Average Error: 2.5 → 0.7
Time: 10.3s
Precision: binary32
\[\left(\left(-1 \leq sinTheta_O \land sinTheta_O \leq 1\right) \land \left(-1 \leq h \land h \leq 1\right)\right) \land \left(0 \leq eta \land eta \leq 10\right)\]
\[\sin^{-1} \left(\frac{h}{\sqrt{eta \cdot eta - \frac{sinTheta_O \cdot sinTheta_O}{\sqrt{1 - sinTheta_O \cdot sinTheta_O}}}}\right) \]
↓
\[\sin^{-1} \left(\frac{h}{eta - 0.5 \cdot {\left(sinTheta_O \cdot {eta}^{-0.5}\right)}^{2}}\right) \]
(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 (pow (* sinTheta_O (pow eta -0.5)) 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 * powf((sinTheta_O * powf(eta, -0.5f)), 2.0f)))));
}

real(4) function code(sintheta_o, h, eta)
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: h
    real(4), intent (in) :: eta
    code = asin((h / sqrt(((eta * eta) - ((sintheta_o * sintheta_o) / sqrt((1.0e0 - (sintheta_o * sintheta_o))))))))
end function
↓
real(4) function code(sintheta_o, h, eta)
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: h
    real(4), intent (in) :: eta
    code = asin((h / (eta - (0.5e0 * ((sintheta_o * (eta ** (-0.5e0))) ** 2.0e0)))))
end function

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(sinTheta_O * (eta ^ Float32(-0.5))) ^ Float32(2.0))))))
end

function tmp = code(sinTheta_O, h, eta)
	tmp = asin((h / sqrt(((eta * eta) - ((sinTheta_O * sinTheta_O) / sqrt((single(1.0) - (sinTheta_O * sinTheta_O))))))));
end
↓
function tmp = code(sinTheta_O, h, eta)
	tmp = asin((h / (eta - (single(0.5) * ((sinTheta_O * (eta ^ single(-0.5))) ^ single(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(sinTheta_O \cdot {eta}^{-0.5}\right)}^{2}}\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 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 egg-rr0.7

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

  4. Applied egg-rr0.7

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

  5. Final simplification0.7

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

Reproduce

herbie shell --seed 2022133 
(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)))))))))