Trowbridge-Reitz Sample, sample surface normal, cosTheta

Percentage Accurate: 99.4% → 99.9%

Time: 30.0s

Alternatives: 15

Speedup: 2.5×

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

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

FPCore

(FPCore (u0 u1 alphax alphay)
 :precision binary32
 (let* ((t_0 (/ (tan (* PI (+ 0.5 (* 2.0 u1)))) (/ alphax alphay))))
   (exp
    (*
     (log1p
      (/
       (/ u0 (- 1.0 u0))
       (+
        (/ (- 1.0 (cos (* 2.0 (atan t_0)))) (* 2.0 (* alphay alphay)))
        (/ (/ 1.0 (+ 1.0 (pow t_0 2.0))) (* alphax alphax)))))
     -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 expf((log1pf(((u0 / (1.0f - u0)) / (((1.0f - cosf((2.0f * atanf(t_0)))) / (2.0f * (alphay * alphay))) + ((1.0f / (1.0f + powf(t_0, 2.0f))) / (alphax * alphax))))) * -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 exp(Float32(log1p(Float32(Float32(u0 / Float32(Float32(1.0) - u0)) / Float32(Float32(Float32(Float32(1.0) - cos(Float32(Float32(2.0) * atan(t_0)))) / Float32(Float32(2.0) * Float32(alphay * alphay))) + Float32(Float32(Float32(1.0) / Float32(Float32(1.0) + (t_0 ^ Float32(2.0)))) / Float32(alphax * alphax))))) * Float32(-0.5)))
end

Derivation

1. Initial program 99.4%

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

3. Simplified 99.4%

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

6. Applied egg-rr 100.0%

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

7. Final simplification 100.0%

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

Alternative 2: 99.8% accurate, 2.1× speedup

Math

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

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

Derivation

1. Initial program 99.4%

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

3. Simplified 99.4%

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

6. Applied egg-rr 100.0%

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

7. Final simplification 100.0%

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

Alternative 3: 99.8% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Simplified 99.4%

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

6. Applied egg-rr 100.0%

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

8. Applied egg-rr 99.9%

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

   1. flip-- N/A

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

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

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

   3. metadata-eval N/A

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

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

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

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

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

   6. +-lowering-+.f32 100.0%

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

10. Applied egg-rr 100.0%

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

11. Final simplification 100.0%

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

Alternative 4: 99.8% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Simplified 99.4%

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

6. Applied egg-rr 100.0%

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

8. Applied egg-rr 99.9%

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

9. Final simplification 99.9%

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

Alternative 5: 98.8% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Simplified 99.4%

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

6. Applied egg-rr 100.0%

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

8. Applied egg-rr 99.9%

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

9. Taylor expanded in alphay around inf

   \[\leadsto \mathsf{pow.f32}\left(\mathsf{+.f32}\left(1, \color{blue}{\left(\frac{{alphay}^{2} \cdot u0}{\left(1 + \frac{{\cos \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} + 2 \cdot u1\right)\right)}^{2}}{{\sin \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} + 2 \cdot u1\right)\right)}^{2}}\right) \cdot \left(1 - u0\right)}\right)}\right), \frac{-1}{2}\right) \]
10. Step-by-step derivation

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

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

    2. *-commutative N/A

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

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

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

    4. unpow2 N/A

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

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

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

    6. *-commutative N/A

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

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

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

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

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

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

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

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

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

11. Simplified 99.3%

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

Alternative 6: 98.2% accurate, 2.6× speedup

Math

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

FPCore

(FPCore (u0 u1 alphax alphay)
 :precision binary32
 (exp
  (*
   -0.5
   (log1p
    (/
     (* 2.0 (* u0 (* alphay alphay)))
     (*
      (- 1.0 u0)
      (-
       1.0
       (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(((2.0f * (u0 * (alphay * alphay))) / ((1.0f - u0) * (1.0f - 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(Float32(2.0) * Float32(u0 * Float32(alphay * alphay))) / Float32(Float32(Float32(1.0) - u0) * Float32(Float32(1.0) - 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 99.4%

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

3. Simplified 99.4%

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

6. Applied egg-rr 100.0%

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

7. Taylor expanded in alphax around inf

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

9. Simplified 98.6%

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

10. Final simplification 98.6%

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

Alternative 7: 98.1% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Simplified 99.4%

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

6. Applied egg-rr 100.0%

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

7. Taylor expanded in alphax around inf

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

9. Simplified 98.6%

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

10. Final simplification 98.6%

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

Alternative 8: 96.7% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Simplified 99.4%

