Trowbridge-Reitz Sample, sample surface normal, cosTheta

Percentage Accurate: 99.3% → 99.3%

Time: 16.7s

Alternatives: 6

Speedup: 1.8×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.3% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.2%

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

   1. cos-atan 99.3%

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

   2. cos-atan 99.3%

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

   3. frac-times 99.3%

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

   4. metadata-eval 99.3%

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

4. Applied egg-rr 99.3%

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

   1. fma-undefine 99.3%

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

   2. *-commutative 99.3%

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

   3. *-commutative 99.3%

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

   4. +-commutative 99.3%

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

   5. associate-*r* 99.3%

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

   6. distribute-rgt-out 99.3%

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

6. Simplified 99.3%

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

   1. *-un-lft-identity 99.3%

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

   2. *-commutative 99.3%

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

   3. associate-*l* 99.3%

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

   4. fma-define 99.3%

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

   5. *-commutative 99.3%

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

8. Applied egg-rr 99.3%

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

   1. *-rgt-identity 99.3%

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

   2. fma-undefine 99.3%

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

   3. *-commutative 99.3%

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

   4. *-commutative 99.3%

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

   5. +-commutative 99.3%

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

   6. associate-*r* 99.3%

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

   7. distribute-rgt-out 99.3%

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

10. Simplified 99.3%

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

    1. *-un-lft-identity 99.3%

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

    2. *-commutative 99.3%

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

    3. associate-*l* 99.3%

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

    4. fma-define 99.3%

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

    5. *-commutative 99.3%

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

12. Applied egg-rr 99.3%

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

    1. *-rgt-identity 99.3%

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

    2. fma-undefine 99.3%

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

    3. *-commutative 99.3%

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

    4. *-commutative 99.3%

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

    5. +-commutative 99.3%

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

    6. associate-*r* 99.3%

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

    7. distribute-rgt-out 99.3%

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

14. Simplified 99.3%

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

15. Taylor expanded in u1 around -inf 99.3%

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

16. Final simplification 99.3%

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

Alternative 2: 98.3% accurate, 2.2× speedup

Math

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

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 (* PI 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 * (((float) M_PI) * 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(Float32(pi) * Float32(0.5)) + Float32(Float32(2.0) * Float32(Float32(pi) * 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) * (single(pi) * u1))))) / alphax))) ^ single(2.0)))))));
end

Derivation

1. Initial program 99.2%

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

   1. cos-atan 99.3%

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

   2. cos-atan 99.3%

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

   3. frac-times 99.3%

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

   4. metadata-eval 99.3%

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

4. Applied egg-rr 99.3%

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

   1. fma-undefine 99.3%

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

   2. *-commutative 99.3%

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

   3. *-commutative 99.3%

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

   4. +-commutative 99.3%

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

   5. associate-*r* 99.3%

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

   6. distribute-rgt-out 99.3%

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

6. Simplified 99.3%

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

7. Taylor expanded in alphay around 0 92.2%

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

9. Applied egg-rr 92.2%

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

10. Taylor expanded in alphax around inf 97.5%

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

11. Final simplification 97.5%

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

Alternative 3: 97.9% accurate, 2.6× speedup

Math

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

FPCore

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

C

float code(float u0, float u1, float alphax, float alphay) {
	return 1.0f / sqrtf((1.0f + ((u0 * ((alphay * alphay) / powf(sinf(atanf((alphay * (tanf((((float) M_PI) * (0.5f + (2.0f * u1)))) / alphax)))), 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(alphay * alphay) / (sin(atan(Float32(alphay * Float32(tan(Float32(Float32(pi) * Float32(Float32(0.5) + Float32(Float32(2.0) * u1)))) / alphax)))) ^ Float32(2.0)))) / Float32(Float32(1.0) - u0)))))
end

