Trowbridge-Reitz Sample, sample surface normal, cosTheta

Percentage Accurate: 99.4% → 99.9%

Time: 25.5s

Alternatives: 8

Speedup: 2.3×

Specification

\[\left(\left(\left(2.328306437 \cdot 10^{-10} \leq u0 \land u0 \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 0.5\right)\right) \land \left(0.0001 \leq alphax \land alphax \leq 1\right)\right) \land \left(0.0001 \leq alphay \land alphay \leq 1\right)\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \pi\right) \cdot u1 + 0.5 \cdot \pi\right)\right)\\ t_1 := \sin t\_0\\ t_2 := \cos t\_0\\ \frac{1}{\sqrt{1 + \frac{\frac{1}{\frac{t\_2 \cdot t\_2}{alphax \cdot alphax} + \frac{t\_1 \cdot t\_1}{alphay \cdot alphay}} \cdot u0}{1 - u0}}} \end{array} \end{array} \]

FPCore

(FPCore (u0 u1 alphax alphay)
 :precision binary32
 (let* ((t_0
         (atan (* (/ alphay alphax) (tan (+ (* (* 2.0 PI) u1) (* 0.5 PI))))))
        (t_1 (sin t_0))
        (t_2 (cos t_0)))
   (/
    1.0
    (sqrt
     (+
      1.0
      (/
       (*
        (/
         1.0
         (+
          (/ (* t_2 t_2) (* alphax alphax))
          (/ (* t_1 t_1) (* alphay alphay))))
        u0)
       (- 1.0 u0)))))))

C

float code(float u0, float u1, float alphax, float alphay) {
	float t_0 = atanf(((alphay / alphax) * tanf((((2.0f * ((float) M_PI)) * u1) + (0.5f * ((float) M_PI))))));
	float t_1 = sinf(t_0);
	float t_2 = cosf(t_0);
	return 1.0f / sqrtf((1.0f + (((1.0f / (((t_2 * t_2) / (alphax * alphax)) + ((t_1 * t_1) / (alphay * alphay)))) * u0) / (1.0f - u0))));
}

Julia

function code(u0, u1, alphax, alphay)
	t_0 = atan(Float32(Float32(alphay / alphax) * tan(Float32(Float32(Float32(Float32(2.0) * Float32(pi)) * u1) + Float32(Float32(0.5) * Float32(pi))))))
	t_1 = sin(t_0)
	t_2 = cos(t_0)
	return Float32(Float32(1.0) / sqrt(Float32(Float32(1.0) + Float32(Float32(Float32(Float32(1.0) / Float32(Float32(Float32(t_2 * t_2) / Float32(alphax * alphax)) + Float32(Float32(t_1 * t_1) / Float32(alphay * alphay)))) * u0) / Float32(Float32(1.0) - u0)))))
end

MATLAB

function tmp = code(u0, u1, alphax, alphay)
	t_0 = atan(((alphay / alphax) * tan((((single(2.0) * single(pi)) * u1) + (single(0.5) * single(pi))))));
	t_1 = sin(t_0);
	t_2 = cos(t_0);
	tmp = single(1.0) / sqrt((single(1.0) + (((single(1.0) / (((t_2 * t_2) / (alphax * alphax)) + ((t_1 * t_1) / (alphay * alphay)))) * u0) / (single(1.0) - u0))));
end

Initial Program: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \pi\right) \cdot u1 + 0.5 \cdot \pi\right)\right)\\ t_1 := \sin t\_0\\ t_2 := \cos t\_0\\ \frac{1}{\sqrt{1 + \frac{\frac{1}{\frac{t\_2 \cdot t\_2}{alphax \cdot alphax} + \frac{t\_1 \cdot t\_1}{alphay \cdot alphay}} \cdot u0}{1 - u0}}} \end{array} \end{array} \]

