UniformSampleCone 2

Percentage Accurate: 98.9% → 98.9%

Time: 22.7s

Alternatives: 20

Speedup: 1.0×

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: 98.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.0%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
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 99.1%

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

3. Simplified 99.1%

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

5. Final simplification 99.1%

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

Alternative 2: 98.9% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.0%

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

3. Simplified 99.1%

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

5. Final simplification 99.1%

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

Alternative 3: 98.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* uy (* 2.0 PI))))
   (fma
    (- maxCos (* ux maxCos))
    (* ux zi)
    (*
     (sqrt
      (- 1.0 (* maxCos (* (- 1.0 ux) (* (* ux ux) (* (- 1.0 ux) maxCos))))))
     (+ (* (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 - (ux * maxCos)), (ux * zi), (sqrtf((1.0f - (maxCos * ((1.0f - ux) * ((ux * ux) * ((1.0f - ux) * maxCos)))))) * ((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(ux * maxCos)), Float32(ux * zi), Float32(sqrt(Float32(Float32(1.0) - Float32(maxCos * Float32(Float32(Float32(1.0) - ux) * Float32(Float32(ux * ux) * Float32(Float32(Float32(1.0) - ux) * maxCos)))))) * Float32(Float32(cos(t_0) * xi) + Float32(sin(t_0) * yi))))
end

Derivation

1. Initial program 99.0%

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

3. Simplified 99.0%

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

5. Taylor expanded in ux around 0 99.1%

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

   1. mul-1-neg 99.1%

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

   2. unsub-neg 99.1%

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

7. Simplified 99.1%

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

8. Final simplification 99.1%

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

Alternative 4: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.0%

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

3. Simplified 99.0%

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

5. Final simplification 99.0%

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

Alternative 5: 98.7% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* 2.0 (* uy PI))))
   (fma maxCos (* ux (* (- 1.0 ux) zi)) (fma 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 fmaf(maxCos, (ux * ((1.0f - ux) * zi)), fmaf(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 fma(maxCos, Float32(ux * Float32(Float32(Float32(1.0) - ux) * zi)), fma(xi, cos(t_0), Float32(yi * sin(t_0))))
end

Derivation

1. Initial program 99.0%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
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 99.1%

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

3. Simplified 99.1%

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

5. Taylor expanded in maxCos around 0 98.9%

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

   1. fma-define 98.9%

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

   2. *-commutative 98.9%

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

   3. fma-define 99.0%

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

7. Simplified 99.0%

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

Alternative 6: 98.7% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* 2.0 (* uy PI))))
   (+
    (* zi (* (- 1.0 ux) (* ux maxCos)))
    (fma 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 (zi * ((1.0f - ux) * (ux * maxCos))) + fmaf(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(zi * Float32(Float32(Float32(1.0) - ux) * Float32(ux * maxCos))) + fma(xi, cos(t_0), Float32(yi * sin(t_0))))
end

Derivation

1. Initial program 99.0%

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

3. Simplified 99.1%

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

5. Taylor expanded in ux around 0 98.9%

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

   1. fma-define 98.9%

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

7. Simplified 98.9%

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

8. Final simplification 98.9%

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

Alternative 7: 98.7% accurate, 2.0× speedup

Math

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

FPCore

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
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 99.1%

