Average Error: 2.7 → 0.7
Time: 8.5s
Precision: binary32
\[\left(\left(-1 \leq sinTheta_O \land sinTheta_O \leq 1\right) \land \left(-1 \leq h \land h \leq 1\right)\right) \land \left(0 \leq eta \land eta \leq 10\right)\]
\[\sin^{-1} \left(\frac{h}{\sqrt{eta \cdot eta - \frac{sinTheta_O \cdot sinTheta_O}{\sqrt{1 - sinTheta_O \cdot sinTheta_O}}}}\right) \]
↓
\[\begin{array}{l} t_0 := sinTheta_O \cdot \sqrt[3]{sinTheta_O}\\ \sin^{-1} \left(\frac{h}{\mathsf{fma}\left(-0.5, t_0 \cdot \frac{\sqrt{t_0}}{eta}, eta\right)}\right) \end{array} \]
(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 (* sinTheta_O (cbrt sinTheta_O))))
   (asin (/ h (fma -0.5 (* t_0 (/ (sqrt t_0) eta)) 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) {
	float t_0 = sinTheta_O * cbrtf(sinTheta_O);
	return asinf((h / fmaf(-0.5f, (t_0 * (sqrtf(t_0) / eta)), 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)
	t_0 = Float32(sinTheta_O * cbrt(sinTheta_O))
	return asin(Float32(h / fma(Float32(-0.5), Float32(t_0 * Float32(sqrt(t_0) / eta)), 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)
↓
\begin{array}{l}
t_0 := sinTheta_O \cdot \sqrt[3]{sinTheta_O}\\
\sin^{-1} \left(\frac{h}{\mathsf{fma}\left(-0.5, t_0 \cdot \frac{\sqrt{t_0}}{eta}, eta\right)}\right)
\end{array}

Error

Bits error versus sinTheta_O

Bits error versus h

Bits error versus eta

Derivation

  1. Initial program 2.7

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

  2. Simplified2.7

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

  3. Taylor expanded in sinTheta_O around 0 0.8

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

  4. Simplified0.7

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

  5. Applied egg-rr0.7

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

  6. Applied egg-rr0.7

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

  7. Applied egg-rr0.7

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

  8. Final simplification0.7

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

Reproduce

herbie shell --seed 2022178 
(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)))))))))