UniformSampleCone 2

Percentage Accurate: 98.9% → 98.9%

Time: 17.8s

Alternatives: 14

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, 1.0× speedup

Math

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

FPCore

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

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 = (uy * 2.0f) * ((float) M_PI);
	return (((cosf(t_2) * t_1) * xi) + ((t_1 * sinf(t_2)) * yi)) + (t_0 * zi);
}

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(uy * Float32(2.0)) * Float32(pi))
	return Float32(Float32(Float32(Float32(cos(t_2) * t_1) * xi) + Float32(Float32(t_1 * sin(t_2)) * yi)) + Float32(t_0 * zi))
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 = (uy * single(2.0)) * single(pi);
	tmp = (((cos(t_2) * t_1) * xi) + ((t_1 * sin(t_2)) * yi)) + (t_0 * 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. Final simplification 99.0%

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

Alternative 2: 98.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* uy (* 2.0 PI))))
   (fma
    (* maxCos (+ (- 2.0 ux) -1.0))
    (* ux zi)
    (*
     (sqrt
      (+ 1.0 (* (* (- 1.0 ux) maxCos) (* (* maxCos (+ ux -1.0)) (* ux ux)))))
     (+ (* 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 = uy * (2.0f * ((float) M_PI));
	return fmaf((maxCos * ((2.0f - ux) + -1.0f)), (ux * zi), (sqrtf((1.0f + (((1.0f - ux) * maxCos) * ((maxCos * (ux + -1.0f)) * (ux * ux))))) * ((xi * cosf(t_0)) + (yi * sinf(t_0)))));
}

Julia

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

Derivation

1. Initial program 99.0%

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

3. Simplified 98.9%

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

   1. expm1-log1p-u 98.9%

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

6. Applied egg-rr 98.9%

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

   1. expm1-undefine 98.9%

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

   2. sub-neg 98.9%

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

   3. log1p-undefine 98.9%

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

   4. rem-exp-log 98.9%

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

   5. associate-+r- 98.9%

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

   6. metadata-eval 98.9%

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

   7. metadata-eval 98.9%

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

8. Simplified 98.9%

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

9. Final simplification 98.9%

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

Alternative 3: 98.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (- 1.0 ux) maxCos)) (t_1 (* uy (* 2.0 PI))))
   (fma
    t_0
    (* ux zi)
    (*
     (sqrt (+ 1.0 (* t_0 (* (* maxCos (+ ux -1.0)) (* ux ux)))))
     (+ (* xi (cos t_1)) (* yi (sin 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 = uy * (2.0f * ((float) M_PI));
	return fmaf(t_0, (ux * zi), (sqrtf((1.0f + (t_0 * ((maxCos * (ux + -1.0f)) * (ux * ux))))) * ((xi * cosf(t_1)) + (yi * sinf(t_1)))));
}

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(xi * cos(t_1)) + Float32(yi * sin(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 98.9%

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

5. Final simplification 98.9%

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

Alternative 4: 98.7% accurate, 1.1× speedup

Math

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

FPCore

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

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 ((ux * ((1.0f - ux) * maxCos)) * zi) + (xi * (sqrtf((1.0f - powf((maxCos * (ux * (1.0f - ux))), 2.0f))) * (cosf(t_0) + (yi * (sinf(t_0) / xi)))));
}

Julia

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

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	t_0 = uy * (single(2.0) * single(pi));
	tmp = ((ux * ((single(1.0) - ux) * maxCos)) * zi) + (xi * (sqrt((single(1.0) - ((maxCos * (ux * (single(1.0) - ux))) ^ single(2.0)))) * (cos(t_0) + (yi * (sin(t_0) / xi)))));
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. add-log-exp 98.0%

