UniformSampleCone 2

Percentage Accurate: 98.9% → 98.9%

Time: 18.9s

Alternatives: 21

Speedup: N/A×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

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%

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

   1. *-commutative N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

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

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

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

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

   6. *-lowering-*.f32 99.2%

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

6. Applied egg-rr 99.2%

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

7. Final simplification 99.2%

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

Alternative 2: 98.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

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%

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

   1. associate-*r* N/A

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

   2. associate-*l* N/A

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

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

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

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

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

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

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

   6. *-lowering-*.f32 99.2%

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

6. Applied egg-rr 99.2%

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

7. Final simplification 99.2%

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

Alternative 3: 98.9% accurate, 1.3× speedup

Math

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

FPCore

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

Julia

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

MATLAB

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

Derivation

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%

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

5. Final simplification 99.2%

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

Alternative 4: 98.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - ux\right) \cdot \left(ux + -1\right)\\ t_1 := -1.3333333333333333 \cdot \left(\left(uy \cdot yi\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right) + \left(xi \cdot -2\right) \cdot \left(\pi \cdot \pi\right)\\ t_2 := \left(2 \cdot uy\right) \cdot \left(\pi \cdot yi\right)\\ t_3 := 2 \cdot \left(uy \cdot \pi\right)\\ \mathbf{if}\;uy \leq 0.010499999858438969:\\ \;\;\;\;xi + \left(\left(t\_2 + maxCos \cdot \left(ux \cdot \left(\left(1 - ux\right) \cdot zi\right) + maxCos \cdot \left(0.5 \cdot \left(\left(ux \cdot ux\right) \cdot \left(t\_0 \cdot \left(xi + t\_2\right)\right) + \left(\left(ux \cdot ux\right) \cdot \left(uy \cdot uy\right)\right) \cdot \left(t\_0 \cdot t\_1\right)\right)\right)\right)\right) + \left(uy \cdot uy\right) \cdot t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;ux \cdot \left(maxCos \cdot zi + \left(\frac{\cos t\_3 \cdot xi}{ux} + \frac{\sin t\_3 \cdot yi}{ux}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (- 1.0 ux) (+ ux -1.0)))
        (t_1
         (+
          (* -1.3333333333333333 (* (* uy yi) (* PI (* PI PI))))
          (* (* xi -2.0) (* PI PI))))
        (t_2 (* (* 2.0 uy) (* PI yi)))
        (t_3 (* 2.0 (* uy PI))))
   (if (<= uy 0.010499999858438969)
     (+
      xi
      (+
       (+
        t_2
        (*
         maxCos
         (+
          (* ux (* (- 1.0 ux) zi))
          (*
           maxCos
           (*
            0.5
            (+
             (* (* ux ux) (* t_0 (+ xi t_2)))
             (* (* (* ux ux) (* uy uy)) (* t_0 t_1))))))))
       (* (* uy uy) t_1)))
     (*
      ux
      (+ (* maxCos zi) (+ (/ (* (cos t_3) xi) ux) (/ (* (sin t_3) yi) ux)))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = (1.0f - ux) * (ux + -1.0f);
	float t_1 = (-1.3333333333333333f * ((uy * yi) * (((float) M_PI) * (((float) M_PI) * ((float) M_PI))))) + ((xi * -2.0f) * (((float) M_PI) * ((float) M_PI)));
	float t_2 = (2.0f * uy) * (((float) M_PI) * yi);
	float t_3 = 2.0f * (uy * ((float) M_PI));
	float tmp;
	if (uy <= 0.010499999858438969f) {
		tmp = xi + ((t_2 + (maxCos * ((ux * ((1.0f - ux) * zi)) + (maxCos * (0.5f * (((ux * ux) * (t_0 * (xi + t_2))) + (((ux * ux) * (uy * uy)) * (t_0 * t_1)))))))) + ((uy * uy) * t_1));
	} else {
		tmp = ux * ((maxCos * zi) + (((cosf(t_3) * xi) / ux) + ((sinf(t_3) * yi) / ux)));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(Float32(1.0) - ux) * Float32(ux + Float32(-1.0)))
	t_1 = Float32(Float32(Float32(-1.3333333333333333) * Float32(Float32(uy * yi) * Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi))))) + Float32(Float32(xi * Float32(-2.0)) * Float32(Float32(pi) * Float32(pi))))
	t_2 = Float32(Float32(Float32(2.0) * uy) * Float32(Float32(pi) * yi))
	t_3 = Float32(Float32(2.0) * Float32(uy * Float32(pi)))
	tmp = Float32(0.0)
	if (uy <= Float32(0.010499999858438969))
		tmp = Float32(xi + Float32(Float32(t_2 + Float32(maxCos * Float32(Float32(ux * Float32(Float32(Float32(1.0) - ux) * zi)) + Float32(maxCos * Float32(Float32(0.5) * Float32(Float32(Float32(ux * ux) * Float32(t_0 * Float32(xi + t_2))) + Float32(Float32(Float32(ux * ux) * Float32(uy * uy)) * Float32(t_0 * t_1)))))))) + Float32(Float32(uy * uy) * t_1)));
	else
		tmp = Float32(ux * Float32(Float32(maxCos * zi) + Float32(Float32(Float32(cos(t_3) * xi) / ux) + Float32(Float32(sin(t_3) * yi) / ux))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(xi, yi, zi, ux, uy, maxCos)
	t_0 = (single(1.0) - ux) * (ux + single(-1.0));
	t_1 = (single(-1.3333333333333333) * ((uy * yi) * (single(pi) * (single(pi) * single(pi))))) + ((xi * single(-2.0)) * (single(pi) * single(pi)));
	t_2 = (single(2.0) * uy) * (single(pi) * yi);
	t_3 = single(2.0) * (uy * single(pi));
	tmp = single(0.0);
	if (uy <= single(0.010499999858438969))
		tmp = xi + ((t_2 + (maxCos * ((ux * ((single(1.0) - ux) * zi)) + (maxCos * (single(0.5) * (((ux * ux) * (t_0 * (xi + t_2))) + (((ux * ux) * (uy * uy)) * (t_0 * t_1)))))))) + ((uy * uy) * t_1));
	else
		tmp = ux * ((maxCos * zi) + (((cos(t_3) * xi) / ux) + ((sin(t_3) * yi) / ux)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.0104999999

   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 99.4%

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

   5. Taylor expanded in uy around 0

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

   7. Simplified 99.5%

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

   8. Taylor expanded in maxCos around 0

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

   10. Simplified 99.4%

       \[\leadsto \color{blue}{xi + \left(\left(\left(2 \cdot uy\right) \cdot \left(\pi \cdot yi\right) + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right) + maxCos \cdot \left(0.5 \cdot \left(\left(ux \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right) \cdot \left(xi + \left(2 \cdot uy\right) \cdot \left(\pi \cdot yi\right)\right)\right) + \left(\left(ux \cdot ux\right) \cdot \left(uy \cdot uy\right)\right) \cdot \left(\left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right) \cdot \left(\left(-2 \cdot xi\right) \cdot \left(\pi \cdot \pi\right) + -1.3333333333333333 \cdot \left(\left(uy \cdot yi\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\right)\right)\right)\right)\right) + \left(\left(-2 \cdot xi\right) \cdot \left(\pi \cdot \pi\right) + -1.3333333333333333 \cdot \left(\left(uy \cdot yi\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right)\right) \cdot \left(uy \cdot uy\right)\right)} \]

