 2
Error	0.6
Cost	6816

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

Alternative 3
Error	0.6
Cost	3552

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

Alternative 4
Error	1.4
Cost	3296

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

Reproduce

herbie shell --seed 2022306 
(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)))))))))