UniformSampleCone 2

Percentage Accurate: 98.9% → 99.0%

Time: 10.3s

Alternatives: 20

Speedup: 1.2×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

2. Taylor expanded in xi around 0

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

   1. +-commutative N/A

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

4. Applied rewrites 99.0%

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

Alternative 2: 98.7% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

2. Taylor expanded in xi around 0

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

   1. +-commutative N/A

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

4. Applied rewrites 99.0%

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

5. Taylor expanded in ux around 0

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

   1. pow2 N/A

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

   2. lift-*.f32 98.7

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

7. Applied rewrites 98.7%

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

Alternative 3: 98.7% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

2. Taylor expanded in maxCos around 0

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

   1. associate-*r* N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lift--.f32 N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f32 N/A

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

4. Applied rewrites 98.7%

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

Alternative 4: 97.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \left(uy \cdot \pi\right)\\ \mathbf{if}\;uy \leq 0.00800000037997961:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{1 - \left(maxCos \cdot maxCos\right) \cdot \left(\left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right)\right)}, xi + uy \cdot \mathsf{fma}\left(2, yi \cdot \pi, uy \cdot \mathsf{fma}\left(-2, xi \cdot \left(\pi \cdot \pi\right), -1.3333333333333333 \cdot \left(uy \cdot \left(yi \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)\right)\right)\right)\right), maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;xi \cdot \left(\cos t\_0 + \frac{yi \cdot \sin t\_0}{xi}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* 2.0 (* uy PI))))
   (if (<= uy 0.00800000037997961)
     (fma
      (sqrt
       (- 1.0 (* (* maxCos maxCos) (* (* ux ux) (* (- 1.0 ux) (- 1.0 ux))))))
      (+
       xi
       (*
        uy
        (fma
         2.0
         (* yi PI)
         (*
          uy
          (fma
           -2.0
           (* xi (* PI PI))
           (* -1.3333333333333333 (* uy (* yi (* (* PI PI) PI)))))))))
      (* maxCos (* ux (* zi (- 1.0 ux)))))
     (* xi (+ (cos t_0) (/ (* yi (sin t_0)) xi))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = 2.0f * (uy * ((float) M_PI));
	float tmp;
	if (uy <= 0.00800000037997961f) {
		tmp = fmaf(sqrtf((1.0f - ((maxCos * maxCos) * ((ux * ux) * ((1.0f - ux) * (1.0f - ux)))))), (xi + (uy * fmaf(2.0f, (yi * ((float) M_PI)), (uy * fmaf(-2.0f, (xi * (((float) M_PI) * ((float) M_PI))), (-1.3333333333333333f * (uy * (yi * ((((float) M_PI) * ((float) M_PI)) * ((float) M_PI)))))))))), (maxCos * (ux * (zi * (1.0f - ux)))));
	} else {
		tmp = xi * (cosf(t_0) + ((yi * sinf(t_0)) / xi));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(2.0) * Float32(uy * Float32(pi)))
	tmp = Float32(0.0)
	if (uy <= Float32(0.00800000037997961))
		tmp = fma(sqrt(Float32(Float32(1.0) - Float32(Float32(maxCos * maxCos) * Float32(Float32(ux * ux) * Float32(Float32(Float32(1.0) - ux) * Float32(Float32(1.0) - ux)))))), Float32(xi + Float32(uy * fma(Float32(2.0), Float32(yi * Float32(pi)), Float32(uy * fma(Float32(-2.0), Float32(xi * Float32(Float32(pi) * Float32(pi))), Float32(Float32(-1.3333333333333333) * Float32(uy * Float32(yi * Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi)))))))))), Float32(maxCos * Float32(ux * Float32(zi * Float32(Float32(1.0) - ux)))));
	else
		tmp = Float32(xi * Float32(cos(t_0) + Float32(Float32(yi * sin(t_0)) / xi)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.00800000038

   1. Initial program 99.3%

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

   2. Taylor expanded in xi around 0

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

      1. +-commutative N/A

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in uy around 0

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

      1. lower-+.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-fma.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lower-fma.f32 N/A

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

   7. Applied rewrites 99.3%

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

   if 0.00800000038 < uy

   1. Initial program 97.8%

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

   2. Taylor expanded in xi around 0

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

      1. +-commutative N/A

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

   4. Applied rewrites 97.8%

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

   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. lower-fma.f32 N/A

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

      2. lower-cos.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lift-PI.f32 N/A

         \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      6. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      7. lower-sin.f32 N/A

         \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      8. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      9. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      10. lift-PI.f32 90.5

          \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]

   7. Applied rewrites 90.5%

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

   8. Taylor expanded in xi around inf

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

      1. lower-*.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. lift-PI.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. lift-cos.f32 N/A

