Trowbridge-Reitz Sample, sample surface normal, cosTheta

Percentage Accurate: 99.4% → 99.9%

Time: 17.2s

Alternatives: 7

Speedup: 1.6×

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: 99.4% 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, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in alphay around 0 99.3%

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

5. Simplified 99.3%

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

   1. *-un-lft-identity 99.3%

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

   2. *-commutative 99.3%

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

   3. associate-*l* 99.3%

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

   4. fma-define 99.3%

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

   5. *-commutative 99.3%

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

7. Applied egg-rr 99.3%

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

   1. *-rgt-identity 99.3%

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

   2. fma-undefine 99.3%

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

   3. *-commutative 99.3%

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

   4. *-commutative 99.3%

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

   5. +-commutative 99.3%

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

   6. associate-*r* 99.3%

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

   7. distribute-rgt-out 99.3%

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

9. Simplified 99.3%

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

    1. *-un-lft-identity 99.3%

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

    2. *-commutative 99.3%

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

    3. associate-*l* 99.3%

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

    4. fma-define 99.3%

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

    5. *-commutative 99.3%

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

11. Applied egg-rr 99.3%

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

    1. *-rgt-identity 99.3%

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

    2. fma-undefine 99.3%

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

    3. *-commutative 99.3%

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

    4. *-commutative 99.3%

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

    5. +-commutative 99.3%

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

    6. associate-*r* 99.3%

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

    7. distribute-rgt-out 99.3%

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

13. Simplified 99.3%

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

15. Applied egg-rr 99.9%

    \[\leadsto \color{blue}{1 \cdot {\left(1 + \frac{\frac{u0}{{\left(\mathsf{hypot}\left(\frac{\cos \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\pi \cdot \mathsf{fma}\left(2, u1, 0.5\right)\right)\right)}{alphax}, \frac{\sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\pi \cdot \mathsf{fma}\left(2, u1, 0.5\right)\right)\right)}{alphay}\right)\right)}^{2}}}{1 - u0}\right)}^{-0.5}} \]
16. Step-by-step derivation

    1. *-lft-identity 99.9%

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

17. Simplified 99.9%

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

Alternative 2: 99.4% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.3%

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

   1. cos-atan 99.4%

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

   2. cos-atan 99.4%

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

   3. frac-times 99.4%

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

   4. metadata-eval 99.4%

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

4. Applied egg-rr 99.4%

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

   1. fma-undefine 99.4%

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

   2. *-commutative 99.4%

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

   3. *-commutative 99.4%

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

   4. +-commutative 99.4%

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

   5. associate-*r* 99.4%

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

   6. distribute-rgt-out 99.4%

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

6. Simplified 99.4%

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

   1. *-un-lft-identity 99.3%

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

   2. *-commutative 99.3%

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

   3. associate-*l* 99.3%

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

   4. fma-define 99.3%

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

   5. *-commutative 99.3%

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

8. Applied egg-rr 99.4%

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

   1. *-rgt-identity 99.3%

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

   2. fma-undefine 99.3%

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

   3. *-commutative 99.3%

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

   4. *-commutative 99.3%

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

   5. +-commutative 99.3%

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

   6. associate-*r* 99.3%

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

   7. distribute-rgt-out 99.3%

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

10. Simplified 99.4%

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

    1. *-un-lft-identity 99.3%

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

    2. *-commutative 99.3%

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

    3. associate-*l* 99.3%

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

    4. fma-define 99.3%

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

    5. *-commutative 99.3%

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

12. Applied egg-rr 99.4%

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

    1. *-rgt-identity 99.3%

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

    2. fma-undefine 99.3%

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

    3. *-commutative 99.3%

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

    4. *-commutative 99.3%

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

    5. +-commutative 99.3%

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

    6. associate-*r* 99.3%

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

    7. distribute-rgt-out 99.3%

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

14. Simplified 99.4%

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

15. Taylor expanded in alphay around 0 99.4%

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

    1. +-commutative 99.4%

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

    2. fma-undefine 99.4%

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

    3. associate-*l/ 99.4%

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

    4. unpow2 99.4%

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

    5. unpow2 99.4%

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

    6. times-frac 99.4%

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

    7. unpow2 99.4%

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

17. Simplified 99.4%

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

18. Final simplification 99.4%

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

Alternative 3: 98.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in alphay around 0 99.3%

