Trowbridge-Reitz Sample, sample surface normal, cosTheta

Percentage Accurate: 99.4% → 99.9%

Time: 23.7s

Alternatives: 12

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, 1.7× speedup

Math

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

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
       (*
        (+
         (/
          (pow (/ 1.0 (sqrt (+ 1.0 (pow (* t_0 (/ alphay alphax)) 2.0)))) 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 / (((powf((1.0f / sqrtf((1.0f + powf((t_0 * (alphay / alphax)), 2.0f)))), 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) / sqrt(Float32(Float32(1.0) + (Float32(t_0 * Float32(alphay / alphax)) ^ Float32(2.0))))) ^ 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. lower-/.f32 N/A

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

10. Applied rewrites 99.9%

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

11. Final simplification 99.9%

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

Alternative 2: 99.8% accurate, 2.2× speedup

Math

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

FPCore

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

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 sqrtf((1.0f / (1.0f + (u0 / ((1.0f - u0) * ((1.0f / ((1.0f + powf(t_0, 2.0f)) * (alphax * alphax))) + ((0.5f - (0.5f * cosf((2.0f * atanf(t_0))))) / (alphay * alphay))))))));
}

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 sqrt(Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(u0 / Float32(Float32(Float32(1.0) - u0) * Float32(Float32(Float32(1.0) / Float32(Float32(Float32(1.0) + (t_0 ^ Float32(2.0))) * Float32(alphax * alphax))) + Float32(Float32(Float32(0.5) - Float32(Float32(0.5) * cos(Float32(Float32(2.0) * atan(t_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

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

8. Applied rewrites 99.9%

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

10. Applied rewrites 99.8%

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

11. Final simplification 99.8%

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

Alternative 3: 99.8% accurate, 2.2× speedup

Math

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

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 t_0) (* alphax alphax)))
        (/ (+ 1.0 (/ 1.0 (- -1.0 t_0))) (* alphay alphay))))))
    -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 + t_0) * (alphax * alphax))) + ((1.0f + (1.0f / (-1.0f - t_0))) / (alphay * alphay)))))), -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(1.0) / Float32(Float32(Float32(1.0) + t_0) * Float32(alphax * alphax))) + Float32(Float32(Float32(1.0) + Float32(Float32(1.0) / Float32(Float32(-1.0) - t_0))) / Float32(alphay * alphay)))))) ^ 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 99.3%

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

6. Applied rewrites 99.8%

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

7. Final simplification 99.8%

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

Alternative 4: 99.3% accurate, 2.6× speedup

Math

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

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 t_0) (* alphax alphax)))
         (/ (+ 1.0 (/ 1.0 (- -1.0 t_0))) (* alphay alphay))))))))))

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 + t_0) * (alphax * alphax))) + ((1.0f + (1.0f / (-1.0f - t_0))) / (alphay * alphay)))))));
}

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(1.0) / Float32(Float32(Float32(1.0) + t_0) * Float32(alphax * alphax))) + Float32(Float32(Float32(1.0) + Float32(Float32(1.0) / Float32(Float32(-1.0) - t_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%

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

7. Applied rewrites 99.3%

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

8. Final simplification 99.3%

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

Alternative 5: 98.1% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(u0, u1, alphax, alphay)
	return exp(Float32(Float32(-0.5) * log1p(Float32(Float32(alphay * Float32(u0 * alphay)) / Float32(Float32(Float32(1.0) - u0) * Float32(Float32(1.0) + Float32(Float32(1.0) / Float32(Float32(-1.0) - (Float32(tan(Float32(Float32(pi) * fma(Float32(2.0), u1, Float32(0.5)))) * Float32(alphay / alphax)) ^ Float32(2.0))))))))))
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. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-*.f32 N/A

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

5. Applied rewrites 97.8%

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

7. Applied rewrites 98.3%

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

8. Final simplification 98.3%

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

Alternative 6: 98.1% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Julia

function code(u0, u1, alphax, alphay)
	return fma(u0, Float32(Float32(alphay * alphay) / Float32(Float32(Float32(1.0) - u0) * Float32(Float32(1.0) + Float32(Float32(1.0) / Float32(Float32(-1.0) - (Float32(tan(Float32(Float32(pi) * fma(Float32(2.0), u1, Float32(0.5)))) * Float32(alphay / alphax)) ^ Float32(2.0))))))), Float32(1.0)) ^ 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 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. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-*.f32 N/A

