Trowbridge-Reitz Sample, sample surface normal, cosTheta

Percentage Accurate: 99.4% → 99.9%

Time: 19.7s

Alternatives: 10

Speedup: 2.0×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.9% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

function code(u0, u1, alphax, alphay)
	t_0 = Float32(Float32(alphay / alphax) * tan(Float32(Float32(pi) * fma(u1, Float32(2.0), Float32(0.5)))))
	return exp(Float32(Float32(-0.5) * log1p(Float32(u0 / Float32(Float32(Float32(1.0) - u0) * fma(Float32(Float32(0.5) / Float32(alphay * alphay)), Float32(Float32(1.0) - cos(Float32(atan(t_0) * Float32(2.0)))), Float32(Float32(-1.0) / Float32(Float32(Float32(-1.0) - (t_0 ^ Float32(2.0))) * Float32(alphax * alphax)))))))))
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 rewrites 99.3%

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

6. Applied rewrites 100.0%

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

7. Final simplification 100.0%

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

Alternative 2: 99.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(u0, u1, alphax, alphay)
	t_0 = Float32(Float32(alphay / alphax) * tan(Float32(Float32(pi) * fma(u1, Float32(2.0), Float32(0.5)))))
	return Float32(Float32(1.0) - Float32(u0 / Float32(Float32(u0 - Float32(1.0)) * fma(Float32(Float32(0.5) / Float32(alphay * alphay)), Float32(Float32(1.0) - cos(Float32(atan(t_0) * Float32(2.0)))), Float32(Float32(-1.0) / Float32(Float32(Float32(-1.0) - (t_0 ^ Float32(2.0))) * Float32(alphax * alphax))))))) ^ 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 rewrites 99.3%

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

6. Applied rewrites 99.9%

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

7. Final simplification 99.9%

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

Alternative 3: 99.3% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

function code(u0, u1, alphax, alphay)
	t_0 = Float32(Float32(alphay / alphax) * tan(Float32(Float32(pi) * fma(u1, Float32(2.0), Float32(0.5)))))
	return Float32(Float32(1.0) / sqrt(Float32(Float32(1.0) - Float32(u0 / Float32(Float32(u0 - Float32(1.0)) * fma(Float32(Float32(0.5) / Float32(alphay * alphay)), Float32(Float32(1.0) - cos(Float32(atan(t_0) * Float32(2.0)))), Float32(Float32(-1.0) / Float32(Float32(Float32(-1.0) - (t_0 ^ Float32(2.0))) * Float32(alphax * alphax)))))))))
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 rewrites 99.3%

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

6. Applied rewrites 99.3%

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

7. Final simplification 99.3%

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

Alternative 4: 98.2% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

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

3. Taylor expanded in alphay around 0

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. lower--.f32 N/A

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

   9. lower-pow.f32 N/A

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

5. Applied rewrites 97.9%

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

7. Applied rewrites 98.4%

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

8. Final simplification 98.4%

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

Alternative 5: 98.2% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

3. Taylor expanded in alphay around 0

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. lower--.f32 N/A

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

   9. lower-pow.f32 N/A

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

5. Applied rewrites 97.9%

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

7. Applied rewrites 98.4%

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

8. Final simplification 98.4%

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

Alternative 6: 98.2% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Julia

function code(u0, u1, alphax, alphay)
	return sqrt(Float32(Float32(1.0) / fma(Float32(Float32(Float32(alphay * alphay) * u0) / Float32(Float32(cos(Float32(atan(Float32(Float32(tan(Float32(Float32(pi) * fma(u1, Float32(2.0), Float32(0.5)))) / alphax) * alphay)) * Float32(2.0))) - Float32(1.0)) * Float32(u0 - Float32(1.0)))), Float32(2.0), Float32(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 rewrites 99.3%

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

5. Taylor expanded in alphax around inf

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

7. Applied rewrites 98.3%

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

8. Final simplification 98.3%

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

Alternative 7: 96.7% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Julia

function code(u0, u1, alphax, alphay)
	return Float32(Float32(1.0) - Float32(Float32(Float32(alphay * alphay) * u0) / Float32(Float32(cos(Float32(atan(Float32(Float32(tan(Float32(Float32(pi) * fma(u1, Float32(2.0), Float32(0.5)))) / alphax) * alphay)) * Float32(2.0))) - Float32(1.0)) * Float32(u0 - Float32(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 rewrites 99.3%

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

5. Taylor expanded in alphay around 0

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 N/A

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

   9. *-commutative N/A

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

7. Applied rewrites 96.5%

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

8. Final simplification 96.5%

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

Alternative 8: 96.7% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u0, u1, alphax, alphay)
	tmp = single(1.0) - (((alphay * alphay) * u0) / ((cos((atan(((tan((single(pi) * single(0.5))) / alphax) * alphay)) * single(2.0))) - single(1.0)) * (u0 - 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 rewrites 99.3%

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

5. Taylor expanded in alphay around 0

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 N/A

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

   9. *-commutative N/A

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

7. Applied rewrites 96.5%

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

8. Taylor expanded in u1 around 0

   \[\leadsto 1 - \frac{u0 \cdot \left(alphay \cdot alphay\right)}{\left(1 - \cos \left(\tan^{-1} \left(alphay \cdot \frac{\tan \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}{alphax}\right) \cdot 2\right)\right) \cdot \left(1 - u0\right)} \]
9. Step-by-step derivation

10. Applied rewrites 96.1%

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

11. Final simplification 96.1%

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

Alternative 9: 95.1% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(u0, u1, alphax, alphay)
	tmp = single(1.0) - (((alphay * alphay) * u0) / (single(1.0) - cos((atan(((tan(((single(0.5) - (single(-2.0) * u1)) * single(pi))) * alphay) / 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 rewrites 99.3%

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

5. Taylor expanded in alphay around 0

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 N/A

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

   9. *-commutative N/A

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

7. Applied rewrites 96.5%

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

8. Taylor expanded in u0 around 0

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

10. Applied rewrites 94.6%

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

11. Final simplification 94.6%

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

Alternative 10: 91.8% accurate, 1436.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in alphay around 0

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. lower--.f32 N/A

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

   9. lower-pow.f32 N/A

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

5. Applied rewrites 97.9%

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

6. Taylor expanded in alphay around 0

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

8. Applied rewrites 90.4%

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

Reproduce

herbie shell --seed 2024235 
(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))))))