Trowbridge-Reitz Sample, sample surface normal, cosTheta

Percentage Accurate: 99.4% → 100.0%

Time: 31.6s

Alternatives: 10

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: 100.0% accurate, 1.8× speedup

Math

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

FPCore

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

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 expf((-0.5f * log1pf(((u0 / (((1.0f / (alphax * alphax)) / (1.0f + powf(t_0, 2.0f))) + ((0.5f + (cosf((2.0f * atanf(t_0))) * -0.5f)) / (alphay * alphay)))) / (1.0f - u0)))));
}

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 exp(Float32(Float32(-0.5) * log1p(Float32(Float32(u0 / Float32(Float32(Float32(Float32(1.0) / Float32(alphax * alphax)) / Float32(Float32(1.0) + (t_0 ^ Float32(2.0)))) + Float32(Float32(Float32(0.5) + Float32(cos(Float32(Float32(2.0) * atan(t_0))) * Float32(-0.5))) / Float32(alphay * alphay)))) / Float32(Float32(1.0) - u0)))))
end

Derivation

1. Initial program 99.5%

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

3. Simplified 99.5%

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

6. Applied egg-rr 100.0%

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

8. Applied egg-rr 100.0%

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

9. Final simplification 100.0%

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

Alternative 2: 99.9% accurate, 2.1× speedup

Math

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

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 (* alphax alphax)) (+ 1.0 (pow t_0 2.0)))
       (/ (+ 0.5 (* (cos (* 2.0 (atan t_0))) -0.5)) (* 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 / (alphax * alphax)) / (1.0f + powf(t_0, 2.0f))) + ((0.5f + (cosf((2.0f * atanf(t_0))) * -0.5f)) / (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(alphax * alphax)) / Float32(Float32(1.0) + (t_0 ^ Float32(2.0)))) + Float32(Float32(Float32(0.5) + Float32(cos(Float32(Float32(2.0) * atan(t_0))) * Float32(-0.5))) / 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) / (alphax * alphax)) / (single(1.0) + (t_0 ^ single(2.0)))) + ((single(0.5) + (cos((single(2.0) * atan(t_0))) * single(-0.5))) / (alphay * alphay))))) ^ single(-0.5);
end

Derivation

1. Initial program 99.5%

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

3. Simplified 99.5%

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

6. Applied egg-rr 100.0%

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

Alternative 3: 98.3% accurate, 2.6× speedup

Math

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

FPCore

(FPCore (u0 u1 alphax alphay)
 :precision binary32
 (exp
  (*
   -0.5
   (log1p
    (/
     (* 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 expf((-0.5f * log1pf(((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 exp(Float32(Float32(-0.5) * log1p(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(tan(Float32(Float32(pi) * Float32(Float32(0.5) + Float32(Float32(2.0) * u1)))) * alphay) / alphax)))))))))))
end

Derivation

1. Initial program 99.5%

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

3. Simplified 99.5%

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

6. Applied egg-rr 100.0%

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

7. Taylor expanded in alphay around 0

   \[\leadsto \mathsf{exp.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\color{blue}{\left(\frac{{alphay}^{2} \cdot u0}{\left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} + 2 \cdot u1\right)\right)}{alphax}\right)\right)\right) \cdot \left(1 - u0\right)}\right)}\right), \frac{-1}{2}\right)\right) \]
8. Step-by-step derivation

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

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left({alphay}^{2} \cdot u0\right), \left(\left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} + 2 \cdot u1\right)\right)}{alphax}\right)\right)\right) \cdot \left(1 - u0\right)\right)\right)\right), \frac{-1}{2}\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(u0 \cdot {alphay}^{2}\right), \left(\left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} + 2 \cdot u1\right)\right)}{alphax}\right)\right)\right) \cdot \left(1 - u0\right)\right)\right)\right), \frac{-1}{2}\right)\right) \]

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

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \left({alphay}^{2}\right)\right), \left(\left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} + 2 \cdot u1\right)\right)}{alphax}\right)\right)\right) \cdot \left(1 - u0\right)\right)\right)\right), \frac{-1}{2}\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \left(alphay \cdot alphay\right)\right), \left(\left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} + 2 \cdot u1\right)\right)}{alphax}\right)\right)\right) \cdot \left(1 - u0\right)\right)\right)\right), \frac{-1}{2}\right)\right) \]

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

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{*.f32}\left(alphay, alphay\right)\right), \left(\left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} + 2 \cdot u1\right)\right)}{alphax}\right)\right)\right) \cdot \left(1 - u0\right)\right)\right)\right), \frac{-1}{2}\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{*.f32}\left(alphay, alphay\right)\right), \left(\left(1 - u0\right) \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} + 2 \cdot u1\right)\right)}{alphax}\right)\right)\right)\right)\right)\right), \frac{-1}{2}\right)\right) \]

