Trowbridge-Reitz Sample, sample surface normal, cosTheta

Percentage Accurate: 99.4% → 99.4%

Time: 1.5min

Alternatives: 8

Speedup: 1.8×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.4% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.3%

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

3. Simplified 99.3%

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

4. Taylor expanded in alphay around 0 99.3%

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

6. Simplified 99.3%

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

7. Final simplification 99.3%

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

Alternative 2: 99.4% accurate, 1.8× speedup

Math

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

FPCore

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

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 1.0f / sqrtf((1.0f + (u0 / ((1.0f - u0) * powf(hypotf((sinf(atanf(t_0)) / alphay), ((1.0f / hypotf(1.0f, t_0)) / alphax)), 2.0f)))));
}

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) / sqrt(Float32(Float32(1.0) + Float32(u0 / Float32(Float32(Float32(1.0) - u0) * (hypot(Float32(sin(atan(t_0)) / alphay), Float32(Float32(Float32(1.0) / hypot(Float32(1.0), t_0)) / alphax)) ^ Float32(2.0)))))))
end

Derivation

1. Initial program 99.3%

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

3. Simplified 99.3%

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

4. Taylor expanded in alphay around 0 99.3%

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

6. Simplified 99.3%

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

   1. cos-atan 99.3%

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

   2. hypot-1-def 99.3%

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

8. Applied egg-rr 99.3%

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

9. Final simplification 99.3%

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

Alternative 3: 99.4% accurate, 1.8× speedup

Math

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

FPCore

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

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 1.0f / sqrtf((1.0f + (u0 / ((1.0f - u0) * powf(hypotf((sinf(atanf(((alphay * t_0) / alphax))) / alphay), ((1.0f / hypotf(1.0f, ((alphay / alphax) * t_0))) / alphax)), 2.0f)))));
}

Julia

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

Derivation

1. Initial program 99.3%

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

3. Simplified 99.3%

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

4. Taylor expanded in alphay around 0 99.3%

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

6. Simplified 99.3%

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

   1. cos-atan 99.3%

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

   2. hypot-1-def 99.3%

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

8. Applied egg-rr 99.3%

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

   1. associate-*l/ 99.3%

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

10. Applied egg-rr 99.3%

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

11. Final simplification 99.3%

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

Alternative 4: 98.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.3%

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

3. Simplified 99.3%

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

4. Taylor expanded in alphay around 0 99.3%

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

6. Simplified 99.3%

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

   1. cos-atan 99.3%

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

   2. hypot-1-def 99.3%

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

8. Applied egg-rr 99.3%

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

9. Taylor expanded in u1 around 0 97.6%

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

    1. *-commutative 97.2%

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

11. Simplified 97.6%

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

12. Final simplification 97.6%

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

Alternative 5: 97.8% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.3%

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

3. Simplified 99.3%

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

4. Taylor expanded in alphay around 0 97.2%

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

6. Simplified 97.2%

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

   1. unpow2 97.2%

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

   2. div-inv 97.2%

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

   3. div-inv 97.2%

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

   4. swap-sqr 97.2%

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

   5. pow2 97.2%

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

8. Applied egg-rr 97.2%

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

9. Final simplification 97.2%

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

Alternative 6: 97.8% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.3%

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

3. Simplified 99.3%

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

4. Taylor expanded in alphay around 0 97.2%

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

6. Simplified 97.2%

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

7. Final simplification 97.2%

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

Alternative 7: 97.7% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

3. Simplified 99.3%

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

4. Taylor expanded in alphay around 0 97.2%

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

6. Simplified 97.2%

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

7. Taylor expanded in u1 around 0 97.2%

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

   1. *-commutative 97.2%

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

9. Simplified 97.2%

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

10. Final simplification 97.2%

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

Alternative 8: 91.5% accurate, 2167.0× 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%

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

3. Simplified 99.3%

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

4. Taylor expanded in alphay around inf 49.9%

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

6. Simplified 49.9%

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

   1. unpow2 49.9%

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

   2. cos-atan 47.1%

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

   3. cos-atan 49.2%

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

   4. frac-times 49.2%

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

   5. metadata-eval 49.2%

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

   6. add-sqr-sqrt 49.2%

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

   7. pow2 49.2%

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

8. Applied egg-rr 49.2%

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

9. Taylor expanded in u0 around 0 90.7%

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

10. Final simplification 90.7%

    \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023293 
(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))))))