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

3. Simplified 99.1%

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

5. Taylor expanded in maxCos around 0 98.9%

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

6. Final simplification 98.9%

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

Alternative 8: 95.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \left(uy \cdot \pi\right)\\ \mathbf{if}\;uy \cdot 2 \leq 0.008999999612569809:\\ \;\;\;\;xi + \left(maxCos \cdot \left(ux \cdot \left(\left(1 - ux\right) \cdot zi\right)\right) + uy \cdot \left(-2 \cdot \left(uy \cdot \left(xi \cdot {\pi}^{2}\right)\right) + 2 \cdot \left(\pi \cdot yi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;yi \cdot \sin t\_0 + xi \cdot \cos t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* 2.0 (* uy PI))))
   (if (<= (* uy 2.0) 0.008999999612569809)
     (+
      xi
      (+
       (* maxCos (* ux (* (- 1.0 ux) zi)))
       (* uy (+ (* -2.0 (* uy (* xi (pow PI 2.0)))) (* 2.0 (* PI yi))))))
     (+ (* yi (sin t_0)) (* xi (cos 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));
	float tmp;
	if ((uy * 2.0f) <= 0.008999999612569809f) {
		tmp = xi + ((maxCos * (ux * ((1.0f - ux) * zi))) + (uy * ((-2.0f * (uy * (xi * powf(((float) M_PI), 2.0f)))) + (2.0f * (((float) M_PI) * yi)))));
	} else {
		tmp = (yi * sinf(t_0)) + (xi * cosf(t_0));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(2.0) * Float32(uy * Float32(pi)))
	tmp = Float32(0.0)
	if (Float32(uy * Float32(2.0)) <= Float32(0.008999999612569809))
		tmp = Float32(xi + Float32(Float32(maxCos * Float32(ux * Float32(Float32(Float32(1.0) - ux) * zi))) + Float32(uy * Float32(Float32(Float32(-2.0) * Float32(uy * Float32(xi * (Float32(pi) ^ Float32(2.0))))) + Float32(Float32(2.0) * Float32(Float32(pi) * yi))))));
	else
		tmp = Float32(Float32(yi * sin(t_0)) + Float32(xi * cos(t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(xi, yi, zi, ux, uy, maxCos)
	t_0 = single(2.0) * (uy * single(pi));
	tmp = single(0.0);
	if ((uy * single(2.0)) <= single(0.008999999612569809))
		tmp = xi + ((maxCos * (ux * ((single(1.0) - ux) * zi))) + (uy * ((single(-2.0) * (uy * (xi * (single(pi) ^ single(2.0))))) + (single(2.0) * (single(pi) * yi)))));
	else
		tmp = (yi * sin(t_0)) + (xi * cos(t_0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 0.00899999961

   1. Initial program 99.2%

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

      1. associate-+l+ 99.2%

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

      2. associate-*l* 99.2%

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

      3. fma-define 99.3%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in maxCos around 0 99.2%

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

      1. fma-define 99.2%

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

      2. *-commutative 99.2%

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

      3. fma-define 99.3%

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

   7. Simplified 99.3%

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

   8. Taylor expanded in uy around 0 97.5%

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

   if 0.00899999961 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 98.1%

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

      1. associate-+l+ 98.1%

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

      2. associate-*l* 98.1%

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

      3. fma-define 98.2%

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

   3. Simplified 98.3%

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

   5. Taylor expanded in zi around inf 97.9%

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

      1. fma-define 98.0%

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

      2. distribute-rgt-out 98.0%

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

   7. Simplified 98.0%

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

   8. Taylor expanded in ux around 0 95.9%

      \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot zi\right) + zi \cdot \left(\frac{xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right)}{zi} + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}{zi}\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 95.9%

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

      2. *-commutative 95.9%

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

      3. distribute-rgt-out 95.9%

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

      4. associate-/l* 95.9%

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

      5. associate-*r/ 95.8%

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

   10. Simplified 95.8%

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

   11. Taylor expanded in zi around 0 89.9%

       \[\leadsto \color{blue}{xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.008999999612569809:\\ \;\;\;\;xi + \left(maxCos \cdot \left(ux \cdot \left(\left(1 - ux\right) \cdot zi\right)\right) + uy \cdot \left(-2 \cdot \left(uy \cdot \left(xi \cdot {\pi}^{2}\right)\right) + 2 \cdot \left(\pi \cdot yi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right) + xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 95.8% accurate, 2.1× speedup

Math

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

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* 2.0 (* uy PI))))
   (+ (+ (* yi (sin t_0)) (* xi (cos 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 ((yi * sinf(t_0)) + (xi * cosf(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(yi * sin(t_0)) + Float32(xi * cos(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 = ((yi * sin(t_0)) + (xi * cos(t_0))) + (maxCos * (ux * zi));
end

