Trowbridge-Reitz Sample, sample surface normal, cosTheta

Percentage Accurate: 99.4% → 99.4%

Time: 14.7s

Alternatives: 8

Speedup: 1.5×

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: 99.4% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

function code(u0, u1, alphax, alphay)
	t_0 = tan(Float32(Float32(pi) * fma(Float32(2.0), u1, Float32(0.5))))
	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) * t_0) ^ Float32(2.0)))) / Float32(alphax * alphax)) + Float32(Float32(sin(atan(Float32(Float32(alphay * t_0) / alphax))) * sin(atan(Float32(Float32(alphay / alphax) * tan(Float32(Float32(u1 * Float32(Float32(pi) * Float32(2.0))) + Float32(Float32(pi) * Float32(0.5)))))))) / Float32(alphay * alphay)))) * u0) / Float32(Float32(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

4. Applied egg-rr 99.3%

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

5. Taylor expanded in alphay around 0

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

   1. lower-atan.f32 N/A

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

   2. lower-/.f32 N/A

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

7. Simplified 99.3%

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

8. Final simplification 99.3%

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

Alternative 2: 99.4% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

4. Applied egg-rr 99.3%

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

5. Taylor expanded in u1 around inf

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

7. Simplified 99.3%

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

Alternative 3: 98.6% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

4. Applied egg-rr 99.3%

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

5. Taylor expanded in alphax around 0

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

7. Simplified 98.6%

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

   1. lift-PI.f32 N/A

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

   2. lift-fma.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-cos.f32 N/A

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

   5. unpow2 N/A

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

   6. lift-cos.f32 N/A

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

   7. lift-cos.f32 N/A

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

   8. sqr-cos-a N/A

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

   9. lower-+.f32 N/A

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

   10. lower-*.f32 N/A

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

   11. cos-2 N/A

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

   12. cos-sum N/A

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

   13. lower-cos.f32 N/A

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

   14. lift-*.f32 N/A

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

   15. lift-*.f32 N/A

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

   16. distribute-rgt-out N/A

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

   17. lower-*.f32 N/A

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

9. Applied egg-rr 98.6%

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

    1. lift-PI.f32 N/A

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

    2. lift-fma.f32 N/A

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

    3. lift-*.f32 N/A

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

    4. lift-tan.f32 N/A

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

    5. lift-*.f32 N/A

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

    6. lift-/.f32 N/A

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

    7. sin-atan N/A

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

    8. lower-/.f32 N/A

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

11. Applied egg-rr 98.6%

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

12. Final simplification 98.6%

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

Alternative 4: 98.6% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

4. Applied egg-rr 99.3%

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

5. Taylor expanded in alphax around 0

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

7. Simplified 98.6%

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

   1. lift-PI.f32 N/A

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

   2. lift-fma.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-cos.f32 N/A

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

   5. unpow2 N/A

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

   6. lift-cos.f32 N/A

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

   7. lift-cos.f32 N/A

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

   8. sqr-cos-a N/A

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

   9. lower-+.f32 N/A

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

   10. lower-*.f32 N/A

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

   11. cos-2 N/A

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

   12. cos-sum N/A

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

   13. lower-cos.f32 N/A

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

   14. lift-*.f32 N/A

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

   15. lift-*.f32 N/A

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

   16. distribute-rgt-out N/A

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

   17. lower-*.f32 N/A

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

9. Applied egg-rr 98.6%

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

10. Final simplification 98.6%

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

Alternative 5: 98.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \mathsf{fma}\left(2, u1, 0.5\right)\\ \frac{1}{\sqrt{1 + \frac{u0}{\left(1 - u0\right) \cdot \left(\frac{{\sin \tan^{-1} \left(\frac{alphay \cdot \tan t\_0}{alphax}\right)}^{2}}{alphay \cdot alphay} + \frac{0.5 + 0.5 \cdot \mathsf{fma}\left(u1, u1 \cdot \left(\left(\pi \cdot \pi\right) \cdot 8\right), -1\right)}{\left(alphay \cdot alphay\right) \cdot {\sin t\_0}^{2}}\right)}}} \end{array} \end{array} \]

FPCore

(FPCore (u0 u1 alphax alphay)
 :precision binary32
 (let* ((t_0 (* PI (fma 2.0 u1 0.5))))
   (/
    1.0
    (sqrt
     (+
      1.0
      (/
       u0
       (*
        (- 1.0 u0)
        (+
         (/
          (pow (sin (atan (/ (* alphay (tan t_0)) alphax))) 2.0)
          (* alphay alphay))
         (/
          (+ 0.5 (* 0.5 (fma u1 (* u1 (* (* PI PI) 8.0)) -1.0)))
          (* (* alphay alphay) (pow (sin t_0) 2.0)))))))))))

C

float code(float u0, float u1, float alphax, float alphay) {
	float t_0 = ((float) M_PI) * fmaf(2.0f, u1, 0.5f);
	return 1.0f / sqrtf((1.0f + (u0 / ((1.0f - u0) * ((powf(sinf(atanf(((alphay * tanf(t_0)) / alphax))), 2.0f) / (alphay * alphay)) + ((0.5f + (0.5f * fmaf(u1, (u1 * ((((float) M_PI) * ((float) M_PI)) * 8.0f)), -1.0f))) / ((alphay * alphay) * powf(sinf(t_0), 2.0f))))))));
}