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

4. Applied egg-rr 98.0%

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

5. Taylor expanded in xi around inf 98.8%

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

7. Simplified 98.7%

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

8. Final simplification 98.7%

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

Alternative 5: 98.8% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* ux (* (- 1.0 ux) maxCos))))
   (+
    (* t_0 zi)
    (+
     (*
      (*
       (cos (* (* uy 2.0) PI))
       (sqrt (+ 1.0 (* t_0 (* ux (* maxCos (+ ux -1.0)))))))
      xi)
     (* yi (sin (* uy (* 2.0 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 (t_0 * zi) + (((cosf(((uy * 2.0f) * ((float) M_PI))) * sqrtf((1.0f + (t_0 * (ux * (maxCos * (ux + -1.0f))))))) * xi) + (yi * sinf((uy * (2.0f * ((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(t_0 * zi) + Float32(Float32(Float32(cos(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) * sqrt(Float32(Float32(1.0) + Float32(t_0 * Float32(ux * Float32(maxCos * Float32(ux + Float32(-1.0)))))))) * xi) + Float32(yi * sin(Float32(uy * Float32(Float32(2.0) * Float32(pi)))))))
end

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	t_0 = ux * ((single(1.0) - ux) * maxCos);
	tmp = (t_0 * zi) + (((cos(((uy * single(2.0)) * single(pi))) * sqrt((single(1.0) + (t_0 * (ux * (maxCos * (ux + single(-1.0)))))))) * xi) + (yi * sin((uy * (single(2.0) * 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 98.7%

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

   1. associate-*r* 98.7%

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

   2. *-commutative 98.7%

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

   3. associate-*l* 98.7%

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

5. Simplified 98.7%

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

6. Final simplification 98.7%

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

Alternative 6: 98.7% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* uy (* 2.0 PI))))
   (+
    (* (* ux (* (- 1.0 ux) maxCos)) 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 = uy * (2.0f * ((float) M_PI));
	return ((ux * ((1.0f - ux) * maxCos)) * zi) + fmaf(xi, cosf(t_0), (yi * sinf(t_0)));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(uy * Float32(Float32(2.0) * Float32(pi)))
	return Float32(Float32(Float32(ux * Float32(Float32(Float32(1.0) - ux) * maxCos)) * 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. Add Preprocessing
3. Step-by-step derivation

   1. add-log-exp 98.0%

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

4. Applied egg-rr 98.0%

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

5. Taylor expanded in ux around 0 98.7%

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

   1. fma-define 98.7%

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

   2. associate-*r* 98.7%

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

   3. *-commutative 98.7%

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

   4. associate-*r* 98.7%

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

   5. associate-*r* 98.7%

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

   6. *-commutative 98.7%

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

   7. associate-*r* 98.7%

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

7. Simplified 98.7%

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

8. Final simplification 98.7%

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

Alternative 7: 98.7% accurate, 2.0× speedup

Math

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

FPCore

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

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	t_0 = single(2.0) * (uy * single(pi));
	tmp = ((ux * ((single(1.0) - ux) * maxCos)) * zi) + ((xi * cos(t_0)) + (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. Add Preprocessing
3. Step-by-step derivation

   1. add-log-exp 98.0%

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

4. Applied egg-rr 98.0%

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

5. Taylor expanded in ux around 0 98.7%

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

6. Final simplification 98.7%

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

Alternative 8: 95.6% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

3. Simplified 98.9%

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

   1. expm1-log1p-u 98.9%

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

6. Applied egg-rr 98.9%

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

   1. expm1-undefine 98.9%

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

   2. sub-neg 98.9%

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

   3. log1p-undefine 98.9%

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

   4. rem-exp-log 98.9%

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

   5. associate-+r- 98.9%

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

   6. metadata-eval 98.9%

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

   7. metadata-eval 98.9%

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

8. Simplified 98.9%

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

   1. associate-*r* 98.9%

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

   2. rem-log-exp 97.9%

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

   3. add-cube-cbrt 98.7%

      \[\leadsto \mathsf{fma}\left(\left(\left(2 - ux\right) + -1\right) \cdot maxCos, ux \cdot zi, \sqrt{1 - \left(maxCos \cdot \left(ux + -1\right)\right) \cdot \left(\left(ux \cdot ux\right) \cdot \left(maxCos \cdot \left(ux + -1\right)\right)\right)} \cdot \left(\color{blue}{\left(\left(\sqrt[3]{\log \left(e^{\cos \left(\left(uy \cdot 2\right) \cdot \pi\right)}\right)} \cdot \sqrt[3]{\log \left(e^{\cos \left(\left(uy \cdot 2\right) \cdot \pi\right)}\right)}\right) \cdot \sqrt[3]{\log \left(e^{\cos \left(\left(uy \cdot 2\right) \cdot \pi\right)}\right)}\right)} \cdot xi + \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot yi\right)\right) \]

