?

Average Accuracy: 92.1% → 98.0%
Time: 19.0s
Precision: binary32
Cost: 16384

?

\[\left(\left(-1 \leq sinTheta_O \land sinTheta_O \leq 1\right) \land \left(-1 \leq h \land h \leq 1\right)\right) \land \left(0 \leq eta \land eta \leq 10\right)\]
\[\sin^{-1} \left(\frac{h}{\sqrt{eta \cdot eta - \frac{sinTheta_O \cdot sinTheta_O}{\sqrt{1 - sinTheta_O \cdot sinTheta_O}}}}\right) \]
↓
\[\sin^{-1} \left(\frac{h}{\mathsf{fma}\left(-0.5, {\left(\sqrt[3]{sinTheta_O}\right)}^{2} \cdot \left(\sqrt[3]{sinTheta_O} \cdot \frac{sinTheta_O}{eta}\right), eta\right)}\right) \]
(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
   (fma
    -0.5
    (* (pow (cbrt sinTheta_O) 2.0) (* (cbrt sinTheta_O) (/ sinTheta_O 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) {
	return asinf((h / fmaf(-0.5f, (powf(cbrtf(sinTheta_O), 2.0f) * (cbrtf(sinTheta_O) * (sinTheta_O / 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)
	return asin(Float32(h / fma(Float32(-0.5), Float32((cbrt(sinTheta_O) ^ Float32(2.0)) * Float32(cbrt(sinTheta_O) * Float32(sinTheta_O / 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)
↓
\sin^{-1} \left(\frac{h}{\mathsf{fma}\left(-0.5, {\left(\sqrt[3]{sinTheta_O}\right)}^{2} \cdot \left(\sqrt[3]{sinTheta_O} \cdot \frac{sinTheta_O}{eta}\right), eta\right)}\right)

Error?

Derivation?

  1. Initial program 92.1%

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

  2. Simplified92.1%

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

    [Start]92.1

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

    fma-neg [=>]92.1

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

    distribute-neg-frac [=>]92.1

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

    distribute-rgt-neg-in [=>]92.1

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

  3. Taylor expanded in sinTheta_O around 0 97.5%

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

  4. Simplified98.0%

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

    [Start]97.5

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

    +-commutative [=>]97.5

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

    fma-def [=>]97.5

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

    unpow2 [=>]97.5

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

    associate-/l* [=>]98.0

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

  5. Applied egg-rr98.0%

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

    [Start]98.0

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

    associate-/r/ [=>]98.0

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

    *-commutative [<=]98.0

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

    add-cube-cbrt [=>]98.0

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

    associate-*l* [=>]98.0

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

    pow2 [=>]98.0

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

  6. Final simplification98.0%

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

Alternatives

Alternative 1
Accuracy98.0%
Cost16288
\[\sin^{-1} \left(\frac{h}{\mathsf{fma}\left(-0.5, \frac{sinTheta_O}{\frac{{e}^{\log eta}}{sinTheta_O}}, eta\right)}\right) \]
Alternative 2
Accuracy98.0%
Cost13088
\[\sin^{-1} \left(\frac{h}{\mathsf{fma}\left(-0.5, e^{\log \left(sinTheta_O \cdot \frac{sinTheta_O}{eta}\right)}, eta\right)}\right) \]
Alternative 3
Accuracy98.0%
Cost3552
\[\sin^{-1} \left(\frac{h}{eta + -0.5 \cdot \left(sinTheta_O \cdot \frac{sinTheta_O}{eta}\right)}\right) \]
Alternative 4
Accuracy95.4%
Cost3296
\[\sin^{-1} \left(\frac{h}{eta}\right) \]

Error

Reproduce?

herbie shell --seed 2023153 
(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)))))))))