UniformSampleCone 2

Percentage Accurate: 98.9% → 98.9%

Time: 26.3s

Alternatives: 23

Speedup: N/A×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := uy \cdot \left(2 \cdot \pi\right)\\ {\left(1 + \left(1 - ux\right) \cdot \left(\left(\left(ux \cdot ux\right) \cdot \left(maxCos \cdot maxCos\right)\right) \cdot \left(ux + -1\right)\right)\right)}^{0.5} \cdot \left(yi \cdot \sin t\_0 + xi \cdot \cos t\_0\right) + \left(1 - ux\right) \cdot \left(maxCos \cdot \left(ux \cdot zi\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* uy (* 2.0 PI))))
   (+
    (*
     (pow
      (+ 1.0 (* (- 1.0 ux) (* (* (* ux ux) (* maxCos maxCos)) (+ ux -1.0))))
      0.5)
     (+ (* yi (sin t_0)) (* xi (cos t_0))))
    (* (- 1.0 ux) (* maxCos (* ux zi))))))

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 (powf((1.0f + ((1.0f - ux) * (((ux * ux) * (maxCos * maxCos)) * (ux + -1.0f)))), 0.5f) * ((yi * sinf(t_0)) + (xi * cosf(t_0)))) + ((1.0f - ux) * (maxCos * (ux * zi)));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(uy * Float32(Float32(2.0) * Float32(pi)))
	return Float32(Float32((Float32(Float32(1.0) + Float32(Float32(Float32(1.0) - ux) * Float32(Float32(Float32(ux * ux) * Float32(maxCos * maxCos)) * Float32(ux + Float32(-1.0))))) ^ Float32(0.5)) * Float32(Float32(yi * sin(t_0)) + Float32(xi * cos(t_0)))) + Float32(Float32(Float32(1.0) - ux) * Float32(maxCos * Float32(ux * zi))))
end

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	t_0 = uy * (single(2.0) * single(pi));
	tmp = (((single(1.0) + ((single(1.0) - ux) * (((ux * ux) * (maxCos * maxCos)) * (ux + single(-1.0))))) ^ single(0.5)) * ((yi * sin(t_0)) + (xi * cos(t_0)))) + ((single(1.0) - ux) * (maxCos * (ux * zi)));
end

Derivation

1. Initial program 99.0%

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

3. Simplified 99.0%

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

6. Applied egg-rr 99.0%

   \[\leadsto \color{blue}{{\left(1 + \left(1 - ux\right) \cdot \left(\left(\left(ux \cdot ux\right) \cdot \left(maxCos \cdot maxCos\right)\right) \cdot \left(ux + -1\right)\right)\right)}^{0.5} \cdot \left(yi \cdot \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) + xi \cdot \cos \left(uy \cdot \left(2 \cdot \pi\right)\right)\right) + \left(1 - ux\right) \cdot \left(maxCos \cdot \left(ux \cdot zi\right)\right)} \]
7. Add Preprocessing

Alternative 2: 98.9% accurate, 1.3× speedup

Math

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

FPCore

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

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

3. Simplified 99.0%

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

5. Final simplification 99.0%

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

Alternative 3: 98.7% accurate, 2.0× speedup

Math

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

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)) (* (* ux maxCos) (* (- 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)) + ((ux * maxCos) * ((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(xi * cos(t_0)) + Float32(Float32(yi * sin(t_0)) + Float32(Float32(ux * maxCos) * 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)) + ((ux * maxCos) * ((single(1.0) - ux) * zi)));
end

Derivation

1. Initial program 99.0%

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

3. Simplified 99.0%

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

5. Taylor expanded in maxCos around 0

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

   1. +-commutative N/A

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

   2. associate-+l+ N/A

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

   3. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(\left(xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right), \color{blue}{\left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right)}\right) \]

   4. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right), \left(\color{blue}{yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right)\right) \]

   5. cos-lowering-cos.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(xi, \mathsf{cos.f32}\left(\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \left(yi \cdot \color{blue}{\sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right)\right) \]

   6. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(xi, \mathsf{cos.f32}\left(\mathsf{*.f32}\left(2, \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \left(yi \cdot \sin \color{blue}{\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right)\right) \]

   7. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(xi, \mathsf{cos.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \mathsf{PI}\left(\right)\right)\right)\right)\right), \left(yi \cdot \sin \left(2 \cdot \color{blue}{\left(uy \cdot \mathsf{PI}\left(\right)\right)}\right) + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right)\right) \]

   8. PI-lowering-PI.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(xi, \mathsf{cos.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \mathsf{PI.f32}\left(\right)\right)\right)\right)\right), \left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right)\right) \]

   9. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(xi, \mathsf{cos.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \mathsf{PI.f32}\left(\right)\right)\right)\right)\right), \mathsf{+.f32}\left(\left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right), \color{blue}{\left(maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right)}\right)\right) \]

7. Simplified 98.9%

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

8. Final simplification 98.9%

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

Alternative 4: 97.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \left(uy \cdot \pi\right)\\ \mathbf{if}\;uy \leq 0.007499999832361937:\\ \;\;\;\;\left(1 - ux\right) \cdot \left(maxCos \cdot \left(ux \cdot zi\right)\right) + \left(xi + uy \cdot \left(yi \cdot \left(2 \cdot \pi\right) + uy \cdot \left(xi \cdot \left(-2 \cdot \left(\pi \cdot \pi\right)\right) + \left(yi \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right) \cdot \left(uy \cdot -1.3333333333333333\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{xi \cdot \cos t\_0 + yi \cdot \sin t\_0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* 2.0 (* uy PI))))
   (if (<= uy 0.007499999832361937)
     (+
      (* (- 1.0 ux) (* maxCos (* ux zi)))
      (+
       xi
       (*
        uy
        (+
         (* yi (* 2.0 PI))
         (*
          uy
          (+
           (* xi (* -2.0 (* PI PI)))
           (* (* yi (* PI (* PI PI))) (* uy -1.3333333333333333))))))))
     (/ 1.0 (/ 1.0 (+ (* 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 <= 0.007499999832361937f) {
		tmp = ((1.0f - ux) * (maxCos * (ux * zi))) + (xi + (uy * ((yi * (2.0f * ((float) M_PI))) + (uy * ((xi * (-2.0f * (((float) M_PI) * ((float) M_PI)))) + ((yi * (((float) M_PI) * (((float) M_PI) * ((float) M_PI)))) * (uy * -1.3333333333333333f)))))));
	} else {
		tmp = 1.0f / (1.0f / ((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(0.007499999832361937))
		tmp = Float32(Float32(Float32(Float32(1.0) - ux) * Float32(maxCos * Float32(ux * zi))) + Float32(xi + Float32(uy * Float32(Float32(yi * Float32(Float32(2.0) * Float32(pi))) + Float32(uy * Float32(Float32(xi * Float32(Float32(-2.0) * Float32(Float32(pi) * Float32(pi)))) + Float32(Float32(yi * Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi)))) * Float32(uy * Float32(-1.3333333333333333)))))))));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) / 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(0.007499999832361937))
		tmp = ((single(1.0) - ux) * (maxCos * (ux * zi))) + (xi + (uy * ((yi * (single(2.0) * single(pi))) + (uy * ((xi * (single(-2.0) * (single(pi) * single(pi)))) + ((yi * (single(pi) * (single(pi) * single(pi)))) * (uy * single(-1.3333333333333333))))))));
	else
		tmp = single(1.0) / (single(1.0) / ((xi * cos(t_0)) + (yi * sin(t_0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.00749999983

   1. Initial program 99.2%

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

   3. Simplified 99.2%

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

   6. Applied egg-rr 99.3%

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

   7. Taylor expanded in ux around 0

      \[\leadsto \mathsf{+.f32}\left(\color{blue}{\left(xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)}, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, ux\right), \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right)\right) \]
   8. Step-by-step derivation

      1. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(\left(xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right), \left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \mathsf{*.f32}\left(\color{blue}{\mathsf{\_.f32}\left(1, ux\right)}, \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right)\right) \]

      2. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right), \left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(\color{blue}{1}, ux\right), \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right)\right) \]

      3. cos-lowering-cos.f32 N/A

         \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(xi, \mathsf{cos.f32}\left(\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, ux\right), \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right)\right) \]

      4. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(xi, \mathsf{cos.f32}\left(\mathsf{*.f32}\left(2, \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, ux\right), \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right)\right) \]

      5. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(xi, \mathsf{cos.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \mathsf{PI}\left(\right)\right)\right)\right)\right), \left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, ux\right), \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right)\right) \]

      6. PI-lowering-PI.f32 N/A

         \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(xi, \mathsf{cos.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \mathsf{PI.f32}\left(\right)\right)\right)\right)\right), \left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, ux\right), \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right)\right) \]

      7. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(xi, \mathsf{cos.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \mathsf{PI.f32}\left(\right)\right)\right)\right)\right), \mathsf{*.f32}\left(yi, \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, \color{blue}{ux}\right), \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right)\right) \]

      8. sin-lowering-sin.f32 N/A

         \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(xi, \mathsf{cos.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \mathsf{PI.f32}\left(\right)\right)\right)\right)\right), \mathsf{*.f32}\left(yi, \mathsf{sin.f32}\left(\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, ux\right), \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right)\right) \]

      9. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(xi, \mathsf{cos.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \mathsf{PI.f32}\left(\right)\right)\right)\right)\right), \mathsf{*.f32}\left(yi, \mathsf{sin.f32}\left(\mathsf{*.f32}\left(2, \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, ux\right), \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right)\right) \]

      10. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(xi, \mathsf{cos.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \mathsf{PI.f32}\left(\right)\right)\right)\right)\right), \mathsf{*.f32}\left(yi, \mathsf{sin.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \mathsf{PI}\left(\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, ux\right), \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right)\right) \]

      11. PI-lowering-PI.f32 99.1%

          \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(xi, \mathsf{cos.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \mathsf{PI.f32}\left(\right)\right)\right)\right)\right), \mathsf{*.f32}\left(yi, \mathsf{sin.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \mathsf{PI.f32}\left(\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, ux\right), \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right)\right) \]

   9. Simplified 99.1%

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

   10. Taylor expanded in uy around 0

       \[\leadsto \mathsf{+.f32}\left(\color{blue}{\left(xi + uy \cdot \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right) + uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right)}, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, ux\right), \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right)\right) \]
   11. Step-by-step derivation

       1. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(xi, \left(uy \cdot \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right) + uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(\color{blue}{\mathsf{\_.f32}\left(1, ux\right)}, \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right)\right) \]

       2. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(uy, \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right) + uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, \color{blue}{ux}\right), \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right)\right) \]

       3. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(uy, \mathsf{+.f32}\left(\left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right), \left(uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, ux\right), \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(uy, \mathsf{+.f32}\left(\left(\left(yi \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right), \left(uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, ux\right), \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right)\right) \]

       5. associate-*r* N/A

          \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(uy, \mathsf{+.f32}\left(\left(yi \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right), \left(uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, ux\right), \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(uy, \mathsf{+.f32}\left(\left(yi \cdot \left(2 \cdot \mathsf{PI}\left(\right)\right)\right), \left(uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, ux\right), \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right)\right) \]

       7. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(uy, \mathsf{+.f32}\left(\mathsf{*.f32}\left(yi, \left(2 \cdot \mathsf{PI}\left(\right)\right)\right), \left(uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, ux\right), \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right)\right) \]

       8. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(uy, \mathsf{+.f32}\left(\mathsf{*.f32}\left(yi, \mathsf{*.f32}\left(2, \mathsf{PI}\left(\right)\right)\right), \left(uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, ux\right), \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right)\right) \]

       9. PI-lowering-PI.f32 N/A

          \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(uy, \mathsf{+.f32}\left(\mathsf{*.f32}\left(yi, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right), \left(uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, ux\right), \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right)\right) \]

       10. *-lowering-*.f32 N/A

           \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(uy, \mathsf{+.f32}\left(\mathsf{*.f32}\left(yi, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right), \mathsf{*.f32}\left(uy, \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, ux\right), \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right)\right) \]

       11. +-lowering-+.f32 N/A

           \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(uy, \mathsf{+.f32}\left(\mathsf{*.f32}\left(yi, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right), \mathsf{*.f32}\left(uy, \mathsf{+.f32}\left(\left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right), \left(\frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, ux\right), \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right)\right) \]

