UniformSampleCone 2

Percentage Accurate: 98.9% → 98.9%

Time: 47.2s

Alternatives: 29

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := ux \cdot \left(1 - ux\right)\\ t_1 := 2 \cdot \left(\pi \cdot uy\right)\\ \mathsf{fma}\left(zi \cdot t\_0, maxCos, {\left(1 + t\_0 \cdot \left(maxCos \cdot \left(ux \cdot \left(maxCos \cdot \left(ux + -1\right)\right)\right)\right)\right)}^{0.5} \cdot \left(xi \cdot \cos t\_1 + yi \cdot \sin t\_1\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* ux (- 1.0 ux))) (t_1 (* 2.0 (* PI uy))))
   (fma
    (* zi t_0)
    maxCos
    (*
     (pow (+ 1.0 (* t_0 (* maxCos (* ux (* maxCos (+ ux -1.0)))))) 0.5)
     (+ (* xi (cos t_1)) (* yi (sin t_1)))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = ux * (1.0f - ux);
	float t_1 = 2.0f * (((float) M_PI) * uy);
	return fmaf((zi * t_0), maxCos, (powf((1.0f + (t_0 * (maxCos * (ux * (maxCos * (ux + -1.0f)))))), 0.5f) * ((xi * cosf(t_1)) + (yi * sinf(t_1)))));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(ux * Float32(Float32(1.0) - ux))
	t_1 = Float32(Float32(2.0) * Float32(Float32(pi) * uy))
	return fma(Float32(zi * t_0), maxCos, Float32((Float32(Float32(1.0) + Float32(t_0 * Float32(maxCos * Float32(ux * Float32(maxCos * Float32(ux + Float32(-1.0))))))) ^ Float32(0.5)) * Float32(Float32(xi * cos(t_1)) + Float32(yi * sin(t_1)))))
end

Derivation

1. Initial program 99.1%

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

3. Simplified 99.1%

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

6. Applied egg-rr 99.1%

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

7. Final simplification 99.1%

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

Alternative 2: 98.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

3. Simplified 99.1%

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

5. Taylor expanded in ux around inf

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

   1. unpow2 N/A

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

   2. associate-*l* N/A

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

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

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

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

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

   5. +-commutative N/A

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

   6. mul-1-neg N/A

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

   7. unsub-neg N/A

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

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

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

   9. /-lowering-/.f32 99.1%

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

7. Simplified 99.1%

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

8. Final simplification 99.1%

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

Alternative 3: 98.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

3. Simplified 99.1%

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

5. Final simplification 99.1%

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

Alternative 4: 98.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.1%

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

3. Simplified 99.1%

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

6. Applied egg-rr 99.1%

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

7. Taylor expanded in maxCos around 0

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

9. Simplified 99.0%

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

10. Final simplification 99.0%

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

Alternative 5: 98.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

3. Simplified 99.1%

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

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

8. Final simplification 98.9%

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

Alternative 6: 94.6% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \left(yi \cdot \sin \left(2 \cdot \left(\pi \cdot uy\right)\right) + zi \cdot \left(\left(ux \cdot \left(1 - ux\right)\right) \cdot maxCos\right)\right) + \left(xi + uy \cdot \left(uy \cdot \left(xi \cdot \left(\left(\pi \cdot \pi\right) \cdot -2\right) + xi \cdot \left(\left(\pi \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right) \cdot \left(uy \cdot \left(uy \cdot 0.6666666666666666\right)\right)\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (+
  (+ (* yi (sin (* 2.0 (* PI uy)))) (* zi (* (* ux (- 1.0 ux)) maxCos)))
  (+
   xi
   (*
    uy
    (*
     uy
     (+
      (* xi (* (* PI PI) -2.0))
      (* xi (* (* PI (* PI (* PI PI))) (* uy (* uy 0.6666666666666666))))))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return ((yi * sinf((2.0f * (((float) M_PI) * uy)))) + (zi * ((ux * (1.0f - ux)) * maxCos))) + (xi + (uy * (uy * ((xi * ((((float) M_PI) * ((float) M_PI)) * -2.0f)) + (xi * ((((float) M_PI) * (((float) M_PI) * (((float) M_PI) * ((float) M_PI)))) * (uy * (uy * 0.6666666666666666f))))))));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return Float32(Float32(Float32(yi * sin(Float32(Float32(2.0) * Float32(Float32(pi) * uy)))) + Float32(zi * Float32(Float32(ux * Float32(Float32(1.0) - ux)) * maxCos))) + Float32(xi + Float32(uy * Float32(uy * Float32(Float32(xi * Float32(Float32(Float32(pi) * Float32(pi)) * Float32(-2.0))) + Float32(xi * Float32(Float32(Float32(pi) * Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi)))) * Float32(uy * Float32(uy * Float32(0.6666666666666666))))))))))
end

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	tmp = ((yi * sin((single(2.0) * (single(pi) * uy)))) + (zi * ((ux * (single(1.0) - ux)) * maxCos))) + (xi + (uy * (uy * ((xi * ((single(pi) * single(pi)) * single(-2.0))) + (xi * ((single(pi) * (single(pi) * (single(pi) * single(pi)))) * (uy * (uy * single(0.6666666666666666)))))))));
end

Derivation

1. Initial program 99.1%

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

3. Simplified 99.1%

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

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

8. Taylor expanded in uy around 0

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

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

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

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

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

   3. unpow2 N/A

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

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

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

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

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

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

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

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

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

   8. unpow2 N/A

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

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

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

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

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

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

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

   12. associate-*r* N/A

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

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

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

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

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

   15. unpow2 N/A

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

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

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

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

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

   18. pow-lowering-pow.f32 N/A

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

10. Simplified 96.5%

    \[\leadsto \color{blue}{\left(xi + \left(uy \cdot uy\right) \cdot \left(-2 \cdot \left(xi \cdot \left(\pi \cdot \pi\right)\right) + \left(0.6666666666666666 \cdot \left(uy \cdot uy\right)\right) \cdot \left(xi \cdot {\pi}^{4}\right)\right)\right)} + \left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right) + zi \cdot \left(maxCos \cdot \left(ux \cdot \left(1 - ux\right)\right)\right)\right) \]
11. Step-by-step derivation

    1. +-commutative N/A

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

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

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

12. Applied egg-rr 96.5%

    \[\leadsto \color{blue}{\left(uy \cdot \left(uy \cdot \left(xi \cdot \left(\left(\pi \cdot \pi\right) \cdot -2\right) + xi \cdot \left(\left(\pi \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right) \cdot \left(uy \cdot \left(uy \cdot 0.6666666666666666\right)\right)\right)\right)\right) + xi\right)} + \left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right) + zi \cdot \left(maxCos \cdot \left(ux \cdot \left(1 - ux\right)\right)\right)\right) \]

13. Final simplification 96.5%

    \[\leadsto \left(yi \cdot \sin \left(2 \cdot \left(\pi \cdot uy\right)\right) + zi \cdot \left(\left(ux \cdot \left(1 - ux\right)\right) \cdot maxCos\right)\right) + \left(xi + uy \cdot \left(uy \cdot \left(xi \cdot \left(\left(\pi \cdot \pi\right) \cdot -2\right) + xi \cdot \left(\left(\pi \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right) \cdot \left(uy \cdot \left(uy \cdot 0.6666666666666666\right)\right)\right)\right)\right)\right) \]
14. Add Preprocessing

Alternative 7: 94.7% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \left(xi + \left(yi \cdot \sin \left(2 \cdot \left(\pi \cdot uy\right)\right) + uy \cdot \left(uy \cdot \left(xi \cdot \left(\left(\pi \cdot \pi\right) \cdot -2\right) + xi \cdot \left(\left(\pi \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right) \cdot \left(uy \cdot \left(uy \cdot 0.6666666666666666\right)\right)\right)\right)\right)\right)\right) + \left(\left(1 - ux\right) \cdot maxCos\right) \cdot \left(zi \cdot ux\right) \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (+
  (+
   xi
   (+
    (* yi (sin (* 2.0 (* PI uy))))
    (*
     uy
     (*
      uy
      (+
       (* xi (* (* PI PI) -2.0))
       (* xi (* (* PI (* PI (* PI PI))) (* uy (* uy 0.6666666666666666)))))))))
  (* (* (- 1.0 ux) maxCos) (* zi ux))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return (xi + ((yi * sinf((2.0f * (((float) M_PI) * uy)))) + (uy * (uy * ((xi * ((((float) M_PI) * ((float) M_PI)) * -2.0f)) + (xi * ((((float) M_PI) * (((float) M_PI) * (((float) M_PI) * ((float) M_PI)))) * (uy * (uy * 0.6666666666666666f))))))))) + (((1.0f - ux) * maxCos) * (zi * ux));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return Float32(Float32(xi + Float32(Float32(yi * sin(Float32(Float32(2.0) * Float32(Float32(pi) * uy)))) + Float32(uy * Float32(uy * Float32(Float32(xi * Float32(Float32(Float32(pi) * Float32(pi)) * Float32(-2.0))) + Float32(xi * Float32(Float32(Float32(pi) * Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi)))) * Float32(uy * Float32(uy * Float32(0.6666666666666666)))))))))) + Float32(Float32(Float32(Float32(1.0) - ux) * maxCos) * Float32(zi * ux)))
end

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	tmp = (xi + ((yi * sin((single(2.0) * (single(pi) * uy)))) + (uy * (uy * ((xi * ((single(pi) * single(pi)) * single(-2.0))) + (xi * ((single(pi) * (single(pi) * (single(pi) * single(pi)))) * (uy * (uy * single(0.6666666666666666)))))))))) + (((single(1.0) - ux) * maxCos) * (zi * ux));
end

Derivation

1. Initial program 99.1%

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

3. Simplified 99.1%

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

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

8. Taylor expanded in uy around 0

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

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

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

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

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

   3. unpow2 N/A

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

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

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

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

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

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

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

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

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

   8. unpow2 N/A

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

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

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

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

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

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

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

   12. associate-*r* N/A

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

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

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

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

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

   15. unpow2 N/A

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

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

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

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

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

   18. pow-lowering-pow.f32 N/A

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

10. Simplified 96.5%

    \[\leadsto \color{blue}{\left(xi + \left(uy \cdot uy\right) \cdot \left(-2 \cdot \left(xi \cdot \left(\pi \cdot \pi\right)\right) + \left(0.6666666666666666 \cdot \left(uy \cdot uy\right)\right) \cdot \left(xi \cdot {\pi}^{4}\right)\right)\right)} + \left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right) + zi \cdot \left(maxCos \cdot \left(ux \cdot \left(1 - ux\right)\right)\right)\right) \]

12. Applied egg-rr 96.5%

    \[\leadsto \color{blue}{\left(xi + \left(uy \cdot \left(uy \cdot \left(xi \cdot \left(\left(\pi \cdot \pi\right) \cdot -2\right) + xi \cdot \left(\left(\pi \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right) \cdot \left(uy \cdot \left(uy \cdot 0.6666666666666666\right)\right)\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(\pi \cdot uy\right)\right)\right)\right) + \left(zi \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot maxCos\right)} \]