Julia

function code(u0, u1, alphax, alphay)
	t_0 = Float32(Float32(pi) * fma(Float32(2.0), u1, Float32(0.5)))
	return Float32(Float32(1.0) / sqrt(Float32(Float32(1.0) + Float32(u0 / Float32(Float32(Float32(1.0) - u0) * Float32(Float32((sin(atan(Float32(Float32(alphay * tan(t_0)) / alphax))) ^ Float32(2.0)) / Float32(alphay * alphay)) + Float32(Float32(Float32(0.5) + Float32(Float32(0.5) * fma(u1, Float32(u1 * Float32(Float32(Float32(pi) * Float32(pi)) * Float32(8.0))), Float32(-1.0)))) / Float32(Float32(alphay * alphay) * (sin(t_0) ^ Float32(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

4. Applied egg-rr 99.3%

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

5. Taylor expanded in alphax around 0

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

7. Simplified 98.6%

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

   1. lift-PI.f32 N/A

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

   2. lift-fma.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-cos.f32 N/A

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

   5. unpow2 N/A

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

   6. lift-cos.f32 N/A

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

   7. lift-cos.f32 N/A

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

   8. sqr-cos-a N/A

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

   9. lower-+.f32 N/A

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

   10. lower-*.f32 N/A

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

   11. cos-2 N/A

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

   12. cos-sum N/A

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

   13. lower-cos.f32 N/A

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

   14. lift-*.f32 N/A

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

   15. lift-*.f32 N/A

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

   16. distribute-rgt-out N/A

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

   17. lower-*.f32 N/A

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

9. Applied egg-rr 98.6%

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

10. Taylor expanded in u1 around 0

    \[\leadsto \frac{1}{\sqrt{1 + \frac{u0}{\left(1 - u0\right) \cdot \left(\frac{\frac{1}{2} + \frac{1}{2} \cdot \color{blue}{\left(\cos \mathsf{PI}\left(\right) + u1 \cdot \left(-8 \cdot \left(u1 \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \cos \mathsf{PI}\left(\right)\right)\right) - 4 \cdot \left(\mathsf{PI}\left(\right) \cdot \sin \mathsf{PI}\left(\right)\right)\right)\right)}}{\left(alphay \cdot alphay\right) \cdot {\sin \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(2, u1, \frac{1}{2}\right)\right)}^{2}} + \frac{{\sin \tan^{-1} \left(\frac{alphay \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(2, u1, \frac{1}{2}\right)\right)}{alphax}\right)}^{2}}{alphay \cdot alphay}\right)}}} \]
11. Step-by-step derivation

    1. cos-PI N/A

       \[\leadsto \frac{1}{\sqrt{1 + \frac{u0}{\left(1 - u0\right) \cdot \left(\frac{\frac{1}{2} + \frac{1}{2} \cdot \left(\color{blue}{-1} + u1 \cdot \left(-8 \cdot \left(u1 \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \cos \mathsf{PI}\left(\right)\right)\right) - 4 \cdot \left(\mathsf{PI}\left(\right) \cdot \sin \mathsf{PI}\left(\right)\right)\right)\right)}{\left(alphay \cdot alphay\right) \cdot {\sin \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(2, u1, \frac{1}{2}\right)\right)}^{2}} + \frac{{\sin \tan^{-1} \left(\frac{alphay \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(2, u1, \frac{1}{2}\right)\right)}{alphax}\right)}^{2}}{alphay \cdot alphay}\right)}}} \]

    2. +-commutative N/A

       \[\leadsto \frac{1}{\sqrt{1 + \frac{u0}{\left(1 - u0\right) \cdot \left(\frac{\frac{1}{2} + \frac{1}{2} \cdot \color{blue}{\left(u1 \cdot \left(-8 \cdot \left(u1 \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \cos \mathsf{PI}\left(\right)\right)\right) - 4 \cdot \left(\mathsf{PI}\left(\right) \cdot \sin \mathsf{PI}\left(\right)\right)\right) + -1\right)}}{\left(alphay \cdot alphay\right) \cdot {\sin \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(2, u1, \frac{1}{2}\right)\right)}^{2}} + \frac{{\sin \tan^{-1} \left(\frac{alphay \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(2, u1, \frac{1}{2}\right)\right)}{alphax}\right)}^{2}}{alphay \cdot alphay}\right)}}} \]

    3. associate-*r* N/A

       \[\leadsto \frac{1}{\sqrt{1 + \frac{u0}{\left(1 - u0\right) \cdot \left(\frac{\frac{1}{2} + \frac{1}{2} \cdot \left(u1 \cdot \left(-8 \cdot \left(u1 \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \cos \mathsf{PI}\left(\right)\right)\right) - \color{blue}{\left(4 \cdot \mathsf{PI}\left(\right)\right) \cdot \sin \mathsf{PI}\left(\right)}\right) + -1\right)}{\left(alphay \cdot alphay\right) \cdot {\sin \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(2, u1, \frac{1}{2}\right)\right)}^{2}} + \frac{{\sin \tan^{-1} \left(\frac{alphay \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(2, u1, \frac{1}{2}\right)\right)}{alphax}\right)}^{2}}{alphay \cdot alphay}\right)}}} \]