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

      7. lower-/.f32 N/A

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

   10. Applied rewrites 90.2%

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

Alternative 5: 97.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(uy + uy\right) \cdot \pi\\ \mathbf{if}\;uy \leq 0.00800000037997961:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{1 - \left(maxCos \cdot maxCos\right) \cdot \left(\left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right)\right)}, xi + uy \cdot \mathsf{fma}\left(2, yi \cdot \pi, uy \cdot \mathsf{fma}\left(-2, xi \cdot \left(\pi \cdot \pi\right), -1.3333333333333333 \cdot \left(uy \cdot \left(yi \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)\right)\right)\right)\right), maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(yi, \sin t\_0, xi \cdot \cos t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (+ uy uy) PI)))
   (if (<= uy 0.00800000037997961)
     (fma
      (sqrt
       (- 1.0 (* (* maxCos maxCos) (* (* ux ux) (* (- 1.0 ux) (- 1.0 ux))))))
      (+
       xi
       (*
        uy
        (fma
         2.0
         (* yi PI)
         (*
          uy
          (fma
           -2.0
           (* xi (* PI PI))
           (* -1.3333333333333333 (* uy (* yi (* (* PI PI) PI)))))))))
      (* maxCos (* ux (* zi (- 1.0 ux)))))
     (fma yi (sin t_0) (* xi (cos t_0))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = (uy + uy) * ((float) M_PI);
	float tmp;
	if (uy <= 0.00800000037997961f) {
		tmp = fmaf(sqrtf((1.0f - ((maxCos * maxCos) * ((ux * ux) * ((1.0f - ux) * (1.0f - ux)))))), (xi + (uy * fmaf(2.0f, (yi * ((float) M_PI)), (uy * fmaf(-2.0f, (xi * (((float) M_PI) * ((float) M_PI))), (-1.3333333333333333f * (uy * (yi * ((((float) M_PI) * ((float) M_PI)) * ((float) M_PI)))))))))), (maxCos * (ux * (zi * (1.0f - ux)))));
	} else {
		tmp = fmaf(yi, sinf(t_0), (xi * cosf(t_0)));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(uy + uy) * Float32(pi))
	tmp = Float32(0.0)
	if (uy <= Float32(0.00800000037997961))
		tmp = fma(sqrt(Float32(Float32(1.0) - Float32(Float32(maxCos * maxCos) * Float32(Float32(ux * ux) * Float32(Float32(Float32(1.0) - ux) * Float32(Float32(1.0) - ux)))))), Float32(xi + Float32(uy * fma(Float32(2.0), Float32(yi * Float32(pi)), Float32(uy * fma(Float32(-2.0), Float32(xi * Float32(Float32(pi) * Float32(pi))), Float32(Float32(-1.3333333333333333) * Float32(uy * Float32(yi * Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi)))))))))), Float32(maxCos * Float32(ux * Float32(zi * Float32(Float32(1.0) - ux)))));
	else
		tmp = fma(yi, sin(t_0), Float32(xi * cos(t_0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.00800000038

   1. Initial program 99.3%

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

   2. Taylor expanded in xi around 0

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

      1. +-commutative N/A

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in uy around 0

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

      1. lower-+.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-fma.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lower-fma.f32 N/A

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

   7. Applied rewrites 99.3%

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

   if 0.00800000038 < uy

   1. Initial program 97.8%

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

   2. Taylor expanded in ux around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-sin.f32 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f32 N/A

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

      6. count-2-rev N/A

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

      7. lower-+.f32 N/A

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

      8. lift-PI.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-cos.f32 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f32 N/A

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

   4. Applied rewrites 90.5%

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

Alternative 6: 95.8% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

2. Taylor expanded in ux around 0

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

   1. associate-*r* N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f32 N/A

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

4. Applied rewrites 95.8%

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

Alternative 7: 91.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;xi \leq -9.99999983775159 \cdot 10^{-18}:\\ \;\;\;\;\mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \left(uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), 2 \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{1 - \left(maxCos \cdot maxCos\right) \cdot \left(ux \cdot ux\right)}, \mathsf{fma}\left(yi, \sin \left(\left(uy + uy\right) \cdot \pi\right), xi\right), maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (if (<= xi -9.99999983775159e-18)
   (fma
    xi
    (cos (* 2.0 (* uy PI)))
    (*
     yi
     (*
      uy
      (fma -1.3333333333333333 (* (* uy uy) (* (* PI PI) PI)) (* 2.0 PI)))))
   (fma
    (sqrt (- 1.0 (* (* maxCos maxCos) (* ux ux))))
    (fma yi (sin (* (+ uy uy) PI)) xi)
    (* maxCos (* ux (* zi (- 1.0 ux)))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float tmp;
	if (xi <= -9.99999983775159e-18f) {
		tmp = fmaf(xi, cosf((2.0f * (uy * ((float) M_PI)))), (yi * (uy * fmaf(-1.3333333333333333f, ((uy * uy) * ((((float) M_PI) * ((float) M_PI)) * ((float) M_PI))), (2.0f * ((float) M_PI))))));
	} else {
		tmp = fmaf(sqrtf((1.0f - ((maxCos * maxCos) * (ux * ux)))), fmaf(yi, sinf(((uy + uy) * ((float) M_PI))), xi), (maxCos * (ux * (zi * (1.0f - ux)))));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	tmp = Float32(0.0)
	if (xi <= Float32(-9.99999983775159e-18))
		tmp = fma(xi, cos(Float32(Float32(2.0) * Float32(uy * Float32(pi)))), Float32(yi * Float32(uy * fma(Float32(-1.3333333333333333), Float32(Float32(uy * uy) * Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi))), Float32(Float32(2.0) * Float32(pi))))));
	else
		tmp = fma(sqrt(Float32(Float32(1.0) - Float32(Float32(maxCos * maxCos) * Float32(ux * ux)))), fma(yi, sin(Float32(Float32(uy + uy) * Float32(pi))), xi), Float32(maxCos * Float32(ux * Float32(zi * Float32(Float32(1.0) - ux)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if xi < -9.99999984e-18

