Trowbridge-Reitz Sample, sample surface normal, cosTheta

Percentage Accurate: 99.4% → 99.9%

Time: 26.1s

Alternatives: 9

Speedup: 2.3×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.9% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.1%

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

4. Applied egg-rr 99.3%

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

6. Applied egg-rr 100.0%

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

7. Final simplification 100.0%

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

Alternative 2: 99.8% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.1%

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

4. Applied egg-rr 99.3%

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

6. Applied egg-rr 100.0%

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

8. Applied egg-rr 99.8%

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

10. Applied egg-rr 99.9%

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

11. Final simplification 99.9%

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

Alternative 3: 99.8% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.1%

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

4. Applied egg-rr 99.3%

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

6. Applied egg-rr 100.0%

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

8. Applied egg-rr 99.8%

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

9. Final simplification 99.8%

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

Alternative 4: 99.3% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.1%

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

4. Applied egg-rr 99.3%

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

6. Applied egg-rr 99.3%

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

7. Final simplification 99.3%

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

Alternative 5: 98.9% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.1%

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

4. Applied egg-rr 99.3%

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

6. Applied egg-rr 100.0%

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

8. Applied egg-rr 99.8%

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

9. Taylor expanded in alphay around inf

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

11. Simplified 98.8%

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

12. Final simplification 98.8%

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

Alternative 6: 98.2% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.1%

   \[\frac{1}{\sqrt{1 + \frac{\frac{1}{\frac{\cos \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \pi\right) \cdot u1 + 0.5 \cdot \pi\right)\right) \cdot \cos \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \pi\right) \cdot u1 + 0.5 \cdot \pi\right)\right)}{alphax \cdot alphax} + \frac{\sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \pi\right) \cdot u1 + 0.5 \cdot \pi\right)\right) \cdot \sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \pi\right) \cdot u1 + 0.5 \cdot \pi\right)\right)}{alphay \cdot alphay}} \cdot u0}{1 - u0}}} \]
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. /-lowering-/.f32 N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

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

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

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

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

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

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

5. Simplified 97.3%

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

7. Applied egg-rr 97.7%

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

8. Final simplification 97.7%

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

Alternative 7: 98.2% accurate, 29.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

   \[\frac{1}{\sqrt{1 + \frac{\frac{1}{\frac{\cos \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \pi\right) \cdot u1 + 0.5 \cdot \pi\right)\right) \cdot \cos \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \pi\right) \cdot u1 + 0.5 \cdot \pi\right)\right)}{alphax \cdot alphax} + \frac{\sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \pi\right) \cdot u1 + 0.5 \cdot \pi\right)\right) \cdot \sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \pi\right) \cdot u1 + 0.5 \cdot \pi\right)\right)}{alphay \cdot alphay}} \cdot u0}{1 - u0}}} \]
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. /-lowering-/.f32 N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

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

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

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

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

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

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

5. Simplified 97.3%

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

7. Applied egg-rr 97.7%

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

8. Taylor expanded in alphax around 0

   \[\leadsto \color{blue}{\sqrt{\frac{1}{1 + \frac{{alphay}^{2} \cdot u0}{1 - u0}}}} \]
9. Step-by-step derivation

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

      \[\leadsto \color{blue}{\sqrt{\frac{1}{1 + \frac{{alphay}^{2} \cdot u0}{1 - u0}}}} \]

   2. /-lowering-/.f32 N/A

      \[\leadsto \sqrt{\color{blue}{\frac{1}{1 + \frac{{alphay}^{2} \cdot u0}{1 - u0}}}} \]

   3. +-lowering-+.f32 N/A

      \[\leadsto \sqrt{\frac{1}{\color{blue}{1 + \frac{{alphay}^{2} \cdot u0}{1 - u0}}}} \]

   4. /-lowering-/.f32 N/A

      \[\leadsto \sqrt{\frac{1}{1 + \color{blue}{\frac{{alphay}^{2} \cdot u0}{1 - u0}}}} \]

