HairBSDF, gamma for a refracted ray

Percentage Accurate: 91.7% → 98.7%

Time: 11.0s

Alternatives: 4

Speedup: 1.5×

Specification

\[\left(\left(-1 \leq sinTheta\_O \land sinTheta\_O \leq 1\right) \land \left(-1 \leq h \land h \leq 1\right)\right) \land \left(0 \leq eta \land eta \leq 10\right)\]

Math

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

FPCore

(FPCore (sinTheta_O h eta)
 :precision binary32
 (asin
  (/
   h
   (sqrt
    (-
     (* eta eta)
     (/
      (* sinTheta_O sinTheta_O)
      (sqrt (- 1.0 (* sinTheta_O sinTheta_O)))))))))

C

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))))))));
}

Fortran

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

Julia

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

MATLAB

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

Initial Program: 91.7% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (sinTheta_O h eta)
 :precision binary32
 (asin
  (/
   h
   (sqrt
    (-
     (* eta eta)
     (/
      (* sinTheta_O sinTheta_O)
      (sqrt (- 1.0 (* sinTheta_O sinTheta_O)))))))))

C

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))))))));
}

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 98.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;sinTheta\_O \cdot sinTheta\_O \leq 0:\\ \;\;\;\;\sin^{-1} \left(\frac{h}{eta}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{h}{\sqrt{\left(eta - sinTheta\_O\right) \cdot \left(sinTheta\_O + eta\right)}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (sinTheta_O h eta)
 :precision binary32
 (if (<= (* sinTheta_O sinTheta_O) 0.0)
   (asin (/ h eta))
   (asin (/ h (sqrt (* (- eta sinTheta_O) (+ sinTheta_O eta)))))))

C

float code(float sinTheta_O, float h, float eta) {
	float tmp;
	if ((sinTheta_O * sinTheta_O) <= 0.0f) {
		tmp = asinf((h / eta));
	} else {
		tmp = asinf((h / sqrtf(((eta - sinTheta_O) * (sinTheta_O + eta)))));
	}
	return tmp;
}

Fortran

real(4) function code(sintheta_o, h, eta)
    real(4), intent (in) :: sintheta_o
    real(4), intent (in) :: h
    real(4), intent (in) :: eta
    real(4) :: tmp
    if ((sintheta_o * sintheta_o) <= 0.0e0) then
        tmp = asin((h / eta))
    else
        tmp = asin((h / sqrt(((eta - sintheta_o) * (sintheta_o + eta)))))
    end if
    code = tmp
end function

Julia

function code(sinTheta_O, h, eta)
	tmp = Float32(0.0)
	if (Float32(sinTheta_O * sinTheta_O) <= Float32(0.0))
		tmp = asin(Float32(h / eta));
	else
		tmp = asin(Float32(h / sqrt(Float32(Float32(eta - sinTheta_O) * Float32(sinTheta_O + eta)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(sinTheta_O, h, eta)
	tmp = single(0.0);
	if ((sinTheta_O * sinTheta_O) <= single(0.0))
		tmp = asin((h / eta));
	else
		tmp = asin((h / sqrt(((eta - sinTheta_O) * (sinTheta_O + eta)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 sinTheta_O sinTheta_O) < 0.0

   1. Initial program 91.3%

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

   3. Taylor expanded in eta around inf

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

      1. lower-/.f32 99.4

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

   5. Applied rewrites 99.4%

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

   if 0.0 < (*.f32 sinTheta_O sinTheta_O)

   1. Initial program 99.3%

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

   3. Taylor expanded in sinTheta_O around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. difference-of-squares N/A

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

      7. lower-*.f32 N/A

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

      8. lower-+.f32 N/A

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

      9. lower--.f32 99.4

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

   5. Applied rewrites 99.4%

      \[\leadsto \sin^{-1} \left(\frac{h}{\sqrt{\color{blue}{\left(eta + sinTheta\_O\right) \cdot \left(eta - sinTheta\_O\right)}}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;sinTheta\_O \cdot sinTheta\_O \leq 0:\\ \;\;\;\;\sin^{-1} \left(\frac{h}{eta}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{h}{\sqrt{\left(eta - sinTheta\_O\right) \cdot \left(sinTheta\_O + eta\right)}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \sin^{-1} \left(\frac{\mathsf{fma}\left(\frac{\frac{sinTheta\_O}{eta} \cdot sinTheta\_O}{eta} \cdot \left(0.5 \cdot h\right), \sqrt{\frac{1}{1 - sinTheta\_O \cdot sinTheta\_O}}, h\right)}{eta}\right) \end{array} \]

