UniformSampleCone 2

Percentage Accurate: 98.9% → 98.7%

Time: 16.5s

Alternatives: 14

Speedup: N/A×

Specification

\[\left(\left(\left(\left(\left(-10000 \leq xi \land xi \leq 10000\right) \land \left(-10000 \leq yi \land yi \leq 10000\right)\right) \land \left(-10000 \leq zi \land zi \leq 10000\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq ux \land ux \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq uy \land uy \leq 1\right)\right) \land \left(0 \leq maxCos \land maxCos \leq 1\right)\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\\ t_1 := \sqrt{1 - t\_0 \cdot t\_0}\\ t_2 := \left(uy \cdot 2\right) \cdot \pi\\ \left(\left(\cos t\_2 \cdot t\_1\right) \cdot xi + \left(\sin t\_2 \cdot t\_1\right) \cdot yi\right) + t\_0 \cdot zi \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (* (- 1.0 ux) maxCos) ux))
        (t_1 (sqrt (- 1.0 (* t_0 t_0))))
        (t_2 (* (* uy 2.0) PI)))
   (+ (+ (* (* (cos t_2) t_1) xi) (* (* (sin t_2) t_1) yi)) (* t_0 zi))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = ((1.0f - ux) * maxCos) * ux;
	float t_1 = sqrtf((1.0f - (t_0 * t_0)));
	float t_2 = (uy * 2.0f) * ((float) M_PI);
	return (((cosf(t_2) * t_1) * xi) + ((sinf(t_2) * t_1) * yi)) + (t_0 * zi);
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(Float32(Float32(1.0) - ux) * maxCos) * ux)
	t_1 = sqrt(Float32(Float32(1.0) - Float32(t_0 * t_0)))
	t_2 = Float32(Float32(uy * Float32(2.0)) * Float32(pi))
	return Float32(Float32(Float32(Float32(cos(t_2) * t_1) * xi) + Float32(Float32(sin(t_2) * t_1) * yi)) + Float32(t_0 * zi))
end

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	t_0 = ((single(1.0) - ux) * maxCos) * ux;
	t_1 = sqrt((single(1.0) - (t_0 * t_0)));
	t_2 = (uy * single(2.0)) * single(pi);
	tmp = (((cos(t_2) * t_1) * xi) + ((sin(t_2) * t_1) * yi)) + (t_0 * zi);
end

Initial Program: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\\ t_1 := \sqrt{1 - t\_0 \cdot t\_0}\\ t_2 := \left(uy \cdot 2\right) \cdot \pi\\ \left(\left(\cos t\_2 \cdot t\_1\right) \cdot xi + \left(\sin t\_2 \cdot t\_1\right) \cdot yi\right) + t\_0 \cdot zi \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (* (- 1.0 ux) maxCos) ux))
        (t_1 (sqrt (- 1.0 (* t_0 t_0))))
        (t_2 (* (* uy 2.0) PI)))
   (+ (+ (* (* (cos t_2) t_1) xi) (* (* (sin t_2) t_1) yi)) (* t_0 zi))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = ((1.0f - ux) * maxCos) * ux;
	float t_1 = sqrtf((1.0f - (t_0 * t_0)));
	float t_2 = (uy * 2.0f) * ((float) M_PI);
	return (((cosf(t_2) * t_1) * xi) + ((sinf(t_2) * t_1) * yi)) + (t_0 * zi);
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(Float32(Float32(1.0) - ux) * maxCos) * ux)
	t_1 = sqrt(Float32(Float32(1.0) - Float32(t_0 * t_0)))
	t_2 = Float32(Float32(uy * Float32(2.0)) * Float32(pi))
	return Float32(Float32(Float32(Float32(cos(t_2) * t_1) * xi) + Float32(Float32(sin(t_2) * t_1) * yi)) + Float32(t_0 * zi))
end

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	t_0 = ((single(1.0) - ux) * maxCos) * ux;
	t_1 = sqrt((single(1.0) - (t_0 * t_0)));
	t_2 = (uy * single(2.0)) * single(pi);
	tmp = (((cos(t_2) * t_1) * xi) + ((sin(t_2) * t_1) * yi)) + (t_0 * zi);
end

Alternative 1: 98.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := uy \cdot \left(2 \cdot \pi\right)\\ \mathsf{fma}\left(maxCos - maxCos \cdot ux, ux \cdot zi, \sqrt{1 - \left(maxCos \cdot \left(1 - ux\right)\right) \cdot \left(maxCos \cdot \left(ux \cdot ux\right)\right)} \cdot \left(\cos t\_0 \cdot xi + \sin t\_0 \cdot yi\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* uy (* 2.0 PI))))
   (fma
    (- maxCos (* maxCos ux))
    (* ux zi)
    (*
     (sqrt (- 1.0 (* (* maxCos (- 1.0 ux)) (* maxCos (* ux ux)))))
     (+ (* (cos t_0) xi) (* (sin t_0) yi))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = uy * (2.0f * ((float) M_PI));
	return fmaf((maxCos - (maxCos * ux)), (ux * zi), (sqrtf((1.0f - ((maxCos * (1.0f - ux)) * (maxCos * (ux * ux))))) * ((cosf(t_0) * xi) + (sinf(t_0) * yi))));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(uy * Float32(Float32(2.0) * Float32(pi)))
	return fma(Float32(maxCos - Float32(maxCos * ux)), Float32(ux * zi), Float32(sqrt(Float32(Float32(1.0) - Float32(Float32(maxCos * Float32(Float32(1.0) - ux)) * Float32(maxCos * Float32(ux * ux))))) * Float32(Float32(cos(t_0) * xi) + Float32(sin(t_0) * yi))))
end

Derivation

1. Initial program 99.1%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

3. Simplified 99.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left(1 - ux\right) \cdot maxCos, ux \cdot zi, \sqrt{1 - \left(maxCos \cdot \left(ux + -1\right)\right) \cdot \left(\left(ux \cdot ux\right) \cdot \left(maxCos \cdot \left(ux + -1\right)\right)\right)} \cdot \left(\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot xi + \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot yi\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in ux around 0 99.1%

   \[\leadsto \mathsf{fma}\left(\color{blue}{maxCos + -1 \cdot \left(maxCos \cdot ux\right)}, ux \cdot zi, \sqrt{1 - \left(maxCos \cdot \left(ux + -1\right)\right) \cdot \left(\left(ux \cdot ux\right) \cdot \left(maxCos \cdot \left(ux + -1\right)\right)\right)} \cdot \left(\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot xi + \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot yi\right)\right) \]

6. Taylor expanded in ux around 0 99.1%

   \[\leadsto \mathsf{fma}\left(maxCos + -1 \cdot \left(maxCos \cdot ux\right), ux \cdot zi, \sqrt{1 - \left(maxCos \cdot \left(ux + -1\right)\right) \cdot \left(\left(ux \cdot ux\right) \cdot \color{blue}{\left(-1 \cdot maxCos\right)}\right)} \cdot \left(\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot xi + \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot yi\right)\right) \]
7. Step-by-step derivation

   1. neg-mul-1 99.1%

      \[\leadsto \mathsf{fma}\left(maxCos + -1 \cdot \left(maxCos \cdot ux\right), ux \cdot zi, \sqrt{1 - \left(maxCos \cdot \left(ux + -1\right)\right) \cdot \left(\left(ux \cdot ux\right) \cdot \color{blue}{\left(-maxCos\right)}\right)} \cdot \left(\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot xi + \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot yi\right)\right) \]

8. Simplified 99.1%

   \[\leadsto \mathsf{fma}\left(maxCos + -1 \cdot \left(maxCos \cdot ux\right), ux \cdot zi, \sqrt{1 - \left(maxCos \cdot \left(ux + -1\right)\right) \cdot \left(\left(ux \cdot ux\right) \cdot \color{blue}{\left(-maxCos\right)}\right)} \cdot \left(\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot xi + \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot yi\right)\right) \]

9. Final simplification 99.1%

   \[\leadsto \mathsf{fma}\left(maxCos - maxCos \cdot ux, ux \cdot zi, \sqrt{1 - \left(maxCos \cdot \left(1 - ux\right)\right) \cdot \left(maxCos \cdot \left(ux \cdot ux\right)\right)} \cdot \left(\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot xi + \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot yi\right)\right) \]
10. Add Preprocessing

Alternative 2: 98.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := uy \cdot \left(2 \cdot \pi\right)\\ \mathsf{fma}\left(maxCos - maxCos \cdot ux, ux \cdot zi, \cos t\_0 \cdot xi + \sin t\_0 \cdot yi\right) \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* uy (* 2.0 PI))))
   (fma
    (- maxCos (* maxCos ux))
    (* ux zi)
    (+ (* (cos t_0) xi) (* (sin t_0) yi)))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = uy * (2.0f * ((float) M_PI));
	return fmaf((maxCos - (maxCos * ux)), (ux * zi), ((cosf(t_0) * xi) + (sinf(t_0) * yi)));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(uy * Float32(Float32(2.0) * Float32(pi)))
	return fma(Float32(maxCos - Float32(maxCos * ux)), Float32(ux * zi), Float32(Float32(cos(t_0) * xi) + Float32(sin(t_0) * yi)))
end

Derivation

1. Initial program 99.1%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

3. Simplified 99.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left(1 - ux\right) \cdot maxCos, ux \cdot zi, \sqrt{1 - \left(maxCos \cdot \left(ux + -1\right)\right) \cdot \left(\left(ux \cdot ux\right) \cdot \left(maxCos \cdot \left(ux + -1\right)\right)\right)} \cdot \left(\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot xi + \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot yi\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in ux around 0 99.1%

   \[\leadsto \mathsf{fma}\left(\color{blue}{maxCos + -1 \cdot \left(maxCos \cdot ux\right)}, ux \cdot zi, \sqrt{1 - \left(maxCos \cdot \left(ux + -1\right)\right) \cdot \left(\left(ux \cdot ux\right) \cdot \left(maxCos \cdot \left(ux + -1\right)\right)\right)} \cdot \left(\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot xi + \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot yi\right)\right) \]

6. Taylor expanded in ux around 0 99.1%

   \[\leadsto \mathsf{fma}\left(maxCos + -1 \cdot \left(maxCos \cdot ux\right), ux \cdot zi, \sqrt{1 - \left(maxCos \cdot \left(ux + -1\right)\right) \cdot \left(\left(ux \cdot ux\right) \cdot \color{blue}{\left(-1 \cdot maxCos\right)}\right)} \cdot \left(\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot xi + \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot yi\right)\right) \]
7. Step-by-step derivation