Derivation

1. Initial program 99.0%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
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 99.1%

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

3. Simplified 99.1%

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

5. Taylor expanded in ux around 0 96.6%

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

6. Final simplification 96.6%

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

Alternative 10: 88.4% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.14000000059604645:\\ \;\;\;\;xi + \left(maxCos \cdot \left(ux \cdot \left(\left(1 - ux\right) \cdot zi\right)\right) + uy \cdot \left(-2 \cdot \left(uy \cdot \left(xi \cdot {\pi}^{2}\right)\right) + 2 \cdot \left(\pi \cdot yi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;zi \cdot \left(ux \cdot maxCos + \left(\frac{xi}{zi} + yi \cdot \frac{\sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}{zi}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (if (<= (* uy 2.0) 0.14000000059604645)
   (+
    xi
    (+
     (* maxCos (* ux (* (- 1.0 ux) zi)))
     (* uy (+ (* -2.0 (* uy (* xi (pow PI 2.0)))) (* 2.0 (* PI yi))))))
   (*
    zi
    (+ (* ux maxCos) (+ (/ xi zi) (* yi (/ (sin (* 2.0 (* uy PI))) zi)))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float tmp;
	if ((uy * 2.0f) <= 0.14000000059604645f) {
		tmp = xi + ((maxCos * (ux * ((1.0f - ux) * zi))) + (uy * ((-2.0f * (uy * (xi * powf(((float) M_PI), 2.0f)))) + (2.0f * (((float) M_PI) * yi)))));
	} else {
		tmp = zi * ((ux * maxCos) + ((xi / zi) + (yi * (sinf((2.0f * (uy * ((float) M_PI)))) / zi))));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	tmp = Float32(0.0)
	if (Float32(uy * Float32(2.0)) <= Float32(0.14000000059604645))
		tmp = Float32(xi + Float32(Float32(maxCos * Float32(ux * Float32(Float32(Float32(1.0) - ux) * zi))) + Float32(uy * Float32(Float32(Float32(-2.0) * Float32(uy * Float32(xi * (Float32(pi) ^ Float32(2.0))))) + Float32(Float32(2.0) * Float32(Float32(pi) * yi))))));
	else
		tmp = Float32(zi * Float32(Float32(ux * maxCos) + Float32(Float32(xi / zi) + Float32(yi * Float32(sin(Float32(Float32(2.0) * Float32(uy * Float32(pi)))) / zi)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(xi, yi, zi, ux, uy, maxCos)
	tmp = single(0.0);
	if ((uy * single(2.0)) <= single(0.14000000059604645))
		tmp = xi + ((maxCos * (ux * ((single(1.0) - ux) * zi))) + (uy * ((single(-2.0) * (uy * (xi * (single(pi) ^ single(2.0))))) + (single(2.0) * (single(pi) * yi)))));
	else
		tmp = zi * ((ux * maxCos) + ((xi / zi) + (yi * (sin((single(2.0) * (uy * single(pi)))) / zi))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 0.140000001

   1. Initial program 99.2%

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

      1. associate-+l+ 99.2%

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

      2. associate-*l* 99.2%

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

      3. fma-define 99.2%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in maxCos around 0 99.1%

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

      1. fma-define 99.2%

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

      2. *-commutative 99.2%

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

      3. fma-define 99.3%

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

   7. Simplified 99.3%

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

   8. Taylor expanded in uy around 0 94.2%

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

   if 0.140000001 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 97.4%

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

      1. associate-+l+ 97.4%

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

      2. associate-*l* 97.4%

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

      3. fma-define 97.4%

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

   3. Simplified 97.4%

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

   5. Taylor expanded in zi around inf 97.1%

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

      1. fma-define 97.2%

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

      2. distribute-rgt-out 97.2%

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

   7. Simplified 97.5%

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

   8. Taylor expanded in ux around 0 92.9%

      \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot zi\right) + zi \cdot \left(\frac{xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right)}{zi} + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}{zi}\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 92.9%