FPCore

(FPCore (sinTheta_O h eta)
 :precision binary32
 (asin
  (/
   (fma
    (* (/ (* (/ sinTheta_O eta) sinTheta_O) eta) (* 0.5 h))
    (sqrt (/ 1.0 (- 1.0 (* sinTheta_O sinTheta_O))))
    h)
   eta)))

C

float code(float sinTheta_O, float h, float eta) {
	return asinf((fmaf(((((sinTheta_O / eta) * sinTheta_O) / eta) * (0.5f * h)), sqrtf((1.0f / (1.0f - (sinTheta_O * sinTheta_O)))), h) / eta));
}

Julia

function code(sinTheta_O, h, eta)
	return asin(Float32(fma(Float32(Float32(Float32(Float32(sinTheta_O / eta) * sinTheta_O) / eta) * Float32(Float32(0.5) * h)), sqrt(Float32(Float32(1.0) / Float32(Float32(1.0) - Float32(sinTheta_O * sinTheta_O)))), h) / eta))
end

Derivation

1. Initial program 94.7%

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

3. Taylor expanded in eta around inf

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

   1. lower-/.f32 N/A

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

5. Applied rewrites 93.8%

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

7. Applied rewrites 97.4%

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

8. Final simplification 97.4%

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

Alternative 3: 97.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sin^{-1} \left(\frac{\mathsf{fma}\left(\frac{\frac{sinTheta\_O}{eta} \cdot sinTheta\_O}{eta} \cdot \left(0.5 \cdot h\right), \sqrt{1}, h\right)}{eta}\right) \end{array} \]

FPCore

(FPCore (sinTheta_O h eta)
 :precision binary32
 (asin
  (/
   (fma (* (/ (* (/ sinTheta_O eta) sinTheta_O) eta) (* 0.5 h)) (sqrt 1.0) h)
   eta)))

C

float code(float sinTheta_O, float h, float eta) {
	return asinf((fmaf(((((sinTheta_O / eta) * sinTheta_O) / eta) * (0.5f * h)), sqrtf(1.0f), h) / eta));
}

Julia

function code(sinTheta_O, h, eta)
	return asin(Float32(fma(Float32(Float32(Float32(Float32(sinTheta_O / eta) * sinTheta_O) / eta) * Float32(Float32(0.5) * h)), sqrt(Float32(1.0)), h) / eta))
end

Derivation

1. Initial program 94.7%

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

3. Taylor expanded in eta around inf

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

   1. lower-/.f32 N/A

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

5. Applied rewrites 93.8%

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

7. Applied rewrites 97.4%

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

8. Taylor expanded in sinTheta_O around 0

   \[\leadsto \sin^{-1} \left(\frac{\mathsf{fma}\left(\left(\frac{1}{2} \cdot h\right) \cdot \frac{\frac{sinTheta\_O}{eta} \cdot sinTheta\_O}{eta}, \sqrt{1}, h\right)}{eta}\right) \]
9. Step-by-step derivation

10. Applied rewrites 97.4%

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

11. Final simplification 97.4%

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

Alternative 4: 95.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \sin^{-1} \left(\frac{h}{eta}\right) \end{array} \]

FPCore

(FPCore (sinTheta_O h eta) :precision binary32 (asin (/ h eta)))

C

float code(float sinTheta_O, float h, float eta) {
	return asinf((h / eta));
}

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 94.7%

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

3. Taylor expanded in eta around inf

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

   1. lower-/.f32 95.1

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

5. Applied rewrites 95.1%

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

Reproduce

herbie shell --seed 2024240 
(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)))))))))