   12. Simplified 99.2%

       \[\leadsto \color{blue}{\left(xi + uy \cdot \left(yi \cdot \left(2 \cdot \pi\right) + uy \cdot \left(xi \cdot \left(-2 \cdot \left(\pi \cdot \pi\right)\right) + \left(yi \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right) \cdot \left(uy \cdot -1.3333333333333333\right)\right)\right)\right)} + \left(1 - ux\right) \cdot \left(maxCos \cdot \left(ux \cdot zi\right)\right) \]

   if 0.00749999983 < uy

   1. Initial program 98.0%

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

   3. Simplified 98.0%

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

   6. Applied egg-rr 98.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{1}{{\left(1 + \left(1 - ux\right) \cdot \left(\left(\left(ux \cdot ux\right) \cdot \left(maxCos \cdot maxCos\right)\right) \cdot \left(ux + -1\right)\right)\right)}^{0.5} \cdot \left(yi \cdot \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) + xi \cdot \cos \left(uy \cdot \left(2 \cdot \pi\right)\right)\right) + \left(1 - ux\right) \cdot \left(maxCos \cdot \left(ux \cdot zi\right)\right)}}} \]

   7. Taylor expanded in ux around 0

      \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(\frac{1}{xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(1, \color{blue}{\left(xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)}\right)\right) \]

      2. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\left(xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right), \color{blue}{\left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)}\right)\right)\right) \]

      3. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right), \left(\color{blue}{yi} \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right) \]

      4. cos-lowering-cos.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(xi, \mathsf{cos.f32}\left(\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(xi, \mathsf{cos.f32}\left(\mathsf{*.f32}\left(2, \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(xi, \mathsf{cos.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \mathsf{PI}\left(\right)\right)\right)\right)\right), \left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right) \]

      7. PI-lowering-PI.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(xi, \mathsf{cos.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \mathsf{PI.f32}\left(\right)\right)\right)\right)\right), \left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(xi, \mathsf{cos.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \mathsf{PI.f32}\left(\right)\right)\right)\right)\right), \mathsf{*.f32}\left(yi, \color{blue}{\sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)\right)\right) \]

      9. sin-lowering-sin.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(xi, \mathsf{cos.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \mathsf{PI.f32}\left(\right)\right)\right)\right)\right), \mathsf{*.f32}\left(yi, \mathsf{sin.f32}\left(\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(xi, \mathsf{cos.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \mathsf{PI.f32}\left(\right)\right)\right)\right)\right), \mathsf{*.f32}\left(yi, \mathsf{sin.f32}\left(\mathsf{*.f32}\left(2, \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(xi, \mathsf{cos.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \mathsf{PI.f32}\left(\right)\right)\right)\right)\right), \mathsf{*.f32}\left(yi, \mathsf{sin.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right)\right) \]

      12. PI-lowering-PI.f32 89.8%

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(\mathsf{*.f32}\left(xi, \mathsf{cos.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \mathsf{PI.f32}\left(\right)\right)\right)\right)\right), \mathsf{*.f32}\left(yi, \mathsf{sin.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \mathsf{PI.f32}\left(\right)\right)\right)\right)\right)\right)\right)\right) \]

   9. Simplified 89.8%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \leq 0.007499999832361937:\\ \;\;\;\;\left(1 - ux\right) \cdot \left(maxCos \cdot \left(ux \cdot zi\right)\right) + \left(xi + uy \cdot \left(yi \cdot \left(2 \cdot \pi\right) + uy \cdot \left(xi \cdot \left(-2 \cdot \left(\pi \cdot \pi\right)\right) + \left(yi \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right) \cdot \left(uy \cdot -1.3333333333333333\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.8% accurate, 2.1× speedup

Math

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

FPCore

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

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 (yi * sinf(t_0)) + ((xi * cosf(t_0)) + (ux * (maxCos * zi)));
}

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

3. Simplified 99.0%

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

5. Taylor expanded in ux around 0

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

   1. +-commutative N/A

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

   2. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(\left(xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right), \color{blue}{\left(maxCos \cdot \left(ux \cdot zi\right)\right)}\right) \]

7. Simplified 96.7%

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

   1. +-commutative N/A

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

   2. associate-+r+ N/A

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

   3. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(\left(maxCos \cdot \left(ux \cdot zi\right) + xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right), \color{blue}{\left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)}\right) \]

9. Applied egg-rr 96.8%

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

10. Final simplification 96.8%

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

Alternative 6: 97.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \left(uy \cdot \pi\right)\\ \mathbf{if}\;uy \leq 0.007499999832361937:\\ \;\;\;\;\left(1 - ux\right) \cdot \left(maxCos \cdot \left(ux \cdot zi\right)\right) + \left(xi + uy \cdot \left(yi \cdot \left(2 \cdot \pi\right) + uy \cdot \left(xi \cdot \left(-2 \cdot \left(\pi \cdot \pi\right)\right) + \left(yi \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right) \cdot \left(uy \cdot -1.3333333333333333\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;xi \cdot \cos t\_0 + yi \cdot \sin t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* 2.0 (* uy PI))))
   (if (<= uy 0.007499999832361937)
     (+
      (* (- 1.0 ux) (* maxCos (* ux zi)))
      (+
       xi
       (*
        uy
        (+
         (* yi (* 2.0 PI))
         (*
          uy
          (+
           (* xi (* -2.0 (* PI PI)))
           (* (* yi (* PI (* PI PI))) (* uy -1.3333333333333333))))))))
     (+ (* 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 <= 0.007499999832361937f) {
		tmp = ((1.0f - ux) * (maxCos * (ux * zi))) + (xi + (uy * ((yi * (2.0f * ((float) M_PI))) + (uy * ((xi * (-2.0f * (((float) M_PI) * ((float) M_PI)))) + ((yi * (((float) M_PI) * (((float) M_PI) * ((float) M_PI)))) * (uy * -1.3333333333333333f)))))));
	} 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(0.007499999832361937))
		tmp = Float32(Float32(Float32(Float32(1.0) - ux) * Float32(maxCos * Float32(ux * zi))) + Float32(xi + Float32(uy * Float32(Float32(yi * Float32(Float32(2.0) * Float32(pi))) + Float32(uy * Float32(Float32(xi * Float32(Float32(-2.0) * Float32(Float32(pi) * Float32(pi)))) + Float32(Float32(yi * Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi)))) * Float32(uy * Float32(-1.3333333333333333)))))))));
	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(0.007499999832361937))
		tmp = ((single(1.0) - ux) * (maxCos * (ux * zi))) + (xi + (uy * ((yi * (single(2.0) * single(pi))) + (uy * ((xi * (single(-2.0) * (single(pi) * single(pi)))) + ((yi * (single(pi) * (single(pi) * single(pi)))) * (uy * single(-1.3333333333333333))))))));
	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 < 0.00749999983

   1. Initial program 99.2%

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

   3. Simplified 99.2%

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

   6. Applied egg-rr 99.3%

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

   7. Taylor expanded in ux around 0

      \[\leadsto \mathsf{+.f32}\left(\color{blue}{\left(xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)}, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, ux\right), \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right)\right) \]
   8. Step-by-step derivation

      1. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(\left(xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right), \left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \mathsf{*.f32}\left(\color{blue}{\mathsf{\_.f32}\left(1, ux\right)}, \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right)\right) \]

      2. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right), \left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(\color{blue}{1}, ux\right), \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right)\right) \]

      3. cos-lowering-cos.f32 N/A

         \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(xi, \mathsf{cos.f32}\left(\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, ux\right), \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right)\right) \]

      4. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(xi, \mathsf{cos.f32}\left(\mathsf{*.f32}\left(2, \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, ux\right), \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right)\right) \]

      5. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(xi, \mathsf{cos.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \mathsf{PI}\left(\right)\right)\right)\right)\right), \left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, ux\right), \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right)\right) \]

      6. PI-lowering-PI.f32 N/A

         \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(xi, \mathsf{cos.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \mathsf{PI.f32}\left(\right)\right)\right)\right)\right), \left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, ux\right), \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right)\right) \]

      7. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(xi, \mathsf{cos.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \mathsf{PI.f32}\left(\right)\right)\right)\right)\right), \mathsf{*.f32}\left(yi, \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, \color{blue}{ux}\right), \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right)\right) \]

      8. sin-lowering-sin.f32 N/A

         \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(xi, \mathsf{cos.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \mathsf{PI.f32}\left(\right)\right)\right)\right)\right), \mathsf{*.f32}\left(yi, \mathsf{sin.f32}\left(\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, ux\right), \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right)\right) \]

      9. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(xi, \mathsf{cos.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \mathsf{PI.f32}\left(\right)\right)\right)\right)\right), \mathsf{*.f32}\left(yi, \mathsf{sin.f32}\left(\mathsf{*.f32}\left(2, \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, ux\right), \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right)\right) \]

      10. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(xi, \mathsf{cos.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \mathsf{PI.f32}\left(\right)\right)\right)\right)\right), \mathsf{*.f32}\left(yi, \mathsf{sin.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \mathsf{PI}\left(\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, ux\right), \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right)\right) \]

      11. PI-lowering-PI.f32 99.1%

          \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(xi, \mathsf{cos.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \mathsf{PI.f32}\left(\right)\right)\right)\right)\right), \mathsf{*.f32}\left(yi, \mathsf{sin.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \mathsf{PI.f32}\left(\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, ux\right), \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right)\right) \]

   9. Simplified 99.1%

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

   10. Taylor expanded in uy around 0

       \[\leadsto \mathsf{+.f32}\left(\color{blue}{\left(xi + uy \cdot \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right) + uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right)}, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, ux\right), \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right)\right) \]
   11. Step-by-step derivation

       1. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(xi, \left(uy \cdot \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right) + uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(\color{blue}{\mathsf{\_.f32}\left(1, ux\right)}, \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right)\right) \]

       2. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(uy, \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right) + uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, \color{blue}{ux}\right), \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right)\right) \]

       3. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(uy, \mathsf{+.f32}\left(\left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right), \left(uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, ux\right), \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(uy, \mathsf{+.f32}\left(\left(\left(yi \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right), \left(uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, ux\right), \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right)\right) \]

       5. associate-*r* N/A

          \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(uy, \mathsf{+.f32}\left(\left(yi \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right), \left(uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, ux\right), \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(uy, \mathsf{+.f32}\left(\left(yi \cdot \left(2 \cdot \mathsf{PI}\left(\right)\right)\right), \left(uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, ux\right), \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right)\right) \]

       7. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(uy, \mathsf{+.f32}\left(\mathsf{*.f32}\left(yi, \left(2 \cdot \mathsf{PI}\left(\right)\right)\right), \left(uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, ux\right), \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right)\right) \]

       8. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(uy, \mathsf{+.f32}\left(\mathsf{*.f32}\left(yi, \mathsf{*.f32}\left(2, \mathsf{PI}\left(\right)\right)\right), \left(uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, ux\right), \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right)\right) \]

       9. PI-lowering-PI.f32 N/A

          \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(uy, \mathsf{+.f32}\left(\mathsf{*.f32}\left(yi, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right), \left(uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, ux\right), \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right)\right) \]

       10. *-lowering-*.f32 N/A

           \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(uy, \mathsf{+.f32}\left(\mathsf{*.f32}\left(yi, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right), \mathsf{*.f32}\left(uy, \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, ux\right), \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right)\right) \]

       11. +-lowering-+.f32 N/A

           \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(uy, \mathsf{+.f32}\left(\mathsf{*.f32}\left(yi, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right), \mathsf{*.f32}\left(uy, \mathsf{+.f32}\left(\left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right), \left(\frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, ux\right), \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right)\right) \]

   12. Simplified 99.2%

       \[\leadsto \color{blue}{\left(xi + uy \cdot \left(yi \cdot \left(2 \cdot \pi\right) + uy \cdot \left(xi \cdot \left(-2 \cdot \left(\pi \cdot \pi\right)\right) + \left(yi \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right) \cdot \left(uy \cdot -1.3333333333333333\right)\right)\right)\right)} + \left(1 - ux\right) \cdot \left(maxCos \cdot \left(ux \cdot zi\right)\right) \]

   if 0.00749999983 < uy

   1. Initial program 98.0%

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

   3. Simplified 98.0%

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

   5. Taylor expanded in ux around 0

      \[\leadsto \color{blue}{xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   6. Step-by-step derivation

