Trowbridge-Reitz Sample, sample surface normal, cosTheta

Average Error: 0.2 → 0.2

Time: 1.7min

Precision: binary32

Cost: 46688

\[\left(\left(\left(2.328306437 \cdot 10^{-10} \leq u0 \land u0 \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 0.5\right)\right) \land \left(0.0001 \leq alphax \land alphax \leq 1\right)\right) \land \left(0.0001 \leq alphay \land alphay \leq 1\right)\]

Math

\[\frac{1}{\sqrt{1 + \frac{\frac{1}{\frac{\cos \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \pi\right) \cdot u1 + 0.5 \cdot \pi\right)\right) \cdot \cos \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \pi\right) \cdot u1 + 0.5 \cdot \pi\right)\right)}{alphax \cdot alphax} + \frac{\sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \pi\right) \cdot u1 + 0.5 \cdot \pi\right)\right) \cdot \sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \pi\right) \cdot u1 + 0.5 \cdot \pi\right)\right)}{alphay \cdot alphay}} \cdot u0}{1 - u0}}} \]

↓

\[\begin{array}{l} t_0 := \tan \left(\pi \cdot \left(0.5 + \left(u1 + u1\right)\right)\right)\\ \frac{1}{\sqrt{1 + \frac{\frac{1}{\frac{\frac{1 + \cos \left(2 \cdot \tan^{-1} \left(t_0 \cdot \frac{alphay}{alphax}\right)\right)}{2}}{alphax \cdot alphax} + \frac{\sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \pi\right) \cdot u1 + 0.5 \cdot \pi\right)\right) \cdot \sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot t_0\right)}{alphay \cdot alphay}} \cdot u0}{1 - u0}}} \end{array} \]

FPCore

(FPCore (u0 u1 alphax alphay)
 :precision binary32
 (/
  1.0
  (sqrt
   (+
    1.0
    (/
     (*
      (/
       1.0
       (+
        (/
         (*
          (cos
           (atan (* (/ alphay alphax) (tan (+ (* (* 2.0 PI) u1) (* 0.5 PI))))))
          (cos
           (atan
            (* (/ alphay alphax) (tan (+ (* (* 2.0 PI) u1) (* 0.5 PI)))))))
         (* alphax alphax))
        (/
         (*
          (sin
           (atan (* (/ alphay alphax) (tan (+ (* (* 2.0 PI) u1) (* 0.5 PI))))))
          (sin
           (atan
            (* (/ alphay alphax) (tan (+ (* (* 2.0 PI) u1) (* 0.5 PI)))))))
         (* alphay alphay))))
      u0)
     (- 1.0 u0))))))

↓

(FPCore (u0 u1 alphax alphay)
 :precision binary32
 (let* ((t_0 (tan (* PI (+ 0.5 (+ u1 u1))))))
   (/
    1.0
    (sqrt
     (+
      1.0
      (/
       (*
        (/
         1.0
         (+
          (/
           (/ (+ 1.0 (cos (* 2.0 (atan (* t_0 (/ alphay alphax)))))) 2.0)
           (* alphax alphax))
          (/
           (*
            (sin
             (atan
              (* (/ alphay alphax) (tan (+ (* (* 2.0 PI) u1) (* 0.5 PI))))))
            (sin (atan (* (/ alphay alphax) t_0))))
           (* alphay alphay))))
        u0)
       (- 1.0 u0)))))))

C

float code(float u0, float u1, float alphax, float alphay) {
	return 1.0f / sqrtf((1.0f + (((1.0f / (((cosf(atanf(((alphay / alphax) * tanf((((2.0f * ((float) M_PI)) * u1) + (0.5f * ((float) M_PI))))))) * cosf(atanf(((alphay / alphax) * tanf((((2.0f * ((float) M_PI)) * u1) + (0.5f * ((float) M_PI)))))))) / (alphax * alphax)) + ((sinf(atanf(((alphay / alphax) * tanf((((2.0f * ((float) M_PI)) * u1) + (0.5f * ((float) M_PI))))))) * sinf(atanf(((alphay / alphax) * tanf((((2.0f * ((float) M_PI)) * u1) + (0.5f * ((float) M_PI)))))))) / (alphay * alphay)))) * u0) / (1.0f - u0))));
}

