Trowbridge-Reitz Sample, sample surface normal, cosTheta

Percentage Accurate: 99.4% → 99.7%

Time: 6.5s

Alternatives: 6

Speedup: 1.7×

Specification

\[\left(\left(\left(\frac{2328306437}{10000000000000000000} \leq u0 \land u0 \leq 1\right) \land \left(\frac{2328306437}{10000000000000000000} \leq u1 \land u1 \leq \frac{1}{2}\right)\right) \land \left(\frac{1}{10000} \leq alphax \land alphax \leq 1\right)\right) \land \left(\frac{1}{10000} \leq alphay \land alphay \leq 1\right)\]

Math

\[\begin{array}{l} t_0 := \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \pi\right) \cdot u1 + \frac{1}{2} \cdot \pi\right)\right)\\ t_1 := \sin t\_0\\ t_2 := \cos t\_0\\ \frac{1}{\sqrt{1 + \frac{\frac{1}{\frac{t\_2 \cdot t\_2}{alphax \cdot alphax} + \frac{t\_1 \cdot t\_1}{alphay \cdot alphay}} \cdot u0}{1 - u0}}} \end{array} \]

FPCore

(FPCore (u0 u1 alphax alphay)
  :precision binary32
  (let* ((t_0
        (atan
         (* (/ alphay alphax) (tan (+ (* (* 2 PI) u1) (* 1/2 PI))))))
       (t_1 (sin t_0))
       (t_2 (cos t_0)))
  (/
   1
   (sqrt
    (+
     1
     (/
      (*
       (/
        1
        (+
         (/ (* t_2 t_2) (* alphax alphax))
         (/ (* t_1 t_1) (* alphay alphay))))
       u0)
      (- 1 u0)))))))

C

float code(float u0, float u1, float alphax, float alphay) {
	float t_0 = atanf(((alphay / alphax) * tanf((((2.0f * ((float) M_PI)) * u1) + (0.5f * ((float) M_PI))))));
	float t_1 = sinf(t_0);
	float t_2 = cosf(t_0);
	return 1.0f / sqrtf((1.0f + (((1.0f / (((t_2 * t_2) / (alphax * alphax)) + ((t_1 * t_1) / (alphay * alphay)))) * u0) / (1.0f - u0))));
}

Julia

function code(u0, u1, alphax, alphay)
	t_0 = atan(Float32(Float32(alphay / alphax) * tan(Float32(Float32(Float32(Float32(2.0) * Float32(pi)) * u1) + Float32(Float32(0.5) * Float32(pi))))))
	t_1 = sin(t_0)
	t_2 = cos(t_0)
	return Float32(Float32(1.0) / sqrt(Float32(Float32(1.0) + Float32(Float32(Float32(Float32(1.0) / Float32(Float32(Float32(t_2 * t_2) / Float32(alphax * alphax)) + Float32(Float32(t_1 * t_1) / Float32(alphay * alphay)))) * u0) / Float32(Float32(1.0) - u0)))))
end

MATLAB

function tmp = code(u0, u1, alphax, alphay)
	t_0 = atan(((alphay / alphax) * tan((((single(2.0) * single(pi)) * u1) + (single(0.5) * single(pi))))));
	t_1 = sin(t_0);
	t_2 = cos(t_0);
	tmp = single(1.0) / sqrt((single(1.0) + (((single(1.0) / (((t_2 * t_2) / (alphax * alphax)) + ((t_1 * t_1) / (alphay * alphay)))) * u0) / (single(1.0) - u0))));
end

Initial Program: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \pi\right) \cdot u1 + \frac{1}{2} \cdot \pi\right)\right)\\ t_1 := \sin t\_0\\ t_2 := \cos t\_0\\ \frac{1}{\sqrt{1 + \frac{\frac{1}{\frac{t\_2 \cdot t\_2}{alphax \cdot alphax} + \frac{t\_1 \cdot t\_1}{alphay \cdot alphay}} \cdot u0}{1 - u0}}} \end{array} \]

FPCore

(FPCore (u0 u1 alphax alphay)
  :precision binary32
  (let* ((t_0
        (atan
         (* (/ alphay alphax) (tan (+ (* (* 2 PI) u1) (* 1/2 PI))))))
       (t_1 (sin t_0))
       (t_2 (cos t_0)))
  (/
   1
   (sqrt
    (+
     1
     (/
      (*
       (/
        1
        (+
         (/ (* t_2 t_2) (* alphax alphax))
         (/ (* t_1 t_1) (* alphay alphay))))
       u0)
      (- 1 u0)))))))