   4. pow3 98.7%

      \[\leadsto \mathsf{fma}\left(\left(\left(2 - ux\right) + -1\right) \cdot maxCos, ux \cdot zi, \sqrt{1 - \left(maxCos \cdot \left(ux + -1\right)\right) \cdot \left(\left(ux \cdot ux\right) \cdot \left(maxCos \cdot \left(ux + -1\right)\right)\right)} \cdot \left(\color{blue}{{\left(\sqrt[3]{\log \left(e^{\cos \left(\left(uy \cdot 2\right) \cdot \pi\right)}\right)}\right)}^{3}} \cdot xi + \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot yi\right)\right) \]

   5. rem-log-exp 98.8%

      \[\leadsto \mathsf{fma}\left(\left(\left(2 - ux\right) + -1\right) \cdot maxCos, ux \cdot zi, \sqrt{1 - \left(maxCos \cdot \left(ux + -1\right)\right) \cdot \left(\left(ux \cdot ux\right) \cdot \left(maxCos \cdot \left(ux + -1\right)\right)\right)} \cdot \left({\left(\sqrt[3]{\color{blue}{\cos \left(\left(uy \cdot 2\right) \cdot \pi\right)}}\right)}^{3} \cdot xi + \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot yi\right)\right) \]

   6. associate-*r* 98.8%

      \[\leadsto \mathsf{fma}\left(\left(\left(2 - ux\right) + -1\right) \cdot maxCos, ux \cdot zi, \sqrt{1 - \left(maxCos \cdot \left(ux + -1\right)\right) \cdot \left(\left(ux \cdot ux\right) \cdot \left(maxCos \cdot \left(ux + -1\right)\right)\right)} \cdot \left({\left(\sqrt[3]{\cos \color{blue}{\left(uy \cdot \left(2 \cdot \pi\right)\right)}}\right)}^{3} \cdot xi + \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot yi\right)\right) \]

10. Applied egg-rr 98.8%

    \[\leadsto \mathsf{fma}\left(\left(\left(2 - ux\right) + -1\right) \cdot maxCos, ux \cdot zi, \sqrt{1 - \left(maxCos \cdot \left(ux + -1\right)\right) \cdot \left(\left(ux \cdot ux\right) \cdot \left(maxCos \cdot \left(ux + -1\right)\right)\right)} \cdot \left(\color{blue}{{\left(\sqrt[3]{\cos \left(uy \cdot \left(2 \cdot \pi\right)\right)}\right)}^{3}} \cdot xi + \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot yi\right)\right) \]

11. Taylor expanded in ux around 0 96.8%

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

12. Final simplification 96.8%

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

Alternative 9: 84.7% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.054999999701976776:\\ \;\;\;\;\left(ux \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right) \cdot zi + \sqrt{1 + \left(1 - ux\right) \cdot \left(\left(ux \cdot maxCos\right) \cdot \left(\left(ux + -1\right) \cdot \left(ux \cdot maxCos\right)\right)\right)} \cdot \left(xi + 2 \cdot \left(\pi \cdot \left(uy \cdot yi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right) + maxCos \cdot \left(ux \cdot \left(\left(1 - ux\right) \cdot zi\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (if (<= uy 0.054999999701976776)
   (+
    (* (* ux (* (- 1.0 ux) maxCos)) zi)
    (*
     (sqrt
      (+ 1.0 (* (- 1.0 ux) (* (* ux maxCos) (* (+ ux -1.0) (* ux maxCos))))))
     (+ xi (* 2.0 (* PI (* uy yi))))))
   (+ (* xi (cos (* 2.0 (* uy PI)))) (* maxCos (* ux (* (- 1.0 ux) zi))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.054999999701976776f) {
		tmp = ((ux * ((1.0f - ux) * maxCos)) * zi) + (sqrtf((1.0f + ((1.0f - ux) * ((ux * maxCos) * ((ux + -1.0f) * (ux * maxCos)))))) * (xi + (2.0f * (((float) M_PI) * (uy * yi)))));
	} else {
		tmp = (xi * cosf((2.0f * (uy * ((float) M_PI))))) + (maxCos * (ux * ((1.0f - ux) * zi)));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.054999999701976776))
		tmp = Float32(Float32(Float32(ux * Float32(Float32(Float32(1.0) - ux) * maxCos)) * zi) + Float32(sqrt(Float32(Float32(1.0) + Float32(Float32(Float32(1.0) - ux) * Float32(Float32(ux * maxCos) * Float32(Float32(ux + Float32(-1.0)) * Float32(ux * maxCos)))))) * Float32(xi + Float32(Float32(2.0) * Float32(Float32(pi) * Float32(uy * yi))))));
	else
		tmp = Float32(Float32(xi * cos(Float32(Float32(2.0) * Float32(uy * Float32(pi))))) + Float32(maxCos * Float32(ux * Float32(Float32(Float32(1.0) - ux) * zi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(xi, yi, zi, ux, uy, maxCos)
	tmp = single(0.0);
	if (uy <= single(0.054999999701976776))
		tmp = ((ux * ((single(1.0) - ux) * maxCos)) * zi) + (sqrt((single(1.0) + ((single(1.0) - ux) * ((ux * maxCos) * ((ux + single(-1.0)) * (ux * maxCos)))))) * (xi + (single(2.0) * (single(pi) * (uy * yi)))));
	else
		tmp = (xi * cos((single(2.0) * (uy * single(pi))))) + (maxCos * (ux * ((single(1.0) - ux) * zi)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.0549999997