   1. neg-mul-1 99.1%

      \[\leadsto \mathsf{fma}\left(maxCos + -1 \cdot \left(maxCos \cdot ux\right), ux \cdot zi, \sqrt{1 - \left(maxCos \cdot \left(ux + -1\right)\right) \cdot \left(\left(ux \cdot ux\right) \cdot \color{blue}{\left(-maxCos\right)}\right)} \cdot \left(\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot xi + \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot yi\right)\right) \]

8. Simplified 99.1%

   \[\leadsto \mathsf{fma}\left(maxCos + -1 \cdot \left(maxCos \cdot ux\right), ux \cdot zi, \sqrt{1 - \left(maxCos \cdot \left(ux + -1\right)\right) \cdot \left(\left(ux \cdot ux\right) \cdot \color{blue}{\left(-maxCos\right)}\right)} \cdot \left(\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot xi + \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot yi\right)\right) \]

9. Taylor expanded in ux around 0 99.1%

   \[\leadsto \mathsf{fma}\left(maxCos + -1 \cdot \left(maxCos \cdot ux\right), ux \cdot zi, \sqrt{1 - \color{blue}{\left(-1 \cdot maxCos\right)} \cdot \left(\left(ux \cdot ux\right) \cdot \left(-maxCos\right)\right)} \cdot \left(\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot xi + \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot yi\right)\right) \]
10. Step-by-step derivation

    1. neg-mul-1 99.1%

       \[\leadsto \mathsf{fma}\left(maxCos + -1 \cdot \left(maxCos \cdot ux\right), ux \cdot zi, \sqrt{1 - \left(maxCos \cdot \left(ux + -1\right)\right) \cdot \left(\left(ux \cdot ux\right) \cdot \color{blue}{\left(-maxCos\right)}\right)} \cdot \left(\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot xi + \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot yi\right)\right) \]

11. Simplified 99.1%

    \[\leadsto \mathsf{fma}\left(maxCos + -1 \cdot \left(maxCos \cdot ux\right), ux \cdot zi, \sqrt{1 - \color{blue}{\left(-maxCos\right)} \cdot \left(\left(ux \cdot ux\right) \cdot \left(-maxCos\right)\right)} \cdot \left(\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot xi + \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot yi\right)\right) \]

12. Taylor expanded in maxCos around 0 99.1%

    \[\leadsto \mathsf{fma}\left(maxCos + -1 \cdot \left(maxCos \cdot ux\right), ux \cdot zi, \color{blue}{1} \cdot \left(\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot xi + \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot yi\right)\right) \]

13. Final simplification 99.1%

    \[\leadsto \mathsf{fma}\left(maxCos - maxCos \cdot ux, ux \cdot zi, \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot xi + \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot yi\right) \]
14. Add Preprocessing

Alternative 3: 98.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \left(uy \cdot \pi\right)\\ \left(xi \cdot \cos t\_0 + yi \cdot \sin t\_0\right) + zi \cdot \left(ux \cdot \left(maxCos \cdot \left(1 - ux\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* 2.0 (* uy PI))))
   (+
    (+ (* xi (cos t_0)) (* yi (sin t_0)))
    (* zi (* ux (* maxCos (- 1.0 ux)))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = 2.0f * (uy * ((float) M_PI));
	return ((xi * cosf(t_0)) + (yi * sinf(t_0))) + (zi * (ux * (maxCos * (1.0f - ux))));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(2.0) * Float32(uy * Float32(pi)))
	return Float32(Float32(Float32(xi * cos(t_0)) + Float32(yi * sin(t_0))) + Float32(zi * Float32(ux * Float32(maxCos * Float32(Float32(1.0) - ux)))))
end

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	t_0 = single(2.0) * (uy * single(pi));
	tmp = ((xi * cos(t_0)) + (yi * sin(t_0))) + (zi * (ux * (maxCos * (single(1.0) - ux))));
end

Derivation

1. Initial program 99.1%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
2. Add Preprocessing
3. Step-by-step derivation

   1. add-log-exp 98.0%

      \[\leadsto \left(\left(\color{blue}{\log \left(e^{\cos \left(\left(uy \cdot 2\right) \cdot \pi\right)}\right)} \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

4. Applied egg-rr 98.0%

   \[\leadsto \left(\left(\color{blue}{\log \left(e^{\cos \left(\left(uy \cdot 2\right) \cdot \pi\right)}\right)} \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

5. Taylor expanded in ux around 0 99.0%

   \[\leadsto \color{blue}{\left(xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

6. Final simplification 99.0%

   \[\leadsto \left(xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) + zi \cdot \left(ux \cdot \left(maxCos \cdot \left(1 - ux\right)\right)\right) \]
7. Add Preprocessing

Alternative 4: 95.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \left(uy \cdot \pi\right)\\ \left(xi \cdot \cos t\_0 + yi \cdot \sin t\_0\right) + maxCos \cdot \left(ux \cdot zi\right) \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* 2.0 (* uy PI))))
   (+ (+ (* xi (cos t_0)) (* yi (sin t_0))) (* maxCos (* ux zi)))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = 2.0f * (uy * ((float) M_PI));
	return ((xi * cosf(t_0)) + (yi * sinf(t_0))) + (maxCos * (ux * zi));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(2.0) * Float32(uy * Float32(pi)))
	return Float32(Float32(Float32(xi * cos(t_0)) + Float32(yi * sin(t_0))) + Float32(maxCos * Float32(ux * zi)))
end

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	t_0 = single(2.0) * (uy * single(pi));
	tmp = ((xi * cos(t_0)) + (yi * sin(t_0))) + (maxCos * (ux * zi));
end

Derivation

1. Initial program 99.1%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
2. Step-by-step derivation

   1. associate-+l+ 99.0%

      \[\leadsto \color{blue}{\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)} \]

   2. associate-*l* 99.0%

      \[\leadsto \color{blue}{\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \left(\sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)} \cdot xi\right)} + \left(\left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right) \]

   3. fma-define 99.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right), \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)} \cdot xi, \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)} \]

3. Simplified 99.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(uy \cdot \left(2 \cdot \pi\right)\right), \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)} \cdot xi, \mathsf{fma}\left(\sin \left(uy \cdot \left(2 \cdot \pi\right)\right), \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)} \cdot yi, \left(1 - ux\right) \cdot \left(\left(ux \cdot maxCos\right) \cdot zi\right)\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in maxCos around 0 99.1%

   \[\leadsto \mathsf{fma}\left(\cos \left(uy \cdot \left(2 \cdot \pi\right)\right), \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)} \cdot xi, \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}\right) \]
6. Step-by-step derivation

   1. expm1-log1p-u 99.0%

      \[\leadsto \mathsf{fma}\left(\cos \left(uy \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot \pi\right)\right)}\right), \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)} \cdot xi, maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]

   2. expm1-undefine 99.0%

      \[\leadsto \mathsf{fma}\left(\cos \left(uy \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(2 \cdot \pi\right)} - 1\right)}\right), \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)} \cdot xi, maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]

7. Applied egg-rr 99.0%

   \[\leadsto \mathsf{fma}\left(\cos \left(uy \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(2 \cdot \pi\right)} - 1\right)}\right), \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)} \cdot xi, maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]
8. Step-by-step derivation

   1. expm1-define 99.0%

      \[\leadsto \mathsf{fma}\left(\cos \left(uy \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot \pi\right)\right)}\right), \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)} \cdot xi, maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]

9. Simplified 99.0%

   \[\leadsto \mathsf{fma}\left(\cos \left(uy \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot \pi\right)\right)}\right), \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)} \cdot xi, maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]

10. Taylor expanded in ux around 0 95.9%

    \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot zi\right) + \left(xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)} \]

11. Final simplification 95.9%

    \[\leadsto \left(xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) + maxCos \cdot \left(ux \cdot zi\right) \]
12. Add Preprocessing

Alternative 5: 90.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \left(uy \cdot \pi\right)\\ xi \cdot \cos t\_0 + yi \cdot \sin t\_0 \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* 2.0 (* uy PI)))) (+ (* xi (cos t_0)) (* yi (sin t_0)))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = 2.0f * (uy * ((float) M_PI));
	return (xi * cosf(t_0)) + (yi * sinf(t_0));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(2.0) * Float32(uy * Float32(pi)))
	return Float32(Float32(xi * cos(t_0)) + Float32(yi * sin(t_0)))
end

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	t_0 = single(2.0) * (uy * single(pi));
	tmp = (xi * cos(t_0)) + (yi * sin(t_0));
end

Derivation

1. Initial program 99.1%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
2. Step-by-step derivation

   1. associate-+l+ 99.0%

      \[\leadsto \color{blue}{\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)} \]

   2. associate-*l* 99.0%

      \[\leadsto \color{blue}{\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \left(\sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)} \cdot xi\right)} + \left(\left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right) \]

   3. fma-define 99.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right), \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)} \cdot xi, \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)} \]

3. Simplified 99.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(uy \cdot \left(2 \cdot \pi\right)\right), \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)} \cdot xi, \mathsf{fma}\left(\sin \left(uy \cdot \left(2 \cdot \pi\right)\right), \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)} \cdot yi, \left(1 - ux\right) \cdot \left(\left(ux \cdot maxCos\right) \cdot zi\right)\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in maxCos around 0 99.1%

   \[\leadsto \mathsf{fma}\left(\cos \left(uy \cdot \left(2 \cdot \pi\right)\right), \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)} \cdot xi, \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}\right) \]
6. Step-by-step derivation

   1. expm1-log1p-u 99.0%

      \[\leadsto \mathsf{fma}\left(\cos \left(uy \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot \pi\right)\right)}\right), \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)} \cdot xi, maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]

   2. expm1-undefine 99.0%

      \[\leadsto \mathsf{fma}\left(\cos \left(uy \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(2 \cdot \pi\right)} - 1\right)}\right), \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)} \cdot xi, maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]

7. Applied egg-rr 99.0%

   \[\leadsto \mathsf{fma}\left(\cos \left(uy \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(2 \cdot \pi\right)} - 1\right)}\right), \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)} \cdot xi, maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]
8. Step-by-step derivation

   1. expm1-define 99.0%

      \[\leadsto \mathsf{fma}\left(\cos \left(uy \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot \pi\right)\right)}\right), \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)} \cdot xi, maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]

9. Simplified 99.0%

   \[\leadsto \mathsf{fma}\left(\cos \left(uy \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot \pi\right)\right)}\right), \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)} \cdot xi, maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]

10. Taylor expanded in ux around 0 91.7%

    \[\leadsto \color{blue}{xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)} \]
11. Add Preprocessing