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

      2. *-commutative 92.9%

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

      3. distribute-rgt-out 92.9%

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

      4. associate-/l* 92.8%

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

      5. associate-*r/ 93.1%

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

   10. Simplified 93.1%

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

   11. Taylor expanded in uy around 0 69.2%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.14000000059604645:\\ \;\;\;\;xi + \left(maxCos \cdot \left(ux \cdot \left(\left(1 - ux\right) \cdot zi\right)\right) + uy \cdot \left(-2 \cdot \left(uy \cdot \left(xi \cdot {\pi}^{2}\right)\right) + 2 \cdot \left(\pi \cdot yi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;zi \cdot \left(ux \cdot maxCos + \left(\frac{xi}{zi} + yi \cdot \frac{\sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}{zi}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 83.9% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.07000000029802322:\\ \;\;\;\;xi + \left(maxCos \cdot \left(ux \cdot \left(\left(1 - ux\right) \cdot zi\right)\right) + 2 \cdot \left(uy \cdot \left(\pi \cdot yi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;ux \cdot \left(zi \cdot \left(maxCos + \frac{yi \cdot \frac{\sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}{zi}}{ux}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (if (<= uy 0.07000000029802322)
   (+ xi (+ (* maxCos (* ux (* (- 1.0 ux) zi))) (* 2.0 (* uy (* PI yi)))))
   (* ux (* zi (+ maxCos (/ (* yi (/ (sin (* 2.0 (* uy PI))) zi)) ux))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.07000000029802322f) {
		tmp = xi + ((maxCos * (ux * ((1.0f - ux) * zi))) + (2.0f * (uy * (((float) M_PI) * yi))));
	} else {
		tmp = ux * (zi * (maxCos + ((yi * (sinf((2.0f * (uy * ((float) M_PI)))) / zi)) / ux)));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.07000000029802322))
		tmp = Float32(xi + Float32(Float32(maxCos * Float32(ux * Float32(Float32(Float32(1.0) - ux) * zi))) + Float32(Float32(2.0) * Float32(uy * Float32(Float32(pi) * yi)))));
	else
		tmp = Float32(ux * Float32(zi * Float32(maxCos + Float32(Float32(yi * Float32(sin(Float32(Float32(2.0) * Float32(uy * Float32(pi)))) / zi)) / ux))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(xi, yi, zi, ux, uy, maxCos)
	tmp = single(0.0);
	if (uy <= single(0.07000000029802322))
		tmp = xi + ((maxCos * (ux * ((single(1.0) - ux) * zi))) + (single(2.0) * (uy * (single(pi) * yi))));
	else
		tmp = ux * (zi * (maxCos + ((yi * (sin((single(2.0) * (uy * single(pi)))) / zi)) / ux)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.0700000003

   1. Initial program 99.2%

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

      1. associate-+l+ 99.2%

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

      2. associate-*l* 99.2%

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

      3. fma-define 99.2%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in maxCos around 0 99.1%

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

      1. fma-define 99.2%

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

      2. *-commutative 99.2%

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

      3. fma-define 99.3%

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

   7. Simplified 99.3%

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

   8. Taylor expanded in uy around 0 89.2%

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

   if 0.0700000003 < uy

   1. Initial program 97.4%

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

      1. associate-+l+ 97.4%

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

      2. associate-*l* 97.4%

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

      3. fma-define 97.4%

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

   3. Simplified 97.4%

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

   5. Taylor expanded in zi around inf 97.1%

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

      1. fma-define 97.2%

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

      2. distribute-rgt-out 97.2%

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

   7. Simplified 97.5%

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

   8. Taylor expanded in ux around 0 92.9%

      \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot zi\right) + zi \cdot \left(\frac{xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right)}{zi} + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}{zi}\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 92.9%