   1. Initial program 99.4%

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

      1. add-log-exp 98.3%

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

   4. Applied egg-rr 98.3%

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

   5. Taylor expanded in uy around 0 89.1%

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

      1. +-commutative 89.1%

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

      2. associate-*r* 89.1%

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

      3. distribute-rgt-out 89.1%

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

   7. Simplified 89.1%

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

      1. associate-*r* 89.1%

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

      2. *-commutative 89.1%

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

      3. *-commutative 89.1%

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

      4. pow2 89.1%

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

      5. associate-*l* 89.1%

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

   9. Applied egg-rr 89.1%

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

   if 0.0549999997 < uy

   1. Initial program 96.2%

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

      1. associate-+l+ 96.2%

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

      2. associate-*l* 96.2%

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

      3. fma-define 96.2%

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

   3. Simplified 96.2%

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

   5. Taylor expanded in uy around 0 51.9%

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

      1. *-commutative 51.9%

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

   7. Simplified 51.9%

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

      1. log1p-expm1-u 51.9%

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

   9. Applied egg-rr 51.9%

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

   10. Taylor expanded in maxCos around 0 51.9%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \leq 0.054999999701976776:\\ \;\;\;\;\left(ux \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right) \cdot zi + \sqrt{1 + \left(1 - ux\right) \cdot \left(\left(ux \cdot maxCos\right) \cdot \left(\left(ux + -1\right) \cdot \left(ux \cdot maxCos\right)\right)\right)} \cdot \left(xi + 2 \cdot \left(\pi \cdot \left(uy \cdot yi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right) + maxCos \cdot \left(ux \cdot \left(\left(1 - ux\right) \cdot zi\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 59.4% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	tmp = (xi * cos((single(2.0) * (uy * single(pi))))) + (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+ 98.9%

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

   2. associate-*l* 98.9%

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

   3. fma-define 98.9%

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

3. Simplified 98.9%

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

5. Taylor expanded in uy around 0 54.7%

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

   1. *-commutative 54.7%

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

7. Simplified 54.7%

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

   1. log1p-expm1-u 54.7%

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

9. Applied egg-rr 54.7%

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

10. Taylor expanded in maxCos around 0 54.7%

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

11. Final simplification 54.7%

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

Alternative 11: 57.2% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	tmp = (xi * cos((single(2.0) * (uy * single(pi))))) + (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+ 98.9%

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

   2. associate-*l* 98.9%

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

   3. fma-define 98.9%

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

3. Simplified 98.9%

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

5. Taylor expanded in uy around 0 54.7%

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

   1. *-commutative 54.7%

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

7. Simplified 54.7%

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

   1. log1p-expm1-u 54.7%

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

9. Applied egg-rr 54.7%

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

10. Taylor expanded in ux around 0 53.3%

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

11. Final simplification 53.3%

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

Alternative 12: 13.7% accurate, 51.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	tmp = maxCos * ((single(1.0) - ux) * (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 98.9%

