Average Error: 2.6 → 0.7
Time: 26.3s
Precision: binary32
\[\left(\left(-1 \leq sinTheta_O \land sinTheta_O \leq 1\right) \land \left(-1 \leq h \land h \leq 1\right)\right) \land \left(0 \leq eta \land eta \leq 10\right)\]
\[\sin^{-1} \left(\frac{h}{\sqrt{eta \cdot eta - \frac{sinTheta_O \cdot sinTheta_O}{\sqrt{1 - sinTheta_O \cdot sinTheta_O}}}}\right) \]
↓
\[\sin^{-1} \left(\frac{h}{eta - \left(0.5 \cdot \frac{sinTheta_O}{\sqrt[3]{eta} \cdot \sqrt[3]{eta}}\right) \cdot \frac{sinTheta_O}{\sqrt[3]{eta}}}\right) \]
\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 - \left(0.5 \cdot \frac{sinTheta_O}{\sqrt[3]{eta} \cdot \sqrt[3]{eta}}\right) \cdot \frac{sinTheta_O}{\sqrt[3]{eta}}}\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 eta) (cbrt eta))))
     (/ sinTheta_O (cbrt 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 / (eta - ((0.5f * (sinTheta_O / (cbrtf(eta) * cbrtf(eta)))) * (sinTheta_O / cbrtf(eta)))));
}

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 2.6

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

  2. Taylor expanded in sinTheta_O around 0 0.9

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

  3. Applied add-cube-cbrt_binary320.9

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

  4. Applied sqr-pow_binary320.9

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

  5. Applied times-frac_binary320.7

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

  6. Applied associate-*r*_binary320.7

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

  7. Simplified0.7

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

  8. Final simplification0.7

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

Reproduce

herbie shell --seed 2022068 
(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)))))))))