↓

float code(float u0, float u1, float alphax, float alphay) {
	float t_0 = tanf((((float) M_PI) * (0.5f + (u1 + u1))));
	return 1.0f / sqrtf((1.0f + (((1.0f / ((((1.0f + cosf((2.0f * atanf((t_0 * (alphay / alphax)))))) / 2.0f) / (alphax * alphax)) + ((sinf(atanf(((alphay / alphax) * tanf((((2.0f * ((float) M_PI)) * u1) + (0.5f * ((float) M_PI))))))) * sinf(atanf(((alphay / alphax) * t_0)))) / (alphay * alphay)))) * u0) / (1.0f - u0))));
}

Julia

function code(u0, u1, alphax, alphay)
	return Float32(Float32(1.0) / sqrt(Float32(Float32(1.0) + Float32(Float32(Float32(Float32(1.0) / Float32(Float32(Float32(cos(atan(Float32(Float32(alphay / alphax) * tan(Float32(Float32(Float32(Float32(2.0) * Float32(pi)) * u1) + Float32(Float32(0.5) * Float32(pi))))))) * cos(atan(Float32(Float32(alphay / alphax) * tan(Float32(Float32(Float32(Float32(2.0) * Float32(pi)) * u1) + Float32(Float32(0.5) * Float32(pi)))))))) / Float32(alphax * alphax)) + Float32(Float32(sin(atan(Float32(Float32(alphay / alphax) * tan(Float32(Float32(Float32(Float32(2.0) * Float32(pi)) * u1) + Float32(Float32(0.5) * Float32(pi))))))) * sin(atan(Float32(Float32(alphay / alphax) * tan(Float32(Float32(Float32(Float32(2.0) * Float32(pi)) * u1) + Float32(Float32(0.5) * Float32(pi)))))))) / Float32(alphay * alphay)))) * u0) / Float32(Float32(1.0) - u0)))))
end

↓

function code(u0, u1, alphax, alphay)
	t_0 = tan(Float32(Float32(pi) * Float32(Float32(0.5) + Float32(u1 + u1))))
	return Float32(Float32(1.0) / sqrt(Float32(Float32(1.0) + Float32(Float32(Float32(Float32(1.0) / Float32(Float32(Float32(Float32(Float32(1.0) + cos(Float32(Float32(2.0) * atan(Float32(t_0 * Float32(alphay / alphax)))))) / Float32(2.0)) / Float32(alphax * alphax)) + Float32(Float32(sin(atan(Float32(Float32(alphay / alphax) * tan(Float32(Float32(Float32(Float32(2.0) * Float32(pi)) * u1) + Float32(Float32(0.5) * Float32(pi))))))) * sin(atan(Float32(Float32(alphay / alphax) * t_0)))) / Float32(alphay * alphay)))) * u0) / Float32(Float32(1.0) - u0)))))
end

MATLAB

function tmp = code(u0, u1, alphax, alphay)
	tmp = single(1.0) / sqrt((single(1.0) + (((single(1.0) / (((cos(atan(((alphay / alphax) * tan((((single(2.0) * single(pi)) * u1) + (single(0.5) * single(pi))))))) * cos(atan(((alphay / alphax) * tan((((single(2.0) * single(pi)) * u1) + (single(0.5) * single(pi)))))))) / (alphax * alphax)) + ((sin(atan(((alphay / alphax) * tan((((single(2.0) * single(pi)) * u1) + (single(0.5) * single(pi))))))) * sin(atan(((alphay / alphax) * tan((((single(2.0) * single(pi)) * u1) + (single(0.5) * single(pi)))))))) / (alphay * alphay)))) * u0) / (single(1.0) - u0))));
end

↓

function tmp = code(u0, u1, alphax, alphay)
	t_0 = tan((single(pi) * (single(0.5) + (u1 + u1))));
	tmp = single(1.0) / sqrt((single(1.0) + (((single(1.0) / ((((single(1.0) + cos((single(2.0) * atan((t_0 * (alphay / alphax)))))) / single(2.0)) / (alphax * alphax)) + ((sin(atan(((alphay / alphax) * tan((((single(2.0) * single(pi)) * u1) + (single(0.5) * single(pi))))))) * sin(atan(((alphay / alphax) * t_0)))) / (alphay * alphay)))) * u0) / (single(1.0) - u0))));
end