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

      \[\leadsto \sqrt{\frac{1}{1 + \frac{\color{blue}{{alphay}^{2} \cdot u0}}{1 - u0}}} \]

   6. unpow2 N/A

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

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

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

   8. --lowering--.f32 97.6

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

10. Simplified 97.6%

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

11. Final simplification 97.6%

    \[\leadsto \sqrt{\frac{1}{1 + \frac{u0 \cdot \left(alphay \cdot alphay\right)}{1 - u0}}} \]
12. Add Preprocessing

Alternative 8: 97.8% accurate, 30.6× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\sqrt{\mathsf{fma}\left(\frac{u0 \cdot alphay}{1 - u0}, alphay, 1\right)}} \end{array} \]

FPCore

(FPCore (u0 u1 alphax alphay)
 :precision binary32
 (/ 1.0 (sqrt (fma (/ (* u0 alphay) (- 1.0 u0)) alphay 1.0))))

C

float code(float u0, float u1, float alphax, float alphay) {
	return 1.0f / sqrtf(fmaf(((u0 * alphay) / (1.0f - u0)), alphay, 1.0f));
}

Julia

function code(u0, u1, alphax, alphay)
	return Float32(Float32(1.0) / sqrt(fma(Float32(Float32(u0 * alphay) / Float32(Float32(1.0) - u0)), alphay, Float32(1.0))))
end

Derivation

1. Initial program 99.1%

   \[\frac{1}{\sqrt{1 + \frac{\frac{1}{\frac{\cos \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \pi\right) \cdot u1 + 0.5 \cdot \pi\right)\right) \cdot \cos \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \pi\right) \cdot u1 + 0.5 \cdot \pi\right)\right)}{alphax \cdot alphax} + \frac{\sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \pi\right) \cdot u1 + 0.5 \cdot \pi\right)\right) \cdot \sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \pi\right) \cdot u1 + 0.5 \cdot \pi\right)\right)}{alphay \cdot alphay}} \cdot u0}{1 - u0}}} \]
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. /-lowering-/.f32 N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

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

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

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

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

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

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

5. Simplified 97.3%

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

7. Applied egg-rr 97.3%

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

8. Taylor expanded in alphay around inf

   \[\leadsto \frac{1}{\sqrt{\mathsf{fma}\left(\frac{alphay \cdot u0}{\color{blue}{1 - u0}}, alphay, 1\right)}} \]
9. Step-by-step derivation

   1. --lowering--.f32 97.2

      \[\leadsto \frac{1}{\sqrt{\mathsf{fma}\left(\frac{alphay \cdot u0}{\color{blue}{1 - u0}}, alphay, 1\right)}} \]

10. Simplified 97.2%

    \[\leadsto \frac{1}{\sqrt{\mathsf{fma}\left(\frac{alphay \cdot u0}{\color{blue}{1 - u0}}, alphay, 1\right)}} \]

11. Final simplification 97.2%

    \[\leadsto \frac{1}{\sqrt{\mathsf{fma}\left(\frac{u0 \cdot alphay}{1 - u0}, alphay, 1\right)}} \]
12. Add Preprocessing

Alternative 9: 91.6% accurate, 1436.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

   \[\frac{1}{\sqrt{1 + \frac{\frac{1}{\frac{\cos \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \pi\right) \cdot u1 + 0.5 \cdot \pi\right)\right) \cdot \cos \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \pi\right) \cdot u1 + 0.5 \cdot \pi\right)\right)}{alphax \cdot alphax} + \frac{\sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \pi\right) \cdot u1 + 0.5 \cdot \pi\right)\right) \cdot \sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \pi\right) \cdot u1 + 0.5 \cdot \pi\right)\right)}{alphay \cdot alphay}} \cdot u0}{1 - u0}}} \]
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. /-lowering-/.f32 N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

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

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

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

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

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

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

5. Simplified 97.3%

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

6. Taylor expanded in alphay around 0

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

8. Simplified 90.0%

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

Reproduce

herbie shell --seed 2024204 
(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))))))