Trowbridge-Reitz Sample, sample surface normal, cosTheta

Percentage Accurate: 99.4% → 99.9%

Time: 28.4s

Alternatives: 9

Speedup: 2.1×

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, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

3. Simplified 99.3%

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

6. Applied egg-rr 99.8%

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

   1. sqr-sin-a N/A

      \[\leadsto \mathsf{pow.f32}\left(\mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\mathsf{/.f32}\left(u0, \mathsf{\_.f32}\left(1, u0\right)\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{/.f32}\left(\mathsf{tan.f32}\left(\mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(2, u1\right)\right)\right)\right), \mathsf{/.f32}\left(alphax, alphay\right)\right), 2\right)\right)\right), \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(\left(\sin \tan^{-1} \left(\frac{\tan \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} + 2 \cdot u1\right)\right)}{\frac{alphax}{alphay}}\right) \cdot \sin \tan^{-1} \left(\frac{\tan \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} + 2 \cdot u1\right)\right)}{\frac{alphax}{alphay}}\right)\right), \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right)\right), \frac{-1}{2}\right) \]

   2. pow2 N/A

      \[\leadsto \mathsf{pow.f32}\left(\mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\mathsf{/.f32}\left(u0, \mathsf{\_.f32}\left(1, u0\right)\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{/.f32}\left(\mathsf{tan.f32}\left(\mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(2, u1\right)\right)\right)\right), \mathsf{/.f32}\left(alphax, alphay\right)\right), 2\right)\right)\right), \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(\left({\sin \tan^{-1} \left(\frac{\tan \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} + 2 \cdot u1\right)\right)}{\frac{alphax}{alphay}}\right)}^{2}\right), \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right)\right), \frac{-1}{2}\right) \]

   3. pow-lowering-pow.f32 N/A

      \[\leadsto \mathsf{pow.f32}\left(\mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\mathsf{/.f32}\left(u0, \mathsf{\_.f32}\left(1, u0\right)\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{/.f32}\left(\mathsf{tan.f32}\left(\mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(2, u1\right)\right)\right)\right), \mathsf{/.f32}\left(alphax, alphay\right)\right), 2\right)\right)\right), \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(\mathsf{pow.f32}\left(\sin \tan^{-1} \left(\frac{\tan \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} + 2 \cdot u1\right)\right)}{\frac{alphax}{alphay}}\right), 2\right), \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right)\right), \frac{-1}{2}\right) \]

   4. sin-lowering-sin.f32 N/A

      \[\leadsto \mathsf{pow.f32}\left(\mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\mathsf{/.f32}\left(u0, \mathsf{\_.f32}\left(1, u0\right)\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{/.f32}\left(\mathsf{tan.f32}\left(\mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(2, u1\right)\right)\right)\right), \mathsf{/.f32}\left(alphax, alphay\right)\right), 2\right)\right)\right), \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(\mathsf{pow.f32}\left(\mathsf{sin.f32}\left(\tan^{-1} \left(\frac{\tan \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} + 2 \cdot u1\right)\right)}{\frac{alphax}{alphay}}\right)\right), 2\right), \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right)\right), \frac{-1}{2}\right) \]

   5. atan-lowering-atan.f32 N/A

      \[\leadsto \mathsf{pow.f32}\left(\mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\mathsf{/.f32}\left(u0, \mathsf{\_.f32}\left(1, u0\right)\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{/.f32}\left(\mathsf{tan.f32}\left(\mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(2, u1\right)\right)\right)\right), \mathsf{/.f32}\left(alphax, alphay\right)\right), 2\right)\right)\right), \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(\mathsf{pow.f32}\left(\mathsf{sin.f32}\left(\mathsf{atan.f32}\left(\left(\frac{\tan \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} + 2 \cdot u1\right)\right)}{\frac{alphax}{alphay}}\right)\right)\right), 2\right), \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right)\right), \frac{-1}{2}\right) \]

   6. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{pow.f32}\left(\mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\mathsf{/.f32}\left(u0, \mathsf{\_.f32}\left(1, u0\right)\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{/.f32}\left(\mathsf{tan.f32}\left(\mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(2, u1\right)\right)\right)\right), \mathsf{/.f32}\left(alphax, alphay\right)\right), 2\right)\right)\right), \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(\mathsf{pow.f32}\left(\mathsf{sin.f32}\left(\mathsf{atan.f32}\left(\mathsf{/.f32}\left(\tan \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} + 2 \cdot u1\right)\right), \left(\frac{alphax}{alphay}\right)\right)\right)\right), 2\right), \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right)\right), \frac{-1}{2}\right) \]