      1. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{+.f32}\left(\left(xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right), \color{blue}{\left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)}\right) \]

      2. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right), \left(\color{blue}{yi} \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      3. cos-lowering-cos.f32 N/A

         \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(xi, \mathsf{cos.f32}\left(\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(xi, \mathsf{cos.f32}\left(\mathsf{*.f32}\left(2, \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(xi, \mathsf{cos.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \mathsf{PI}\left(\right)\right)\right)\right)\right), \left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      6. PI-lowering-PI.f32 N/A

         \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(xi, \mathsf{cos.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \mathsf{PI.f32}\left(\right)\right)\right)\right)\right), \left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(xi, \mathsf{cos.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \mathsf{PI.f32}\left(\right)\right)\right)\right)\right), \mathsf{*.f32}\left(yi, \color{blue}{\sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right) \]

      8. sin-lowering-sin.f32 N/A

         \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(xi, \mathsf{cos.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \mathsf{PI.f32}\left(\right)\right)\right)\right)\right), \mathsf{*.f32}\left(yi, \mathsf{sin.f32}\left(\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(xi, \mathsf{cos.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \mathsf{PI.f32}\left(\right)\right)\right)\right)\right), \mathsf{*.f32}\left(yi, \mathsf{sin.f32}\left(\mathsf{*.f32}\left(2, \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(xi, \mathsf{cos.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \mathsf{PI.f32}\left(\right)\right)\right)\right)\right), \mathsf{*.f32}\left(yi, \mathsf{sin.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \mathsf{PI}\left(\right)\right)\right)\right)\right)\right) \]

      11. PI-lowering-PI.f32 89.8%

          \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(xi, \mathsf{cos.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \mathsf{PI.f32}\left(\right)\right)\right)\right)\right), \mathsf{*.f32}\left(yi, \mathsf{sin.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \mathsf{PI.f32}\left(\right)\right)\right)\right)\right)\right) \]

   7. Simplified 89.8%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \leq 0.007499999832361937:\\ \;\;\;\;\left(1 - ux\right) \cdot \left(maxCos \cdot \left(ux \cdot zi\right)\right) + \left(xi + uy \cdot \left(yi \cdot \left(2 \cdot \pi\right) + uy \cdot \left(xi \cdot \left(-2 \cdot \left(\pi \cdot \pi\right)\right) + \left(yi \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right) \cdot \left(uy \cdot -1.3333333333333333\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 90.0% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

3. Simplified 99.0%

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

5. Taylor expanded in ux around 0

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

   1. +-commutative N/A

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

   2. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(\left(xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right), \color{blue}{\left(maxCos \cdot \left(ux \cdot zi\right)\right)}\right) \]

7. Simplified 96.7%

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

8. Taylor expanded in uy around 0

   \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(\color{blue}{\left(xi + -2 \cdot \left({uy}^{2} \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)}, \mathsf{*.f32}\left(yi, \mathsf{sin.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \mathsf{PI.f32}\left(\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right) \]
9. Step-by-step derivation

   1. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(xi, \left(-2 \cdot \left({uy}^{2} \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)\right), \mathsf{*.f32}\left(yi, \mathsf{sin.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \mathsf{PI.f32}\left(\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right) \]

   2. associate-*r* N/A

      \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(xi, \left(\left(-2 \cdot {uy}^{2}\right) \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \mathsf{*.f32}\left(yi, \mathsf{sin.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \mathsf{PI.f32}\left(\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right) \]

   3. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(\left(-2 \cdot {uy}^{2}\right), \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \mathsf{*.f32}\left(yi, \mathsf{sin.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \mathsf{PI.f32}\left(\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right) \]

   4. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \left({uy}^{2}\right)\right), \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \mathsf{*.f32}\left(yi, \mathsf{sin.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \mathsf{PI.f32}\left(\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \left(uy \cdot uy\right)\right), \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \mathsf{*.f32}\left(yi, \mathsf{sin.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \mathsf{PI.f32}\left(\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right) \]

   6. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \mathsf{*.f32}\left(yi, \mathsf{sin.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \mathsf{PI.f32}\left(\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right) \]

   7. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \mathsf{*.f32}\left(xi, \left({\mathsf{PI}\left(\right)}^{2}\right)\right)\right)\right), \mathsf{*.f32}\left(yi, \mathsf{sin.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \mathsf{PI.f32}\left(\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \mathsf{*.f32}\left(xi, \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \mathsf{*.f32}\left(yi, \mathsf{sin.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \mathsf{PI.f32}\left(\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right) \]

   9. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \mathsf{*.f32}\left(xi, \mathsf{*.f32}\left(\mathsf{PI}\left(\right), \mathsf{PI}\left(\right)\right)\right)\right)\right), \mathsf{*.f32}\left(yi, \mathsf{sin.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \mathsf{PI.f32}\left(\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right) \]

   10. PI-lowering-PI.f32 N/A

       \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \mathsf{*.f32}\left(xi, \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{PI}\left(\right)\right)\right)\right)\right), \mathsf{*.f32}\left(yi, \mathsf{sin.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \mathsf{PI.f32}\left(\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right) \]

   11. PI-lowering-PI.f32 92.1%

       \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \mathsf{*.f32}\left(xi, \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{PI.f32}\left(\right)\right)\right)\right)\right), \mathsf{*.f32}\left(yi, \mathsf{sin.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \mathsf{PI.f32}\left(\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right) \]

10. Simplified 92.1%

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

11. Final simplification 92.1%

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

Alternative 8: 88.9% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \left(1 - ux\right) \cdot \left(maxCos \cdot \left(ux \cdot zi\right)\right) + \left(xi + uy \cdot \left(yi \cdot \left(2 \cdot \pi\right) + uy \cdot \left(xi \cdot \left(-2 \cdot \left(\pi \cdot \pi\right)\right) + \left(yi \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right) \cdot \left(uy \cdot -1.3333333333333333\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (+
  (* (- 1.0 ux) (* maxCos (* ux zi)))
  (+
   xi
   (*
    uy
    (+
     (* yi (* 2.0 PI))
     (*
      uy
      (+
       (* xi (* -2.0 (* PI PI)))
       (* (* yi (* PI (* PI PI))) (* uy -1.3333333333333333)))))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return ((1.0f - ux) * (maxCos * (ux * zi))) + (xi + (uy * ((yi * (2.0f * ((float) M_PI))) + (uy * ((xi * (-2.0f * (((float) M_PI) * ((float) M_PI)))) + ((yi * (((float) M_PI) * (((float) M_PI) * ((float) M_PI)))) * (uy * -1.3333333333333333f)))))));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return Float32(Float32(Float32(Float32(1.0) - ux) * Float32(maxCos * Float32(ux * zi))) + Float32(xi + Float32(uy * Float32(Float32(yi * Float32(Float32(2.0) * Float32(pi))) + Float32(uy * Float32(Float32(xi * Float32(Float32(-2.0) * Float32(Float32(pi) * Float32(pi)))) + Float32(Float32(yi * Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi)))) * Float32(uy * Float32(-1.3333333333333333)))))))))
end

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	tmp = ((single(1.0) - ux) * (maxCos * (ux * zi))) + (xi + (uy * ((yi * (single(2.0) * single(pi))) + (uy * ((xi * (single(-2.0) * (single(pi) * single(pi)))) + ((yi * (single(pi) * (single(pi) * single(pi)))) * (uy * single(-1.3333333333333333))))))));
end

Derivation

1. Initial program 99.0%

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

3. Simplified 99.0%

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

6. Applied egg-rr 99.0%

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

7. Taylor expanded in ux around 0

   \[\leadsto \mathsf{+.f32}\left(\color{blue}{\left(xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)}, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, ux\right), \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right)\right) \]
8. Step-by-step derivation

   1. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(\left(xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right), \left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \mathsf{*.f32}\left(\color{blue}{\mathsf{\_.f32}\left(1, ux\right)}, \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right)\right) \]

   2. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right), \left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(\color{blue}{1}, ux\right), \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right)\right) \]

   3. cos-lowering-cos.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(xi, \mathsf{cos.f32}\left(\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, ux\right), \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right)\right) \]

   4. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(xi, \mathsf{cos.f32}\left(\mathsf{*.f32}\left(2, \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, ux\right), \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right)\right) \]

   5. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(xi, \mathsf{cos.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \mathsf{PI}\left(\right)\right)\right)\right)\right), \left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, ux\right), \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right)\right) \]

   6. PI-lowering-PI.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(xi, \mathsf{cos.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \mathsf{PI.f32}\left(\right)\right)\right)\right)\right), \left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, ux\right), \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right)\right) \]

   7. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(xi, \mathsf{cos.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \mathsf{PI.f32}\left(\right)\right)\right)\right)\right), \mathsf{*.f32}\left(yi, \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, \color{blue}{ux}\right), \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right)\right) \]

   8. sin-lowering-sin.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(xi, \mathsf{cos.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \mathsf{PI.f32}\left(\right)\right)\right)\right)\right), \mathsf{*.f32}\left(yi, \mathsf{sin.f32}\left(\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, ux\right), \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right)\right) \]

   9. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(xi, \mathsf{cos.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \mathsf{PI.f32}\left(\right)\right)\right)\right)\right), \mathsf{*.f32}\left(yi, \mathsf{sin.f32}\left(\mathsf{*.f32}\left(2, \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, ux\right), \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right)\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(xi, \mathsf{cos.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \mathsf{PI.f32}\left(\right)\right)\right)\right)\right), \mathsf{*.f32}\left(yi, \mathsf{sin.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \mathsf{PI}\left(\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, ux\right), \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right)\right) \]

   11. PI-lowering-PI.f32 98.9%

       \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(xi, \mathsf{cos.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \mathsf{PI.f32}\left(\right)\right)\right)\right)\right), \mathsf{*.f32}\left(yi, \mathsf{sin.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \mathsf{PI.f32}\left(\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, ux\right), \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right)\right) \]

9. Simplified 98.9%

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

10. Taylor expanded in uy around 0

    \[\leadsto \mathsf{+.f32}\left(\color{blue}{\left(xi + uy \cdot \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right) + uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right)}, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, ux\right), \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right)\right) \]
11. Step-by-step derivation

    1. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(xi, \left(uy \cdot \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right) + uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(\color{blue}{\mathsf{\_.f32}\left(1, ux\right)}, \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right)\right) \]

    2. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(uy, \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right) + uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, \color{blue}{ux}\right), \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right)\right) \]

    3. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(uy, \mathsf{+.f32}\left(\left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right), \left(uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, ux\right), \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right)\right) \]

    4. *-commutative N/A

       \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(uy, \mathsf{+.f32}\left(\left(\left(yi \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right), \left(uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, ux\right), \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right)\right) \]

    5. associate-*r* N/A

       \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(uy, \mathsf{+.f32}\left(\left(yi \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right), \left(uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, ux\right), \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right)\right) \]

    6. *-commutative N/A

       \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(uy, \mathsf{+.f32}\left(\left(yi \cdot \left(2 \cdot \mathsf{PI}\left(\right)\right)\right), \left(uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, ux\right), \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right)\right) \]

    7. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(uy, \mathsf{+.f32}\left(\mathsf{*.f32}\left(yi, \left(2 \cdot \mathsf{PI}\left(\right)\right)\right), \left(uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, ux\right), \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right)\right) \]

    8. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(uy, \mathsf{+.f32}\left(\mathsf{*.f32}\left(yi, \mathsf{*.f32}\left(2, \mathsf{PI}\left(\right)\right)\right), \left(uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, ux\right), \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right)\right) \]

    9. PI-lowering-PI.f32 N/A

       \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(uy, \mathsf{+.f32}\left(\mathsf{*.f32}\left(yi, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right), \left(uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, ux\right), \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right)\right) \]

    10. *-lowering-*.f32 N/A

        \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(uy, \mathsf{+.f32}\left(\mathsf{*.f32}\left(yi, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right), \mathsf{*.f32}\left(uy, \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, ux\right), \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right)\right) \]

    11. +-lowering-+.f32 N/A

        \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(uy, \mathsf{+.f32}\left(\mathsf{*.f32}\left(yi, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right), \mathsf{*.f32}\left(uy, \mathsf{+.f32}\left(\left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right), \left(\frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, ux\right), \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right)\right) \]