C

float code(float u0, float u1, float alphax, float alphay) {
	float t_0 = atanf(((alphay / alphax) * tanf((((2.0f * ((float) M_PI)) * u1) + (0.5f * ((float) M_PI))))));
	float t_1 = sinf(t_0);
	float t_2 = cosf(t_0);
	return 1.0f / sqrtf((1.0f + (((1.0f / (((t_2 * t_2) / (alphax * alphax)) + ((t_1 * t_1) / (alphay * alphay)))) * u0) / (1.0f - u0))));
}

Julia

function code(u0, u1, alphax, alphay)
	t_0 = atan(Float32(Float32(alphay / alphax) * tan(Float32(Float32(Float32(Float32(2.0) * Float32(pi)) * u1) + Float32(Float32(0.5) * Float32(pi))))))
	t_1 = sin(t_0)
	t_2 = cos(t_0)
	return Float32(Float32(1.0) / sqrt(Float32(Float32(1.0) + Float32(Float32(Float32(Float32(1.0) / Float32(Float32(Float32(t_2 * t_2) / Float32(alphax * alphax)) + Float32(Float32(t_1 * t_1) / Float32(alphay * alphay)))) * u0) / Float32(Float32(1.0) - u0)))))
end

MATLAB

function tmp = code(u0, u1, alphax, alphay)
	t_0 = atan(((alphay / alphax) * tan((((single(2.0) * single(pi)) * u1) + (single(0.5) * single(pi))))));
	t_1 = sin(t_0);
	t_2 = cos(t_0);
	tmp = single(1.0) / sqrt((single(1.0) + (((single(1.0) / (((t_2 * t_2) / (alphax * alphax)) + ((t_1 * t_1) / (alphay * alphay)))) * u0) / (single(1.0) - u0))));
end

Alternative 1: 99.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(-2 \cdot \tan^{-1} \left(\tan \left(\pi \cdot \left(\left(u1 + u1\right) + \frac{1}{2}\right)\right) \cdot \frac{alphay}{alphax}\right)\right)\\ {\left(\frac{u0}{\left(\frac{1 - t\_0}{\left(alphay + alphay\right) \cdot alphay} - \frac{-1 - t\_0}{\left(alphax + alphax\right) \cdot alphax}\right) \cdot \left(1 - u0\right)} - -1\right)}^{\frac{-1}{2}} \end{array} \]

FPCore

(FPCore (u0 u1 alphax alphay)
  :precision binary32
  (let* ((t_0
        (cos
         (*
          -2
          (atan
           (* (tan (* PI (+ (+ u1 u1) 1/2))) (/ alphay alphax)))))))
  (pow
   (-
    (/
     u0
     (*
      (-
       (/ (- 1 t_0) (* (+ alphay alphay) alphay))
       (/ (- -1 t_0) (* (+ alphax alphax) alphax)))
      (- 1 u0)))
    -1)
   -1/2)))

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Applied rewrites 99.7%

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

5. Applied rewrites 99.7%

   \[\leadsto {\left(\frac{u0}{\color{blue}{\left(\frac{1 - \cos \left(-2 \cdot \tan^{-1} \left(\tan \left(\pi \cdot \left(\left(u1 + u1\right) + \frac{1}{2}\right)\right) \cdot \frac{alphay}{alphax}\right)\right)}{\left(alphay + alphay\right) \cdot alphay} - \frac{-1 - \cos \left(-2 \cdot \tan^{-1} \left(\tan \left(\pi \cdot \left(\left(u1 + u1\right) + \frac{1}{2}\right)\right) \cdot \frac{alphay}{alphax}\right)\right)}{\left(alphax + alphax\right) \cdot alphax}\right)} \cdot \left(1 - u0\right)} - -1\right)}^{\frac{-1}{2}} \]
6. Add Preprocessing

Alternative 2: 99.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(-2 \cdot \tan^{-1} \left(\tan \left(\pi \cdot \left(\left(u1 + u1\right) + \frac{1}{2}\right)\right) \cdot \frac{alphay}{alphax}\right)\right)\\ \frac{1}{\sqrt{\frac{u0}{\left(\frac{1 - t\_0}{\left(alphay + alphay\right) \cdot alphay} - \frac{-1 - t\_0}{\left(alphax + alphax\right) \cdot alphax}\right) \cdot \left(1 - u0\right)} - -1}} \end{array} \]

FPCore

(FPCore (u0 u1 alphax alphay)
  :precision binary32
  (let* ((t_0
        (cos
         (*
          -2
          (atan
           (* (tan (* PI (+ (+ u1 u1) 1/2))) (/ alphay alphax)))))))
  (/
   1
   (sqrt
    (-
     (/
      u0
      (*
       (-
        (/ (- 1 t_0) (* (+ alphay alphay) alphay))
        (/ (- -1 t_0) (* (+ alphax alphax) alphax)))
       (- 1 u0)))
     -1)))))

