Trowbridge-Reitz Sample, sample surface normal, cosTheta

Percentage Accurate: 99.4% → 99.7%

Time: 16.9s

Alternatives: 12

Speedup: 1.9×

Specification

\[\left(\left(\left(\frac{2328306437}{10000000000000000000} \leq u0 \land u0 \leq 1\right) \land \left(\frac{2328306437}{10000000000000000000} \leq u1 \land u1 \leq \frac{1}{2}\right)\right) \land \left(\frac{1}{10000} \leq alphax \land alphax \leq 1\right)\right) \land \left(\frac{1}{10000} \leq alphay \land alphay \leq 1\right)\]

Math

\[\begin{array}{l} t_0 := \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \pi\right) \cdot u1 + \frac{1}{2} \cdot \pi\right)\right)\\ t_1 := \sin t\_0\\ t_2 := \cos t\_0\\ \frac{1}{\sqrt{1 + \frac{\frac{1}{\frac{t\_2 \cdot t\_2}{alphax \cdot alphax} + \frac{t\_1 \cdot t\_1}{alphay \cdot alphay}} \cdot u0}{1 - u0}}} \end{array} \]

FPCore

(FPCore (u0 u1 alphax alphay)
  :precision binary32
  (let* ((t_0
        (atan
         (* (/ alphay alphax) (tan (+ (* (* 2 PI) u1) (* 1/2 PI))))))
       (t_1 (sin t_0))
       (t_2 (cos t_0)))
  (/
   1
   (sqrt
    (+
     1
     (/
      (*
       (/
        1
        (+
         (/ (* t_2 t_2) (* alphax alphax))
         (/ (* t_1 t_1) (* alphay alphay))))
       u0)
      (- 1 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

\[\begin{array}{l} t_0 := \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \pi\right) \cdot u1 + \frac{1}{2} \cdot \pi\right)\right)\\ t_1 := \sin t\_0\\ t_2 := \cos t\_0\\ \frac{1}{\sqrt{1 + \frac{\frac{1}{\frac{t\_2 \cdot t\_2}{alphax \cdot alphax} + \frac{t\_1 \cdot t\_1}{alphay \cdot alphay}} \cdot u0}{1 - u0}}} \end{array} \]

FPCore

(FPCore (u0 u1 alphax alphay)
  :precision binary32
  (let* ((t_0
        (atan
         (* (/ alphay alphax) (tan (+ (* (* 2 PI) u1) (* 1/2 PI))))))
       (t_1 (sin t_0))
       (t_2 (cos t_0)))
  (/
   1
   (sqrt
    (+
     1
     (/
      (*
       (/
        1
        (+
         (/ (* t_2 t_2) (* alphax alphax))
         (/ (* t_1 t_1) (* alphay alphay))))
       u0)
      (- 1 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: 99.7% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (u0 u1 alphax alphay)
  :precision binary32
  (let* ((t_0
        (cos
         (*
          -2
          (atan
           (*
            (/ alphay alphax)
            (tan (- (* (+ PI PI) u1) (* -1/2 PI)))))))))
  (pow
   (-
    (/
     u0
     (*
      (+
       (/
        (-
         1
         (cos
          (*
           (atan
            (*
             (tan (- (* u1 (+ PI PI)) (* -1/2 PI)))
             (/ alphay alphax)))
           2)))
        (* 2 (* alphay alphay)))
       (/ (* (- 1 (/ -1 t_0)) t_0) (* 2 (* alphax alphax))))
      (- 1 u0)))
    -1)
   -1/2)))

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Applied rewrites 99.7%

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

5. Applied rewrites 99.7%

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

Alternative 2: 99.7% accurate, 1.7× speedup

Math

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

FPCore

(FPCore (u0 u1 alphax alphay)
  :precision binary32
  (let* ((t_0
        (cos
         (*
          (atan
           (*
            (tan (- (* u1 (+ PI PI)) (* -1/2 PI)))
            (/ alphay alphax)))
          2))))
  (pow
   (-
    (/
     u0
     (*
      (+
       (/ (- 1 t_0) (* 2 (* alphay alphay)))
       (/ (- t_0 -1) (* 2 (* alphax alphax))))
      (- 1 u0)))
    -1)
   -1/2)))