12. Simplified 90.5%

    \[\leadsto \color{blue}{\left(xi + uy \cdot \left(yi \cdot \left(2 \cdot \pi\right) + uy \cdot \left(xi \cdot \left(-2 \cdot \left(\pi \cdot \pi\right)\right) + \left(yi \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right) \cdot \left(uy \cdot -1.3333333333333333\right)\right)\right)\right)} + \left(1 - ux\right) \cdot \left(maxCos \cdot \left(ux \cdot zi\right)\right) \]

13. Final simplification 90.5%

    \[\leadsto \left(1 - ux\right) \cdot \left(maxCos \cdot \left(ux \cdot zi\right)\right) + \left(xi + uy \cdot \left(yi \cdot \left(2 \cdot \pi\right) + uy \cdot \left(xi \cdot \left(-2 \cdot \left(\pi \cdot \pi\right)\right) + \left(yi \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right) \cdot \left(uy \cdot -1.3333333333333333\right)\right)\right)\right) \]
14. Add Preprocessing

Alternative 9: 86.2% accurate, 12.5× speedup

Math

\[\begin{array}{l} \\ \left(xi + maxCos \cdot \left(ux \cdot zi\right)\right) + uy \cdot \left(2 \cdot \left(yi \cdot \pi\right) + uy \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(xi \cdot -2\right) + -1.3333333333333333 \cdot \left(\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(yi \cdot uy\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (+
  (+ xi (* maxCos (* ux zi)))
  (*
   uy
   (+
    (* 2.0 (* yi PI))
    (*
     uy
     (+
      (* (* PI PI) (* xi -2.0))
      (* -1.3333333333333333 (* (* PI (* PI PI)) (* yi uy)))))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return (xi + (maxCos * (ux * zi))) + (uy * ((2.0f * (yi * ((float) M_PI))) + (uy * (((((float) M_PI) * ((float) M_PI)) * (xi * -2.0f)) + (-1.3333333333333333f * ((((float) M_PI) * (((float) M_PI) * ((float) M_PI))) * (yi * uy)))))));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return Float32(Float32(xi + Float32(maxCos * Float32(ux * zi))) + Float32(uy * Float32(Float32(Float32(2.0) * Float32(yi * Float32(pi))) + Float32(uy * Float32(Float32(Float32(Float32(pi) * Float32(pi)) * Float32(xi * Float32(-2.0))) + Float32(Float32(-1.3333333333333333) * Float32(Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi))) * Float32(yi * uy))))))))
end

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	tmp = (xi + (maxCos * (ux * zi))) + (uy * ((single(2.0) * (yi * single(pi))) + (uy * (((single(pi) * single(pi)) * (xi * single(-2.0))) + (single(-1.3333333333333333) * ((single(pi) * (single(pi) * single(pi))) * (yi * uy)))))));
end

Derivation

1. Initial program 99.0%

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

3. Simplified 99.0%

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

5. Taylor expanded in ux around 0

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

   1. +-commutative N/A

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

   2. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(\left(xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right), \color{blue}{\left(maxCos \cdot \left(ux \cdot zi\right)\right)}\right) \]

7. Simplified 96.7%

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

8. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{xi + \left(maxCos \cdot \left(ux \cdot zi\right) + uy \cdot \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right) + uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right)} \]
9. Step-by-step derivation

   1. associate-+r+ N/A

      \[\leadsto \left(xi + maxCos \cdot \left(ux \cdot zi\right)\right) + \color{blue}{uy \cdot \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right) + uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)} \]

   2. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(\left(xi + maxCos \cdot \left(ux \cdot zi\right)\right), \color{blue}{\left(uy \cdot \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right) + uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right)}\right) \]

   3. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(xi, \left(maxCos \cdot \left(ux \cdot zi\right)\right)\right), \left(\color{blue}{uy} \cdot \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right) + uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right)\right) \]

   4. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(maxCos, \left(ux \cdot zi\right)\right)\right), \left(uy \cdot \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right) + uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right)\right) \]

   5. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right), \left(uy \cdot \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right) + uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right)\right) \]

   6. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right), \mathsf{*.f32}\left(uy, \color{blue}{\left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right) + uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)}\right)\right) \]

   7. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right), \mathsf{*.f32}\left(uy, \mathsf{+.f32}\left(\left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{\left(uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)}\right)\right)\right) \]

   8. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right), \mathsf{*.f32}\left(uy, \mathsf{+.f32}\left(\mathsf{*.f32}\left(2, \left(yi \cdot \mathsf{PI}\left(\right)\right)\right), \left(\color{blue}{uy} \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right)\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right), \mathsf{*.f32}\left(uy, \mathsf{+.f32}\left(\mathsf{*.f32}\left(2, \left(\mathsf{PI}\left(\right) \cdot yi\right)\right), \left(uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right)\right)\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right), \mathsf{*.f32}\left(uy, \mathsf{+.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(\mathsf{PI}\left(\right), yi\right)\right), \left(uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right)\right)\right) \]

   11. PI-lowering-PI.f32 N/A

       \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right), \mathsf{*.f32}\left(uy, \mathsf{+.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), yi\right)\right), \left(uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right)\right)\right) \]

10. Simplified 88.6%

    \[\leadsto \color{blue}{\left(xi + maxCos \cdot \left(ux \cdot zi\right)\right) + uy \cdot \left(2 \cdot \left(\pi \cdot yi\right) + uy \cdot \left(\left(-2 \cdot xi\right) \cdot \left(\pi \cdot \pi\right) + -1.3333333333333333 \cdot \left(\left(uy \cdot yi\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\right)} \]

11. Final simplification 88.6%

    \[\leadsto \left(xi + maxCos \cdot \left(ux \cdot zi\right)\right) + uy \cdot \left(2 \cdot \left(yi \cdot \pi\right) + uy \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(xi \cdot -2\right) + -1.3333333333333333 \cdot \left(\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(yi \cdot uy\right)\right)\right)\right) \]
12. Add Preprocessing

Alternative 10: 85.0% accurate, 14.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

3. Simplified 99.0%

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

5. Taylor expanded in uy around 0

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

7. Simplified 87.9%

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

8. Taylor expanded in maxCos around 0

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

   1. +-commutative N/A

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

   2. +-commutative N/A

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

   3. associate-+l+ N/A

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

   4. +-commutative N/A

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

   5. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(\left(uy \cdot \left(-2 \cdot \left(uy \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right), \color{blue}{\left(xi + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right)}\right) \]

10. Simplified 87.7%

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

11. Taylor expanded in xi around inf

    \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(uy, \color{blue}{\left(xi \cdot \left(-2 \cdot \left(uy \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 2 \cdot \frac{yi \cdot \mathsf{PI}\left(\right)}{xi}\right)\right)}\right), \mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(\mathsf{*.f32}\left(maxCos, ux\right), \mathsf{*.f32}\left(zi, \mathsf{\_.f32}\left(1, ux\right)\right)\right)\right)\right) \]
12. Step-by-step derivation

    1. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(xi, \left(-2 \cdot \left(uy \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 2 \cdot \frac{yi \cdot \mathsf{PI}\left(\right)}{xi}\right)\right)\right), \mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(\mathsf{*.f32}\left(maxCos, ux\right), \mathsf{*.f32}\left(zi, \mathsf{\_.f32}\left(1, ux\right)\right)\right)\right)\right) \]

    2. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(xi, \mathsf{+.f32}\left(\left(-2 \cdot \left(uy \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right), \left(2 \cdot \frac{yi \cdot \mathsf{PI}\left(\right)}{xi}\right)\right)\right)\right), \mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(\mathsf{*.f32}\left(maxCos, ux\right), \mathsf{*.f32}\left(zi, \mathsf{\_.f32}\left(1, ux\right)\right)\right)\right)\right) \]

    3. *-commutative N/A

       \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(xi, \mathsf{+.f32}\left(\left(\left(uy \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot -2\right), \left(2 \cdot \frac{yi \cdot \mathsf{PI}\left(\right)}{xi}\right)\right)\right)\right), \mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(\mathsf{*.f32}\left(maxCos, ux\right), \mathsf{*.f32}\left(zi, \mathsf{\_.f32}\left(1, ux\right)\right)\right)\right)\right) \]

    4. associate-*l* N/A

       \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(xi, \mathsf{+.f32}\left(\left(uy \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot -2\right)\right), \left(2 \cdot \frac{yi \cdot \mathsf{PI}\left(\right)}{xi}\right)\right)\right)\right), \mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(\mathsf{*.f32}\left(maxCos, ux\right), \mathsf{*.f32}\left(zi, \mathsf{\_.f32}\left(1, ux\right)\right)\right)\right)\right) \]

    5. *-commutative N/A

       \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(xi, \mathsf{+.f32}\left(\left(uy \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right), \left(2 \cdot \frac{yi \cdot \mathsf{PI}\left(\right)}{xi}\right)\right)\right)\right), \mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(\mathsf{*.f32}\left(maxCos, ux\right), \mathsf{*.f32}\left(zi, \mathsf{\_.f32}\left(1, ux\right)\right)\right)\right)\right) \]

    6. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(xi, \mathsf{+.f32}\left(\mathsf{*.f32}\left(uy, \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right), \left(2 \cdot \frac{yi \cdot \mathsf{PI}\left(\right)}{xi}\right)\right)\right)\right), \mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(\mathsf{*.f32}\left(maxCos, ux\right), \mathsf{*.f32}\left(zi, \mathsf{\_.f32}\left(1, ux\right)\right)\right)\right)\right) \]

    7. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(xi, \mathsf{+.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(-2, \left({\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \left(2 \cdot \frac{yi \cdot \mathsf{PI}\left(\right)}{xi}\right)\right)\right)\right), \mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(\mathsf{*.f32}\left(maxCos, ux\right), \mathsf{*.f32}\left(zi, \mathsf{\_.f32}\left(1, ux\right)\right)\right)\right)\right) \]

    8. unpow2 N/A

       \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(xi, \mathsf{+.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(-2, \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right), \left(2 \cdot \frac{yi \cdot \mathsf{PI}\left(\right)}{xi}\right)\right)\right)\right), \mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(\mathsf{*.f32}\left(maxCos, ux\right), \mathsf{*.f32}\left(zi, \mathsf{\_.f32}\left(1, ux\right)\right)\right)\right)\right) \]

    9. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(xi, \mathsf{+.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(\mathsf{PI}\left(\right), \mathsf{PI}\left(\right)\right)\right)\right), \left(2 \cdot \frac{yi \cdot \mathsf{PI}\left(\right)}{xi}\right)\right)\right)\right), \mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(\mathsf{*.f32}\left(maxCos, ux\right), \mathsf{*.f32}\left(zi, \mathsf{\_.f32}\left(1, ux\right)\right)\right)\right)\right) \]

    10. PI-lowering-PI.f32 N/A

        \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(xi, \mathsf{+.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{PI}\left(\right)\right)\right)\right), \left(2 \cdot \frac{yi \cdot \mathsf{PI}\left(\right)}{xi}\right)\right)\right)\right), \mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(\mathsf{*.f32}\left(maxCos, ux\right), \mathsf{*.f32}\left(zi, \mathsf{\_.f32}\left(1, ux\right)\right)\right)\right)\right) \]

    11. PI-lowering-PI.f32 N/A

        \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(xi, \mathsf{+.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{PI.f32}\left(\right)\right)\right)\right), \left(2 \cdot \frac{yi \cdot \mathsf{PI}\left(\right)}{xi}\right)\right)\right)\right), \mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(\mathsf{*.f32}\left(maxCos, ux\right), \mathsf{*.f32}\left(zi, \mathsf{\_.f32}\left(1, ux\right)\right)\right)\right)\right) \]

    12. *-lowering-*.f32 N/A

        \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(xi, \mathsf{+.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{*.f32}\left(2, \left(\frac{yi \cdot \mathsf{PI}\left(\right)}{xi}\right)\right)\right)\right)\right), \mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(\mathsf{*.f32}\left(maxCos, ux\right), \mathsf{*.f32}\left(zi, \mathsf{\_.f32}\left(1, ux\right)\right)\right)\right)\right) \]

