HairBSDF, gamma for a refracted ray

Percentage Accurate: 92.4% → 98.8%

Time: 10.2s

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: 92.4% 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: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} sinTheta_O_m = \left|sinTheta\_O\right| \\ \begin{array}{l} \mathbf{if}\;sinTheta\_O\_m \leq 1.3000000155431645 \cdot 10^{-23}:\\ \;\;\;\;\sin^{-1} \left(\frac{h}{eta}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{h}{\sqrt{eta \cdot eta - \frac{sinTheta\_O\_m \cdot sinTheta\_O\_m}{\sqrt{1 - sinTheta\_O\_m \cdot sinTheta\_O\_m}}}}\right)\\ \end{array} \end{array} \]

FPCore

sinTheta_O_m = (fabs.f32 sinTheta_O)
(FPCore (sinTheta_O_m h eta)
 :precision binary32
 (if (<= sinTheta_O_m 1.3000000155431645e-23)
   (asin (/ h eta))
   (asin
    (/
     h
     (sqrt
      (-
       (* eta eta)
       (/
        (* sinTheta_O_m sinTheta_O_m)
        (sqrt (- 1.0 (* sinTheta_O_m sinTheta_O_m))))))))))

C

sinTheta_O_m = fabs(sinTheta_O);
float code(float sinTheta_O_m, float h, float eta) {
	float tmp;
	if (sinTheta_O_m <= 1.3000000155431645e-23f) {
		tmp = asinf((h / eta));
	} else {
		tmp = asinf((h / sqrtf(((eta * eta) - ((sinTheta_O_m * sinTheta_O_m) / sqrtf((1.0f - (sinTheta_O_m * sinTheta_O_m))))))));
	}
	return tmp;
}

Fortran

sinTheta_O_m = abs(sintheta_o)
real(4) function code(sintheta_o_m, h, eta)
    real(4), intent (in) :: sintheta_o_m
    real(4), intent (in) :: h
    real(4), intent (in) :: eta
    real(4) :: tmp
    if (sintheta_o_m <= 1.3000000155431645e-23) then
        tmp = asin((h / eta))
    else
        tmp = asin((h / sqrt(((eta * eta) - ((sintheta_o_m * sintheta_o_m) / sqrt((1.0e0 - (sintheta_o_m * sintheta_o_m))))))))
    end if
    code = tmp
end function

Julia

sinTheta_O_m = abs(sinTheta_O)
function code(sinTheta_O_m, h, eta)
	tmp = Float32(0.0)
	if (sinTheta_O_m <= Float32(1.3000000155431645e-23))
		tmp = asin(Float32(h / eta));
	else
		tmp = asin(Float32(h / sqrt(Float32(Float32(eta * eta) - Float32(Float32(sinTheta_O_m * sinTheta_O_m) / sqrt(Float32(Float32(1.0) - Float32(sinTheta_O_m * sinTheta_O_m))))))));
	end
	return tmp
end

MATLAB

sinTheta_O_m = abs(sinTheta_O);
function tmp_2 = code(sinTheta_O_m, h, eta)
	tmp = single(0.0);
	if (sinTheta_O_m <= single(1.3000000155431645e-23))
		tmp = asin((h / eta));
	else
		tmp = asin((h / sqrt(((eta * eta) - ((sinTheta_O_m * sinTheta_O_m) / sqrt((single(1.0) - (sinTheta_O_m * sinTheta_O_m))))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if sinTheta_O < 1.30000002e-23

   1. Initial program 89.6%

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

   3. Taylor expanded in sinTheta_O around 0

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

      1. lower-/.f32 97.0

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

   5. Applied rewrites 97.0%

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

   if 1.30000002e-23 < sinTheta_O

   1. Initial program 99.7%

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

Alternative 2: 98.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} sinTheta_O_m = \left|sinTheta\_O\right| \\ \begin{array}{l} t_0 := \frac{sinTheta\_O\_m}{\sqrt{\cosh \sinh^{-1} sinTheta\_O\_m}}\\ \sin^{-1} \left(\frac{h}{\sqrt{eta - t\_0} \cdot \sqrt{eta + t\_0}}\right) \end{array} \end{array} \]

FPCore

sinTheta_O_m = (fabs.f32 sinTheta_O)
(FPCore (sinTheta_O_m h eta)
 :precision binary32
 (let* ((t_0 (/ sinTheta_O_m (sqrt (cosh (asinh sinTheta_O_m))))))
   (asin (/ h (* (sqrt (- eta t_0)) (sqrt (+ eta t_0)))))))

