Average Error: 2.6 → 0.6
Time: 8.4s
Precision: binary32
\[\left(\left(-1 \leq sinTheta_O \land sinTheta_O \leq 1\right) \land \left(-1 \leq h \land h \leq 1\right)\right) \land \left(0 \leq eta \land eta \leq 10\right)\]
\[\sin^{-1} \left(\frac{h}{\sqrt{eta \cdot eta - \frac{sinTheta_O \cdot sinTheta_O}{\sqrt{1 - sinTheta_O \cdot sinTheta_O}}}}\right) \]
↓
\[\sin^{-1} \left(\frac{h}{\mathsf{fma}\left(\frac{sinTheta_O}{\frac{eta}{sinTheta_O}}, -0.5, eta\right)}\right) \]
(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 (fma (/ sinTheta_O (/ eta sinTheta_O)) -0.5 eta))))
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 / fmaf((sinTheta_O / (eta / sinTheta_O)), -0.5f, eta)));
}

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 / fma(Float32(sinTheta_O / Float32(eta / sinTheta_O)), Float32(-0.5), eta)))
end

\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(\frac{sinTheta_O}{\frac{eta}{sinTheta_O}}, -0.5, eta\right)}\right)

Error

Bits error versus sinTheta_O

Bits error versus h

Bits error versus eta

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. 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. Applied egg-rr1.1

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

  4. Taylor expanded in h around 0 0.8

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

  5. Simplified0.6

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

  6. Final simplification0.6

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

Reproduce

herbie shell --seed 2022150 
(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)))))))))