    13. /-lowering-/.f32 N/A

        \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(xi, \mathsf{+.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{*.f32}\left(2, \mathsf{/.f32}\left(\left(yi \cdot \mathsf{PI}\left(\right)\right), xi\right)\right)\right)\right)\right), \mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(\mathsf{*.f32}\left(maxCos, ux\right), \mathsf{*.f32}\left(zi, \mathsf{\_.f32}\left(1, ux\right)\right)\right)\right)\right) \]

    14. *-lowering-*.f32 N/A

        \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(xi, \mathsf{+.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{*.f32}\left(2, \mathsf{/.f32}\left(\mathsf{*.f32}\left(yi, \mathsf{PI}\left(\right)\right), xi\right)\right)\right)\right)\right), \mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(\mathsf{*.f32}\left(maxCos, ux\right), \mathsf{*.f32}\left(zi, \mathsf{\_.f32}\left(1, ux\right)\right)\right)\right)\right) \]

    15. PI-lowering-PI.f32 87.7%

        \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(xi, \mathsf{+.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{*.f32}\left(2, \mathsf{/.f32}\left(\mathsf{*.f32}\left(yi, \mathsf{PI.f32}\left(\right)\right), xi\right)\right)\right)\right)\right), \mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(\mathsf{*.f32}\left(maxCos, ux\right), \mathsf{*.f32}\left(zi, \mathsf{\_.f32}\left(1, ux\right)\right)\right)\right)\right) \]

13. Simplified 87.7%

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

14. Final simplification 87.7%

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

Alternative 11: 85.4% accurate, 15.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

3. Simplified 99.0%

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

5. Taylor expanded in uy around 0

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

7. Simplified 87.9%

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

8. Taylor expanded in maxCos around 0

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

   1. +-commutative N/A

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

   2. +-commutative N/A

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

   3. associate-+l+ N/A

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

   4. +-commutative N/A

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

   5. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(\left(uy \cdot \left(-2 \cdot \left(uy \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right), \color{blue}{\left(xi + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right)}\right) \]

10. Simplified 87.7%

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

11. Final simplification 87.7%

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

Alternative 12: 82.8% accurate, 18.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

3. Simplified 99.0%

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

5. Taylor expanded in ux around 0

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

   1. +-commutative N/A

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

   2. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(\left(xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right), \color{blue}{\left(maxCos \cdot \left(ux \cdot zi\right)\right)}\right) \]

7. Simplified 96.7%

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

8. Taylor expanded in uy around 0

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

   1. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(xi, \color{blue}{\left(maxCos \cdot \left(ux \cdot zi\right) + uy \cdot \left(-2 \cdot \left(uy \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right)}\right) \]

   2. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(xi, \mathsf{+.f32}\left(\left(maxCos \cdot \left(ux \cdot zi\right)\right), \color{blue}{\left(uy \cdot \left(-2 \cdot \left(uy \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right)}\right)\right) \]

   3. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(xi, \mathsf{+.f32}\left(\mathsf{*.f32}\left(maxCos, \left(ux \cdot zi\right)\right), \left(\color{blue}{uy} \cdot \left(-2 \cdot \left(uy \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right) \]

   4. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(xi, \mathsf{+.f32}\left(\mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right), \left(uy \cdot \left(-2 \cdot \left(uy \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right) \]

   5. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(xi, \mathsf{+.f32}\left(\mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right), \mathsf{*.f32}\left(uy, \color{blue}{\left(-2 \cdot \left(uy \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)\right) \]

   6. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(xi, \mathsf{+.f32}\left(\mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right), \mathsf{*.f32}\left(uy, \mathsf{+.f32}\left(\left(-2 \cdot \left(uy \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \color{blue}{\left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)\right)\right) \]

   7. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(xi, \mathsf{+.f32}\left(\mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right), \mathsf{*.f32}\left(uy, \mathsf{+.f32}\left(\mathsf{*.f32}\left(-2, \left(uy \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \left(\color{blue}{2} \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right) \]

   8. associate-*r* N/A

      \[\leadsto \mathsf{+.f32}\left(xi, \mathsf{+.f32}\left(\mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right), \mathsf{*.f32}\left(uy, \mathsf{+.f32}\left(\mathsf{*.f32}\left(-2, \left(\left(uy \cdot xi\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right), \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right) \]

   9. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(xi, \mathsf{+.f32}\left(\mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right), \mathsf{*.f32}\left(uy, \mathsf{+.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(\left(uy \cdot xi\right), \left({\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{+.f32}\left(xi, \mathsf{+.f32}\left(\mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right), \mathsf{*.f32}\left(uy, \mathsf{+.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(\mathsf{*.f32}\left(uy, xi\right), \left({\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{+.f32}\left(xi, \mathsf{+.f32}\left(\mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right), \mathsf{*.f32}\left(uy, \mathsf{+.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(\mathsf{*.f32}\left(uy, xi\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right), \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right) \]

   12. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{+.f32}\left(xi, \mathsf{+.f32}\left(\mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right), \mathsf{*.f32}\left(uy, \mathsf{+.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(\mathsf{*.f32}\left(uy, xi\right), \mathsf{*.f32}\left(\mathsf{PI}\left(\right), \mathsf{PI}\left(\right)\right)\right)\right), \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right) \]

   13. PI-lowering-PI.f32 N/A

       \[\leadsto \mathsf{+.f32}\left(xi, \mathsf{+.f32}\left(\mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right), \mathsf{*.f32}\left(uy, \mathsf{+.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(\mathsf{*.f32}\left(uy, xi\right), \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{PI}\left(\right)\right)\right)\right), \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right) \]

   14. PI-lowering-PI.f32 N/A

       \[\leadsto \mathsf{+.f32}\left(xi, \mathsf{+.f32}\left(\mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right), \mathsf{*.f32}\left(uy, \mathsf{+.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(\mathsf{*.f32}\left(uy, xi\right), \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{PI.f32}\left(\right)\right)\right)\right), \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right) \]

   15. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{+.f32}\left(xi, \mathsf{+.f32}\left(\mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right), \mathsf{*.f32}\left(uy, \mathsf{+.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(\mathsf{*.f32}\left(uy, xi\right), \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{*.f32}\left(2, \color{blue}{\left(yi \cdot \mathsf{PI}\left(\right)\right)}\right)\right)\right)\right)\right) \]

   16. *-commutative N/A

       \[\leadsto \mathsf{+.f32}\left(xi, \mathsf{+.f32}\left(\mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right), \mathsf{*.f32}\left(uy, \mathsf{+.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(\mathsf{*.f32}\left(uy, xi\right), \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{*.f32}\left(2, \left(\mathsf{PI}\left(\right) \cdot \color{blue}{yi}\right)\right)\right)\right)\right)\right) \]

10. Simplified 85.9%

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

11. Final simplification 85.9%

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

Alternative 13: 59.5% accurate, 20.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := xi + \left(-2 \cdot \left(uy \cdot uy\right)\right) \cdot \left(xi \cdot \left(\pi \cdot \pi\right)\right)\\ \mathbf{if}\;xi \leq -1.4999999523982838 \cdot 10^{-22}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;xi \leq 4.720000108707208 \cdot 10^{-19}:\\ \;\;\;\;2 \cdot \left(yi \cdot \left(uy \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ xi (* (* -2.0 (* uy uy)) (* xi (* PI PI))))))
   (if (<= xi -1.4999999523982838e-22)
     t_0
     (if (<= xi 4.720000108707208e-19) (* 2.0 (* yi (* uy PI))) t_0))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = xi + ((-2.0f * (uy * uy)) * (xi * (((float) M_PI) * ((float) M_PI))));
	float tmp;
	if (xi <= -1.4999999523982838e-22f) {
		tmp = t_0;
	} else if (xi <= 4.720000108707208e-19f) {
		tmp = 2.0f * (yi * (uy * ((float) M_PI)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(xi + Float32(Float32(Float32(-2.0) * Float32(uy * uy)) * Float32(xi * Float32(Float32(pi) * Float32(pi)))))
	tmp = Float32(0.0)
	if (xi <= Float32(-1.4999999523982838e-22))
		tmp = t_0;
	elseif (xi <= Float32(4.720000108707208e-19))
		tmp = Float32(Float32(2.0) * Float32(yi * Float32(uy * Float32(pi))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(xi, yi, zi, ux, uy, maxCos)
	t_0 = xi + ((single(-2.0) * (uy * uy)) * (xi * (single(pi) * single(pi))));
	tmp = single(0.0);
	if (xi <= single(-1.4999999523982838e-22))
		tmp = t_0;
	elseif (xi <= single(4.720000108707208e-19))
		tmp = single(2.0) * (yi * (uy * single(pi)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if xi < -1.5e-22 or 4.7200001e-19 < xi

   1. Initial program 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in uy around 0

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

   7. Simplified 89.0%

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

   8. Taylor expanded in maxCos around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{+.f32}\left(\left(uy \cdot \left(-2 \cdot \left(uy \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right), \color{blue}{\left(xi + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right)}\right) \]

   10. Simplified 88.7%

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

   11. Taylor expanded in uy around inf

       \[\leadsto \mathsf{+.f32}\left(\color{blue}{\left(-2 \cdot \left({uy}^{2} \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)}, \mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(\mathsf{*.f32}\left(maxCos, ux\right), \mathsf{*.f32}\left(zi, \mathsf{\_.f32}\left(1, ux\right)\right)\right)\right)\right) \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{+.f32}\left(\left(-2 \cdot \left(\left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {uy}^{2}\right)\right), \mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(\mathsf{*.f32}\left(maxCos, ux\right), \mathsf{*.f32}\left(zi, \mathsf{\_.f32}\left(1, ux\right)\right)\right)\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{+.f32}\left(\left(\left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot {uy}^{2}\right), \mathsf{+.f32}\left(\color{blue}{xi}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(maxCos, ux\right), \mathsf{*.f32}\left(zi, \mathsf{\_.f32}\left(1, ux\right)\right)\right)\right)\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{+.f32}\left(\left(\left(\left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot -2\right) \cdot {uy}^{2}\right), \mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(\mathsf{*.f32}\left(maxCos, ux\right), \mathsf{*.f32}\left(zi, \mathsf{\_.f32}\left(1, ux\right)\right)\right)\right)\right) \]

       4. associate-*l* N/A

          \[\leadsto \mathsf{+.f32}\left(\left(\left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \left(-2 \cdot {uy}^{2}\right)\right), \mathsf{+.f32}\left(\color{blue}{xi}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(maxCos, ux\right), \mathsf{*.f32}\left(zi, \mathsf{\_.f32}\left(1, ux\right)\right)\right)\right)\right) \]

       5. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(\left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right), \left(-2 \cdot {uy}^{2}\right)\right), \mathsf{+.f32}\left(\color{blue}{xi}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(maxCos, ux\right), \mathsf{*.f32}\left(zi, \mathsf{\_.f32}\left(1, ux\right)\right)\right)\right)\right) \]

       6. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(xi, \left({\mathsf{PI}\left(\right)}^{2}\right)\right), \left(-2 \cdot {uy}^{2}\right)\right), \mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(\mathsf{*.f32}\left(maxCos, ux\right), \mathsf{*.f32}\left(zi, \mathsf{\_.f32}\left(1, ux\right)\right)\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(xi, \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right), \left(-2 \cdot {uy}^{2}\right)\right), \mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(\mathsf{*.f32}\left(maxCos, ux\right), \mathsf{*.f32}\left(zi, \mathsf{\_.f32}\left(1, ux\right)\right)\right)\right)\right) \]

       8. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(xi, \mathsf{*.f32}\left(\mathsf{PI}\left(\right), \mathsf{PI}\left(\right)\right)\right), \left(-2 \cdot {uy}^{2}\right)\right), \mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(\mathsf{*.f32}\left(maxCos, ux\right), \mathsf{*.f32}\left(zi, \mathsf{\_.f32}\left(1, ux\right)\right)\right)\right)\right) \]

       9. PI-lowering-PI.f32 N/A

          \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(xi, \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{PI}\left(\right)\right)\right), \left(-2 \cdot {uy}^{2}\right)\right), \mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(\mathsf{*.f32}\left(maxCos, ux\right), \mathsf{*.f32}\left(zi, \mathsf{\_.f32}\left(1, ux\right)\right)\right)\right)\right) \]

