Trowbridge-Reitz Sample, sample surface normal, cosTheta

Percentage Accurate: 99.3% → 99.9%

Time: 1.1min

Alternatives: 2

Speedup: N/A×

Specification

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

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \pi\right) \cdot u1 + 0.5 \cdot \pi\right)\right)\\ t_1 := \sin t\_0\\ t_2 := \cos t\_0\\ \frac{1}{\sqrt{1 + \frac{\frac{1}{\frac{t\_2 \cdot t\_2}{alphax \cdot alphax} + \frac{t\_1 \cdot t\_1}{alphay \cdot alphay}} \cdot u0}{1 - u0}}} \end{array} \end{array} \]

FPCore

(FPCore (u0 u1 alphax alphay)
 :precision binary32
 (let* ((t_0
         (atan (* (/ alphay alphax) (tan (+ (* (* 2.0 PI) u1) (* 0.5 PI))))))
        (t_1 (sin t_0))
        (t_2 (cos t_0)))
   (/
    1.0
    (sqrt
     (+
      1.0
      (/
       (*
        (/
         1.0
         (+
          (/ (* t_2 t_2) (* alphax alphax))
          (/ (* t_1 t_1) (* alphay alphay))))
        u0)
       (- 1.0 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: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \pi\right) \cdot u1 + 0.5 \cdot \pi\right)\right)\\ t_1 := \sin t\_0\\ t_2 := \cos t\_0\\ \frac{1}{\sqrt{1 + \frac{\frac{1}{\frac{t\_2 \cdot t\_2}{alphax \cdot alphax} + \frac{t\_1 \cdot t\_1}{alphay \cdot alphay}} \cdot u0}{1 - u0}}} \end{array} \end{array} \]

FPCore

(FPCore (u0 u1 alphax alphay)
 :precision binary32
 (let* ((t_0
         (atan (* (/ alphay alphax) (tan (+ (* (* 2.0 PI) u1) (* 0.5 PI))))))
        (t_1 (sin t_0))
        (t_2 (cos t_0)))
   (/
    1.0
    (sqrt
     (+
      1.0
      (/
       (*
        (/
         1.0
         (+
          (/ (* t_2 t_2) (* alphax alphax))
          (/ (* t_1 t_1) (* alphay alphay))))
        u0)
       (- 1.0 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: 99.9% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1} \left(\tan \left(\left(\mathsf{fma}\left(\pi \cdot 2, u1, \pi \cdot 0.5\right) + \pi\right) + \pi\right) \cdot \frac{alphay}{alphax}\right)\\ t_1 := \frac{\sin t\_0}{alphay}\\ t_2 := \frac{\cos t\_0}{alphax}\\ \sqrt{\frac{1}{\frac{\frac{u0}{\mathsf{fma}\left(t\_1, t\_1, t\_2 \cdot t\_2\right)}}{1 - u0} + 1}} \end{array} \end{array} \]

FPCore

(FPCore (u0 u1 alphax alphay)
 :precision binary32
 (let* ((t_0
         (atan
          (*
           (tan (+ (+ (fma (* PI 2.0) u1 (* PI 0.5)) PI) PI))
           (/ alphay alphax))))
        (t_1 (/ (sin t_0) alphay))
        (t_2 (/ (cos t_0) alphax)))
   (sqrt (/ 1.0 (+ (/ (/ u0 (fma t_1 t_1 (* t_2 t_2))) (- 1.0 u0)) 1.0)))))

C

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

Julia

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in u1 around 0

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

5. Applied rewrites 100.0%

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

Alternative 2: 99.9% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1} \left(\tan \left(\left(\left(\mathsf{fma}\left(\pi \cdot 2, u1, \pi \cdot 0.5\right) + \pi\right) + \pi\right) + \pi\right) \cdot \frac{alphay}{alphax}\right)\\ t_1 := \frac{\cos t\_0}{alphax}\\ t_2 := t\_1 \cdot t\_1\\ t_3 := \frac{\sin t\_0}{alphay}\\ t_4 := t\_3 \cdot t\_3\\ \sqrt{\frac{1}{\frac{\frac{u0}{\frac{{t\_4}^{3} + {t\_2}^{3}}{t\_4 \cdot t\_4 + \left(t\_2 \cdot t\_2 - t\_4 \cdot t\_2\right)}}}{1 - u0} + 1}} \end{array} \end{array} \]