13. Final simplification 96.5%

    \[\leadsto \left(xi + \left(yi \cdot \sin \left(2 \cdot \left(\pi \cdot uy\right)\right) + uy \cdot \left(uy \cdot \left(xi \cdot \left(\left(\pi \cdot \pi\right) \cdot -2\right) + xi \cdot \left(\left(\pi \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right) \cdot \left(uy \cdot \left(uy \cdot 0.6666666666666666\right)\right)\right)\right)\right)\right)\right) + \left(\left(1 - ux\right) \cdot maxCos\right) \cdot \left(zi \cdot ux\right) \]
14. Add Preprocessing

Alternative 8: 92.4% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.011420000344514847:\\ \;\;\;\;\left(xi + \left(ux \cdot maxCos\right) \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + uy \cdot \left(2 \cdot \left(\pi \cdot yi\right) + uy \cdot \left(-2 \cdot \left(xi \cdot \left(\pi \cdot \pi\right)\right) + -1.3333333333333333 \cdot \left(\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(uy \cdot yi\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(yi \cdot \sin \left(2 \cdot \left(\pi \cdot uy\right)\right) + xi \cdot \left(1 + uy \cdot \left(-2 \cdot \left(uy \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\right) + maxCos \cdot \left(zi \cdot ux\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (if (<= uy 0.011420000344514847)
   (+
    (+ xi (* (* ux maxCos) (* zi (- 1.0 ux))))
    (*
     uy
     (+
      (* 2.0 (* PI yi))
      (*
       uy
       (+
        (* -2.0 (* xi (* PI PI)))
        (* -1.3333333333333333 (* (* PI (* PI PI)) (* uy yi))))))))
   (+
    (+
     (* yi (sin (* 2.0 (* PI uy))))
     (* xi (+ 1.0 (* uy (* -2.0 (* uy (* PI PI)))))))
    (* maxCos (* zi ux)))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.011420000344514847f) {
		tmp = (xi + ((ux * maxCos) * (zi * (1.0f - ux)))) + (uy * ((2.0f * (((float) M_PI) * yi)) + (uy * ((-2.0f * (xi * (((float) M_PI) * ((float) M_PI)))) + (-1.3333333333333333f * ((((float) M_PI) * (((float) M_PI) * ((float) M_PI))) * (uy * yi)))))));
	} else {
		tmp = ((yi * sinf((2.0f * (((float) M_PI) * uy)))) + (xi * (1.0f + (uy * (-2.0f * (uy * (((float) M_PI) * ((float) M_PI)))))))) + (maxCos * (zi * ux));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.011420000344514847))
		tmp = Float32(Float32(xi + Float32(Float32(ux * maxCos) * Float32(zi * Float32(Float32(1.0) - ux)))) + Float32(uy * Float32(Float32(Float32(2.0) * Float32(Float32(pi) * yi)) + Float32(uy * Float32(Float32(Float32(-2.0) * Float32(xi * Float32(Float32(pi) * Float32(pi)))) + Float32(Float32(-1.3333333333333333) * Float32(Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi))) * Float32(uy * yi))))))));
	else
		tmp = Float32(Float32(Float32(yi * sin(Float32(Float32(2.0) * Float32(Float32(pi) * uy)))) + Float32(xi * Float32(Float32(1.0) + Float32(uy * Float32(Float32(-2.0) * Float32(uy * Float32(Float32(pi) * Float32(pi)))))))) + Float32(maxCos * Float32(zi * ux)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(xi, yi, zi, ux, uy, maxCos)
	tmp = single(0.0);
	if (uy <= single(0.011420000344514847))
		tmp = (xi + ((ux * maxCos) * (zi * (single(1.0) - ux)))) + (uy * ((single(2.0) * (single(pi) * yi)) + (uy * ((single(-2.0) * (xi * (single(pi) * single(pi)))) + (single(-1.3333333333333333) * ((single(pi) * (single(pi) * single(pi))) * (uy * yi)))))));
	else
		tmp = ((yi * sin((single(2.0) * (single(pi) * uy)))) + (xi * (single(1.0) + (uy * (single(-2.0) * (uy * (single(pi) * single(pi)))))))) + (maxCos * (zi * ux));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.0114200003

   1. Initial program 99.3%

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

   3. Simplified 99.3%

      \[\leadsto \color{blue}{\sqrt{1 + \left(1 - ux\right) \cdot \left(ux \cdot \left(maxCos \cdot \left(ux \cdot \left(maxCos \cdot \left(ux + -1\right)\right)\right)\right)\right)} \cdot \left(\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot yi + \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot xi\right) + \left(ux \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right) \cdot zi} \]
   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 99.3%

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

   8. Taylor expanded in uy around 0

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

      1. associate-+r+ N/A

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

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

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

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

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

      4. associate-*r* N/A

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

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

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

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

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

      7. *-commutative N/A

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

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

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

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

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

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

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

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

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

   10. Simplified 99.1%

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

   if 0.0114200003 < uy

   1. Initial program 98.2%

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

   3. Simplified 98.2%

      \[\leadsto \color{blue}{\sqrt{1 + \left(1 - ux\right) \cdot \left(ux \cdot \left(maxCos \cdot \left(ux \cdot \left(maxCos \cdot \left(ux + -1\right)\right)\right)\right)\right)} \cdot \left(\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot yi + \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot xi\right) + \left(ux \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right) \cdot zi} \]
   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. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

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

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

   7. Simplified 95.6%

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

   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. *-lft-identity N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

      5. distribute-rgt-in N/A

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

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

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

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

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. associate-*l* N/A

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

      15. *-commutative N/A

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

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

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

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

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

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

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

      19. unpow2 N/A

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

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

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

   10. Simplified 76.9%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \leq 0.011420000344514847:\\ \;\;\;\;\left(xi + \left(ux \cdot maxCos\right) \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + uy \cdot \left(2 \cdot \left(\pi \cdot yi\right) + uy \cdot \left(-2 \cdot \left(xi \cdot \left(\pi \cdot \pi\right)\right) + -1.3333333333333333 \cdot \left(\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(uy \cdot yi\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(yi \cdot \sin \left(2 \cdot \left(\pi \cdot uy\right)\right) + xi \cdot \left(1 + uy \cdot \left(-2 \cdot \left(uy \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\right) + maxCos \cdot \left(zi \cdot ux\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 92.8% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.1%

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

3. Simplified 99.1%

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

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

8. Taylor expanded in uy around 0

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

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

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

   2. associate-*r* N/A

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

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

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

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

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

   5. unpow2 N/A

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

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

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

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

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

   8. unpow2 N/A

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

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

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

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

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

   11. PI-lowering-PI.f32 94.8%

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

10. Simplified 94.8%

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

11. Final simplification 94.8%

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

Alternative 10: 91.5% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.05000000074505806:\\ \;\;\;\;\left(xi + \left(ux \cdot maxCos\right) \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + uy \cdot \left(2 \cdot \left(\pi \cdot yi\right) + uy \cdot \left(-2 \cdot \left(xi \cdot \left(\pi \cdot \pi\right)\right) + -1.3333333333333333 \cdot \left(\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(uy \cdot yi\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;xi + \left(yi \cdot \sin \left(2 \cdot \left(\pi \cdot uy\right)\right) + zi \cdot \left(\left(ux \cdot \left(1 - ux\right)\right) \cdot maxCos\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (if (<= uy 0.05000000074505806)
   (+
    (+ xi (* (* ux maxCos) (* zi (- 1.0 ux))))
    (*
     uy
     (+
      (* 2.0 (* PI yi))
      (*
       uy
       (+
        (* -2.0 (* xi (* PI PI)))
        (* -1.3333333333333333 (* (* PI (* PI PI)) (* uy yi))))))))
   (+
    xi
    (+ (* yi (sin (* 2.0 (* PI uy)))) (* zi (* (* ux (- 1.0 ux)) maxCos))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.05000000074505806f) {
		tmp = (xi + ((ux * maxCos) * (zi * (1.0f - ux)))) + (uy * ((2.0f * (((float) M_PI) * yi)) + (uy * ((-2.0f * (xi * (((float) M_PI) * ((float) M_PI)))) + (-1.3333333333333333f * ((((float) M_PI) * (((float) M_PI) * ((float) M_PI))) * (uy * yi)))))));
	} else {
		tmp = xi + ((yi * sinf((2.0f * (((float) M_PI) * uy)))) + (zi * ((ux * (1.0f - ux)) * maxCos)));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.05000000074505806))
		tmp = Float32(Float32(xi + Float32(Float32(ux * maxCos) * Float32(zi * Float32(Float32(1.0) - ux)))) + Float32(uy * Float32(Float32(Float32(2.0) * Float32(Float32(pi) * yi)) + Float32(uy * Float32(Float32(Float32(-2.0) * Float32(xi * Float32(Float32(pi) * Float32(pi)))) + Float32(Float32(-1.3333333333333333) * Float32(Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi))) * Float32(uy * yi))))))));
	else
		tmp = Float32(xi + Float32(Float32(yi * sin(Float32(Float32(2.0) * Float32(Float32(pi) * uy)))) + Float32(zi * Float32(Float32(ux * Float32(Float32(1.0) - ux)) * maxCos))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(xi, yi, zi, ux, uy, maxCos)
	tmp = single(0.0);
	if (uy <= single(0.05000000074505806))
		tmp = (xi + ((ux * maxCos) * (zi * (single(1.0) - ux)))) + (uy * ((single(2.0) * (single(pi) * yi)) + (uy * ((single(-2.0) * (xi * (single(pi) * single(pi)))) + (single(-1.3333333333333333) * ((single(pi) * (single(pi) * single(pi))) * (uy * yi)))))));
	else
		tmp = xi + ((yi * sin((single(2.0) * (single(pi) * uy)))) + (zi * ((ux * (single(1.0) - ux)) * maxCos)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.0500000007

   1. Initial program 99.3%

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

   3. Simplified 99.3%

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

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

   8. Taylor expanded in uy around 0

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

      1. associate-+r+ N/A

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

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

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

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

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

      4. associate-*r* N/A

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

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

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

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

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

      7. *-commutative N/A

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

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

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

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

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

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

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

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

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

   10. Simplified 97.9%

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

   if 0.0500000007 < 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%

      \[\leadsto \color{blue}{\sqrt{1 + \left(1 - ux\right) \cdot \left(ux \cdot \left(maxCos \cdot \left(ux \cdot \left(maxCos \cdot \left(ux + -1\right)\right)\right)\right)\right)} \cdot \left(\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot yi + \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot xi\right) + \left(ux \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right) \cdot zi} \]
   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 97.9%