   if 0.0104999999 < 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%

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

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

      3. associate-*r* N/A

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

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

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

   7. Simplified 96.0%

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

   8. Taylor expanded in ux around inf

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   10. Simplified 96.1%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \leq 0.010499999858438969:\\ \;\;\;\;xi + \left(\left(\left(2 \cdot uy\right) \cdot \left(\pi \cdot yi\right) + maxCos \cdot \left(ux \cdot \left(\left(1 - ux\right) \cdot zi\right) + maxCos \cdot \left(0.5 \cdot \left(\left(ux \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right) \cdot \left(xi + \left(2 \cdot uy\right) \cdot \left(\pi \cdot yi\right)\right)\right) + \left(\left(ux \cdot ux\right) \cdot \left(uy \cdot uy\right)\right) \cdot \left(\left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right) \cdot \left(-1.3333333333333333 \cdot \left(\left(uy \cdot yi\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right) + \left(xi \cdot -2\right) \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\right)\right)\right) + \left(uy \cdot uy\right) \cdot \left(-1.3333333333333333 \cdot \left(\left(uy \cdot yi\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right) + \left(xi \cdot -2\right) \cdot \left(\pi \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;ux \cdot \left(maxCos \cdot zi + \left(\frac{\cos \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot xi}{ux} + \frac{\sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot yi}{ux}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - ux\right) \cdot \left(ux + -1\right)\\ t_1 := 2 \cdot \left(uy \cdot \pi\right)\\ t_2 := -1.3333333333333333 \cdot \left(\left(uy \cdot yi\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right) + \left(xi \cdot -2\right) \cdot \left(\pi \cdot \pi\right)\\ t_3 := \left(2 \cdot uy\right) \cdot \left(\pi \cdot yi\right)\\ \mathbf{if}\;uy \leq 0.010499999858438969:\\ \;\;\;\;xi + \left(\left(t\_3 + maxCos \cdot \left(ux \cdot \left(\left(1 - ux\right) \cdot zi\right) + maxCos \cdot \left(0.5 \cdot \left(\left(ux \cdot ux\right) \cdot \left(t\_0 \cdot \left(xi + t\_3\right)\right) + \left(\left(ux \cdot ux\right) \cdot \left(uy \cdot uy\right)\right) \cdot \left(t\_0 \cdot t\_2\right)\right)\right)\right)\right) + \left(uy \cdot uy\right) \cdot t\_2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos t\_1 \cdot xi + \sin t\_1 \cdot yi\right) + maxCos \cdot \left(ux \cdot zi\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (- 1.0 ux) (+ ux -1.0)))
        (t_1 (* 2.0 (* uy PI)))
        (t_2
         (+
          (* -1.3333333333333333 (* (* uy yi) (* PI (* PI PI))))
          (* (* xi -2.0) (* PI PI))))
        (t_3 (* (* 2.0 uy) (* PI yi))))
   (if (<= uy 0.010499999858438969)
     (+
      xi
      (+
       (+
        t_3
        (*
         maxCos
         (+
          (* ux (* (- 1.0 ux) zi))
          (*
           maxCos
           (*
            0.5
            (+
             (* (* ux ux) (* t_0 (+ xi t_3)))
             (* (* (* ux ux) (* uy uy)) (* t_0 t_2))))))))
       (* (* uy uy) t_2)))
     (+ (+ (* (cos t_1) xi) (* (sin t_1) yi)) (* maxCos (* ux zi))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = (1.0f - ux) * (ux + -1.0f);
	float t_1 = 2.0f * (uy * ((float) M_PI));
	float t_2 = (-1.3333333333333333f * ((uy * yi) * (((float) M_PI) * (((float) M_PI) * ((float) M_PI))))) + ((xi * -2.0f) * (((float) M_PI) * ((float) M_PI)));
	float t_3 = (2.0f * uy) * (((float) M_PI) * yi);
	float tmp;
	if (uy <= 0.010499999858438969f) {
		tmp = xi + ((t_3 + (maxCos * ((ux * ((1.0f - ux) * zi)) + (maxCos * (0.5f * (((ux * ux) * (t_0 * (xi + t_3))) + (((ux * ux) * (uy * uy)) * (t_0 * t_2)))))))) + ((uy * uy) * t_2));
	} else {
		tmp = ((cosf(t_1) * xi) + (sinf(t_1) * yi)) + (maxCos * (ux * zi));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(Float32(1.0) - ux) * Float32(ux + Float32(-1.0)))
	t_1 = Float32(Float32(2.0) * Float32(uy * Float32(pi)))
	t_2 = Float32(Float32(Float32(-1.3333333333333333) * Float32(Float32(uy * yi) * Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi))))) + Float32(Float32(xi * Float32(-2.0)) * Float32(Float32(pi) * Float32(pi))))
	t_3 = Float32(Float32(Float32(2.0) * uy) * Float32(Float32(pi) * yi))
	tmp = Float32(0.0)
	if (uy <= Float32(0.010499999858438969))
		tmp = Float32(xi + Float32(Float32(t_3 + Float32(maxCos * Float32(Float32(ux * Float32(Float32(Float32(1.0) - ux) * zi)) + Float32(maxCos * Float32(Float32(0.5) * Float32(Float32(Float32(ux * ux) * Float32(t_0 * Float32(xi + t_3))) + Float32(Float32(Float32(ux * ux) * Float32(uy * uy)) * Float32(t_0 * t_2)))))))) + Float32(Float32(uy * uy) * t_2)));
	else
		tmp = Float32(Float32(Float32(cos(t_1) * xi) + Float32(sin(t_1) * yi)) + Float32(maxCos * Float32(ux * zi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(xi, yi, zi, ux, uy, maxCos)
	t_0 = (single(1.0) - ux) * (ux + single(-1.0));
	t_1 = single(2.0) * (uy * single(pi));
	t_2 = (single(-1.3333333333333333) * ((uy * yi) * (single(pi) * (single(pi) * single(pi))))) + ((xi * single(-2.0)) * (single(pi) * single(pi)));
	t_3 = (single(2.0) * uy) * (single(pi) * yi);
	tmp = single(0.0);
	if (uy <= single(0.010499999858438969))
		tmp = xi + ((t_3 + (maxCos * ((ux * ((single(1.0) - ux) * zi)) + (maxCos * (single(0.5) * (((ux * ux) * (t_0 * (xi + t_3))) + (((ux * ux) * (uy * uy)) * (t_0 * t_2)))))))) + ((uy * uy) * t_2));
	else
		tmp = ((cos(t_1) * xi) + (sin(t_1) * yi)) + (maxCos * (ux * zi));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.0104999999