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

      2. *-commutative 92.9%

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

      3. distribute-rgt-out 92.9%

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

      4. associate-/l* 92.8%

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

      5. associate-*r/ 93.1%

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

   10. Simplified 93.1%

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

   11. Taylor expanded in ux around inf 92.8%

       \[\leadsto \color{blue}{ux \cdot \left(maxCos \cdot zi + \frac{zi \cdot \left(\frac{xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right)}{zi} + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}{zi}\right)}{ux}\right)} \]
   12. Step-by-step derivation

       1. *-commutative 92.8%

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

       2. associate-/l* 92.7%

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

       3. distribute-lft-out 92.7%

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

   13. Simplified 92.8%

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

   14. Taylor expanded in xi around 0 67.5%

       \[\leadsto ux \cdot \left(zi \cdot \left(maxCos + \frac{\color{blue}{\frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}{zi}}}{ux}\right)\right) \]
   15. Step-by-step derivation

       1. associate-/l* 67.6%

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

   16. Simplified 67.6%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \leq 0.07000000029802322:\\ \;\;\;\;xi + \left(maxCos \cdot \left(ux \cdot \left(\left(1 - ux\right) \cdot zi\right)\right) + 2 \cdot \left(uy \cdot \left(\pi \cdot yi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;ux \cdot \left(zi \cdot \left(maxCos + \frac{yi \cdot \frac{\sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}{zi}}{ux}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 84.1% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.07000000029802322:\\ \;\;\;\;xi + \left(maxCos \cdot \left(ux \cdot \left(\left(1 - ux\right) \cdot zi\right)\right) + 2 \cdot \left(uy \cdot \left(\pi \cdot yi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;zi \cdot \left(ux \cdot maxCos + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}{zi}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (if (<= uy 0.07000000029802322)
   (+ xi (+ (* maxCos (* ux (* (- 1.0 ux) zi))) (* 2.0 (* uy (* PI yi)))))
   (* zi (+ (* ux maxCos) (/ (* yi (sin (* 2.0 (* uy PI)))) zi)))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.07000000029802322f) {
		tmp = xi + ((maxCos * (ux * ((1.0f - ux) * zi))) + (2.0f * (uy * (((float) M_PI) * yi))));
	} else {
		tmp = zi * ((ux * maxCos) + ((yi * sinf((2.0f * (uy * ((float) M_PI))))) / zi));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.07000000029802322))
		tmp = Float32(xi + Float32(Float32(maxCos * Float32(ux * Float32(Float32(Float32(1.0) - ux) * zi))) + Float32(Float32(2.0) * Float32(uy * Float32(Float32(pi) * yi)))));
	else
		tmp = Float32(zi * Float32(Float32(ux * maxCos) + Float32(Float32(yi * sin(Float32(Float32(2.0) * Float32(uy * Float32(pi))))) / zi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(xi, yi, zi, ux, uy, maxCos)
	tmp = single(0.0);
	if (uy <= single(0.07000000029802322))
		tmp = xi + ((maxCos * (ux * ((single(1.0) - ux) * zi))) + (single(2.0) * (uy * (single(pi) * yi))));
	else
		tmp = zi * ((ux * maxCos) + ((yi * sin((single(2.0) * (uy * single(pi))))) / zi));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.0700000003

   1. Initial program 99.2%

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

      1. associate-+l+ 99.2%

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

      2. associate-*l* 99.2%

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

      3. fma-define 99.2%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in maxCos around 0 99.1%

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

      1. fma-define 99.2%

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

      2. *-commutative 99.2%

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

      3. fma-define 99.3%

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

   7. Simplified 99.3%

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

   8. Taylor expanded in uy around 0 89.2%

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

   if 0.0700000003 < uy

   1. Initial program 97.4%

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

      1. associate-+l+ 97.4%

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

      2. associate-*l* 97.4%

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

      3. fma-define 97.4%

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

   3. Simplified 97.4%

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

   5. Taylor expanded in zi around inf 97.1%

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

      1. fma-define 97.2%

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

      2. distribute-rgt-out 97.2%

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

   7. Simplified 97.5%

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

   8. Taylor expanded in ux around 0 92.9%

      \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot zi\right) + zi \cdot \left(\frac{xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right)}{zi} + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}{zi}\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 92.9%