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Applied rewrites 99.7%

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

Alternative 3: 99.5% accurate, 1.8× speedup

Math

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

FPCore

(FPCore (u0 u1 alphax alphay)
  :precision binary32
  (let* ((t_0 (* (* 2 alphax) alphax)))
  (sqrt
   (/
    (- 1 u0)
    (-
     (- 1 u0)
     (/
      u0
      (-
       (/
        (-
         (*
          1/2
          (cos
           (*
            (atan
             (*
              (tan (- (* (+ PI PI) u1) (* -1/2 PI)))
              (/ alphay alphax)))
            -2)))
         1/2)
        (* alphay alphay))
       (/
        (-
         (*
          (cos
           (*
            (atan
             (*
              (tan (- (* u1 (+ PI PI)) (* -1/2 PI)))
              (/ alphay alphax)))
            2))
          t_0)
         (* t_0 -1))
        (* t_0 t_0)))))))))

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Applied rewrites 99.0%

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

5. Applied rewrites 99.5%

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

7. Applied rewrites 99.4%

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

Alternative 4: 99.4% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (u0 u1 alphax alphay)
  :precision binary32
  (let* ((t_0
        (cos
         (*
          (atan
           (*
            (tan (- (* (+ PI PI) u1) (* -1/2 PI)))
            (/ alphay alphax)))
          -2))))
  (sqrt
   (/
    (- 1 u0)
    (-
     (- 1 u0)
     (/
      u0
      (-
       (/ (- (* 1/2 t_0) 1/2) (* alphay alphay))
       (/ (- t_0 -1) (* (* alphax alphax) 2)))))))))

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Applied rewrites 99.0%

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

5. Applied rewrites 99.5%

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

Alternative 5: 98.3% accurate, 2.1× speedup

Math

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

FPCore

(FPCore (u0 u1 alphax alphay)
  :precision binary32
  (let* ((t_0 (- (* 2 (* u1 PI)) (* -1/2 PI))))
  (pow
   (-
    (*
     2
     (/
      (* (pow alphay 2) u0)
      (*
       (- 1 u0)
       (-
        1
        (cos
         (*
          2
          (atan (/ (* alphay (sin t_0)) (* alphax (cos t_0))))))))))
    -1)
   -1/2)))

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Applied rewrites 99.7%

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

4. Taylor expanded in alphax around inf

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

   1. lower-*.f32 N/A

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

   2. lower-/.f32 N/A

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

6. Applied rewrites 98.3%

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

Alternative 6: 98.1% accurate, 2.3× speedup

Math

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

FPCore

(FPCore (u0 u1 alphax alphay)
  :precision binary32
  (let* ((t_0 (- (* 2 (* u1 PI)) (* -1/2 PI))))
  (sqrt
   (/
    (- 1 u0)
    (-
     (+
      1
      (*
       -1
       (/
        (* (pow alphay 2) u0)
        (-
         (*
          1/2
          (cos
           (*
            -2
            (atan (/ (* alphay (sin t_0)) (* alphax (cos t_0)))))))
         1/2))))
     u0)))))

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Applied rewrites 99.0%

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

5. Applied rewrites 99.5%

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

6. Taylor expanded in alphay around 0

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

8. Applied rewrites 98.1%

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

Alternative 7: 98.1% accurate, 2.3× speedup

Math

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

FPCore

(FPCore (u0 u1 alphax alphay)
  :precision binary32
  (let* ((t_0 (- (* 2 (* u1 PI)) (* -1/2 PI))))
  (sqrt
   (/
    (- 1 u0)
    (-
     (- 1 u0)
     (/
      (* (pow alphay 2) u0)
      (-
       (*
        1/2
        (cos
         (* -2 (atan (/ (* alphay (sin t_0)) (* alphax (cos t_0)))))))
       1/2)))))))

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Applied rewrites 99.0%

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

5. Applied rewrites 99.5%

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

6. Taylor expanded in alphay around 0

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-pow.f32 N/A

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

8. Applied rewrites 98.1%

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

Alternative 8: 97.0% accurate, 2.8× speedup

Math

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

FPCore

(FPCore (u0 u1 alphax alphay)
  :precision binary32
  (/
 1
 (-
  1
  (*
   -1/2
   (/
    (* (* alphay alphay) u0)
    (*
     (-
      1/2
      (*
       1/2
       (cos
        (*
         2
         (atan
          (*
           alphay
           (/
            (sin (* PI 1/2))
            (* (cos (- (* -1/2 PI) (* u1 (+ PI PI)))) alphax))))))))
     (- 1 u0)))))))

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

2. Taylor expanded in alphay around 0

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

   1. lower-+.f32 N/A

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

4. Applied rewrites 96.6%

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

5. Taylor expanded in u1 around 0

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

   1. lower-sin.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-PI.f32 97.0%

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

7. Applied rewrites 97.0%

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

9. Applied rewrites 97.0%

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

Alternative 9: 96.5% accurate, 3.4× speedup

Math

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

FPCore

(FPCore (u0 u1 alphax alphay)
  :precision binary32
  (/
 1
 (+
  1
  (*
   1/2
   (/
    1
    (/
     (* u0 (- (/ 1 u0) 1))
     (/
      (* (* alphay alphay) u0)
      (-
       1/2
       (*
        1/2
        (cos
         (*
          -2
          (atan
           (*
            (/ alphay alphax)
            (tan (- (* (+ PI PI) u1) (* -1/2 PI))))))))))))))))