   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 99.4%

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

   5. Taylor expanded in uy around 0

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

   7. Simplified 99.5%

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

   8. Taylor expanded in maxCos around 0

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

   10. Simplified 99.4%

       \[\leadsto \color{blue}{xi + \left(\left(\left(2 \cdot uy\right) \cdot \left(\pi \cdot yi\right) + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right) + maxCos \cdot \left(0.5 \cdot \left(\left(ux \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right) \cdot \left(xi + \left(2 \cdot uy\right) \cdot \left(\pi \cdot yi\right)\right)\right) + \left(\left(ux \cdot ux\right) \cdot \left(uy \cdot uy\right)\right) \cdot \left(\left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right) \cdot \left(\left(-2 \cdot xi\right) \cdot \left(\pi \cdot \pi\right) + -1.3333333333333333 \cdot \left(\left(uy \cdot yi\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\right)\right)\right)\right)\right) + \left(\left(-2 \cdot xi\right) \cdot \left(\pi \cdot \pi\right) + -1.3333333333333333 \cdot \left(\left(uy \cdot yi\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right)\right) \cdot \left(uy \cdot uy\right)\right)} \]

   if 0.0104999999 < 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%

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

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

      3. associate-*r* N/A

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

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

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

   7. Simplified 96.0%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \leq 0.010499999858438969:\\ \;\;\;\;xi + \left(\left(\left(2 \cdot uy\right) \cdot \left(\pi \cdot yi\right) + maxCos \cdot \left(ux \cdot \left(\left(1 - ux\right) \cdot zi\right) + maxCos \cdot \left(0.5 \cdot \left(\left(ux \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right) \cdot \left(xi + \left(2 \cdot uy\right) \cdot \left(\pi \cdot yi\right)\right)\right) + \left(\left(ux \cdot ux\right) \cdot \left(uy \cdot uy\right)\right) \cdot \left(\left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right) \cdot \left(-1.3333333333333333 \cdot \left(\left(uy \cdot yi\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right) + \left(xi \cdot -2\right) \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\right)\right)\right) + \left(uy \cdot uy\right) \cdot \left(-1.3333333333333333 \cdot \left(\left(uy \cdot yi\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right) + \left(xi \cdot -2\right) \cdot \left(\pi \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot xi + \sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot yi\right) + maxCos \cdot \left(ux \cdot zi\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

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%

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

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

   4. associate-*r* N/A

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

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

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

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

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

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

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

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

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

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

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

   12. *-commutative N/A

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

   13. associate-*r* N/A

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

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

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

   15. associate-*r* N/A

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

   16. *-commutative N/A

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

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

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

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

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

   19. 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. Final simplification 99.0%

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

Alternative 7: 97.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - ux\right) \cdot \left(ux + -1\right)\\ t_1 := 2 \cdot \left(uy \cdot \pi\right)\\ t_2 := -1.3333333333333333 \cdot \left(\left(uy \cdot yi\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right) + \left(xi \cdot -2\right) \cdot \left(\pi \cdot \pi\right)\\ t_3 := \left(2 \cdot uy\right) \cdot \left(\pi \cdot yi\right)\\ \mathbf{if}\;uy \leq 0.010499999858438969:\\ \;\;\;\;xi + \left(\left(t\_3 + maxCos \cdot \left(ux \cdot \left(\left(1 - ux\right) \cdot zi\right) + maxCos \cdot \left(0.5 \cdot \left(\left(ux \cdot ux\right) \cdot \left(t\_0 \cdot \left(xi + t\_3\right)\right) + \left(\left(ux \cdot ux\right) \cdot \left(uy \cdot uy\right)\right) \cdot \left(t\_0 \cdot t\_2\right)\right)\right)\right)\right) + \left(uy \cdot uy\right) \cdot t\_2\right)\\ \mathbf{else}:\\ \;\;\;\;\cos t\_1 \cdot xi + \sin t\_1 \cdot yi\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (- 1.0 ux) (+ ux -1.0)))
        (t_1 (* 2.0 (* uy PI)))
        (t_2
         (+
          (* -1.3333333333333333 (* (* uy yi) (* PI (* PI PI))))
          (* (* xi -2.0) (* PI PI))))
        (t_3 (* (* 2.0 uy) (* PI yi))))
   (if (<= uy 0.010499999858438969)
     (+
      xi
      (+
       (+
        t_3
        (*
         maxCos
         (+
          (* ux (* (- 1.0 ux) zi))
          (*
           maxCos
           (*
            0.5
            (+
             (* (* ux ux) (* t_0 (+ xi t_3)))
             (* (* (* ux ux) (* uy uy)) (* t_0 t_2))))))))
       (* (* uy uy) t_2)))
     (+ (* (cos t_1) xi) (* (sin t_1) yi)))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = (1.0f - ux) * (ux + -1.0f);
	float t_1 = 2.0f * (uy * ((float) M_PI));
	float t_2 = (-1.3333333333333333f * ((uy * yi) * (((float) M_PI) * (((float) M_PI) * ((float) M_PI))))) + ((xi * -2.0f) * (((float) M_PI) * ((float) M_PI)));
	float t_3 = (2.0f * uy) * (((float) M_PI) * yi);
	float tmp;
	if (uy <= 0.010499999858438969f) {
		tmp = xi + ((t_3 + (maxCos * ((ux * ((1.0f - ux) * zi)) + (maxCos * (0.5f * (((ux * ux) * (t_0 * (xi + t_3))) + (((ux * ux) * (uy * uy)) * (t_0 * t_2)))))))) + ((uy * uy) * t_2));
	} else {
		tmp = (cosf(t_1) * xi) + (sinf(t_1) * yi);
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(Float32(1.0) - ux) * Float32(ux + Float32(-1.0)))
	t_1 = Float32(Float32(2.0) * Float32(uy * Float32(pi)))
	t_2 = Float32(Float32(Float32(-1.3333333333333333) * Float32(Float32(uy * yi) * Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi))))) + Float32(Float32(xi * Float32(-2.0)) * Float32(Float32(pi) * Float32(pi))))
	t_3 = Float32(Float32(Float32(2.0) * uy) * Float32(Float32(pi) * yi))
	tmp = Float32(0.0)
	if (uy <= Float32(0.010499999858438969))
		tmp = Float32(xi + Float32(Float32(t_3 + Float32(maxCos * Float32(Float32(ux * Float32(Float32(Float32(1.0) - ux) * zi)) + Float32(maxCos * Float32(Float32(0.5) * Float32(Float32(Float32(ux * ux) * Float32(t_0 * Float32(xi + t_3))) + Float32(Float32(Float32(ux * ux) * Float32(uy * uy)) * Float32(t_0 * t_2)))))))) + Float32(Float32(uy * uy) * t_2)));
	else
		tmp = Float32(Float32(cos(t_1) * xi) + Float32(sin(t_1) * yi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(xi, yi, zi, ux, uy, maxCos)
	t_0 = (single(1.0) - ux) * (ux + single(-1.0));
	t_1 = single(2.0) * (uy * single(pi));
	t_2 = (single(-1.3333333333333333) * ((uy * yi) * (single(pi) * (single(pi) * single(pi))))) + ((xi * single(-2.0)) * (single(pi) * single(pi)));
	t_3 = (single(2.0) * uy) * (single(pi) * yi);
	tmp = single(0.0);
	if (uy <= single(0.010499999858438969))
		tmp = xi + ((t_3 + (maxCos * ((ux * ((single(1.0) - ux) * zi)) + (maxCos * (single(0.5) * (((ux * ux) * (t_0 * (xi + t_3))) + (((ux * ux) * (uy * uy)) * (t_0 * t_2)))))))) + ((uy * uy) * t_2));
	else
		tmp = (cos(t_1) * xi) + (sin(t_1) * yi);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.0104999999