FPCore

(FPCore (u0 u1 alphax alphay)
 :precision binary32
 (let* ((t_0
         (atan (* (/ alphay alphax) (tan (+ (* (* 2.0 PI) u1) (* 0.5 PI))))))
        (t_1 (sin t_0))
        (t_2 (cos t_0)))
   (/
    1.0
    (sqrt
     (+
      1.0
      (/
       (*
        (/
         1.0
         (+
          (/ (* t_2 t_2) (* alphax alphax))
          (/ (* t_1 t_1) (* alphay alphay))))
        u0)
       (- 1.0 u0)))))))

C

float code(float u0, float u1, float alphax, float alphay) {
	float t_0 = atanf(((alphay / alphax) * tanf((((2.0f * ((float) M_PI)) * u1) + (0.5f * ((float) M_PI))))));
	float t_1 = sinf(t_0);
	float t_2 = cosf(t_0);
	return 1.0f / sqrtf((1.0f + (((1.0f / (((t_2 * t_2) / (alphax * alphax)) + ((t_1 * t_1) / (alphay * alphay)))) * u0) / (1.0f - u0))));
}

Julia

function code(u0, u1, alphax, alphay)
	t_0 = atan(Float32(Float32(alphay / alphax) * tan(Float32(Float32(Float32(Float32(2.0) * Float32(pi)) * u1) + Float32(Float32(0.5) * Float32(pi))))))
	t_1 = sin(t_0)
	t_2 = cos(t_0)
	return Float32(Float32(1.0) / sqrt(Float32(Float32(1.0) + Float32(Float32(Float32(Float32(1.0) / Float32(Float32(Float32(t_2 * t_2) / Float32(alphax * alphax)) + Float32(Float32(t_1 * t_1) / Float32(alphay * alphay)))) * u0) / Float32(Float32(1.0) - u0)))))
end

MATLAB

function tmp = code(u0, u1, alphax, alphay)
	t_0 = atan(((alphay / alphax) * tan((((single(2.0) * single(pi)) * u1) + (single(0.5) * single(pi))))));
	t_1 = sin(t_0);
	t_2 = cos(t_0);
	tmp = single(1.0) / sqrt((single(1.0) + (((single(1.0) / (((t_2 * t_2) / (alphax * alphax)) + ((t_1 * t_1) / (alphay * alphay)))) * u0) / (single(1.0) - u0))));
end

Alternative 1: 99.9% accurate, 1.9× speedup

Math

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

FPCore

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

C

float code(float u0, float u1, float alphax, float alphay) {
	float t_0 = powf(((alphay / alphax) * tanf((fmaf(2.0f, u1, 0.5f) * ((float) M_PI)))), 2.0f);
	return expf((log1pf((u0 / ((1.0f - u0) * ((1.0f / (alphax * fmaf(t_0, alphax, alphax))) + ((1.0f + (1.0f / (-1.0f - t_0))) / (alphay * alphay)))))) * -0.5f));
}

Julia

function code(u0, u1, alphax, alphay)
	t_0 = Float32(Float32(alphay / alphax) * tan(Float32(fma(Float32(2.0), u1, Float32(0.5)) * Float32(pi)))) ^ Float32(2.0)
	return exp(Float32(log1p(Float32(u0 / Float32(Float32(Float32(1.0) - u0) * Float32(Float32(Float32(1.0) / Float32(alphax * fma(t_0, alphax, alphax))) + Float32(Float32(Float32(1.0) + Float32(Float32(1.0) / Float32(Float32(-1.0) - t_0))) / Float32(alphay * alphay)))))) * Float32(-0.5)))
end

Derivation

1. Initial program 99.2%

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

4. Applied rewrites 99.3%

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

6. Applied rewrites 100.0%

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

   1. lift-*.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. +-commutative N/A

