HairBSDF, gamma for a refracted ray

Average Error: 2.6 → 0.5

Time: 21.0s

Precision: binary32

Cost: 6820

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

Math

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

↓

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

FPCore

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

↓

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

↓

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(((sinTheta_O + eta) * (eta - sinTheta_O)))));
	}
	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
    code = asin((h / sqrt(((eta * eta) - ((sintheta_o * sintheta_o) / sqrt((1.0e0 - (sintheta_o * sintheta_o))))))))
end function

↓

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(((sintheta_o + eta) * (eta - sintheta_o)))))
    end if
    code = tmp
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

↓

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(sinTheta_O + eta) * Float32(eta - sinTheta_O)))));
	end
	return tmp
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

↓

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(((sinTheta_O + eta) * (eta - sinTheta_O)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 4.4

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

   2. Taylor expanded in eta around inf 0.6

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

   if 0.0 < (*.f32 sinTheta_O sinTheta_O)

   1. Initial program 0.2

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

   2. Applied egg-rr 0.2

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

   3. Taylor expanded in sinTheta_O around 0 0.4

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

   4. Simplified 0.4

      \[\leadsto \sin^{-1} \left(\frac{h}{\sqrt{\color{blue}{eta \cdot eta - sinTheta_O \cdot sinTheta_O}}}\right) \]
      Proof
      (-.f32 (*.f32 eta eta) (*.f32 sinTheta_O sinTheta_O)): 0 points increase in error, 0 points decrease in error
      (-.f32 (Rewrite<= unpow2_binary32 (pow.f32 eta 2)) (*.f32 sinTheta_O sinTheta_O)): 0 points increase in error, 0 points decrease in error
      (-.f32 (pow.f32 eta 2) (Rewrite<= unpow2_binary32 (pow.f32 sinTheta_O 2))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= unsub-neg_binary32 (+.f32 (pow.f32 eta 2) (neg.f32 (pow.f32 sinTheta_O 2)))): 0 points increase in error, 0 points decrease in error
      (+.f32 (pow.f32 eta 2) (Rewrite<= mul-1-neg_binary32 (*.f32 -1 (pow.f32 sinTheta_O 2)))): 0 points increase in error, 0 points decrease in error

   5. Applied egg-rr 0.3

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

4. Final simplification 0.5

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

Alternatives

Alternative 1
Error	0.5
Cost	19872

\[\begin{array}{l} t_0 := {\left(1 - sinTheta_O \cdot sinTheta_O\right)}^{-0.25}\\ \sin^{-1} \left(\frac{h}{\sqrt{\mathsf{fma}\left(sinTheta_O, t_0, eta\right)} \cdot \sqrt{eta - sinTheta_O \cdot t_0}}\right) \end{array} \]

Alternative 2
Error	0.5
Cost	9888

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

Alternative 3
Error	0.7
Cost	6688

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

Alternative 4
Error	1.6
Cost	3296

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

Reproduce

herbie shell --seed 2022291 
(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)))))))))