Average Error: 2.5 → 0.7
Time: 7.1s
Precision: binary32
\[\left(\left(-1 \leq sinTheta_O \land sinTheta_O \leq 1\right) \land \left(-1 \leq h \land h \leq 1\right)\right) \land \left(0 \leq eta \land eta \leq 10\right)\]
\[\sin^{-1} \left(\frac{h}{\sqrt{eta \cdot eta - \frac{sinTheta_O \cdot sinTheta_O}{\sqrt{1 - sinTheta_O \cdot sinTheta_O}}}}\right) \]
↓
\[\begin{array}{l} t_0 := \sqrt[3]{\log \left(sinTheta_O \cdot \frac{sinTheta_O}{eta}\right)}\\ \sin^{-1} \left(\frac{h}{eta + e^{t_0 \cdot {t_0}^{2}} \cdot -0.5}\right) \end{array} \]
(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 (cbrt (log (* sinTheta_O (/ sinTheta_O eta))))))
   (asin (/ h (+ eta (* (exp (* t_0 (pow t_0 2.0))) -0.5))))))
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 = cbrtf(logf((sinTheta_O * (sinTheta_O / eta))));
	return asinf((h / (eta + (expf((t_0 * powf(t_0, 2.0f))) * -0.5f))));
}

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 = cbrt(log(Float32(sinTheta_O * Float32(sinTheta_O / eta))))
	return asin(Float32(h / Float32(eta + Float32(exp(Float32(t_0 * (t_0 ^ Float32(2.0)))) * Float32(-0.5)))))
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 := \sqrt[3]{\log \left(sinTheta_O \cdot \frac{sinTheta_O}{eta}\right)}\\
\sin^{-1} \left(\frac{h}{eta + e^{t_0 \cdot {t_0}^{2}} \cdot -0.5}\right)
\end{array}

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 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-rr0.7

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

  4. Applied egg-rr0.7

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

  5. Final simplification0.7

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

Reproduce

herbie shell --seed 2022162 
(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)))))))))