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

6. Applied egg-rr 100.0%

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

7. Taylor expanded in alphay around 0

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

   1. mul-1-neg N/A

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

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

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

   3. associate-/r* N/A

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

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

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

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

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

   6. *-commutative N/A

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

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

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

   8. unpow2 N/A

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

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

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

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

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

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

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

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

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

9. Simplified 97.2%

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

10. Final simplification 97.2%

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

Alternative 9: 96.5% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Simplified 99.4%

   \[\leadsto \color{blue}{\frac{1}{\sqrt{1 + \frac{\frac{u0}{1 - u0}}{\frac{\sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\pi \cdot \left(0.5 + 2 \cdot u1\right)\right)\right) \cdot \sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\pi \cdot \left(0.5 + 2 \cdot u1\right)\right)\right)}{alphay \cdot alphay} + \frac{\cos \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\pi \cdot \left(0.5 + 2 \cdot u1\right)\right)\right) \cdot \cos \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\pi \cdot \left(0.5 + 2 \cdot u1\right)\right)\right)}{alphax \cdot alphax}}}}} \]
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 97.1%

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

9. Applied egg-rr 97.1%

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

10. Final simplification 97.1%

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

Alternative 10: 96.5% accurate, 5.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Simplified 99.4%

   \[\leadsto \color{blue}{\frac{1}{\sqrt{1 + \frac{\frac{u0}{1 - u0}}{\frac{\sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\pi \cdot \left(0.5 + 2 \cdot u1\right)\right)\right) \cdot \sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\pi \cdot \left(0.5 + 2 \cdot u1\right)\right)\right)}{alphay \cdot alphay} + \frac{\cos \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\pi \cdot \left(0.5 + 2 \cdot u1\right)\right)\right) \cdot \cos \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\pi \cdot \left(0.5 + 2 \cdot u1\right)\right)\right)}{alphax \cdot alphax}}}}} \]
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 97.1%

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

9. Applied egg-rr 97.1%

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

10. Final simplification 97.1%

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

Alternative 11: 96.5% accurate, 5.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Simplified 99.4%

   \[\leadsto \color{blue}{\frac{1}{\sqrt{1 + \frac{\frac{u0}{1 - u0}}{\frac{\sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\pi \cdot \left(0.5 + 2 \cdot u1\right)\right)\right) \cdot \sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\pi \cdot \left(0.5 + 2 \cdot u1\right)\right)\right)}{alphay \cdot alphay} + \frac{\cos \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\pi \cdot \left(0.5 + 2 \cdot u1\right)\right)\right) \cdot \cos \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\pi \cdot \left(0.5 + 2 \cdot u1\right)\right)\right)}{alphax \cdot alphax}}}}} \]
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 97.1%

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

9. Applied egg-rr 97.1%

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

10. Final simplification 97.1%

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

Alternative 12: 96.5% accurate, 5.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Simplified 99.4%

   \[\leadsto \color{blue}{\frac{1}{\sqrt{1 + \frac{\frac{u0}{1 - u0}}{\frac{\sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\pi \cdot \left(0.5 + 2 \cdot u1\right)\right)\right) \cdot \sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\pi \cdot \left(0.5 + 2 \cdot u1\right)\right)\right)}{alphay \cdot alphay} + \frac{\cos \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\pi \cdot \left(0.5 + 2 \cdot u1\right)\right)\right) \cdot \cos \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\pi \cdot \left(0.5 + 2 \cdot u1\right)\right)\right)}{alphax \cdot alphax}}}}} \]
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 96.9%

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

9. Applied egg-rr 96.9%

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

10. Final simplification 96.9%

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

Alternative 13: 96.6% accurate, 105.8× speedup

Math

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

FPCore

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

C

float code(float u0, float u1, float alphax, float alphay) {
	return 1.0f + ((u0 * (0.5f * (alphay * alphay))) / (u0 + -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 + ((u0 * (0.5e0 * (alphay * alphay))) / (u0 + (-1.0e0)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Simplified 99.4%

   \[\leadsto \color{blue}{\frac{1}{\sqrt{1 + \frac{\frac{u0}{1 - u0}}{\frac{\sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\pi \cdot \left(0.5 + 2 \cdot u1\right)\right)\right) \cdot \sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\pi \cdot \left(0.5 + 2 \cdot u1\right)\right)\right)}{alphay \cdot alphay} + \frac{\cos \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\pi \cdot \left(0.5 + 2 \cdot u1\right)\right)\right) \cdot \cos \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\pi \cdot \left(0.5 + 2 \cdot u1\right)\right)\right)}{alphax \cdot alphax}}}}} \]
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 96.9%

