Trowbridge-Reitz Sample, sample surface normal, cosTheta

Percentage Accurate: 99.4% → 99.9%

Time: 24.6s

Alternatives: 11

Speedup: 2.0×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in u1 around 0

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

   4. lower--.f32 N/A

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

5. Applied rewrites 99.4%

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

6. Taylor expanded in u1 around 0

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

8. Applied rewrites 99.9%

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

Alternative 2: 99.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan \left(\pi \cdot \mathsf{fma}\left(2, u1, 0.5\right)\right) \cdot \frac{alphay}{alphax}\\ e^{\mathsf{log1p}\left(\frac{u0}{\left(1 - u0\right) \cdot \mathsf{fma}\left(1 - \cos \left(2 \cdot \tan^{-1} t\_0\right), \frac{0.5}{alphay \cdot alphay}, \frac{1}{alphax \cdot \mathsf{fma}\left(alphax, {t\_0}^{2}, alphax\right)}\right)}\right) \cdot -0.5} \end{array} \end{array} \]

FPCore

(FPCore (u0 u1 alphax alphay)
 :precision binary32
 (let* ((t_0 (* (tan (* PI (fma 2.0 u1 0.5))) (/ alphay alphax))))
   (exp
    (*
     (log1p
      (/
       u0
       (*
        (- 1.0 u0)
        (fma
         (- 1.0 (cos (* 2.0 (atan t_0))))
         (/ 0.5 (* alphay alphay))
         (/ 1.0 (* alphax (fma alphax (pow t_0 2.0) alphax)))))))
     -0.5))))

C

float code(float u0, float u1, float alphax, float alphay) {
	float t_0 = tanf((((float) M_PI) * fmaf(2.0f, u1, 0.5f))) * (alphay / alphax);
	return expf((log1pf((u0 / ((1.0f - u0) * fmaf((1.0f - cosf((2.0f * atanf(t_0)))), (0.5f / (alphay * alphay)), (1.0f / (alphax * fmaf(alphax, powf(t_0, 2.0f), alphax))))))) * -0.5f));
}

Julia

function code(u0, u1, alphax, alphay)
	t_0 = Float32(tan(Float32(Float32(pi) * fma(Float32(2.0), u1, Float32(0.5)))) * Float32(alphay / alphax))
	return exp(Float32(log1p(Float32(u0 / Float32(Float32(Float32(1.0) - u0) * fma(Float32(Float32(1.0) - cos(Float32(Float32(2.0) * atan(t_0)))), Float32(Float32(0.5) / Float32(alphay * alphay)), Float32(Float32(1.0) / Float32(alphax * fma(alphax, (t_0 ^ Float32(2.0)), alphax))))))) * Float32(-0.5)))
end

Derivation

1. Initial program 99.4%

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

4. Applied rewrites 99.2%

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

6. Applied rewrites 99.8%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{u0}{\left(1 - u0\right) \cdot \mathsf{fma}\left(1 - \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\pi \cdot \mathsf{fma}\left(2, u1, 0.5\right)\right)\right)\right), \frac{0.5}{alphay \cdot alphay}, \frac{1}{alphax \cdot \left(alphax \cdot \left(1 + {\left(\frac{alphay}{alphax} \cdot \tan \left(\pi \cdot \mathsf{fma}\left(2, u1, 0.5\right)\right)\right)}^{2}\right)\right)}\right)}\right) \cdot -0.5}} \]
7. Step-by-step derivation

8. Applied rewrites 99.8%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{u0}{\left(1 - u0\right) \cdot \mathsf{fma}\left(1 - \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\mathsf{fma}\left(2, u1, 0.5\right) \cdot \pi\right)\right)\right), \frac{0.5}{alphay \cdot alphay}, \frac{1}{alphax \cdot \mathsf{fma}\left(alphax, {\left(\frac{alphay}{alphax} \cdot \tan \left(\mathsf{fma}\left(2, u1, 0.5\right) \cdot \pi\right)\right)}^{2}, alphax\right)}\right)}\right) \cdot -0.5}} \]

9. Final simplification 99.8%

   \[\leadsto e^{\mathsf{log1p}\left(\frac{u0}{\left(1 - u0\right) \cdot \mathsf{fma}\left(1 - \cos \left(2 \cdot \tan^{-1} \left(\tan \left(\pi \cdot \mathsf{fma}\left(2, u1, 0.5\right)\right) \cdot \frac{alphay}{alphax}\right)\right), \frac{0.5}{alphay \cdot alphay}, \frac{1}{alphax \cdot \mathsf{fma}\left(alphax, {\left(\tan \left(\pi \cdot \mathsf{fma}\left(2, u1, 0.5\right)\right) \cdot \frac{alphay}{alphax}\right)}^{2}, alphax\right)}\right)}\right) \cdot -0.5} \]
10. Add Preprocessing

Alternative 3: 99.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.4%

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

4. Applied rewrites 99.2%

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

6. Applied rewrites 99.8%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{u0}{\left(1 - u0\right) \cdot \mathsf{fma}\left(1 - \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\pi \cdot \mathsf{fma}\left(2, u1, 0.5\right)\right)\right)\right), \frac{0.5}{alphay \cdot alphay}, \frac{1}{alphax \cdot \left(alphax \cdot \left(1 + {\left(\frac{alphay}{alphax} \cdot \tan \left(\pi \cdot \mathsf{fma}\left(2, u1, 0.5\right)\right)\right)}^{2}\right)\right)}\right)}\right) \cdot -0.5}} \]