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

   8. Taylor expanded in uy around 0

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

   10. Simplified 71.5%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \leq 0.05000000074505806:\\ \;\;\;\;\left(xi + \left(ux \cdot maxCos\right) \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + uy \cdot \left(2 \cdot \left(\pi \cdot yi\right) + uy \cdot \left(-2 \cdot \left(xi \cdot \left(\pi \cdot \pi\right)\right) + -1.3333333333333333 \cdot \left(\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(uy \cdot yi\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;xi + \left(yi \cdot \sin \left(2 \cdot \left(\pi \cdot uy\right)\right) + zi \cdot \left(\left(ux \cdot \left(1 - ux\right)\right) \cdot maxCos\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 91.3% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.05000000074505806:\\ \;\;\;\;\left(xi + \left(ux \cdot maxCos\right) \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + uy \cdot \left(2 \cdot \left(\pi \cdot yi\right) + uy \cdot \left(-2 \cdot \left(xi \cdot \left(\pi \cdot \pi\right)\right) + -1.3333333333333333 \cdot \left(\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(uy \cdot yi\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;maxCos \cdot \left(zi \cdot ux\right) + \left(xi + yi \cdot \sin \left(2 \cdot \left(\pi \cdot uy\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (if (<= uy 0.05000000074505806)
   (+
    (+ xi (* (* ux maxCos) (* zi (- 1.0 ux))))
    (*
     uy
     (+
      (* 2.0 (* PI yi))
      (*
       uy
       (+
        (* -2.0 (* xi (* PI PI)))
        (* -1.3333333333333333 (* (* PI (* PI PI)) (* uy yi))))))))
   (+ (* maxCos (* zi ux)) (+ xi (* yi (sin (* 2.0 (* PI uy))))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.05000000074505806f) {
		tmp = (xi + ((ux * maxCos) * (zi * (1.0f - ux)))) + (uy * ((2.0f * (((float) M_PI) * yi)) + (uy * ((-2.0f * (xi * (((float) M_PI) * ((float) M_PI)))) + (-1.3333333333333333f * ((((float) M_PI) * (((float) M_PI) * ((float) M_PI))) * (uy * yi)))))));
	} else {
		tmp = (maxCos * (zi * ux)) + (xi + (yi * sinf((2.0f * (((float) M_PI) * uy)))));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.05000000074505806))
		tmp = Float32(Float32(xi + Float32(Float32(ux * maxCos) * Float32(zi * Float32(Float32(1.0) - ux)))) + Float32(uy * Float32(Float32(Float32(2.0) * Float32(Float32(pi) * yi)) + Float32(uy * Float32(Float32(Float32(-2.0) * Float32(xi * Float32(Float32(pi) * Float32(pi)))) + Float32(Float32(-1.3333333333333333) * Float32(Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi))) * Float32(uy * yi))))))));
	else
		tmp = Float32(Float32(maxCos * Float32(zi * ux)) + Float32(xi + Float32(yi * sin(Float32(Float32(2.0) * Float32(Float32(pi) * uy))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(xi, yi, zi, ux, uy, maxCos)
	tmp = single(0.0);
	if (uy <= single(0.05000000074505806))
		tmp = (xi + ((ux * maxCos) * (zi * (single(1.0) - ux)))) + (uy * ((single(2.0) * (single(pi) * yi)) + (uy * ((single(-2.0) * (xi * (single(pi) * single(pi)))) + (single(-1.3333333333333333) * ((single(pi) * (single(pi) * single(pi))) * (uy * yi)))))));
	else
		tmp = (maxCos * (zi * ux)) + (xi + (yi * sin((single(2.0) * (single(pi) * uy)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.0500000007

   1. Initial program 99.3%

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

   3. Simplified 99.3%

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

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

   8. Taylor expanded in uy around 0

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

      1. associate-+r+ N/A

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

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

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

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

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

      4. associate-*r* N/A

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

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

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

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

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

      7. *-commutative N/A

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

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

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

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

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

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

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

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

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

   10. Simplified 97.9%

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

   if 0.0500000007 < 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%

      \[\leadsto \color{blue}{\sqrt{1 + \left(1 - ux\right) \cdot \left(ux \cdot \left(maxCos \cdot \left(ux \cdot \left(maxCos \cdot \left(ux + -1\right)\right)\right)\right)\right)} \cdot \left(\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot yi + \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot xi\right) + \left(ux \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right) \cdot zi} \]
   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. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

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

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

   7. Simplified 95.2%

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

   8. Taylor expanded in uy around 0

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

   10. Simplified 69.9%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \leq 0.05000000074505806:\\ \;\;\;\;\left(xi + \left(ux \cdot maxCos\right) \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + uy \cdot \left(2 \cdot \left(\pi \cdot yi\right) + uy \cdot \left(-2 \cdot \left(xi \cdot \left(\pi \cdot \pi\right)\right) + -1.3333333333333333 \cdot \left(\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(uy \cdot yi\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;maxCos \cdot \left(zi \cdot ux\right) + \left(xi + yi \cdot \sin \left(2 \cdot \left(\pi \cdot uy\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 90.9% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.08749999850988388:\\ \;\;\;\;\left(xi + \left(ux \cdot maxCos\right) \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + uy \cdot \left(2 \cdot \left(\pi \cdot yi\right) + uy \cdot \left(-2 \cdot \left(xi \cdot \left(\pi \cdot \pi\right)\right) + -1.3333333333333333 \cdot \left(\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(uy \cdot yi\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;maxCos \cdot \left(zi \cdot ux\right) + yi \cdot \sin \left(\pi \cdot \left(2 \cdot uy\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (if (<= uy 0.08749999850988388)
   (+
    (+ xi (* (* ux maxCos) (* zi (- 1.0 ux))))
    (*
     uy
     (+
      (* 2.0 (* PI yi))
      (*
       uy
       (+
        (* -2.0 (* xi (* PI PI)))
        (* -1.3333333333333333 (* (* PI (* PI PI)) (* uy yi))))))))
   (+ (* maxCos (* zi ux)) (* yi (sin (* PI (* 2.0 uy)))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.08749999850988388f) {
		tmp = (xi + ((ux * maxCos) * (zi * (1.0f - ux)))) + (uy * ((2.0f * (((float) M_PI) * yi)) + (uy * ((-2.0f * (xi * (((float) M_PI) * ((float) M_PI)))) + (-1.3333333333333333f * ((((float) M_PI) * (((float) M_PI) * ((float) M_PI))) * (uy * yi)))))));
	} else {
		tmp = (maxCos * (zi * ux)) + (yi * sinf((((float) M_PI) * (2.0f * uy))));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.08749999850988388))
		tmp = Float32(Float32(xi + Float32(Float32(ux * maxCos) * Float32(zi * Float32(Float32(1.0) - ux)))) + Float32(uy * Float32(Float32(Float32(2.0) * Float32(Float32(pi) * yi)) + Float32(uy * Float32(Float32(Float32(-2.0) * Float32(xi * Float32(Float32(pi) * Float32(pi)))) + Float32(Float32(-1.3333333333333333) * Float32(Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi))) * Float32(uy * yi))))))));
	else
		tmp = Float32(Float32(maxCos * Float32(zi * ux)) + Float32(yi * sin(Float32(Float32(pi) * Float32(Float32(2.0) * uy)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(xi, yi, zi, ux, uy, maxCos)
	tmp = single(0.0);
	if (uy <= single(0.08749999850988388))
		tmp = (xi + ((ux * maxCos) * (zi * (single(1.0) - ux)))) + (uy * ((single(2.0) * (single(pi) * yi)) + (uy * ((single(-2.0) * (xi * (single(pi) * single(pi)))) + (single(-1.3333333333333333) * ((single(pi) * (single(pi) * single(pi))) * (uy * yi)))))));
	else
		tmp = (maxCos * (zi * ux)) + (yi * sin((single(pi) * (single(2.0) * uy))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.0874999985

   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 99.3%

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

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

   8. Taylor expanded in uy around 0

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

      1. associate-+r+ N/A

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

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

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

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

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

      4. associate-*r* N/A

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

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

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

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

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

      7. *-commutative N/A

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

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

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

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

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

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

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

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

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

   10. Simplified 96.9%

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

   if 0.0874999985 < uy

   1. Initial program 97.9%

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

   3. Simplified 97.9%

      \[\leadsto \color{blue}{\sqrt{1 + \left(1 - ux\right) \cdot \left(ux \cdot \left(maxCos \cdot \left(ux \cdot \left(maxCos \cdot \left(ux + -1\right)\right)\right)\right)\right)} \cdot \left(\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot yi + \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot xi\right) + \left(ux \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right) \cdot zi} \]
   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. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

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

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

   7. Simplified 94.6%

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

   8. Taylor expanded in xi around 0

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

      1. +-commutative N/A

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

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

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

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

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

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

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

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

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

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

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

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

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

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

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

      11. *-lowering-*.f32 65.2%

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

   10. Simplified 65.2%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \leq 0.08749999850988388:\\ \;\;\;\;\left(xi + \left(ux \cdot maxCos\right) \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + uy \cdot \left(2 \cdot \left(\pi \cdot yi\right) + uy \cdot \left(-2 \cdot \left(xi \cdot \left(\pi \cdot \pi\right)\right) + -1.3333333333333333 \cdot \left(\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(uy \cdot yi\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;maxCos \cdot \left(zi \cdot ux\right) + yi \cdot \sin \left(\pi \cdot \left(2 \cdot uy\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 88.2% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.0006600000197067857:\\ \;\;\;\;\left(xi + \left(ux \cdot maxCos\right) \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + yi \cdot \left(\pi \cdot \left(2 \cdot uy\right) + \frac{\left(\pi \cdot \pi\right) \cdot \left(-2 \cdot \left(xi \cdot \left(uy \cdot uy\right)\right)\right)}{yi}\right)\\ \mathbf{else}:\\ \;\;\;\;maxCos \cdot \left(zi \cdot ux\right) + \left(xi + uy \cdot \left(2 \cdot \left(\pi \cdot yi\right) + uy \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(xi \cdot -2\right) + \left(\pi \cdot \left(yi \cdot \left(\pi \cdot \pi\right)\right)\right) \cdot \left(uy \cdot -1.3333333333333333\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (if (<= uy 0.0006600000197067857)
   (+
    (+ xi (* (* ux maxCos) (* zi (- 1.0 ux))))
    (*
     yi
     (+ (* PI (* 2.0 uy)) (/ (* (* PI PI) (* -2.0 (* xi (* uy uy)))) yi))))
   (+
    (* maxCos (* zi ux))
    (+
     xi
     (*
      uy
      (+
       (* 2.0 (* PI yi))
       (*
        uy
        (+
         (* (* PI PI) (* xi -2.0))
         (* (* PI (* yi (* PI PI))) (* uy -1.3333333333333333))))))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.0006600000197067857f) {
		tmp = (xi + ((ux * maxCos) * (zi * (1.0f - ux)))) + (yi * ((((float) M_PI) * (2.0f * uy)) + (((((float) M_PI) * ((float) M_PI)) * (-2.0f * (xi * (uy * uy)))) / yi)));
	} else {
		tmp = (maxCos * (zi * ux)) + (xi + (uy * ((2.0f * (((float) M_PI) * yi)) + (uy * (((((float) M_PI) * ((float) M_PI)) * (xi * -2.0f)) + ((((float) M_PI) * (yi * (((float) M_PI) * ((float) M_PI)))) * (uy * -1.3333333333333333f)))))));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.0006600000197067857))
		tmp = Float32(Float32(xi + Float32(Float32(ux * maxCos) * Float32(zi * Float32(Float32(1.0) - ux)))) + Float32(yi * Float32(Float32(Float32(pi) * Float32(Float32(2.0) * uy)) + Float32(Float32(Float32(Float32(pi) * Float32(pi)) * Float32(Float32(-2.0) * Float32(xi * Float32(uy * uy)))) / yi))));
	else
		tmp = Float32(Float32(maxCos * Float32(zi * ux)) + Float32(xi + Float32(uy * Float32(Float32(Float32(2.0) * Float32(Float32(pi) * yi)) + Float32(uy * Float32(Float32(Float32(Float32(pi) * Float32(pi)) * Float32(xi * Float32(-2.0))) + Float32(Float32(Float32(pi) * Float32(yi * Float32(Float32(pi) * Float32(pi)))) * Float32(uy * Float32(-1.3333333333333333)))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(xi, yi, zi, ux, uy, maxCos)
	tmp = single(0.0);
	if (uy <= single(0.0006600000197067857))
		tmp = (xi + ((ux * maxCos) * (zi * (single(1.0) - ux)))) + (yi * ((single(pi) * (single(2.0) * uy)) + (((single(pi) * single(pi)) * (single(-2.0) * (xi * (uy * uy)))) / yi)));
	else
		tmp = (maxCos * (zi * ux)) + (xi + (uy * ((single(2.0) * (single(pi) * yi)) + (uy * (((single(pi) * single(pi)) * (xi * single(-2.0))) + ((single(pi) * (yi * (single(pi) * single(pi)))) * (uy * single(-1.3333333333333333))))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if uy < 6.6000002e-4