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

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{*.f32}\left(alphay, alphay\right)\right), \mathsf{*.f32}\left(\left(1 - u0\right), \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} + 2 \cdot u1\right)\right)}{alphax}\right)\right)\right)\right)\right)\right), \frac{-1}{2}\right)\right) \]

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

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{*.f32}\left(alphay, alphay\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u0\right), \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} + 2 \cdot u1\right)\right)}{alphax}\right)\right)\right)\right)\right)\right), \frac{-1}{2}\right)\right) \]

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

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{*.f32}\left(alphay, alphay\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u0\right), \mathsf{+.f32}\left(\frac{1}{2}, \left(\frac{-1}{2} \cdot \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} + 2 \cdot u1\right)\right)}{alphax}\right)\right)\right)\right)\right)\right)\right), \frac{-1}{2}\right)\right) \]

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

       \[\leadsto \mathsf{exp.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{*.f32}\left(alphay, alphay\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u0\right), \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\frac{-1}{2}, \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} + 2 \cdot u1\right)\right)}{alphax}\right)\right)\right)\right)\right)\right)\right), \frac{-1}{2}\right)\right) \]

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

       \[\leadsto \mathsf{exp.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{*.f32}\left(alphay, alphay\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u0\right), \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\frac{-1}{2}, \mathsf{cos.f32}\left(\left(2 \cdot \tan^{-1} \left(\frac{alphay \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} + 2 \cdot u1\right)\right)}{alphax}\right)\right)\right)\right)\right)\right)\right)\right), \frac{-1}{2}\right)\right) \]

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

       \[\leadsto \mathsf{exp.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{*.f32}\left(alphay, alphay\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u0\right), \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\frac{-1}{2}, \mathsf{cos.f32}\left(\mathsf{*.f32}\left(2, \tan^{-1} \left(\frac{alphay \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} + 2 \cdot u1\right)\right)}{alphax}\right)\right)\right)\right)\right)\right)\right)\right), \frac{-1}{2}\right)\right) \]

9. Simplified 98.3%

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

10. Final simplification 98.3%

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

Alternative 4: 98.2% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.5%

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

6. Applied egg-rr 100.0%

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

7. Taylor expanded in alphax around inf

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

9. Simplified 98.2%

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

10. Final simplification 98.2%

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

Alternative 5: 98.2% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.5%

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

6. Applied egg-rr 100.0%

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

7. Taylor expanded in alphay around 0

   \[\leadsto \mathsf{exp.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\color{blue}{\left(\frac{{alphay}^{2} \cdot u0}{\left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} + 2 \cdot u1\right)\right)}{alphax}\right)\right)\right) \cdot \left(1 - u0\right)}\right)}\right), \frac{-1}{2}\right)\right) \]
8. Step-by-step derivation

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

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left({alphay}^{2} \cdot u0\right), \left(\left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} + 2 \cdot u1\right)\right)}{alphax}\right)\right)\right) \cdot \left(1 - u0\right)\right)\right)\right), \frac{-1}{2}\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(u0 \cdot {alphay}^{2}\right), \left(\left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} + 2 \cdot u1\right)\right)}{alphax}\right)\right)\right) \cdot \left(1 - u0\right)\right)\right)\right), \frac{-1}{2}\right)\right) \]

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

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \left({alphay}^{2}\right)\right), \left(\left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} + 2 \cdot u1\right)\right)}{alphax}\right)\right)\right) \cdot \left(1 - u0\right)\right)\right)\right), \frac{-1}{2}\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \left(alphay \cdot alphay\right)\right), \left(\left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} + 2 \cdot u1\right)\right)}{alphax}\right)\right)\right) \cdot \left(1 - u0\right)\right)\right)\right), \frac{-1}{2}\right)\right) \]

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

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{*.f32}\left(alphay, alphay\right)\right), \left(\left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} + 2 \cdot u1\right)\right)}{alphax}\right)\right)\right) \cdot \left(1 - u0\right)\right)\right)\right), \frac{-1}{2}\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{*.f32}\left(alphay, alphay\right)\right), \left(\left(1 - u0\right) \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} + 2 \cdot u1\right)\right)}{alphax}\right)\right)\right)\right)\right)\right), \frac{-1}{2}\right)\right) \]

