Trowbridge-Reitz Sample, sample surface normal, cosTheta

Percentage Accurate: 99.4% → 99.4%

Time: 1.6min

Alternatives: 6

Speedup: N/A×

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: 99.4% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in u1 around 0 99.3%

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

5. Simplified 99.3%

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

6. Final simplification 99.3%

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

Alternative 2: 98.3% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in u1 around 0 99.3%

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

5. Simplified 99.3%

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

6. Taylor expanded in u1 around 0 98.1%

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

   1. *-commutative 97.7%

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

8. Simplified 98.1%

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

9. Final simplification 98.1%

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

Alternative 3: 97.7% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in alphay around 0 97.7%

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

   1. associate-/l* 97.7%

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

   2. associate-*r* 97.7%

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

   3. distribute-rgt-out 97.7%

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

5. Simplified 97.7%

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

6. Taylor expanded in u1 around 0 97.7%

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

   1. *-commutative 97.7%

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

8. Simplified 97.7%

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

9. Final simplification 97.7%

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

Alternative 4: 49.0% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in alphay around inf 49.6%

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

   1. associate-/l* 49.6%

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

   2. associate-*r* 49.6%

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

   3. distribute-rgt-out 49.5%

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

5. Simplified 49.5%

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

   1. unpow2 49.5%

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

   2. cos-atan 47.2%

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

   3. cos-atan 49.0%

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

   4. frac-times 49.0%

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

   5. metadata-eval 49.0%

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

   6. add-sqr-sqrt 49.0%

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

7. Applied egg-rr 49.0%

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

   1. associate-*r/ 49.0%

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

   2. associate-*l/ 49.0%

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

   3. *-commutative 49.0%

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

9. Simplified 49.0%

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

    1. expm1-log1p-u 49.6%

       \[\leadsto \frac{1}{\sqrt{1 + \frac{\frac{{alphax}^{2}}{\frac{1}{1 + {\left(\tan \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\pi\right)\right)} \cdot \mathsf{fma}\left(2, u1, 0.5\right)\right) \cdot \frac{alphay}{alphax}\right)}^{2}}} \cdot u0}{1 - u0}}} \]

    2. expm1-undefine 49.6%

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

    3. log1p-undefine 49.6%

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

    4. rem-exp-log 49.6%

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

    5. +-commutative 49.6%

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

11. Applied egg-rr 49.6%

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

12. Final simplification 49.6%

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

Alternative 5: 48.5% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in alphay around inf 49.6%

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

   1. associate-/l* 49.6%

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

   2. associate-*r* 49.6%

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

   3. distribute-rgt-out 49.5%

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

5. Simplified 49.5%

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

   1. unpow2 49.5%

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

   2. cos-atan 47.2%

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

   3. cos-atan 49.0%

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

   4. frac-times 49.0%

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

   5. metadata-eval 49.0%

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

   6. add-sqr-sqrt 49.0%

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

7. Applied egg-rr 49.0%

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

   1. associate-*r/ 49.0%

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

   2. associate-*l/ 49.0%

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

   3. *-commutative 49.0%

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

9. Simplified 49.0%

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

10. Final simplification 49.0%

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

Alternative 6: 6.3% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\sqrt{1 + \frac{u0 \cdot \frac{{alphax}^{2}}{\frac{1}{1 + {\left(\frac{1}{0} \cdot \frac{alphay}{alphax}\right)}^{2}}}}{1 - u0}}} \end{array} \]

FPCore

(FPCore (u0 u1 alphax alphay)
 :precision binary32
 (/
  1.0
  (sqrt
   (+
    1.0
    (/
     (*
      u0
      (/
       (pow alphax 2.0)
       (/ 1.0 (+ 1.0 (pow (* (/ 1.0 0.0) (/ alphay alphax)) 2.0)))))
     (- 1.0 u0))))))

C

float code(float u0, float u1, float alphax, float alphay) {
	return 1.0f / sqrtf((1.0f + ((u0 * (powf(alphax, 2.0f) / (1.0f / (1.0f + powf(((1.0f / 0.0f) * (alphay / alphax)), 2.0f))))) / (1.0f - u0))));
}

