UniformSampleCone 2

Percentage Accurate: 99.0% → 99.0%

Time: 19.7s

Alternatives: 13

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: 99.0% 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: 99.0% accurate, 0.7× speedup

Math

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

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 -1.0) (* ux maxCos)))))))
   (fma
    (cos t_0)
    (* t_1 xi)
    (fma (sin t_0) (* t_1 yi) (* (- 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 + -1.0f) * (ux * maxCos)))));
	return fmaf(cosf(t_0), (t_1 * xi), fmaf(sinf(t_0), (t_1 * yi), ((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 + Float32(-1.0)) * Float32(ux * maxCos)))))
	return fma(cos(t_0), Float32(t_1 * xi), fma(sin(t_0), Float32(t_1 * yi), 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 99.0%

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

3. Simplified 99.0%

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

5. Final simplification 99.0%

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

Alternative 2: 99.0% accurate, 0.9× speedup

Math

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

FPCore

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

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

Julia

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

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

   1. add-exp-log 47.3%

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

   2. *-commutative 47.3%

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

   3. associate-*r* 47.3%

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

6. Applied egg-rr 47.3%

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

   1. +-commutative 47.3%

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

   2. rem-exp-log 99.0%

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

   3. *-commutative 99.0%

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

   4. fma-define 99.0%

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

   5. *-commutative 99.0%

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

   6. associate-*r* 99.0%

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

8. Applied egg-rr 99.0%

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

9. Final simplification 99.0%

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

Alternative 3: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

3. Final simplification 99.0%

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

Alternative 4: 99.0% accurate, 1.0× speedup

Math

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

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 (* t_0 (* (* maxCos (+ ux -1.0)) (* ux ux)))))
     (+ (* (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 + (t_0 * ((maxCos * (ux + -1.0f)) * (ux * ux))))) * ((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(t_0 * Float32(Float32(maxCos * Float32(ux + Float32(-1.0))) * Float32(ux * ux))))) * 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%

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

5. Final simplification 99.0%

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

Alternative 5: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return fma(cos(Float32(uy * Float32(Float32(2.0) * Float32(pi)))), Float32(sqrt(Float32(Float32(1.0) + Float32(Float32(Float32(Float32(1.0) - ux) * Float32(ux * maxCos)) * Float32(Float32(ux + Float32(-1.0)) * Float32(ux * maxCos))))) * xi), Float32(Float32(maxCos * Float32(ux * Float32(Float32(Float32(1.0) - ux) * zi))) + Float32(yi * sin(Float32(Float32(2.0) * Float32(uy * Float32(pi)))))))
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 99.0%

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

3. Simplified 99.0%

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

5. Taylor expanded in maxCos around 0 98.8%

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

6. Final simplification 98.8%

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

Alternative 6: 98.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

3. Taylor expanded in ux around 0 98.8%

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

4. Final simplification 98.8%

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

Alternative 7: 98.8% 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%

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

3. Simplified 99.0%

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

   1. expm1-log1p-u 98.6%

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

   2. expm1-undefine 98.6%

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

6. Applied egg-rr 98.6%

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

   1. expm1-define 98.6%

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

8. Simplified 98.6%

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

   1. log1p-expm1-u 98.6%

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

   2. *-commutative 98.6%

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

   3. expm1-log1p-u 99.0%

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

   4. *-commutative 99.0%

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

   5. associate-*l* 99.0%

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

10. Applied egg-rr 99.0%

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

11. Taylor expanded in maxCos around 0 98.8%

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

12. Final simplification 98.8%

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

Alternative 8: 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%

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

3. Simplified 99.0%

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

   1. expm1-log1p-u 98.6%

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

   2. expm1-undefine 98.6%

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

6. Applied egg-rr 98.6%

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

   1. expm1-define 98.6%

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

8. Simplified 98.6%

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

   1. log1p-expm1-u 98.6%

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

   2. *-commutative 98.6%

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

   3. expm1-log1p-u 99.0%

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

   4. *-commutative 99.0%

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

   5. associate-*l* 99.0%

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

10. Applied egg-rr 99.0%

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

11. Taylor expanded in ux around 0 95.8%

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

12. Final simplification 95.8%

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

Alternative 9: 52.3% accurate, 35.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	tmp = xi + (zi * (ux * (ux * ((maxCos / 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. Add Preprocessing
3. Step-by-step derivation