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

   4. distribute-rgt-in N/A

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

   5. *-lft-identity N/A

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

   6. lower-fma.f32 100.0

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

   7. lift-PI.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 N/A

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

   11. lift-PI.f32 100.0

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

8. Applied rewrites 100.0%

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

9. Final simplification 100.0%

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

Alternative 2: 99.8% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.2%

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

4. Applied rewrites 99.3%

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

6. Applied rewrites 100.0%

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

8. Applied rewrites 99.9%

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

9. Final simplification 99.9%

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

Alternative 3: 99.3% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.2%

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

4. Applied rewrites 99.3%

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

6. Applied rewrites 99.3%

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

7. Final simplification 99.3%

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

Alternative 4: 98.9% accurate, 2.9× speedup

Math

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

FPCore

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

C

float code(float u0, float u1, float alphax, float alphay) {
	return expf((-0.5f * log1pf((u0 / ((1.0f - u0) * ((1.0f / (alphax * (alphax * (1.0f + powf(((alphay / alphax) * tanf((fmaf(2.0f, u1, 0.5f) * ((float) M_PI)))), 2.0f))))) + (1.0f / (alphay * alphay))))))));
}

Julia

function code(u0, u1, alphax, alphay)
	return exp(Float32(Float32(-0.5) * log1p(Float32(u0 / Float32(Float32(Float32(1.0) - u0) * Float32(Float32(Float32(1.0) / Float32(alphax * Float32(alphax * Float32(Float32(1.0) + (Float32(Float32(alphay / alphax) * tan(Float32(fma(Float32(2.0), u1, Float32(0.5)) * Float32(pi)))) ^ Float32(2.0)))))) + Float32(Float32(1.0) / Float32(alphay * alphay))))))))
end

Derivation

1. Initial program 99.2%

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

4. Applied rewrites 99.3%

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

6. Applied rewrites 100.0%

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

7. Taylor expanded in alphay around inf

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

9. Applied rewrites 99.1%

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

10. Final simplification 99.1%

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

Alternative 5: 98.2% accurate, 3.0× speedup

Math

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

FPCore

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

C

float code(float u0, float u1, float alphax, float alphay) {
	return expf((-0.5f * log1pf(((u0 * (alphay * alphay)) / ((1.0f - u0) * (1.0f + (1.0f / (-1.0f - powf(((alphay / alphax) * tanf((fmaf(2.0f, u1, 0.5f) * ((float) M_PI)))), 2.0f)))))))));
}

Julia

function code(u0, u1, alphax, alphay)
	return exp(Float32(Float32(-0.5) * log1p(Float32(Float32(u0 * Float32(alphay * alphay)) / Float32(Float32(Float32(1.0) - u0) * Float32(Float32(1.0) + Float32(Float32(1.0) / Float32(Float32(-1.0) - (Float32(Float32(alphay / alphax) * tan(Float32(fma(Float32(2.0), u1, Float32(0.5)) * Float32(pi)))) ^ Float32(2.0))))))))))
end

Derivation

1. Initial program 99.2%

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

3. Taylor expanded in alphay around 0

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

   1. lower-/.f32 N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-*.f32 N/A

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

5. Applied rewrites 97.2%

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

7. Applied rewrites 97.7%

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

8. Final simplification 97.7%

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

Alternative 6: 98.1% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.2%

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

3. Taylor expanded in alphay around 0

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

   1. lower-/.f32 N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-*.f32 N/A

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

5. Applied rewrites 97.2%

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

7. Applied rewrites 97.6%

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

8. Final simplification 97.6%

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

Alternative 7: 97.7% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.2%

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

3. Taylor expanded in alphay around 0

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

   1. lower-/.f32 N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-*.f32 N/A

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

5. Applied rewrites 97.2%

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

7. Applied rewrites 97.2%

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

8. Final simplification 97.2%

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

Alternative 8: 91.6% accurate, 1436.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (u0 u1 alphax alphay) :precision binary32 1.0)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.2%

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

3. Taylor expanded in alphay around 0

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

   1. lower-/.f32 N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-*.f32 N/A

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

5. Applied rewrites 97.2%

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

6. Taylor expanded in alphay around 0

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

8. Applied rewrites 90.7%

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

Reproduce

herbie shell --seed 2024219 
(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))))))