   1. Initial program 99.2%

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

   2. Taylor expanded in xi around 0

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

      1. +-commutative N/A

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

   4. Applied rewrites 99.2%

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

   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. lower-fma.f32 N/A

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

      2. lower-cos.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lift-PI.f32 N/A

         \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      6. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      7. lower-sin.f32 N/A

         \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      8. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      9. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      10. lift-PI.f32 96.2

          \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]

   7. Applied rewrites 96.2%

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

   8. Taylor expanded in uy around 0

      \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \left(uy \cdot \left(\frac{-4}{3} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]
   9. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \left(uy \cdot \left(\frac{-4}{3} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      2. lower-fma.f32 N/A

         \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, {uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}, 2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      3. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, {uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}, 2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot {\mathsf{PI}\left(\right)}^{3}, 2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      5. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot {\mathsf{PI}\left(\right)}^{3}, 2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      6. unpow3 N/A

         \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right), 2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      7. pow2 N/A

         \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \mathsf{PI}\left(\right)\right), 2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      8. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \mathsf{PI}\left(\right)\right), 2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right), 2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      10. lift-*.f32 N/A

          \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right), 2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      11. lift-PI.f32 N/A

          \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right), 2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      12. lift-PI.f32 N/A

          \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \mathsf{PI}\left(\right)\right), 2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      13. lift-PI.f32 N/A

          \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), 2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      14. lower-*.f32 N/A

          \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), 2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      15. lift-PI.f32 93.4

          \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \left(uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), 2 \cdot \pi\right)\right)\right) \]

   10. Applied rewrites 93.4%

       \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \left(uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), 2 \cdot \pi\right)\right)\right) \]

   if -9.99999984e-18 < xi

   1. Initial program 98.9%

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

   2. Taylor expanded in xi around 0

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

      1. +-commutative N/A

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

   4. Applied rewrites 98.9%

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

   5. Taylor expanded in ux around 0

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

      1. pow2 N/A

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

      2. lift-*.f32 98.7

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

   7. Applied rewrites 98.7%

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

   8. Taylor expanded in uy around 0

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

   10. Applied rewrites 89.8%

       \[\leadsto \mathsf{fma}\left(\sqrt{1 - \left(maxCos \cdot maxCos\right) \cdot \left(ux \cdot ux\right)}, \mathsf{fma}\left(yi, \sin \left(\left(uy + uy\right) \cdot \pi\right), xi\right), maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 90.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.001550000044517219:\\ \;\;\;\;\mathsf{fma}\left(maxCos \cdot ux, zi \cdot \left(1 - ux\right), \mathsf{fma}\left(uy, \mathsf{fma}\left(-2, uy \cdot \left(xi \cdot \left(\pi \cdot \pi\right)\right), 2 \cdot \left(yi \cdot \pi\right)\right), xi \cdot \sqrt{1 - \left(maxCos \cdot maxCos\right) \cdot \left(\left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right)\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \left(uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), 2 \cdot \pi\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (if (<= uy 0.001550000044517219)
   (fma
    (* maxCos ux)
    (* zi (- 1.0 ux))
    (fma
     uy
     (fma -2.0 (* uy (* xi (* PI PI))) (* 2.0 (* yi PI)))
     (*
      xi
      (sqrt
       (-
        1.0
        (* (* maxCos maxCos) (* (* ux ux) (* (- 1.0 ux) (- 1.0 ux)))))))))
   (fma
    xi
    (cos (* 2.0 (* uy PI)))
    (*
     yi
     (*
      uy
      (fma -1.3333333333333333 (* (* uy uy) (* (* PI PI) PI)) (* 2.0 PI)))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.001550000044517219f) {
		tmp = fmaf((maxCos * ux), (zi * (1.0f - ux)), fmaf(uy, fmaf(-2.0f, (uy * (xi * (((float) M_PI) * ((float) M_PI)))), (2.0f * (yi * ((float) M_PI)))), (xi * sqrtf((1.0f - ((maxCos * maxCos) * ((ux * ux) * ((1.0f - ux) * (1.0f - ux)))))))));
	} else {
		tmp = fmaf(xi, cosf((2.0f * (uy * ((float) M_PI)))), (yi * (uy * fmaf(-1.3333333333333333f, ((uy * uy) * ((((float) M_PI) * ((float) M_PI)) * ((float) M_PI))), (2.0f * ((float) M_PI))))));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.001550000044517219))
		tmp = fma(Float32(maxCos * ux), Float32(zi * Float32(Float32(1.0) - ux)), fma(uy, fma(Float32(-2.0), Float32(uy * Float32(xi * Float32(Float32(pi) * Float32(pi)))), Float32(Float32(2.0) * Float32(yi * Float32(pi)))), Float32(xi * sqrt(Float32(Float32(1.0) - Float32(Float32(maxCos * maxCos) * Float32(Float32(ux * ux) * Float32(Float32(Float32(1.0) - ux) * Float32(Float32(1.0) - ux)))))))));
	else
		tmp = fma(xi, cos(Float32(Float32(2.0) * Float32(uy * Float32(pi)))), Float32(yi * Float32(uy * fma(Float32(-1.3333333333333333), Float32(Float32(uy * uy) * Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi))), Float32(Float32(2.0) * Float32(pi))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.00155000004