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Applied rewrites 99.2%

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

5. Applied rewrites 99.2%

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

Alternative 3: 98.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} t_0 := \pi \cdot \left(\frac{1}{2} + 2 \cdot u1\right)\\ {\left(2 \cdot \frac{{alphay}^{2} \cdot u0}{\left(1 - u0\right) \cdot \left(1 - \cos \left(-2 \cdot \tan^{-1} \left(\frac{alphay \cdot \sin t\_0}{alphax \cdot \cos t\_0}\right)\right)\right)} - -1\right)}^{\frac{-1}{2}} \end{array} \]

FPCore

(FPCore (u0 u1 alphax alphay)
  :precision binary32
  (let* ((t_0 (* PI (+ 1/2 (* 2 u1)))))
  (pow
   (-
    (*
     2
     (/
      (* (pow alphay 2) u0)
      (*
       (- 1 u0)
       (-
        1
        (cos
         (*
          -2
          (atan (/ (* alphay (sin t_0)) (* alphax (cos t_0))))))))))
    -1)
   -1/2)))

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Applied rewrites 99.7%

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

5. Applied rewrites 99.7%

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

6. Taylor expanded in alphax around inf

   \[\leadsto {\left(\color{blue}{2 \cdot \frac{{alphay}^{2} \cdot u0}{\left(1 - u0\right) \cdot \left(1 - \cos \left(-2 \cdot \tan^{-1} \left(\frac{alphay \cdot \sin \left(\pi \cdot \left(\frac{1}{2} + 2 \cdot u1\right)\right)}{alphax \cdot \cos \left(\pi \cdot \left(\frac{1}{2} + 2 \cdot u1\right)\right)}\right)\right)\right)}} - -1\right)}^{\frac{-1}{2}} \]
7. Step-by-step derivation

   1. lower-*.f32 N/A

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

   2. lower-/.f32 N/A

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

8. Applied rewrites 98.0%

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

Alternative 4: 97.6% accurate, 2.4× speedup

Math

\[\begin{array}{l} t_0 := \pi \cdot \left(\frac{1}{2} + 2 \cdot u1\right)\\ \frac{1}{\sqrt{2 \cdot \frac{{alphay}^{2} \cdot u0}{\left(1 - u0\right) \cdot \left(1 - \cos \left(-2 \cdot \tan^{-1} \left(\frac{alphay \cdot \sin t\_0}{alphax \cdot \cos t\_0}\right)\right)\right)} - -1}} \end{array} \]

FPCore

(FPCore (u0 u1 alphax alphay)
  :precision binary32
  (let* ((t_0 (* PI (+ 1/2 (* 2 u1)))))
  (/
   1
   (sqrt
    (-
     (*
      2
      (/
       (* (pow alphay 2) u0)
       (*
        (- 1 u0)
        (-
         1
         (cos
          (*
           -2
           (atan (/ (* alphay (sin t_0)) (* alphax (cos t_0))))))))))
     -1)))))

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Applied rewrites 99.2%

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

5. Applied rewrites 99.2%

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

6. Taylor expanded in alphax around inf

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

   1. lower-*.f32 N/A

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

   2. lower-/.f32 N/A

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

8. Applied rewrites 97.6%

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

Alternative 5: 96.8% accurate, 2.4× speedup

Math

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

FPCore

(FPCore (u0 u1 alphax alphay)
  :precision binary32
  (+
 1
 (*
  -1
  (/
   (* (pow alphay 2) u0)
   (*
    (- 1 u0)
    (-
     1
     (cos
      (*
       2
       (atan
        (/
         (* alphay (sin (- (* -1/2 PI))))
         (* alphax (cos (- (* 2 (* u1 PI)) (* -1/2 PI))))))))))))))

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Applied rewrites 99.2%

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

4. Taylor expanded in alphay around 0

   \[\leadsto \color{blue}{1 + -1 \cdot \frac{{alphay}^{2} \cdot u0}{\left(1 - u0\right) \cdot \left(1 - \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay \cdot \sin \left(2 \cdot \left(u1 \cdot \pi\right) - \frac{-1}{2} \cdot \pi\right)}{alphax \cdot \cos \left(2 \cdot \left(u1 \cdot \pi\right) - \frac{-1}{2} \cdot \pi\right)}\right)\right)\right)}} \]
5. Step-by-step derivation

   1. lower-+.f32 N/A

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

6. Applied rewrites 96.3%

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

7. Taylor expanded in u1 around 0

   \[\leadsto 1 + -1 \cdot \frac{{alphay}^{2} \cdot u0}{\left(1 - u0\right) \cdot \left(1 - \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay \cdot \sin \left(\mathsf{neg}\left(\frac{-1}{2} \cdot \pi\right)\right)}{alphax \cdot \cos \left(2 \cdot \left(u1 \cdot \pi\right) - \frac{-1}{2} \cdot \pi\right)}\right)\right)\right)} \]
8. Step-by-step derivation