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

   1. expm1-log1p-u 98.9%

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

6. Applied egg-rr 98.9%

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

   1. expm1-undefine 98.9%

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

   2. sub-neg 98.9%

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

   3. log1p-undefine 98.9%

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

   4. rem-exp-log 98.9%

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

   5. associate-+r- 98.9%

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

   6. metadata-eval 98.9%

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

   7. metadata-eval 98.9%

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

8. Simplified 98.9%

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

   1. associate-*r* 98.9%

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

   2. rem-log-exp 97.9%

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

   3. add-cube-cbrt 98.7%

      \[\leadsto \mathsf{fma}\left(\left(\left(2 - ux\right) + -1\right) \cdot maxCos, ux \cdot zi, \sqrt{1 - \left(maxCos \cdot \left(ux + -1\right)\right) \cdot \left(\left(ux \cdot ux\right) \cdot \left(maxCos \cdot \left(ux + -1\right)\right)\right)} \cdot \left(\color{blue}{\left(\left(\sqrt[3]{\log \left(e^{\cos \left(\left(uy \cdot 2\right) \cdot \pi\right)}\right)} \cdot \sqrt[3]{\log \left(e^{\cos \left(\left(uy \cdot 2\right) \cdot \pi\right)}\right)}\right) \cdot \sqrt[3]{\log \left(e^{\cos \left(\left(uy \cdot 2\right) \cdot \pi\right)}\right)}\right)} \cdot xi + \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot yi\right)\right) \]

   4. pow3 98.7%

      \[\leadsto \mathsf{fma}\left(\left(\left(2 - ux\right) + -1\right) \cdot maxCos, ux \cdot zi, \sqrt{1 - \left(maxCos \cdot \left(ux + -1\right)\right) \cdot \left(\left(ux \cdot ux\right) \cdot \left(maxCos \cdot \left(ux + -1\right)\right)\right)} \cdot \left(\color{blue}{{\left(\sqrt[3]{\log \left(e^{\cos \left(\left(uy \cdot 2\right) \cdot \pi\right)}\right)}\right)}^{3}} \cdot xi + \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot yi\right)\right) \]

   5. rem-log-exp 98.8%

      \[\leadsto \mathsf{fma}\left(\left(\left(2 - ux\right) + -1\right) \cdot maxCos, ux \cdot zi, \sqrt{1 - \left(maxCos \cdot \left(ux + -1\right)\right) \cdot \left(\left(ux \cdot ux\right) \cdot \left(maxCos \cdot \left(ux + -1\right)\right)\right)} \cdot \left({\left(\sqrt[3]{\color{blue}{\cos \left(\left(uy \cdot 2\right) \cdot \pi\right)}}\right)}^{3} \cdot xi + \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot yi\right)\right) \]

   6. associate-*r* 98.8%

      \[\leadsto \mathsf{fma}\left(\left(\left(2 - ux\right) + -1\right) \cdot maxCos, ux \cdot zi, \sqrt{1 - \left(maxCos \cdot \left(ux + -1\right)\right) \cdot \left(\left(ux \cdot ux\right) \cdot \left(maxCos \cdot \left(ux + -1\right)\right)\right)} \cdot \left({\left(\sqrt[3]{\cos \color{blue}{\left(uy \cdot \left(2 \cdot \pi\right)\right)}}\right)}^{3} \cdot xi + \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot yi\right)\right) \]

10. Applied egg-rr 98.8%

    \[\leadsto \mathsf{fma}\left(\left(\left(2 - ux\right) + -1\right) \cdot maxCos, ux \cdot zi, \sqrt{1 - \left(maxCos \cdot \left(ux + -1\right)\right) \cdot \left(\left(ux \cdot ux\right) \cdot \left(maxCos \cdot \left(ux + -1\right)\right)\right)} \cdot \left(\color{blue}{{\left(\sqrt[3]{\cos \left(uy \cdot \left(2 \cdot \pi\right)\right)}\right)}^{3}} \cdot xi + \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot yi\right)\right) \]

