HairBSDF, gamma for a refracted ray

Percentage Accurate: 91.9% → 97.8%
Time: 12.5s
Alternatives: 2
Speedup: 3.1×

Specification

?
\[\left(\left(-1 \leq sinTheta\_O \land sinTheta\_O \leq 1\right) \land \left(-1 \leq h \land h \leq 1\right)\right) \land \left(0 \leq eta \land eta \leq 10\right)\]
\[\begin{array}{l} \\ \sin^{-1} \left(\frac{h}{\sqrt{eta \cdot eta - \frac{sinTheta\_O \cdot sinTheta\_O}{\sqrt{1 - sinTheta\_O \cdot sinTheta\_O}}}}\right) \end{array} \]
(FPCore (sinTheta_O h eta)
 :precision binary32
 (asin
  (/
   h
   (sqrt
    (-
     (* eta eta)
     (/
      (* sinTheta_O sinTheta_O)
      (sqrt (- 1.0 (* sinTheta_O sinTheta_O)))))))))
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))))))));
}

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

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 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

\begin{array}{l}

\\
\sin^{-1} \left(\frac{h}{\sqrt{eta \cdot eta - \frac{sinTheta\_O \cdot sinTheta\_O}{\sqrt{1 - sinTheta\_O \cdot sinTheta\_O}}}}\right)
\end{array}

Sampling outcomes in binary32 precision:

Local Percentage Accuracy vs ?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Accuracy vs Speed?

Herbie found 2 alternatives:

AlternativeAccuracySpeedup
The accuracy (vertical axis) and speed (horizontal axis) of each alternatives. Up and to the right is better. The red square shows the initial program, and each blue circle shows an alternative.The line shows the best available speed-accuracy tradeoffs.

Initial Program: 91.9% accurate, 1.0× speedup?

\[\begin{array}{l} \\ \sin^{-1} \left(\frac{h}{\sqrt{eta \cdot eta - \frac{sinTheta\_O \cdot sinTheta\_O}{\sqrt{1 - sinTheta\_O \cdot sinTheta\_O}}}}\right) \end{array} \]
(FPCore (sinTheta_O h eta)
 :precision binary32
 (asin
  (/
   h
   (sqrt
    (-
     (* eta eta)
     (/
      (* sinTheta_O sinTheta_O)
      (sqrt (- 1.0 (* sinTheta_O sinTheta_O)))))))))
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))))))));
}

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

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 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

\begin{array}{l}

\\
\sin^{-1} \left(\frac{h}{\sqrt{eta \cdot eta - \frac{sinTheta\_O \cdot sinTheta\_O}{\sqrt{1 - sinTheta\_O \cdot sinTheta\_O}}}}\right)
\end{array}

Alternative 1: 97.8% accurate, 2.8× speedup?

\[\begin{array}{l} \\ \sin^{-1} \left(\frac{h}{eta - 0.5 \cdot \left(sinTheta\_O \cdot \frac{sinTheta\_O}{eta}\right)}\right) \end{array} \]
(FPCore (sinTheta_O h eta)
 :precision binary32
 (asin (/ h (- eta (* 0.5 (* sinTheta_O (/ sinTheta_O eta)))))))
float code(float sinTheta_O, float h, float eta) {
	return asinf((h / (eta - (0.5f * (sinTheta_O * (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 / (eta - (0.5e0 * (sintheta_o * (sintheta_o / eta))))))
end function

function code(sinTheta_O, h, eta)
	return asin(Float32(h / Float32(eta - Float32(Float32(0.5) * Float32(sinTheta_O * Float32(sinTheta_O / eta))))))
end

function tmp = code(sinTheta_O, h, eta)
	tmp = asin((h / (eta - (single(0.5) * (sinTheta_O * (sinTheta_O / eta))))));
end

\begin{array}{l}

\\
\sin^{-1} \left(\frac{h}{eta - 0.5 \cdot \left(sinTheta\_O \cdot \frac{sinTheta\_O}{eta}\right)}\right)
\end{array}
Derivation

  1. Initial program 89.2%

    \[\sin^{-1} \left(\frac{h}{\sqrt{eta \cdot eta - \frac{sinTheta\_O \cdot sinTheta\_O}{\sqrt{1 - sinTheta\_O \cdot sinTheta\_O}}}}\right) \]
  2. Add Preprocessing

  3. Taylor expanded in sinTheta_O around 0 97.2%

    \[\leadsto \sin^{-1} \left(\frac{h}{\color{blue}{eta + -0.5 \cdot \frac{{sinTheta\_O}^{2}}{eta}}}\right) \]
  4. Step-by-step derivation

    1. associate-*r/97.2%

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

    2. *-commutative97.2%

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

  5. Simplified97.2%

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

  6. Taylor expanded in sinTheta_O around inf 97.2%

    \[\leadsto \color{blue}{\sin^{-1} \left(\frac{h}{eta - 0.5 \cdot \frac{{sinTheta\_O}^{2}}{eta}}\right)} \]
  7. Step-by-step derivation

    1. unpow297.2%

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

    2. *-un-lft-identity97.2%

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

    3. times-frac97.8%

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

  8. Applied egg-rr97.8%

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

  9. Final simplification97.8%

    \[\leadsto \sin^{-1} \left(\frac{h}{eta - 0.5 \cdot \left(sinTheta\_O \cdot \frac{sinTheta\_O}{eta}\right)}\right) \]
  10. Add Preprocessing

Alternative 2: 95.3% accurate, 3.1× speedup?

\[\begin{array}{l} \\ \sin^{-1} \left(\frac{h}{eta}\right) \end{array} \]
(FPCore (sinTheta_O h eta) :precision binary32 (asin (/ h eta)))
float code(float sinTheta_O, float h, float eta) {
	return asinf((h / 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 / eta))
end function

function code(sinTheta_O, h, eta)
	return asin(Float32(h / eta))
end

function tmp = code(sinTheta_O, h, eta)
	tmp = asin((h / eta));
end

\begin{array}{l}

\\
\sin^{-1} \left(\frac{h}{eta}\right)
\end{array}
Derivation

  1. Initial program 89.2%

    \[\sin^{-1} \left(\frac{h}{\sqrt{eta \cdot eta - \frac{sinTheta\_O \cdot sinTheta\_O}{\sqrt{1 - sinTheta\_O \cdot sinTheta\_O}}}}\right) \]
  2. Add Preprocessing

  3. Taylor expanded in eta around inf 95.5%

    \[\leadsto \sin^{-1} \left(\frac{h}{\color{blue}{eta}}\right) \]
  4. Add Preprocessing

Reproduce

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herbie shell --seed 2024107 
(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)))))))))