UniformSampleCone 2

Percentage Accurate: 98.9% → 99.0%

Time: 20.1s

Alternatives: 22

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: 99.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.7%

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

   1. associate-+l+ 98.7%

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

   2. associate-*l* 98.7%

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

   3. fma-define 98.7%

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

3. Simplified 98.8%

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

5. Final simplification 98.8%

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

Alternative 2: 98.7% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (fma
  (* (- 1.0 ux) maxCos)
  (* ux zi)
  (*
   (sqrt
    (+ 1.0 (* maxCos (* (- 1.0 ux) (* (* ux ux) (* maxCos (+ ux -1.0)))))))
   (+
    (* (cos (* uy (* 2.0 PI))) xi)
    (* yi (sin (* uy (expm1 (log1p (* 2.0 PI))))))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return fmaf(((1.0f - ux) * maxCos), (ux * zi), (sqrtf((1.0f + (maxCos * ((1.0f - ux) * ((ux * ux) * (maxCos * (ux + -1.0f))))))) * ((cosf((uy * (2.0f * ((float) M_PI)))) * xi) + (yi * sinf((uy * expm1f(log1pf((2.0f * ((float) M_PI))))))))));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return fma(Float32(Float32(Float32(1.0) - ux) * maxCos), Float32(ux * zi), Float32(sqrt(Float32(Float32(1.0) + Float32(maxCos * Float32(Float32(Float32(1.0) - ux) * Float32(Float32(ux * ux) * Float32(maxCos * Float32(ux + Float32(-1.0)))))))) * Float32(Float32(cos(Float32(uy * Float32(Float32(2.0) * Float32(pi)))) * xi) + Float32(yi * sin(Float32(uy * expm1(log1p(Float32(Float32(2.0) * Float32(pi))))))))))
end

Derivation

1. Initial program 98.7%

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

3. Simplified 98.7%

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

   1. expm1-log1p-u 98.8%

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

   2. expm1-undefine 98.8%

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

6. Applied egg-rr 98.8%

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

   1. expm1-define 98.8%

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

8. Simplified 98.8%

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

9. Final simplification 98.8%

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

Alternative 3: 98.9% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.7%

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

3. Simplified 98.7%

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

5. Final simplification 98.7%

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

Alternative 4: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

3. Final simplification 98.7%

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

Alternative 5: 98.7% accurate, 1.0× speedup

Math

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

FPCore

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

   1. associate-+l+ 98.7%

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

   2. associate-*l* 98.7%

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

   3. fma-define 98.7%

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

3. Simplified 98.8%

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

5. Taylor expanded in xi around inf 98.7%

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

7. Simplified 98.7%

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

8. Final simplification 98.7%

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

Alternative 6: 98.7% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* uy (* 2.0 PI))))
   (+
    (* zi (* (- 1.0 ux) (* ux maxCos)))
    (*
     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 (zi * ((1.0f - ux) * (ux * maxCos))) + (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(zi * Float32(Float32(Float32(1.0) - ux) * Float32(ux * maxCos))) + 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 = (zi * ((single(1.0) - ux) * (ux * maxCos))) + (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 98.7%

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

3. Simplified 98.7%

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

5. Taylor expanded in xi around inf 98.7%

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

   1. distribute-rgt-out 98.7%

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

7. Simplified 98.7%

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

8. Final simplification 98.7%

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

Alternative 7: 98.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
2. 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-*r* 98.7%

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

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%

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

Alternative 8: 98.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.7%

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

3. Simplified 98.7%

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

5. Taylor expanded in ux around 0 98.5%

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

   1. fma-define 98.5%

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

   2. associate-*r* 98.5%

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

   3. *-commutative 98.5%

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

   4. associate-*r* 98.5%

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

   5. associate-*r* 98.5%

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

   6. *-commutative 98.5%

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

   7. associate-*r* 98.5%

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

7. Simplified 98.5%

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

8. Taylor expanded in ux around 0 98.6%

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

   1. mul-1-neg 98.6%

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

   2. unsub-neg 98.6%

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

10. Simplified 98.6%

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

11. Final simplification 98.6%

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

Alternative 9: 98.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.7%

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

3. Simplified 98.7%

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

5. Taylor expanded in ux around 0 98.5%

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

6. Final simplification 98.5%

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

Alternative 10: 98.7% accurate, 2.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.7%