MATLAB

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

Derivation

1. Initial program 99.2%

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

   1. cos-atan 99.3%

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

   2. cos-atan 99.3%

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

   3. frac-times 99.3%

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

   4. metadata-eval 99.3%

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

4. Applied egg-rr 99.3%

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

   1. fma-undefine 99.3%

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

   2. *-commutative 99.3%

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

   3. *-commutative 99.3%

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

   4. +-commutative 99.3%

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

   5. associate-*r* 99.3%

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

   6. distribute-rgt-out 99.3%

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

6. Simplified 99.3%

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

7. Taylor expanded in alphay around 0 92.2%

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

8. Taylor expanded in alphax around inf 97.0%

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

   1. associate-/l* 97.0%

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

   2. associate-*r* 97.0%

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

   3. *-commutative 97.0%

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

   4. distribute-rgt-out 97.0%

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

   5. *-commutative 97.0%

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

10. Simplified 97.0%

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

    1. pow2 97.0%

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

12. Applied egg-rr 97.0%

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

13. Final simplification 97.0%

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

Alternative 4: 88.1% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\sqrt{1 + u0 \cdot {alphax}^{2}}} \end{array} \]

FPCore

(FPCore (u0 u1 alphax alphay)
 :precision binary32
 (/ 1.0 (sqrt (+ 1.0 (* u0 (pow alphax 2.0))))))

C

float code(float u0, float u1, float alphax, float alphay) {
	return 1.0f / sqrtf((1.0f + (u0 * powf(alphax, 2.0f))));
}

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 / sqrt((1.0e0 + (u0 * (alphax ** 2.0e0))))
end function

Julia

function code(u0, u1, alphax, alphay)
	return Float32(Float32(1.0) / sqrt(Float32(Float32(1.0) + Float32(u0 * (alphax ^ Float32(2.0))))))
end

MATLAB

function tmp = code(u0, u1, alphax, alphay)
	tmp = single(1.0) / sqrt((single(1.0) + (u0 * (alphax ^ single(2.0)))));
end

Derivation

1. Initial program 99.2%

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

   1. cos-atan 99.3%

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

   2. cos-atan 99.3%

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

   3. frac-times 99.3%

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

   4. metadata-eval 99.3%

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

4. Applied egg-rr 99.3%

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

   1. fma-undefine 99.3%

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

   2. *-commutative 99.3%

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

   3. *-commutative 99.3%

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

   4. +-commutative 99.3%

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

   5. associate-*r* 99.3%

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

   6. distribute-rgt-out 99.3%

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

6. Simplified 99.3%

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

7. Taylor expanded in alphay around 0 92.2%

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

8. Taylor expanded in alphax around 0 86.8%

   \[\leadsto \frac{1}{\sqrt{1 + \frac{\color{blue}{{alphax}^{2}} \cdot u0}{1 - u0}}} \]

9. Taylor expanded in u0 around 0 86.9%

   \[\leadsto \frac{1}{\sqrt{1 + \color{blue}{{alphax}^{2} \cdot u0}}} \]

10. Final simplification 86.9%

    \[\leadsto \frac{1}{\sqrt{1 + u0 \cdot {alphax}^{2}}} \]
11. Add Preprocessing

Alternative 5: 88.1% accurate, 12.7× speedup

Math

\[\begin{array}{l} \\ 1 + \left(u0 \cdot {alphax}^{2}\right) \cdot -0.5 \end{array} \]

FPCore

(FPCore (u0 u1 alphax alphay)
 :precision binary32
 (+ 1.0 (* (* u0 (pow alphax 2.0)) -0.5)))

C

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

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 + ((u0 * (alphax ** 2.0e0)) * (-0.5e0))
end function

Julia

function code(u0, u1, alphax, alphay)
	return Float32(Float32(1.0) + Float32(Float32(u0 * (alphax ^ Float32(2.0))) * Float32(-0.5)))
end

MATLAB

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

Derivation

1. Initial program 99.2%

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

   1. cos-atan 99.3%

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

   2. cos-atan 99.3%

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

   3. frac-times 99.3%

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

   4. metadata-eval 99.3%

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

4. Applied egg-rr 99.3%

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

   1. fma-undefine 99.3%

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

   2. *-commutative 99.3%

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

   3. *-commutative 99.3%

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

   4. +-commutative 99.3%

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

   5. associate-*r* 99.3%

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

   6. distribute-rgt-out 99.3%

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

6. Simplified 99.3%

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

7. Taylor expanded in alphay around 0 92.2%

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

8. Taylor expanded in alphax around 0 86.8%

   \[\leadsto \frac{1}{\sqrt{1 + \frac{\color{blue}{{alphax}^{2}} \cdot u0}{1 - u0}}} \]