       10. PI-lowering-PI.f32 N/A

           \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(xi, \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{PI.f32}\left(\right)\right)\right), \left(-2 \cdot {uy}^{2}\right)\right), \mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(\mathsf{*.f32}\left(maxCos, ux\right), \mathsf{*.f32}\left(zi, \mathsf{\_.f32}\left(1, ux\right)\right)\right)\right)\right) \]

       11. *-lowering-*.f32 N/A

           \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(xi, \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{PI.f32}\left(\right)\right)\right), \mathsf{*.f32}\left(-2, \left({uy}^{2}\right)\right)\right), \mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(\mathsf{*.f32}\left(maxCos, ux\right), \mathsf{*.f32}\left(zi, \mathsf{\_.f32}\left(1, ux\right)\right)\right)\right)\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(xi, \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{PI.f32}\left(\right)\right)\right), \mathsf{*.f32}\left(-2, \left(uy \cdot uy\right)\right)\right), \mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(\mathsf{*.f32}\left(maxCos, ux\right), \mathsf{*.f32}\left(zi, \mathsf{\_.f32}\left(1, ux\right)\right)\right)\right)\right) \]

       13. *-lowering-*.f32 76.5%

           \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(xi, \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{PI.f32}\left(\right)\right)\right), \mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right)\right), \mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(\mathsf{*.f32}\left(maxCos, ux\right), \mathsf{*.f32}\left(zi, \mathsf{\_.f32}\left(1, ux\right)\right)\right)\right)\right) \]

   13. Simplified 76.5%

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

   14. Taylor expanded in maxCos around 0

       \[\leadsto \color{blue}{xi + -2 \cdot \left({uy}^{2} \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
   15. Step-by-step derivation

       1. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{+.f32}\left(xi, \color{blue}{\left(-2 \cdot \left({uy}^{2} \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)}\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{+.f32}\left(xi, \left(\left(-2 \cdot {uy}^{2}\right) \cdot \color{blue}{\left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)}\right)\right) \]

       3. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(\left(-2 \cdot {uy}^{2}\right), \color{blue}{\left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)}\right)\right) \]

       4. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \left({uy}^{2}\right)\right), \left(\color{blue}{xi} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \left(uy \cdot uy\right)\right), \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right) \]

       6. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right) \]

       7. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \mathsf{*.f32}\left(xi, \color{blue}{\left({\mathsf{PI}\left(\right)}^{2}\right)}\right)\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \mathsf{*.f32}\left(xi, \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right)\right) \]

       9. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \mathsf{*.f32}\left(xi, \mathsf{*.f32}\left(\mathsf{PI}\left(\right), \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right)\right) \]

       10. PI-lowering-PI.f32 N/A

           \[\leadsto \mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \mathsf{*.f32}\left(xi, \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

       11. PI-lowering-PI.f32 73.5%

           \[\leadsto \mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \mathsf{*.f32}\left(xi, \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{PI.f32}\left(\right)\right)\right)\right)\right) \]

   16. Simplified 73.5%

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

   if -1.5e-22 < xi < 4.7200001e-19

   1. Initial program 98.6%

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

   3. Simplified 98.6%

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

   5. Taylor expanded in uy around 0

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

   7. Simplified 86.1%

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

   8. Taylor expanded in maxCos around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{+.f32}\left(\left(uy \cdot \left(-2 \cdot \left(uy \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right), \color{blue}{\left(xi + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right)}\right) \]

   10. Simplified 86.0%

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

   11. Taylor expanded in yi around inf

       \[\leadsto \color{blue}{2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   12. Step-by-step derivation

       1. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(2, \color{blue}{\left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{*.f32}\left(2, \left(\left(yi \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{uy}\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{*.f32}\left(2, \left(yi \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot uy\right)}\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{*.f32}\left(2, \left(yi \cdot \left(uy \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right) \]

       5. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(2, \mathsf{*.f32}\left(yi, \color{blue}{\left(uy \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \]

       6. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(2, \mathsf{*.f32}\left(yi, \mathsf{*.f32}\left(uy, \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right) \]

       7. PI-lowering-PI.f32 51.9%

          \[\leadsto \mathsf{*.f32}\left(2, \mathsf{*.f32}\left(yi, \mathsf{*.f32}\left(uy, \mathsf{PI.f32}\left(\right)\right)\right)\right) \]

   13. Simplified 51.9%

       \[\leadsto \color{blue}{2 \cdot \left(yi \cdot \left(uy \cdot \pi\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 59.5% accurate, 20.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := xi \cdot \left(1 + \left(\pi \cdot \pi\right) \cdot \left(-2 \cdot \left(uy \cdot uy\right)\right)\right)\\ \mathbf{if}\;xi \leq -1.4999999523982838 \cdot 10^{-22}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;xi \leq 4.720000108707208 \cdot 10^{-19}:\\ \;\;\;\;2 \cdot \left(yi \cdot \left(uy \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* xi (+ 1.0 (* (* PI PI) (* -2.0 (* uy uy)))))))
   (if (<= xi -1.4999999523982838e-22)
     t_0
     (if (<= xi 4.720000108707208e-19) (* 2.0 (* yi (* uy PI))) t_0))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = xi * (1.0f + ((((float) M_PI) * ((float) M_PI)) * (-2.0f * (uy * uy))));
	float tmp;
	if (xi <= -1.4999999523982838e-22f) {
		tmp = t_0;
	} else if (xi <= 4.720000108707208e-19f) {
		tmp = 2.0f * (yi * (uy * ((float) M_PI)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(xi * Float32(Float32(1.0) + Float32(Float32(Float32(pi) * Float32(pi)) * Float32(Float32(-2.0) * Float32(uy * uy)))))
	tmp = Float32(0.0)
	if (xi <= Float32(-1.4999999523982838e-22))
		tmp = t_0;
	elseif (xi <= Float32(4.720000108707208e-19))
		tmp = Float32(Float32(2.0) * Float32(yi * Float32(uy * Float32(pi))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(xi, yi, zi, ux, uy, maxCos)
	t_0 = xi * (single(1.0) + ((single(pi) * single(pi)) * (single(-2.0) * (uy * uy))));
	tmp = single(0.0);
	if (xi <= single(-1.4999999523982838e-22))
		tmp = t_0;
	elseif (xi <= single(4.720000108707208e-19))
		tmp = single(2.0) * (yi * (uy * single(pi)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if xi < -1.5e-22 or 4.7200001e-19 < xi

   1. Initial program 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in uy around 0

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

   7. Simplified 89.0%

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

   8. Taylor expanded in maxCos around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{+.f32}\left(\left(uy \cdot \left(-2 \cdot \left(uy \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right), \color{blue}{\left(xi + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right)}\right) \]

   10. Simplified 88.7%

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

   11. Taylor expanded in xi around inf

       \[\leadsto \color{blue}{xi \cdot \left(1 + -2 \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
   12. Step-by-step derivation

       1. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(xi, \color{blue}{\left(1 + -2 \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}\right) \]

       2. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(xi, \mathsf{+.f32}\left(1, \color{blue}{\left(-2 \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}\right)\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{*.f32}\left(xi, \mathsf{+.f32}\left(1, \left(-2 \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \color{blue}{{uy}^{2}}\right)\right)\right)\right) \]

       4. associate-*r* N/A

          \[\leadsto \mathsf{*.f32}\left(xi, \mathsf{+.f32}\left(1, \left(\left(-2 \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \color{blue}{{uy}^{2}}\right)\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f32}\left(xi, \mathsf{+.f32}\left(1, \left(\left({\mathsf{PI}\left(\right)}^{2} \cdot -2\right) \cdot {\color{blue}{uy}}^{2}\right)\right)\right) \]

       6. associate-*l* N/A

          \[\leadsto \mathsf{*.f32}\left(xi, \mathsf{+.f32}\left(1, \left({\mathsf{PI}\left(\right)}^{2} \cdot \color{blue}{\left(-2 \cdot {uy}^{2}\right)}\right)\right)\right) \]

       7. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(xi, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\left({\mathsf{PI}\left(\right)}^{2}\right), \color{blue}{\left(-2 \cdot {uy}^{2}\right)}\right)\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f32}\left(xi, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), \left(\color{blue}{-2} \cdot {uy}^{2}\right)\right)\right)\right) \]

       9. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(xi, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{PI}\left(\right), \mathsf{PI}\left(\right)\right), \left(\color{blue}{-2} \cdot {uy}^{2}\right)\right)\right)\right) \]

       10. PI-lowering-PI.f32 N/A

           \[\leadsto \mathsf{*.f32}\left(xi, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{PI}\left(\right)\right), \left(-2 \cdot {uy}^{2}\right)\right)\right)\right) \]

       11. PI-lowering-PI.f32 N/A

           \[\leadsto \mathsf{*.f32}\left(xi, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{PI.f32}\left(\right)\right), \left(-2 \cdot {uy}^{2}\right)\right)\right)\right) \]

       12. *-lowering-*.f32 N/A

           \[\leadsto \mathsf{*.f32}\left(xi, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{PI.f32}\left(\right)\right), \mathsf{*.f32}\left(-2, \color{blue}{\left({uy}^{2}\right)}\right)\right)\right)\right) \]

       13. unpow2 N/A

           \[\leadsto \mathsf{*.f32}\left(xi, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{PI.f32}\left(\right)\right), \mathsf{*.f32}\left(-2, \left(uy \cdot \color{blue}{uy}\right)\right)\right)\right)\right) \]

       14. *-lowering-*.f32 73.3%

           \[\leadsto \mathsf{*.f32}\left(xi, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{PI.f32}\left(\right)\right), \mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, \color{blue}{uy}\right)\right)\right)\right)\right) \]

   13. Simplified 73.3%

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

   if -1.5e-22 < xi < 4.7200001e-19

   1. Initial program 98.6%

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

   3. Simplified 98.6%

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

   5. Taylor expanded in uy around 0

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

   7. Simplified 86.1%

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

   8. Taylor expanded in maxCos around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{+.f32}\left(\left(uy \cdot \left(-2 \cdot \left(uy \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right), \color{blue}{\left(xi + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right)}\right) \]

   10. Simplified 86.0%

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

   11. Taylor expanded in yi around inf

       \[\leadsto \color{blue}{2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   12. Step-by-step derivation

       1. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(2, \color{blue}{\left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{*.f32}\left(2, \left(\left(yi \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{uy}\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{*.f32}\left(2, \left(yi \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot uy\right)}\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{*.f32}\left(2, \left(yi \cdot \left(uy \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right) \]

       5. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(2, \mathsf{*.f32}\left(yi, \color{blue}{\left(uy \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \]

       6. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(2, \mathsf{*.f32}\left(yi, \mathsf{*.f32}\left(uy, \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right) \]

       7. PI-lowering-PI.f32 51.9%

          \[\leadsto \mathsf{*.f32}\left(2, \mathsf{*.f32}\left(yi, \mathsf{*.f32}\left(uy, \mathsf{PI.f32}\left(\right)\right)\right)\right) \]

   13. Simplified 51.9%

       \[\leadsto \color{blue}{2 \cdot \left(yi \cdot \left(uy \cdot \pi\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 61.3% accurate, 21.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \left(yi \cdot \left(uy \cdot \pi\right)\right)\\ \mathbf{if}\;yi \leq -9.9999998245167 \cdot 10^{-15}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;yi \leq 1.9999999920083944 \cdot 10^{-11}:\\ \;\;\;\;xi + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) \cdot zi\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* 2.0 (* yi (* uy PI)))))
   (if (<= yi -9.9999998245167e-15)
     t_0
     (if (<= yi 1.9999999920083944e-11)
       (+ xi (* (* ux maxCos) (* (- 1.0 ux) zi)))
       t_0))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = 2.0f * (yi * (uy * ((float) M_PI)));
	float tmp;
	if (yi <= -9.9999998245167e-15f) {
		tmp = t_0;
	} else if (yi <= 1.9999999920083944e-11f) {
		tmp = xi + ((ux * maxCos) * ((1.0f - ux) * zi));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(2.0) * Float32(yi * Float32(uy * Float32(pi))))
	tmp = Float32(0.0)
	if (yi <= Float32(-9.9999998245167e-15))
		tmp = t_0;
	elseif (yi <= Float32(1.9999999920083944e-11))
		tmp = Float32(xi + Float32(Float32(ux * maxCos) * Float32(Float32(Float32(1.0) - ux) * zi)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(xi, yi, zi, ux, uy, maxCos)
	t_0 = single(2.0) * (yi * (uy * single(pi)));
	tmp = single(0.0);
	if (yi <= single(-9.9999998245167e-15))
		tmp = t_0;
	elseif (yi <= single(1.9999999920083944e-11))
		tmp = xi + ((ux * maxCos) * ((single(1.0) - ux) * zi));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if yi < -9.99999982e-15 or 1.99999999e-11 < yi