   7. tan-lowering-tan.f32 N/A

      \[\leadsto \mathsf{pow.f32}\left(\mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\mathsf{/.f32}\left(u0, \mathsf{\_.f32}\left(1, u0\right)\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{/.f32}\left(\mathsf{tan.f32}\left(\mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(2, u1\right)\right)\right)\right), \mathsf{/.f32}\left(alphax, alphay\right)\right), 2\right)\right)\right), \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(\mathsf{pow.f32}\left(\mathsf{sin.f32}\left(\mathsf{atan.f32}\left(\mathsf{/.f32}\left(\mathsf{tan.f32}\left(\left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} + 2 \cdot u1\right)\right)\right), \left(\frac{alphax}{alphay}\right)\right)\right)\right), 2\right), \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right)\right), \frac{-1}{2}\right) \]

   8. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{pow.f32}\left(\mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\mathsf{/.f32}\left(u0, \mathsf{\_.f32}\left(1, u0\right)\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{/.f32}\left(\mathsf{tan.f32}\left(\mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(2, u1\right)\right)\right)\right), \mathsf{/.f32}\left(alphax, alphay\right)\right), 2\right)\right)\right), \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(\mathsf{pow.f32}\left(\mathsf{sin.f32}\left(\mathsf{atan.f32}\left(\mathsf{/.f32}\left(\mathsf{tan.f32}\left(\mathsf{*.f32}\left(\mathsf{PI}\left(\right), \left(\frac{1}{2} + 2 \cdot u1\right)\right)\right), \left(\frac{alphax}{alphay}\right)\right)\right)\right), 2\right), \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right)\right), \frac{-1}{2}\right) \]

   9. PI-lowering-PI.f32 N/A

      \[\leadsto \mathsf{pow.f32}\left(\mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\mathsf{/.f32}\left(u0, \mathsf{\_.f32}\left(1, u0\right)\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{/.f32}\left(\mathsf{tan.f32}\left(\mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(2, u1\right)\right)\right)\right), \mathsf{/.f32}\left(alphax, alphay\right)\right), 2\right)\right)\right), \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(\mathsf{pow.f32}\left(\mathsf{sin.f32}\left(\mathsf{atan.f32}\left(\mathsf{/.f32}\left(\mathsf{tan.f32}\left(\mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \left(\frac{1}{2} + 2 \cdot u1\right)\right)\right), \left(\frac{alphax}{alphay}\right)\right)\right)\right), 2\right), \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right)\right), \frac{-1}{2}\right) \]

   10. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{pow.f32}\left(\mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\mathsf{/.f32}\left(u0, \mathsf{\_.f32}\left(1, u0\right)\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{/.f32}\left(\mathsf{tan.f32}\left(\mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(2, u1\right)\right)\right)\right), \mathsf{/.f32}\left(alphax, alphay\right)\right), 2\right)\right)\right), \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(\mathsf{pow.f32}\left(\mathsf{sin.f32}\left(\mathsf{atan.f32}\left(\mathsf{/.f32}\left(\mathsf{tan.f32}\left(\mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{+.f32}\left(\frac{1}{2}, \left(2 \cdot u1\right)\right)\right)\right), \left(\frac{alphax}{alphay}\right)\right)\right)\right), 2\right), \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right)\right), \frac{-1}{2}\right) \]

   11. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{pow.f32}\left(\mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\mathsf{/.f32}\left(u0, \mathsf{\_.f32}\left(1, u0\right)\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{/.f32}\left(\mathsf{tan.f32}\left(\mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(2, u1\right)\right)\right)\right), \mathsf{/.f32}\left(alphax, alphay\right)\right), 2\right)\right)\right), \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(\mathsf{pow.f32}\left(\mathsf{sin.f32}\left(\mathsf{atan.f32}\left(\mathsf{/.f32}\left(\mathsf{tan.f32}\left(\mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(2, u1\right)\right)\right)\right), \left(\frac{alphax}{alphay}\right)\right)\right)\right), 2\right), \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right)\right), \frac{-1}{2}\right) \]