   1. Initial program 99.3%

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

   2. Taylor expanded in uy around 0

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

   4. Applied rewrites 98.6%

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

   5. Taylor expanded in ux around 0

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

      1. lower-fma.f32 N/A

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

      2. pow2 N/A

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

      3. lift-*.f32 N/A

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

      4. lift-PI.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. lift-*.f32 N/A

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

      7. lift-*.f32 N/A

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

      8. lift-*.f32 N/A

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

      9. lift-PI.f32 N/A

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

      10. lift-*.f32 98.5

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

   7. Applied rewrites 98.5%

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

   if 0.00155000004 < uy

   1. Initial program 98.1%

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

   2. Taylor expanded in xi around 0

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

      1. +-commutative N/A

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

   4. Applied rewrites 98.1%

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

   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. lower-fma.f32 N/A

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

      2. lower-cos.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lift-PI.f32 N/A

         \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      6. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      7. lower-sin.f32 N/A

         \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      8. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      9. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      10. lift-PI.f32 90.5

          \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]

   7. Applied rewrites 90.5%

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

   8. Taylor expanded in uy around 0

      \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \left(uy \cdot \left(\frac{-4}{3} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]
   9. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \left(uy \cdot \left(\frac{-4}{3} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      2. lower-fma.f32 N/A

         \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, {uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}, 2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      3. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, {uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}, 2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot {\mathsf{PI}\left(\right)}^{3}, 2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      5. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot {\mathsf{PI}\left(\right)}^{3}, 2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      6. unpow3 N/A

         \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right), 2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      7. pow2 N/A

         \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \mathsf{PI}\left(\right)\right), 2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      8. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \mathsf{PI}\left(\right)\right), 2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      9. pow2 N/A

         \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right), 2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      10. lift-*.f32 N/A

          \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right), 2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      11. lift-PI.f32 N/A

          \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right), 2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      12. lift-PI.f32 N/A

          \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \mathsf{PI}\left(\right)\right), 2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      13. lift-PI.f32 N/A

          \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), 2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      14. lower-*.f32 N/A

          \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), 2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      15. lift-PI.f32 75.3

          \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \left(uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), 2 \cdot \pi\right)\right)\right) \]