   1. Initial program 99.4%

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

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\sqrt{1 + \left(1 - ux\right) \cdot \left(ux \cdot \left(maxCos \cdot \left(ux \cdot \left(maxCos \cdot \left(ux + -1\right)\right)\right)\right)\right)} \cdot \left(\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot yi + \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot xi\right) + \left(ux \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right) \cdot zi} \]
   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 99.4%

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

   8. Taylor expanded in uy 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. associate-+r+ N/A

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

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

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

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

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

      4. associate-*r* N/A

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

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

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

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

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

      7. *-commutative N/A

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

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

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

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

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

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

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

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

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

   10. Simplified 98.9%

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

   11. Taylor expanded in yi around inf

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

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

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

       2. +-commutative N/A

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

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

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

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

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

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

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

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

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

       9. *-commutative N/A

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

       10. associate-*l/ N/A

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

       11. associate-*r* N/A

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

       12. *-commutative N/A

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

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

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

   13. Simplified 99.0%

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

   if 6.6000002e-4 < uy

   1. Initial program 98.2%

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

   3. Simplified 98.2%

      \[\leadsto \color{blue}{\sqrt{1 + \left(1 - ux\right) \cdot \left(ux \cdot \left(maxCos \cdot \left(ux \cdot \left(maxCos \cdot \left(ux + -1\right)\right)\right)\right)\right)} \cdot \left(\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot yi + \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot xi\right) + \left(ux \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right) \cdot zi} \]
   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. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

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

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

   7. Simplified 96.2%

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

   8. Taylor expanded in uy around 0

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

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

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

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

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

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

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

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

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

      5. *-commutative N/A

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

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

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

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

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

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

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

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

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

   10. Simplified 66.4%

       \[\leadsto \color{blue}{\left(xi + uy \cdot \left(2 \cdot \left(\pi \cdot yi\right) + uy \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(xi \cdot -2\right) + \left(\pi \cdot \left(\left(\pi \cdot \pi\right) \cdot yi\right)\right) \cdot \left(uy \cdot -1.3333333333333333\right)\right)\right)\right)} + maxCos \cdot \left(ux \cdot zi\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \leq 0.0006600000197067857:\\ \;\;\;\;\left(xi + \left(ux \cdot maxCos\right) \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + yi \cdot \left(\pi \cdot \left(2 \cdot uy\right) + \frac{\left(\pi \cdot \pi\right) \cdot \left(-2 \cdot \left(xi \cdot \left(uy \cdot uy\right)\right)\right)}{yi}\right)\\ \mathbf{else}:\\ \;\;\;\;maxCos \cdot \left(zi \cdot ux\right) + \left(xi + uy \cdot \left(2 \cdot \left(\pi \cdot yi\right) + uy \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(xi \cdot -2\right) + \left(\pi \cdot \left(yi \cdot \left(\pi \cdot \pi\right)\right)\right) \cdot \left(uy \cdot -1.3333333333333333\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 89.1% accurate, 11.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

3. Simplified 99.1%

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

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

8. Taylor expanded in uy around 0

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

   1. associate-+r+ N/A

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

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

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

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

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

   4. associate-*r* N/A

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

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

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

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

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

   7. *-commutative N/A

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

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

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

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

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

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

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

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

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

10. Simplified 89.6%

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

11. Final simplification 89.6%

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

Alternative 15: 85.5% accurate, 14.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

3. Simplified 99.1%

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

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

8. Taylor expanded in uy 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. associate-+r+ N/A

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

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

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

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

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

   4. associate-*r* N/A

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

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

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

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

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

   7. *-commutative N/A

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

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

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

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

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

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

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

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

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

10. Simplified 85.9%

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

11. Taylor expanded in yi around inf

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

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

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

    2. +-commutative N/A

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

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

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

    4. associate-*r* N/A

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

    5. *-commutative N/A

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

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

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

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

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

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

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

    9. *-commutative N/A

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

    10. associate-*l/ N/A

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

    11. associate-*r* N/A

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

    12. *-commutative N/A

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

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

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

13. Simplified 86.0%

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

14. Final simplification 86.0%

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

Alternative 16: 85.5% accurate, 14.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

3. Simplified 99.1%

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

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

8. Taylor expanded in uy 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. associate-+r+ N/A

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

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

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

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

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

   4. associate-*r* N/A

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

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

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

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

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

   7. *-commutative N/A

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

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

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

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

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

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

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

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

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

10. Simplified 85.9%

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

11. Taylor expanded in ux around 0

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

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

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

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

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

    3. +-commutative N/A

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

    4. mul-1-neg N/A

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

    5. unsub-neg N/A

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

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

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

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

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

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

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

    9. *-lowering-*.f32 85.9%

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

13. Simplified 85.9%

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

14. Final simplification 85.9%

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

Alternative 17: 85.5% accurate, 15.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

3. Simplified 99.1%

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

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

8. Taylor expanded in uy 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. associate-+r+ N/A

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

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

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

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

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

   4. associate-*r* N/A

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

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

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

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

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

   7. *-commutative N/A

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

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

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

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

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

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

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

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

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

10. Simplified 85.9%

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

11. Final simplification 85.9%

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

Alternative 18: 61.1% accurate, 18.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(ux \cdot ux\right) \cdot \frac{2 \cdot \left(uy \cdot \left(\pi \cdot yi\right)\right)}{ux \cdot ux}\\ \mathbf{if}\;yi \leq -3.99999992980668 \cdot 10^{-13}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;yi \leq 1.000000013351432 \cdot 10^{-10}:\\ \;\;\;\;xi + \left(1 - ux\right) \cdot \left(maxCos \cdot \left(zi \cdot ux\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (* ux ux) (/ (* 2.0 (* uy (* PI yi))) (* ux ux)))))
   (if (<= yi -3.99999992980668e-13)
     t_0
     (if (<= yi 1.000000013351432e-10)
       (+ xi (* (- 1.0 ux) (* maxCos (* zi ux))))
       t_0))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = (ux * ux) * ((2.0f * (uy * (((float) M_PI) * yi))) / (ux * ux));
	float tmp;
	if (yi <= -3.99999992980668e-13f) {
		tmp = t_0;
	} else if (yi <= 1.000000013351432e-10f) {
		tmp = xi + ((1.0f - ux) * (maxCos * (zi * ux)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(ux * ux) * Float32(Float32(Float32(2.0) * Float32(uy * Float32(Float32(pi) * yi))) / Float32(ux * ux)))
	tmp = Float32(0.0)
	if (yi <= Float32(-3.99999992980668e-13))
		tmp = t_0;
	elseif (yi <= Float32(1.000000013351432e-10))
		tmp = Float32(xi + Float32(Float32(Float32(1.0) - ux) * Float32(maxCos * Float32(zi * ux))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(xi, yi, zi, ux, uy, maxCos)
	t_0 = (ux * ux) * ((single(2.0) * (uy * (single(pi) * yi))) / (ux * ux));
	tmp = single(0.0);
	if (yi <= single(-3.99999992980668e-13))
		tmp = t_0;
	elseif (yi <= single(1.000000013351432e-10))
		tmp = xi + ((single(1.0) - ux) * (maxCos * (zi * ux)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if yi < -3.99999993e-13 or 1.00000001e-10 < yi

   1. Initial program 98.8%

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

   3. Simplified 98.8%

      \[\leadsto \color{blue}{\sqrt{1 + \left(1 - ux\right) \cdot \left(ux \cdot \left(maxCos \cdot \left(ux \cdot \left(maxCos \cdot \left(ux + -1\right)\right)\right)\right)\right)} \cdot \left(\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot yi + \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot xi\right) + \left(ux \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right) \cdot zi} \]
   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 98.5%

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

   8. Taylor expanded in uy 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. associate-+r+ N/A

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

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

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

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

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

      4. associate-*r* N/A

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

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

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

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

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

      7. *-commutative N/A

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

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

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

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

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

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

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

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

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

   10. Simplified 82.5%

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

   11. Taylor expanded in ux around -inf

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

   13. Simplified 82.1%

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

   14. Taylor expanded in yi around inf

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, ux\right), \color{blue}{\left(2 \cdot \frac{uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)}{{ux}^{2}}\right)}\right) \]
   15. Step-by-step derivation

       1. associate-*r/ N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

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

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

       6. *-commutative N/A

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

       7. associate-*r* N/A

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

       8. *-commutative N/A

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

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

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

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

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

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

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

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

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

       13. unpow2 N/A

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

       14. *-lowering-*.f32 56.4%

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

   16. Simplified 56.4%

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

   if -3.99999993e-13 < yi < 1.00000001e-10

   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%

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

   5. Taylor expanded in ux around 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      11. PI-lowering-PI.f32 99.0%

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

   7. Simplified 99.0%

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

   8. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{xi + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)} \]
   9. 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. associate-*r* N/A

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

      4. associate-*r* N/A

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

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

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

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

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

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

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

      8. --lowering--.f32 68.2%

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

   10. Simplified 68.2%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;yi \leq -3.99999992980668 \cdot 10^{-13}:\\ \;\;\;\;\left(ux \cdot ux\right) \cdot \frac{2 \cdot \left(uy \cdot \left(\pi \cdot yi\right)\right)}{ux \cdot ux}\\ \mathbf{elif}\;yi \leq 1.000000013351432 \cdot 10^{-10}:\\ \;\;\;\;xi + \left(1 - ux\right) \cdot \left(maxCos \cdot \left(zi \cdot ux\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(ux \cdot ux\right) \cdot \frac{2 \cdot \left(uy \cdot \left(\pi \cdot yi\right)\right)}{ux \cdot ux}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 83.0% accurate, 18.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.1%

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

3. Simplified 99.1%

   \[\leadsto \color{blue}{\sqrt{1 + \left(1 - ux\right) \cdot \left(ux \cdot \left(maxCos \cdot \left(ux \cdot \left(maxCos \cdot \left(ux + -1\right)\right)\right)\right)\right)} \cdot \left(\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot yi + \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot xi\right) + \left(ux \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right) \cdot zi} \]
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. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