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

5. Applied rewrites 97.8%

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

7. Applied rewrites 98.2%

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

9. Applied rewrites 98.2%

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

10. Final simplification 98.2%

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

Alternative 7: 98.1% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Julia

function code(u0, u1, alphax, alphay)
	return fma(alphay, Float32(Float32(u0 * alphay) / Float32(Float32(Float32(1.0) - u0) * Float32(Float32(1.0) + Float32(Float32(1.0) / Float32(Float32(-1.0) - (Float32(tan(Float32(Float32(pi) * fma(Float32(2.0), u1, Float32(0.5)))) * Float32(alphay / alphax)) ^ Float32(2.0))))))), Float32(1.0)) ^ 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 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. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-*.f32 N/A

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

5. Applied rewrites 97.8%

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

7. Applied rewrites 98.2%

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

8. Final simplification 98.2%

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

Alternative 8: 98.1% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Julia

function code(u0, u1, alphax, alphay)
	return fma(Float32(Float32(alphay * alphay) / Float32(Float32(Float32(1.0) - u0) + Float32(Float32(Float32(1.0) - u0) / Float32(Float32(-1.0) - (Float32(tan(Float32(Float32(pi) * fma(Float32(2.0), u1, Float32(0.5)))) * Float32(alphay / alphax)) ^ Float32(2.0)))))), u0, Float32(1.0)) ^ 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 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. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-*.f32 N/A

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

5. Applied rewrites 97.8%

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

7. Applied rewrites 98.2%

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

9. Applied rewrites 98.2%

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

11. Applied rewrites 98.2%

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

12. Final simplification 98.2%

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

Alternative 9: 98.1% accurate, 30.6× speedup

Math

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

FPCore

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

C

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

Julia

function code(u0, u1, alphax, alphay)
	return sqrt(Float32(Float32(1.0) / fma(u0, Float32(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. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-*.f32 N/A

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

5. Applied rewrites 97.8%

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

7. Applied rewrites 98.2%

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

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 98.1

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

10. Applied rewrites 98.1%

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

    1. lift-*.f32 N/A

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

    2. lift--.f32 N/A

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

    3. lift-/.f32 N/A

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

    4. lift-/.f32 N/A

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

    5. associate-*r/ N/A

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

    6. *-commutative N/A

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

    7. associate-/l* N/A

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

    8. lower-fma.f32 N/A

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

    9. lower-/.f32 98.1

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

12. Applied rewrites 98.1%

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

Alternative 10: 96.5% accurate, 43.5× speedup

Math

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

FPCore

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

C

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

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 + (((-0.5e0) * (u0 * (alphay * alphay))) / (1.0e0 - u0))
end function

Julia

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

MATLAB

function tmp = code(u0, u1, alphax, alphay)
	tmp = single(1.0) + ((single(-0.5) * (u0 * (alphay * alphay))) / (single(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 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. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-*.f32 N/A

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

5. Applied rewrites 97.8%

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

7. Applied rewrites 98.2%

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

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 98.1

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

10. Applied rewrites 98.1%

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

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. lower-+.f32 N/A

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

    2. associate-*r/ N/A

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

    3. associate-*r* N/A

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

    4. lower-/.f32 N/A

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

    5. associate-*r* N/A

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

    6. lower-*.f32 N/A

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

    7. lower-*.f32 N/A

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

    8. unpow2 N/A

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

    9. lower-*.f32 N/A

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

    10. lower--.f32 96.7

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

13. Applied rewrites 96.7%

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

14. Final simplification 96.7%

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

Alternative 11: 95.0% accurate, 75.6× speedup

Math

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

FPCore

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

C

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

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 + ((-0.5e0) * (u0 * (alphay * alphay)))
end function

Julia

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

MATLAB

function tmp = code(u0, u1, alphax, alphay)
	tmp = single(1.0) + (single(-0.5) * (u0 * (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 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. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-*.f32 N/A

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

5. Applied rewrites 97.8%

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

7. Applied rewrites 98.2%

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

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 98.1

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

10. Applied rewrites 98.1%

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

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. lower-+.f32 N/A

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

    3. associate-*r* N/A

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

    4. lower-*.f32 N/A

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

    5. lower-*.f32 N/A

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

    6. unpow2 N/A

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

    7. lower-*.f32 95.3

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

13. Applied rewrites 95.3%

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

14. Final simplification 95.3%

    \[\leadsto 1 + -0.5 \cdot \left(u0 \cdot \left(alphay \cdot alphay\right)\right) \]
15. Add Preprocessing

Alternative 12: 91.7% 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

4. Applied rewrites 99.3%

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

5. Taylor expanded in u0 around 0

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

7. Applied rewrites 91.9%

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

Reproduce

herbie shell --seed 2024220 
(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))))))