UniformSampleCone 2

Percentage Accurate: 99.0% → 99.0%

Time: 20.8s

Alternatives: 14

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 99.0% 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, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.0%

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

4. Applied rewrites 99.1%

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

5. Final simplification 99.1%

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

Alternative 2: 98.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in maxCos around 0

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

   1. lower-fma.f32 N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower--.f32 N/A

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

   6. lower-fma.f32 N/A

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

   7. lower-cos.f32 N/A

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

   8. associate-*r* N/A

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

   9. lower-*.f32 N/A

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

   10. lower-*.f32 N/A

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

   11. lower-PI.f32 N/A

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

   12. lower-*.f32 N/A

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

   13. lower-sin.f32 N/A

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

   14. associate-*r* N/A

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

   15. lower-*.f32 N/A

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

   16. lower-*.f32 N/A

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

   17. lower-PI.f32 98.9

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

5. Applied rewrites 98.9%

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

6. Final simplification 98.9%

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

Alternative 3: 95.8% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in ux around 0

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

   1. +-commutative N/A

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

   2. associate-+l+ N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-cos.f32 N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lower-PI.f32 N/A

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

   9. lower-fma.f32 N/A

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

   10. lower-sin.f32 N/A

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

   11. associate-*r* N/A

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

   12. lower-*.f32 N/A

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

   13. lower-*.f32 N/A

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

   14. lower-PI.f32 N/A

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

   15. lower-*.f32 N/A

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

   16. lower-*.f32 95.6

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

5. Applied rewrites 95.6%

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

6. Final simplification 95.6%

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

Alternative 4: 97.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := uy \cdot \left(\pi + \pi\right)\\ t_1 := \sqrt{\mathsf{fma}\left(maxCos, \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right) \cdot \left(maxCos \cdot \left(ux \cdot ux\right)\right), 1\right)}\\ \mathbf{if}\;2 \cdot uy \leq 0.031950000673532486:\\ \;\;\;\;\mathsf{fma}\left(\left(1 - ux\right) \cdot zi, ux \cdot maxCos, \mathsf{fma}\left(uy, \mathsf{fma}\left(2, t\_1 \cdot \left(\pi \cdot yi\right), t\_1 \cdot \left(uy \cdot \mathsf{fma}\left(xi, \left(\pi \cdot \pi\right) \cdot -2, \left(uy \cdot yi\right) \cdot \left(\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot -1.3333333333333333\right)\right)\right)\right), xi \cdot t\_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos t\_0, xi, yi \cdot \sin t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* uy (+ PI PI)))
        (t_1
         (sqrt
          (fma
           maxCos
           (* (* (- 1.0 ux) (+ ux -1.0)) (* maxCos (* ux ux)))
           1.0))))
   (if (<= (* 2.0 uy) 0.031950000673532486)
     (fma
      (* (- 1.0 ux) zi)
      (* ux maxCos)
      (fma
       uy
       (fma
        2.0
        (* t_1 (* PI yi))
        (*
         t_1
         (*
          uy
          (fma
           xi
           (* (* PI PI) -2.0)
           (* (* uy yi) (* (* PI (* PI PI)) -1.3333333333333333))))))
       (* xi t_1)))
     (fma (cos t_0) xi (* yi (sin t_0))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = uy * (((float) M_PI) + ((float) M_PI));
	float t_1 = sqrtf(fmaf(maxCos, (((1.0f - ux) * (ux + -1.0f)) * (maxCos * (ux * ux))), 1.0f));
	float tmp;
	if ((2.0f * uy) <= 0.031950000673532486f) {
		tmp = fmaf(((1.0f - ux) * zi), (ux * maxCos), fmaf(uy, fmaf(2.0f, (t_1 * (((float) M_PI) * yi)), (t_1 * (uy * fmaf(xi, ((((float) M_PI) * ((float) M_PI)) * -2.0f), ((uy * yi) * ((((float) M_PI) * (((float) M_PI) * ((float) M_PI))) * -1.3333333333333333f)))))), (xi * t_1)));
	} else {
		tmp = fmaf(cosf(t_0), xi, (yi * sinf(t_0)));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(uy * Float32(Float32(pi) + Float32(pi)))
	t_1 = sqrt(fma(maxCos, Float32(Float32(Float32(Float32(1.0) - ux) * Float32(ux + Float32(-1.0))) * Float32(maxCos * Float32(ux * ux))), Float32(1.0)))
	tmp = Float32(0.0)
	if (Float32(Float32(2.0) * uy) <= Float32(0.031950000673532486))
		tmp = fma(Float32(Float32(Float32(1.0) - ux) * zi), Float32(ux * maxCos), fma(uy, fma(Float32(2.0), Float32(t_1 * Float32(Float32(pi) * yi)), Float32(t_1 * Float32(uy * fma(xi, Float32(Float32(Float32(pi) * Float32(pi)) * Float32(-2.0)), Float32(Float32(uy * yi) * Float32(Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi))) * Float32(-1.3333333333333333))))))), Float32(xi * t_1)));
	else
		tmp = fma(cos(t_0), xi, Float32(yi * sin(t_0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 0.0319500007