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

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

7. Simplified 95.2%

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

8. Taylor expanded in uy around 0

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

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

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

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

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

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

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

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

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

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

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

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

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

   11. unpow2 N/A

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

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

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

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

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

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

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

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

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

   16. *-commutative N/A

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

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

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

10. Simplified 82.6%

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

11. Final simplification 82.6%

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

Alternative 20: 61.1% accurate, 21.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \left(uy \cdot \left(\pi \cdot yi\right)\right)\\ \mathbf{if}\;yi \leq -3.99999992980668 \cdot 10^{-13}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;yi \leq 1.000000013351432 \cdot 10^{-10}:\\ \;\;\;\;xi + \left(1 - ux\right) \cdot \left(maxCos \cdot \left(zi \cdot ux\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* 2.0 (* uy (* PI yi)))))
   (if (<= yi -3.99999992980668e-13)
     t_0
     (if (<= yi 1.000000013351432e-10)
       (+ xi (* (- 1.0 ux) (* maxCos (* zi ux))))
       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) * yi));
	float tmp;
	if (yi <= -3.99999992980668e-13f) {
		tmp = t_0;
	} else if (yi <= 1.000000013351432e-10f) {
		tmp = xi + ((1.0f - ux) * (maxCos * (zi * ux)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(2.0) * Float32(uy * Float32(Float32(pi) * yi)))
	tmp = Float32(0.0)
	if (yi <= Float32(-3.99999992980668e-13))
		tmp = t_0;
	elseif (yi <= Float32(1.000000013351432e-10))
		tmp = Float32(xi + Float32(Float32(Float32(1.0) - ux) * Float32(maxCos * Float32(zi * ux))));
	else
		tmp = 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) * yi));
	tmp = single(0.0);
	if (yi <= single(-3.99999992980668e-13))
		tmp = t_0;
	elseif (yi <= single(1.000000013351432e-10))
		tmp = xi + ((single(1.0) - ux) * (maxCos * (zi * ux)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if yi < -3.99999993e-13 or 1.00000001e-10 < yi

   1. Initial program 98.8%

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

   3. Simplified 98.8%

      \[\leadsto \color{blue}{\sqrt{1 + \left(1 - ux\right) \cdot \left(ux \cdot \left(maxCos \cdot \left(ux \cdot \left(maxCos \cdot \left(ux + -1\right)\right)\right)\right)\right)} \cdot \left(\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot yi + \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot xi\right) + \left(ux \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right) \cdot zi} \]
   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 98.5%

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

   8. Taylor expanded in uy 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. associate-+r+ N/A

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

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

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

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

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

      4. associate-*r* N/A

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

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

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

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

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

      7. *-commutative N/A

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

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

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

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

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

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

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

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

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

   10. Simplified 82.5%

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

   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. *-lowering-*.f32 N/A

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

       3. *-commutative N/A

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

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

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

       5. PI-lowering-PI.f32 56.4%

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

   13. Simplified 56.4%

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

   if -3.99999993e-13 < yi < 1.00000001e-10

   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%

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

   5. Taylor expanded in ux around 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      11. PI-lowering-PI.f32 99.0%

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

   7. Simplified 99.0%

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

   8. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{xi + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)} \]
   9. 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. associate-*r* N/A

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

      4. associate-*r* N/A

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

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

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

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

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

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

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

      8. --lowering--.f32 68.2%

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

   10. Simplified 68.2%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;yi \leq -3.99999992980668 \cdot 10^{-13}:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\pi \cdot yi\right)\right)\\ \mathbf{elif}\;yi \leq 1.000000013351432 \cdot 10^{-10}:\\ \;\;\;\;xi + \left(1 - ux\right) \cdot \left(maxCos \cdot \left(zi \cdot ux\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\pi \cdot yi\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 61.1% accurate, 21.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \left(uy \cdot \left(\pi \cdot yi\right)\right)\\ \mathbf{if}\;yi \leq -3.99999992980668 \cdot 10^{-13}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;yi \leq 1.000000013351432 \cdot 10^{-10}:\\ \;\;\;\;xi + zi \cdot \left(ux \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* 2.0 (* uy (* PI yi)))))
   (if (<= yi -3.99999992980668e-13)
     t_0
     (if (<= yi 1.000000013351432e-10)
       (+ xi (* zi (* ux (* (- 1.0 ux) maxCos))))
       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) * yi));
	float tmp;
	if (yi <= -3.99999992980668e-13f) {
		tmp = t_0;
	} else if (yi <= 1.000000013351432e-10f) {
		tmp = xi + (zi * (ux * ((1.0f - ux) * maxCos)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(2.0) * Float32(uy * Float32(Float32(pi) * yi)))
	tmp = Float32(0.0)
	if (yi <= Float32(-3.99999992980668e-13))
		tmp = t_0;
	elseif (yi <= Float32(1.000000013351432e-10))
		tmp = Float32(xi + Float32(zi * Float32(ux * Float32(Float32(Float32(1.0) - ux) * maxCos))));
	else
		tmp = 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) * yi));
	tmp = single(0.0);
	if (yi <= single(-3.99999992980668e-13))
		tmp = t_0;
	elseif (yi <= single(1.000000013351432e-10))
		tmp = xi + (zi * (ux * ((single(1.0) - ux) * maxCos)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if yi < -3.99999993e-13 or 1.00000001e-10 < yi

   1. Initial program 98.8%

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

   3. Simplified 98.8%

      \[\leadsto \color{blue}{\sqrt{1 + \left(1 - ux\right) \cdot \left(ux \cdot \left(maxCos \cdot \left(ux \cdot \left(maxCos \cdot \left(ux + -1\right)\right)\right)\right)\right)} \cdot \left(\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot yi + \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot xi\right) + \left(ux \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right) \cdot zi} \]
   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 98.5%

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

   8. Taylor expanded in uy 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. associate-+r+ N/A

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

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

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

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

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

      4. associate-*r* N/A

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

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

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

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

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

      7. *-commutative N/A

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

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

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

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

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

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

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

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

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

   10. Simplified 82.5%

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

   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. *-lowering-*.f32 N/A

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

       3. *-commutative N/A

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

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

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

       5. PI-lowering-PI.f32 56.4%

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

   13. Simplified 56.4%

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

   if -3.99999993e-13 < yi < 1.00000001e-10

   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%

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

   5. Taylor expanded in ux around 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      11. PI-lowering-PI.f32 99.0%

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

   7. Simplified 99.0%

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

   8. Taylor expanded in uy around 0

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

   10. Simplified 68.2%

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

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;yi \leq -3.99999992980668 \cdot 10^{-13}:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\pi \cdot yi\right)\right)\\ \mathbf{elif}\;yi \leq 1.000000013351432 \cdot 10^{-10}:\\ \;\;\;\;xi + zi \cdot \left(ux \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\pi \cdot yi\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 61.1% accurate, 21.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \left(uy \cdot \left(\pi \cdot yi\right)\right)\\ \mathbf{if}\;yi \leq -3.99999992980668 \cdot 10^{-13}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;yi \leq 1.000000013351432 \cdot 10^{-10}:\\ \;\;\;\;xi + \left(ux \cdot maxCos\right) \cdot \left(zi \cdot \left(1 - ux\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* 2.0 (* uy (* PI yi)))))
   (if (<= yi -3.99999992980668e-13)
     t_0
     (if (<= yi 1.000000013351432e-10)
       (+ xi (* (* ux maxCos) (* zi (- 1.0 ux))))
       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) * yi));
	float tmp;
	if (yi <= -3.99999992980668e-13f) {
		tmp = t_0;
	} else if (yi <= 1.000000013351432e-10f) {
		tmp = xi + ((ux * maxCos) * (zi * (1.0f - ux)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(2.0) * Float32(uy * Float32(Float32(pi) * yi)))
	tmp = Float32(0.0)
	if (yi <= Float32(-3.99999992980668e-13))
		tmp = t_0;
	elseif (yi <= Float32(1.000000013351432e-10))
		tmp = Float32(xi + Float32(Float32(ux * maxCos) * Float32(zi * Float32(Float32(1.0) - ux))));
	else
		tmp = 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) * yi));
	tmp = single(0.0);
	if (yi <= single(-3.99999992980668e-13))
		tmp = t_0;
	elseif (yi <= single(1.000000013351432e-10))
		tmp = xi + ((ux * maxCos) * (zi * (single(1.0) - ux)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if yi < -3.99999993e-13 or 1.00000001e-10 < yi

   1. Initial program 98.8%

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

   3. Simplified 98.8%

      \[\leadsto \color{blue}{\sqrt{1 + \left(1 - ux\right) \cdot \left(ux \cdot \left(maxCos \cdot \left(ux \cdot \left(maxCos \cdot \left(ux + -1\right)\right)\right)\right)\right)} \cdot \left(\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot yi + \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot xi\right) + \left(ux \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right) \cdot zi} \]
   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 98.5%

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

   8. Taylor expanded in uy 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. associate-+r+ N/A

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

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

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

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

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

      4. associate-*r* N/A

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

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

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

      6. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(\mathsf{*.f32}\left(maxCos, ux\right), \left(zi \cdot \left(1 - ux\right)\right)\right)\right), \left(uy \cdot \left(-2 \cdot \left(uy \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(\mathsf{*.f32}\left(maxCos, ux\right), \left(\left(1 - ux\right) \cdot zi\right)\right)\right), \left(uy \cdot \left(-2 \cdot \left(uy \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(\mathsf{*.f32}\left(maxCos, ux\right), \mathsf{*.f32}\left(\left(1 - ux\right), zi\right)\right)\right), \left(uy \cdot \left(-2 \cdot \left(uy \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      9. --lowering--.f32 N/A

         \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(\mathsf{*.f32}\left(maxCos, ux\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, ux\right), zi\right)\right)\right), \left(uy \cdot \left(-2 \cdot \left(uy \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(\mathsf{*.f32}\left(maxCos, ux\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, ux\right), zi\right)\right)\right), \mathsf{*.f32}\left(uy, \color{blue}{\left(-2 \cdot \left(uy \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right) \]

      11. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(\mathsf{*.f32}\left(maxCos, ux\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, ux\right), zi\right)\right)\right), \mathsf{*.f32}\left(uy, \mathsf{+.f32}\left(\left(-2 \cdot \left(uy \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \color{blue}{\left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)\right) \]

   10. Simplified 82.5%

       \[\leadsto \color{blue}{\left(xi + \left(maxCos \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot zi\right)\right) + uy \cdot \left(\left(-2 \cdot uy\right) \cdot \left(xi \cdot \left(\pi \cdot \pi\right)\right) + 2 \cdot \left(\pi \cdot yi\right)\right)} \]

   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. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \color{blue}{\left(yi \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \left(\mathsf{PI}\left(\right) \cdot \color{blue}{yi}\right)\right)\right) \]

       4. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(\mathsf{PI}\left(\right), \color{blue}{yi}\right)\right)\right) \]

       5. PI-lowering-PI.f32 56.4%

          \[\leadsto \mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), yi\right)\right)\right) \]

