Trowbridge-Reitz Sample, sample surface normal, cosTheta

Percentage Accurate: 99.4% → 99.9%

Time: 23.4s

Alternatives: 8

Speedup: 2.2×

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, 1.9× speedup

Math

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

FPCore

(FPCore (u0 u1 alphax alphay)
 :precision binary32
 (let* ((t_0 (tan (* PI (fma 2.0 u1 0.5)))))
   (sqrt
    (/
     1.0
     (+
      1.0
      (/
       u0
       (*
        (+
         (/
          (/ 1.0 (+ 1.0 (pow (* t_0 (/ alphay alphax)) 2.0)))
          (* alphax alphax))
         (/
          (pow (sin (atan (/ (* t_0 alphay) alphax))) 2.0)
          (* alphay alphay)))
        (- 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 sqrtf((1.0f / (1.0f + (u0 / ((((1.0f / (1.0f + powf((t_0 * (alphay / alphax)), 2.0f))) / (alphax * alphax)) + (powf(sinf(atanf(((t_0 * alphay) / alphax))), 2.0f) / (alphay * alphay))) * (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 sqrt(Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(u0 / Float32(Float32(Float32(Float32(Float32(1.0) / Float32(Float32(1.0) + (Float32(t_0 * Float32(alphay / alphax)) ^ Float32(2.0)))) / Float32(alphax * alphax)) + Float32((sin(atan(Float32(Float32(t_0 * alphay) / alphax))) ^ 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

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

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

8. Applied rewrites 99.9%

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

   1. lift-PI.f32 N/A

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

   2. lift-fma.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-tan.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-/.f32 N/A

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

   7. cos-atan N/A

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

   8. inv-pow N/A

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

   9. pow-pow N/A

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

10. Applied rewrites 99.9%

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

11. Final simplification 99.9%

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

Alternative 2: 99.9% accurate, 1.9× speedup

Math

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

FPCore

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

C

float code(float u0, float u1, float alphax, float alphay) {
	float t_0 = powf((tanf((((float) M_PI) * fmaf(2.0f, u1, 0.5f))) * (alphay / alphax)), 2.0f);
	return expf((log1pf((u0 / ((1.0f - u0) * (((1.0f + (1.0f / (-1.0f - t_0))) / (alphay * alphay)) + (1.0f / (alphax * (alphax * (1.0f + t_0)))))))) * -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)) ^ Float32(2.0)
	return exp(Float32(log1p(Float32(u0 / Float32(Float32(Float32(1.0) - u0) * Float32(Float32(Float32(Float32(1.0) + Float32(Float32(1.0) / Float32(Float32(-1.0) - t_0))) / Float32(alphay * alphay)) + Float32(Float32(1.0) / Float32(alphax * Float32(alphax * Float32(Float32(1.0) + t_0)))))))) * 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 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%

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

7. Applied rewrites 99.7%

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

8. Final simplification 99.7%

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

Alternative 3: 99.8% accurate, 2.2× speedup

Math

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

FPCore

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

C

float code(float u0, float u1, float alphax, float alphay) {
	float t_0 = powf((tanf((((float) M_PI) * fmaf(2.0f, u1, 0.5f))) * (alphay / alphax)), 2.0f);
	return powf((1.0f + (u0 / ((1.0f - u0) * (((1.0f + (1.0f / (-1.0f - t_0))) / (alphay * alphay)) + (1.0f / (alphax * (alphax * (1.0f + t_0)))))))), -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)) ^ Float32(2.0)
	return Float32(Float32(1.0) + Float32(u0 / Float32(Float32(Float32(1.0) - u0) * Float32(Float32(Float32(Float32(1.0) + Float32(Float32(1.0) / Float32(Float32(-1.0) - t_0))) / Float32(alphay * alphay)) + Float32(Float32(1.0) / Float32(alphax * Float32(alphax * Float32(Float32(1.0) + t_0)))))))) ^ 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 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%

