Trowbridge-Reitz Sample, sample surface normal, cosTheta

Percentage Accurate: 99.3% → 99.9%

Time: 28.9s

Alternatives: 8

Speedup: 2.1×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 99.3% 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, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Simplified 99.4%

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

6. Applied egg-rr 99.4%

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

8. Applied egg-rr 99.9%

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

Alternative 2: 98.8% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Simplified 99.4%

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

6. Applied egg-rr 99.9%

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

7. Taylor expanded in u1 around 0

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

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

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

   2. PI-lowering-PI.f32 98.7%

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

9. Simplified 98.7%

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

10. Final simplification 98.7%

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

Alternative 3: 98.1% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Simplified 99.4%

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

5. Taylor expanded in alphax around inf

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

7. Simplified 98.3%

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

8. Final simplification 98.3%

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

Alternative 4: 98.1% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Simplified 99.4%

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

6. Applied egg-rr 99.9%

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

7. Taylor expanded in alphay around 0

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

9. Simplified 93.4%

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

10. Taylor expanded in alphax around inf

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

12. Simplified 98.0%

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

13. Final simplification 98.0%

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

Alternative 5: 98.1% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Simplified 99.4%

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

6. Applied egg-rr 99.9%

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

7. Taylor expanded in alphay around 0

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

9. Simplified 98.0%

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

Alternative 6: 96.4% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Simplified 99.4%

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

5. Taylor expanded in alphay around 0

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

7. Simplified 96.5%

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

8. Taylor expanded in u1 around 0

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

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

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

   2. PI-lowering-PI.f32 96.5%

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

10. Simplified 96.5%

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

11. Final simplification 96.5%

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

Alternative 7: 96.5% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Simplified 99.4%

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

6. Applied egg-rr 99.9%

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

7. Taylor expanded in alphay around 0

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

9. Simplified 96.1%

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

10. Taylor expanded in u1 around 0

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

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

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

    2. PI-lowering-PI.f32 96.4%

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

12. Simplified 96.4%

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

13. Final simplification 96.4%

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

Alternative 8: 91.4% accurate, 1375.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Simplified 99.4%

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

5. Taylor expanded in u0 around 0

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

7. Simplified 92.1%

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

Reproduce

herbie shell --seed 2024159 
(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))))))