   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 99.4%

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

   5. Taylor expanded in uy around 0

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

   7. Simplified 99.5%

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

   8. Taylor expanded in maxCos around 0

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

   10. Simplified 99.4%

       \[\leadsto \color{blue}{xi + \left(\left(\left(2 \cdot uy\right) \cdot \left(\pi \cdot yi\right) + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right) + maxCos \cdot \left(0.5 \cdot \left(\left(ux \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right) \cdot \left(xi + \left(2 \cdot uy\right) \cdot \left(\pi \cdot yi\right)\right)\right) + \left(\left(ux \cdot ux\right) \cdot \left(uy \cdot uy\right)\right) \cdot \left(\left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right) \cdot \left(\left(-2 \cdot xi\right) \cdot \left(\pi \cdot \pi\right) + -1.3333333333333333 \cdot \left(\left(uy \cdot yi\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\right)\right)\right)\right)\right) + \left(\left(-2 \cdot xi\right) \cdot \left(\pi \cdot \pi\right) + -1.3333333333333333 \cdot \left(\left(uy \cdot yi\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right)\right) \cdot \left(uy \cdot uy\right)\right)} \]

   if 0.0104999999 < 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%

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

   5. Taylor expanded in ux around 0

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

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

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

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

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

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

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

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

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

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

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

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

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

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

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

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

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

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

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

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

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

      19. PI-lowering-PI.f32 94.2%

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

   7. Simplified 94.2%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \leq 0.010499999858438969:\\ \;\;\;\;xi + \left(\left(\left(2 \cdot uy\right) \cdot \left(\pi \cdot yi\right) + maxCos \cdot \left(ux \cdot \left(\left(1 - ux\right) \cdot zi\right) + maxCos \cdot \left(0.5 \cdot \left(\left(ux \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right) \cdot \left(xi + \left(2 \cdot uy\right) \cdot \left(\pi \cdot yi\right)\right)\right) + \left(\left(ux \cdot ux\right) \cdot \left(uy \cdot uy\right)\right) \cdot \left(\left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right) \cdot \left(-1.3333333333333333 \cdot \left(\left(uy \cdot yi\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right) + \left(xi \cdot -2\right) \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\right)\right)\right) + \left(uy \cdot uy\right) \cdot \left(-1.3333333333333333 \cdot \left(\left(uy \cdot yi\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right) + \left(xi \cdot -2\right) \cdot \left(\pi \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot xi + \sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot yi\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 93.6% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \left(\pi \cdot \pi\right)\\ t_1 := \left(1 - ux\right) \cdot \left(ux + -1\right)\\ t_2 := -1.3333333333333333 \cdot \left(\left(uy \cdot yi\right) \cdot t\_0\right) + \left(xi \cdot -2\right) \cdot \left(\pi \cdot \pi\right)\\ t_3 := \left(2 \cdot uy\right) \cdot \left(\pi \cdot yi\right)\\ \mathbf{if}\;uy \leq 0.012000000104308128:\\ \;\;\;\;xi + \left(\left(t\_3 + maxCos \cdot \left(ux \cdot \left(\left(1 - ux\right) \cdot zi\right) + maxCos \cdot \left(0.5 \cdot \left(\left(ux \cdot ux\right) \cdot \left(t\_1 \cdot \left(xi + t\_3\right)\right) + \left(\left(ux \cdot ux\right) \cdot \left(uy \cdot uy\right)\right) \cdot \left(t\_1 \cdot t\_2\right)\right)\right)\right)\right) + \left(uy \cdot uy\right) \cdot t\_2\right)\\ \mathbf{else}:\\ \;\;\;\;maxCos \cdot \left(ux \cdot zi\right) + \left(\cos \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot xi + yi \cdot \left(uy \cdot \left(t\_0 \cdot \left(\left(uy \cdot uy\right) \cdot -1.3333333333333333\right) + 2 \cdot \pi\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* PI (* PI PI)))
        (t_1 (* (- 1.0 ux) (+ ux -1.0)))
        (t_2
         (+
          (* -1.3333333333333333 (* (* uy yi) t_0))
          (* (* xi -2.0) (* PI PI))))
        (t_3 (* (* 2.0 uy) (* PI yi))))
   (if (<= uy 0.012000000104308128)
     (+
      xi
      (+
       (+
        t_3
        (*
         maxCos
         (+
          (* ux (* (- 1.0 ux) zi))
          (*
           maxCos
           (*
            0.5
            (+
             (* (* ux ux) (* t_1 (+ xi t_3)))
             (* (* (* ux ux) (* uy uy)) (* t_1 t_2))))))))
       (* (* uy uy) t_2)))
     (+
      (* maxCos (* ux zi))
      (+
       (* (cos (* 2.0 (* uy PI))) xi)
       (*
        yi
        (* uy (+ (* t_0 (* (* uy uy) -1.3333333333333333)) (* 2.0 PI)))))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = ((float) M_PI) * (((float) M_PI) * ((float) M_PI));
	float t_1 = (1.0f - ux) * (ux + -1.0f);
	float t_2 = (-1.3333333333333333f * ((uy * yi) * t_0)) + ((xi * -2.0f) * (((float) M_PI) * ((float) M_PI)));
	float t_3 = (2.0f * uy) * (((float) M_PI) * yi);
	float tmp;
	if (uy <= 0.012000000104308128f) {
		tmp = xi + ((t_3 + (maxCos * ((ux * ((1.0f - ux) * zi)) + (maxCos * (0.5f * (((ux * ux) * (t_1 * (xi + t_3))) + (((ux * ux) * (uy * uy)) * (t_1 * t_2)))))))) + ((uy * uy) * t_2));
	} else {
		tmp = (maxCos * (ux * zi)) + ((cosf((2.0f * (uy * ((float) M_PI)))) * xi) + (yi * (uy * ((t_0 * ((uy * uy) * -1.3333333333333333f)) + (2.0f * ((float) M_PI))))));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi)))
	t_1 = Float32(Float32(Float32(1.0) - ux) * Float32(ux + Float32(-1.0)))
	t_2 = Float32(Float32(Float32(-1.3333333333333333) * Float32(Float32(uy * yi) * t_0)) + Float32(Float32(xi * Float32(-2.0)) * Float32(Float32(pi) * Float32(pi))))
	t_3 = Float32(Float32(Float32(2.0) * uy) * Float32(Float32(pi) * yi))
	tmp = Float32(0.0)
	if (uy <= Float32(0.012000000104308128))
		tmp = Float32(xi + Float32(Float32(t_3 + Float32(maxCos * Float32(Float32(ux * Float32(Float32(Float32(1.0) - ux) * zi)) + Float32(maxCos * Float32(Float32(0.5) * Float32(Float32(Float32(ux * ux) * Float32(t_1 * Float32(xi + t_3))) + Float32(Float32(Float32(ux * ux) * Float32(uy * uy)) * Float32(t_1 * t_2)))))))) + Float32(Float32(uy * uy) * t_2)));
	else
		tmp = Float32(Float32(maxCos * Float32(ux * zi)) + Float32(Float32(cos(Float32(Float32(2.0) * Float32(uy * Float32(pi)))) * xi) + Float32(yi * Float32(uy * Float32(Float32(t_0 * Float32(Float32(uy * uy) * Float32(-1.3333333333333333))) + Float32(Float32(2.0) * Float32(pi)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(xi, yi, zi, ux, uy, maxCos)
	t_0 = single(pi) * (single(pi) * single(pi));
	t_1 = (single(1.0) - ux) * (ux + single(-1.0));
	t_2 = (single(-1.3333333333333333) * ((uy * yi) * t_0)) + ((xi * single(-2.0)) * (single(pi) * single(pi)));
	t_3 = (single(2.0) * uy) * (single(pi) * yi);
	tmp = single(0.0);
	if (uy <= single(0.012000000104308128))
		tmp = xi + ((t_3 + (maxCos * ((ux * ((single(1.0) - ux) * zi)) + (maxCos * (single(0.5) * (((ux * ux) * (t_1 * (xi + t_3))) + (((ux * ux) * (uy * uy)) * (t_1 * t_2)))))))) + ((uy * uy) * t_2));
	else
		tmp = (maxCos * (ux * zi)) + ((cos((single(2.0) * (uy * single(pi)))) * xi) + (yi * (uy * ((t_0 * ((uy * uy) * single(-1.3333333333333333))) + (single(2.0) * single(pi))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.0120000001