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

      2. *-commutative 92.9%

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

      3. distribute-rgt-out 92.9%

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

      4. associate-/l* 92.8%

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

      5. associate-*r/ 93.1%

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

   10. Simplified 93.1%

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

   11. Taylor expanded in xi around 0 67.5%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \leq 0.07000000029802322:\\ \;\;\;\;xi + \left(maxCos \cdot \left(ux \cdot \left(\left(1 - ux\right) \cdot zi\right)\right) + 2 \cdot \left(uy \cdot \left(\pi \cdot yi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;zi \cdot \left(ux \cdot maxCos + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}{zi}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 84.9% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ zi \cdot \left(ux \cdot maxCos + \left(\frac{xi}{zi} + yi \cdot \frac{\sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}{zi}\right)\right) \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
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 99.1%

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

3. Simplified 99.1%

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

5. Taylor expanded in zi around inf 98.4%

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

   1. fma-define 98.4%

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

   2. distribute-rgt-out 98.4%

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

7. Simplified 98.4%

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

8. Taylor expanded in ux around 0 95.9%

   \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot zi\right) + zi \cdot \left(\frac{xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right)}{zi} + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}{zi}\right)} \]
9. Step-by-step derivation

   1. associate-*r* 95.9%

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

   2. *-commutative 95.9%

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

   3. distribute-rgt-out 95.9%

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

   4. associate-/l* 95.7%

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

   5. associate-*r/ 95.7%

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

10. Simplified 95.7%

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

11. Taylor expanded in uy around 0 86.8%

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

12. Final simplification 86.8%

    \[\leadsto zi \cdot \left(ux \cdot maxCos + \left(\frac{xi}{zi} + yi \cdot \frac{\sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}{zi}\right)\right) \]
13. Add Preprocessing

Alternative 14: 81.6% accurate, 24.3× speedup

Math

\[\begin{array}{l} \\ xi + \left(maxCos \cdot \left(ux \cdot \left(\left(1 - ux\right) \cdot zi\right)\right) + 2 \cdot \left(uy \cdot \left(\pi \cdot yi\right)\right)\right) \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
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 99.1%

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

3. Simplified 99.1%

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

5. Taylor expanded in maxCos around 0 98.9%

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

   1. fma-define 98.9%

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

   2. *-commutative 98.9%

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

   3. fma-define 99.0%

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

7. Simplified 99.0%

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

8. Taylor expanded in uy around 0 83.7%

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

9. Final simplification 83.7%

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

Alternative 15: 78.6% accurate, 27.1× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(uy \cdot \left(\pi \cdot yi\right)\right) + zi \cdot \left(ux \cdot maxCos + \frac{xi}{zi}\right) \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (+ (* 2.0 (* uy (* PI yi))) (* zi (+ (* ux maxCos) (/ xi zi)))))

C

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

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return Float32(Float32(Float32(2.0) * Float32(uy * Float32(Float32(pi) * yi))) + Float32(zi * Float32(Float32(ux * maxCos) + Float32(xi / zi))))
end

MATLAB

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

Derivation

1. Initial program 99.0%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
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 99.1%

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

3. Simplified 99.1%

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

5. Taylor expanded in zi around inf 98.4%

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

   1. fma-define 98.4%

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

   2. distribute-rgt-out 98.4%

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

7. Simplified 98.4%

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

8. Taylor expanded in ux around 0 95.9%

   \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot zi\right) + zi \cdot \left(\frac{xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right)}{zi} + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}{zi}\right)} \]
9. Step-by-step derivation