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

   1. associate-+l+ 98.7%

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

   2. associate-*l* 98.7%

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

   3. fma-define 98.7%

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

3. Simplified 98.8%

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

5. Taylor expanded in maxCos around 0 98.5%

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

6. Final simplification 98.5%

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

Alternative 11: 95.7% 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 98.7%

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

   1. associate-+l+ 98.7%

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

   2. associate-*l* 98.7%

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

   3. fma-define 98.7%

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

3. Simplified 98.8%

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

5. Taylor expanded in ux around 0 96.6%

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

6. Final simplification 96.6%

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

Alternative 12: 96.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \left(uy \cdot \pi\right)\\ \mathbf{if}\;uy \leq 0.0008500000112690032:\\ \;\;\;\;zi \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right) + \left(xi + uy \cdot \left(-2 \cdot \left(uy \cdot \left(xi \cdot {\pi}^{2}\right)\right) + 2 \cdot \left(\pi \cdot yi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;xi \cdot \cos t\_0 + yi \cdot \sin t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* 2.0 (* uy PI))))
   (if (<= uy 0.0008500000112690032)
     (+
      (* zi (* (- 1.0 ux) (* ux maxCos)))
      (+ xi (* uy (+ (* -2.0 (* uy (* xi (pow PI 2.0)))) (* 2.0 (* PI yi))))))
     (+ (* 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));
	float tmp;
	if (uy <= 0.0008500000112690032f) {
		tmp = (zi * ((1.0f - ux) * (ux * maxCos))) + (xi + (uy * ((-2.0f * (uy * (xi * powf(((float) M_PI), 2.0f)))) + (2.0f * (((float) M_PI) * yi)))));
	} else {
		tmp = (xi * cosf(t_0)) + (yi * sinf(t_0));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(2.0) * Float32(uy * Float32(pi)))
	tmp = Float32(0.0)
	if (uy <= Float32(0.0008500000112690032))
		tmp = Float32(Float32(zi * Float32(Float32(Float32(1.0) - ux) * Float32(ux * maxCos))) + Float32(xi + Float32(uy * Float32(Float32(Float32(-2.0) * Float32(uy * Float32(xi * (Float32(pi) ^ Float32(2.0))))) + Float32(Float32(2.0) * Float32(Float32(pi) * yi))))));
	else
		tmp = Float32(Float32(xi * cos(t_0)) + Float32(yi * sin(t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(xi, yi, zi, ux, uy, maxCos)
	t_0 = single(2.0) * (uy * single(pi));
	tmp = single(0.0);
	if (uy <= single(0.0008500000112690032))
		tmp = (zi * ((single(1.0) - ux) * (ux * maxCos))) + (xi + (uy * ((single(-2.0) * (uy * (xi * (single(pi) ^ single(2.0))))) + (single(2.0) * (single(pi) * yi)))));
	else
		tmp = (xi * cos(t_0)) + (yi * sin(t_0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if uy < 8.50000011e-4