   10. Applied rewrites 75.3%

       \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \left(uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), 2 \cdot \pi\right)\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 90.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.001550000044517219:\\ \;\;\;\;\mathsf{fma}\left(maxCos \cdot ux, zi \cdot \left(1 - ux\right), \mathsf{fma}\left(uy, \mathsf{fma}\left(-2, uy \cdot \left(xi \cdot \left(\pi \cdot \pi\right)\right), 2 \cdot \left(yi \cdot \pi\right)\right), xi \cdot \sqrt{1 - \left(maxCos \cdot maxCos\right) \cdot \left(\left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right)\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(xi, 1 + -2 \cdot \left(\left(uy \cdot uy\right) \cdot \left(\pi \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (if (<= uy 0.001550000044517219)
   (fma
    (* maxCos ux)
    (* zi (- 1.0 ux))
    (fma
     uy
     (fma -2.0 (* uy (* xi (* PI PI))) (* 2.0 (* yi PI)))
     (*
      xi
      (sqrt
       (-
        1.0
        (* (* maxCos maxCos) (* (* ux ux) (* (- 1.0 ux) (- 1.0 ux)))))))))
   (fma
    xi
    (+ 1.0 (* -2.0 (* (* uy uy) (* PI PI))))
    (* yi (sin (* 2.0 (* uy PI)))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.001550000044517219f) {
		tmp = fmaf((maxCos * ux), (zi * (1.0f - ux)), fmaf(uy, fmaf(-2.0f, (uy * (xi * (((float) M_PI) * ((float) M_PI)))), (2.0f * (yi * ((float) M_PI)))), (xi * sqrtf((1.0f - ((maxCos * maxCos) * ((ux * ux) * ((1.0f - ux) * (1.0f - ux)))))))));
	} else {
		tmp = fmaf(xi, (1.0f + (-2.0f * ((uy * uy) * (((float) M_PI) * ((float) M_PI))))), (yi * sinf((2.0f * (uy * ((float) M_PI))))));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.001550000044517219))
		tmp = fma(Float32(maxCos * ux), Float32(zi * Float32(Float32(1.0) - ux)), fma(uy, fma(Float32(-2.0), Float32(uy * Float32(xi * Float32(Float32(pi) * Float32(pi)))), Float32(Float32(2.0) * Float32(yi * Float32(pi)))), Float32(xi * sqrt(Float32(Float32(1.0) - Float32(Float32(maxCos * maxCos) * Float32(Float32(ux * ux) * Float32(Float32(Float32(1.0) - ux) * Float32(Float32(1.0) - ux)))))))));
	else
		tmp = fma(xi, Float32(Float32(1.0) + Float32(Float32(-2.0) * Float32(Float32(uy * uy) * Float32(Float32(pi) * Float32(pi))))), Float32(yi * sin(Float32(Float32(2.0) * Float32(uy * Float32(pi))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.00155000004

   1. Initial program 99.3%

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

   2. Taylor expanded in uy around 0

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

   4. Applied rewrites 98.6%

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

   5. Taylor expanded in ux around 0

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

      1. lower-fma.f32 N/A

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

      2. pow2 N/A

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

      3. lift-*.f32 N/A

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

      4. lift-PI.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. lift-*.f32 N/A

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

      7. lift-*.f32 N/A

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

      8. lift-*.f32 N/A

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

      9. lift-PI.f32 N/A

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

      10. lift-*.f32 98.5

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

   7. Applied rewrites 98.5%

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

   if 0.00155000004 < uy

   1. Initial program 98.1%

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

   2. Taylor expanded in xi around 0

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

      1. +-commutative N/A

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

   4. Applied rewrites 98.1%

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

   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. lower-fma.f32 N/A

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

      2. lower-cos.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lift-PI.f32 N/A

         \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      6. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      7. lower-sin.f32 N/A

         \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      8. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      9. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      10. lift-PI.f32 90.5

          \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]

   7. Applied rewrites 90.5%

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

   8. Taylor expanded in uy around 0

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

      1. lower-+.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(xi, 1 + -2 \cdot \left(\left(uy \cdot uy\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]

      5. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(xi, 1 + -2 \cdot \left(\left(uy \cdot uy\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]

      6. pow2 N/A

         \[\leadsto \mathsf{fma}\left(xi, 1 + -2 \cdot \left(\left(uy \cdot uy\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]

      7. lift-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(xi, 1 + -2 \cdot \left(\left(uy \cdot uy\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]

      8. lift-PI.f32 N/A

         \[\leadsto \mathsf{fma}\left(xi, 1 + -2 \cdot \left(\left(uy \cdot uy\right) \cdot \left(\pi \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]

      9. lift-PI.f32 72.1

         \[\leadsto \mathsf{fma}\left(xi, 1 + -2 \cdot \left(\left(uy \cdot uy\right) \cdot \left(\pi \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]

   10. Applied rewrites 72.1%

       \[\leadsto \mathsf{fma}\left(xi, 1 + \color{blue}{-2 \cdot \left(\left(uy \cdot uy\right) \cdot \left(\pi \cdot \pi\right)\right)}, yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 90.6% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.001550000044517219:\\ \;\;\;\;xi + \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), uy \cdot \mathsf{fma}\left(-2, uy \cdot \left(xi \cdot \left(\pi \cdot \pi\right)\right), 2 \cdot \left(yi \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(xi, 1 + -2 \cdot \left(\left(uy \cdot uy\right) \cdot \left(\pi \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (if (<= uy 0.001550000044517219)
   (+
    xi
    (fma
     maxCos
     (* ux (* zi (- 1.0 ux)))
     (* uy (fma -2.0 (* uy (* xi (* PI PI))) (* 2.0 (* yi PI))))))
   (fma
    xi
    (+ 1.0 (* -2.0 (* (* uy uy) (* PI PI))))
    (* yi (sin (* 2.0 (* uy PI)))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.001550000044517219f) {
		tmp = xi + fmaf(maxCos, (ux * (zi * (1.0f - ux))), (uy * fmaf(-2.0f, (uy * (xi * (((float) M_PI) * ((float) M_PI)))), (2.0f * (yi * ((float) M_PI))))));
	} else {
		tmp = fmaf(xi, (1.0f + (-2.0f * ((uy * uy) * (((float) M_PI) * ((float) M_PI))))), (yi * sinf((2.0f * (uy * ((float) M_PI))))));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.001550000044517219))
		tmp = Float32(xi + fma(maxCos, Float32(ux * Float32(zi * Float32(Float32(1.0) - ux))), Float32(uy * fma(Float32(-2.0), Float32(uy * Float32(xi * Float32(Float32(pi) * Float32(pi)))), Float32(Float32(2.0) * Float32(yi * Float32(pi)))))));
	else
		tmp = fma(xi, Float32(Float32(1.0) + Float32(Float32(-2.0) * Float32(Float32(uy * uy) * Float32(Float32(pi) * Float32(pi))))), Float32(yi * sin(Float32(Float32(2.0) * Float32(uy * Float32(pi))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.00155000004