FPCore

(FPCore (u0 u1 alphax alphay)
 :precision binary32
 (let* ((t_0
         (atan
          (*
           (tan (+ (+ (+ (fma (* PI 2.0) u1 (* PI 0.5)) PI) PI) PI))
           (/ alphay alphax))))
        (t_1 (/ (cos t_0) alphax))
        (t_2 (* t_1 t_1))
        (t_3 (/ (sin t_0) alphay))
        (t_4 (* t_3 t_3)))
   (sqrt
    (/
     1.0
     (+
      (/
       (/
        u0
        (/
         (+ (pow t_4 3.0) (pow t_2 3.0))
         (+ (* t_4 t_4) (- (* t_2 t_2) (* t_4 t_2)))))
       (- 1.0 u0))
      1.0)))))

C

float code(float u0, float u1, float alphax, float alphay) {
	float t_0 = atanf((tanf((((fmaf((((float) M_PI) * 2.0f), u1, (((float) M_PI) * 0.5f)) + ((float) M_PI)) + ((float) M_PI)) + ((float) M_PI))) * (alphay / alphax)));
	float t_1 = cosf(t_0) / alphax;
	float t_2 = t_1 * t_1;
	float t_3 = sinf(t_0) / alphay;
	float t_4 = t_3 * t_3;
	return sqrtf((1.0f / (((u0 / ((powf(t_4, 3.0f) + powf(t_2, 3.0f)) / ((t_4 * t_4) + ((t_2 * t_2) - (t_4 * t_2))))) / (1.0f - u0)) + 1.0f)));
}

Julia

function code(u0, u1, alphax, alphay)
	t_0 = atan(Float32(tan(Float32(Float32(Float32(fma(Float32(Float32(pi) * Float32(2.0)), u1, Float32(Float32(pi) * Float32(0.5))) + Float32(pi)) + Float32(pi)) + Float32(pi))) * Float32(alphay / alphax)))
	t_1 = Float32(cos(t_0) / alphax)
	t_2 = Float32(t_1 * t_1)
	t_3 = Float32(sin(t_0) / alphay)
	t_4 = Float32(t_3 * t_3)
	return sqrt(Float32(Float32(1.0) / Float32(Float32(Float32(u0 / Float32(Float32((t_4 ^ Float32(3.0)) + (t_2 ^ Float32(3.0))) / Float32(Float32(t_4 * t_4) + Float32(Float32(t_2 * t_2) - Float32(t_4 * t_2))))) / Float32(Float32(1.0) - u0)) + Float32(1.0))))
end