8. Applied rewrites 99.7%

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

9. Final simplification 99.7%

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

Alternative 4: 99.3% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.4%

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

4. Applied rewrites 99.2%

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

5. Final simplification 99.2%

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

Alternative 5: 99.3% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.4%

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

4. Applied rewrites 99.2%

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

6. Applied rewrites 99.8%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{u0}{\left(1 - u0\right) \cdot \mathsf{fma}\left(1 - \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\pi \cdot \mathsf{fma}\left(2, u1, 0.5\right)\right)\right)\right), \frac{0.5}{alphay \cdot alphay}, \frac{1}{alphax \cdot \left(alphax \cdot \left(1 + {\left(\frac{alphay}{alphax} \cdot \tan \left(\pi \cdot \mathsf{fma}\left(2, u1, 0.5\right)\right)\right)}^{2}\right)\right)}\right)}\right) \cdot -0.5}} \]

8. Applied rewrites 99.2%

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

9. Final simplification 99.2%

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

Alternative 6: 98.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ e^{-0.5 \cdot \mathsf{log1p}\left(\frac{u0 \cdot \left(alphay \cdot alphay\right)}{\left(1 - u0\right) \cdot \left(0.5 + \cos \left(2 \cdot \tan^{-1} \left(\tan \left(\pi \cdot \mathsf{fma}\left(2, u1, 0.5\right)\right) \cdot \frac{alphay}{alphax}\right)\right) \cdot -0.5\right)}\right)} \end{array} \]

FPCore

(FPCore (u0 u1 alphax alphay)
 :precision binary32
 (exp
  (*
   -0.5
   (log1p
    (/
     (* u0 (* alphay alphay))
     (*
      (- 1.0 u0)
      (+
       0.5
       (*
        (cos
         (* 2.0 (atan (* (tan (* PI (fma 2.0 u1 0.5))) (/ alphay alphax)))))
        -0.5))))))))

C

float code(float u0, float u1, float alphax, float alphay) {
	return expf((-0.5f * log1pf(((u0 * (alphay * alphay)) / ((1.0f - u0) * (0.5f + (cosf((2.0f * atanf((tanf((((float) M_PI) * fmaf(2.0f, u1, 0.5f))) * (alphay / alphax))))) * -0.5f)))))));
}

Julia

function code(u0, u1, alphax, alphay)
	return exp(Float32(Float32(-0.5) * log1p(Float32(Float32(u0 * Float32(alphay * alphay)) / Float32(Float32(Float32(1.0) - u0) * Float32(Float32(0.5) + Float32(cos(Float32(Float32(2.0) * atan(Float32(tan(Float32(Float32(pi) * fma(Float32(2.0), u1, Float32(0.5)))) * Float32(alphay / alphax))))) * Float32(-0.5))))))))
end

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in alphay around 0

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

5. Applied rewrites 97.1%

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

7. Applied rewrites 97.6%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{u0 \cdot \left(alphay \cdot alphay\right)}{\left(1 - u0\right) \cdot \left(0.5 + -0.5 \cdot \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\pi \cdot \mathsf{fma}\left(2, u1, 0.5\right)\right)\right)\right)\right)}\right) \cdot -0.5}} \]

8. Final simplification 97.6%

   \[\leadsto e^{-0.5 \cdot \mathsf{log1p}\left(\frac{u0 \cdot \left(alphay \cdot alphay\right)}{\left(1 - u0\right) \cdot \left(0.5 + \cos \left(2 \cdot \tan^{-1} \left(\tan \left(\pi \cdot \mathsf{fma}\left(2, u1, 0.5\right)\right) \cdot \frac{alphay}{alphax}\right)\right) \cdot -0.5\right)}\right)} \]
9. Add Preprocessing

Alternative 7: 98.2% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.4%

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

4. Applied rewrites 99.2%

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

5. Taylor expanded in alphax around inf

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

7. Applied rewrites 97.5%

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

8. Final simplification 97.5%

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

Alternative 8: 97.8% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in alphay around 0

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

5. Applied rewrites 97.1%

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

7. Applied rewrites 97.1%

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

9. Applied rewrites 97.1%

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

10. Taylor expanded in u1 around 0

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

12. Applied rewrites 97.1%

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

13. Final simplification 97.1%

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

Alternative 9: 96.6% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.4%

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

4. Applied rewrites 99.2%

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

5. Taylor expanded in alphay around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. associate-/l* N/A

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

   4. distribute-rgt-neg-in N/A

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

   5. mul-1-neg N/A

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

7. Applied rewrites 96.1%

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

8. Taylor expanded in u1 around 0

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

10. Applied rewrites 96.2%

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

11. Final simplification 96.2%

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

Alternative 10: 95.0% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.4%

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

4. Applied rewrites 99.2%

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

5. Taylor expanded in alphay around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. associate-/l* N/A

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

   4. distribute-rgt-neg-in N/A

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

   5. mul-1-neg N/A

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

7. Applied rewrites 96.1%

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

8. Taylor expanded in u0 around 0

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

10. Applied rewrites 94.4%

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

11. Final simplification 94.4%

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

Alternative 11: 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 99.4%

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

3. Taylor expanded in u1 around 0

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

   4. lower--.f32 N/A

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

5. Applied rewrites 99.4%

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

6. Taylor expanded in alphay around 0

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

8. Applied rewrites 91.7%

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

Reproduce

herbie shell --seed 2024216 
(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))))))