   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 99.4%

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

   5. Taylor expanded in uy around 0

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

   7. Simplified 99.4%

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

   8. Taylor expanded in maxCos around 0

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

   10. Simplified 99.4%

       \[\leadsto \color{blue}{xi + \left(\left(\left(2 \cdot uy\right) \cdot \left(\pi \cdot yi\right) + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right) + maxCos \cdot \left(0.5 \cdot \left(\left(ux \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right) \cdot \left(xi + \left(2 \cdot uy\right) \cdot \left(\pi \cdot yi\right)\right)\right) + \left(\left(ux \cdot ux\right) \cdot \left(uy \cdot uy\right)\right) \cdot \left(\left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right) \cdot \left(\left(-2 \cdot xi\right) \cdot \left(\pi \cdot \pi\right) + -1.3333333333333333 \cdot \left(\left(uy \cdot yi\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\right)\right)\right)\right)\right) + \left(\left(-2 \cdot xi\right) \cdot \left(\pi \cdot \pi\right) + -1.3333333333333333 \cdot \left(\left(uy \cdot yi\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right)\right) \cdot \left(uy \cdot uy\right)\right)} \]

   if 0.0120000001 < 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%

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

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

      3. associate-*r* N/A

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

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

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

   7. Simplified 96.0%

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

   8. Taylor expanded in uy around 0

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

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

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

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

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

      3. associate-*r* N/A

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

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

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

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

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

      6. unpow2 N/A

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

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

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

      8. cube-mult N/A

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

      9. unpow2 N/A

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

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

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

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

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

      12. unpow2 N/A

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

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

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

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

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

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

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

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

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

      17. PI-lowering-PI.f32 70.1%

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

   10. Simplified 70.1%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \leq 0.012000000104308128:\\ \;\;\;\;xi + \left(\left(\left(2 \cdot uy\right) \cdot \left(\pi \cdot yi\right) + maxCos \cdot \left(ux \cdot \left(\left(1 - ux\right) \cdot zi\right) + maxCos \cdot \left(0.5 \cdot \left(\left(ux \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right) \cdot \left(xi + \left(2 \cdot uy\right) \cdot \left(\pi \cdot yi\right)\right)\right) + \left(\left(ux \cdot ux\right) \cdot \left(uy \cdot uy\right)\right) \cdot \left(\left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right) \cdot \left(-1.3333333333333333 \cdot \left(\left(uy \cdot yi\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right) + \left(xi \cdot -2\right) \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\right)\right)\right) + \left(uy \cdot uy\right) \cdot \left(-1.3333333333333333 \cdot \left(\left(uy \cdot yi\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right) + \left(xi \cdot -2\right) \cdot \left(\pi \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;maxCos \cdot \left(ux \cdot zi\right) + \left(\cos \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot xi + yi \cdot \left(uy \cdot \left(\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(\left(uy \cdot uy\right) \cdot -1.3333333333333333\right) + 2 \cdot \pi\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 91.8% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - ux\right) \cdot \left(ux + -1\right)\\ t_1 := -1.3333333333333333 \cdot \left(\left(uy \cdot yi\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right) + \left(xi \cdot -2\right) \cdot \left(\pi \cdot \pi\right)\\ t_2 := \left(2 \cdot uy\right) \cdot \left(\pi \cdot yi\right)\\ \mathbf{if}\;uy \leq 0.05999999865889549:\\ \;\;\;\;xi + \left(\left(t\_2 + maxCos \cdot \left(ux \cdot \left(\left(1 - ux\right) \cdot zi\right) + maxCos \cdot \left(0.5 \cdot \left(\left(ux \cdot ux\right) \cdot \left(t\_0 \cdot \left(xi + t\_2\right)\right) + \left(\left(ux \cdot ux\right) \cdot \left(uy \cdot uy\right)\right) \cdot \left(t\_0 \cdot t\_1\right)\right)\right)\right)\right) + \left(uy \cdot uy\right) \cdot t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;maxCos \cdot \left(ux \cdot zi\right) + \left(xi + \sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot yi\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (- 1.0 ux) (+ ux -1.0)))
        (t_1
         (+
          (* -1.3333333333333333 (* (* uy yi) (* PI (* PI PI))))
          (* (* xi -2.0) (* PI PI))))
        (t_2 (* (* 2.0 uy) (* PI yi))))
   (if (<= uy 0.05999999865889549)
     (+
      xi
      (+
       (+
        t_2
        (*
         maxCos
         (+
          (* ux (* (- 1.0 ux) zi))
          (*
           maxCos
           (*
            0.5
            (+
             (* (* ux ux) (* t_0 (+ xi t_2)))
             (* (* (* ux ux) (* uy uy)) (* t_0 t_1))))))))
       (* (* uy uy) t_1)))
     (+ (* maxCos (* ux zi)) (+ xi (* (sin (* 2.0 (* uy PI))) yi))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = (1.0f - ux) * (ux + -1.0f);
	float t_1 = (-1.3333333333333333f * ((uy * yi) * (((float) M_PI) * (((float) M_PI) * ((float) M_PI))))) + ((xi * -2.0f) * (((float) M_PI) * ((float) M_PI)));
	float t_2 = (2.0f * uy) * (((float) M_PI) * yi);
	float tmp;
	if (uy <= 0.05999999865889549f) {
		tmp = xi + ((t_2 + (maxCos * ((ux * ((1.0f - ux) * zi)) + (maxCos * (0.5f * (((ux * ux) * (t_0 * (xi + t_2))) + (((ux * ux) * (uy * uy)) * (t_0 * t_1)))))))) + ((uy * uy) * t_1));
	} else {
		tmp = (maxCos * (ux * zi)) + (xi + (sinf((2.0f * (uy * ((float) M_PI)))) * yi));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(Float32(1.0) - ux) * Float32(ux + Float32(-1.0)))
	t_1 = Float32(Float32(Float32(-1.3333333333333333) * Float32(Float32(uy * yi) * Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi))))) + Float32(Float32(xi * Float32(-2.0)) * Float32(Float32(pi) * Float32(pi))))
	t_2 = Float32(Float32(Float32(2.0) * uy) * Float32(Float32(pi) * yi))
	tmp = Float32(0.0)
	if (uy <= Float32(0.05999999865889549))
		tmp = Float32(xi + Float32(Float32(t_2 + Float32(maxCos * Float32(Float32(ux * Float32(Float32(Float32(1.0) - ux) * zi)) + Float32(maxCos * Float32(Float32(0.5) * Float32(Float32(Float32(ux * ux) * Float32(t_0 * Float32(xi + t_2))) + Float32(Float32(Float32(ux * ux) * Float32(uy * uy)) * Float32(t_0 * t_1)))))))) + Float32(Float32(uy * uy) * t_1)));
	else
		tmp = Float32(Float32(maxCos * Float32(ux * zi)) + Float32(xi + Float32(sin(Float32(Float32(2.0) * Float32(uy * Float32(pi)))) * yi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(xi, yi, zi, ux, uy, maxCos)
	t_0 = (single(1.0) - ux) * (ux + single(-1.0));
	t_1 = (single(-1.3333333333333333) * ((uy * yi) * (single(pi) * (single(pi) * single(pi))))) + ((xi * single(-2.0)) * (single(pi) * single(pi)));
	t_2 = (single(2.0) * uy) * (single(pi) * yi);
	tmp = single(0.0);
	if (uy <= single(0.05999999865889549))
		tmp = xi + ((t_2 + (maxCos * ((ux * ((single(1.0) - ux) * zi)) + (maxCos * (single(0.5) * (((ux * ux) * (t_0 * (xi + t_2))) + (((ux * ux) * (uy * uy)) * (t_0 * t_1)))))))) + ((uy * uy) * t_1));
	else
		tmp = (maxCos * (ux * zi)) + (xi + (sin((single(2.0) * (uy * single(pi)))) * yi));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.0599999987