   13. Simplified 56.4%

       \[\leadsto \color{blue}{2 \cdot \left(uy \cdot \left(\pi \cdot yi\right)\right)} \]

   if -3.99999993e-13 < yi < 1.00000001e-10

   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%

      \[\leadsto \color{blue}{\sqrt{1 + \left(1 - ux\right) \cdot \left(ux \cdot \left(maxCos \cdot \left(ux \cdot \left(maxCos \cdot \left(ux + -1\right)\right)\right)\right)\right)} \cdot \left(\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot yi + \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot xi\right) + \left(ux \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right) \cdot zi} \]
   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 99.1%

      \[\leadsto \color{blue}{xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right) + \left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right) + zi \cdot \left(maxCos \cdot \left(ux \cdot \left(1 - ux\right)\right)\right)\right)} \]

   8. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{xi + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)} \]
   9. Step-by-step derivation

      1. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{+.f32}\left(xi, \color{blue}{\left(maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right)}\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{+.f32}\left(xi, \left(\left(maxCos \cdot ux\right) \cdot \color{blue}{\left(zi \cdot \left(1 - ux\right)\right)}\right)\right) \]

      3. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(\left(maxCos \cdot ux\right), \color{blue}{\left(zi \cdot \left(1 - ux\right)\right)}\right)\right) \]

      4. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(\mathsf{*.f32}\left(maxCos, ux\right), \left(\color{blue}{zi} \cdot \left(1 - ux\right)\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(\mathsf{*.f32}\left(maxCos, ux\right), \left(\left(1 - ux\right) \cdot \color{blue}{zi}\right)\right)\right) \]

      6. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(\mathsf{*.f32}\left(maxCos, ux\right), \mathsf{*.f32}\left(\left(1 - ux\right), \color{blue}{zi}\right)\right)\right) \]

      7. --lowering--.f32 68.1%

         \[\leadsto \mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(\mathsf{*.f32}\left(maxCos, ux\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, ux\right), zi\right)\right)\right) \]

   10. Simplified 68.1%

       \[\leadsto \color{blue}{xi + \left(maxCos \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot zi\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;yi \leq -3.99999992980668 \cdot 10^{-13}:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\pi \cdot yi\right)\right)\\ \mathbf{elif}\;yi \leq 1.000000013351432 \cdot 10^{-10}:\\ \;\;\;\;xi + \left(ux \cdot maxCos\right) \cdot \left(zi \cdot \left(1 - ux\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\pi \cdot yi\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 81.3% accurate, 24.3× speedup

Math

\[\begin{array}{l} \\ \left(1 - ux\right) \cdot \left(maxCos \cdot \left(zi \cdot ux\right)\right) + \left(xi + 2 \cdot \left(uy \cdot \left(\pi \cdot yi\right)\right)\right) \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (+ (* (- 1.0 ux) (* maxCos (* zi ux))) (+ xi (* 2.0 (* uy (* PI yi))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return ((1.0f - ux) * (maxCos * (zi * ux))) + (xi + (2.0f * (uy * (((float) M_PI) * yi))));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return Float32(Float32(Float32(Float32(1.0) - ux) * Float32(maxCos * Float32(zi * ux))) + Float32(xi + Float32(Float32(2.0) * Float32(uy * Float32(Float32(pi) * yi)))))
end

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	tmp = ((single(1.0) - ux) * (maxCos * (zi * ux))) + (xi + (single(2.0) * (uy * (single(pi) * yi))));
end

Derivation

1. Initial program 99.1%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

3. Simplified 99.1%

   \[\leadsto \color{blue}{\sqrt{1 + \left(1 - ux\right) \cdot \left(ux \cdot \left(maxCos \cdot \left(ux \cdot \left(maxCos \cdot \left(ux + -1\right)\right)\right)\right)\right)} \cdot \left(\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot yi + \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot xi\right) + \left(ux \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right) \cdot zi} \]
4. Add Preprocessing

5. Taylor expanded in ux around 0

   \[\leadsto \mathsf{+.f32}\left(\color{blue}{\left(xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)}, \mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, ux\right), maxCos\right)\right), zi\right)\right) \]
6. Step-by-step derivation

   1. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(\left(xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right), \left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \mathsf{*.f32}\left(\color{blue}{\mathsf{*.f32}\left(ux, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, ux\right), maxCos\right)\right)}, zi\right)\right) \]

   2. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right), \left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(\color{blue}{ux}, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, ux\right), maxCos\right)\right), zi\right)\right) \]

   3. cos-lowering-cos.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(xi, \mathsf{cos.f32}\left(\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, ux\right), maxCos\right)\right), zi\right)\right) \]

   4. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(xi, \mathsf{cos.f32}\left(\mathsf{*.f32}\left(2, \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, ux\right), maxCos\right)\right), zi\right)\right) \]

   5. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(xi, \mathsf{cos.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \mathsf{PI}\left(\right)\right)\right)\right)\right), \left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, ux\right), maxCos\right)\right), zi\right)\right) \]

   6. PI-lowering-PI.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(xi, \mathsf{cos.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \mathsf{PI.f32}\left(\right)\right)\right)\right)\right), \left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, ux\right), maxCos\right)\right), zi\right)\right) \]

   7. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(xi, \mathsf{cos.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \mathsf{PI.f32}\left(\right)\right)\right)\right)\right), \mathsf{*.f32}\left(yi, \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, \color{blue}{\mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, ux\right), maxCos\right)}\right), zi\right)\right) \]

   8. sin-lowering-sin.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(xi, \mathsf{cos.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \mathsf{PI.f32}\left(\right)\right)\right)\right)\right), \mathsf{*.f32}\left(yi, \mathsf{sin.f32}\left(\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, ux\right), \color{blue}{maxCos}\right)\right), zi\right)\right) \]

   9. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(xi, \mathsf{cos.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \mathsf{PI.f32}\left(\right)\right)\right)\right)\right), \mathsf{*.f32}\left(yi, \mathsf{sin.f32}\left(\mathsf{*.f32}\left(2, \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, ux\right), maxCos\right)\right), zi\right)\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(xi, \mathsf{cos.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \mathsf{PI.f32}\left(\right)\right)\right)\right)\right), \mathsf{*.f32}\left(yi, \mathsf{sin.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \mathsf{PI}\left(\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, ux\right), maxCos\right)\right), zi\right)\right) \]

   11. PI-lowering-PI.f32 98.9%

       \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(xi, \mathsf{cos.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \mathsf{PI.f32}\left(\right)\right)\right)\right)\right), \mathsf{*.f32}\left(yi, \mathsf{sin.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \mathsf{PI.f32}\left(\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, ux\right), maxCos\right)\right), zi\right)\right) \]

7. Simplified 98.9%

   \[\leadsto \color{blue}{\left(xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)} + \left(ux \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right) \cdot zi \]

8. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{xi + \left(2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right) + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right)} \]
9. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \left(2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right) + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right) + \color{blue}{xi} \]

   2. +-commutative N/A

      \[\leadsto \left(maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + 2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right) + xi \]

   3. associate-+l+ N/A

      \[\leadsto maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + \color{blue}{\left(2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right) + xi\right)} \]

   4. *-commutative N/A

      \[\leadsto maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + \left(\left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right) \cdot 2 + xi\right) \]

   5. associate-*r* N/A

      \[\leadsto maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + \left(uy \cdot \left(\left(yi \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right) + xi\right) \]

   6. *-commutative N/A

      \[\leadsto maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + \left(uy \cdot \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right) + xi\right) \]

   7. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(\left(maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right), \color{blue}{\left(uy \cdot \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right) + xi\right)}\right) \]

   8. associate-*r* N/A

      \[\leadsto \mathsf{+.f32}\left(\left(maxCos \cdot \left(\left(ux \cdot zi\right) \cdot \left(1 - ux\right)\right)\right), \left(uy \cdot \color{blue}{\left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)} + xi\right)\right) \]

   9. associate-*r* N/A

      \[\leadsto \mathsf{+.f32}\left(\left(\left(maxCos \cdot \left(ux \cdot zi\right)\right) \cdot \left(1 - ux\right)\right), \left(\color{blue}{uy \cdot \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)} + xi\right)\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(\left(maxCos \cdot \left(ux \cdot zi\right)\right), \left(1 - ux\right)\right), \left(\color{blue}{uy \cdot \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)} + xi\right)\right) \]

   11. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(maxCos, \left(ux \cdot zi\right)\right), \left(1 - ux\right)\right), \left(\color{blue}{uy} \cdot \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right) + xi\right)\right) \]

   12. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right), \left(1 - ux\right)\right), \left(uy \cdot \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right) + xi\right)\right) \]

   13. --lowering--.f32 N/A

       \[\leadsto \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right), \mathsf{\_.f32}\left(1, ux\right)\right), \left(uy \cdot \color{blue}{\left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)} + xi\right)\right) \]

10. Simplified 82.4%

    \[\leadsto \color{blue}{\left(maxCos \cdot \left(ux \cdot zi\right)\right) \cdot \left(1 - ux\right) + \left(xi + 2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right)\right)} \]

11. Final simplification 82.4%

    \[\leadsto \left(1 - ux\right) \cdot \left(maxCos \cdot \left(zi \cdot ux\right)\right) + \left(xi + 2 \cdot \left(uy \cdot \left(\pi \cdot yi\right)\right)\right) \]
12. Add Preprocessing

Alternative 24: 81.3% accurate, 24.3× speedup

Math

\[\begin{array}{l} \\ xi + \left(\left(ux \cdot maxCos\right) \cdot \left(zi \cdot \left(1 - ux\right)\right) + 2 \cdot \left(uy \cdot \left(\pi \cdot yi\right)\right)\right) \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (+ xi (+ (* (* ux maxCos) (* zi (- 1.0 ux))) (* 2.0 (* uy (* PI yi))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return xi + (((ux * maxCos) * (zi * (1.0f - ux))) + (2.0f * (uy * (((float) M_PI) * yi))));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return Float32(xi + Float32(Float32(Float32(ux * maxCos) * Float32(zi * Float32(Float32(1.0) - ux))) + Float32(Float32(2.0) * Float32(uy * Float32(Float32(pi) * yi)))))
end

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	tmp = xi + (((ux * maxCos) * (zi * (single(1.0) - ux))) + (single(2.0) * (uy * (single(pi) * yi))));
end

Derivation

1. Initial program 99.1%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

3. Simplified 99.1%

   \[\leadsto \color{blue}{\sqrt{1 + \left(1 - ux\right) \cdot \left(ux \cdot \left(maxCos \cdot \left(ux \cdot \left(maxCos \cdot \left(ux + -1\right)\right)\right)\right)\right)} \cdot \left(\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot yi + \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot xi\right) + \left(ux \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right) \cdot zi} \]
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%

   \[\leadsto \color{blue}{xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right) + \left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right) + zi \cdot \left(maxCos \cdot \left(ux \cdot \left(1 - ux\right)\right)\right)\right)} \]

8. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{xi + \left(2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right) + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right)} \]
9. Step-by-step derivation

   1. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(xi, \color{blue}{\left(2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right) + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right)}\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{+.f32}\left(xi, \left(maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + \color{blue}{2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right) \]

   3. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(xi, \mathsf{+.f32}\left(\left(maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right), \color{blue}{\left(2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right)}\right)\right) \]