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

5. Simplified 99.3%

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

   1. *-un-lft-identity 99.3%

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

   2. *-commutative 99.3%

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

   3. associate-*l* 99.3%

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

   4. fma-define 99.3%

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

   5. *-commutative 99.3%

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

7. Applied egg-rr 99.3%

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

   1. *-rgt-identity 99.3%

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

   2. fma-undefine 99.3%

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

   3. *-commutative 99.3%

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

   4. *-commutative 99.3%

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

   5. +-commutative 99.3%

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

   6. associate-*r* 99.3%

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

   7. distribute-rgt-out 99.3%

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

9. Simplified 99.3%

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

    1. *-un-lft-identity 99.3%

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

    2. *-commutative 99.3%

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

    3. associate-*l* 99.3%

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

    4. fma-define 99.3%

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

    5. *-commutative 99.3%

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

11. Applied egg-rr 99.3%

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

    1. *-rgt-identity 99.3%

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

    2. fma-undefine 99.3%

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

    3. *-commutative 99.3%

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

    4. *-commutative 99.3%

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

    5. +-commutative 99.3%

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

    6. associate-*r* 99.3%

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

    7. distribute-rgt-out 99.3%

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

13. Simplified 99.3%

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

14. Taylor expanded in alphax around inf 98.3%

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

    1. times-frac 98.3%

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

16. Simplified 98.3%

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

Alternative 4: 98.1% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in alphay around 0 99.3%

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

5. Simplified 99.3%

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

   1. *-un-lft-identity 99.3%

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

   2. *-commutative 99.3%

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

   3. associate-*l* 99.3%

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

   4. fma-define 99.3%

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

   5. *-commutative 99.3%

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

7. Applied egg-rr 99.3%

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

   1. *-rgt-identity 99.3%

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

   2. fma-undefine 99.3%

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

   3. *-commutative 99.3%

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

   4. *-commutative 99.3%

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

   5. +-commutative 99.3%

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

   6. associate-*r* 99.3%

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

   7. distribute-rgt-out 99.3%

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

9. Simplified 99.3%

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

    1. *-un-lft-identity 99.3%

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

    2. *-commutative 99.3%

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

    3. associate-*l* 99.3%

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

    4. fma-define 99.3%

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

    5. *-commutative 99.3%

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

11. Applied egg-rr 99.3%

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

    1. *-rgt-identity 99.3%

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

    2. fma-undefine 99.3%

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

    3. *-commutative 99.3%

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

    4. *-commutative 99.3%

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

    5. +-commutative 99.3%

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

    6. associate-*r* 99.3%

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

    7. distribute-rgt-out 99.3%

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

13. Simplified 99.3%

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

14. Taylor expanded in alphax around inf 98.3%

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

15. Final simplification 98.3%

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

Alternative 5: 96.4% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in alphay around 0 99.3%