   12. /-lowering-/.f32 99.9%

       \[\leadsto \mathsf{pow.f32}\left(\mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\mathsf{/.f32}\left(u0, \mathsf{\_.f32}\left(1, u0\right)\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{/.f32}\left(\mathsf{tan.f32}\left(\mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(2, u1\right)\right)\right)\right), \mathsf{/.f32}\left(alphax, alphay\right)\right), 2\right)\right)\right), \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(\mathsf{pow.f32}\left(\mathsf{sin.f32}\left(\mathsf{atan.f32}\left(\mathsf{/.f32}\left(\mathsf{tan.f32}\left(\mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(2, u1\right)\right)\right)\right), \mathsf{/.f32}\left(alphax, alphay\right)\right)\right)\right), 2\right), \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right)\right), \frac{-1}{2}\right) \]

8. Applied egg-rr 99.9%

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

Alternative 2: 99.8% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

3. Simplified 99.3%

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

6. Applied egg-rr 99.8%

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

Alternative 3: 98.8% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

3. Simplified 99.3%

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

6. Applied egg-rr 99.8%

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

7. Taylor expanded in u1 around 0

   \[\leadsto \mathsf{pow.f32}\left(\mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\mathsf{/.f32}\left(u0, \mathsf{\_.f32}\left(1, u0\right)\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{/.f32}\left(\mathsf{tan.f32}\left(\mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(2, u1\right)\right)\right)\right), \mathsf{/.f32}\left(alphax, alphay\right)\right), 2\right)\right)\right), \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\frac{1}{2}, \mathsf{cos.f32}\left(\mathsf{*.f32}\left(2, \mathsf{atan.f32}\left(\mathsf{/.f32}\left(\mathsf{tan.f32}\left(\color{blue}{\left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}\right), \mathsf{/.f32}\left(alphax, alphay\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right)\right), \frac{-1}{2}\right) \]
8. Step-by-step derivation

   1. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{pow.f32}\left(\mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\mathsf{/.f32}\left(u0, \mathsf{\_.f32}\left(1, u0\right)\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{/.f32}\left(\mathsf{tan.f32}\left(\mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(2, u1\right)\right)\right)\right), \mathsf{/.f32}\left(alphax, alphay\right)\right), 2\right)\right)\right), \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\frac{1}{2}, \mathsf{cos.f32}\left(\mathsf{*.f32}\left(2, \mathsf{atan.f32}\left(\mathsf{/.f32}\left(\mathsf{tan.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{PI}\left(\right)\right)\right), \mathsf{/.f32}\left(alphax, alphay\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right)\right), \frac{-1}{2}\right) \]

   2. PI-lowering-PI.f32 98.2%

      \[\leadsto \mathsf{pow.f32}\left(\mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\mathsf{/.f32}\left(u0, \mathsf{\_.f32}\left(1, u0\right)\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{/.f32}\left(\mathsf{tan.f32}\left(\mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(2, u1\right)\right)\right)\right), \mathsf{/.f32}\left(alphax, alphay\right)\right), 2\right)\right)\right), \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\frac{1}{2}, \mathsf{cos.f32}\left(\mathsf{*.f32}\left(2, \mathsf{atan.f32}\left(\mathsf{/.f32}\left(\mathsf{tan.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{PI.f32}\left(\right)\right)\right), \mathsf{/.f32}\left(alphax, alphay\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right)\right), \frac{-1}{2}\right) \]

9. Simplified 98.2%

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

10. Final simplification 98.2%

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

Alternative 4: 98.2% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

3. Simplified 99.3%

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

5. Taylor expanded in alphax around inf

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

7. Simplified 98.0%

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

9. Applied egg-rr 98.0%

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

10. Final simplification 98.0%

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

Alternative 5: 98.2% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

3. Simplified 99.3%

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

6. Applied egg-rr 99.8%

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

7. Taylor expanded in u1 around 0

   \[\leadsto \mathsf{pow.f32}\left(\mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\mathsf{/.f32}\left(u0, \mathsf{\_.f32}\left(1, u0\right)\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{/.f32}\left(\mathsf{tan.f32}\left(\mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(2, u1\right)\right)\right)\right), \mathsf{/.f32}\left(alphax, alphay\right)\right), 2\right)\right)\right), \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\frac{1}{2}, \mathsf{cos.f32}\left(\mathsf{*.f32}\left(2, \mathsf{atan.f32}\left(\mathsf{/.f32}\left(\mathsf{tan.f32}\left(\color{blue}{\left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}\right), \mathsf{/.f32}\left(alphax, alphay\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right)\right), \frac{-1}{2}\right) \]
8. Step-by-step derivation