Alternative 6: 66.6% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;xi \leq -1.0000000036274937 \cdot 10^{-15}:\\ \;\;\;\;xi + zi \cdot \left(ux \cdot \left(maxCos \cdot \left(1 - ux\right)\right)\right)\\ \mathbf{elif}\;xi \leq 4.1999998667151947 \cdot 10^{-20}:\\ \;\;\;\;yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right) + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;xi \cdot \sqrt{1 + \left(-1 + ux\right) \cdot \left(\left(maxCos \cdot ux\right) \cdot \left(\left(maxCos \cdot ux\right) \cdot \left(1 - ux\right)\right)\right)} + zi \cdot \left(ux \cdot \left(maxCos \cdot \left(-1 + \left(2 - ux\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (if (<= xi -1.0000000036274937e-15)
   (+ xi (* zi (* ux (* maxCos (- 1.0 ux)))))
   (if (<= xi 4.1999998667151947e-20)
     (+ (* yi (sin (* 2.0 (* uy PI)))) (* maxCos (* ux (* zi (- 1.0 ux)))))
     (+
      (*
       xi
       (sqrt
        (+
         1.0
         (* (+ -1.0 ux) (* (* maxCos ux) (* (* maxCos ux) (- 1.0 ux)))))))
      (* zi (* ux (* maxCos (+ -1.0 (- 2.0 ux)))))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float tmp;
	if (xi <= -1.0000000036274937e-15f) {
		tmp = xi + (zi * (ux * (maxCos * (1.0f - ux))));
	} else if (xi <= 4.1999998667151947e-20f) {
		tmp = (yi * sinf((2.0f * (uy * ((float) M_PI))))) + (maxCos * (ux * (zi * (1.0f - ux))));
	} else {
		tmp = (xi * sqrtf((1.0f + ((-1.0f + ux) * ((maxCos * ux) * ((maxCos * ux) * (1.0f - ux))))))) + (zi * (ux * (maxCos * (-1.0f + (2.0f - ux)))));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	tmp = Float32(0.0)
	if (xi <= Float32(-1.0000000036274937e-15))
		tmp = Float32(xi + Float32(zi * Float32(ux * Float32(maxCos * Float32(Float32(1.0) - ux)))));
	elseif (xi <= Float32(4.1999998667151947e-20))
		tmp = Float32(Float32(yi * sin(Float32(Float32(2.0) * Float32(uy * Float32(pi))))) + Float32(maxCos * Float32(ux * Float32(zi * Float32(Float32(1.0) - ux)))));
	else
		tmp = Float32(Float32(xi * sqrt(Float32(Float32(1.0) + Float32(Float32(Float32(-1.0) + ux) * Float32(Float32(maxCos * ux) * Float32(Float32(maxCos * ux) * Float32(Float32(1.0) - ux))))))) + Float32(zi * Float32(ux * Float32(maxCos * Float32(Float32(-1.0) + Float32(Float32(2.0) - ux))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(xi, yi, zi, ux, uy, maxCos)
	tmp = single(0.0);
	if (xi <= single(-1.0000000036274937e-15))
		tmp = xi + (zi * (ux * (maxCos * (single(1.0) - ux))));
	elseif (xi <= single(4.1999998667151947e-20))
		tmp = (yi * sin((single(2.0) * (uy * single(pi))))) + (maxCos * (ux * (zi * (single(1.0) - ux))));
	else
		tmp = (xi * sqrt((single(1.0) + ((single(-1.0) + ux) * ((maxCos * ux) * ((maxCos * ux) * (single(1.0) - ux))))))) + (zi * (ux * (maxCos * (single(-1.0) + (single(2.0) - ux)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if xi < -1e-15

   1. Initial program 99.4%

      \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-log-exp 97.8%

         \[\leadsto \left(\left(\color{blue}{\log \left(e^{\cos \left(\left(uy \cdot 2\right) \cdot \pi\right)}\right)} \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   4. Applied egg-rr 97.8%

      \[\leadsto \left(\left(\color{blue}{\log \left(e^{\cos \left(\left(uy \cdot 2\right) \cdot \pi\right)}\right)} \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   5. Taylor expanded in uy around 0 73.1%

      \[\leadsto \color{blue}{xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
   6. Step-by-step derivation

      1. *-commutative 73.1%

         \[\leadsto xi \cdot \sqrt{1 - \color{blue}{\left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right) \cdot {maxCos}^{2}}} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

      2. *-commutative 73.1%

         \[\leadsto xi \cdot \sqrt{1 - \color{blue}{{maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

      3. associate-*r* 73.1%

         \[\leadsto xi \cdot \sqrt{1 - \color{blue}{\left({maxCos}^{2} \cdot {ux}^{2}\right) \cdot {\left(1 - ux\right)}^{2}}} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

      4. unpow2 73.1%

         \[\leadsto xi \cdot \sqrt{1 - \left(\color{blue}{\left(maxCos \cdot maxCos\right)} \cdot {ux}^{2}\right) \cdot {\left(1 - ux\right)}^{2}} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

      5. unpow2 73.1%

         \[\leadsto xi \cdot \sqrt{1 - \left(\left(maxCos \cdot maxCos\right) \cdot \color{blue}{\left(ux \cdot ux\right)}\right) \cdot {\left(1 - ux\right)}^{2}} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

      6. swap-sqr 73.1%

         \[\leadsto xi \cdot \sqrt{1 - \color{blue}{\left(\left(maxCos \cdot ux\right) \cdot \left(maxCos \cdot ux\right)\right)} \cdot {\left(1 - ux\right)}^{2}} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

      7. unpow2 73.1%

         \[\leadsto xi \cdot \sqrt{1 - \left(\left(maxCos \cdot ux\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \color{blue}{\left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right)}} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

      8. swap-sqr 73.1%

         \[\leadsto xi \cdot \sqrt{1 - \color{blue}{\left(\left(maxCos \cdot ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(\left(maxCos \cdot ux\right) \cdot \left(1 - ux\right)\right)}} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

      9. unpow2 73.1%

         \[\leadsto xi \cdot \sqrt{1 - \color{blue}{{\left(\left(maxCos \cdot ux\right) \cdot \left(1 - ux\right)\right)}^{2}}} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   7. Simplified 73.1%

      \[\leadsto \color{blue}{xi \cdot \sqrt{1 - {\left(\left(maxCos \cdot ux\right) \cdot \left(1 - ux\right)\right)}^{2}}} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   8. Taylor expanded in maxCos around 0 73.1%

      \[\leadsto \color{blue}{xi} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   if -1e-15 < xi < 4.19999987e-20

   1. Initial program 98.7%

      \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
   2. Step-by-step derivation

      1. associate-+l+ 98.7%

         \[\leadsto \color{blue}{\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)} \]

      2. associate-*l* 98.7%

         \[\leadsto \color{blue}{\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \left(\sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)} \cdot xi\right)} + \left(\left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right) \]

      3. fma-define 98.7%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right), \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)} \cdot xi, \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)} \]

   3. Simplified 98.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(uy \cdot \left(2 \cdot \pi\right)\right), \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)} \cdot xi, \mathsf{fma}\left(\sin \left(uy \cdot \left(2 \cdot \pi\right)\right), \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)} \cdot yi, \left(1 - ux\right) \cdot \left(\left(ux \cdot maxCos\right) \cdot zi\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in maxCos around 0 98.8%

      \[\leadsto \mathsf{fma}\left(\cos \left(uy \cdot \left(2 \cdot \pi\right)\right), \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)} \cdot xi, \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}\right) \]
   6. Step-by-step derivation

      1. expm1-log1p-u 98.7%

         \[\leadsto \mathsf{fma}\left(\cos \left(uy \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot \pi\right)\right)}\right), \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)} \cdot xi, maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]

      2. expm1-undefine 98.7%

         \[\leadsto \mathsf{fma}\left(\cos \left(uy \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(2 \cdot \pi\right)} - 1\right)}\right), \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)} \cdot xi, maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]

   7. Applied egg-rr 98.7%

      \[\leadsto \mathsf{fma}\left(\cos \left(uy \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(2 \cdot \pi\right)} - 1\right)}\right), \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)} \cdot xi, maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]
   8. Step-by-step derivation

      1. expm1-define 98.7%

         \[\leadsto \mathsf{fma}\left(\cos \left(uy \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot \pi\right)\right)}\right), \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)} \cdot xi, maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]

   9. Simplified 98.7%

      \[\leadsto \mathsf{fma}\left(\cos \left(uy \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot \pi\right)\right)}\right), \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)} \cdot xi, maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]

   10. Taylor expanded in xi around 0 68.9%

       \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)} \]

   if 4.19999987e-20 < xi

   1. Initial program 99.4%

      \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-log-exp 97.6%

         \[\leadsto \left(\left(\color{blue}{\log \left(e^{\cos \left(\left(uy \cdot 2\right) \cdot \pi\right)}\right)} \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   4. Applied egg-rr 97.6%

      \[\leadsto \left(\left(\color{blue}{\log \left(e^{\cos \left(\left(uy \cdot 2\right) \cdot \pi\right)}\right)} \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   5. Taylor expanded in uy around 0 67.6%

      \[\leadsto \color{blue}{xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
   6. Step-by-step derivation

      1. *-commutative 67.6%

         \[\leadsto xi \cdot \sqrt{1 - \color{blue}{\left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right) \cdot {maxCos}^{2}}} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

      2. *-commutative 67.6%

         \[\leadsto xi \cdot \sqrt{1 - \color{blue}{{maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

      3. associate-*r* 67.6%

         \[\leadsto xi \cdot \sqrt{1 - \color{blue}{\left({maxCos}^{2} \cdot {ux}^{2}\right) \cdot {\left(1 - ux\right)}^{2}}} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

      4. unpow2 67.6%

         \[\leadsto xi \cdot \sqrt{1 - \left(\color{blue}{\left(maxCos \cdot maxCos\right)} \cdot {ux}^{2}\right) \cdot {\left(1 - ux\right)}^{2}} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

      5. unpow2 67.6%

         \[\leadsto xi \cdot \sqrt{1 - \left(\left(maxCos \cdot maxCos\right) \cdot \color{blue}{\left(ux \cdot ux\right)}\right) \cdot {\left(1 - ux\right)}^{2}} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

      6. swap-sqr 67.6%

         \[\leadsto xi \cdot \sqrt{1 - \color{blue}{\left(\left(maxCos \cdot ux\right) \cdot \left(maxCos \cdot ux\right)\right)} \cdot {\left(1 - ux\right)}^{2}} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

      7. unpow2 67.6%

         \[\leadsto xi \cdot \sqrt{1 - \left(\left(maxCos \cdot ux\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \color{blue}{\left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right)}} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

      8. swap-sqr 67.6%

         \[\leadsto xi \cdot \sqrt{1 - \color{blue}{\left(\left(maxCos \cdot ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(\left(maxCos \cdot ux\right) \cdot \left(1 - ux\right)\right)}} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