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

2. Taylor expanded in alphay around 0

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

   1. lower-+.f32 N/A

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

4. Applied rewrites 96.6%

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

6. Applied rewrites 96.5%

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

7. Taylor expanded in u0 around inf

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

   1. lower-*.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lower-/.f32 96.5%

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

9. Applied rewrites 96.5%

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

Alternative 10: 96.5% accurate, 3.6× speedup

Math

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

FPCore

(FPCore (u0 u1 alphax alphay)
  :precision binary32
  (/
 1
 (+
  1
  (/
   1/2
   (*
    (/ (- 1 u0) (* (* alphay alphay) u0))
    (-
     1/2
     (*
      1/2
      (cos
       (*
        (atan
         (* (tan (- (* (+ PI PI) u1) (* -1/2 PI))) (/ alphay alphax)))
        -2)))))))))

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

2. Taylor expanded in alphay around 0

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

   1. lower-+.f32 N/A

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

4. Applied rewrites 96.6%

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

6. Applied rewrites 96.5%

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

8. Applied rewrites 96.5%

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

Alternative 11: 96.5% accurate, 3.6× speedup

Math

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

FPCore

(FPCore (u0 u1 alphax alphay)
  :precision binary32
  (/
 1
 (-
  (*
   (/
    (* (* alphay alphay) u0)
    (*
     (-
      1/2
      (*
       1/2
       (cos
        (*
         -2
         (atan
          (*
           (/ alphay alphax)
           (tan (- (* (+ PI PI) u1) (* -1/2 PI)))))))))
     (- 1 u0)))
   1/2)
  -1)))

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

2. Taylor expanded in alphay around 0

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

   1. lower-+.f32 N/A

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

4. Applied rewrites 96.6%

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

6. Applied rewrites 96.5%

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

Alternative 12: 91.5% accurate, 51.3× speedup

Math

\[\sqrt{\frac{1 - u0}{1 - u0}} \]

FPCore

(FPCore (u0 u1 alphax alphay)
  :precision binary32
  (sqrt (/ (- 1 u0) (- 1 u0))))

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(4) function code(u0, u1, alphax, alphay)
use fmin_fmax_functions
    real(4), intent (in) :: u0
    real(4), intent (in) :: u1
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    code = sqrt(((1.0e0 - u0) / (1.0e0 - u0)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Applied rewrites 99.0%

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

4. Taylor expanded in alphax around 0

   \[\leadsto \frac{1}{\frac{\sqrt{\color{blue}{1 - u0}}}{\sqrt{1 - u0}}} \]
5. Step-by-step derivation

   1. lower--.f32 91.5%

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

6. Applied rewrites 91.5%

   \[\leadsto \frac{1}{\frac{\sqrt{\color{blue}{1 - u0}}}{\sqrt{1 - u0}}} \]
7. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. lift-/.f32 N/A

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

8. Applied rewrites 91.5%

   \[\leadsto \color{blue}{\sqrt{\frac{1 - u0}{1 - u0}}} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025271 -o generate:evaluate
(FPCore (u0 u1 alphax alphay)
  :name "Trowbridge-Reitz Sample, sample surface normal, cosTheta"
  :precision binary32
  :pre (and (and (and (and (<= 2328306437/10000000000000000000 u0) (<= u0 1)) (and (<= 2328306437/10000000000000000000 u1) (<= u1 1/2))) (and (<= 1/10000 alphax) (<= alphax 1))) (and (<= 1/10000 alphay) (<= alphay 1)))
  (/ 1 (sqrt (+ 1 (/ (* (/ 1 (+ (/ (* (cos (atan (* (/ alphay alphax) (tan (+ (* (* 2 PI) u1) (* 1/2 PI)))))) (cos (atan (* (/ alphay alphax) (tan (+ (* (* 2 PI) u1) (* 1/2 PI))))))) (* alphax alphax)) (/ (* (sin (atan (* (/ alphay alphax) (tan (+ (* (* 2 PI) u1) (* 1/2 PI)))))) (sin (atan (* (/ alphay alphax) (tan (+ (* (* 2 PI) u1) (* 1/2 PI))))))) (* alphay alphay)))) u0) (- 1 u0))))))