11. Taylor expanded in zi around inf 11.7%

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

    1. associate-*r* 11.8%

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

13. Simplified 11.8%

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

14. Final simplification 11.8%

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

Alternative 13: 13.7% 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%

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

3. Simplified 98.9%

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

   1. expm1-log1p-u 98.9%

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

6. Applied egg-rr 98.9%

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

   1. expm1-undefine 98.9%

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

   2. sub-neg 98.9%

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

   3. log1p-undefine 98.9%

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

   4. rem-exp-log 98.9%

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

   5. associate-+r- 98.9%

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

   6. metadata-eval 98.9%

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

   7. metadata-eval 98.9%

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

8. Simplified 98.9%

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

   1. associate-*r* 98.9%

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

   2. rem-log-exp 97.9%

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

   3. add-cube-cbrt 98.7%

      \[\leadsto \mathsf{fma}\left(\left(\left(2 - ux\right) + -1\right) \cdot maxCos, ux \cdot zi, \sqrt{1 - \left(maxCos \cdot \left(ux + -1\right)\right) \cdot \left(\left(ux \cdot ux\right) \cdot \left(maxCos \cdot \left(ux + -1\right)\right)\right)} \cdot \left(\color{blue}{\left(\left(\sqrt[3]{\log \left(e^{\cos \left(\left(uy \cdot 2\right) \cdot \pi\right)}\right)} \cdot \sqrt[3]{\log \left(e^{\cos \left(\left(uy \cdot 2\right) \cdot \pi\right)}\right)}\right) \cdot \sqrt[3]{\log \left(e^{\cos \left(\left(uy \cdot 2\right) \cdot \pi\right)}\right)}\right)} \cdot xi + \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot yi\right)\right) \]

   4. pow3 98.7%

      \[\leadsto \mathsf{fma}\left(\left(\left(2 - ux\right) + -1\right) \cdot maxCos, ux \cdot zi, \sqrt{1 - \left(maxCos \cdot \left(ux + -1\right)\right) \cdot \left(\left(ux \cdot ux\right) \cdot \left(maxCos \cdot \left(ux + -1\right)\right)\right)} \cdot \left(\color{blue}{{\left(\sqrt[3]{\log \left(e^{\cos \left(\left(uy \cdot 2\right) \cdot \pi\right)}\right)}\right)}^{3}} \cdot xi + \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot yi\right)\right) \]

   5. rem-log-exp 98.8%

      \[\leadsto \mathsf{fma}\left(\left(\left(2 - ux\right) + -1\right) \cdot maxCos, ux \cdot zi, \sqrt{1 - \left(maxCos \cdot \left(ux + -1\right)\right) \cdot \left(\left(ux \cdot ux\right) \cdot \left(maxCos \cdot \left(ux + -1\right)\right)\right)} \cdot \left({\left(\sqrt[3]{\color{blue}{\cos \left(\left(uy \cdot 2\right) \cdot \pi\right)}}\right)}^{3} \cdot xi + \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot yi\right)\right) \]

   6. associate-*r* 98.8%

      \[\leadsto \mathsf{fma}\left(\left(\left(2 - ux\right) + -1\right) \cdot maxCos, ux \cdot zi, \sqrt{1 - \left(maxCos \cdot \left(ux + -1\right)\right) \cdot \left(\left(ux \cdot ux\right) \cdot \left(maxCos \cdot \left(ux + -1\right)\right)\right)} \cdot \left({\left(\sqrt[3]{\cos \color{blue}{\left(uy \cdot \left(2 \cdot \pi\right)\right)}}\right)}^{3} \cdot xi + \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot yi\right)\right) \]

10. Applied egg-rr 98.8%

    \[\leadsto \mathsf{fma}\left(\left(\left(2 - ux\right) + -1\right) \cdot maxCos, ux \cdot zi, \sqrt{1 - \left(maxCos \cdot \left(ux + -1\right)\right) \cdot \left(\left(ux \cdot ux\right) \cdot \left(maxCos \cdot \left(ux + -1\right)\right)\right)} \cdot \left(\color{blue}{{\left(\sqrt[3]{\cos \left(uy \cdot \left(2 \cdot \pi\right)\right)}\right)}^{3}} \cdot xi + \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot yi\right)\right) \]