   1. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{pow.f32}\left(\mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\mathsf{/.f32}\left(u0, \mathsf{\_.f32}\left(1, u0\right)\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{/.f32}\left(\mathsf{tan.f32}\left(\mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(2, u1\right)\right)\right)\right), \mathsf{/.f32}\left(alphax, alphay\right)\right), 2\right)\right)\right), \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\frac{1}{2}, \mathsf{cos.f32}\left(\mathsf{*.f32}\left(2, \mathsf{atan.f32}\left(\mathsf{/.f32}\left(\mathsf{tan.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{PI}\left(\right)\right)\right), \mathsf{/.f32}\left(alphax, alphay\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right)\right), \frac{-1}{2}\right) \]

   2. PI-lowering-PI.f32 98.2%

      \[\leadsto \mathsf{pow.f32}\left(\mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\mathsf{/.f32}\left(u0, \mathsf{\_.f32}\left(1, u0\right)\right), \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{/.f32}\left(\mathsf{tan.f32}\left(\mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(2, u1\right)\right)\right)\right), \mathsf{/.f32}\left(alphax, alphay\right)\right), 2\right)\right)\right), \mathsf{*.f32}\left(alphax, alphax\right)\right), \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\frac{1}{2}, \mathsf{cos.f32}\left(\mathsf{*.f32}\left(2, \mathsf{atan.f32}\left(\mathsf{/.f32}\left(\mathsf{tan.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{PI.f32}\left(\right)\right)\right), \mathsf{/.f32}\left(alphax, alphay\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(alphay, alphay\right)\right)\right)\right)\right), \frac{-1}{2}\right) \]

9. Simplified 98.2%

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

10. Taylor expanded in alphay around 0

    \[\leadsto \mathsf{pow.f32}\left(\color{blue}{\left(1 + \frac{{alphay}^{2} \cdot u0}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay \cdot \tan \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}{alphax}\right)\right)\right) \cdot \left(1 - u0\right)}\right)}, \frac{-1}{2}\right) \]
11. Step-by-step derivation

    1. associate-/l* N/A

       \[\leadsto \mathsf{pow.f32}\left(\left(1 + {alphay}^{2} \cdot \frac{u0}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay \cdot \tan \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}{alphax}\right)\right)\right) \cdot \left(1 - u0\right)}\right), \frac{-1}{2}\right) \]

    2. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{pow.f32}\left(\mathsf{+.f32}\left(1, \left({alphay}^{2} \cdot \frac{u0}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay \cdot \tan \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}{alphax}\right)\right)\right) \cdot \left(1 - u0\right)}\right)\right), \frac{-1}{2}\right) \]

    3. associate-/l* N/A

       \[\leadsto \mathsf{pow.f32}\left(\mathsf{+.f32}\left(1, \left(\frac{{alphay}^{2} \cdot u0}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay \cdot \tan \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}{alphax}\right)\right)\right) \cdot \left(1 - u0\right)}\right)\right), \frac{-1}{2}\right) \]

    4. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{pow.f32}\left(\mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\left({alphay}^{2} \cdot u0\right), \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay \cdot \tan \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}{alphax}\right)\right)\right) \cdot \left(1 - u0\right)\right)\right)\right), \frac{-1}{2}\right) \]

    5. *-commutative N/A

       \[\leadsto \mathsf{pow.f32}\left(\mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\left(u0 \cdot {alphay}^{2}\right), \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay \cdot \tan \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}{alphax}\right)\right)\right) \cdot \left(1 - u0\right)\right)\right)\right), \frac{-1}{2}\right) \]

    6. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{pow.f32}\left(\mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \left({alphay}^{2}\right)\right), \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay \cdot \tan \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}{alphax}\right)\right)\right) \cdot \left(1 - u0\right)\right)\right)\right), \frac{-1}{2}\right) \]