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

5. Simplified 99.3%

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

   1. *-un-lft-identity 99.3%

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

   2. *-commutative 99.3%

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

   3. associate-*l* 99.3%

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

   4. fma-define 99.3%

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

   5. *-commutative 99.3%

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

7. Applied egg-rr 99.3%

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

   1. *-rgt-identity 99.3%

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

   2. fma-undefine 99.3%

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

   3. *-commutative 99.3%

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

   4. *-commutative 99.3%

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

   5. +-commutative 99.3%

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

   6. associate-*r* 99.3%

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

   7. distribute-rgt-out 99.3%

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

9. Simplified 99.3%

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

    1. *-un-lft-identity 99.3%

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

    2. *-commutative 99.3%

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

    3. associate-*l* 99.3%

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

    4. fma-define 99.3%

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

    5. *-commutative 99.3%

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

11. Applied egg-rr 99.3%

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

    1. *-rgt-identity 99.3%

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

    2. fma-undefine 99.3%

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

    3. *-commutative 99.3%

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

    4. *-commutative 99.3%

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

    5. +-commutative 99.3%

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

    6. associate-*r* 99.3%

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

    7. distribute-rgt-out 99.3%

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

13. Simplified 99.3%

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

14. Taylor expanded in alphay around 0 95.2%

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

15. Final simplification 95.2%

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

Alternative 6: 96.4% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in alphay around 0 99.3%

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

5. Simplified 99.3%

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

   1. *-un-lft-identity 99.3%

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

   2. *-commutative 99.3%

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

   3. associate-*l* 99.3%

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

   4. fma-define 99.3%

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

   5. *-commutative 99.3%

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

7. Applied egg-rr 99.3%

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

   1. *-rgt-identity 99.3%

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

   2. fma-undefine 99.3%

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

   3. *-commutative 99.3%

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

   4. *-commutative 99.3%

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

   5. +-commutative 99.3%

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

   6. associate-*r* 99.3%

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

   7. distribute-rgt-out 99.3%

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

9. Simplified 99.3%

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

    1. *-un-lft-identity 99.3%

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

    2. *-commutative 99.3%

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

    3. associate-*l* 99.3%

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

    4. fma-define 99.3%

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

    5. *-commutative 99.3%

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

11. Applied egg-rr 99.3%

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

    1. *-rgt-identity 99.3%

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

    2. fma-undefine 99.3%

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

    3. *-commutative 99.3%

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

    4. *-commutative 99.3%

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

    5. +-commutative 99.3%

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

    6. associate-*r* 99.3%

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

    7. distribute-rgt-out 99.3%

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

13. Simplified 99.3%

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

14. Taylor expanded in alphay around 0 94.9%

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

15. Final simplification 94.9%

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

Alternative 7: 91.3% accurate, 1375.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (u0 u1 alphax alphay) :precision binary32 1.0)

C

float code(float u0, float u1, float alphax, float alphay) {
	return 1.0f;
}

Fortran

real(4) function code(u0, u1, alphax, alphay)
    real(4), intent (in) :: u0
    real(4), intent (in) :: u1
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    code = 1.0e0
end function

Julia

function code(u0, u1, alphax, alphay)
	return Float32(1.0)
end

MATLAB

function tmp = code(u0, u1, alphax, alphay)
	tmp = single(1.0);
end

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in alphay around 0 99.3%

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

5. Simplified 99.3%

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

   1. *-un-lft-identity 99.3%

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

   2. *-commutative 99.3%

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

   3. associate-*l* 99.3%

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

   4. fma-define 99.3%

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

   5. *-commutative 99.3%

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

7. Applied egg-rr 99.3%

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

   1. *-rgt-identity 99.3%

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

   2. fma-undefine 99.3%

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

   3. *-commutative 99.3%

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

   4. *-commutative 99.3%

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

   5. +-commutative 99.3%

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

   6. associate-*r* 99.3%

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

   7. distribute-rgt-out 99.3%

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

9. Simplified 99.3%

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

    1. *-un-lft-identity 99.3%

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

    2. *-commutative 99.3%

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

    3. associate-*l* 99.3%

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

    4. fma-define 99.3%

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

    5. *-commutative 99.3%

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

11. Applied egg-rr 99.3%

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

    1. *-rgt-identity 99.3%

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

    2. fma-undefine 99.3%

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

    3. *-commutative 99.3%

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

    4. *-commutative 99.3%

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

    5. +-commutative 99.3%

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

    6. associate-*r* 99.3%

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

    7. distribute-rgt-out 99.3%

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

13. Simplified 99.3%

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

14. Taylor expanded in alphay around 0 89.0%

    \[\leadsto \color{blue}{1} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024166 
(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))))))