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

9. Applied egg-rr 96.9%

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

10. Taylor expanded in alphay around inf

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

    1. associate-*r/ N/A

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

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

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

    3. associate-*r* N/A

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

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

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

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

       \[\leadsto \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \left({alphay}^{2}\right)\right), u0\right), \left(1 - u0\right)\right)\right) \]

    6. unpow2 N/A

       \[\leadsto \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \left(alphay \cdot alphay\right)\right), u0\right), \left(1 - u0\right)\right)\right) \]

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

       \[\leadsto \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(alphay, alphay\right)\right), u0\right), \left(1 - u0\right)\right)\right) \]

    8. --lowering--.f32 96.7%

       \[\leadsto \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(alphay, alphay\right)\right), u0\right), \mathsf{\_.f32}\left(1, \color{blue}{u0}\right)\right)\right) \]

12. Simplified 96.7%

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

13. Final simplification 96.7%

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

Alternative 14: 95.1% accurate, 152.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(u0, u1, alphax, alphay)
    real(4), intent (in) :: u0
    real(4), intent (in) :: u1
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    code = 1.0e0 - (0.5e0 * (u0 * (alphay * alphay)))
end function

Julia

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

MATLAB

function tmp = code(u0, u1, alphax, alphay)
	tmp = single(1.0) - (single(0.5) * (u0 * (alphay * alphay)));
end

Derivation

1. Initial program 99.4%

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

3. Simplified 99.4%

   \[\leadsto \color{blue}{\frac{1}{\sqrt{1 + \frac{\frac{u0}{1 - u0}}{\frac{\sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\pi \cdot \left(0.5 + 2 \cdot u1\right)\right)\right) \cdot \sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\pi \cdot \left(0.5 + 2 \cdot u1\right)\right)\right)}{alphay \cdot alphay} + \frac{\cos \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\pi \cdot \left(0.5 + 2 \cdot u1\right)\right)\right) \cdot \cos \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\pi \cdot \left(0.5 + 2 \cdot u1\right)\right)\right)}{alphax \cdot alphax}}}}} \]
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 96.9%

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

9. Applied egg-rr 96.9%

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

10. Taylor expanded in alphay around inf

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

    1. associate-*r/ N/A

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

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

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

    3. associate-*r* N/A

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

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

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

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

       \[\leadsto \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \left({alphay}^{2}\right)\right), u0\right), \left(1 - u0\right)\right)\right) \]

    6. unpow2 N/A

       \[\leadsto \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \left(alphay \cdot alphay\right)\right), u0\right), \left(1 - u0\right)\right)\right) \]

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

       \[\leadsto \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(alphay, alphay\right)\right), u0\right), \left(1 - u0\right)\right)\right) \]

    8. --lowering--.f32 96.7%

       \[\leadsto \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(alphay, alphay\right)\right), u0\right), \mathsf{\_.f32}\left(1, \color{blue}{u0}\right)\right)\right) \]

12. Simplified 96.7%

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

13. Taylor expanded in u0 around 0

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

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

       \[\leadsto \mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\frac{1}{2}, \color{blue}{\left({alphay}^{2} \cdot u0\right)}\right)\right) \]

    2. *-commutative N/A

       \[\leadsto \mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\frac{1}{2}, \left(u0 \cdot \color{blue}{{alphay}^{2}}\right)\right)\right) \]

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

       \[\leadsto \mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(u0, \color{blue}{\left({alphay}^{2}\right)}\right)\right)\right) \]

    4. unpow2 N/A

       \[\leadsto \mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(u0, \left(alphay \cdot \color{blue}{alphay}\right)\right)\right)\right) \]

    5. *-lowering-*.f32 95.3%

       \[\leadsto \mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(u0, \mathsf{*.f32}\left(alphay, \color{blue}{alphay}\right)\right)\right)\right) \]

15. Simplified 95.3%

    \[\leadsto 1 - \color{blue}{0.5 \cdot \left(u0 \cdot \left(alphay \cdot alphay\right)\right)} \]
16. Add Preprocessing

Alternative 15: 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 99.4%

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

3. Simplified 99.4%

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

5. Taylor expanded in u0 around 0

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

7. Simplified 91.6%

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

Reproduce

herbie shell --seed 2024288 
(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))))))