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in u1 around 0

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

5. Applied rewrites 100.0%

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

7. Applied rewrites 99.9%

   \[\leadsto \sqrt{\frac{1}{\frac{\frac{u0}{\frac{{\left(\frac{\sin \tan^{-1} \left(\tan \left(\left(\left(\mathsf{fma}\left(\pi \cdot 2, u1, \pi \cdot 0.5\right) + \pi\right) + \pi\right) + \pi\right) \cdot \frac{alphay}{alphax}\right)}{alphay} \cdot \frac{\sin \tan^{-1} \left(\tan \left(\left(\left(\mathsf{fma}\left(\pi \cdot 2, u1, \pi \cdot 0.5\right) + \pi\right) + \pi\right) + \pi\right) \cdot \frac{alphay}{alphax}\right)}{alphay}\right)}^{3} + {\left(\frac{\cos \tan^{-1} \left(\tan \left(\left(\left(\mathsf{fma}\left(\pi \cdot 2, u1, \pi \cdot 0.5\right) + \pi\right) + \pi\right) + \pi\right) \cdot \frac{alphay}{alphax}\right)}{alphax} \cdot \frac{\cos \tan^{-1} \left(\tan \left(\left(\left(\mathsf{fma}\left(\pi \cdot 2, u1, \pi \cdot 0.5\right) + \pi\right) + \pi\right) + \pi\right) \cdot \frac{alphay}{alphax}\right)}{alphax}\right)}^{3}}{\left(\frac{\sin \tan^{-1} \left(\tan \left(\left(\left(\mathsf{fma}\left(\pi \cdot 2, u1, \pi \cdot 0.5\right) + \pi\right) + \pi\right) + \pi\right) \cdot \frac{alphay}{alphax}\right)}{alphay} \cdot \frac{\sin \tan^{-1} \left(\tan \left(\left(\left(\mathsf{fma}\left(\pi \cdot 2, u1, \pi \cdot 0.5\right) + \pi\right) + \pi\right) + \pi\right) \cdot \frac{alphay}{alphax}\right)}{alphay}\right) \cdot \left(\frac{\sin \tan^{-1} \left(\tan \left(\left(\left(\mathsf{fma}\left(\pi \cdot 2, u1, \pi \cdot 0.5\right) + \pi\right) + \pi\right) + \pi\right) \cdot \frac{alphay}{alphax}\right)}{alphay} \cdot \frac{\sin \tan^{-1} \left(\tan \left(\left(\left(\mathsf{fma}\left(\pi \cdot 2, u1, \pi \cdot 0.5\right) + \pi\right) + \pi\right) + \pi\right) \cdot \frac{alphay}{alphax}\right)}{alphay}\right) + \left(\left(\frac{\cos \tan^{-1} \left(\tan \left(\left(\left(\mathsf{fma}\left(\pi \cdot 2, u1, \pi \cdot 0.5\right) + \pi\right) + \pi\right) + \pi\right) \cdot \frac{alphay}{alphax}\right)}{alphax} \cdot \frac{\cos \tan^{-1} \left(\tan \left(\left(\left(\mathsf{fma}\left(\pi \cdot 2, u1, \pi \cdot 0.5\right) + \pi\right) + \pi\right) + \pi\right) \cdot \frac{alphay}{alphax}\right)}{alphax}\right) \cdot \left(\frac{\cos \tan^{-1} \left(\tan \left(\left(\left(\mathsf{fma}\left(\pi \cdot 2, u1, \pi \cdot 0.5\right) + \pi\right) + \pi\right) + \pi\right) \cdot \frac{alphay}{alphax}\right)}{alphax} \cdot \frac{\cos \tan^{-1} \left(\tan \left(\left(\left(\mathsf{fma}\left(\pi \cdot 2, u1, \pi \cdot 0.5\right) + \pi\right) + \pi\right) + \pi\right) \cdot \frac{alphay}{alphax}\right)}{alphax}\right) - \left(\frac{\sin \tan^{-1} \left(\tan \left(\left(\left(\mathsf{fma}\left(\pi \cdot 2, u1, \pi \cdot 0.5\right) + \pi\right) + \pi\right) + \pi\right) \cdot \frac{alphay}{alphax}\right)}{alphay} \cdot \frac{\sin \tan^{-1} \left(\tan \left(\left(\left(\mathsf{fma}\left(\pi \cdot 2, u1, \pi \cdot 0.5\right) + \pi\right) + \pi\right) + \pi\right) \cdot \frac{alphay}{alphax}\right)}{alphay}\right) \cdot \left(\frac{\cos \tan^{-1} \left(\tan \left(\left(\left(\mathsf{fma}\left(\pi \cdot 2, u1, \pi \cdot 0.5\right) + \pi\right) + \pi\right) + \pi\right) \cdot \frac{alphay}{alphax}\right)}{alphax} \cdot \frac{\cos \tan^{-1} \left(\tan \left(\left(\left(\mathsf{fma}\left(\pi \cdot 2, u1, \pi \cdot 0.5\right) + \pi\right) + \pi\right) + \pi\right) \cdot \frac{alphay}{alphax}\right)}{alphax}\right)\right)}}}{1 - u0} + 1}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025065 
(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))))))