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

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{*.f32}\left(alphay, alphay\right)\right), \mathsf{*.f32}\left(\left(1 - u0\right), \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} + 2 \cdot u1\right)\right)}{alphax}\right)\right)\right)\right)\right)\right), \frac{-1}{2}\right)\right) \]

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

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{*.f32}\left(alphay, alphay\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u0\right), \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} + 2 \cdot u1\right)\right)}{alphax}\right)\right)\right)\right)\right)\right), \frac{-1}{2}\right)\right) \]

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

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{*.f32}\left(alphay, alphay\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u0\right), \mathsf{+.f32}\left(\frac{1}{2}, \left(\frac{-1}{2} \cdot \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} + 2 \cdot u1\right)\right)}{alphax}\right)\right)\right)\right)\right)\right)\right), \frac{-1}{2}\right)\right) \]

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

       \[\leadsto \mathsf{exp.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{*.f32}\left(alphay, alphay\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u0\right), \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\frac{-1}{2}, \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} + 2 \cdot u1\right)\right)}{alphax}\right)\right)\right)\right)\right)\right)\right), \frac{-1}{2}\right)\right) \]

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

       \[\leadsto \mathsf{exp.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{*.f32}\left(alphay, alphay\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u0\right), \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\frac{-1}{2}, \mathsf{cos.f32}\left(\left(2 \cdot \tan^{-1} \left(\frac{alphay \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} + 2 \cdot u1\right)\right)}{alphax}\right)\right)\right)\right)\right)\right)\right)\right), \frac{-1}{2}\right)\right) \]

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

       \[\leadsto \mathsf{exp.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{*.f32}\left(alphay, alphay\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u0\right), \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\frac{-1}{2}, \mathsf{cos.f32}\left(\mathsf{*.f32}\left(2, \tan^{-1} \left(\frac{alphay \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} + 2 \cdot u1\right)\right)}{alphax}\right)\right)\right)\right)\right)\right)\right)\right), \frac{-1}{2}\right)\right) \]

9. Simplified 98.3%

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

    1. exp-prod N/A

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

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

       \[\leadsto \mathsf{pow.f32}\left(\left(e^{\log \left(1 + \frac{u0 \cdot \left(alphay \cdot alphay\right)}{\left(1 - u0\right) \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} + 2 \cdot u1\right)\right)}{alphax}\right)\right)\right)}\right)}\right), \color{blue}{\frac{-1}{2}}\right) \]

11. Applied egg-rr 98.2%

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

12. Final simplification 98.2%

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

Alternative 6: 96.5% accurate, 4.1× speedup

Math

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

FPCore

(FPCore (u0 u1 alphax alphay)
 :precision binary32
 (/
  1.0
  (+
   1.0
   (/
    (* -0.5 (/ (* alphay alphay) (- (/ -1.0 u0) -1.0)))
    (+
     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 1.0f / (1.0f + ((-0.5f * ((alphay * alphay) / ((-1.0f / u0) - -1.0f))) / (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(1.0) + Float32(Float32(Float32(-0.5) * Float32(Float32(alphay * alphay) / Float32(Float32(Float32(-1.0) / u0) - Float32(-1.0)))) / Float32(Float32(0.5) + Float32(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) / (single(1.0) + ((single(-0.5) * ((alphay * alphay) / ((single(-1.0) / u0) - single(-1.0)))) / (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 99.5%

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

3. Simplified 99.5%

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

5. Taylor expanded in alphay around 0

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

7. Simplified 97.3%

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

9. Applied egg-rr 97.3%

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

10. Final simplification 97.3%

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

Alternative 7: 96.5% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.5%

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

5. Taylor expanded in alphay around 0

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

7. Simplified 97.3%

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

8. Taylor expanded in u1 around 0

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

   1. *-commutative N/A

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

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

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

   3. PI-lowering-PI.f32 97.3%

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

10. Simplified 97.3%

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

12. Applied egg-rr 97.3%

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

13. Final simplification 97.3%

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

Alternative 8: 96.5% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.5%

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

5. Taylor expanded in alphay around 0

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

7. Simplified 97.3%

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

8. Taylor expanded in u1 around 0

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

   1. *-commutative N/A

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

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

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

   3. PI-lowering-PI.f32 97.2%

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

10. Simplified 97.2%

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

    1. +-commutative N/A

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

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

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

12. Applied egg-rr 97.2%

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

13. Final simplification 97.2%

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

Alternative 9: 94.9% accurate, 4.2× speedup

Math

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

FPCore

(FPCore (u0 u1 alphax alphay)
 :precision binary32
 (+
  1.0
  (/
   (* -0.5 (* u0 (* alphay alphay)))
   (+
    0.5
    (*
     -0.5
     (cos
      (*
       2.0
       (atan (/ (* (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))) / (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(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(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 99.5%