   1. Initial program 99.0%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in uy around 0

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

   7. Simplified 86.3%

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

   8. Taylor expanded in maxCos around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{+.f32}\left(\left(uy \cdot \left(-2 \cdot \left(uy \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right), \color{blue}{\left(xi + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right)}\right) \]

   10. Simplified 85.8%

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

   11. Taylor expanded in yi around inf

       \[\leadsto \color{blue}{2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   12. Step-by-step derivation

       1. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(2, \color{blue}{\left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{*.f32}\left(2, \left(\left(yi \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{uy}\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{*.f32}\left(2, \left(yi \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot uy\right)}\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{*.f32}\left(2, \left(yi \cdot \left(uy \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right) \]

       5. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(2, \mathsf{*.f32}\left(yi, \color{blue}{\left(uy \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \]

       6. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(2, \mathsf{*.f32}\left(yi, \mathsf{*.f32}\left(uy, \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right) \]

       7. PI-lowering-PI.f32 56.5%

          \[\leadsto \mathsf{*.f32}\left(2, \mathsf{*.f32}\left(yi, \mathsf{*.f32}\left(uy, \mathsf{PI.f32}\left(\right)\right)\right)\right) \]

   13. Simplified 56.5%

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

   if -9.99999982e-15 < yi < 1.99999999e-11

   1. Initial program 98.9%

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

   3. Simplified 98.9%

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

   6. Applied egg-rr 99.0%

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

   7. Taylor expanded in ux around 0

      \[\leadsto \mathsf{+.f32}\left(\color{blue}{\left(xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)}, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, ux\right), \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right)\right) \]
   8. Step-by-step derivation

      1. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(\left(xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right), \left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \mathsf{*.f32}\left(\color{blue}{\mathsf{\_.f32}\left(1, ux\right)}, \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right)\right) \]

      2. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right), \left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(\color{blue}{1}, ux\right), \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right)\right) \]

      3. cos-lowering-cos.f32 N/A

         \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(xi, \mathsf{cos.f32}\left(\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, ux\right), \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right)\right) \]

      4. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(xi, \mathsf{cos.f32}\left(\mathsf{*.f32}\left(2, \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, ux\right), \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right)\right) \]

      5. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(xi, \mathsf{cos.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \mathsf{PI}\left(\right)\right)\right)\right)\right), \left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, ux\right), \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right)\right) \]

      6. PI-lowering-PI.f32 N/A

         \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(xi, \mathsf{cos.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \mathsf{PI.f32}\left(\right)\right)\right)\right)\right), \left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, ux\right), \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right)\right) \]

      7. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(xi, \mathsf{cos.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \mathsf{PI.f32}\left(\right)\right)\right)\right)\right), \mathsf{*.f32}\left(yi, \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, \color{blue}{ux}\right), \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right)\right) \]

      8. sin-lowering-sin.f32 N/A

         \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(xi, \mathsf{cos.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \mathsf{PI.f32}\left(\right)\right)\right)\right)\right), \mathsf{*.f32}\left(yi, \mathsf{sin.f32}\left(\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, ux\right), \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right)\right) \]

      9. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(xi, \mathsf{cos.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \mathsf{PI.f32}\left(\right)\right)\right)\right)\right), \mathsf{*.f32}\left(yi, \mathsf{sin.f32}\left(\mathsf{*.f32}\left(2, \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, ux\right), \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right)\right) \]

      10. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(xi, \mathsf{cos.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \mathsf{PI.f32}\left(\right)\right)\right)\right)\right), \mathsf{*.f32}\left(yi, \mathsf{sin.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \mathsf{PI}\left(\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, ux\right), \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right)\right) \]

      11. PI-lowering-PI.f32 99.0%

          \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(xi, \mathsf{cos.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \mathsf{PI.f32}\left(\right)\right)\right)\right)\right), \mathsf{*.f32}\left(yi, \mathsf{sin.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \mathsf{PI.f32}\left(\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, ux\right), \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right)\right) \]

   9. Simplified 99.0%

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

   10. Taylor expanded in uy around 0

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

       1. +-commutative N/A

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

       2. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{+.f32}\left(\left(maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right), \color{blue}{xi}\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{+.f32}\left(\left(\left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) \cdot maxCos\right), xi\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{+.f32}\left(\left(\left(\left(zi \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot maxCos\right), xi\right) \]

       5. associate-*l* N/A

          \[\leadsto \mathsf{+.f32}\left(\left(\left(zi \cdot \left(1 - ux\right)\right) \cdot \left(ux \cdot maxCos\right)\right), xi\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{+.f32}\left(\left(\left(zi \cdot \left(1 - ux\right)\right) \cdot \left(maxCos \cdot ux\right)\right), xi\right) \]

       7. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(\left(zi \cdot \left(1 - ux\right)\right), \left(maxCos \cdot ux\right)\right), xi\right) \]

       8. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(zi, \left(1 - ux\right)\right), \left(maxCos \cdot ux\right)\right), xi\right) \]

       9. --lowering--.f32 N/A

          \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(zi, \mathsf{\_.f32}\left(1, ux\right)\right), \left(maxCos \cdot ux\right)\right), xi\right) \]

       10. *-lowering-*.f32 68.6%

           \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(zi, \mathsf{\_.f32}\left(1, ux\right)\right), \mathsf{*.f32}\left(maxCos, ux\right)\right), xi\right) \]

   12. Simplified 68.6%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;yi \leq -9.9999998245167 \cdot 10^{-15}:\\ \;\;\;\;2 \cdot \left(yi \cdot \left(uy \cdot \pi\right)\right)\\ \mathbf{elif}\;yi \leq 1.9999999920083944 \cdot 10^{-11}:\\ \;\;\;\;xi + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) \cdot zi\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(yi \cdot \left(uy \cdot \pi\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 81.3% accurate, 24.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

3. Simplified 99.0%

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

5. Taylor expanded in uy around 0

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

7. Simplified 87.9%

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

8. Taylor expanded in maxCos around 0

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

   1. +-commutative N/A

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

   2. +-commutative N/A

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

   3. associate-+l+ N/A

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

   4. +-commutative N/A

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

   5. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(\left(uy \cdot \left(-2 \cdot \left(uy \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right), \color{blue}{\left(xi + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right)}\right) \]

10. Simplified 87.7%

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

11. Taylor expanded in uy around 0

    \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(uy, \color{blue}{\left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)}\right), \mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(\mathsf{*.f32}\left(maxCos, ux\right), \mathsf{*.f32}\left(zi, \mathsf{\_.f32}\left(1, ux\right)\right)\right)\right)\right) \]
12. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(uy, \left(\left(yi \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)\right), \mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(\mathsf{*.f32}\left(maxCos, ux\right), \mathsf{*.f32}\left(zi, \mathsf{\_.f32}\left(1, ux\right)\right)\right)\right)\right) \]

    2. associate-*r* N/A

       \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(uy, \left(yi \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right)\right), \mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(\mathsf{*.f32}\left(maxCos, ux\right), \mathsf{*.f32}\left(zi, \mathsf{\_.f32}\left(1, ux\right)\right)\right)\right)\right) \]

    3. *-commutative N/A

       \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(uy, \left(yi \cdot \left(2 \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(\mathsf{*.f32}\left(maxCos, ux\right), \mathsf{*.f32}\left(zi, \mathsf{\_.f32}\left(1, ux\right)\right)\right)\right)\right) \]

    4. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(yi, \left(2 \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(\mathsf{*.f32}\left(maxCos, ux\right), \mathsf{*.f32}\left(zi, \mathsf{\_.f32}\left(1, ux\right)\right)\right)\right)\right) \]

    5. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(yi, \mathsf{*.f32}\left(2, \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(\mathsf{*.f32}\left(maxCos, ux\right), \mathsf{*.f32}\left(zi, \mathsf{\_.f32}\left(1, ux\right)\right)\right)\right)\right) \]

    6. PI-lowering-PI.f32 82.6%

       \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(yi, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(\mathsf{*.f32}\left(maxCos, ux\right), \mathsf{*.f32}\left(zi, \mathsf{\_.f32}\left(1, ux\right)\right)\right)\right)\right) \]

13. Simplified 82.6%

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

14. Final simplification 82.6%

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

Alternative 17: 81.3% accurate, 24.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

3. Simplified 99.0%

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

5. Taylor expanded in uy around 0

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

7. Simplified 87.9%

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

8. Taylor expanded in maxCos around 0

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

   1. +-commutative N/A

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

   2. +-commutative N/A

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

   3. associate-+l+ N/A

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

   4. +-commutative N/A

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

   5. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(\left(uy \cdot \left(-2 \cdot \left(uy \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right), \color{blue}{\left(xi + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right)}\right) \]

10. Simplified 87.7%

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

11. Taylor expanded in uy around 0

    \[\leadsto \mathsf{+.f32}\left(\color{blue}{\left(2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right)}, \mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(\mathsf{*.f32}\left(maxCos, ux\right), \mathsf{*.f32}\left(zi, \mathsf{\_.f32}\left(1, ux\right)\right)\right)\right)\right) \]
12. Step-by-step derivation

    1. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(2, \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{+.f32}\left(\color{blue}{xi}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(maxCos, ux\right), \mathsf{*.f32}\left(zi, \mathsf{\_.f32}\left(1, ux\right)\right)\right)\right)\right) \]

    2. *-commutative N/A

       \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(2, \left(\left(yi \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)\right), \mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(\mathsf{*.f32}\left(maxCos, ux\right), \mathsf{*.f32}\left(zi, \mathsf{\_.f32}\left(1, ux\right)\right)\right)\right)\right) \]

    3. associate-*l* N/A

       \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(2, \left(yi \cdot \left(\mathsf{PI}\left(\right) \cdot uy\right)\right)\right), \mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(\mathsf{*.f32}\left(maxCos, ux\right), \mathsf{*.f32}\left(zi, \mathsf{\_.f32}\left(1, ux\right)\right)\right)\right)\right) \]

    4. *-commutative N/A

       \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(2, \left(yi \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(\mathsf{*.f32}\left(maxCos, ux\right), \mathsf{*.f32}\left(zi, \mathsf{\_.f32}\left(1, ux\right)\right)\right)\right)\right) \]

    5. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(yi, \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(\mathsf{*.f32}\left(maxCos, ux\right), \mathsf{*.f32}\left(zi, \mathsf{\_.f32}\left(1, ux\right)\right)\right)\right)\right) \]

    6. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(yi, \mathsf{*.f32}\left(uy, \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(\mathsf{*.f32}\left(maxCos, ux\right), \mathsf{*.f32}\left(zi, \mathsf{\_.f32}\left(1, ux\right)\right)\right)\right)\right) \]

    7. PI-lowering-PI.f32 82.6%

       \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(yi, \mathsf{*.f32}\left(uy, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(\mathsf{*.f32}\left(maxCos, ux\right), \mathsf{*.f32}\left(zi, \mathsf{\_.f32}\left(1, ux\right)\right)\right)\right)\right) \]

13. Simplified 82.6%

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

14. Final simplification 82.6%

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

Alternative 18: 77.6% accurate, 24.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

3. Simplified 99.0%

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

5. Taylor expanded in uy around 0

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

7. Simplified 87.9%

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

8. Taylor expanded in maxCos around 0

   \[\leadsto \color{blue}{xi + uy \cdot \left(-2 \cdot \left(uy \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)} \]
9. Step-by-step derivation

   1. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(xi, \color{blue}{\left(uy \cdot \left(-2 \cdot \left(uy \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right)}\right) \]

   2. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(uy, \color{blue}{\left(-2 \cdot \left(uy \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right) \]