   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%

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

   5. Taylor expanded in uy around 0

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

   7. Simplified 97.4%

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

   8. Taylor expanded in maxCos around 0

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

   10. Simplified 97.3%

       \[\leadsto \color{blue}{xi + \left(\left(\left(2 \cdot uy\right) \cdot \left(\pi \cdot yi\right) + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right) + maxCos \cdot \left(0.5 \cdot \left(\left(ux \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right) \cdot \left(xi + \left(2 \cdot uy\right) \cdot \left(\pi \cdot yi\right)\right)\right) + \left(\left(ux \cdot ux\right) \cdot \left(uy \cdot uy\right)\right) \cdot \left(\left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right) \cdot \left(\left(-2 \cdot xi\right) \cdot \left(\pi \cdot \pi\right) + -1.3333333333333333 \cdot \left(\left(uy \cdot yi\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\right)\right)\right)\right)\right) + \left(\left(-2 \cdot xi\right) \cdot \left(\pi \cdot \pi\right) + -1.3333333333333333 \cdot \left(\left(uy \cdot yi\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right)\right) \cdot \left(uy \cdot uy\right)\right)} \]

   if 0.0599999987 < 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%

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

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

      3. associate-*r* N/A

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

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

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

   7. Simplified 96.0%

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

   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 65.1%

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

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \leq 0.05999999865889549:\\ \;\;\;\;xi + \left(\left(\left(2 \cdot uy\right) \cdot \left(\pi \cdot yi\right) + maxCos \cdot \left(ux \cdot \left(\left(1 - ux\right) \cdot zi\right) + maxCos \cdot \left(0.5 \cdot \left(\left(ux \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right) \cdot \left(xi + \left(2 \cdot uy\right) \cdot \left(\pi \cdot yi\right)\right)\right) + \left(\left(ux \cdot ux\right) \cdot \left(uy \cdot uy\right)\right) \cdot \left(\left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right) \cdot \left(-1.3333333333333333 \cdot \left(\left(uy \cdot yi\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right) + \left(xi \cdot -2\right) \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\right)\right)\right) + \left(uy \cdot uy\right) \cdot \left(-1.3333333333333333 \cdot \left(\left(uy \cdot yi\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right) + \left(xi \cdot -2\right) \cdot \left(\pi \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;maxCos \cdot \left(ux \cdot zi\right) + \left(xi + \sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot yi\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 89.7% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - ux\right) \cdot \left(ux + -1\right)\\ t_1 := -1.3333333333333333 \cdot \left(\left(uy \cdot yi\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right) + \left(xi \cdot -2\right) \cdot \left(\pi \cdot \pi\right)\\ t_2 := \left(2 \cdot uy\right) \cdot \left(\pi \cdot yi\right)\\ xi + \left(\left(t\_2 + maxCos \cdot \left(ux \cdot \left(\left(1 - ux\right) \cdot zi\right) + maxCos \cdot \left(0.5 \cdot \left(\left(ux \cdot ux\right) \cdot \left(t\_0 \cdot \left(xi + t\_2\right)\right) + \left(\left(ux \cdot ux\right) \cdot \left(uy \cdot uy\right)\right) \cdot \left(t\_0 \cdot t\_1\right)\right)\right)\right)\right) + \left(uy \cdot uy\right) \cdot t\_1\right) \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (- 1.0 ux) (+ ux -1.0)))
        (t_1
         (+
          (* -1.3333333333333333 (* (* uy yi) (* PI (* PI PI))))
          (* (* xi -2.0) (* PI PI))))
        (t_2 (* (* 2.0 uy) (* PI yi))))
   (+
    xi
    (+
     (+
      t_2
      (*
       maxCos
       (+
        (* ux (* (- 1.0 ux) zi))
        (*
         maxCos
         (*
          0.5
          (+
           (* (* ux ux) (* t_0 (+ xi t_2)))
           (* (* (* ux ux) (* uy uy)) (* t_0 t_1))))))))
     (* (* uy uy) t_1)))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = (1.0f - ux) * (ux + -1.0f);
	float t_1 = (-1.3333333333333333f * ((uy * yi) * (((float) M_PI) * (((float) M_PI) * ((float) M_PI))))) + ((xi * -2.0f) * (((float) M_PI) * ((float) M_PI)));
	float t_2 = (2.0f * uy) * (((float) M_PI) * yi);
	return xi + ((t_2 + (maxCos * ((ux * ((1.0f - ux) * zi)) + (maxCos * (0.5f * (((ux * ux) * (t_0 * (xi + t_2))) + (((ux * ux) * (uy * uy)) * (t_0 * t_1)))))))) + ((uy * uy) * t_1));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(Float32(1.0) - ux) * Float32(ux + Float32(-1.0)))
	t_1 = Float32(Float32(Float32(-1.3333333333333333) * Float32(Float32(uy * yi) * Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi))))) + Float32(Float32(xi * Float32(-2.0)) * Float32(Float32(pi) * Float32(pi))))
	t_2 = Float32(Float32(Float32(2.0) * uy) * Float32(Float32(pi) * yi))
	return Float32(xi + Float32(Float32(t_2 + Float32(maxCos * Float32(Float32(ux * Float32(Float32(Float32(1.0) - ux) * zi)) + Float32(maxCos * Float32(Float32(0.5) * Float32(Float32(Float32(ux * ux) * Float32(t_0 * Float32(xi + t_2))) + Float32(Float32(Float32(ux * ux) * Float32(uy * uy)) * Float32(t_0 * t_1)))))))) + Float32(Float32(uy * uy) * t_1)))
end

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	t_0 = (single(1.0) - ux) * (ux + single(-1.0));
	t_1 = (single(-1.3333333333333333) * ((uy * yi) * (single(pi) * (single(pi) * single(pi))))) + ((xi * single(-2.0)) * (single(pi) * single(pi)));
	t_2 = (single(2.0) * uy) * (single(pi) * yi);
	tmp = xi + ((t_2 + (maxCos * ((ux * ((single(1.0) - ux) * zi)) + (maxCos * (single(0.5) * (((ux * ux) * (t_0 * (xi + t_2))) + (((ux * ux) * (uy * uy)) * (t_0 * t_1)))))))) + ((uy * uy) * t_1));
end

Derivation

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%

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

5. Taylor expanded in uy around 0

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

7. Simplified 90.5%

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

8. Taylor expanded in maxCos around 0

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

10. Simplified 90.4%

    \[\leadsto \color{blue}{xi + \left(\left(\left(2 \cdot uy\right) \cdot \left(\pi \cdot yi\right) + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right) + maxCos \cdot \left(0.5 \cdot \left(\left(ux \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right) \cdot \left(xi + \left(2 \cdot uy\right) \cdot \left(\pi \cdot yi\right)\right)\right) + \left(\left(ux \cdot ux\right) \cdot \left(uy \cdot uy\right)\right) \cdot \left(\left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right) \cdot \left(\left(-2 \cdot xi\right) \cdot \left(\pi \cdot \pi\right) + -1.3333333333333333 \cdot \left(\left(uy \cdot yi\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\right)\right)\right)\right)\right) + \left(\left(-2 \cdot xi\right) \cdot \left(\pi \cdot \pi\right) + -1.3333333333333333 \cdot \left(\left(uy \cdot yi\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right)\right) \cdot \left(uy \cdot uy\right)\right)} \]