    7. unpow2 N/A

       \[\leadsto \mathsf{pow.f32}\left(\mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \left(alphay \cdot alphay\right)\right), \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay \cdot \tan \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}{alphax}\right)\right)\right) \cdot \left(1 - u0\right)\right)\right)\right), \frac{-1}{2}\right) \]

    8. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{pow.f32}\left(\mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{*.f32}\left(alphay, alphay\right)\right), \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay \cdot \tan \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}{alphax}\right)\right)\right) \cdot \left(1 - u0\right)\right)\right)\right), \frac{-1}{2}\right) \]

    9. *-commutative N/A

       \[\leadsto \mathsf{pow.f32}\left(\mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{*.f32}\left(alphay, alphay\right)\right), \left(\left(1 - u0\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay \cdot \tan \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}{alphax}\right)\right)\right)\right)\right)\right), \frac{-1}{2}\right) \]

    10. *-lowering-*.f32 N/A

        \[\leadsto \mathsf{pow.f32}\left(\mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{*.f32}\left(alphay, alphay\right)\right), \mathsf{*.f32}\left(\left(1 - u0\right), \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay \cdot \tan \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}{alphax}\right)\right)\right)\right)\right)\right), \frac{-1}{2}\right) \]

    11. --lowering--.f32 N/A

        \[\leadsto \mathsf{pow.f32}\left(\mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{*.f32}\left(alphay, alphay\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u0\right), \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay \cdot \tan \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}{alphax}\right)\right)\right)\right)\right)\right), \frac{-1}{2}\right) \]

    12. cancel-sign-sub-inv N/A

        \[\leadsto \mathsf{pow.f32}\left(\mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{*.f32}\left(alphay, alphay\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u0\right), \left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay \cdot \tan \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}{alphax}\right)\right)\right)\right)\right)\right), \frac{-1}{2}\right) \]

12. Simplified 97.8%

    \[\leadsto {\color{blue}{\left(1 + \frac{u0 \cdot \left(alphay \cdot alphay\right)}{\left(1 - u0\right) \cdot \left(0.5 + -0.5 \cdot \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay \cdot \tan \left(\pi \cdot 0.5\right)}{alphax}\right)\right)\right)}\right)}}^{-0.5} \]
13. Add Preprocessing

Alternative 6: 96.6% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

3. Simplified 99.3%

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

6. Applied egg-rr 99.9%

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

7. Taylor expanded in alphay around 0

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

9. Simplified 96.7%

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

10. Final simplification 96.7%

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

Alternative 7: 96.6% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

3. Simplified 99.3%

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

6. Applied egg-rr 99.9%

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

7. Taylor expanded in alphay around 0

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

9. Simplified 96.7%

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

10. Taylor expanded in u1 around 0

    \[\leadsto \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(u0, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u0\right), \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\frac{-1}{2}, \mathsf{cos.f32}\left(\mathsf{*.f32}\left(2, \mathsf{atan.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(alphay, \mathsf{tan.f32}\left(\color{blue}{\left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}\right)\right), alphax\right)\right)\right)\right)\right)\right)\right)\right)\right) \]
11. Step-by-step derivation

    1. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(u0, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u0\right), \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\frac{-1}{2}, \mathsf{cos.f32}\left(\mathsf{*.f32}\left(2, \mathsf{atan.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(alphay, \mathsf{tan.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{PI}\left(\right)\right)\right)\right), alphax\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

    2. PI-lowering-PI.f32 96.5%

       \[\leadsto \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(u0, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u0\right), \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\frac{-1}{2}, \mathsf{cos.f32}\left(\mathsf{*.f32}\left(2, \mathsf{atan.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(alphay, \mathsf{tan.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{PI.f32}\left(\right)\right)\right)\right), alphax\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

12. Simplified 96.5%

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

13. Final simplification 96.5%

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

Alternative 8: 95.0% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

3. Simplified 99.3%

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

6. Applied egg-rr 99.9%

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

7. Taylor expanded in alphay around 0

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

9. Simplified 96.7%

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

10. Taylor expanded in u1 around 0

    \[\leadsto \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(u0, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u0\right), \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\frac{-1}{2}, \mathsf{cos.f32}\left(\mathsf{*.f32}\left(2, \mathsf{atan.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(alphay, \mathsf{tan.f32}\left(\color{blue}{\left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}\right)\right), alphax\right)\right)\right)\right)\right)\right)\right)\right)\right) \]
11. Step-by-step derivation