   1. Initial program 99.1%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in ux around 0 98.8%

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

      1. fma-define 98.8%

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

      2. associate-*r* 98.8%

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

      3. *-commutative 98.8%

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

      4. associate-*r* 98.8%

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

      5. associate-*r* 98.8%

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

      6. *-commutative 98.8%

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

      7. associate-*r* 98.8%

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

   7. Simplified 98.8%

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

   8. Taylor expanded in uy around 0 98.4%

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

   if 8.50000011e-4 < uy

   1. Initial program 98.0%

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

   3. Simplified 98.0%

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

   5. Taylor expanded in ux around 0 97.9%

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

      1. fma-define 98.0%

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

      2. associate-*r* 98.0%

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

      3. *-commutative 98.0%

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

      4. associate-*r* 98.0%

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

      5. associate-*r* 98.0%

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

      6. *-commutative 98.0%

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

      7. associate-*r* 98.0%

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

   7. Simplified 98.0%

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

      1. expm1-log1p-u 90.9%

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

      2. expm1-undefine 40.1%

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

   9. Applied egg-rr 40.1%

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

       1. expm1-define 90.9%

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

       2. fma-define 90.8%

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

       3. +-commutative 90.8%

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

       4. fma-define 90.9%

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

   11. Simplified 90.9%

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

   12. Taylor expanded in ux around -inf 97.6%

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

   13. Taylor expanded in ux around 0 92.2%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \leq 0.0008500000112690032:\\ \;\;\;\;zi \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right) + \left(xi + uy \cdot \left(-2 \cdot \left(uy \cdot \left(xi \cdot {\pi}^{2}\right)\right) + 2 \cdot \left(\pi \cdot yi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 88.6% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := zi \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)\\ \mathbf{if}\;uy \leq 0.02800000086426735:\\ \;\;\;\;t\_0 + \left(xi + uy \cdot \left(-2 \cdot \left(uy \cdot \left(xi \cdot {\pi}^{2}\right)\right) + 2 \cdot \left(\pi \cdot yi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 + xi \cdot \cos \left(\pi \cdot \left(uy \cdot 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* zi (* (- 1.0 ux) (* ux maxCos)))))
   (if (<= uy 0.02800000086426735)
     (+
      t_0
      (+ xi (* uy (+ (* -2.0 (* uy (* xi (pow PI 2.0)))) (* 2.0 (* PI yi))))))
     (+ t_0 (* xi (cos (* PI (* uy 2.0))))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = zi * ((1.0f - ux) * (ux * maxCos));
	float tmp;
	if (uy <= 0.02800000086426735f) {
		tmp = t_0 + (xi + (uy * ((-2.0f * (uy * (xi * powf(((float) M_PI), 2.0f)))) + (2.0f * (((float) M_PI) * yi)))));
	} else {
		tmp = t_0 + (xi * cosf((((float) M_PI) * (uy * 2.0f))));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(zi * Float32(Float32(Float32(1.0) - ux) * Float32(ux * maxCos)))
	tmp = Float32(0.0)
	if (uy <= Float32(0.02800000086426735))
		tmp = Float32(t_0 + Float32(xi + Float32(uy * Float32(Float32(Float32(-2.0) * Float32(uy * Float32(xi * (Float32(pi) ^ Float32(2.0))))) + Float32(Float32(2.0) * Float32(Float32(pi) * yi))))));
	else
		tmp = Float32(t_0 + Float32(xi * cos(Float32(Float32(pi) * Float32(uy * Float32(2.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(xi, yi, zi, ux, uy, maxCos)
	t_0 = zi * ((single(1.0) - ux) * (ux * maxCos));
	tmp = single(0.0);
	if (uy <= single(0.02800000086426735))
		tmp = t_0 + (xi + (uy * ((single(-2.0) * (uy * (xi * (single(pi) ^ single(2.0))))) + (single(2.0) * (single(pi) * yi)))));
	else
		tmp = t_0 + (xi * cos((single(pi) * (uy * single(2.0)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.0280000009