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

3. Simplified 99.5%

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

6. Applied egg-rr 100.0%

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

7. Taylor expanded in alphay around 0

   \[\leadsto \mathsf{exp.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\color{blue}{\left(\frac{{alphay}^{2} \cdot u0}{\left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} + 2 \cdot u1\right)\right)}{alphax}\right)\right)\right) \cdot \left(1 - u0\right)}\right)}\right), \frac{-1}{2}\right)\right) \]
8. Step-by-step derivation

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

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left({alphay}^{2} \cdot u0\right), \left(\left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} + 2 \cdot u1\right)\right)}{alphax}\right)\right)\right) \cdot \left(1 - u0\right)\right)\right)\right), \frac{-1}{2}\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\left(u0 \cdot {alphay}^{2}\right), \left(\left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} + 2 \cdot u1\right)\right)}{alphax}\right)\right)\right) \cdot \left(1 - u0\right)\right)\right)\right), \frac{-1}{2}\right)\right) \]

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

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \left({alphay}^{2}\right)\right), \left(\left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} + 2 \cdot u1\right)\right)}{alphax}\right)\right)\right) \cdot \left(1 - u0\right)\right)\right)\right), \frac{-1}{2}\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \left(alphay \cdot alphay\right)\right), \left(\left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} + 2 \cdot u1\right)\right)}{alphax}\right)\right)\right) \cdot \left(1 - u0\right)\right)\right)\right), \frac{-1}{2}\right)\right) \]

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

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{*.f32}\left(alphay, alphay\right)\right), \left(\left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} + 2 \cdot u1\right)\right)}{alphax}\right)\right)\right) \cdot \left(1 - u0\right)\right)\right)\right), \frac{-1}{2}\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{*.f32}\left(alphay, alphay\right)\right), \left(\left(1 - u0\right) \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} + 2 \cdot u1\right)\right)}{alphax}\right)\right)\right)\right)\right)\right), \frac{-1}{2}\right)\right) \]

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

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{*.f32}\left(alphay, alphay\right)\right), \mathsf{*.f32}\left(\left(1 - u0\right), \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} + 2 \cdot u1\right)\right)}{alphax}\right)\right)\right)\right)\right)\right), \frac{-1}{2}\right)\right) \]

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

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{*.f32}\left(alphay, alphay\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u0\right), \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} + 2 \cdot u1\right)\right)}{alphax}\right)\right)\right)\right)\right)\right), \frac{-1}{2}\right)\right) \]

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

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{*.f32}\left(alphay, alphay\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u0\right), \mathsf{+.f32}\left(\frac{1}{2}, \left(\frac{-1}{2} \cdot \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} + 2 \cdot u1\right)\right)}{alphax}\right)\right)\right)\right)\right)\right)\right), \frac{-1}{2}\right)\right) \]

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

       \[\leadsto \mathsf{exp.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{*.f32}\left(alphay, alphay\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u0\right), \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\frac{-1}{2}, \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} + 2 \cdot u1\right)\right)}{alphax}\right)\right)\right)\right)\right)\right)\right), \frac{-1}{2}\right)\right) \]

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

       \[\leadsto \mathsf{exp.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{*.f32}\left(alphay, alphay\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u0\right), \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\frac{-1}{2}, \mathsf{cos.f32}\left(\left(2 \cdot \tan^{-1} \left(\frac{alphay \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} + 2 \cdot u1\right)\right)}{alphax}\right)\right)\right)\right)\right)\right)\right)\right), \frac{-1}{2}\right)\right) \]

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

       \[\leadsto \mathsf{exp.f32}\left(\mathsf{*.f32}\left(\mathsf{log1p.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(u0, \mathsf{*.f32}\left(alphay, alphay\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, u0\right), \mathsf{+.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\frac{-1}{2}, \mathsf{cos.f32}\left(\mathsf{*.f32}\left(2, \tan^{-1} \left(\frac{alphay \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} + 2 \cdot u1\right)\right)}{alphax}\right)\right)\right)\right)\right)\right)\right)\right), \frac{-1}{2}\right)\right) \]

9. Simplified 98.3%

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

10. Taylor expanded in u0 around 0

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

12. Simplified 95.7%

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

13. Final simplification 95.7%

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

Alternative 10: 91.3% 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 99.5%

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

3. Simplified 99.5%

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

5. Taylor expanded in u0 around 0

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

7. Simplified 92.9%

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

Reproduce

herbie shell --seed 2024164 
(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))))))