Average Error: 2.5 → 0.6
Time: 13.9s
Precision: binary32
\[\left(\left(-1 \leq sinTheta_O \land sinTheta_O \leq 1\right) \land \left(-1 \leq h \land h \leq 1\right)\right) \land \left(0 \leq eta \land eta \leq 10\right)\]
\[\sin^{-1} \left(\frac{h}{\sqrt{eta \cdot eta - \frac{sinTheta_O \cdot sinTheta_O}{\sqrt{1 - sinTheta_O \cdot sinTheta_O}}}}\right) \]
↓
\[\sin^{-1} \left(\frac{h}{eta - 0.5 \cdot \left(\left(\left(sinTheta_O \cdot \sqrt[3]{sinTheta_O}\right) \cdot \frac{{\left(\sqrt[3]{sinTheta_O}\right)}^{2}}{eta}\right) \cdot \sqrt{\frac{1}{1 - {sinTheta_O}^{2}}}\right)}\right) \]
(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
   (-
    eta
    (*
     0.5
     (*
      (* (* sinTheta_O (cbrt sinTheta_O)) (/ (pow (cbrt sinTheta_O) 2.0) eta))
      (sqrt (/ 1.0 (- 1.0 (pow sinTheta_O 2.0))))))))))
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 / (eta - (0.5f * (((sinTheta_O * cbrtf(sinTheta_O)) * (powf(cbrtf(sinTheta_O), 2.0f) / eta)) * sqrtf((1.0f / (1.0f - powf(sinTheta_O, 2.0f)))))))));
}

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 / Float32(eta - Float32(Float32(0.5) * Float32(Float32(Float32(sinTheta_O * cbrt(sinTheta_O)) * Float32((cbrt(sinTheta_O) ^ Float32(2.0)) / eta)) * sqrt(Float32(Float32(1.0) / Float32(Float32(1.0) - (sinTheta_O ^ Float32(2.0))))))))))
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}{eta - 0.5 \cdot \left(\left(\left(sinTheta_O \cdot \sqrt[3]{sinTheta_O}\right) \cdot \frac{{\left(\sqrt[3]{sinTheta_O}\right)}^{2}}{eta}\right) \cdot \sqrt{\frac{1}{1 - {sinTheta_O}^{2}}}\right)}\right)

Error

Bits error versus sinTheta_O

Bits error versus h

Bits error versus eta

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 2.5

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

  2. Taylor expanded in eta around inf 0.8

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

  3. Applied *-un-lft-identity_binary320.8

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

  4. Applied add-cube-cbrt_binary320.8

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

  5. Applied unpow-prod-down_binary320.8

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

  6. Applied times-frac_binary320.6

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

  7. Simplified0.6

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

  8. Final simplification0.6

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

Reproduce

herbie shell --seed 2022130 
(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)))))))))