9. Taylor expanded in u0 around 0 86.9%

   \[\leadsto \color{blue}{1 + -0.5 \cdot \left({alphax}^{2} \cdot u0\right)} \]
10. Step-by-step derivation

    1. *-commutative 86.9%

       \[\leadsto 1 + \color{blue}{\left({alphax}^{2} \cdot u0\right) \cdot -0.5} \]

11. Simplified 86.9%

    \[\leadsto \color{blue}{1 + \left({alphax}^{2} \cdot u0\right) \cdot -0.5} \]

12. Final simplification 86.9%

    \[\leadsto 1 + \left(u0 \cdot {alphax}^{2}\right) \cdot -0.5 \]
13. Add Preprocessing

Alternative 6: 16.7% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \frac{\sqrt{\frac{1}{u0} + -1}}{alphax} \end{array} \]

FPCore

(FPCore (u0 u1 alphax alphay)
 :precision binary32
 (/ (sqrt (+ (/ 1.0 u0) -1.0)) alphax))

C

float code(float u0, float u1, float alphax, float alphay) {
	return sqrtf(((1.0f / u0) + -1.0f)) / alphax;
}

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 = sqrt(((1.0e0 / u0) + (-1.0e0))) / alphax
end function

Julia

function code(u0, u1, alphax, alphay)
	return Float32(sqrt(Float32(Float32(Float32(1.0) / u0) + Float32(-1.0))) / alphax)
end

MATLAB

function tmp = code(u0, u1, alphax, alphay)
	tmp = sqrt(((single(1.0) / u0) + single(-1.0))) / alphax;
end

Derivation

1. Initial program 99.2%

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

   1. cos-atan 99.3%

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

   2. cos-atan 99.3%

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

   3. frac-times 99.3%

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

   4. metadata-eval 99.3%

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

4. Applied egg-rr 99.3%

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

   1. fma-undefine 99.3%

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

   2. *-commutative 99.3%

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

   3. *-commutative 99.3%

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

   4. +-commutative 99.3%

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

   5. associate-*r* 99.3%

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

   6. distribute-rgt-out 99.3%

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

6. Simplified 99.3%

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

7. Taylor expanded in alphay around 0 92.2%

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

8. Taylor expanded in alphax around 0 86.8%

   \[\leadsto \frac{1}{\sqrt{1 + \frac{\color{blue}{{alphax}^{2}} \cdot u0}{1 - u0}}} \]

9. Taylor expanded in alphax around inf 16.8%

   \[\leadsto \color{blue}{\frac{1}{alphax} \cdot \sqrt{\frac{1 - u0}{u0}}} \]
10. Step-by-step derivation

    1. associate-*l/ 16.8%

       \[\leadsto \color{blue}{\frac{1 \cdot \sqrt{\frac{1 - u0}{u0}}}{alphax}} \]

    2. *-lft-identity 16.8%

       \[\leadsto \frac{\color{blue}{\sqrt{\frac{1 - u0}{u0}}}}{alphax} \]

    3. div-sub 16.8%

       \[\leadsto \frac{\sqrt{\color{blue}{\frac{1}{u0} - \frac{u0}{u0}}}}{alphax} \]

    4. sub-neg 16.8%

       \[\leadsto \frac{\sqrt{\color{blue}{\frac{1}{u0} + \left(-\frac{u0}{u0}\right)}}}{alphax} \]

    5. *-inverses 16.8%

       \[\leadsto \frac{\sqrt{\frac{1}{u0} + \left(-\color{blue}{1}\right)}}{alphax} \]

    6. metadata-eval 16.8%

       \[\leadsto \frac{\sqrt{\frac{1}{u0} + \color{blue}{-1}}}{alphax} \]

11. Simplified 16.8%

    \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{u0} + -1}}{alphax}} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024163 
(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))))))