      9. unpow2 67.6%

         \[\leadsto xi \cdot \sqrt{1 - \color{blue}{{\left(\left(maxCos \cdot ux\right) \cdot \left(1 - ux\right)\right)}^{2}}} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   7. Simplified 67.6%

      \[\leadsto \color{blue}{xi \cdot \sqrt{1 - {\left(\left(maxCos \cdot ux\right) \cdot \left(1 - ux\right)\right)}^{2}}} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
   8. Step-by-step derivation

      1. expm1-log1p-u 67.6%

         \[\leadsto xi \cdot \sqrt{1 - {\left(\left(maxCos \cdot ux\right) \cdot \left(1 - ux\right)\right)}^{2}} + \left(\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - ux\right)\right)} \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   9. Applied egg-rr 67.6%

      \[\leadsto xi \cdot \sqrt{1 - {\left(\left(maxCos \cdot ux\right) \cdot \left(1 - ux\right)\right)}^{2}} + \left(\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - ux\right)\right)} \cdot maxCos\right) \cdot ux\right) \cdot zi \]
   10. Step-by-step derivation

       1. expm1-undefine 67.6%

          \[\leadsto xi \cdot \sqrt{1 - {\left(\left(maxCos \cdot ux\right) \cdot \left(1 - ux\right)\right)}^{2}} + \left(\left(\color{blue}{\left(e^{\mathsf{log1p}\left(1 - ux\right)} - 1\right)} \cdot maxCos\right) \cdot ux\right) \cdot zi \]

       2. sub-neg 67.6%

          \[\leadsto xi \cdot \sqrt{1 - {\left(\left(maxCos \cdot ux\right) \cdot \left(1 - ux\right)\right)}^{2}} + \left(\left(\color{blue}{\left(e^{\mathsf{log1p}\left(1 - ux\right)} + \left(-1\right)\right)} \cdot maxCos\right) \cdot ux\right) \cdot zi \]

       3. log1p-undefine 67.7%

          \[\leadsto xi \cdot \sqrt{1 - {\left(\left(maxCos \cdot ux\right) \cdot \left(1 - ux\right)\right)}^{2}} + \left(\left(\left(e^{\color{blue}{\log \left(1 + \left(1 - ux\right)\right)}} + \left(-1\right)\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

       4. rem-exp-log 67.7%

          \[\leadsto xi \cdot \sqrt{1 - {\left(\left(maxCos \cdot ux\right) \cdot \left(1 - ux\right)\right)}^{2}} + \left(\left(\left(\color{blue}{\left(1 + \left(1 - ux\right)\right)} + \left(-1\right)\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

       5. associate-+r- 67.7%

          \[\leadsto xi \cdot \sqrt{1 - {\left(\left(maxCos \cdot ux\right) \cdot \left(1 - ux\right)\right)}^{2}} + \left(\left(\left(\color{blue}{\left(\left(1 + 1\right) - ux\right)} + \left(-1\right)\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

       6. metadata-eval 67.7%

          \[\leadsto xi \cdot \sqrt{1 - {\left(\left(maxCos \cdot ux\right) \cdot \left(1 - ux\right)\right)}^{2}} + \left(\left(\left(\left(\color{blue}{2} - ux\right) + \left(-1\right)\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

       7. metadata-eval 67.7%

          \[\leadsto xi \cdot \sqrt{1 - {\left(\left(maxCos \cdot ux\right) \cdot \left(1 - ux\right)\right)}^{2}} + \left(\left(\left(\left(2 - ux\right) + \color{blue}{-1}\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   11. Simplified 67.7%

       \[\leadsto xi \cdot \sqrt{1 - {\left(\left(maxCos \cdot ux\right) \cdot \left(1 - ux\right)\right)}^{2}} + \left(\left(\color{blue}{\left(\left(2 - ux\right) + -1\right)} \cdot maxCos\right) \cdot ux\right) \cdot zi \]
   12. Step-by-step derivation

       1. *-commutative 67.7%

          \[\leadsto xi \cdot \sqrt{1 - {\left(\color{blue}{\left(ux \cdot maxCos\right)} \cdot \left(1 - ux\right)\right)}^{2}} + \left(\left(\left(\left(2 - ux\right) + -1\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

       2. *-commutative 67.7%

          \[\leadsto xi \cdot \sqrt{1 - {\color{blue}{\left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)}}^{2}} + \left(\left(\left(\left(2 - ux\right) + -1\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

       3. pow2 67.7%

          \[\leadsto xi \cdot \sqrt{1 - \color{blue}{\left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)}} + \left(\left(\left(\left(2 - ux\right) + -1\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

       4. associate-*l* 67.7%

          \[\leadsto xi \cdot \sqrt{1 - \color{blue}{\left(1 - ux\right) \cdot \left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)\right)}} + \left(\left(\left(\left(2 - ux\right) + -1\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

       5. *-commutative 67.7%

          \[\leadsto xi \cdot \sqrt{1 - \left(1 - ux\right) \cdot \left(\color{blue}{\left(maxCos \cdot ux\right)} \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)\right)} + \left(\left(\left(\left(2 - ux\right) + -1\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

       6. *-commutative 67.7%

          \[\leadsto xi \cdot \sqrt{1 - \left(1 - ux\right) \cdot \left(\left(maxCos \cdot ux\right) \cdot \color{blue}{\left(\left(ux \cdot maxCos\right) \cdot \left(1 - ux\right)\right)}\right)} + \left(\left(\left(\left(2 - ux\right) + -1\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

       7. *-commutative 67.7%

          \[\leadsto xi \cdot \sqrt{1 - \left(1 - ux\right) \cdot \left(\left(maxCos \cdot ux\right) \cdot \left(\color{blue}{\left(maxCos \cdot ux\right)} \cdot \left(1 - ux\right)\right)\right)} + \left(\left(\left(\left(2 - ux\right) + -1\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

       8. associate-*l* 67.7%

          \[\leadsto xi \cdot \sqrt{1 - \left(1 - ux\right) \cdot \left(\left(maxCos \cdot ux\right) \cdot \color{blue}{\left(maxCos \cdot \left(ux \cdot \left(1 - ux\right)\right)\right)}\right)} + \left(\left(\left(\left(2 - ux\right) + -1\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   13. Applied egg-rr 67.7%

       \[\leadsto xi \cdot \sqrt{1 - \color{blue}{\left(1 - ux\right) \cdot \left(\left(maxCos \cdot ux\right) \cdot \left(maxCos \cdot \left(ux \cdot \left(1 - ux\right)\right)\right)\right)}} + \left(\left(\left(\left(2 - ux\right) + -1\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
   14. Step-by-step derivation

       1. associate-*r* 67.7%

          \[\leadsto xi \cdot \sqrt{1 - \left(1 - ux\right) \cdot \left(\left(maxCos \cdot ux\right) \cdot \color{blue}{\left(\left(maxCos \cdot ux\right) \cdot \left(1 - ux\right)\right)}\right)} + \left(\left(\left(\left(2 - ux\right) + -1\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   15. Simplified 67.7%

       \[\leadsto xi \cdot \sqrt{1 - \color{blue}{\left(1 - ux\right) \cdot \left(\left(maxCos \cdot ux\right) \cdot \left(\left(maxCos \cdot ux\right) \cdot \left(1 - ux\right)\right)\right)}} + \left(\left(\left(\left(2 - ux\right) + -1\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;xi \leq -1.0000000036274937 \cdot 10^{-15}:\\ \;\;\;\;xi + zi \cdot \left(ux \cdot \left(maxCos \cdot \left(1 - ux\right)\right)\right)\\ \mathbf{elif}\;xi \leq 4.1999998667151947 \cdot 10^{-20}:\\ \;\;\;\;yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right) + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;xi \cdot \sqrt{1 + \left(-1 + ux\right) \cdot \left(\left(maxCos \cdot ux\right) \cdot \left(\left(maxCos \cdot ux\right) \cdot \left(1 - ux\right)\right)\right)} + zi \cdot \left(ux \cdot \left(maxCos \cdot \left(-1 + \left(2 - ux\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 66.6% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := zi \cdot \left(ux \cdot \left(maxCos \cdot \left(1 - ux\right)\right)\right)\\ \mathbf{if}\;xi \leq -1.0000000036274937 \cdot 10^{-15}:\\ \;\;\;\;xi + t\_0\\ \mathbf{elif}\;xi \leq 4.1999998667151947 \cdot 10^{-20}:\\ \;\;\;\;yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right) + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \left(xi + -0.5 \cdot \left(xi \cdot {\left(maxCos \cdot ux\right)}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* zi (* ux (* maxCos (- 1.0 ux))))))
   (if (<= xi -1.0000000036274937e-15)
     (+ xi t_0)
     (if (<= xi 4.1999998667151947e-20)
       (+ (* yi (sin (* 2.0 (* uy PI)))) (* maxCos (* ux (* zi (- 1.0 ux)))))
       (+ t_0 (+ xi (* -0.5 (* xi (pow (* maxCos ux) 2.0)))))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = zi * (ux * (maxCos * (1.0f - ux)));
	float tmp;
	if (xi <= -1.0000000036274937e-15f) {
		tmp = xi + t_0;
	} else if (xi <= 4.1999998667151947e-20f) {
		tmp = (yi * sinf((2.0f * (uy * ((float) M_PI))))) + (maxCos * (ux * (zi * (1.0f - ux))));
	} else {
		tmp = t_0 + (xi + (-0.5f * (xi * powf((maxCos * ux), 2.0f))));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(zi * Float32(ux * Float32(maxCos * Float32(Float32(1.0) - ux))))
	tmp = Float32(0.0)
	if (xi <= Float32(-1.0000000036274937e-15))
		tmp = Float32(xi + t_0);
	elseif (xi <= Float32(4.1999998667151947e-20))
		tmp = Float32(Float32(yi * sin(Float32(Float32(2.0) * Float32(uy * Float32(pi))))) + Float32(maxCos * Float32(ux * Float32(zi * Float32(Float32(1.0) - ux)))));
	else
		tmp = Float32(t_0 + Float32(xi + Float32(Float32(-0.5) * Float32(xi * (Float32(maxCos * ux) ^ Float32(2.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(xi, yi, zi, ux, uy, maxCos)
	t_0 = zi * (ux * (maxCos * (single(1.0) - ux)));
	tmp = single(0.0);
	if (xi <= single(-1.0000000036274937e-15))
		tmp = xi + t_0;
	elseif (xi <= single(4.1999998667151947e-20))
		tmp = (yi * sin((single(2.0) * (uy * single(pi))))) + (maxCos * (ux * (zi * (single(1.0) - ux))));
	else
		tmp = t_0 + (xi + (single(-0.5) * (xi * ((maxCos * ux) ^ single(2.0)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if xi < -1e-15