   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. Add Preprocessing

   3. Taylor expanded in uy around 0

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

   5. Applied rewrites 99.2%

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

   7. Applied rewrites 99.4%

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

   if 0.0319500007 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 97.5%

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

   3. Taylor expanded in ux around 0

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

      1. lower-fma.f32 N/A

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

      2. lower-cos.f32 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-PI.f32 N/A

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

      7. lower-*.f32 N/A

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

      8. lower-sin.f32 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f32 N/A

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

      11. lower-*.f32 N/A

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

      12. lower-PI.f32 91.7

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

   5. Applied rewrites 91.7%

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

   7. Applied rewrites 91.7%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.031950000673532486:\\ \;\;\;\;\mathsf{fma}\left(\left(1 - ux\right) \cdot zi, ux \cdot maxCos, \mathsf{fma}\left(uy, \mathsf{fma}\left(2, \sqrt{\mathsf{fma}\left(maxCos, \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right) \cdot \left(maxCos \cdot \left(ux \cdot ux\right)\right), 1\right)} \cdot \left(\pi \cdot yi\right), \sqrt{\mathsf{fma}\left(maxCos, \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right) \cdot \left(maxCos \cdot \left(ux \cdot ux\right)\right), 1\right)} \cdot \left(uy \cdot \mathsf{fma}\left(xi, \left(\pi \cdot \pi\right) \cdot -2, \left(uy \cdot yi\right) \cdot \left(\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot -1.3333333333333333\right)\right)\right)\right), xi \cdot \sqrt{\mathsf{fma}\left(maxCos, \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right) \cdot \left(maxCos \cdot \left(ux \cdot ux\right)\right), 1\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(uy \cdot \left(\pi + \pi\right)\right), xi, yi \cdot \sin \left(uy \cdot \left(\pi + \pi\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := uy \cdot \left(\pi + \pi\right)\\ t_1 := \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)}\\ \mathbf{if}\;2 \cdot uy \leq 0.031950000673532486:\\ \;\;\;\;\mathsf{fma}\left(maxCos, ux \cdot \left(\left(1 - ux\right) \cdot zi\right), \mathsf{fma}\left(uy, \mathsf{fma}\left(uy, t\_1 \cdot \mathsf{fma}\left(-2, xi \cdot \left(\pi \cdot \pi\right), -1.3333333333333333 \cdot \left(uy \cdot \left(yi \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\right), t\_1 \cdot \left(2 \cdot \left(\pi \cdot yi\right)\right)\right), xi \cdot t\_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos t\_0, xi, yi \cdot \sin t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* uy (+ PI PI)))
        (t_1
         (sqrt
          (fma
           (* maxCos maxCos)
           (* (* ux ux) (* (- 1.0 ux) (+ ux -1.0)))
           1.0))))
   (if (<= (* 2.0 uy) 0.031950000673532486)
     (fma
      maxCos
      (* ux (* (- 1.0 ux) zi))
      (fma
       uy
       (fma
        uy
        (*
         t_1
         (fma
          -2.0
          (* xi (* PI PI))
          (* -1.3333333333333333 (* uy (* yi (* PI (* PI PI)))))))
        (* t_1 (* 2.0 (* PI yi))))
       (* xi t_1)))
     (fma (cos t_0) xi (* yi (sin t_0))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = uy * (((float) M_PI) + ((float) M_PI));
	float t_1 = sqrtf(fmaf((maxCos * maxCos), ((ux * ux) * ((1.0f - ux) * (ux + -1.0f))), 1.0f));
	float tmp;
	if ((2.0f * uy) <= 0.031950000673532486f) {
		tmp = fmaf(maxCos, (ux * ((1.0f - ux) * zi)), fmaf(uy, fmaf(uy, (t_1 * fmaf(-2.0f, (xi * (((float) M_PI) * ((float) M_PI))), (-1.3333333333333333f * (uy * (yi * (((float) M_PI) * (((float) M_PI) * ((float) M_PI)))))))), (t_1 * (2.0f * (((float) M_PI) * yi)))), (xi * t_1)));
	} else {
		tmp = fmaf(cosf(t_0), xi, (yi * sinf(t_0)));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(uy * Float32(Float32(pi) + Float32(pi)))
	t_1 = sqrt(fma(Float32(maxCos * maxCos), Float32(Float32(ux * ux) * Float32(Float32(Float32(1.0) - ux) * Float32(ux + Float32(-1.0)))), Float32(1.0)))
	tmp = Float32(0.0)
	if (Float32(Float32(2.0) * uy) <= Float32(0.031950000673532486))
		tmp = fma(maxCos, Float32(ux * Float32(Float32(Float32(1.0) - ux) * zi)), fma(uy, fma(uy, Float32(t_1 * fma(Float32(-2.0), Float32(xi * Float32(Float32(pi) * Float32(pi))), Float32(Float32(-1.3333333333333333) * Float32(uy * Float32(yi * Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi)))))))), Float32(t_1 * Float32(Float32(2.0) * Float32(Float32(pi) * yi)))), Float32(xi * t_1)));
	else
		tmp = fma(cos(t_0), xi, Float32(yi * sin(t_0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 0.0319500007