   1. Initial program 99.1%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in ux around 0 98.8%

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

      1. fma-define 98.9%

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

      2. associate-*r* 98.9%

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

      3. *-commutative 98.9%

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

      4. associate-*r* 98.9%

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

      5. associate-*r* 98.9%

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

      6. *-commutative 98.9%

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

      7. associate-*r* 98.9%

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

   7. Simplified 98.9%

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

   8. Taylor expanded in uy around 0 95.4%

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

   if 0.0280000009 < uy

   1. Initial program 96.8%

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

   3. Simplified 96.8%

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

   5. Taylor expanded in ux around 0 96.8%

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

      1. fma-define 96.8%

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

      2. associate-*r* 96.8%

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

      3. *-commutative 96.8%

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

      4. associate-*r* 96.8%

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

      5. associate-*r* 96.8%

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

      6. *-commutative 96.8%

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

      7. associate-*r* 96.8%

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

   7. Simplified 96.8%

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

   8. Taylor expanded in xi around inf 63.5%

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

      1. associate-*r* 63.5%

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

   10. Simplified 63.5%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \leq 0.02800000086426735:\\ \;\;\;\;zi \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right) + \left(xi + uy \cdot \left(-2 \cdot \left(uy \cdot \left(xi \cdot {\pi}^{2}\right)\right) + 2 \cdot \left(\pi \cdot yi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;zi \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right) + xi \cdot \cos \left(\pi \cdot \left(uy \cdot 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 85.2% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := zi \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right)\\ \mathbf{if}\;uy \leq 0.024000000208616257:\\ \;\;\;\;t\_0 + \left(xi + 2 \cdot \left(uy \cdot \left(\pi \cdot yi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 + xi \cdot \cos \left(\pi \cdot \left(uy \cdot 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* zi (* (- 1.0 ux) (* ux maxCos)))))
   (if (<= uy 0.024000000208616257)
     (+ t_0 (+ xi (* 2.0 (* uy (* PI yi)))))
     (+ t_0 (* xi (cos (* PI (* uy 2.0))))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = zi * ((1.0f - ux) * (ux * maxCos));
	float tmp;
	if (uy <= 0.024000000208616257f) {
		tmp = t_0 + (xi + (2.0f * (uy * (((float) M_PI) * yi))));
	} else {
		tmp = t_0 + (xi * cosf((((float) M_PI) * (uy * 2.0f))));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(zi * Float32(Float32(Float32(1.0) - ux) * Float32(ux * maxCos)))
	tmp = Float32(0.0)
	if (uy <= Float32(0.024000000208616257))
		tmp = Float32(t_0 + Float32(xi + Float32(Float32(2.0) * Float32(uy * Float32(Float32(pi) * yi)))));
	else
		tmp = Float32(t_0 + Float32(xi * cos(Float32(Float32(pi) * Float32(uy * Float32(2.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(xi, yi, zi, ux, uy, maxCos)
	t_0 = zi * ((single(1.0) - ux) * (ux * maxCos));
	tmp = single(0.0);
	if (uy <= single(0.024000000208616257))
		tmp = t_0 + (xi + (single(2.0) * (uy * (single(pi) * yi))));
	else
		tmp = t_0 + (xi * cos((single(pi) * (uy * single(2.0)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.0240000002