   1. Initial program 99.4%

      \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-log-exp 97.8%

         \[\leadsto \left(\left(\color{blue}{\log \left(e^{\cos \left(\left(uy \cdot 2\right) \cdot \pi\right)}\right)} \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   4. Applied egg-rr 97.8%

      \[\leadsto \left(\left(\color{blue}{\log \left(e^{\cos \left(\left(uy \cdot 2\right) \cdot \pi\right)}\right)} \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   5. Taylor expanded in uy around 0 73.1%

      \[\leadsto \color{blue}{xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
   6. Step-by-step derivation

      1. *-commutative 73.1%

         \[\leadsto xi \cdot \sqrt{1 - \color{blue}{\left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right) \cdot {maxCos}^{2}}} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

      2. *-commutative 73.1%

         \[\leadsto xi \cdot \sqrt{1 - \color{blue}{{maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

      3. associate-*r* 73.1%

         \[\leadsto xi \cdot \sqrt{1 - \color{blue}{\left({maxCos}^{2} \cdot {ux}^{2}\right) \cdot {\left(1 - ux\right)}^{2}}} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

      4. unpow2 73.1%

         \[\leadsto xi \cdot \sqrt{1 - \left(\color{blue}{\left(maxCos \cdot maxCos\right)} \cdot {ux}^{2}\right) \cdot {\left(1 - ux\right)}^{2}} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

      5. unpow2 73.1%

         \[\leadsto xi \cdot \sqrt{1 - \left(\left(maxCos \cdot maxCos\right) \cdot \color{blue}{\left(ux \cdot ux\right)}\right) \cdot {\left(1 - ux\right)}^{2}} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

      6. swap-sqr 73.1%

         \[\leadsto xi \cdot \sqrt{1 - \color{blue}{\left(\left(maxCos \cdot ux\right) \cdot \left(maxCos \cdot ux\right)\right)} \cdot {\left(1 - ux\right)}^{2}} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

      7. unpow2 73.1%

         \[\leadsto xi \cdot \sqrt{1 - \left(\left(maxCos \cdot ux\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \color{blue}{\left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right)}} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

      8. swap-sqr 73.1%

         \[\leadsto xi \cdot \sqrt{1 - \color{blue}{\left(\left(maxCos \cdot ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(\left(maxCos \cdot ux\right) \cdot \left(1 - ux\right)\right)}} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

      9. unpow2 73.1%

         \[\leadsto xi \cdot \sqrt{1 - \color{blue}{{\left(\left(maxCos \cdot ux\right) \cdot \left(1 - ux\right)\right)}^{2}}} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   7. Simplified 73.1%

      \[\leadsto \color{blue}{xi \cdot \sqrt{1 - {\left(\left(maxCos \cdot ux\right) \cdot \left(1 - ux\right)\right)}^{2}}} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   8. Taylor expanded in maxCos around 0 73.1%

      \[\leadsto \color{blue}{xi} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   if -1e-15 < xi < 4.19999987e-20

   1. Initial program 98.7%

      \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
   2. Step-by-step derivation

      1. associate-+l+ 98.7%

         \[\leadsto \color{blue}{\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)} \]

      2. associate-*l* 98.7%

         \[\leadsto \color{blue}{\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \left(\sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)} \cdot xi\right)} + \left(\left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right) \]

      3. fma-define 98.7%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right), \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)} \cdot xi, \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)} \]

   3. Simplified 98.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(uy \cdot \left(2 \cdot \pi\right)\right), \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)} \cdot xi, \mathsf{fma}\left(\sin \left(uy \cdot \left(2 \cdot \pi\right)\right), \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)} \cdot yi, \left(1 - ux\right) \cdot \left(\left(ux \cdot maxCos\right) \cdot zi\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in maxCos around 0 98.8%

      \[\leadsto \mathsf{fma}\left(\cos \left(uy \cdot \left(2 \cdot \pi\right)\right), \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)} \cdot xi, \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}\right) \]
   6. Step-by-step derivation

      1. expm1-log1p-u 98.7%

         \[\leadsto \mathsf{fma}\left(\cos \left(uy \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot \pi\right)\right)}\right), \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)} \cdot xi, maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]

      2. expm1-undefine 98.7%

         \[\leadsto \mathsf{fma}\left(\cos \left(uy \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(2 \cdot \pi\right)} - 1\right)}\right), \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)} \cdot xi, maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]

   7. Applied egg-rr 98.7%

      \[\leadsto \mathsf{fma}\left(\cos \left(uy \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(2 \cdot \pi\right)} - 1\right)}\right), \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)} \cdot xi, maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]
   8. Step-by-step derivation

      1. expm1-define 98.7%

         \[\leadsto \mathsf{fma}\left(\cos \left(uy \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot \pi\right)\right)}\right), \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)} \cdot xi, maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]

   9. Simplified 98.7%

      \[\leadsto \mathsf{fma}\left(\cos \left(uy \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot \pi\right)\right)}\right), \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)} \cdot xi, maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]

   10. Taylor expanded in xi around 0 68.9%

       \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)} \]

   if 4.19999987e-20 < xi

   1. Initial program 99.4%

      \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-log-exp 97.6%

         \[\leadsto \left(\left(\color{blue}{\log \left(e^{\cos \left(\left(uy \cdot 2\right) \cdot \pi\right)}\right)} \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   4. Applied egg-rr 97.6%

      \[\leadsto \left(\left(\color{blue}{\log \left(e^{\cos \left(\left(uy \cdot 2\right) \cdot \pi\right)}\right)} \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   5. Taylor expanded in uy around 0 67.6%

      \[\leadsto \color{blue}{xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
   6. Step-by-step derivation

      1. *-commutative 67.6%

         \[\leadsto xi \cdot \sqrt{1 - \color{blue}{\left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right) \cdot {maxCos}^{2}}} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

      2. *-commutative 67.6%

         \[\leadsto xi \cdot \sqrt{1 - \color{blue}{{maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

      3. associate-*r* 67.6%

         \[\leadsto xi \cdot \sqrt{1 - \color{blue}{\left({maxCos}^{2} \cdot {ux}^{2}\right) \cdot {\left(1 - ux\right)}^{2}}} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

      4. unpow2 67.6%

         \[\leadsto xi \cdot \sqrt{1 - \left(\color{blue}{\left(maxCos \cdot maxCos\right)} \cdot {ux}^{2}\right) \cdot {\left(1 - ux\right)}^{2}} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

      5. unpow2 67.6%

         \[\leadsto xi \cdot \sqrt{1 - \left(\left(maxCos \cdot maxCos\right) \cdot \color{blue}{\left(ux \cdot ux\right)}\right) \cdot {\left(1 - ux\right)}^{2}} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

      6. swap-sqr 67.6%

         \[\leadsto xi \cdot \sqrt{1 - \color{blue}{\left(\left(maxCos \cdot ux\right) \cdot \left(maxCos \cdot ux\right)\right)} \cdot {\left(1 - ux\right)}^{2}} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

      7. unpow2 67.6%

         \[\leadsto xi \cdot \sqrt{1 - \left(\left(maxCos \cdot ux\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \color{blue}{\left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right)}} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

      8. swap-sqr 67.6%

         \[\leadsto xi \cdot \sqrt{1 - \color{blue}{\left(\left(maxCos \cdot ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(\left(maxCos \cdot ux\right) \cdot \left(1 - ux\right)\right)}} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

      9. unpow2 67.6%

         \[\leadsto xi \cdot \sqrt{1 - \color{blue}{{\left(\left(maxCos \cdot ux\right) \cdot \left(1 - ux\right)\right)}^{2}}} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   7. Simplified 67.6%

      \[\leadsto \color{blue}{xi \cdot \sqrt{1 - {\left(\left(maxCos \cdot ux\right) \cdot \left(1 - ux\right)\right)}^{2}}} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   8. Taylor expanded in ux around 0 67.6%

      \[\leadsto \color{blue}{\left(xi + -0.5 \cdot \left({maxCos}^{2} \cdot \left({ux}^{2} \cdot xi\right)\right)\right)} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
   9. Step-by-step derivation

      1. associate-*r* 67.6%

         \[\leadsto \left(xi + -0.5 \cdot \color{blue}{\left(\left({maxCos}^{2} \cdot {ux}^{2}\right) \cdot xi\right)}\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

      2. unpow2 67.6%

         \[\leadsto \left(xi + -0.5 \cdot \left(\left(\color{blue}{\left(maxCos \cdot maxCos\right)} \cdot {ux}^{2}\right) \cdot xi\right)\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

      3. unpow2 67.6%

         \[\leadsto \left(xi + -0.5 \cdot \left(\left(\left(maxCos \cdot maxCos\right) \cdot \color{blue}{\left(ux \cdot ux\right)}\right) \cdot xi\right)\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

      4. swap-sqr 67.6%

         \[\leadsto \left(xi + -0.5 \cdot \left(\color{blue}{\left(\left(maxCos \cdot ux\right) \cdot \left(maxCos \cdot ux\right)\right)} \cdot xi\right)\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

      5. unpow2 67.6%

         \[\leadsto \left(xi + -0.5 \cdot \left(\color{blue}{{\left(maxCos \cdot ux\right)}^{2}} \cdot xi\right)\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   10. Simplified 67.6%

       \[\leadsto \color{blue}{\left(xi + -0.5 \cdot \left({\left(maxCos \cdot ux\right)}^{2} \cdot xi\right)\right)} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;xi \leq -1.0000000036274937 \cdot 10^{-15}:\\ \;\;\;\;xi + zi \cdot \left(ux \cdot \left(maxCos \cdot \left(1 - ux\right)\right)\right)\\ \mathbf{elif}\;xi \leq 4.1999998667151947 \cdot 10^{-20}:\\ \;\;\;\;yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right) + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;zi \cdot \left(ux \cdot \left(maxCos \cdot \left(1 - ux\right)\right)\right) + \left(xi + -0.5 \cdot \left(xi \cdot {\left(maxCos \cdot ux\right)}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 66.6% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;xi \leq -1.0000000036274937 \cdot 10^{-15} \lor \neg \left(xi \leq 4.1999998667151947 \cdot 10^{-20}\right):\\ \;\;\;\;xi + zi \cdot \left(ux \cdot \left(maxCos \cdot \left(1 - ux\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right) + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (if (or (<= xi -1.0000000036274937e-15) (not (<= xi 4.1999998667151947e-20)))
   (+ xi (* zi (* ux (* maxCos (- 1.0 ux)))))
   (+ (* yi (sin (* 2.0 (* uy PI)))) (* maxCos (* ux (* zi (- 1.0 ux)))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float tmp;
	if ((xi <= -1.0000000036274937e-15f) || !(xi <= 4.1999998667151947e-20f)) {
		tmp = xi + (zi * (ux * (maxCos * (1.0f - ux))));
	} else {
		tmp = (yi * sinf((2.0f * (uy * ((float) M_PI))))) + (maxCos * (ux * (zi * (1.0f - ux))));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	tmp = Float32(0.0)
	if ((xi <= Float32(-1.0000000036274937e-15)) || !(xi <= Float32(4.1999998667151947e-20)))
		tmp = Float32(xi + Float32(zi * Float32(ux * Float32(maxCos * Float32(Float32(1.0) - ux)))));
	else
		tmp = Float32(Float32(yi * sin(Float32(Float32(2.0) * Float32(uy * Float32(pi))))) + Float32(maxCos * Float32(ux * Float32(zi * Float32(Float32(1.0) - ux)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(xi, yi, zi, ux, uy, maxCos)
	tmp = single(0.0);
	if ((xi <= single(-1.0000000036274937e-15)) || ~((xi <= single(4.1999998667151947e-20))))
		tmp = xi + (zi * (ux * (maxCos * (single(1.0) - ux))));
	else
		tmp = (yi * sin((single(2.0) * (uy * single(pi))))) + (maxCos * (ux * (zi * (single(1.0) - ux))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if xi < -1e-15 or 4.19999987e-20 < xi