   4. associate-*r* N/A

      \[\leadsto \mathsf{+.f32}\left(xi, \mathsf{+.f32}\left(\left(\left(maxCos \cdot ux\right) \cdot \left(zi \cdot \left(1 - ux\right)\right)\right), \left(\color{blue}{2} \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right) \]

   5. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(xi, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\left(maxCos \cdot ux\right), \left(zi \cdot \left(1 - ux\right)\right)\right), \left(\color{blue}{2} \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right) \]

   6. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(xi, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(maxCos, ux\right), \left(zi \cdot \left(1 - ux\right)\right)\right), \left(2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{+.f32}\left(xi, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(maxCos, ux\right), \left(\left(1 - ux\right) \cdot zi\right)\right), \left(2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right) \]

   8. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(xi, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(maxCos, ux\right), \mathsf{*.f32}\left(\left(1 - ux\right), zi\right)\right), \left(2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right) \]

   9. --lowering--.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(xi, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(maxCos, ux\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, ux\right), zi\right)\right), \left(2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{+.f32}\left(xi, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(maxCos, ux\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, ux\right), zi\right)\right), \mathsf{*.f32}\left(2, \color{blue}{\left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)\right) \]

   11. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{+.f32}\left(xi, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(maxCos, ux\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, ux\right), zi\right)\right), \mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \color{blue}{\left(yi \cdot \mathsf{PI}\left(\right)\right)}\right)\right)\right)\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{+.f32}\left(xi, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(maxCos, ux\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, ux\right), zi\right)\right), \mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \left(\mathsf{PI}\left(\right) \cdot \color{blue}{yi}\right)\right)\right)\right)\right) \]

   13. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{+.f32}\left(xi, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(maxCos, ux\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, ux\right), zi\right)\right), \mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(\mathsf{PI}\left(\right), \color{blue}{yi}\right)\right)\right)\right)\right) \]

   14. PI-lowering-PI.f32 82.4%

       \[\leadsto \mathsf{+.f32}\left(xi, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(maxCos, ux\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, ux\right), zi\right)\right), \mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), yi\right)\right)\right)\right)\right) \]

10. Simplified 82.4%

    \[\leadsto \color{blue}{xi + \left(\left(maxCos \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot zi\right) + 2 \cdot \left(uy \cdot \left(\pi \cdot yi\right)\right)\right)} \]

11. Final simplification 82.4%

    \[\leadsto xi + \left(\left(ux \cdot maxCos\right) \cdot \left(zi \cdot \left(1 - ux\right)\right) + 2 \cdot \left(uy \cdot \left(\pi \cdot yi\right)\right)\right) \]
12. Add Preprocessing

Alternative 25: 59.4% accurate, 27.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \left(uy \cdot \left(\pi \cdot yi\right)\right)\\ \mathbf{if}\;yi \leq -3.99999992980668 \cdot 10^{-13}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;yi \leq 1.000000013351432 \cdot 10^{-10}:\\ \;\;\;\;xi + maxCos \cdot \left(zi \cdot ux\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* 2.0 (* uy (* PI yi)))))
   (if (<= yi -3.99999992980668e-13)
     t_0
     (if (<= yi 1.000000013351432e-10) (+ xi (* maxCos (* zi ux))) 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) * yi));
	float tmp;
	if (yi <= -3.99999992980668e-13f) {
		tmp = t_0;
	} else if (yi <= 1.000000013351432e-10f) {
		tmp = xi + (maxCos * (zi * ux));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(2.0) * Float32(uy * Float32(Float32(pi) * yi)))
	tmp = Float32(0.0)
	if (yi <= Float32(-3.99999992980668e-13))
		tmp = t_0;
	elseif (yi <= Float32(1.000000013351432e-10))
		tmp = Float32(xi + Float32(maxCos * Float32(zi * ux)));
	else
		tmp = 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) * yi));
	tmp = single(0.0);
	if (yi <= single(-3.99999992980668e-13))
		tmp = t_0;
	elseif (yi <= single(1.000000013351432e-10))
		tmp = xi + (maxCos * (zi * ux));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if yi < -3.99999993e-13 or 1.00000001e-10 < yi

   1. Initial program 98.8%

      \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   3. Simplified 98.8%

      \[\leadsto \color{blue}{\sqrt{1 + \left(1 - ux\right) \cdot \left(ux \cdot \left(maxCos \cdot \left(ux \cdot \left(maxCos \cdot \left(ux + -1\right)\right)\right)\right)\right)} \cdot \left(\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot yi + \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot xi\right) + \left(ux \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right) \cdot zi} \]
   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 98.5%

      \[\leadsto \color{blue}{xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right) + \left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right) + zi \cdot \left(maxCos \cdot \left(ux \cdot \left(1 - ux\right)\right)\right)\right)} \]

   8. Taylor expanded in uy 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. associate-+r+ N/A

         \[\leadsto \left(xi + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right) + \color{blue}{uy \cdot \left(-2 \cdot \left(uy \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      2. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{+.f32}\left(\left(xi + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right), \color{blue}{\left(uy \cdot \left(-2 \cdot \left(uy \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right)}\right) \]

      3. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(xi, \left(maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right)\right), \left(\color{blue}{uy} \cdot \left(-2 \cdot \left(uy \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(xi, \left(\left(maxCos \cdot ux\right) \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right), \left(uy \cdot \left(-2 \cdot \left(uy \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(\left(maxCos \cdot ux\right), \left(zi \cdot \left(1 - ux\right)\right)\right)\right), \left(uy \cdot \left(-2 \cdot \left(uy \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(\mathsf{*.f32}\left(maxCos, ux\right), \left(zi \cdot \left(1 - ux\right)\right)\right)\right), \left(uy \cdot \left(-2 \cdot \left(uy \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(\mathsf{*.f32}\left(maxCos, ux\right), \left(\left(1 - ux\right) \cdot zi\right)\right)\right), \left(uy \cdot \left(-2 \cdot \left(uy \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(\mathsf{*.f32}\left(maxCos, ux\right), \mathsf{*.f32}\left(\left(1 - ux\right), zi\right)\right)\right), \left(uy \cdot \left(-2 \cdot \left(uy \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      9. --lowering--.f32 N/A

         \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(\mathsf{*.f32}\left(maxCos, ux\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, ux\right), zi\right)\right)\right), \left(uy \cdot \left(-2 \cdot \left(uy \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(\mathsf{*.f32}\left(maxCos, ux\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, ux\right), zi\right)\right)\right), \mathsf{*.f32}\left(uy, \color{blue}{\left(-2 \cdot \left(uy \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right) \]

      11. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(\mathsf{*.f32}\left(maxCos, ux\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, ux\right), zi\right)\right)\right), \mathsf{*.f32}\left(uy, \mathsf{+.f32}\left(\left(-2 \cdot \left(uy \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \color{blue}{\left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)\right) \]

   10. Simplified 82.5%

       \[\leadsto \color{blue}{\left(xi + \left(maxCos \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot zi\right)\right) + uy \cdot \left(\left(-2 \cdot uy\right) \cdot \left(xi \cdot \left(\pi \cdot \pi\right)\right) + 2 \cdot \left(\pi \cdot yi\right)\right)} \]

   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. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \color{blue}{\left(yi \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \left(\mathsf{PI}\left(\right) \cdot \color{blue}{yi}\right)\right)\right) \]

       4. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(\mathsf{PI}\left(\right), \color{blue}{yi}\right)\right)\right) \]

       5. PI-lowering-PI.f32 56.4%

          \[\leadsto \mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), yi\right)\right)\right) \]

   13. Simplified 56.4%

       \[\leadsto \color{blue}{2 \cdot \left(uy \cdot \left(\pi \cdot yi\right)\right)} \]

   if -3.99999993e-13 < yi < 1.00000001e-10

   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%

      \[\leadsto \color{blue}{\sqrt{1 + \left(1 - ux\right) \cdot \left(ux \cdot \left(maxCos \cdot \left(ux \cdot \left(maxCos \cdot \left(ux + -1\right)\right)\right)\right)\right)} \cdot \left(\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot yi + \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot xi\right) + \left(ux \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right) \cdot zi} \]
   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. associate-*r* N/A

         \[\leadsto \left(xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \left(maxCos \cdot ux\right) \cdot \color{blue}{zi} \]

      3. *-commutative N/A

         \[\leadsto \left(xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \left(ux \cdot maxCos\right) \cdot zi \]

      4. associate-*r* N/A

         \[\leadsto \left(xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) + ux \cdot \color{blue}{\left(maxCos \cdot zi\right)} \]

      5. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{+.f32}\left(\left(xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right), \color{blue}{\left(ux \cdot \left(maxCos \cdot zi\right)\right)}\right) \]

   7. Simplified 94.3%

      \[\leadsto \color{blue}{\left(xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) + maxCos \cdot \left(ux \cdot zi\right)} \]

   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 65.4%

         \[\leadsto \mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, \color{blue}{zi}\right)\right)\right) \]

   10. Simplified 65.4%

       \[\leadsto \color{blue}{xi + maxCos \cdot \left(ux \cdot zi\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;yi \leq -3.99999992980668 \cdot 10^{-13}:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\pi \cdot yi\right)\right)\\ \mathbf{elif}\;yi \leq 1.000000013351432 \cdot 10^{-10}:\\ \;\;\;\;xi + maxCos \cdot \left(zi \cdot ux\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\pi \cdot yi\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 26: 78.8% accurate, 30.7× speedup

Math

\[\begin{array}{l} \\ xi + \left(maxCos \cdot \left(zi \cdot ux\right) + 2 \cdot \left(uy \cdot \left(\pi \cdot yi\right)\right)\right) \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (+ xi (+ (* maxCos (* zi ux)) (* 2.0 (* uy (* PI yi))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return xi + ((maxCos * (zi * ux)) + (2.0f * (uy * (((float) M_PI) * yi))));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return Float32(xi + Float32(Float32(maxCos * Float32(zi * ux)) + Float32(Float32(2.0) * Float32(uy * Float32(Float32(pi) * yi)))))
end

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	tmp = xi + ((maxCos * (zi * ux)) + (single(2.0) * (uy * (single(pi) * yi))));
end

Derivation

1. Initial program 99.1%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

3. Simplified 99.1%

   \[\leadsto \color{blue}{\sqrt{1 + \left(1 - ux\right) \cdot \left(ux \cdot \left(maxCos \cdot \left(ux \cdot \left(maxCos \cdot \left(ux + -1\right)\right)\right)\right)\right)} \cdot \left(\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot yi + \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot xi\right) + \left(ux \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right) \cdot zi} \]
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. associate-*r* N/A

      \[\leadsto \left(xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \left(maxCos \cdot ux\right) \cdot \color{blue}{zi} \]

   3. *-commutative N/A

      \[\leadsto \left(xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \left(ux \cdot maxCos\right) \cdot zi \]

   4. associate-*r* N/A

      \[\leadsto \left(xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) + ux \cdot \color{blue}{\left(maxCos \cdot zi\right)} \]

   5. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(\left(xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right), \color{blue}{\left(ux \cdot \left(maxCos \cdot zi\right)\right)}\right) \]

7. Simplified 95.2%

   \[\leadsto \color{blue}{\left(xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) + maxCos \cdot \left(ux \cdot zi\right)} \]