C

sinTheta_O_m = fabs(sinTheta_O);
float code(float sinTheta_O_m, float h, float eta) {
	float t_0 = sinTheta_O_m / sqrtf(coshf(asinhf(sinTheta_O_m)));
	return asinf((h / (sqrtf((eta - t_0)) * sqrtf((eta + t_0)))));
}

Julia

sinTheta_O_m = abs(sinTheta_O)
function code(sinTheta_O_m, h, eta)
	t_0 = Float32(sinTheta_O_m / sqrt(cosh(asinh(sinTheta_O_m))))
	return asin(Float32(h / Float32(sqrt(Float32(eta - t_0)) * sqrt(Float32(eta + t_0)))))
end

MATLAB

sinTheta_O_m = abs(sinTheta_O);
function tmp = code(sinTheta_O_m, h, eta)
	t_0 = sinTheta_O_m / sqrt(cosh(asinh(sinTheta_O_m)));
	tmp = asin((h / (sqrt((eta - t_0)) * sqrt((eta + t_0)))));
end

Derivation

1. Initial program 91.9%

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

4. Applied rewrites 91.8%

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

6. Applied rewrites 98.0%

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

8. Applied rewrites 98.6%

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

9. Final simplification 98.6%

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

Alternative 3: 97.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} sinTheta_O_m = \left|sinTheta\_O\right| \\ \begin{array}{l} t_0 := \sqrt{\sin \tan^{-1} sinTheta\_O\_m \cdot sinTheta\_O\_m}\\ \sin^{-1} \left(\frac{h}{\sqrt{eta + t\_0} \cdot \sqrt{eta - t\_0}}\right) \end{array} \end{array} \]

FPCore

sinTheta_O_m = (fabs.f32 sinTheta_O)
(FPCore (sinTheta_O_m h eta)
 :precision binary32
 (let* ((t_0 (sqrt (* (sin (atan sinTheta_O_m)) sinTheta_O_m))))
   (asin (/ h (* (sqrt (+ eta t_0)) (sqrt (- eta t_0)))))))

C

sinTheta_O_m = fabs(sinTheta_O);
float code(float sinTheta_O_m, float h, float eta) {
	float t_0 = sqrtf((sinf(atanf(sinTheta_O_m)) * sinTheta_O_m));
	return asinf((h / (sqrtf((eta + t_0)) * sqrtf((eta - t_0)))));
}

Fortran

sinTheta_O_m = abs(sintheta_o)
real(4) function code(sintheta_o_m, h, eta)
    real(4), intent (in) :: sintheta_o_m
    real(4), intent (in) :: h
    real(4), intent (in) :: eta
    real(4) :: t_0
    t_0 = sqrt((sin(atan(sintheta_o_m)) * sintheta_o_m))
    code = asin((h / (sqrt((eta + t_0)) * sqrt((eta - t_0)))))
end function

Julia

sinTheta_O_m = abs(sinTheta_O)
function code(sinTheta_O_m, h, eta)
	t_0 = sqrt(Float32(sin(atan(sinTheta_O_m)) * sinTheta_O_m))
	return asin(Float32(h / Float32(sqrt(Float32(eta + t_0)) * sqrt(Float32(eta - t_0)))))
end

MATLAB

sinTheta_O_m = abs(sinTheta_O);
function tmp = code(sinTheta_O_m, h, eta)
	t_0 = sqrt((sin(atan(sinTheta_O_m)) * sinTheta_O_m));
	tmp = asin((h / (sqrt((eta + t_0)) * sqrt((eta - t_0)))));
end

Derivation

1. Initial program 91.9%

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

4. Applied rewrites 91.8%

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

6. Applied rewrites 98.0%

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

Alternative 4: 95.2% accurate, 1.5× speedup

Math

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

FPCore

sinTheta_O_m = (fabs.f32 sinTheta_O)
(FPCore (sinTheta_O_m h eta) :precision binary32 (asin (/ h eta)))

C

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

Fortran

sinTheta_O_m = abs(sintheta_o)
real(4) function code(sintheta_o_m, h, eta)
    real(4), intent (in) :: sintheta_o_m
    real(4), intent (in) :: h
    real(4), intent (in) :: eta
    code = asin((h / eta))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 91.9%

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

3. Taylor expanded in sinTheta_O around 0

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

   1. lower-/.f32 95.9

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

5. Applied rewrites 95.9%

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

Reproduce

herbie shell --seed 2024339 
(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)))))))))