Derivation

1. Initial program 0.2

   \[\frac{1}{\sqrt{1 + \frac{\frac{1}{\frac{\cos \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \pi\right) \cdot u1 + 0.5 \cdot \pi\right)\right) \cdot \cos \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \pi\right) \cdot u1 + 0.5 \cdot \pi\right)\right)}{alphax \cdot alphax} + \frac{\sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \pi\right) \cdot u1 + 0.5 \cdot \pi\right)\right) \cdot \sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \pi\right) \cdot u1 + 0.5 \cdot \pi\right)\right)}{alphay \cdot alphay}} \cdot u0}{1 - u0}}} \]

2. Applied egg-rr 0.2

   \[\leadsto \frac{1}{\sqrt{1 + \frac{\frac{1}{\frac{\cos \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \pi\right) \cdot u1 + 0.5 \cdot \pi\right)\right) \cdot \cos \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \pi\right) \cdot u1 + 0.5 \cdot \pi\right)\right)}{alphax \cdot alphax} + \frac{\sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \pi\right) \cdot u1 + 0.5 \cdot \pi\right)\right) \cdot \sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \color{blue}{\left(\tan \left(\pi \cdot \left(0.5 + \left(u1 + u1\right)\right)\right) + 0\right)}\right)}{alphay \cdot alphay}} \cdot u0}{1 - u0}}} \]

3. Simplified 0.2

   \[\leadsto \frac{1}{\sqrt{1 + \frac{\frac{1}{\frac{\cos \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \pi\right) \cdot u1 + 0.5 \cdot \pi\right)\right) \cdot \cos \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \pi\right) \cdot u1 + 0.5 \cdot \pi\right)\right)}{alphax \cdot alphax} + \frac{\sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \pi\right) \cdot u1 + 0.5 \cdot \pi\right)\right) \cdot \sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \color{blue}{\tan \left(\pi \cdot \left(0.5 + \left(u1 + u1\right)\right)\right)}\right)}{alphay \cdot alphay}} \cdot u0}{1 - u0}}} \]
   Proof

   [Start] 0.2	\[ \frac{1}{\sqrt{1 + \frac{\frac{1}{\frac{\cos \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \pi\right) \cdot u1 + 0.5 \cdot \pi\right)\right) \cdot \cos \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \pi\right) \cdot u1 + 0.5 \cdot \pi\right)\right)}{alphax \cdot alphax} + \frac{\sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \pi\right) \cdot u1 + 0.5 \cdot \pi\right)\right) \cdot \sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \left(\tan \left(\pi \cdot \left(0.5 + \left(u1 + u1\right)\right)\right) + 0\right)\right)}{alphay \cdot alphay}} \cdot u0}{1 - u0}}} \]
   rational_best-simplify-3 [=>] 0.2	\[ \frac{1}{\sqrt{1 + \frac{\frac{1}{\frac{\cos \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \pi\right) \cdot u1 + 0.5 \cdot \pi\right)\right) \cdot \cos \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \pi\right) \cdot u1 + 0.5 \cdot \pi\right)\right)}{alphax \cdot alphax} + \frac{\sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \pi\right) \cdot u1 + 0.5 \cdot \pi\right)\right) \cdot \sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \color{blue}{\left(0 + \tan \left(\pi \cdot \left(0.5 + \left(u1 + u1\right)\right)\right)\right)}\right)}{alphay \cdot alphay}} \cdot u0}{1 - u0}}} \]
   rational_best-simplify-6 [=>] 0.2	\[ \frac{1}{\sqrt{1 + \frac{\frac{1}{\frac{\cos \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \pi\right) \cdot u1 + 0.5 \cdot \pi\right)\right) \cdot \cos \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \pi\right) \cdot u1 + 0.5 \cdot \pi\right)\right)}{alphax \cdot alphax} + \frac{\sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \pi\right) \cdot u1 + 0.5 \cdot \pi\right)\right) \cdot \sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \color{blue}{\tan \left(\pi \cdot \left(0.5 + \left(u1 + u1\right)\right)\right)}\right)}{alphay \cdot alphay}} \cdot u0}{1 - u0}}} \]

4. Applied egg-rr 0.2

   \[\leadsto \frac{1}{\sqrt{1 + \frac{\frac{1}{\frac{\color{blue}{0.5 + \frac{\cos \left(2 \cdot \tan^{-1} \left(alphay \cdot \frac{\tan \left(\pi \cdot \left(0.5 + \left(u1 + u1\right)\right)\right)}{alphax}\right)\right)}{2}}}{alphax \cdot alphax} + \frac{\sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \pi\right) \cdot u1 + 0.5 \cdot \pi\right)\right) \cdot \sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\pi \cdot \left(0.5 + \left(u1 + u1\right)\right)\right)\right)}{alphay \cdot alphay}} \cdot u0}{1 - u0}}} \]

