Trowbridge-Reitz Sample, sample surface normal, cosTheta

Percentage Accurate: 99.4% → 99.4%

Time: 37.4s

Alternatives: 7

Speedup: 1.4×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.3%

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

4. Applied egg-rr 99.3%

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

   1. log1p-undefine 99.3%

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

   2. rem-exp-log 99.3%

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

   3. +-commutative 99.3%

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

   4. div-inv 99.3%

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

   5. div-inv 99.3%

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

   6. div-inv 99.3%

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

   7. clear-num 99.3%

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

6. Applied egg-rr 99.3%

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

7. Final simplification 99.3%

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

Alternative 2: 99.4% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

4. Applied egg-rr 99.3%

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

   1. fma-undefine 99.3%

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

   2. distribute-rgt-in 99.3%

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

6. Applied egg-rr 99.3%

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

7. Final simplification 99.3%

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

Alternative 3: 98.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.3%

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

4. Applied egg-rr 99.3%

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

5. Taylor expanded in alphax around inf 98.0%

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

   1. add-log-exp 98.0%

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

   2. metadata-eval 98.0%

      \[\leadsto \frac{1}{\sqrt{1 + \frac{\log \left(e^{\frac{1}{\color{blue}{0} + \frac{\sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \pi\right) \cdot u1 + 0.5 \cdot \pi\right)\right) \cdot \sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \pi\right) \cdot u1 + 0.5 \cdot \pi\right)\right)}{alphay \cdot alphay}}}\right) \cdot u0}{1 - u0}}} \]

   3. +-lft-identity 98.0%

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

7. Applied egg-rr 98.0%

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

   1. pow1/2 98.0%

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

   2. pow-flip 98.5%

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

9. Applied egg-rr 98.5%

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

10. Final simplification 98.5%

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

Alternative 4: 98.2% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

4. Applied egg-rr 99.3%

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

5. Taylor expanded in alphax around inf 98.0%

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

   1. add-log-exp 98.0%

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

   2. metadata-eval 98.0%

      \[\leadsto \frac{1}{\sqrt{1 + \frac{\log \left(e^{\frac{1}{\color{blue}{0} + \frac{\sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \pi\right) \cdot u1 + 0.5 \cdot \pi\right)\right) \cdot \sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \pi\right) \cdot u1 + 0.5 \cdot \pi\right)\right)}{alphay \cdot alphay}}}\right) \cdot u0}{1 - u0}}} \]

   3. +-lft-identity 98.0%

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

7. Applied egg-rr 98.0%

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

8. Taylor expanded in u1 around -inf 98.4%

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

9. Final simplification 98.4%

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

Alternative 5: 97.9% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

4. Applied egg-rr 99.3%

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

5. Taylor expanded in alphax around inf 98.0%

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

   1. inv-pow 98.0%

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

   2. metadata-eval 98.0%

      \[\leadsto \frac{1}{\sqrt{1 + \frac{{\left(\color{blue}{0} + \frac{\sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \pi\right) \cdot u1 + 0.5 \cdot \pi\right)\right) \cdot \sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \pi\right) \cdot u1 + 0.5 \cdot \pi\right)\right)}{alphay \cdot alphay}\right)}^{-1} \cdot u0}{1 - u0}}} \]

   3. +-lft-identity 98.0%

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

   4. times-frac 98.0%

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

7. Applied egg-rr 98.0%

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

   1. pow-sqr 98.0%

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

   2. metadata-eval 98.0%

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

9. Simplified 98.0%

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

10. Taylor expanded in u1 around -inf 98.0%

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

11. Final simplification 98.0%

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

Alternative 6: 95.1% accurate, 2.6× speedup

Math

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

FPCore

(FPCore (u0 u1 alphax alphay)
 :precision binary32
 (+
  1.0
  (*
   -0.5
   (/
    (* u0 (pow alphay 2.0))
    (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 1.0f + (-0.5f * ((u0 * powf(alphay, 2.0f)) / powf(sinf(atanf(((alphay * tanf((((float) M_PI) * (0.5f + (2.0f * u1))))) / alphax))), 2.0f)));
}

Julia

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

Derivation

1. Initial program 99.3%

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

4. Applied egg-rr 99.3%

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

5. Taylor expanded in alphax around inf 98.0%

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

   1. add-log-exp 98.0%

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

   2. metadata-eval 98.0%

      \[\leadsto \frac{1}{\sqrt{1 + \frac{\log \left(e^{\frac{1}{\color{blue}{0} + \frac{\sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \pi\right) \cdot u1 + 0.5 \cdot \pi\right)\right) \cdot \sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \pi\right) \cdot u1 + 0.5 \cdot \pi\right)\right)}{alphay \cdot alphay}}}\right) \cdot u0}{1 - u0}}} \]

   3. +-lft-identity 98.0%

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

7. Applied egg-rr 98.0%

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

8. Taylor expanded in u0 around 0 94.5%

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

9. Final simplification 94.5%

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

Alternative 7: 91.6% accurate, 1375.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

4. Applied egg-rr 99.3%

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

5. Taylor expanded in alphax around inf 98.0%

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

   1. inv-pow 98.0%

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

   2. metadata-eval 98.0%

      \[\leadsto \frac{1}{\sqrt{1 + \frac{{\left(\color{blue}{0} + \frac{\sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \pi\right) \cdot u1 + 0.5 \cdot \pi\right)\right) \cdot \sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \pi\right) \cdot u1 + 0.5 \cdot \pi\right)\right)}{alphay \cdot alphay}\right)}^{-1} \cdot u0}{1 - u0}}} \]

   3. +-lft-identity 98.0%

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

   4. times-frac 98.0%

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

7. Applied egg-rr 98.0%

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

   1. pow-sqr 98.0%

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

   2. metadata-eval 98.0%

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

9. Simplified 98.0%

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

10. Taylor expanded in alphay around 0 90.5%

    \[\leadsto \frac{1}{\color{blue}{1}} \]

11. Final simplification 90.5%

    \[\leadsto 1 \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024110 
(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))))))