   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. Add Preprocessing

   3. Taylor expanded in zi around inf

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

      1. lower-*.f32 N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lower--.f32 15.2

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

   5. Applied rewrites 15.2%

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

   6. Taylor expanded in uy around 0

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

   8. Applied rewrites 99.3%

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

   if 0.0319500007 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 97.5%

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

   3. Taylor expanded in ux around 0

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

      1. lower-fma.f32 N/A

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

      2. lower-cos.f32 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-PI.f32 N/A

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

      7. lower-*.f32 N/A

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

      8. lower-sin.f32 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f32 N/A

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

      11. lower-*.f32 N/A

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

      12. lower-PI.f32 91.7

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

   5. Applied rewrites 91.7%

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

   7. Applied rewrites 91.7%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.031950000673532486:\\ \;\;\;\;\mathsf{fma}\left(maxCos, ux \cdot \left(\left(1 - ux\right) \cdot zi\right), \mathsf{fma}\left(uy, \mathsf{fma}\left(uy, \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)} \cdot \mathsf{fma}\left(-2, xi \cdot \left(\pi \cdot \pi\right), -1.3333333333333333 \cdot \left(uy \cdot \left(yi \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\right), \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)} \cdot \left(2 \cdot \left(\pi \cdot yi\right)\right)\right), xi \cdot \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(uy \cdot \left(\pi + \pi\right)\right), xi, yi \cdot \sin \left(uy \cdot \left(\pi + \pi\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := uy \cdot \left(\pi + \pi\right)\\ \mathbf{if}\;2 \cdot uy \leq 0.031950000673532486:\\ \;\;\;\;\mathsf{fma}\left(uy, \mathsf{fma}\left(uy, \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot yi\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), -2 \cdot \left(xi \cdot \left(\pi \cdot \pi\right)\right)\right), 2 \cdot \left(\pi \cdot yi\right)\right), xi\right) + zi \cdot \left(ux \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos t\_0, xi, yi \cdot \sin t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* uy (+ PI PI))))
   (if (<= (* 2.0 uy) 0.031950000673532486)
     (+
      (fma
       uy
       (fma
        uy
        (fma
         -1.3333333333333333
         (* (* uy yi) (* PI (* PI PI)))
         (* -2.0 (* xi (* PI PI))))
        (* 2.0 (* PI yi)))
       xi)
      (* zi (* ux (* (- 1.0 ux) maxCos))))
     (fma (cos t_0) xi (* yi (sin t_0))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = uy * (((float) M_PI) + ((float) M_PI));
	float tmp;
	if ((2.0f * uy) <= 0.031950000673532486f) {
		tmp = fmaf(uy, fmaf(uy, fmaf(-1.3333333333333333f, ((uy * yi) * (((float) M_PI) * (((float) M_PI) * ((float) M_PI)))), (-2.0f * (xi * (((float) M_PI) * ((float) M_PI))))), (2.0f * (((float) M_PI) * yi))), xi) + (zi * (ux * ((1.0f - ux) * maxCos)));
	} else {
		tmp = fmaf(cosf(t_0), xi, (yi * sinf(t_0)));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(uy * Float32(Float32(pi) + Float32(pi)))
	tmp = Float32(0.0)
	if (Float32(Float32(2.0) * uy) <= Float32(0.031950000673532486))
		tmp = Float32(fma(uy, fma(uy, fma(Float32(-1.3333333333333333), Float32(Float32(uy * yi) * Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi)))), Float32(Float32(-2.0) * Float32(xi * Float32(Float32(pi) * Float32(pi))))), Float32(Float32(2.0) * Float32(Float32(pi) * yi))), xi) + Float32(zi * Float32(ux * Float32(Float32(Float32(1.0) - ux) * maxCos))));
	else
		tmp = fma(cos(t_0), xi, Float32(yi * sin(t_0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 0.0319500007