   1. lower-sin.f32 N/A

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

   2. lower-neg.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 96.8%

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

9. Applied rewrites 96.8%

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

Alternative 6: 96.3% accurate, 4.0× speedup

Math

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

FPCore

(FPCore (u0 u1 alphax alphay)
  :precision binary32
  (-
 1
 (/
  (* u0 (* alphay alphay))
  (*
   (- 1 (cos (* (atan (* (tan (* 1/2 PI)) (/ alphay alphax))) 2)))
   (- 1 u0)))))

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Applied rewrites 99.2%

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

4. Taylor expanded in alphay around 0

   \[\leadsto \color{blue}{1 + -1 \cdot \frac{{alphay}^{2} \cdot u0}{\left(1 - u0\right) \cdot \left(1 - \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay \cdot \sin \left(2 \cdot \left(u1 \cdot \pi\right) - \frac{-1}{2} \cdot \pi\right)}{alphax \cdot \cos \left(2 \cdot \left(u1 \cdot \pi\right) - \frac{-1}{2} \cdot \pi\right)}\right)\right)\right)}} \]
5. Step-by-step derivation

   1. lower-+.f32 N/A

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

6. Applied rewrites 96.3%

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

7. Taylor expanded in u1 around 0

   \[\leadsto 1 + -1 \cdot \frac{{alphay}^{2} \cdot u0}{\left(1 - u0\right) \cdot \left(1 - \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay \cdot \sin \left(\frac{1}{2} \cdot \pi\right)}{alphax \cdot \cos \left(2 \cdot \left(u1 \cdot \pi\right) - \frac{-1}{2} \cdot \pi\right)}\right)\right)\right)} \]
8. Step-by-step derivation

   1. lower-*.f32 N/A

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

   2. lower-PI.f32 96.8%

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

9. Applied rewrites 96.8%

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

10. Taylor expanded in u1 around 0

    \[\leadsto 1 + -1 \cdot \frac{{alphay}^{2} \cdot u0}{\left(1 - u0\right) \cdot \left(1 - \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay \cdot \sin \left(\frac{1}{2} \cdot \pi\right)}{alphax \cdot \cos \left(\frac{1}{2} \cdot \pi\right)}\right)\right)\right)} \]
11. Step-by-step derivation

    1. lower-*.f32 N/A

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

    2. lower-PI.f32 96.3%

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

12. Applied rewrites 96.3%

    \[\leadsto 1 + -1 \cdot \frac{{alphay}^{2} \cdot u0}{\left(1 - u0\right) \cdot \left(1 - \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay \cdot \sin \left(\frac{1}{2} \cdot \pi\right)}{alphax \cdot \cos \left(\frac{1}{2} \cdot \pi\right)}\right)\right)\right)} \]
13. Step-by-step derivation

    1. lift-+.f32 N/A

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

    2. add-flip N/A

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

    3. lower--.f32 N/A

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

14. Applied rewrites 96.3%

    \[\leadsto 1 - \color{blue}{\frac{u0 \cdot \left(alphay \cdot alphay\right)}{\left(1 - \cos \left(\tan^{-1} \left(\tan \left(\frac{1}{2} \cdot \pi\right) \cdot \frac{alphay}{alphax}\right) \cdot 2\right)\right) \cdot \left(1 - u0\right)}} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2025285 -o generate:evaluate
(FPCore (u0 u1 alphax alphay)
  :name "Trowbridge-Reitz Sample, sample surface normal, cosTheta"
  :precision binary32
  :pre (and (and (and (and (<= 2328306437/10000000000000000000 u0) (<= u0 1)) (and (<= 2328306437/10000000000000000000 u1) (<= u1 1/2))) (and (<= 1/10000 alphax) (<= alphax 1))) (and (<= 1/10000 alphay) (<= alphay 1)))
  (/ 1 (sqrt (+ 1 (/ (* (/ 1 (+ (/ (* (cos (atan (* (/ alphay alphax) (tan (+ (* (* 2 PI) u1) (* 1/2 PI)))))) (cos (atan (* (/ alphay alphax) (tan (+ (* (* 2 PI) u1) (* 1/2 PI))))))) (* alphax alphax)) (/ (* (sin (atan (* (/ alphay alphax) (tan (+ (* (* 2 PI) u1) (* 1/2 PI)))))) (sin (atan (* (/ alphay alphax) (tan (+ (* (* 2 PI) u1) (* 1/2 PI))))))) (* alphay alphay)))) u0) (- 1 u0))))))