   3. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(uy, \mathsf{+.f32}\left(\left(-2 \cdot \left(uy \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \color{blue}{\left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)\right) \]

   4. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(uy, \mathsf{+.f32}\left(\mathsf{*.f32}\left(-2, \left(uy \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \left(\color{blue}{2} \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right) \]

   5. associate-*r* N/A

      \[\leadsto \mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(uy, \mathsf{+.f32}\left(\mathsf{*.f32}\left(-2, \left(\left(uy \cdot xi\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right), \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right) \]

   6. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(uy, \mathsf{+.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(\left(uy \cdot xi\right), \left({\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right) \]

   7. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(uy, \mathsf{+.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(\mathsf{*.f32}\left(uy, xi\right), \left({\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(uy, \mathsf{+.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(\mathsf{*.f32}\left(uy, xi\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right), \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right) \]

   9. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(uy, \mathsf{+.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(\mathsf{*.f32}\left(uy, xi\right), \mathsf{*.f32}\left(\mathsf{PI}\left(\right), \mathsf{PI}\left(\right)\right)\right)\right), \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right) \]

   10. PI-lowering-PI.f32 N/A

       \[\leadsto \mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(uy, \mathsf{+.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(\mathsf{*.f32}\left(uy, xi\right), \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{PI}\left(\right)\right)\right)\right), \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right) \]

   11. PI-lowering-PI.f32 N/A

       \[\leadsto \mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(uy, \mathsf{+.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(\mathsf{*.f32}\left(uy, xi\right), \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{PI.f32}\left(\right)\right)\right)\right), \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right) \]

   12. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(uy, \mathsf{+.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(\mathsf{*.f32}\left(uy, xi\right), \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{*.f32}\left(2, \color{blue}{\left(yi \cdot \mathsf{PI}\left(\right)\right)}\right)\right)\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(uy, \mathsf{+.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(\mathsf{*.f32}\left(uy, xi\right), \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{*.f32}\left(2, \left(\mathsf{PI}\left(\right) \cdot \color{blue}{yi}\right)\right)\right)\right)\right) \]

   14. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(uy, \mathsf{+.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(\mathsf{*.f32}\left(uy, xi\right), \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{*.f32}\left(2, \mathsf{*.f32}\left(\mathsf{PI}\left(\right), \color{blue}{yi}\right)\right)\right)\right)\right) \]

   15. PI-lowering-PI.f32 80.9%

       \[\leadsto \mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(uy, \mathsf{+.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(\mathsf{*.f32}\left(uy, xi\right), \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{*.f32}\left(2, \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), yi\right)\right)\right)\right)\right) \]

10. Simplified 80.9%

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

11. Final simplification 80.9%

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

Alternative 19: 59.5% accurate, 27.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \left(yi \cdot \left(uy \cdot \pi\right)\right)\\ \mathbf{if}\;yi \leq -9.9999998245167 \cdot 10^{-15}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;yi \leq 1.9999999920083944 \cdot 10^{-11}:\\ \;\;\;\;xi + maxCos \cdot \left(ux \cdot zi\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* 2.0 (* yi (* uy PI)))))
   (if (<= yi -9.9999998245167e-15)
     t_0
     (if (<= yi 1.9999999920083944e-11) (+ xi (* maxCos (* ux zi))) t_0))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = 2.0f * (yi * (uy * ((float) M_PI)));
	float tmp;
	if (yi <= -9.9999998245167e-15f) {
		tmp = t_0;
	} else if (yi <= 1.9999999920083944e-11f) {
		tmp = xi + (maxCos * (ux * zi));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(2.0) * Float32(yi * Float32(uy * Float32(pi))))
	tmp = Float32(0.0)
	if (yi <= Float32(-9.9999998245167e-15))
		tmp = t_0;
	elseif (yi <= Float32(1.9999999920083944e-11))
		tmp = Float32(xi + Float32(maxCos * Float32(ux * zi)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(xi, yi, zi, ux, uy, maxCos)
	t_0 = single(2.0) * (yi * (uy * single(pi)));
	tmp = single(0.0);
	if (yi <= single(-9.9999998245167e-15))
		tmp = t_0;
	elseif (yi <= single(1.9999999920083944e-11))
		tmp = xi + (maxCos * (ux * zi));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if yi < -9.99999982e-15 or 1.99999999e-11 < yi

   1. Initial program 99.0%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in uy around 0

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

   7. Simplified 86.3%

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

   8. Taylor expanded in maxCos around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{+.f32}\left(\left(uy \cdot \left(-2 \cdot \left(uy \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right), \color{blue}{\left(xi + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right)}\right) \]

   10. Simplified 85.8%

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

   11. Taylor expanded in yi around inf

       \[\leadsto \color{blue}{2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   12. Step-by-step derivation

       1. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(2, \color{blue}{\left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{*.f32}\left(2, \left(\left(yi \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{uy}\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{*.f32}\left(2, \left(yi \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot uy\right)}\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{*.f32}\left(2, \left(yi \cdot \left(uy \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right) \]

       5. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(2, \mathsf{*.f32}\left(yi, \color{blue}{\left(uy \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \]

       6. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(2, \mathsf{*.f32}\left(yi, \mathsf{*.f32}\left(uy, \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right) \]

       7. PI-lowering-PI.f32 56.5%

          \[\leadsto \mathsf{*.f32}\left(2, \mathsf{*.f32}\left(yi, \mathsf{*.f32}\left(uy, \mathsf{PI.f32}\left(\right)\right)\right)\right) \]

   13. Simplified 56.5%

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

   if -9.99999982e-15 < yi < 1.99999999e-11

   1. Initial program 98.9%

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

   3. Simplified 98.9%

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

   5. Taylor expanded in ux around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{+.f32}\left(\left(xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right), \color{blue}{\left(maxCos \cdot \left(ux \cdot zi\right)\right)}\right) \]

   7. Simplified 95.8%

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

   8. Taylor expanded in uy around 0

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

      1. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{+.f32}\left(xi, \color{blue}{\left(maxCos \cdot \left(ux \cdot zi\right)\right)}\right) \]

      2. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(maxCos, \color{blue}{\left(ux \cdot zi\right)}\right)\right) \]

      3. *-lowering-*.f32 66.4%

         \[\leadsto \mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, \color{blue}{zi}\right)\right)\right) \]

   10. Simplified 66.4%

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

Alternative 20: 78.8% accurate, 30.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

3. Simplified 99.0%

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

5. Taylor expanded in ux around 0

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

   1. +-commutative N/A

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

   2. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(\left(xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right), \color{blue}{\left(maxCos \cdot \left(ux \cdot zi\right)\right)}\right) \]

7. Simplified 96.7%

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

8. Taylor expanded in uy around 0

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

   1. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(xi, \color{blue}{\left(2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right) + maxCos \cdot \left(ux \cdot zi\right)\right)}\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{+.f32}\left(xi, \left(maxCos \cdot \left(ux \cdot zi\right) + \color{blue}{2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right) \]

   3. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(xi, \mathsf{+.f32}\left(\left(maxCos \cdot \left(ux \cdot zi\right)\right), \color{blue}{\left(2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right)}\right)\right) \]

   4. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(xi, \mathsf{+.f32}\left(\mathsf{*.f32}\left(maxCos, \left(ux \cdot zi\right)\right), \left(\color{blue}{2} \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right) \]

   5. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(xi, \mathsf{+.f32}\left(\mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right), \left(2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right) \]

   6. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(xi, \mathsf{+.f32}\left(\mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right), \mathsf{*.f32}\left(2, \color{blue}{\left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)\right) \]

   7. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(xi, \mathsf{+.f32}\left(\mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right), \mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \color{blue}{\left(yi \cdot \mathsf{PI}\left(\right)\right)}\right)\right)\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{+.f32}\left(xi, \mathsf{+.f32}\left(\mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right), \mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \left(\mathsf{PI}\left(\right) \cdot \color{blue}{yi}\right)\right)\right)\right)\right) \]

   9. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(xi, \mathsf{+.f32}\left(\mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right), \mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(\mathsf{PI}\left(\right), \color{blue}{yi}\right)\right)\right)\right)\right) \]

   10. PI-lowering-PI.f32 80.8%

       \[\leadsto \mathsf{+.f32}\left(xi, \mathsf{+.f32}\left(\mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right), \mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), yi\right)\right)\right)\right)\right) \]

10. Simplified 80.8%

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

11. Final simplification 80.8%

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

Alternative 21: 31.7% accurate, 65.9× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(yi \cdot \left(uy \cdot \pi\right)\right) \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

3. Simplified 99.0%

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

5. Taylor expanded in uy around 0

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

7. Simplified 87.9%

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

8. Taylor expanded in maxCos around 0

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

   1. +-commutative N/A

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

   2. +-commutative N/A

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

   3. associate-+l+ N/A

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

   4. +-commutative N/A

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

   5. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(\left(uy \cdot \left(-2 \cdot \left(uy \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right), \color{blue}{\left(xi + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right)}\right) \]

10. Simplified 87.7%

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

11. Taylor expanded in yi around inf

    \[\leadsto \color{blue}{2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)} \]
12. Step-by-step derivation

    1. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(2, \color{blue}{\left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

    2. *-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(2, \left(\left(yi \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{uy}\right)\right) \]

    3. associate-*l* N/A

       \[\leadsto \mathsf{*.f32}\left(2, \left(yi \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot uy\right)}\right)\right) \]

    4. *-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(2, \left(yi \cdot \left(uy \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right) \]

    5. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(2, \mathsf{*.f32}\left(yi, \color{blue}{\left(uy \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \]

    6. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(2, \mathsf{*.f32}\left(yi, \mathsf{*.f32}\left(uy, \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right) \]

    7. PI-lowering-PI.f32 30.1%

       \[\leadsto \mathsf{*.f32}\left(2, \mathsf{*.f32}\left(yi, \mathsf{*.f32}\left(uy, \mathsf{PI.f32}\left(\right)\right)\right)\right) \]

13. Simplified 30.1%

    \[\leadsto \color{blue}{2 \cdot \left(yi \cdot \left(uy \cdot \pi\right)\right)} \]
14. Add Preprocessing

Alternative 22: 12.1% accurate, 92.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

3. Simplified 99.0%

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

5. Taylor expanded in zi around inf

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

   1. associate-*r* N/A

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

   2. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\left(maxCos \cdot ux\right), \color{blue}{\left(zi \cdot \left(1 - ux\right)\right)}\right) \]

   3. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(maxCos, ux\right), \left(\color{blue}{zi} \cdot \left(1 - ux\right)\right)\right) \]

   4. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(maxCos, ux\right), \mathsf{*.f32}\left(zi, \color{blue}{\left(1 - ux\right)}\right)\right) \]

   5. --lowering--.f32 12.7%

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(maxCos, ux\right), \mathsf{*.f32}\left(zi, \mathsf{\_.f32}\left(1, \color{blue}{ux}\right)\right)\right) \]

7. Simplified 12.7%

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

8. Taylor expanded in ux around 0

   \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(maxCos, ux\right), \color{blue}{zi}\right) \]
9. Step-by-step derivation

10. Simplified 11.9%

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

11. Final simplification 11.9%

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

Alternative 23: 12.1% accurate, 92.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

3. Simplified 99.0%

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

5. Taylor expanded in zi around inf

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

   1. associate-*r* N/A

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

   2. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\left(maxCos \cdot ux\right), \color{blue}{\left(zi \cdot \left(1 - ux\right)\right)}\right) \]

   3. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(maxCos, ux\right), \left(\color{blue}{zi} \cdot \left(1 - ux\right)\right)\right) \]

   4. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(maxCos, ux\right), \mathsf{*.f32}\left(zi, \color{blue}{\left(1 - ux\right)}\right)\right) \]

   5. --lowering--.f32 12.7%

      \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(maxCos, ux\right), \mathsf{*.f32}\left(zi, \mathsf{\_.f32}\left(1, \color{blue}{ux}\right)\right)\right) \]

7. Simplified 12.7%

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

8. Taylor expanded in ux around 0

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

   1. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(maxCos, \color{blue}{\left(ux \cdot zi\right)}\right) \]

   2. *-lowering-*.f32 11.9%

      \[\leadsto \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, \color{blue}{zi}\right)\right) \]

10. Simplified 11.9%

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

Reproduce

herbie shell --seed 2024191 
(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)))