   1. Initial program 99.3%

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

   2. Taylor expanded in uy around 0

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

   4. Applied rewrites 98.6%

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

   5. Taylor expanded in maxCos around 0

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

      1. lower-+.f32 N/A

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

      2. lower-fma.f32 N/A

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

   7. Applied rewrites 98.3%

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

   if 0.00155000004 < uy

   1. Initial program 98.1%

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

   2. Taylor expanded in xi around 0

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

      1. +-commutative N/A

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

   4. Applied rewrites 98.1%

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

   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. lower-fma.f32 N/A

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

      2. lower-cos.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lift-PI.f32 N/A

         \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      6. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      7. lower-sin.f32 N/A

         \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      8. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      9. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      10. lift-PI.f32 90.5

          \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]

   7. Applied rewrites 90.5%

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

   8. Taylor expanded in uy around 0

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

      1. lower-+.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(xi, 1 + -2 \cdot \left(\left(uy \cdot uy\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]

      5. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(xi, 1 + -2 \cdot \left(\left(uy \cdot uy\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]

      6. pow2 N/A

         \[\leadsto \mathsf{fma}\left(xi, 1 + -2 \cdot \left(\left(uy \cdot uy\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]

      7. lift-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(xi, 1 + -2 \cdot \left(\left(uy \cdot uy\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]

      8. lift-PI.f32 N/A

         \[\leadsto \mathsf{fma}\left(xi, 1 + -2 \cdot \left(\left(uy \cdot uy\right) \cdot \left(\pi \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]

      9. lift-PI.f32 72.1

         \[\leadsto \mathsf{fma}\left(xi, 1 + -2 \cdot \left(\left(uy \cdot uy\right) \cdot \left(\pi \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]

   10. Applied rewrites 72.1%

       \[\leadsto \mathsf{fma}\left(xi, 1 + \color{blue}{-2 \cdot \left(\left(uy \cdot uy\right) \cdot \left(\pi \cdot \pi\right)\right)}, yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 88.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\\ \mathbf{if}\;uy \leq 0.04500000178813934:\\ \;\;\;\;xi + \mathsf{fma}\left(maxCos, t\_0, uy \cdot \mathsf{fma}\left(-2, uy \cdot \left(xi \cdot \left(\pi \cdot \pi\right)\right), 2 \cdot \left(yi \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(maxCos, t\_0, yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* ux (* zi (- 1.0 ux)))))
   (if (<= uy 0.04500000178813934)
     (+
      xi
      (fma
       maxCos
       t_0
       (* uy (fma -2.0 (* uy (* xi (* PI PI))) (* 2.0 (* yi PI))))))
     (fma maxCos t_0 (* yi (sin (* 2.0 (* uy PI))))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = ux * (zi * (1.0f - ux));
	float tmp;
	if (uy <= 0.04500000178813934f) {
		tmp = xi + fmaf(maxCos, t_0, (uy * fmaf(-2.0f, (uy * (xi * (((float) M_PI) * ((float) M_PI)))), (2.0f * (yi * ((float) M_PI))))));
	} else {
		tmp = fmaf(maxCos, t_0, (yi * sinf((2.0f * (uy * ((float) M_PI))))));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(ux * Float32(zi * Float32(Float32(1.0) - ux)))
	tmp = Float32(0.0)
	if (uy <= Float32(0.04500000178813934))
		tmp = Float32(xi + fma(maxCos, t_0, Float32(uy * fma(Float32(-2.0), Float32(uy * Float32(xi * Float32(Float32(pi) * Float32(pi)))), Float32(Float32(2.0) * Float32(yi * Float32(pi)))))));
	else
		tmp = fma(maxCos, t_0, Float32(yi * sin(Float32(Float32(2.0) * Float32(uy * Float32(pi))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.0450000018