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

7. Applied rewrites 99.6%

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

8. Final simplification 99.6%

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

Alternative 4: 99.3% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

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)) ^ Float32(2.0)
	return Float32(Float32(1.0) / sqrt(Float32(Float32(1.0) + Float32(u0 / Float32(Float32(Float32(1.0) - u0) * Float32(Float32(Float32(Float32(1.0) + Float32(Float32(1.0) / Float32(Float32(-1.0) - t_0))) / Float32(alphay * alphay)) + Float32(Float32(1.0) / Float32(alphax * Float32(alphax * Float32(Float32(1.0) + t_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%

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

7. Applied rewrites 99.1%

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

8. Final simplification 99.1%

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

Alternative 5: 98.2% accurate, 30.6× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{1}{\mathsf{fma}\left(alphay \cdot alphay, \frac{u0}{1 - u0}, 1\right)}} \end{array} \]

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in 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 97.5%

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

8. Taylor expanded in alphax around 0

   \[\leadsto \color{blue}{\sqrt{\frac{1}{1 + \frac{{alphay}^{2} \cdot u0}{1 - u0}}}} \]
9. Step-by-step derivation

   1. lower-sqrt.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. +-commutative N/A

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

   4. associate-/l* N/A

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

   5. lower-fma.f32 N/A

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

   6. unpow2 N/A

      \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(\color{blue}{alphay \cdot alphay}, \frac{u0}{1 - u0}, 1\right)}} \]

   7. lower-*.f32 N/A

      \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(\color{blue}{alphay \cdot alphay}, \frac{u0}{1 - u0}, 1\right)}} \]

   8. lower-/.f32 N/A

      \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(alphay \cdot alphay, \color{blue}{\frac{u0}{1 - u0}}, 1\right)}} \]

   9. lower--.f32 97.6

      \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(alphay \cdot alphay, \frac{u0}{\color{blue}{1 - u0}}, 1\right)}} \]

10. Applied rewrites 97.6%

    \[\leadsto \color{blue}{\sqrt{\frac{1}{\mathsf{fma}\left(alphay \cdot alphay, \frac{u0}{1 - u0}, 1\right)}}} \]
11. Add Preprocessing

Alternative 6: 96.6% accurate, 46.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-0.5, \frac{u0 \cdot \left(alphay \cdot alphay\right)}{1 - u0}, 1\right) \end{array} \]

FPCore

(FPCore (u0 u1 alphax alphay)
 :precision binary32
 (fma -0.5 (/ (* u0 (* alphay alphay)) (- 1.0 u0)) 1.0))

C

float code(float u0, float u1, float alphax, float alphay) {
	return fmaf(-0.5f, ((u0 * (alphay * alphay)) / (1.0f - u0)), 1.0f);
}

Julia

function code(u0, u1, alphax, alphay)
	return fma(Float32(-0.5), Float32(Float32(u0 * Float32(alphay * alphay)) / Float32(Float32(1.0) - u0)), Float32(1.0))
end

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in 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 97.5%

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

8. Taylor expanded in alphax around 0

   \[\leadsto \color{blue}{\sqrt{\frac{1}{1 + \frac{{alphay}^{2} \cdot u0}{1 - u0}}}} \]
9. Step-by-step derivation

   1. lower-sqrt.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. +-commutative N/A

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

   4. associate-/l* N/A

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

   5. lower-fma.f32 N/A

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

   6. unpow2 N/A

      \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(\color{blue}{alphay \cdot alphay}, \frac{u0}{1 - u0}, 1\right)}} \]

   7. lower-*.f32 N/A

      \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(\color{blue}{alphay \cdot alphay}, \frac{u0}{1 - u0}, 1\right)}} \]

   8. lower-/.f32 N/A

      \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(alphay \cdot alphay, \color{blue}{\frac{u0}{1 - u0}}, 1\right)}} \]

   9. lower--.f32 97.6

      \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(alphay \cdot alphay, \frac{u0}{\color{blue}{1 - u0}}, 1\right)}} \]

