HairBSDF, gamma for a refracted ray

Average Error: 7.4 → 0.5

Time: 2.1min

Precision: binary64

Cost: 26176

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

Math

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

↓

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

FPCore

(FPCore (sinTheta_O h eta)
 :precision binary64
 (asin
  (/
   h
   (sqrt
    (-
     (* eta eta)
     (/
      (* sinTheta_O sinTheta_O)
      (sqrt (- 1.0 (* sinTheta_O sinTheta_O)))))))))

↓

(FPCore (sinTheta_O h eta)
 :precision binary64
 (asin (/ h (fma -0.5 (* (/ sinTheta_O (fabs eta)) sinTheta_O) (fabs eta)))))

C

double code(double sinTheta_O, double h, double eta) {
	return asin((h / sqrt(((eta * eta) - ((sinTheta_O * sinTheta_O) / sqrt((1.0 - (sinTheta_O * sinTheta_O))))))));
}

↓

double code(double sinTheta_O, double h, double eta) {
	return asin((h / fma(-0.5, ((sinTheta_O / fabs(eta)) * sinTheta_O), fabs(eta))));
}

Julia

function code(sinTheta_O, h, eta)
	return asin(Float64(h / sqrt(Float64(Float64(eta * eta) - Float64(Float64(sinTheta_O * sinTheta_O) / sqrt(Float64(1.0 - Float64(sinTheta_O * sinTheta_O))))))))
end

↓

function code(sinTheta_O, h, eta)
	return asin(Float64(h / fma(-0.5, Float64(Float64(sinTheta_O / abs(eta)) * sinTheta_O), abs(eta))))
end

Wolfram

code[sinTheta$95$O_, h_, eta_] := N[ArcSin[N[(h / N[Sqrt[N[(N[(eta * eta), $MachinePrecision] - N[(N[(sinTheta$95$O * sinTheta$95$O), $MachinePrecision] / N[Sqrt[N[(1.0 - N[(sinTheta$95$O * sinTheta$95$O), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

↓

code[sinTheta$95$O_, h_, eta_] := N[ArcSin[N[(h / N[(-0.5 * N[(N[(sinTheta$95$O / N[Abs[eta], $MachinePrecision]), $MachinePrecision] * sinTheta$95$O), $MachinePrecision] + N[Abs[eta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 7.4

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

2. Taylor expanded in sinTheta_O around 0 7.7

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

3. Simplified 0.6

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

4. Applied egg-rr 0.5

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

Alternatives

Alternative 1
Error	0.6
Cost	13760

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

Alternative 2
Error	1.1
Cost	6592

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

Reproduce

herbie shell --seed 2023033 
(FPCore (sinTheta_O h eta)
  :name "HairBSDF, gamma for a refracted ray"
  :precision binary64
  :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)))))))))