   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. Add Preprocessing

   3. Taylor expanded in uy around 0

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in maxCos around 0

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

   8. Applied rewrites 99.1%

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

   if 0.0319500007 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 97.5%

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

   3. Taylor expanded in ux around 0

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

      1. lower-fma.f32 N/A

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

      2. lower-cos.f32 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-PI.f32 N/A

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

      7. lower-*.f32 N/A

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

      8. lower-sin.f32 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f32 N/A

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

      11. lower-*.f32 N/A

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

      12. lower-PI.f32 91.7

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

   5. Applied rewrites 91.7%

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

   7. Applied rewrites 91.7%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.031950000673532486:\\ \;\;\;\;\mathsf{fma}\left(uy, \mathsf{fma}\left(uy, \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot yi\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), -2 \cdot \left(xi \cdot \left(\pi \cdot \pi\right)\right)\right), 2 \cdot \left(\pi \cdot yi\right)\right), xi\right) + zi \cdot \left(ux \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(uy \cdot \left(\pi + \pi\right)\right), xi, yi \cdot \sin \left(uy \cdot \left(\pi + \pi\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 89.5% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in zi around inf

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

5. Applied rewrites 98.3%

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

6. Taylor expanded in uy around 0

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

8. Applied rewrites 90.3%

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

9. Final simplification 90.3%

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

Alternative 8: 92.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \left(\pi \cdot \pi\right)\\ \mathbf{if}\;2 \cdot uy \leq 0.031950000673532486:\\ \;\;\;\;\mathsf{fma}\left(uy, \mathsf{fma}\left(uy, \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot yi\right) \cdot t\_0, -2 \cdot \left(xi \cdot \left(\pi \cdot \pi\right)\right)\right), 2 \cdot \left(\pi \cdot yi\right)\right), xi\right) + zi \cdot \left(ux \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(xi, \cos \left(\pi \cdot \left(2 \cdot uy\right)\right), yi \cdot \left(uy \cdot \mathsf{fma}\left(-1.3333333333333333 \cdot \left(uy \cdot uy\right), t\_0, 2 \cdot \pi\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* PI (* PI PI))))
   (if (<= (* 2.0 uy) 0.031950000673532486)
     (+
      (fma
       uy
       (fma
        uy
        (fma -1.3333333333333333 (* (* uy yi) t_0) (* -2.0 (* xi (* PI PI))))
        (* 2.0 (* PI yi)))
       xi)
      (* zi (* ux (* (- 1.0 ux) maxCos))))
     (fma
      xi
      (cos (* PI (* 2.0 uy)))
      (* yi (* uy (fma (* -1.3333333333333333 (* uy uy)) t_0 (* 2.0 PI))))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = ((float) M_PI) * (((float) M_PI) * ((float) M_PI));
	float tmp;
	if ((2.0f * uy) <= 0.031950000673532486f) {
		tmp = fmaf(uy, fmaf(uy, fmaf(-1.3333333333333333f, ((uy * yi) * t_0), (-2.0f * (xi * (((float) M_PI) * ((float) M_PI))))), (2.0f * (((float) M_PI) * yi))), xi) + (zi * (ux * ((1.0f - ux) * maxCos)));
	} else {
		tmp = fmaf(xi, cosf((((float) M_PI) * (2.0f * uy))), (yi * (uy * fmaf((-1.3333333333333333f * (uy * uy)), t_0, (2.0f * ((float) M_PI))))));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi)))
	tmp = Float32(0.0)
	if (Float32(Float32(2.0) * uy) <= Float32(0.031950000673532486))
		tmp = Float32(fma(uy, fma(uy, fma(Float32(-1.3333333333333333), Float32(Float32(uy * yi) * t_0), Float32(Float32(-2.0) * Float32(xi * Float32(Float32(pi) * Float32(pi))))), Float32(Float32(2.0) * Float32(Float32(pi) * yi))), xi) + Float32(zi * Float32(ux * Float32(Float32(Float32(1.0) - ux) * maxCos))));
	else
		tmp = fma(xi, cos(Float32(Float32(pi) * Float32(Float32(2.0) * uy))), Float32(yi * Float32(uy * fma(Float32(Float32(-1.3333333333333333) * Float32(uy * uy)), t_0, Float32(Float32(2.0) * Float32(pi))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 0.0319500007