5. Simplified 0.2

   \[\leadsto \frac{1}{\sqrt{1 + \frac{\frac{1}{\frac{\color{blue}{\frac{1 + \cos \left(2 \cdot \tan^{-1} \left(\tan \left(\pi \cdot \left(0.5 + \left(u1 + u1\right)\right)\right) \cdot \frac{alphay}{alphax}\right)\right)}{2}}}{alphax \cdot alphax} + \frac{\sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \pi\right) \cdot u1 + 0.5 \cdot \pi\right)\right) \cdot \sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\pi \cdot \left(0.5 + \left(u1 + u1\right)\right)\right)\right)}{alphay \cdot alphay}} \cdot u0}{1 - u0}}} \]
   Proof

   [Start] 0.2	\[ \frac{1}{\sqrt{1 + \frac{\frac{1}{\frac{0.5 + \frac{\cos \left(2 \cdot \tan^{-1} \left(alphay \cdot \frac{\tan \left(\pi \cdot \left(0.5 + \left(u1 + u1\right)\right)\right)}{alphax}\right)\right)}{2}}{alphax \cdot alphax} + \frac{\sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \pi\right) \cdot u1 + 0.5 \cdot \pi\right)\right) \cdot \sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\pi \cdot \left(0.5 + \left(u1 + u1\right)\right)\right)\right)}{alphay \cdot alphay}} \cdot u0}{1 - u0}}} \]
   metadata-eval [<=] 0.2	\[ \frac{1}{\sqrt{1 + \frac{\frac{1}{\frac{\color{blue}{\frac{1}{2}} + \frac{\cos \left(2 \cdot \tan^{-1} \left(alphay \cdot \frac{\tan \left(\pi \cdot \left(0.5 + \left(u1 + u1\right)\right)\right)}{alphax}\right)\right)}{2}}{alphax \cdot alphax} + \frac{\sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \pi\right) \cdot u1 + 0.5 \cdot \pi\right)\right) \cdot \sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\pi \cdot \left(0.5 + \left(u1 + u1\right)\right)\right)\right)}{alphay \cdot alphay}} \cdot u0}{1 - u0}}} \]
   rational_best-simplify-65 [<=] 0.2	\[ \frac{1}{\sqrt{1 + \frac{\frac{1}{\frac{\color{blue}{\frac{1 + \cos \left(2 \cdot \tan^{-1} \left(alphay \cdot \frac{\tan \left(\pi \cdot \left(0.5 + \left(u1 + u1\right)\right)\right)}{alphax}\right)\right)}{2}}}{alphax \cdot alphax} + \frac{\sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \pi\right) \cdot u1 + 0.5 \cdot \pi\right)\right) \cdot \sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\pi \cdot \left(0.5 + \left(u1 + u1\right)\right)\right)\right)}{alphay \cdot alphay}} \cdot u0}{1 - u0}}} \]
   rational_best-simplify-55 [=>] 0.2	\[ \frac{1}{\sqrt{1 + \frac{\frac{1}{\frac{\frac{1 + \cos \left(2 \cdot \tan^{-1} \color{blue}{\left(\tan \left(\pi \cdot \left(0.5 + \left(u1 + u1\right)\right)\right) \cdot \frac{alphay}{alphax}\right)}\right)}{2}}{alphax \cdot alphax} + \frac{\sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \pi\right) \cdot u1 + 0.5 \cdot \pi\right)\right) \cdot \sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\pi \cdot \left(0.5 + \left(u1 + u1\right)\right)\right)\right)}{alphay \cdot alphay}} \cdot u0}{1 - u0}}} \]

6. Final simplification 0.2

   \[\leadsto \frac{1}{\sqrt{1 + \frac{\frac{1}{\frac{\frac{1 + \cos \left(2 \cdot \tan^{-1} \left(\tan \left(\pi \cdot \left(0.5 + \left(u1 + u1\right)\right)\right) \cdot \frac{alphay}{alphax}\right)\right)}{2}}{alphax \cdot alphax} + \frac{\sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \pi\right) \cdot u1 + 0.5 \cdot \pi\right)\right) \cdot \sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\pi \cdot \left(0.5 + \left(u1 + u1\right)\right)\right)\right)}{alphay \cdot alphay}} \cdot u0}{1 - u0}}} \]

