Average Error: 2.5 → 0.4
Time: 29.1s
Precision: binary32
\[-1 \leq sinTheta_O \land sinTheta_O \leq 1 \land -1 \leq h \land h \leq 1 \land 0 \leq eta \land eta \leq 10\]
\[\sin^{-1} \left(\frac{h}{\sqrt{eta \cdot eta - \frac{sinTheta_O \cdot sinTheta_O}{\sqrt{1 - sinTheta_O \cdot sinTheta_O}}}}\right) \]
↓
\[\begin{array}{l} t_0 := 1 - sinTheta_O \cdot sinTheta_O\\ \sin^{-1} \left(\frac{h \cdot {\left(\mathsf{fma}\left(sinTheta_O, {\left(\frac{1}{t_0}\right)}^{0.25}, eta\right)\right)}^{-0.5}}{\sqrt{eta - \frac{sinTheta_O}{\sqrt{\sqrt{t_0}}}}}\right) \end{array} \]
\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 := 1 - sinTheta_O \cdot sinTheta_O\\
\sin^{-1} \left(\frac{h \cdot {\left(\mathsf{fma}\left(sinTheta_O, {\left(\frac{1}{t_0}\right)}^{0.25}, eta\right)\right)}^{-0.5}}{\sqrt{eta - \frac{sinTheta_O}{\sqrt{\sqrt{t_0}}}}}\right)
\end{array}

(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 (- 1.0 (* sinTheta_O sinTheta_O))))
   (asin
    (/
     (* h (pow (fma sinTheta_O (pow (/ 1.0 t_0) 0.25) eta) -0.5))
     (sqrt (- eta (/ sinTheta_O (sqrt (sqrt t_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) {
	float t_0 = 1.0f - (sinTheta_O * sinTheta_O);
	return asinf((h * powf(fmaf(sinTheta_O, powf((1.0f / t_0), 0.25f), eta), -0.5f)) / sqrtf(eta - (sinTheta_O / sqrtf(sqrtf(t_0)))));
}

Error

Bits error versus sinTheta_O

Bits error versus h

Bits error versus eta

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. Applied add-sqr-sqrt_binary322.6

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

  3. Applied times-frac_binary322.5

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

  4. Applied difference-of-squares_binary322.5

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

  5. Applied sqrt-prod_binary320.5

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

  6. Applied associate-/r*_binary320.5

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

  7. Taylor expanded in h around 0 0.5

    \[\leadsto \sin^{-1} \left(\frac{\color{blue}{\sqrt{\frac{1}{eta + sinTheta_O \cdot {\left(\frac{1}{1 - {sinTheta_O}^{2}}\right)}^{0.25}}} \cdot h}}{\sqrt{eta - \frac{sinTheta_O}{\sqrt{\sqrt{1 - sinTheta_O \cdot sinTheta_O}}}}}\right) \]

  8. Simplified0.5

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

  9. Applied inv-pow_binary320.5

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

  10. Applied sqrt-pow1_binary320.4

    \[\leadsto \sin^{-1} \left(\frac{h \cdot \color{blue}{{\left(\mathsf{fma}\left(sinTheta_O, {\left(\frac{1}{1 - sinTheta_O \cdot sinTheta_O}\right)}^{0.25}, eta\right)\right)}^{\left(\frac{-1}{2}\right)}}}{\sqrt{eta - \frac{sinTheta_O}{\sqrt{\sqrt{1 - sinTheta_O \cdot sinTheta_O}}}}}\right) \]

  11. Final simplification0.4

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

Reproduce

herbie shell --seed 2021225 
(FPCore (sinTheta_O h eta)
  :name "HairBSDF, gamma for a refracted ray"
  :precision binary32
  :pre (and (<= -1.0 sinTheta_O 1.0) (<= -1.0 h 1.0) (<= 0.0 eta 10.0))
  (asin (/ h (sqrt (- (* eta eta) (/ (* sinTheta_O sinTheta_O) (sqrt (- 1.0 (* sinTheta_O sinTheta_O)))))))))