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 79.1%

       \[\leadsto \mathsf{+.f32}\left(xi, \mathsf{+.f32}\left(\mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right), \mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), yi\right)\right)\right)\right)\right) \]

10. Simplified 79.1%

    \[\leadsto \color{blue}{xi + \left(maxCos \cdot \left(ux \cdot zi\right) + 2 \cdot \left(uy \cdot \left(\pi \cdot yi\right)\right)\right)} \]

11. Final simplification 79.1%

    \[\leadsto xi + \left(maxCos \cdot \left(zi \cdot ux\right) + 2 \cdot \left(uy \cdot \left(\pi \cdot yi\right)\right)\right) \]
12. Add Preprocessing

Alternative 27: 32.7% accurate, 65.9× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(uy \cdot \left(\pi \cdot yi\right)\right) \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos) :precision binary32 (* 2.0 (* uy (* PI yi))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return 2.0f * (uy * (((float) M_PI) * yi));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return Float32(Float32(2.0) * Float32(uy * Float32(Float32(pi) * yi)))
end

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	tmp = single(2.0) * (uy * (single(pi) * yi));
end

Derivation

1. Initial program 99.1%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

3. Simplified 99.1%

   \[\leadsto \color{blue}{\sqrt{1 + \left(1 - ux\right) \cdot \left(ux \cdot \left(maxCos \cdot \left(ux \cdot \left(maxCos \cdot \left(ux + -1\right)\right)\right)\right)\right)} \cdot \left(\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot yi + \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot xi\right) + \left(ux \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right) \cdot zi} \]
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%

   \[\leadsto \color{blue}{xi \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right) + \left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right) + zi \cdot \left(maxCos \cdot \left(ux \cdot \left(1 - ux\right)\right)\right)\right)} \]

8. Taylor expanded in uy 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. associate-+r+ N/A

      \[\leadsto \left(xi + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right) + \color{blue}{uy \cdot \left(-2 \cdot \left(uy \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)} \]

   2. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(\left(xi + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right), \color{blue}{\left(uy \cdot \left(-2 \cdot \left(uy \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right)}\right) \]

   3. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(xi, \left(maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right)\right), \left(\color{blue}{uy} \cdot \left(-2 \cdot \left(uy \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   4. associate-*r* N/A

      \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(xi, \left(\left(maxCos \cdot ux\right) \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right), \left(uy \cdot \left(-2 \cdot \left(uy \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   5. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(\left(maxCos \cdot ux\right), \left(zi \cdot \left(1 - ux\right)\right)\right)\right), \left(uy \cdot \left(-2 \cdot \left(uy \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   6. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(\mathsf{*.f32}\left(maxCos, ux\right), \left(zi \cdot \left(1 - ux\right)\right)\right)\right), \left(uy \cdot \left(-2 \cdot \left(uy \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(\mathsf{*.f32}\left(maxCos, ux\right), \left(\left(1 - ux\right) \cdot zi\right)\right)\right), \left(uy \cdot \left(-2 \cdot \left(uy \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   8. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(\mathsf{*.f32}\left(maxCos, ux\right), \mathsf{*.f32}\left(\left(1 - ux\right), zi\right)\right)\right), \left(uy \cdot \left(-2 \cdot \left(uy \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   9. --lowering--.f32 N/A

      \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(\mathsf{*.f32}\left(maxCos, ux\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, ux\right), zi\right)\right)\right), \left(uy \cdot \left(-2 \cdot \left(uy \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(\mathsf{*.f32}\left(maxCos, ux\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, ux\right), zi\right)\right)\right), \mathsf{*.f32}\left(uy, \color{blue}{\left(-2 \cdot \left(uy \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right) \]

   11. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(xi, \mathsf{*.f32}\left(\mathsf{*.f32}\left(maxCos, ux\right), \mathsf{*.f32}\left(\mathsf{\_.f32}\left(1, ux\right), zi\right)\right)\right), \mathsf{*.f32}\left(uy, \mathsf{+.f32}\left(\left(-2 \cdot \left(uy \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \color{blue}{\left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)\right) \]

10. Simplified 85.9%

    \[\leadsto \color{blue}{\left(xi + \left(maxCos \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot zi\right)\right) + uy \cdot \left(\left(-2 \cdot uy\right) \cdot \left(xi \cdot \left(\pi \cdot \pi\right)\right) + 2 \cdot \left(\pi \cdot yi\right)\right)} \]

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. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \color{blue}{\left(yi \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \]

    3. *-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \left(\mathsf{PI}\left(\right) \cdot \color{blue}{yi}\right)\right)\right) \]

    4. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(\mathsf{PI}\left(\right), \color{blue}{yi}\right)\right)\right) \]

    5. PI-lowering-PI.f32 29.9%

       \[\leadsto \mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), yi\right)\right)\right) \]

13. Simplified 29.9%

    \[\leadsto \color{blue}{2 \cdot \left(uy \cdot \left(\pi \cdot yi\right)\right)} \]
14. Add Preprocessing

Alternative 28: 11.7% accurate, 92.2× speedup

Math

\[\begin{array}{l} \\ ux \cdot \left(zi \cdot maxCos\right) \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos) :precision binary32 (* ux (* zi maxCos)))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return ux * (zi * 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 = ux * (zi * maxcos)
end function

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return Float32(ux * Float32(zi * maxCos))
end

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	tmp = ux * (zi * maxCos);
end

Derivation

1. Initial program 99.1%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

3. Simplified 99.1%

   \[\leadsto \color{blue}{\sqrt{1 + \left(1 - ux\right) \cdot \left(ux \cdot \left(maxCos \cdot \left(ux \cdot \left(maxCos \cdot \left(ux + -1\right)\right)\right)\right)\right)} \cdot \left(\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot yi + \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot xi\right) + \left(ux \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right) \cdot zi} \]
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. *-commutative N/A

      \[\leadsto \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) \cdot \color{blue}{maxCos} \]

   2. associate-*r* N/A

      \[\leadsto \left(\left(ux \cdot zi\right) \cdot \left(1 - ux\right)\right) \cdot maxCos \]

   3. *-commutative N/A

      \[\leadsto \left(\left(zi \cdot ux\right) \cdot \left(1 - ux\right)\right) \cdot maxCos \]

   4. associate-*l* N/A

      \[\leadsto \left(zi \cdot \left(ux \cdot \left(1 - ux\right)\right)\right) \cdot maxCos \]

   5. associate-*r* N/A

      \[\leadsto zi \cdot \color{blue}{\left(\left(ux \cdot \left(1 - ux\right)\right) \cdot maxCos\right)} \]

   6. *-commutative N/A

      \[\leadsto zi \cdot \left(maxCos \cdot \color{blue}{\left(ux \cdot \left(1 - ux\right)\right)}\right) \]

   7. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(zi, \color{blue}{\left(maxCos \cdot \left(ux \cdot \left(1 - ux\right)\right)\right)}\right) \]

   8. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(zi, \mathsf{*.f32}\left(maxCos, \color{blue}{\left(ux \cdot \left(1 - ux\right)\right)}\right)\right) \]

   9. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(zi, \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, \color{blue}{\left(1 - ux\right)}\right)\right)\right) \]

   10. --lowering--.f32 14.5%

       \[\leadsto \mathsf{*.f32}\left(zi, \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(1, \color{blue}{ux}\right)\right)\right)\right) \]

7. Simplified 14.5%

   \[\leadsto \color{blue}{zi \cdot \left(maxCos \cdot \left(ux \cdot \left(1 - ux\right)\right)\right)} \]

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 12.7%

      \[\leadsto \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, \color{blue}{zi}\right)\right) \]

10. Simplified 12.7%

    \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot zi\right)} \]
11. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto maxCos \cdot \left(zi \cdot \color{blue}{ux}\right) \]

    2. associate-*r* N/A

       \[\leadsto \left(maxCos \cdot zi\right) \cdot \color{blue}{ux} \]

    3. *-commutative N/A

       \[\leadsto \left(zi \cdot maxCos\right) \cdot ux \]

    4. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\left(zi \cdot maxCos\right), \color{blue}{ux}\right) \]

    5. *-lowering-*.f32 12.7%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{*.f32}\left(zi, maxCos\right), ux\right) \]

12. Applied egg-rr 12.7%

    \[\leadsto \color{blue}{\left(zi \cdot maxCos\right) \cdot ux} \]

13. Final simplification 12.7%

    \[\leadsto ux \cdot \left(zi \cdot maxCos\right) \]
14. Add Preprocessing

Alternative 29: 11.7% accurate, 92.2× speedup

Math

\[\begin{array}{l} \\ maxCos \cdot \left(zi \cdot ux\right) \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos) :precision binary32 (* maxCos (* zi ux)))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return maxCos * (zi * ux);
}

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 * (zi * ux)
end function

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return Float32(maxCos * Float32(zi * ux))
end

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	tmp = maxCos * (zi * ux);
end

Derivation

1. Initial program 99.1%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

3. Simplified 99.1%

   \[\leadsto \color{blue}{\sqrt{1 + \left(1 - ux\right) \cdot \left(ux \cdot \left(maxCos \cdot \left(ux \cdot \left(maxCos \cdot \left(ux + -1\right)\right)\right)\right)\right)} \cdot \left(\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot yi + \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot xi\right) + \left(ux \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right) \cdot zi} \]
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. *-commutative N/A

      \[\leadsto \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) \cdot \color{blue}{maxCos} \]

   2. associate-*r* N/A

      \[\leadsto \left(\left(ux \cdot zi\right) \cdot \left(1 - ux\right)\right) \cdot maxCos \]

   3. *-commutative N/A

      \[\leadsto \left(\left(zi \cdot ux\right) \cdot \left(1 - ux\right)\right) \cdot maxCos \]

   4. associate-*l* N/A

      \[\leadsto \left(zi \cdot \left(ux \cdot \left(1 - ux\right)\right)\right) \cdot maxCos \]

   5. associate-*r* N/A

      \[\leadsto zi \cdot \color{blue}{\left(\left(ux \cdot \left(1 - ux\right)\right) \cdot maxCos\right)} \]

   6. *-commutative N/A

      \[\leadsto zi \cdot \left(maxCos \cdot \color{blue}{\left(ux \cdot \left(1 - ux\right)\right)}\right) \]

   7. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(zi, \color{blue}{\left(maxCos \cdot \left(ux \cdot \left(1 - ux\right)\right)\right)}\right) \]

   8. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(zi, \mathsf{*.f32}\left(maxCos, \color{blue}{\left(ux \cdot \left(1 - ux\right)\right)}\right)\right) \]

   9. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(zi, \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, \color{blue}{\left(1 - ux\right)}\right)\right)\right) \]

   10. --lowering--.f32 14.5%

       \[\leadsto \mathsf{*.f32}\left(zi, \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(1, \color{blue}{ux}\right)\right)\right)\right) \]

7. Simplified 14.5%

   \[\leadsto \color{blue}{zi \cdot \left(maxCos \cdot \left(ux \cdot \left(1 - ux\right)\right)\right)} \]

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 12.7%

      \[\leadsto \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, \color{blue}{zi}\right)\right) \]

10. Simplified 12.7%

    \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot zi\right)} \]

11. Final simplification 12.7%

    \[\leadsto maxCos \cdot \left(zi \cdot ux\right) \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024141 
(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)))