11. Final simplification 90.4%

    \[\leadsto xi + \left(\left(\left(2 \cdot uy\right) \cdot \left(\pi \cdot yi\right) + maxCos \cdot \left(ux \cdot \left(\left(1 - ux\right) \cdot zi\right) + maxCos \cdot \left(0.5 \cdot \left(\left(ux \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right) \cdot \left(xi + \left(2 \cdot uy\right) \cdot \left(\pi \cdot yi\right)\right)\right) + \left(\left(ux \cdot ux\right) \cdot \left(uy \cdot uy\right)\right) \cdot \left(\left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right) \cdot \left(-1.3333333333333333 \cdot \left(\left(uy \cdot yi\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right) + \left(xi \cdot -2\right) \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\right)\right)\right) + \left(uy \cdot uy\right) \cdot \left(-1.3333333333333333 \cdot \left(\left(uy \cdot yi\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right) + \left(xi \cdot -2\right) \cdot \left(\pi \cdot \pi\right)\right)\right) \]
12. Add Preprocessing

Alternative 11: 89.5% accurate, 11.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

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%

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

5. Taylor expanded in uy around 0

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

7. Simplified 90.5%

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

8. Taylor expanded in maxCos around 0

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

   1. associate-+r+ N/A

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

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

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

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

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

   4. associate-*r* N/A

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

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

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

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

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

   7. *-commutative N/A

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

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

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

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

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

   10. *-commutative N/A

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

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

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

10. Simplified 90.2%

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

11. Taylor expanded in ux around 0

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

13. Simplified 90.3%

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

14. Final simplification 90.3%

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

Alternative 12: 86.0% accurate, 15.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

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%

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

5. Taylor expanded in uy around 0

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

7. Simplified 90.5%

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

8. Taylor expanded in maxCos around 0

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

   1. associate-+r+ N/A

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

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

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

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

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

   4. associate-*r* N/A

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

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

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

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

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

   7. *-commutative N/A

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

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

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

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

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

   10. *-commutative N/A

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

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

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

10. Simplified 90.2%

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

11. Taylor expanded in uy around 0

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

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

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

    2. +-commutative N/A

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

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

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

13. Simplified 86.1%

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

14. Final simplification 86.1%

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

Alternative 13: 60.3% accurate, 20.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := maxCos \cdot \left(ux \cdot zi\right) + \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot yi\\ \mathbf{if}\;yi \leq -1.2499999968440534 \cdot 10^{-7}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;yi \leq 3.99999987306209 \cdot 10^{-20}:\\ \;\;\;\;xi + zi \cdot \left(ux \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (* maxCos (* ux zi)) (* (* 2.0 (* uy PI)) yi))))
   (if (<= yi -1.2499999968440534e-7)
     t_0
     (if (<= yi 3.99999987306209e-20)
       (+ 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 = (maxCos * (ux * zi)) + ((2.0f * (uy * ((float) M_PI))) * yi);
	float tmp;
	if (yi <= -1.2499999968440534e-7f) {
		tmp = t_0;
	} else if (yi <= 3.99999987306209e-20f) {
		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(maxCos * Float32(ux * zi)) + Float32(Float32(Float32(2.0) * Float32(uy * Float32(pi))) * yi))
	tmp = Float32(0.0)
	if (yi <= Float32(-1.2499999968440534e-7))
		tmp = t_0;
	elseif (yi <= Float32(3.99999987306209e-20))
		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 = (maxCos * (ux * zi)) + ((single(2.0) * (uy * single(pi))) * yi);
	tmp = single(0.0);
	if (yi <= single(-1.2499999968440534e-7))
		tmp = t_0;
	elseif (yi <= single(3.99999987306209e-20))
		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 < -1.25e-7 or 3.99999987e-20 < yi

   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%

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

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

      3. associate-*r* N/A

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

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

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

   7. Simplified 96.1%

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

   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. *-lowering-*.f32 N/A

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

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

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

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

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

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

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

      9. *-lowering-*.f32 66.5%

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

   10. Simplified 66.5%

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

   11. Taylor expanded in uy around 0

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

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

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

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

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

       3. PI-lowering-PI.f32 52.7%

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

   13. Simplified 52.7%

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

   if -1.25e-7 < yi < 3.99999987e-20

   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%

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

   5. Taylor expanded in uy around 0

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

   7. Simplified 91.7%

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

   8. Taylor expanded in maxCos around 0

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

      1. associate-+r+ N/A

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

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

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

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

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

      4. associate-*r* N/A

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

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

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

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

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

      7. *-commutative N/A

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

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

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

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

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

      10. *-commutative N/A

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

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

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

   10. Simplified 91.7%

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

   11. Taylor expanded in uy around 0

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

   13. Simplified 72.4%

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

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;yi \leq -1.2499999968440534 \cdot 10^{-7}:\\ \;\;\;\;maxCos \cdot \left(ux \cdot zi\right) + \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot yi\\ \mathbf{elif}\;yi \leq 3.99999987306209 \cdot 10^{-20}:\\ \;\;\;\;xi + zi \cdot \left(ux \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right)\\ \mathbf{else}:\\ \;\;\;\;maxCos \cdot \left(ux \cdot zi\right) + \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot yi\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 60.3% accurate, 20.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := maxCos \cdot \left(ux \cdot zi\right) + 2 \cdot \left(uy \cdot \left(\pi \cdot yi\right)\right)\\ \mathbf{if}\;yi \leq -1.2499999968440534 \cdot 10^{-7}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;yi \leq 3.99999987306209 \cdot 10^{-20}:\\ \;\;\;\;xi + zi \cdot \left(ux \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (* maxCos (* ux zi)) (* 2.0 (* uy (* PI yi))))))
   (if (<= yi -1.2499999968440534e-7)
     t_0
     (if (<= yi 3.99999987306209e-20)
       (+ 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 = (maxCos * (ux * zi)) + (2.0f * (uy * (((float) M_PI) * yi)));
	float tmp;
	if (yi <= -1.2499999968440534e-7f) {
		tmp = t_0;
	} else if (yi <= 3.99999987306209e-20f) {
		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(maxCos * Float32(ux * zi)) + Float32(Float32(2.0) * Float32(uy * Float32(Float32(pi) * yi))))
	tmp = Float32(0.0)
	if (yi <= Float32(-1.2499999968440534e-7))
		tmp = t_0;
	elseif (yi <= Float32(3.99999987306209e-20))
		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 = (maxCos * (ux * zi)) + (single(2.0) * (uy * (single(pi) * yi)));
	tmp = single(0.0);
	if (yi <= single(-1.2499999968440534e-7))
		tmp = t_0;
	elseif (yi <= single(3.99999987306209e-20))
		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 < -1.25e-7 or 3.99999987e-20 < yi