   1. Initial program 99.2%

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

   2. Taylor expanded in uy around 0

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

   4. Applied rewrites 94.8%

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

   5. Taylor expanded in maxCos around 0

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

      1. lower-+.f32 N/A

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

      2. lower-fma.f32 N/A

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

   7. Applied rewrites 94.6%

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

   if 0.0450000018 < uy

   1. Initial program 97.3%

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

   2. Taylor expanded in xi around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f32 N/A

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

   4. Applied rewrites 49.8%

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

   5. Taylor expanded in maxCos around 0

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

      1. lower-fma.f32 N/A

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

      2. lift--.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      3. lift-*.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      6. lower-sin.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      7. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      8. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      9. lift-PI.f32 49.8

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

   7. Applied rewrites 49.8%

      \[\leadsto \mathsf{fma}\left(maxCos, \color{blue}{ux \cdot \left(zi \cdot \left(1 - ux\right)\right)}, yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 88.1% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

2. Taylor expanded in xi around 0

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

   1. +-commutative N/A

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

4. Applied rewrites 99.0%

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

5. Taylor expanded in uy around 0

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

7. Applied rewrites 88.4%

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

Alternative 13: 87.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.04500000178813934:\\ \;\;\;\;xi + \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), uy \cdot \mathsf{fma}\left(-2, uy \cdot \left(xi \cdot \left(\pi \cdot \pi\right)\right), 2 \cdot \left(yi \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(maxCos, ux \cdot zi, yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (if (<= uy 0.04500000178813934)
   (+
    xi
    (fma
     maxCos
     (* ux (* zi (- 1.0 ux)))
     (* uy (fma -2.0 (* uy (* xi (* PI PI))) (* 2.0 (* yi PI))))))
   (fma maxCos (* ux zi) (* yi (sin (* 2.0 (* uy PI)))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.04500000178813934f) {
		tmp = xi + fmaf(maxCos, (ux * (zi * (1.0f - ux))), (uy * fmaf(-2.0f, (uy * (xi * (((float) M_PI) * ((float) M_PI)))), (2.0f * (yi * ((float) M_PI))))));
	} else {
		tmp = fmaf(maxCos, (ux * zi), (yi * sinf((2.0f * (uy * ((float) M_PI))))));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.04500000178813934))
		tmp = Float32(xi + fma(maxCos, Float32(ux * Float32(zi * Float32(Float32(1.0) - ux))), Float32(uy * fma(Float32(-2.0), Float32(uy * Float32(xi * Float32(Float32(pi) * Float32(pi)))), Float32(Float32(2.0) * Float32(yi * Float32(pi)))))));
	else
		tmp = fma(maxCos, Float32(ux * zi), Float32(yi * sin(Float32(Float32(2.0) * Float32(uy * Float32(pi))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.0450000018

   1. Initial program 99.2%

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

   2. Taylor expanded in uy around 0

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

   4. Applied rewrites 94.8%

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

   5. Taylor expanded in maxCos around 0

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

      1. lower-+.f32 N/A

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

      2. lower-fma.f32 N/A

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

   7. Applied rewrites 94.6%

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

   if 0.0450000018 < uy

   1. Initial program 97.3%

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

   2. Taylor expanded in xi around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f32 N/A

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

   4. Applied rewrites 49.8%

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

   5. Taylor expanded in ux around 0

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

      1. lower-fma.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-sin.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lift-PI.f32 48.3

         \[\leadsto \mathsf{fma}\left(maxCos, ux \cdot zi, yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]

   7. Applied rewrites 48.3%

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

Alternative 14: 85.8% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

2. Taylor expanded in uy around 0

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

4. Applied rewrites 86.0%

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

5. Taylor expanded in maxCos around 0

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

   1. lower-+.f32 N/A

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

   2. lower-fma.f32 N/A

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

7. Applied rewrites 85.8%

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

Alternative 15: 83.1% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

2. Taylor expanded in uy around 0

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

4. Applied rewrites 86.0%

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

5. Taylor expanded in ux around 0

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

   1. lower-+.f32 N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-*.f32 N/A

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

7. Applied rewrites 83.1%

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

Alternative 16: 81.5% accurate, 6.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

2. Taylor expanded in xi around 0

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

   1. +-commutative N/A

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

4. Applied rewrites 99.0%

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

5. Taylor expanded in uy around 0

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

7. Applied rewrites 81.6%

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

8. Taylor expanded in maxCos around 0

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

   1. lower-+.f32 N/A

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

   2. lower-fma.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-PI.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift--.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. lift-*.f32 81.5

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

10. Applied rewrites 81.5%

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

Alternative 17: 78.9% accurate, 7.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

2. Taylor expanded in xi around 0

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

   1. +-commutative N/A

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

4. Applied rewrites 99.0%

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

5. Taylor expanded in uy around 0

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

7. Applied rewrites 81.6%

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

8. Taylor expanded in ux around 0

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

   1. lower-+.f32 N/A

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

   2. lower-fma.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-PI.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-*.f32 78.9