   1. associate-*r* 95.9%

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

   2. *-commutative 95.9%

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

   3. distribute-rgt-out 95.9%

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

   4. associate-/l* 95.7%

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

   5. associate-*r/ 95.7%

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

10. Simplified 95.7%

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

11. Taylor expanded in uy around 0 81.2%

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

12. Final simplification 81.2%

    \[\leadsto 2 \cdot \left(uy \cdot \left(\pi \cdot yi\right)\right) + zi \cdot \left(ux \cdot maxCos + \frac{xi}{zi}\right) \]
13. Add Preprocessing

Alternative 16: 51.5% accurate, 41.9× speedup

Math

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

FPCore

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

C

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

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 * ((1.0e0 - ux) * (ux * maxcos)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

3. Simplified 99.1%

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

5. Taylor expanded in xi around inf 58.9%

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

   1. *-commutative 58.9%

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

7. Simplified 58.9%

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

8. Taylor expanded in maxCos around 0 58.7%

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

9. Taylor expanded in uy around 0 52.2%

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

10. Final simplification 52.2%

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

Alternative 17: 49.1% accurate, 51.2× speedup

Math

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

FPCore

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

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return zi * ((ux * maxCos) + (xi / 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 = zi * ((ux * maxcos) + (xi / zi))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
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 99.1%

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

3. Simplified 99.1%

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

5. Taylor expanded in zi around inf 98.4%

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

   1. fma-define 98.4%

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

   2. distribute-rgt-out 98.4%

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

7. Simplified 98.4%

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

8. Taylor expanded in ux around 0 95.9%

   \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot zi\right) + zi \cdot \left(\frac{xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right)}{zi} + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}{zi}\right)} \]
9. Step-by-step derivation

   1. associate-*r* 95.9%

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

   2. *-commutative 95.9%

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

   3. distribute-rgt-out 95.9%

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

   4. associate-/l* 95.7%

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

   5. associate-*r/ 95.7%

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

10. Simplified 95.7%

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

11. Taylor expanded in uy around 0 50.4%

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

12. Final simplification 50.4%

    \[\leadsto zi \cdot \left(ux \cdot maxCos + \frac{xi}{zi}\right) \]
13. Add Preprocessing

Alternative 18: 13.6% accurate, 51.2× speedup

Math

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

FPCore

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

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return maxCos * (ux * ((1.0f - 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 * ((1.0e0 - ux) * zi))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
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 99.1%

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

3. Simplified 99.1%

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

5. Taylor expanded in zi around inf 13.4%

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

6. Final simplification 13.4%

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

Alternative 19: 12.1% accurate, 92.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
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 99.1%

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

3. Simplified 99.1%

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

5. Taylor expanded in zi around inf 98.4%

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

   1. fma-define 98.4%

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

   2. distribute-rgt-out 98.4%

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

7. Simplified 98.4%

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

8. Taylor expanded in ux around 0 95.9%

   \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot zi\right) + zi \cdot \left(\frac{xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right)}{zi} + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}{zi}\right)} \]
9. Step-by-step derivation

   1. associate-*r* 95.9%

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

   2. *-commutative 95.9%

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

   3. distribute-rgt-out 95.9%

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

   4. associate-/l* 95.7%

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

   5. associate-*r/ 95.7%

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

10. Simplified 95.7%

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

11. Taylor expanded in ux around inf 95.7%

    \[\leadsto \color{blue}{ux \cdot \left(maxCos \cdot zi + \frac{zi \cdot \left(\frac{xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right)}{zi} + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}{zi}\right)}{ux}\right)} \]
12. Step-by-step derivation

    1. *-commutative 95.7%

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

    2. associate-/l* 92.3%

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

    3. distribute-lft-out 92.3%

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

13. Simplified 92.5%

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

14. Taylor expanded in maxCos around inf 12.3%

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

15. Final simplification 12.3%

    \[\leadsto ux \cdot \left(maxCos \cdot zi\right) \]
16. Add Preprocessing

Alternative 20: 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 99.0%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
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 99.1%

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

3. Simplified 99.1%

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

5. Taylor expanded in zi around inf 13.4%

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

6. Taylor expanded in ux around 0 12.3%

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

Reproduce

herbie shell --seed 2024137 
(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)))