   1. Initial program 99.1%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in ux around 0 98.8%

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

      1. fma-define 98.9%

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

      2. associate-*r* 98.9%

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

      3. *-commutative 98.9%

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

      4. associate-*r* 98.9%

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

      5. associate-*r* 98.9%

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

      6. *-commutative 98.9%

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

      7. associate-*r* 98.9%

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

   7. Simplified 98.9%

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

   8. Taylor expanded in uy around 0 91.4%

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

      1. +-commutative 91.4%

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

   10. Simplified 91.4%

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

   if 0.0240000002 < uy

   1. Initial program 97.0%

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

   3. Simplified 97.0%

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

   5. Taylor expanded in ux around 0 97.0%

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

      1. fma-define 97.0%

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

      2. associate-*r* 97.0%

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

      3. *-commutative 97.0%

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

      4. associate-*r* 97.0%

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

      5. associate-*r* 97.0%

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

      6. *-commutative 97.0%

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

      7. associate-*r* 97.0%

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

   7. Simplified 97.0%

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

   8. Taylor expanded in xi around inf 62.7%

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

      1. associate-*r* 62.7%

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

   10. Simplified 62.7%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \leq 0.024000000208616257:\\ \;\;\;\;zi \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right) + \left(xi + 2 \cdot \left(uy \cdot \left(\pi \cdot yi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;zi \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right) + xi \cdot \cos \left(\pi \cdot \left(uy \cdot 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 84.4% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.024000000208616257:\\ \;\;\;\;zi \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right) + \left(xi + 2 \cdot \left(uy \cdot \left(\pi \cdot yi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;xi \cdot \cos \left(\pi \cdot \left(uy \cdot 2\right)\right) + zi \cdot \left(ux \cdot \left(ux \cdot maxCos\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (if (<= uy 0.024000000208616257)
   (+ (* zi (* (- 1.0 ux) (* ux maxCos))) (+ xi (* 2.0 (* uy (* PI yi)))))
   (+ (* xi (cos (* PI (* uy 2.0)))) (* zi (* ux (* ux maxCos))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.024000000208616257f) {
		tmp = (zi * ((1.0f - ux) * (ux * maxCos))) + (xi + (2.0f * (uy * (((float) M_PI) * yi))));
	} else {
		tmp = (xi * cosf((((float) M_PI) * (uy * 2.0f)))) + (zi * (ux * (ux * maxCos)));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.024000000208616257))
		tmp = Float32(Float32(zi * Float32(Float32(Float32(1.0) - ux) * Float32(ux * maxCos))) + Float32(xi + Float32(Float32(2.0) * Float32(uy * Float32(Float32(pi) * yi)))));
	else
		tmp = Float32(Float32(xi * cos(Float32(Float32(pi) * Float32(uy * Float32(2.0))))) + Float32(zi * Float32(ux * Float32(ux * maxCos))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(xi, yi, zi, ux, uy, maxCos)
	tmp = single(0.0);
	if (uy <= single(0.024000000208616257))
		tmp = (zi * ((single(1.0) - ux) * (ux * maxCos))) + (xi + (single(2.0) * (uy * (single(pi) * yi))));
	else
		tmp = (xi * cos((single(pi) * (uy * single(2.0))))) + (zi * (ux * (ux * maxCos)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.0240000002

   1. Initial program 99.1%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in ux around 0 98.8%

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

      1. fma-define 98.9%

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

      2. associate-*r* 98.9%

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

      3. *-commutative 98.9%

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

      4. associate-*r* 98.9%

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

      5. associate-*r* 98.9%

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

      6. *-commutative 98.9%

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

      7. associate-*r* 98.9%

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

   7. Simplified 98.9%

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

   8. Taylor expanded in uy around 0 91.4%

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

      1. +-commutative 91.4%

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

   10. Simplified 91.4%

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

   if 0.0240000002 < uy

   1. Initial program 97.0%

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

   3. Simplified 97.0%

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

   5. Taylor expanded in ux around 0 97.0%

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

      1. fma-define 97.0%

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

      2. associate-*r* 97.0%

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

      3. *-commutative 97.0%

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

      4. associate-*r* 97.0%

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

      5. associate-*r* 97.0%

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

      6. *-commutative 97.0%

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

      7. associate-*r* 97.0%

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

   7. Simplified 97.0%

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

   8. Taylor expanded in ux around inf 90.0%

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

      1. neg-mul-1 90.0%

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

   10. Simplified 90.0%

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

   11. Taylor expanded in xi around inf 57.9%

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

       1. associate-*r* 62.7%

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

   13. Simplified 57.9%

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

       1. add-sqr-sqrt -0.0%

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

       2. sqrt-unprod 58.6%

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

       3. sqr-neg 58.6%

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

       4. sqrt-prod 58.6%

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

       5. add-sqr-sqrt 58.6%

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

       6. pow1 58.6%

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

   15. Applied egg-rr 58.6%

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

       1. unpow1 58.6%

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

   17. Simplified 58.6%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \leq 0.024000000208616257:\\ \;\;\;\;zi \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot maxCos\right)\right) + \left(xi + 2 \cdot \left(uy \cdot \left(\pi \cdot yi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;xi \cdot \cos \left(\pi \cdot \left(uy \cdot 2\right)\right) + zi \cdot \left(ux \cdot \left(ux \cdot maxCos\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 82.1% accurate, 24.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