   1. Initial program 99.4%

      \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-log-exp 97.7%

         \[\leadsto \left(\left(\color{blue}{\log \left(e^{\cos \left(\left(uy \cdot 2\right) \cdot \pi\right)}\right)} \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   4. Applied egg-rr 97.7%

      \[\leadsto \left(\left(\color{blue}{\log \left(e^{\cos \left(\left(uy \cdot 2\right) \cdot \pi\right)}\right)} \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   5. Taylor expanded in uy around 0 70.6%

      \[\leadsto \color{blue}{xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
   6. Step-by-step derivation

      1. *-commutative 70.6%

         \[\leadsto xi \cdot \sqrt{1 - \color{blue}{\left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right) \cdot {maxCos}^{2}}} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

      2. *-commutative 70.6%

         \[\leadsto xi \cdot \sqrt{1 - \color{blue}{{maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

      3. associate-*r* 70.6%

         \[\leadsto xi \cdot \sqrt{1 - \color{blue}{\left({maxCos}^{2} \cdot {ux}^{2}\right) \cdot {\left(1 - ux\right)}^{2}}} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

      4. unpow2 70.6%

         \[\leadsto xi \cdot \sqrt{1 - \left(\color{blue}{\left(maxCos \cdot maxCos\right)} \cdot {ux}^{2}\right) \cdot {\left(1 - ux\right)}^{2}} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

      5. unpow2 70.6%

         \[\leadsto xi \cdot \sqrt{1 - \left(\left(maxCos \cdot maxCos\right) \cdot \color{blue}{\left(ux \cdot ux\right)}\right) \cdot {\left(1 - ux\right)}^{2}} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

      6. swap-sqr 70.6%

         \[\leadsto xi \cdot \sqrt{1 - \color{blue}{\left(\left(maxCos \cdot ux\right) \cdot \left(maxCos \cdot ux\right)\right)} \cdot {\left(1 - ux\right)}^{2}} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

      7. unpow2 70.6%

         \[\leadsto xi \cdot \sqrt{1 - \left(\left(maxCos \cdot ux\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \color{blue}{\left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right)}} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

      8. swap-sqr 70.6%

         \[\leadsto xi \cdot \sqrt{1 - \color{blue}{\left(\left(maxCos \cdot ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(\left(maxCos \cdot ux\right) \cdot \left(1 - ux\right)\right)}} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

      9. unpow2 70.6%

         \[\leadsto xi \cdot \sqrt{1 - \color{blue}{{\left(\left(maxCos \cdot ux\right) \cdot \left(1 - ux\right)\right)}^{2}}} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   7. Simplified 70.6%

      \[\leadsto \color{blue}{xi \cdot \sqrt{1 - {\left(\left(maxCos \cdot ux\right) \cdot \left(1 - ux\right)\right)}^{2}}} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   8. Taylor expanded in maxCos around 0 70.6%

      \[\leadsto \color{blue}{xi} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   if -1e-15 < xi < 4.19999987e-20

   1. Initial program 98.7%

      \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
   2. Step-by-step derivation

      1. associate-+l+ 98.7%

         \[\leadsto \color{blue}{\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)} \]

      2. associate-*l* 98.7%

         \[\leadsto \color{blue}{\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \left(\sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)} \cdot xi\right)} + \left(\left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right) \]

      3. fma-define 98.7%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right), \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)} \cdot xi, \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)} \]

   3. Simplified 98.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(uy \cdot \left(2 \cdot \pi\right)\right), \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)} \cdot xi, \mathsf{fma}\left(\sin \left(uy \cdot \left(2 \cdot \pi\right)\right), \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)} \cdot yi, \left(1 - ux\right) \cdot \left(\left(ux \cdot maxCos\right) \cdot zi\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in maxCos around 0 98.8%

      \[\leadsto \mathsf{fma}\left(\cos \left(uy \cdot \left(2 \cdot \pi\right)\right), \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)} \cdot xi, \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}\right) \]
   6. Step-by-step derivation

      1. expm1-log1p-u 98.7%

         \[\leadsto \mathsf{fma}\left(\cos \left(uy \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot \pi\right)\right)}\right), \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)} \cdot xi, maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]

      2. expm1-undefine 98.7%

         \[\leadsto \mathsf{fma}\left(\cos \left(uy \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(2 \cdot \pi\right)} - 1\right)}\right), \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)} \cdot xi, maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]

   7. Applied egg-rr 98.7%

      \[\leadsto \mathsf{fma}\left(\cos \left(uy \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(2 \cdot \pi\right)} - 1\right)}\right), \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)} \cdot xi, maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]
   8. Step-by-step derivation

      1. expm1-define 98.7%

         \[\leadsto \mathsf{fma}\left(\cos \left(uy \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot \pi\right)\right)}\right), \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)} \cdot xi, maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]

   9. Simplified 98.7%

      \[\leadsto \mathsf{fma}\left(\cos \left(uy \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot \pi\right)\right)}\right), \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)} \cdot xi, maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]

   10. Taylor expanded in xi around 0 68.9%

       \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;xi \leq -1.0000000036274937 \cdot 10^{-15} \lor \neg \left(xi \leq 4.1999998667151947 \cdot 10^{-20}\right):\\ \;\;\;\;xi + zi \cdot \left(ux \cdot \left(maxCos \cdot \left(1 - ux\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right) + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 65.0% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;yi \leq -9.000000180878209 \cdot 10^{-13} \lor \neg \left(yi \leq 2.00000009162741 \cdot 10^{-18}\right):\\ \;\;\;\;yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;xi + zi \cdot \left(ux \cdot \left(maxCos \cdot \left(1 - ux\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (if (or (<= yi -9.000000180878209e-13) (not (<= yi 2.00000009162741e-18)))
   (* yi (sin (* 2.0 (* uy PI))))
   (+ xi (* zi (* ux (* maxCos (- 1.0 ux)))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float tmp;
	if ((yi <= -9.000000180878209e-13f) || !(yi <= 2.00000009162741e-18f)) {
		tmp = yi * sinf((2.0f * (uy * ((float) M_PI))));
	} else {
		tmp = xi + (zi * (ux * (maxCos * (1.0f - ux))));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	tmp = Float32(0.0)
	if ((yi <= Float32(-9.000000180878209e-13)) || !(yi <= Float32(2.00000009162741e-18)))
		tmp = Float32(yi * sin(Float32(Float32(2.0) * Float32(uy * Float32(pi)))));
	else
		tmp = Float32(xi + Float32(zi * Float32(ux * Float32(maxCos * Float32(Float32(1.0) - ux)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(xi, yi, zi, ux, uy, maxCos)
	tmp = single(0.0);
	if ((yi <= single(-9.000000180878209e-13)) || ~((yi <= single(2.00000009162741e-18))))
		tmp = yi * sin((single(2.0) * (uy * single(pi))));
	else
		tmp = xi + (zi * (ux * (maxCos * (single(1.0) - ux))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if yi < -9.00000018e-13 or 2.00000009e-18 < yi

   1. Initial program 98.7%

      \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
   2. Step-by-step derivation

      1. associate-+l+ 98.7%

         \[\leadsto \color{blue}{\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)} \]

      2. associate-*l* 98.7%

         \[\leadsto \color{blue}{\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \left(\sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)} \cdot xi\right)} + \left(\left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right) \]

      3. fma-define 98.7%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right), \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)} \cdot xi, \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)} \]

   3. Simplified 98.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(uy \cdot \left(2 \cdot \pi\right)\right), \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)} \cdot xi, \mathsf{fma}\left(\sin \left(uy \cdot \left(2 \cdot \pi\right)\right), \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)} \cdot yi, \left(1 - ux\right) \cdot \left(\left(ux \cdot maxCos\right) \cdot zi\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in maxCos around 0 98.7%

      \[\leadsto \mathsf{fma}\left(\cos \left(uy \cdot \left(2 \cdot \pi\right)\right), \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)} \cdot xi, \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}\right) \]
   6. Step-by-step derivation

      1. expm1-log1p-u 98.7%

         \[\leadsto \mathsf{fma}\left(\cos \left(uy \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot \pi\right)\right)}\right), \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)} \cdot xi, maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]

      2. expm1-undefine 98.7%

         \[\leadsto \mathsf{fma}\left(\cos \left(uy \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(2 \cdot \pi\right)} - 1\right)}\right), \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)} \cdot xi, maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]

   7. Applied egg-rr 98.7%

      \[\leadsto \mathsf{fma}\left(\cos \left(uy \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(2 \cdot \pi\right)} - 1\right)}\right), \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)} \cdot xi, maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]
   8. Step-by-step derivation

      1. expm1-define 98.7%

         \[\leadsto \mathsf{fma}\left(\cos \left(uy \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot \pi\right)\right)}\right), \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)} \cdot xi, maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]

   9. Simplified 98.7%

      \[\leadsto \mathsf{fma}\left(\cos \left(uy \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot \pi\right)\right)}\right), \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)} \cdot xi, maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]

   10. Taylor expanded in yi around inf 63.9%

       \[\leadsto \color{blue}{yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)} \]

   if -9.00000018e-13 < yi < 2.00000009e-18

   1. Initial program 99.3%

      \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-log-exp 97.8%

         \[\leadsto \left(\left(\color{blue}{\log \left(e^{\cos \left(\left(uy \cdot 2\right) \cdot \pi\right)}\right)} \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   4. Applied egg-rr 97.8%

      \[\leadsto \left(\left(\color{blue}{\log \left(e^{\cos \left(\left(uy \cdot 2\right) \cdot \pi\right)}\right)} \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   5. Taylor expanded in uy around 0 70.1%

      \[\leadsto \color{blue}{xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
   6. Step-by-step derivation