Fortran

real(4) function code(u0, u1, alphax, alphay)
    real(4), intent (in) :: u0
    real(4), intent (in) :: u1
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    code = 1.0e0 / sqrt((1.0e0 + ((u0 * ((alphax ** 2.0e0) / (1.0e0 / (1.0e0 + (((1.0e0 / 0.0e0) * (alphay / alphax)) ** 2.0e0))))) / (1.0e0 - u0))))
end function

Julia

function code(u0, u1, alphax, alphay)
	return Float32(Float32(1.0) / sqrt(Float32(Float32(1.0) + Float32(Float32(u0 * Float32((alphax ^ Float32(2.0)) / Float32(Float32(1.0) / Float32(Float32(1.0) + (Float32(Float32(Float32(1.0) / Float32(0.0)) * Float32(alphay / alphax)) ^ Float32(2.0)))))) / Float32(Float32(1.0) - u0)))))
end

MATLAB

function tmp = code(u0, u1, alphax, alphay)
	tmp = single(1.0) / sqrt((single(1.0) + ((u0 * ((alphax ^ single(2.0)) / (single(1.0) / (single(1.0) + (((single(1.0) / single(0.0)) * (alphay / alphax)) ^ single(2.0)))))) / (single(1.0) - u0))));
end

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in alphay around inf 49.6%

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

   1. associate-/l* 49.6%

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

   2. associate-*r* 49.6%

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

   3. distribute-rgt-out 49.5%

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

5. Simplified 49.5%

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

   1. unpow2 49.5%

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

   2. cos-atan 47.2%

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

   3. cos-atan 49.0%

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

   4. frac-times 49.0%

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

   5. metadata-eval 49.0%

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

   6. add-sqr-sqrt 49.0%

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

7. Applied egg-rr 49.0%

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

   1. expm1-log1p-u 49.6%

      \[\leadsto \frac{1}{\sqrt{1 + \frac{\frac{{alphax}^{2}}{\frac{1}{1 + {\left(\tan \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\pi\right)\right)} \cdot \mathsf{fma}\left(2, u1, 0.5\right)\right) \cdot \frac{alphay}{alphax}\right)}^{2}}} \cdot u0}{1 - u0}}} \]

   2. expm1-undefine 49.6%

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

   3. log1p-undefine 49.6%

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

   4. rem-exp-log 49.6%

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

   5. +-commutative 49.6%

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

9. Applied egg-rr 49.6%

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

10. Taylor expanded in u1 around 0 20.2%

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

    1. *-commutative 20.2%

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

    2. metadata-eval 20.2%

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

    3. associate-/l* 20.2%

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

    4. *-rgt-identity 20.2%

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

    5. sin-PI/2 20.2%

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

    6. *-commutative 20.2%

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

    7. metadata-eval 20.2%

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

    8. associate-/l* 20.2%

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

    9. *-rgt-identity 20.2%

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

    10. cos-PI/2 6.3%

        \[\leadsto \frac{1}{\sqrt{1 + \frac{\frac{{alphax}^{2}}{\frac{1}{1 + {\left(\frac{1}{\color{blue}{0}} \cdot \frac{alphay}{alphax}\right)}^{2}}} \cdot u0}{1 - u0}}} \]

12. Simplified 6.3%

    \[\leadsto \frac{1}{\sqrt{1 + \frac{\frac{{alphax}^{2}}{\frac{1}{1 + {\left(\color{blue}{\frac{1}{0}} \cdot \frac{alphay}{alphax}\right)}^{2}}} \cdot u0}{1 - u0}}} \]

13. Final simplification 6.3%

    \[\leadsto \frac{1}{\sqrt{1 + \frac{u0 \cdot \frac{{alphax}^{2}}{\frac{1}{1 + {\left(\frac{1}{0} \cdot \frac{alphay}{alphax}\right)}^{2}}}}{1 - u0}}} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024053 
(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))))))