?

Average Error: 2.6 → 0.6
Time: 18.1s
Precision: binary32
Cost: 3552

?

\[\left(\left(-1 \leq sinTheta_O \land sinTheta_O \leq 1\right) \land \left(-1 \leq h \land h \leq 1\right)\right) \land \left(0 \leq eta \land eta \leq 10\right)\]
\[\sin^{-1} \left(\frac{h}{\sqrt{eta \cdot eta - \frac{sinTheta_O \cdot sinTheta_O}{\sqrt{1 - sinTheta_O \cdot sinTheta_O}}}}\right) \]
↓
\[\sin^{-1} \left(\frac{h}{eta + sinTheta_O \cdot \left(-0.5 \cdot \frac{sinTheta_O}{eta}\right)}\right) \]
(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 (* sinTheta_O (* -0.5 (/ sinTheta_O 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 + (sinTheta_O * (-0.5f * (sinTheta_O / eta))))));
}

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 + (sintheta_o * ((-0.5e0) * (sintheta_o / eta))))))
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(sinTheta_O * Float32(Float32(-0.5) * Float32(sinTheta_O / eta))))))
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 + (sinTheta_O * (single(-0.5) * (sinTheta_O / 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)
↓
\sin^{-1} \left(\frac{h}{eta + sinTheta_O \cdot \left(-0.5 \cdot \frac{sinTheta_O}{eta}\right)}\right)

Error?

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 2.7

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

  3. Simplified2.7

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

    [Start]2.7

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

    unpow2 [=>]2.7

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

    mul-1-neg [=>]2.7

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

    unsub-neg [=>]2.7

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

    unpow2 [=>]2.7

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

  4. Applied egg-rr0.5

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

  5. Simplified0.5

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

    [Start]0.5

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

    +-commutative [=>]0.5

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

  6. Taylor expanded in eta around inf 0.8

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

  7. Simplified0.6

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

    [Start]0.8

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

    associate-+r+ [=>]0.8

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

  8. Taylor expanded in sinTheta_O around 0 0.8

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

  9. Simplified0.6

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

    [Start]0.8

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

    associate-*r/ [=>]0.8

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

    unpow2 [=>]0.8

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

    associate-*r* [=>]0.8

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

    metadata-eval [<=]0.8

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

    associate-*r* [<=]0.8

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

    neg-mul-1 [<=]0.8

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

    associate-*l/ [<=]0.6

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

    *-commutative [=>]0.6

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

    remove-double-neg [<=]0.6

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

    neg-mul-1 [=>]0.6

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

    times-frac [=>]0.6

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

    metadata-eval [=>]0.6

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

    neg-mul-1 [=>]0.6

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

    mul-1-neg [<=]0.6

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

    times-frac [=>]0.6

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

    metadata-eval [=>]0.6

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

    *-commutative [<=]0.6

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

    *-rgt-identity [=>]0.6

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

  10. Final simplification0.6

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

Alternatives

Alternative 1
Error0.8
Cost3552
\[\sin^{-1} \left(\frac{h}{eta + -0.5 \cdot \frac{sinTheta_O \cdot sinTheta_O}{eta}}\right) \]
Alternative 2
Error1.5
Cost3424
\[\sin^{-1} \left(\frac{h}{sinTheta_O + \left(eta - sinTheta_O\right)}\right) \]
Alternative 3
Error1.5
Cost3296
\[\sin^{-1} \left(\frac{h}{eta}\right) \]

Error

Reproduce?

herbie shell --seed 2023053 
(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)))))))))