Alternatives

Alternative 1
Error	0.2
Cost	43456

\[\begin{array}{l} t_0 := \tan \left(\pi \cdot \left(0.5 + \left(u1 + u1\right)\right)\right)\\ t_1 := \sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot t_0\right)\\ \frac{1}{\sqrt{1 + \frac{\frac{1}{\frac{\frac{1 + \cos \left(2 \cdot \tan^{-1} \left(t_0 \cdot \frac{alphay}{alphax}\right)\right)}{2}}{alphax \cdot alphax} + \frac{t_1 \cdot t_1}{alphay \cdot alphay}} \cdot u0}{1 - u0}}} \end{array} \]

Alternative 2
Error	0.3
Cost	30368

\[\begin{array}{l} t_0 := \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\pi \cdot \left(2 \cdot u1 + 0.5\right)\right)\right)\right)\\ \frac{1}{\sqrt{1 + \frac{\frac{u0}{\frac{\frac{1 - t_0}{alphay \cdot alphay} + \frac{1 + t_0}{alphax \cdot alphax}}{2}}}{1 - u0}}} \end{array} \]

Alternative 3
Error	0.6
Cost	30240

\[\frac{1}{\sqrt{1 + \frac{\frac{u0}{\frac{\frac{1 - \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(0.5 \cdot \pi\right)\right)\right)}{alphay \cdot alphay} + \frac{1 + \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\pi \cdot \left(2 \cdot u1 + 0.5\right)\right)\right)\right)}{alphax \cdot alphax}}{2}}}{1 - u0}}} \]

Alternative 4
Error	2.5
Cost	30176

\[\frac{1}{\sqrt{1 + \frac{\frac{u0}{\frac{\frac{1 - \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(0.5 \cdot \pi\right)\right)\right)}{alphay \cdot alphay} + \frac{1 + \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(2 \cdot \left(u1 \cdot \pi\right)\right)\right)\right)}{alphax \cdot alphax}}{2}}}{1 - u0}}} \]

Alternative 5
Error	2.5
Cost	26976

\[\frac{1}{\sqrt{1 + \frac{\frac{u0}{\frac{\frac{1 - \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(0.5 \cdot \pi\right)\right)\right)}{alphay \cdot alphay} + \frac{1 + \cos \left(2 \cdot \tan^{-1} \left(\left(\pi \cdot \left(u1 \cdot alphay\right)\right) \cdot \frac{2}{alphax}\right)\right)}{alphax \cdot alphax}}{2}}}{1 - u0}}} \]

Alternative 6
Error	3.8
Cost	23840

\[\begin{array}{l} t_0 := \cos \left(2 \cdot \tan^{-1} \left(\left(\pi \cdot \left(u1 \cdot alphay\right)\right) \cdot \frac{2}{alphax}\right)\right)\\ \frac{1}{\sqrt{1 + \frac{\frac{u0}{\frac{\frac{1 - t_0}{alphay \cdot alphay} + \frac{1 + t_0}{alphax \cdot alphax}}{2}}}{1 - u0}}} \end{array} \]

Reproduce

herbie shell --seed 2023099 
(FPCore (u0 u1 alphax alphay)
  :name "Trowbridge-Reitz Sample, sample surface normal, cosTheta"
  :precision binary32
  :pre (and (and (and (and (<= 2.328306437e-10 u0) (<= u0 1.0)) (and (<= 2.328306437e-10 u1) (<= u1 0.5))) (and (<= 0.0001 alphax) (<= alphax 1.0))) (and (<= 0.0001 alphay) (<= alphay 1.0)))
  (/ 1.0 (sqrt (+ 1.0 (/ (* (/ 1.0 (+ (/ (* (cos (atan (* (/ alphay alphax) (tan (+ (* (* 2.0 PI) u1) (* 0.5 PI)))))) (cos (atan (* (/ alphay alphax) (tan (+ (* (* 2.0 PI) u1) (* 0.5 PI))))))) (* alphax alphax)) (/ (* (sin (atan (* (/ alphay alphax) (tan (+ (* (* 2.0 PI) u1) (* 0.5 PI)))))) (sin (atan (* (/ alphay alphax) (tan (+ (* (* 2.0 PI) u1) (* 0.5 PI))))))) (* alphay alphay)))) u0) (- 1.0 u0))))))