10. Applied rewrites 97.6%

    \[\leadsto \color{blue}{\sqrt{\frac{1}{\mathsf{fma}\left(alphay \cdot alphay, \frac{u0}{1 - u0}, 1\right)}}} \]

11. Taylor expanded in alphay around 0

    \[\leadsto \color{blue}{1 + \frac{-1}{2} \cdot \frac{{alphay}^{2} \cdot u0}{1 - u0}} \]
12. Step-by-step derivation

    1. +-commutative N/A

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

    2. lower-fma.f32 N/A

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

    3. lower-/.f32 N/A

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

    4. lower-*.f32 N/A

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

    5. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{\color{blue}{\left(alphay \cdot alphay\right)} \cdot u0}{1 - u0}, 1\right) \]

    6. lower-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \frac{\color{blue}{\left(alphay \cdot alphay\right)} \cdot u0}{1 - u0}, 1\right) \]

    7. lower--.f32 96.2

       \[\leadsto \mathsf{fma}\left(-0.5, \frac{\left(alphay \cdot alphay\right) \cdot u0}{\color{blue}{1 - u0}}, 1\right) \]

13. Applied rewrites 96.2%

    \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{\left(alphay \cdot alphay\right) \cdot u0}{1 - u0}, 1\right)} \]

14. Final simplification 96.2%

    \[\leadsto \mathsf{fma}\left(-0.5, \frac{u0 \cdot \left(alphay \cdot alphay\right)}{1 - u0}, 1\right) \]
15. Add Preprocessing

Alternative 7: 95.1% accurate, 84.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-0.5, u0 \cdot \left(alphay \cdot alphay\right), 1\right) \end{array} \]

FPCore

(FPCore (u0 u1 alphax alphay)
 :precision binary32
 (fma -0.5 (* u0 (* alphay alphay)) 1.0))

C

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

Julia

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in 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 97.5%

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

8. Taylor expanded in alphax around 0

   \[\leadsto \color{blue}{\sqrt{\frac{1}{1 + \frac{{alphay}^{2} \cdot u0}{1 - u0}}}} \]
9. Step-by-step derivation

   1. lower-sqrt.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. +-commutative N/A

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

   4. associate-/l* N/A

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

   5. lower-fma.f32 N/A

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

   6. unpow2 N/A

      \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(\color{blue}{alphay \cdot alphay}, \frac{u0}{1 - u0}, 1\right)}} \]

   7. lower-*.f32 N/A

      \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(\color{blue}{alphay \cdot alphay}, \frac{u0}{1 - u0}, 1\right)}} \]

   8. lower-/.f32 N/A

      \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(alphay \cdot alphay, \color{blue}{\frac{u0}{1 - u0}}, 1\right)}} \]

   9. lower--.f32 97.6

      \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(alphay \cdot alphay, \frac{u0}{\color{blue}{1 - u0}}, 1\right)}} \]

10. Applied rewrites 97.6%

    \[\leadsto \color{blue}{\sqrt{\frac{1}{\mathsf{fma}\left(alphay \cdot alphay, \frac{u0}{1 - u0}, 1\right)}}} \]

11. Taylor expanded in u0 around 0

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

    1. associate-*r* N/A

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

    2. +-commutative N/A

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

    3. associate-*r* N/A

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

    4. lower-fma.f32 N/A

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

    5. lower-*.f32 N/A

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

    6. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{\left(alphay \cdot alphay\right)} \cdot u0, 1\right) \]

    7. lower-*.f32 94.6

       \[\leadsto \mathsf{fma}\left(-0.5, \color{blue}{\left(alphay \cdot alphay\right)} \cdot u0, 1\right) \]

13. Applied rewrites 94.6%

    \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \left(alphay \cdot alphay\right) \cdot u0, 1\right)} \]

14. Final simplification 94.6%

    \[\leadsto \mathsf{fma}\left(-0.5, u0 \cdot \left(alphay \cdot alphay\right), 1\right) \]
15. 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 99.4%

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