HairBSDF, gamma for a refracted ray

Percentage Accurate: 92.0% → 91.9%
Time: 10.5s
Alternatives: 2
Speedup: N/A×

Specification

?
\[\left(\left(-1 \leq sinTheta\_O \land sinTheta\_O \leq 1\right) \land \left(-1 \leq h \land h \leq 1\right)\right) \land \left(0 \leq eta \land eta \leq 10\right)\]
\[\sin^{-1} \left(\frac{h}{\sqrt{eta \cdot eta - \frac{sinTheta\_O \cdot sinTheta\_O}{\sqrt{1 - sinTheta\_O \cdot sinTheta\_O}}}}\right) \]
(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)
use fmin_fmax_functions
    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

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

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: 92.0% accurate, 1.0× speedup?

\[\sin^{-1} \left(\frac{h}{\sqrt{eta \cdot eta - \frac{sinTheta\_O \cdot sinTheta\_O}{\sqrt{1 - sinTheta\_O \cdot sinTheta\_O}}}}\right) \]
(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)
use fmin_fmax_functions
    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

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

Alternative 1: 91.9% accurate, 1.2× speedup?

\[\sin^{-1} \left(\frac{h}{\sqrt{eta \cdot eta - \frac{sinTheta\_O \cdot sinTheta\_O}{\sqrt{1}}}}\right) \]
(FPCore (sinTheta_O h eta)
  :precision binary32
  (asin
 (/
  h
  (sqrt (- (* eta eta) (/ (* sinTheta_O sinTheta_O) (sqrt 1.0)))))))
float code(float sinTheta_O, float h, float eta) {
	return asinf((h / sqrtf(((eta * eta) - ((sinTheta_O * sinTheta_O) / sqrtf(1.0f))))));
}

real(4) function code(sintheta_o, h, eta)
use fmin_fmax_functions
    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))))))
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(1.0)))))))
end

function tmp = code(sinTheta_O, h, eta)
	tmp = asin((h / sqrt(((eta * eta) - ((sinTheta_O * sinTheta_O) / sqrt(single(1.0)))))));
end

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

  1. Initial program 92.0%

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

  2. Taylor expanded in sinTheta_O around 0

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

    1. Applied rewrites91.9%

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

    Alternative 2: 9.0% accurate, 3.1× speedup?

    \[\sin^{-1} \left(\frac{0}{-eta}\right) \]
    (FPCore (sinTheta_O h eta)
  :precision binary32
  (asin (/ 0.0 (- eta))))
    float code(float sinTheta_O, float h, float eta) {
	return asinf((0.0f / -eta));
}

    real(4) function code(sintheta_o, h, eta)
use fmin_fmax_functions
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: h
    real(4), intent (in) :: eta
    code = asin((0.0e0 / -eta))
end function

    function code(sinTheta_O, h, eta)
	return asin(Float32(Float32(0.0) / Float32(-eta)))
end

    function tmp = code(sinTheta_O, h, eta)
	tmp = asin((single(0.0) / -eta));
end

    \sin^{-1} \left(\frac{0}{-eta}\right)
    Derivation

    1. Initial program 92.0%

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

    2. Taylor expanded in eta around -inf

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

      1. lower-*.f325.9%

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

    4. Applied rewrites5.9%

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

    5. Taylor expanded in undef-var around zero

      \[\leadsto \sin^{-1} \left(\frac{\color{blue}{0}}{-1 \cdot eta}\right) \]
    6. Step-by-step derivation

      1. Applied rewrites9.0%

        \[\leadsto \sin^{-1} \left(\frac{\color{blue}{0}}{-1 \cdot eta}\right) \]
      2. Step-by-step derivation

        1. lift-*.f32N/A

          \[\leadsto \sin^{-1} \left(\frac{0}{-1 \cdot \color{blue}{eta}}\right) \]

        2. mul-1-negN/A

          \[\leadsto \sin^{-1} \left(\frac{0}{\mathsf{neg}\left(eta\right)}\right) \]

        3. lower-neg.f329.0%

          \[\leadsto \sin^{-1} \left(\frac{0}{-eta}\right) \]

      3. Applied rewrites9.0%

        \[\leadsto \sin^{-1} \left(\frac{0}{-eta}\right) \]
      4. Add Preprocessing

      Reproduce

      ?
      herbie shell --seed 2025322 
(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)))))))))