      1. *-commutative 70.1%

         \[\leadsto xi \cdot \sqrt{1 - \color{blue}{\left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right) \cdot {maxCos}^{2}}} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

      2. *-commutative 70.1%

         \[\leadsto xi \cdot \sqrt{1 - \color{blue}{{maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

      3. associate-*r* 70.1%

         \[\leadsto xi \cdot \sqrt{1 - \color{blue}{\left({maxCos}^{2} \cdot {ux}^{2}\right) \cdot {\left(1 - ux\right)}^{2}}} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

      4. unpow2 70.1%

         \[\leadsto xi \cdot \sqrt{1 - \left(\color{blue}{\left(maxCos \cdot maxCos\right)} \cdot {ux}^{2}\right) \cdot {\left(1 - ux\right)}^{2}} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

      5. unpow2 70.1%

         \[\leadsto xi \cdot \sqrt{1 - \left(\left(maxCos \cdot maxCos\right) \cdot \color{blue}{\left(ux \cdot ux\right)}\right) \cdot {\left(1 - ux\right)}^{2}} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

      6. swap-sqr 70.1%

         \[\leadsto xi \cdot \sqrt{1 - \color{blue}{\left(\left(maxCos \cdot ux\right) \cdot \left(maxCos \cdot ux\right)\right)} \cdot {\left(1 - ux\right)}^{2}} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

      7. unpow2 70.1%

         \[\leadsto xi \cdot \sqrt{1 - \left(\left(maxCos \cdot ux\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \color{blue}{\left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right)}} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

      8. swap-sqr 70.1%

         \[\leadsto xi \cdot \sqrt{1 - \color{blue}{\left(\left(maxCos \cdot ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(\left(maxCos \cdot ux\right) \cdot \left(1 - ux\right)\right)}} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

      9. unpow2 70.1%

         \[\leadsto xi \cdot \sqrt{1 - \color{blue}{{\left(\left(maxCos \cdot ux\right) \cdot \left(1 - ux\right)\right)}^{2}}} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   7. Simplified 70.1%

      \[\leadsto \color{blue}{xi \cdot \sqrt{1 - {\left(\left(maxCos \cdot ux\right) \cdot \left(1 - ux\right)\right)}^{2}}} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   8. Taylor expanded in maxCos around 0 70.1%

      \[\leadsto \color{blue}{xi} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;yi \leq -9.000000180878209 \cdot 10^{-13} \lor \neg \left(yi \leq 2.00000009162741 \cdot 10^{-18}\right):\\ \;\;\;\;yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;xi + zi \cdot \left(ux \cdot \left(maxCos \cdot \left(1 - ux\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 51.5% accurate, 41.9× speedup

Math

\[\begin{array}{l} \\ xi + zi \cdot \left(ux \cdot \left(maxCos \cdot \left(1 - ux\right)\right)\right) \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (+ xi (* zi (* ux (* maxCos (- 1.0 ux))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return xi + (zi * (ux * (maxCos * (1.0f - ux))));
}

Fortran

real(4) function code(xi, yi, zi, ux, uy, maxcos)
    real(4), intent (in) :: xi
    real(4), intent (in) :: yi
    real(4), intent (in) :: zi
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = xi + (zi * (ux * (maxcos * (1.0e0 - ux))))
end function

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return Float32(xi + Float32(zi * Float32(ux * Float32(maxCos * Float32(Float32(1.0) - ux)))))
end

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	tmp = xi + (zi * (ux * (maxCos * (single(1.0) - ux))));
end

Derivation

1. Initial program 99.1%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
2. Add Preprocessing
3. Step-by-step derivation

   1. add-log-exp 98.0%

      \[\leadsto \left(\left(\color{blue}{\log \left(e^{\cos \left(\left(uy \cdot 2\right) \cdot \pi\right)}\right)} \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

4. Applied egg-rr 98.0%

   \[\leadsto \left(\left(\color{blue}{\log \left(e^{\cos \left(\left(uy \cdot 2\right) \cdot \pi\right)}\right)} \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

5. Taylor expanded in uy around 0 51.5%

   \[\leadsto \color{blue}{xi \cdot \sqrt{1 - {maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
6. Step-by-step derivation

   1. *-commutative 51.5%

      \[\leadsto xi \cdot \sqrt{1 - \color{blue}{\left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right) \cdot {maxCos}^{2}}} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   2. *-commutative 51.5%

      \[\leadsto xi \cdot \sqrt{1 - \color{blue}{{maxCos}^{2} \cdot \left({ux}^{2} \cdot {\left(1 - ux\right)}^{2}\right)}} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   3. associate-*r* 51.5%

      \[\leadsto xi \cdot \sqrt{1 - \color{blue}{\left({maxCos}^{2} \cdot {ux}^{2}\right) \cdot {\left(1 - ux\right)}^{2}}} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   4. unpow2 51.5%

      \[\leadsto xi \cdot \sqrt{1 - \left(\color{blue}{\left(maxCos \cdot maxCos\right)} \cdot {ux}^{2}\right) \cdot {\left(1 - ux\right)}^{2}} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   5. unpow2 51.5%

      \[\leadsto xi \cdot \sqrt{1 - \left(\left(maxCos \cdot maxCos\right) \cdot \color{blue}{\left(ux \cdot ux\right)}\right) \cdot {\left(1 - ux\right)}^{2}} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   6. swap-sqr 51.5%

      \[\leadsto xi \cdot \sqrt{1 - \color{blue}{\left(\left(maxCos \cdot ux\right) \cdot \left(maxCos \cdot ux\right)\right)} \cdot {\left(1 - ux\right)}^{2}} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   7. unpow2 51.5%

      \[\leadsto xi \cdot \sqrt{1 - \left(\left(maxCos \cdot ux\right) \cdot \left(maxCos \cdot ux\right)\right) \cdot \color{blue}{\left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right)}} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   8. swap-sqr 51.5%

      \[\leadsto xi \cdot \sqrt{1 - \color{blue}{\left(\left(maxCos \cdot ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(\left(maxCos \cdot ux\right) \cdot \left(1 - ux\right)\right)}} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   9. unpow2 51.5%

      \[\leadsto xi \cdot \sqrt{1 - \color{blue}{{\left(\left(maxCos \cdot ux\right) \cdot \left(1 - ux\right)\right)}^{2}}} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

7. Simplified 51.5%

   \[\leadsto \color{blue}{xi \cdot \sqrt{1 - {\left(\left(maxCos \cdot ux\right) \cdot \left(1 - ux\right)\right)}^{2}}} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

8. Taylor expanded in maxCos around 0 51.5%

   \[\leadsto \color{blue}{xi} + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

9. Final simplification 51.5%

   \[\leadsto xi + zi \cdot \left(ux \cdot \left(maxCos \cdot \left(1 - ux\right)\right)\right) \]
10. Add Preprocessing

Alternative 11: 13.5% accurate, 41.9× speedup

Math

\[\begin{array}{l} \\ ux \cdot \left(maxCos \cdot zi - maxCos \cdot \left(ux \cdot zi\right)\right) \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (* ux (- (* maxCos zi) (* maxCos (* ux zi)))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return ux * ((maxCos * zi) - (maxCos * (ux * zi)));
}

Fortran

real(4) function code(xi, yi, zi, ux, uy, maxcos)
    real(4), intent (in) :: xi
    real(4), intent (in) :: yi
    real(4), intent (in) :: zi
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = ux * ((maxcos * zi) - (maxcos * (ux * zi)))
end function

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return Float32(ux * Float32(Float32(maxCos * zi) - Float32(maxCos * Float32(ux * zi))))
end

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	tmp = ux * ((maxCos * zi) - (maxCos * (ux * zi)));
end

Derivation

1. Initial program 99.1%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
2. Step-by-step derivation

   1. associate-+l+ 99.0%

      \[\leadsto \color{blue}{\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)} \]

   2. associate-*l* 99.0%

      \[\leadsto \color{blue}{\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \left(\sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)} \cdot xi\right)} + \left(\left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right) \]

   3. fma-define 99.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right), \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)} \cdot xi, \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)} \]

3. Simplified 99.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(uy \cdot \left(2 \cdot \pi\right)\right), \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)} \cdot xi, \mathsf{fma}\left(\sin \left(uy \cdot \left(2 \cdot \pi\right)\right), \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)} \cdot yi, \left(1 - ux\right) \cdot \left(\left(ux \cdot maxCos\right) \cdot zi\right)\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in maxCos around 0 99.1%

   \[\leadsto \mathsf{fma}\left(\cos \left(uy \cdot \left(2 \cdot \pi\right)\right), \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)} \cdot xi, \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}\right) \]
6. Step-by-step derivation

   1. expm1-log1p-u 99.0%

      \[\leadsto \mathsf{fma}\left(\cos \left(uy \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot \pi\right)\right)}\right), \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)} \cdot xi, maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]

   2. expm1-undefine 99.0%

      \[\leadsto \mathsf{fma}\left(\cos \left(uy \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(2 \cdot \pi\right)} - 1\right)}\right), \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)} \cdot xi, maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]

7. Applied egg-rr 99.0%

   \[\leadsto \mathsf{fma}\left(\cos \left(uy \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(2 \cdot \pi\right)} - 1\right)}\right), \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)} \cdot xi, maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]
8. Step-by-step derivation

   1. expm1-define 99.0%

      \[\leadsto \mathsf{fma}\left(\cos \left(uy \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot \pi\right)\right)}\right), \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)} \cdot xi, maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]

9. Simplified 99.0%

   \[\leadsto \mathsf{fma}\left(\cos \left(uy \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot \pi\right)\right)}\right), \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)} \cdot xi, maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]

10. Taylor expanded in zi around inf 12.4%

    \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)} \]

11. Taylor expanded in ux around 0 12.5%

    \[\leadsto \color{blue}{ux \cdot \left(-1 \cdot \left(maxCos \cdot \left(ux \cdot zi\right)\right) + maxCos \cdot zi\right)} \]
12. Step-by-step derivation

    1. +-commutative 12.5%

       \[\leadsto ux \cdot \color{blue}{\left(maxCos \cdot zi + -1 \cdot \left(maxCos \cdot \left(ux \cdot zi\right)\right)\right)} \]

    2. mul-1-neg 12.5%

       \[\leadsto ux \cdot \left(maxCos \cdot zi + \color{blue}{\left(-maxCos \cdot \left(ux \cdot zi\right)\right)}\right) \]