   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%

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

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

      3. associate-*r* N/A

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

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

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

   7. Simplified 96.1%

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

   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. *-lowering-*.f32 N/A

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

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

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

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

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

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

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

      9. *-lowering-*.f32 66.5%

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

   10. Simplified 66.5%

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

   11. Taylor expanded in uy around 0

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

       1. +-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

       7. *-commutative N/A

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

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

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

       9. PI-lowering-PI.f32 52.6%

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

   13. Simplified 52.6%

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

   if -1.25e-7 < yi < 3.99999987e-20

   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%

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

   5. Taylor expanded in uy around 0

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

   7. Simplified 91.7%

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

   8. Taylor expanded in maxCos around 0

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

      1. associate-+r+ N/A

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

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

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

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

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

      4. associate-*r* N/A

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

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

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

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

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

      7. *-commutative N/A

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

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

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

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

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

      10. *-commutative N/A

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

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

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

   10. Simplified 91.7%

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

   11. Taylor expanded in uy around 0

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

   13. Simplified 72.4%

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

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;yi \leq -1.2499999968440534 \cdot 10^{-7}:\\ \;\;\;\;maxCos \cdot \left(ux \cdot zi\right) + 2 \cdot \left(uy \cdot \left(\pi \cdot yi\right)\right)\\ \mathbf{elif}\;yi \leq 3.99999987306209 \cdot 10^{-20}:\\ \;\;\;\;xi + zi \cdot \left(ux \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right)\\ \mathbf{else}:\\ \;\;\;\;maxCos \cdot \left(ux \cdot zi\right) + 2 \cdot \left(uy \cdot \left(\pi \cdot yi\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 81.9% accurate, 24.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

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%

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

5. Taylor expanded in uy around 0

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

7. Simplified 90.5%

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

8. Taylor expanded in maxCos around 0

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

   1. associate-+r+ N/A

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

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

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

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

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

   4. associate-*r* N/A

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

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

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

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

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

   7. *-commutative N/A

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

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

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

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

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

   10. *-commutative N/A

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

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

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

10. Simplified 90.2%

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

11. Taylor expanded in uy around 0

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

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

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

    2. *-commutative N/A

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

    3. *-commutative N/A

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

    4. associate-*l* N/A

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

    5. *-commutative N/A

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

    6. associate-*r* N/A

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

    7. *-commutative N/A

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

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

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

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

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

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

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

    11. PI-lowering-PI.f32 82.9%

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

13. Simplified 82.9%

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

14. Final simplification 82.9%

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

Alternative 16: 79.0% accurate, 30.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

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%

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

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

   3. associate-*r* N/A

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

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

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

7. Simplified 94.4%

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

   1. add-cube-cbrt N/A

      \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(xi, \mathsf{cos.f32}\left(\mathsf{*.f32}\left(2, \left(uy \cdot \left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right)\right)\right)\right), \mathsf{*.f32}\left(yi, \mathsf{sin.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \mathsf{PI.f32}\left(\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right) \]

   2. associate-*r* N/A

      \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(xi, \mathsf{cos.f32}\left(\mathsf{*.f32}\left(2, \left(\left(uy \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right)\right)\right), \mathsf{*.f32}\left(yi, \mathsf{sin.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \mathsf{PI.f32}\left(\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(xi, \mathsf{cos.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(\left(uy \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right), \left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)\right)\right)\right)\right), \mathsf{*.f32}\left(yi, \mathsf{sin.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \mathsf{PI.f32}\left(\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(xi, \mathsf{cos.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(\mathsf{*.f32}\left(uy, \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right), \left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)\right)\right)\right)\right), \mathsf{*.f32}\left(yi, \mathsf{sin.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \mathsf{PI.f32}\left(\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right) \]

   5. pow2 N/A

      \[\leadsto \mathsf{+.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(xi, \mathsf{cos.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(\mathsf{*.f32}\left(uy, \left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}\right)\right), \left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)\right)\right)\right)\right), \mathsf{*.f32}\left(yi, \mathsf{sin.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \mathsf{PI.f32}\left(\right)\right)\right)\right)\right)\right), \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, zi\right)\right)\right) \]

   6. pow1/3 N/A

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

   7. pow-pow N/A

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

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

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

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

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

   10. metadata-eval N/A

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

   11. pow1/3 N/A

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

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

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

   13. PI-lowering-PI.f32 94.4%

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

9. Applied egg-rr 94.4%

   \[\leadsto \left(xi \cdot \cos \left(2 \cdot \color{blue}{\left(\left(uy \cdot {\pi}^{0.6666666666666666}\right) \cdot {\pi}^{0.3333333333333333}\right)}\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) + maxCos \cdot \left(ux \cdot zi\right) \]

10. Taylor expanded in uy around 0

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

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

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

    2. *-commutative N/A

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

    3. *-commutative N/A

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

    4. associate-*l* N/A

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

    5. *-commutative N/A

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

    6. associate-*r* N/A

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

    7. *-commutative N/A

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

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

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

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

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

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

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

    11. PI-lowering-PI.f32 78.4%

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

12. Simplified 78.4%

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

13. Final simplification 78.4%

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

Alternative 17: 52.0% accurate, 41.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

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%

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

5. Taylor expanded in uy around 0

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

7. Simplified 90.5%

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

8. Taylor expanded in maxCos around 0

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

   1. associate-+r+ N/A

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

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

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

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

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

   4. associate-*r* N/A

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

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

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

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

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

   7. *-commutative N/A

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

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

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

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

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

   10. *-commutative N/A

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

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

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

10. Simplified 90.2%

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

11. Taylor expanded in uy around 0

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

13. Simplified 54.9%

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

14. Final simplification 54.9%

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

Alternative 18: 52.0% accurate, 41.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

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%

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

5. Taylor expanded in uy around 0

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

7. Simplified 90.5%

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

8. Taylor expanded in maxCos around 0

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

   1. associate-+r+ N/A

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

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

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

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

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

   4. associate-*r* N/A

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

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

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

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

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

   7. *-commutative N/A

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

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

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

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

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

   10. *-commutative N/A

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

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

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

10. Simplified 90.2%

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

11. Taylor expanded in uy around 0

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

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

    4. associate-*l* N/A

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

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

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

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

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

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

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

    8. --lowering--.f32 54.9%

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

13. Simplified 54.9%

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

14. Final simplification 54.9%

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

Alternative 19: 49.7% accurate, 65.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.2%

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

3. Simplified 99.2%

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

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

   3. associate-*r* N/A

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

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

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

7. Simplified 94.4%

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

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 51.5%

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

10. Simplified 51.5%

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

Alternative 20: 12.1% accurate, 92.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

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%

   \[\leadsto \color{blue}{\sqrt{1 + \left(1 - ux\right) \cdot \left(\left(ux \cdot ux\right) \cdot \left(maxCos \cdot \left(maxCos \cdot \left(ux + -1\right)\right)\right)\right)} \cdot \left(\sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot yi + \cos \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot xi\right) + \left(ux \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right) \cdot zi} \]
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 15.5%

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

7. Simplified 15.5%

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

8. Taylor expanded in ux around 0

   \[\leadsto \mathsf{*.f32}\left(zi, \color{blue}{\left(maxCos \cdot ux\right)}\right) \]
9. Step-by-step derivation

   1. *-lowering-*.f32 13.0%

      \[\leadsto \mathsf{*.f32}\left(zi, \mathsf{*.f32}\left(maxCos, \color{blue}{ux}\right)\right) \]

10. Simplified 13.0%

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

11. Final simplification 13.0%

    \[\leadsto \left(ux \cdot maxCos\right) \cdot zi \]
12. Add Preprocessing

Alternative 21: 12.1% accurate, 92.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.2%

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

3. Simplified 99.2%

   \[\leadsto \color{blue}{\sqrt{1 + \left(1 - ux\right) \cdot \left(\left(ux \cdot ux\right) \cdot \left(maxCos \cdot \left(maxCos \cdot \left(ux + -1\right)\right)\right)\right)} \cdot \left(\sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot yi + \cos \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot xi\right) + \left(ux \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right) \cdot zi} \]
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 15.5%

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

7. Simplified 15.5%

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

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 13.0%

      \[\leadsto \mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, \color{blue}{zi}\right)\right) \]

10. Simplified 13.0%

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

Reproduce

herbie shell --seed 2024160 
(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)))