3. Simplified 98.7%

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

5. Taylor expanded in ux around 0 98.5%

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

   1. fma-define 98.5%

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

   2. associate-*r* 98.5%

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

   3. *-commutative 98.5%

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

   4. associate-*r* 98.5%

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

   5. associate-*r* 98.5%

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

   6. *-commutative 98.5%

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

   7. associate-*r* 98.5%

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

7. Simplified 98.5%

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

8. Taylor expanded in uy around 0 80.5%

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

   1. +-commutative 80.5%

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

10. Simplified 80.5%

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

11. Final simplification 80.5%

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

Alternative 17: 82.1% accurate, 24.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

3. Simplified 98.7%

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

5. Taylor expanded in ux around 0 98.5%

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

   1. fma-define 98.5%

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

   2. associate-*r* 98.5%

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

   3. *-commutative 98.5%

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

   4. associate-*r* 98.5%

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

   5. associate-*r* 98.5%

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

   6. *-commutative 98.5%

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

   7. associate-*r* 98.5%

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

7. Simplified 98.5%

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

   1. expm1-log1p-u 92.1%

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

   2. expm1-undefine 37.6%

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

9. Applied egg-rr 37.6%

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

    1. expm1-define 92.1%

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

    2. fma-define 92.1%

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

    3. +-commutative 92.1%

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

    4. fma-define 92.1%

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

11. Simplified 92.1%

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

12. Taylor expanded in uy around 0 80.4%

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

13. Final simplification 80.4%

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

Alternative 18: 74.5% accurate, 27.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

3. Simplified 98.7%

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

5. Taylor expanded in ux around 0 98.5%

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

   1. fma-define 98.5%

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

   2. associate-*r* 98.5%

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

   3. *-commutative 98.5%

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

   4. associate-*r* 98.5%

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

   5. associate-*r* 98.5%

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

   6. *-commutative 98.5%

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

   7. associate-*r* 98.5%

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

7. Simplified 98.5%

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

8. Taylor expanded in ux around inf 91.0%

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

   1. neg-mul-1 91.0%

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

10. Simplified 91.0%

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

11. Taylor expanded in uy around 0 73.9%

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

    1. +-commutative 80.5%

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

13. Simplified 73.9%

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

14. Final simplification 73.9%

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

Alternative 19: 51.5% accurate, 41.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

3. Simplified 98.7%

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

5. Taylor expanded in ux around 0 98.5%

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

   1. fma-define 98.5%

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

   2. associate-*r* 98.5%

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

   3. *-commutative 98.5%

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

   4. associate-*r* 98.5%

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

   5. associate-*r* 98.5%

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

   6. *-commutative 98.5%

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

   7. associate-*r* 98.5%

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

7. Simplified 98.5%

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

8. Taylor expanded in uy around 0 52.1%

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

9. Final simplification 52.1%

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

Alternative 20: 45.4% accurate, 51.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.7%

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

3. Simplified 98.7%

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

5. Taylor expanded in ux around 0 98.5%

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

   1. fma-define 98.5%

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

   2. associate-*r* 98.5%

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

   3. *-commutative 98.5%

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

   4. associate-*r* 98.5%

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

   5. associate-*r* 98.5%

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

   6. *-commutative 98.5%

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

   7. associate-*r* 98.5%

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

7. Simplified 98.5%

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

8. Taylor expanded in ux around inf 91.0%

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

   1. neg-mul-1 91.0%

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

10. Simplified 91.0%

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

11. Taylor expanded in uy around 0 46.8%

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

12. Final simplification 46.8%

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

Alternative 21: 13.3% 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 98.7%

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

   1. associate-+l+ 98.7%

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

   2. associate-*l* 98.7%

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

   3. fma-define 98.7%

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

3. Simplified 98.8%

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

5. Taylor expanded in zi around inf 12.5%

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

6. Final simplification 12.5%

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

Alternative 22: 11.9% 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 98.7%

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

   1. associate-+l+ 98.7%

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

   2. associate-*l* 98.7%

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

   3. fma-define 98.7%

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

3. Simplified 98.8%

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

5. Taylor expanded in zi around inf 12.5%

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

6. Taylor expanded in ux around 0 11.6%

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

Reproduce

herbie shell --seed 2024139 
(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)))