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

10. Applied rewrites 78.9%

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

Alternative 18: 74.1% accurate, 12.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

2. Taylor expanded in xi around 0

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

   1. +-commutative N/A

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

4. Applied rewrites 99.0%

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

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. lower-fma.f32 N/A

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

   2. lower-cos.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lift-PI.f32 N/A

      \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   6. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   7. lower-sin.f32 N/A

      \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   8. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   9. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   10. lift-PI.f32 90.5

       \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]

7. Applied rewrites 90.5%

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

8. Taylor expanded in uy around 0

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

   1. lower-+.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lift-*.f32 N/A

      \[\leadsto xi + 2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right) \]

   5. lift-PI.f32 74.1

      \[\leadsto xi + 2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right) \]

10. Applied rewrites 74.1%

    \[\leadsto xi + \color{blue}{2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right)} \]
11. Add Preprocessing

Alternative 19: 55.3% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right)\\ \mathbf{if}\;yi \leq -9.999999717180685 \cdot 10^{-10}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;yi \leq 2.0000000072549875 \cdot 10^{-15}:\\ \;\;\;\;xi\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* 2.0 (* uy (* yi PI)))))
   (if (<= yi -9.999999717180685e-10)
     t_0
     (if (<= yi 2.0000000072549875e-15) xi t_0))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = 2.0f * (uy * (yi * ((float) M_PI)));
	float tmp;
	if (yi <= -9.999999717180685e-10f) {
		tmp = t_0;
	} else if (yi <= 2.0000000072549875e-15f) {
		tmp = xi;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(2.0) * Float32(uy * Float32(yi * Float32(pi))))
	tmp = Float32(0.0)
	if (yi <= Float32(-9.999999717180685e-10))
		tmp = t_0;
	elseif (yi <= Float32(2.0000000072549875e-15))
		tmp = xi;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(xi, yi, zi, ux, uy, maxCos)
	t_0 = single(2.0) * (uy * (yi * single(pi)));
	tmp = single(0.0);
	if (yi <= single(-9.999999717180685e-10))
		tmp = t_0;
	elseif (yi <= single(2.0000000072549875e-15))
		tmp = xi;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if yi < -9.99999972e-10 or 2.00000001e-15 < yi

   1. Initial program 98.6%

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

   2. Taylor expanded in xi around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f32 N/A

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

   4. Applied rewrites 65.4%

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

   5. Taylor expanded in ux around 0

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

      1. lower-*.f32 N/A

         \[\leadsto yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \]

      2. lower-sin.f32 N/A

         \[\leadsto yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \]

      3. lower-*.f32 N/A

         \[\leadsto yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \]

      4. lower-*.f32 N/A

         \[\leadsto yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \]

      5. lift-PI.f32 63.0

         \[\leadsto yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \]

   7. Applied rewrites 63.0%

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

   8. Taylor expanded in uy around 0

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

      1. lower-*.f32 N/A

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

      2. lift-*.f32 N/A

         \[\leadsto 2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right) \]

      3. lift-PI.f32 N/A

         \[\leadsto 2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right) \]

      4. lift-*.f32 51.8

         \[\leadsto 2 \cdot \left(uy \cdot \left(yi \cdot \pi\right)\right) \]

   10. Applied rewrites 51.8%

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

   if -9.99999972e-10 < yi < 2.00000001e-15

   1. Initial program 99.1%

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

   2. Taylor expanded in xi around 0

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

      1. +-commutative N/A

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

   4. Applied rewrites 99.1%

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

   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. lower-fma.f32 N/A

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

      2. lower-cos.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lift-PI.f32 N/A

         \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      6. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      7. lower-sin.f32 N/A

         \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      8. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      9. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      10. lift-PI.f32 87.5

          \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]

   7. Applied rewrites 87.5%

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

   8. Taylor expanded in uy around 0

      \[\leadsto xi \]
   9. Step-by-step derivation

   10. Applied rewrites 57.5%

       \[\leadsto xi \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 46.4% accurate, 155.4× speedup

Math

\[\begin{array}{l} \\ xi \end{array} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(4) function code(xi, yi, zi, ux, uy, maxcos)
use fmin_fmax_functions
    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
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

2. Taylor expanded in xi around 0

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

   1. +-commutative N/A

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

4. Applied rewrites 99.0%

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

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. lower-fma.f32 N/A

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

   2. lower-cos.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lift-PI.f32 N/A

      \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   6. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   7. lower-sin.f32 N/A

      \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   8. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   9. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   10. lift-PI.f32 90.5

       \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \]

7. Applied rewrites 90.5%

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

8. Taylor expanded in uy around 0

   \[\leadsto xi \]
9. Step-by-step derivation

10. Applied rewrites 46.4%

    \[\leadsto xi \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025110 
(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)))