    4. sin-PI N/A

       \[\leadsto \frac{1}{\sqrt{1 + \frac{u0}{\left(1 - u0\right) \cdot \left(\frac{\frac{1}{2} + \frac{1}{2} \cdot \left(u1 \cdot \left(-8 \cdot \left(u1 \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \cos \mathsf{PI}\left(\right)\right)\right) - \left(4 \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{0}\right) + -1\right)}{\left(alphay \cdot alphay\right) \cdot {\sin \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(2, u1, \frac{1}{2}\right)\right)}^{2}} + \frac{{\sin \tan^{-1} \left(\frac{alphay \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(2, u1, \frac{1}{2}\right)\right)}{alphax}\right)}^{2}}{alphay \cdot alphay}\right)}}} \]

    5. mul0-rgt N/A

       \[\leadsto \frac{1}{\sqrt{1 + \frac{u0}{\left(1 - u0\right) \cdot \left(\frac{\frac{1}{2} + \frac{1}{2} \cdot \left(u1 \cdot \left(-8 \cdot \left(u1 \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \cos \mathsf{PI}\left(\right)\right)\right) - \color{blue}{0}\right) + -1\right)}{\left(alphay \cdot alphay\right) \cdot {\sin \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(2, u1, \frac{1}{2}\right)\right)}^{2}} + \frac{{\sin \tan^{-1} \left(\frac{alphay \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(2, u1, \frac{1}{2}\right)\right)}{alphax}\right)}^{2}}{alphay \cdot alphay}\right)}}} \]

    6. --rgt-identity N/A

       \[\leadsto \frac{1}{\sqrt{1 + \frac{u0}{\left(1 - u0\right) \cdot \left(\frac{\frac{1}{2} + \frac{1}{2} \cdot \left(u1 \cdot \color{blue}{\left(-8 \cdot \left(u1 \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \cos \mathsf{PI}\left(\right)\right)\right)\right)} + -1\right)}{\left(alphay \cdot alphay\right) \cdot {\sin \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(2, u1, \frac{1}{2}\right)\right)}^{2}} + \frac{{\sin \tan^{-1} \left(\frac{alphay \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(2, u1, \frac{1}{2}\right)\right)}{alphax}\right)}^{2}}{alphay \cdot alphay}\right)}}} \]

    7. lower-fma.f32 N/A

       \[\leadsto \frac{1}{\sqrt{1 + \frac{u0}{\left(1 - u0\right) \cdot \left(\frac{\frac{1}{2} + \frac{1}{2} \cdot \color{blue}{\mathsf{fma}\left(u1, -8 \cdot \left(u1 \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \cos \mathsf{PI}\left(\right)\right)\right), -1\right)}}{\left(alphay \cdot alphay\right) \cdot {\sin \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(2, u1, \frac{1}{2}\right)\right)}^{2}} + \frac{{\sin \tan^{-1} \left(\frac{alphay \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(2, u1, \frac{1}{2}\right)\right)}{alphax}\right)}^{2}}{alphay \cdot alphay}\right)}}} \]

12. Simplified 97.9%

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

13. Final simplification 97.9%

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

Alternative 6: 97.8% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

4. Applied egg-rr 99.3%

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

5. Taylor expanded in alphax around inf

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

   1. lower-sqrt.f32 N/A

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

   2. +-commutative N/A

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

   3. associate-/l* N/A

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

   4. lower-fma.f32 N/A

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

7. Simplified 97.2%

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

8. Final simplification 97.2%

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

Alternative 7: 97.8% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Julia

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

4. Applied egg-rr 99.3%

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

5. Taylor expanded in alphax around inf

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

   1. lower-sqrt.f32 N/A

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

   2. +-commutative N/A

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

   3. associate-/l* N/A

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

   4. lower-fma.f32 N/A

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

7. Simplified 97.2%

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

8. Taylor expanded in u1 around 0

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

   1. lower-*.f32 N/A

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

   2. lower-PI.f32 97.2

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

10. Simplified 97.2%

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

12. Applied egg-rr 97.2%

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

13. Final simplification 97.2%

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

Alternative 8: 91.6% accurate, 1436.0× 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 99.2%

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

4. Applied egg-rr 99.3%

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

5. Taylor expanded in alphay around 0

   \[\leadsto \frac{1}{\color{blue}{1}} \]
6. Step-by-step derivation

7. Simplified 90.7%

   \[\leadsto \frac{1}{\color{blue}{1}} \]
8. Step-by-step derivation

   1. metadata-eval 90.7

      \[\leadsto \color{blue}{1} \]

9. Applied egg-rr 90.7%

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

Reproduce

herbie shell --seed 2024219 
(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))))))