    1. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(u0, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u0\right), \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\frac{-1}{2}, \mathsf{cos.f32}\left(\mathsf{*.f32}\left(2, \mathsf{atan.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(alphay, \mathsf{tan.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{PI}\left(\right)\right)\right)\right), alphax\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

    2. PI-lowering-PI.f32 96.5%

       \[\leadsto \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(u0, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u0\right), \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\frac{-1}{2}, \mathsf{cos.f32}\left(\mathsf{*.f32}\left(2, \mathsf{atan.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(alphay, \mathsf{tan.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{PI.f32}\left(\right)\right)\right)\right), alphax\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

12. Simplified 96.5%

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

13. Taylor expanded in u0 around 0

    \[\leadsto \mathsf{+.f32}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot \frac{{alphay}^{2} \cdot u0}{\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay \cdot \tan \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}{alphax}\right)\right)}\right)}\right) \]
14. Step-by-step derivation

    1. metadata-eval N/A

       \[\leadsto \mathsf{+.f32}\left(1, \left(\frac{-1}{2} \cdot \frac{{alphay}^{2} \cdot u0}{\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \color{blue}{\left(2 \cdot \tan^{-1} \left(\frac{alphay \cdot \tan \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}{alphax}\right)\right)}}\right)\right) \]

    2. cancel-sign-sub-inv N/A

       \[\leadsto \mathsf{+.f32}\left(1, \left(\frac{-1}{2} \cdot \frac{{alphay}^{2} \cdot u0}{\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay \cdot \tan \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}{alphax}\right)\right)}}\right)\right) \]

    3. associate-*r/ N/A

       \[\leadsto \mathsf{+.f32}\left(1, \left(\frac{\frac{-1}{2} \cdot \left({alphay}^{2} \cdot u0\right)}{\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay \cdot \tan \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}{alphax}\right)\right)}}\right)\right) \]

    4. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\left(\frac{-1}{2} \cdot \left({alphay}^{2} \cdot u0\right)\right), \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay \cdot \tan \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}{alphax}\right)\right)\right)}\right)\right) \]

    5. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{-1}{2}, \left({alphay}^{2} \cdot u0\right)\right), \left(\color{blue}{\frac{1}{2}} - \frac{1}{2} \cdot \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay \cdot \tan \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}{alphax}\right)\right)\right)\right)\right) \]

    6. *-commutative N/A

       \[\leadsto \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{-1}{2}, \left(u0 \cdot {alphay}^{2}\right)\right), \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay \cdot \tan \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}{alphax}\right)\right)\right)\right)\right) \]

    7. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(u0, \left({alphay}^{2}\right)\right)\right), \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay \cdot \tan \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}{alphax}\right)\right)\right)\right)\right) \]

    8. unpow2 N/A

       \[\leadsto \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(u0, \left(alphay \cdot alphay\right)\right)\right), \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay \cdot \tan \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}{alphax}\right)\right)\right)\right)\right) \]

    9. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(u0, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right), \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay \cdot \tan \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}{alphax}\right)\right)\right)\right)\right) \]

    10. cancel-sign-sub-inv N/A

        \[\leadsto \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(u0, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right), \left(\frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay \cdot \tan \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}{alphax}\right)\right)}\right)\right)\right) \]

    11. metadata-eval N/A

        \[\leadsto \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{-1}{2}, \mathsf{*.f32}\left(u0, \mathsf{*.f32}\left(alphay, alphay\right)\right)\right), \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \color{blue}{\left(2 \cdot \tan^{-1} \left(\frac{alphay \cdot \tan \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}{alphax}\right)\right)}\right)\right)\right) \]

15. Simplified 94.3%

    \[\leadsto 1 + \color{blue}{\frac{-0.5 \cdot \left(u0 \cdot \left(alphay \cdot alphay\right)\right)}{0.5 + -0.5 \cdot \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay \cdot \tan \left(\pi \cdot 0.5\right)}{alphax}\right)\right)}} \]
16. Add Preprocessing

Alternative 9: 91.6% accurate, 1375.0× 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 99.3%

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

3. Simplified 99.3%

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

5. Taylor expanded in u0 around 0

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

7. Simplified 90.3%

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

Reproduce

herbie shell --seed 2024161 
(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))))))