   1. associate-*r* 99.0%

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

   2. add-sqr-sqrt 95.7%

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

   3. pow2 95.7%

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

   4. associate-*r* 95.7%

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

4. Applied egg-rr 95.7%

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

5. Taylor expanded in uy around 0 54.8%

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

   1. associate-*r* 54.8%

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

   2. unpow2 54.8%

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

   3. unpow2 54.8%

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

   4. swap-sqr 54.8%

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

   5. unpow2 54.8%

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

   6. swap-sqr 54.8%

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

   7. *-commutative 54.8%

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

   8. *-commutative 54.8%

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

   9. unpow2 54.8%

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

   10. *-commutative 54.8%

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

7. Simplified 54.8%

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

8. Taylor expanded in maxCos around 0 54.8%

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

9. Taylor expanded in ux around inf 54.8%

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

    1. neg-mul-1 54.8%

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

    2. +-commutative 54.8%

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

    3. unsub-neg 54.8%

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

11. Simplified 54.8%

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

12. Final simplification 54.8%

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

Alternative 10: 52.3% accurate, 41.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	tmp = xi + (zi * (ux * (maxCos - (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. Add Preprocessing
3. Step-by-step derivation

   1. associate-*r* 99.0%

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

   2. add-sqr-sqrt 95.7%

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

   3. pow2 95.7%

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

   4. associate-*r* 95.7%

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

4. Applied egg-rr 95.7%

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

5. Taylor expanded in uy around 0 54.8%

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

   1. associate-*r* 54.8%

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

   2. unpow2 54.8%

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

   3. unpow2 54.8%

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

   4. swap-sqr 54.8%

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

   5. unpow2 54.8%

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

   6. swap-sqr 54.8%

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

   7. *-commutative 54.8%

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

   8. *-commutative 54.8%

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

   9. unpow2 54.8%

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

   10. *-commutative 54.8%

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

7. Simplified 54.8%

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

8. Taylor expanded in maxCos around 0 54.8%

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

9. Taylor expanded in ux around 0 54.8%

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

    1. mul-1-neg 54.8%

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

    2. unsub-neg 54.8%

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

11. Simplified 54.8%

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

12. Final simplification 54.8%

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

Alternative 11: 52.3% accurate, 41.9× speedup

Math

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

FPCore

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

C

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

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

Julia

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

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	tmp = xi - (maxCos * (ux * (zi * (ux + single(-1.0)))));
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. Add Preprocessing
3. Step-by-step derivation

   1. associate-*r* 99.0%

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

   2. add-sqr-sqrt 95.7%

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

   3. pow2 95.7%

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

   4. associate-*r* 95.7%

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

4. Applied egg-rr 95.7%

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

5. Taylor expanded in uy around 0 54.8%

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

   1. associate-*r* 54.8%

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

   2. unpow2 54.8%

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

   3. unpow2 54.8%

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

   4. swap-sqr 54.8%

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

   5. unpow2 54.8%

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

   6. swap-sqr 54.8%

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

   7. *-commutative 54.8%

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

   8. *-commutative 54.8%

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

   9. unpow2 54.8%

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

   10. *-commutative 54.8%

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

7. Simplified 54.8%

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

8. Taylor expanded in maxCos around 0 54.8%

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

9. Taylor expanded in maxCos around 0 54.8%

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

    1. *-commutative 54.8%

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

11. Simplified 54.8%

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

12. Final simplification 54.8%

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

Alternative 12: 50.2% accurate, 65.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	tmp = xi + (zi * (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. Add Preprocessing
3. Step-by-step derivation

   1. associate-*r* 99.0%

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

   2. add-sqr-sqrt 95.7%

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

   3. pow2 95.7%

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

   4. associate-*r* 95.7%

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

4. Applied egg-rr 95.7%

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

5. Taylor expanded in uy around 0 54.8%

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

   1. associate-*r* 54.8%

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

   2. unpow2 54.8%

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

   3. unpow2 54.8%

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

   4. swap-sqr 54.8%

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

   5. unpow2 54.8%

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

   6. swap-sqr 54.8%

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

   7. *-commutative 54.8%

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

   8. *-commutative 54.8%

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

   9. unpow2 54.8%

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

   10. *-commutative 54.8%

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

7. Simplified 54.8%

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

8. Taylor expanded in maxCos around 0 54.8%

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

9. Taylor expanded in ux around 0 52.8%

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

10. Final simplification 52.8%

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

Alternative 13: 50.2% accurate, 65.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	tmp = xi + (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. Add Preprocessing
3. Step-by-step derivation

   1. associate-*r* 99.0%

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

   2. add-sqr-sqrt 95.7%

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

   3. pow2 95.7%

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

   4. associate-*r* 95.7%

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

4. Applied egg-rr 95.7%

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

5. Taylor expanded in uy around 0 54.8%

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

   1. associate-*r* 54.8%

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

   2. unpow2 54.8%

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

   3. unpow2 54.8%

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

   4. swap-sqr 54.8%

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

   5. unpow2 54.8%

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

   6. swap-sqr 54.8%

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

   7. *-commutative 54.8%

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

   8. *-commutative 54.8%

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

   9. unpow2 54.8%

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

   10. *-commutative 54.8%

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

7. Simplified 54.8%

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

8. Taylor expanded in maxCos around 0 54.8%

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

9. Taylor expanded in ux around 0 52.8%

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

Reproduce

herbie shell --seed 2024151 
(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)))