    3. unsub-neg 12.5%

       \[\leadsto ux \cdot \color{blue}{\left(maxCos \cdot zi - maxCos \cdot \left(ux \cdot zi\right)\right)} \]

13. Simplified 12.5%

    \[\leadsto \color{blue}{ux \cdot \left(maxCos \cdot zi - maxCos \cdot \left(ux \cdot zi\right)\right)} \]
14. Add Preprocessing

Alternative 12: 13.5% accurate, 51.2× speedup

Math

\[\begin{array}{l} \\ maxCos \cdot \left(ux \cdot \left(zi - ux \cdot zi\right)\right) \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (* maxCos (* ux (- zi (* ux zi)))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return maxCos * (ux * (zi - (ux * zi)));
}

Fortran

real(4) function code(xi, yi, zi, ux, uy, maxcos)
    real(4), intent (in) :: xi
    real(4), intent (in) :: yi
    real(4), intent (in) :: zi
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = maxcos * (ux * (zi - (ux * zi)))
end function

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return Float32(maxCos * Float32(ux * Float32(zi - Float32(ux * zi))))
end

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	tmp = maxCos * (ux * (zi - (ux * zi)));
end

Derivation

1. Initial program 99.1%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
2. Step-by-step derivation

   1. associate-+l+ 99.0%

      \[\leadsto \color{blue}{\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)} \]

   2. associate-*l* 99.0%

      \[\leadsto \color{blue}{\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \left(\sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)} \cdot xi\right)} + \left(\left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right) \]

   3. fma-define 99.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right), \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)} \cdot xi, \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)} \]

3. Simplified 99.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(uy \cdot \left(2 \cdot \pi\right)\right), \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)} \cdot xi, \mathsf{fma}\left(\sin \left(uy \cdot \left(2 \cdot \pi\right)\right), \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)} \cdot yi, \left(1 - ux\right) \cdot \left(\left(ux \cdot maxCos\right) \cdot zi\right)\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in maxCos around 0 99.1%

   \[\leadsto \mathsf{fma}\left(\cos \left(uy \cdot \left(2 \cdot \pi\right)\right), \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)} \cdot xi, \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}\right) \]
6. Step-by-step derivation

   1. expm1-log1p-u 99.0%

      \[\leadsto \mathsf{fma}\left(\cos \left(uy \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot \pi\right)\right)}\right), \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)} \cdot xi, maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]

   2. expm1-undefine 99.0%

      \[\leadsto \mathsf{fma}\left(\cos \left(uy \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(2 \cdot \pi\right)} - 1\right)}\right), \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)} \cdot xi, maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]

7. Applied egg-rr 99.0%

   \[\leadsto \mathsf{fma}\left(\cos \left(uy \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(2 \cdot \pi\right)} - 1\right)}\right), \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)} \cdot xi, maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]
8. Step-by-step derivation

   1. expm1-define 99.0%

      \[\leadsto \mathsf{fma}\left(\cos \left(uy \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot \pi\right)\right)}\right), \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)} \cdot xi, maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]

9. Simplified 99.0%

   \[\leadsto \mathsf{fma}\left(\cos \left(uy \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot \pi\right)\right)}\right), \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)} \cdot xi, maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]

10. Taylor expanded in zi around inf 12.4%

    \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)} \]

11. Taylor expanded in ux around 0 12.4%

    \[\leadsto maxCos \cdot \left(ux \cdot \color{blue}{\left(zi + -1 \cdot \left(ux \cdot zi\right)\right)}\right) \]
12. Step-by-step derivation

    1. mul-1-neg 12.4%

       \[\leadsto maxCos \cdot \left(ux \cdot \left(zi + \color{blue}{\left(-ux \cdot zi\right)}\right)\right) \]

    2. unsub-neg 12.4%

       \[\leadsto maxCos \cdot \left(ux \cdot \color{blue}{\left(zi - ux \cdot zi\right)}\right) \]

13. Simplified 12.4%

    \[\leadsto maxCos \cdot \left(ux \cdot \color{blue}{\left(zi - ux \cdot zi\right)}\right) \]
14. Add Preprocessing

Alternative 13: 13.5% accurate, 51.2× speedup

Math

\[\begin{array}{l} \\ maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (* maxCos (* ux (* zi (- 1.0 ux)))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return maxCos * (ux * (zi * (1.0f - ux)));
}

Fortran

real(4) function code(xi, yi, zi, ux, uy, maxcos)
    real(4), intent (in) :: xi
    real(4), intent (in) :: yi
    real(4), intent (in) :: zi
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = maxcos * (ux * (zi * (1.0e0 - ux)))
end function

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return Float32(maxCos * Float32(ux * Float32(zi * Float32(Float32(1.0) - ux))))
end

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	tmp = maxCos * (ux * (zi * (single(1.0) - ux)));
end

Derivation

1. Initial program 99.1%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
2. Step-by-step derivation

   1. associate-+l+ 99.0%

      \[\leadsto \color{blue}{\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)} \]

   2. associate-*l* 99.0%

      \[\leadsto \color{blue}{\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \left(\sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)} \cdot xi\right)} + \left(\left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right) \]

   3. fma-define 99.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right), \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)} \cdot xi, \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)} \]

3. Simplified 99.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(uy \cdot \left(2 \cdot \pi\right)\right), \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)} \cdot xi, \mathsf{fma}\left(\sin \left(uy \cdot \left(2 \cdot \pi\right)\right), \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)} \cdot yi, \left(1 - ux\right) \cdot \left(\left(ux \cdot maxCos\right) \cdot zi\right)\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in maxCos around 0 99.1%

   \[\leadsto \mathsf{fma}\left(\cos \left(uy \cdot \left(2 \cdot \pi\right)\right), \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)} \cdot xi, \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}\right) \]
6. Step-by-step derivation

   1. expm1-log1p-u 99.0%

      \[\leadsto \mathsf{fma}\left(\cos \left(uy \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot \pi\right)\right)}\right), \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)} \cdot xi, maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]

   2. expm1-undefine 99.0%

      \[\leadsto \mathsf{fma}\left(\cos \left(uy \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(2 \cdot \pi\right)} - 1\right)}\right), \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)} \cdot xi, maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]

7. Applied egg-rr 99.0%

   \[\leadsto \mathsf{fma}\left(\cos \left(uy \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(2 \cdot \pi\right)} - 1\right)}\right), \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)} \cdot xi, maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]
8. Step-by-step derivation

   1. expm1-define 99.0%

      \[\leadsto \mathsf{fma}\left(\cos \left(uy \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot \pi\right)\right)}\right), \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)} \cdot xi, maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]

9. Simplified 99.0%

   \[\leadsto \mathsf{fma}\left(\cos \left(uy \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot \pi\right)\right)}\right), \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)} \cdot xi, maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]

10. Taylor expanded in zi around inf 12.4%

    \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)} \]
11. Add Preprocessing

Alternative 14: 12.1% accurate, 92.2× speedup

Math

\[\begin{array}{l} \\ maxCos \cdot \left(ux \cdot zi\right) \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos) :precision binary32 (* maxCos (* ux zi)))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return maxCos * (ux * zi);
}

Fortran

real(4) function code(xi, yi, zi, ux, uy, maxcos)
    real(4), intent (in) :: xi
    real(4), intent (in) :: yi
    real(4), intent (in) :: zi
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = maxcos * (ux * zi)
end function

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return Float32(maxCos * Float32(ux * zi))
end

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	tmp = maxCos * (ux * zi);
end

Derivation

1. Initial program 99.1%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
2. Step-by-step derivation

   1. associate-+l+ 99.0%

      \[\leadsto \color{blue}{\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)} \]

   2. associate-*l* 99.0%

      \[\leadsto \color{blue}{\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \left(\sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)} \cdot xi\right)} + \left(\left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right) \]

   3. fma-define 99.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right), \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)} \cdot xi, \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\right)} \]

3. Simplified 99.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(uy \cdot \left(2 \cdot \pi\right)\right), \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)} \cdot xi, \mathsf{fma}\left(\sin \left(uy \cdot \left(2 \cdot \pi\right)\right), \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)} \cdot yi, \left(1 - ux\right) \cdot \left(\left(ux \cdot maxCos\right) \cdot zi\right)\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in maxCos around 0 99.1%

   \[\leadsto \mathsf{fma}\left(\cos \left(uy \cdot \left(2 \cdot \pi\right)\right), \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)} \cdot xi, \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}\right) \]
6. Step-by-step derivation

   1. expm1-log1p-u 99.0%

      \[\leadsto \mathsf{fma}\left(\cos \left(uy \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot \pi\right)\right)}\right), \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)} \cdot xi, maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]

   2. expm1-undefine 99.0%

      \[\leadsto \mathsf{fma}\left(\cos \left(uy \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(2 \cdot \pi\right)} - 1\right)}\right), \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)} \cdot xi, maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]

7. Applied egg-rr 99.0%

   \[\leadsto \mathsf{fma}\left(\cos \left(uy \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(2 \cdot \pi\right)} - 1\right)}\right), \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)} \cdot xi, maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]
8. Step-by-step derivation

   1. expm1-define 99.0%

      \[\leadsto \mathsf{fma}\left(\cos \left(uy \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot \pi\right)\right)}\right), \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)} \cdot xi, maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]

9. Simplified 99.0%

   \[\leadsto \mathsf{fma}\left(\cos \left(uy \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot \pi\right)\right)}\right), \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)} \cdot xi, maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]

10. Taylor expanded in zi around inf 12.4%

    \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)} \]

11. Taylor expanded in ux around 0 11.0%

    \[\leadsto maxCos \cdot \left(ux \cdot \color{blue}{zi}\right) \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024170 
(FPCore (xi yi zi ux uy maxCos)
  :name "UniformSampleCone 2"
  :precision binary32
  :pre (and (and (and (and (and (and (<= -10000.0 xi) (<= xi 10000.0)) (and (<= -10000.0 yi) (<= yi 10000.0))) (and (<= -10000.0 zi) (<= zi 10000.0))) (and (<= 2.328306437e-10 ux) (<= ux 1.0))) (and (<= 2.328306437e-10 uy) (<= uy 1.0))) (and (<= 0.0 maxCos) (<= maxCos 1.0)))
  (+ (+ (* (* (cos (* (* uy 2.0) PI)) (sqrt (- 1.0 (* (* (* (- 1.0 ux) maxCos) ux) (* (* (- 1.0 ux) maxCos) ux))))) xi) (* (* (sin (* (* uy 2.0) PI)) (sqrt (- 1.0 (* (* (* (- 1.0 ux) maxCos) ux) (* (* (- 1.0 ux) maxCos) ux))))) yi)) (* (* (* (- 1.0 ux) maxCos) ux) zi)))