11. Taylor expanded in zi around inf 11.7%

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

12. Final simplification 11.7%

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

Alternative 14: 12.3% 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%

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

3. Simplified 98.9%

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

   1. expm1-log1p-u 98.9%

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

6. Applied egg-rr 98.9%

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

   1. expm1-undefine 98.9%

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

   2. sub-neg 98.9%

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

   3. log1p-undefine 98.9%

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

   4. rem-exp-log 98.9%

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

   5. associate-+r- 98.9%

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

   6. metadata-eval 98.9%

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

   7. metadata-eval 98.9%

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

8. Simplified 98.9%

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

   1. associate-*r* 98.9%

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

   2. rem-log-exp 97.9%

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

   3. add-cube-cbrt 98.7%

      \[\leadsto \mathsf{fma}\left(\left(\left(2 - ux\right) + -1\right) \cdot maxCos, ux \cdot zi, \sqrt{1 - \left(maxCos \cdot \left(ux + -1\right)\right) \cdot \left(\left(ux \cdot ux\right) \cdot \left(maxCos \cdot \left(ux + -1\right)\right)\right)} \cdot \left(\color{blue}{\left(\left(\sqrt[3]{\log \left(e^{\cos \left(\left(uy \cdot 2\right) \cdot \pi\right)}\right)} \cdot \sqrt[3]{\log \left(e^{\cos \left(\left(uy \cdot 2\right) \cdot \pi\right)}\right)}\right) \cdot \sqrt[3]{\log \left(e^{\cos \left(\left(uy \cdot 2\right) \cdot \pi\right)}\right)}\right)} \cdot xi + \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot yi\right)\right) \]

   4. pow3 98.7%

      \[\leadsto \mathsf{fma}\left(\left(\left(2 - ux\right) + -1\right) \cdot maxCos, ux \cdot zi, \sqrt{1 - \left(maxCos \cdot \left(ux + -1\right)\right) \cdot \left(\left(ux \cdot ux\right) \cdot \left(maxCos \cdot \left(ux + -1\right)\right)\right)} \cdot \left(\color{blue}{{\left(\sqrt[3]{\log \left(e^{\cos \left(\left(uy \cdot 2\right) \cdot \pi\right)}\right)}\right)}^{3}} \cdot xi + \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot yi\right)\right) \]

   5. rem-log-exp 98.8%

      \[\leadsto \mathsf{fma}\left(\left(\left(2 - ux\right) + -1\right) \cdot maxCos, ux \cdot zi, \sqrt{1 - \left(maxCos \cdot \left(ux + -1\right)\right) \cdot \left(\left(ux \cdot ux\right) \cdot \left(maxCos \cdot \left(ux + -1\right)\right)\right)} \cdot \left({\left(\sqrt[3]{\color{blue}{\cos \left(\left(uy \cdot 2\right) \cdot \pi\right)}}\right)}^{3} \cdot xi + \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot yi\right)\right) \]

   6. associate-*r* 98.8%

      \[\leadsto \mathsf{fma}\left(\left(\left(2 - ux\right) + -1\right) \cdot maxCos, ux \cdot zi, \sqrt{1 - \left(maxCos \cdot \left(ux + -1\right)\right) \cdot \left(\left(ux \cdot ux\right) \cdot \left(maxCos \cdot \left(ux + -1\right)\right)\right)} \cdot \left({\left(\sqrt[3]{\cos \color{blue}{\left(uy \cdot \left(2 \cdot \pi\right)\right)}}\right)}^{3} \cdot xi + \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot yi\right)\right) \]

10. Applied egg-rr 98.8%

    \[\leadsto \mathsf{fma}\left(\left(\left(2 - ux\right) + -1\right) \cdot maxCos, ux \cdot zi, \sqrt{1 - \left(maxCos \cdot \left(ux + -1\right)\right) \cdot \left(\left(ux \cdot ux\right) \cdot \left(maxCos \cdot \left(ux + -1\right)\right)\right)} \cdot \left(\color{blue}{{\left(\sqrt[3]{\cos \left(uy \cdot \left(2 \cdot \pi\right)\right)}\right)}^{3}} \cdot xi + \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot yi\right)\right) \]

11. Taylor expanded in zi around inf 11.7%

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

12. Taylor expanded in ux around 0 11.1%

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

Reproduce

herbie shell --seed 2024172 
(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)))