   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. Add Preprocessing

   3. Taylor expanded in uy around 0

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in maxCos around 0

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

   8. Applied rewrites 99.1%

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

   if 0.0319500007 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 97.5%

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

   3. Taylor expanded in ux around 0

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

      1. lower-fma.f32 N/A

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

      2. lower-cos.f32 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-PI.f32 N/A

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

      7. lower-*.f32 N/A

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

      8. lower-sin.f32 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f32 N/A

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

      11. lower-*.f32 N/A

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

      12. lower-PI.f32 91.7

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

   5. Applied rewrites 91.7%

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

   6. Taylor expanded in uy around 0

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

   8. Applied rewrites 68.4%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.031950000673532486:\\ \;\;\;\;\mathsf{fma}\left(uy, \mathsf{fma}\left(uy, \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot yi\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), -2 \cdot \left(xi \cdot \left(\pi \cdot \pi\right)\right)\right), 2 \cdot \left(\pi \cdot yi\right)\right), xi\right) + zi \cdot \left(ux \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(xi, \cos \left(\pi \cdot \left(2 \cdot uy\right)\right), yi \cdot \left(uy \cdot \mathsf{fma}\left(-1.3333333333333333 \cdot \left(uy \cdot uy\right), \pi \cdot \left(\pi \cdot \pi\right), 2 \cdot \pi\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 86.7% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.0006000000284984708:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)}, \mathsf{fma}\left(2, uy \cdot \left(\pi \cdot yi\right), xi\right), maxCos \cdot \left(ux \cdot \left(\left(1 - ux\right) \cdot zi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(uy, \mathsf{fma}\left(uy, \mathsf{fma}\left(-2, xi \cdot \left(\pi \cdot \pi\right), -1.3333333333333333 \cdot \left(\left(uy \cdot yi\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right)\right), 2 \cdot \left(\pi \cdot yi\right)\right), xi\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (if (<= (* 2.0 uy) 0.0006000000284984708)
   (fma
    (sqrt (fma (* maxCos maxCos) (* (* ux ux) (* (- 1.0 ux) (+ ux -1.0))) 1.0))
    (fma 2.0 (* uy (* PI yi)) xi)
    (* maxCos (* ux (* (- 1.0 ux) zi))))
   (fma
    uy
    (fma
     uy
     (fma
      -2.0
      (* xi (* PI PI))
      (* -1.3333333333333333 (* (* uy yi) (* PI (* PI PI)))))
     (* 2.0 (* PI yi)))
    xi)))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float tmp;
	if ((2.0f * uy) <= 0.0006000000284984708f) {
		tmp = fmaf(sqrtf(fmaf((maxCos * maxCos), ((ux * ux) * ((1.0f - ux) * (ux + -1.0f))), 1.0f)), fmaf(2.0f, (uy * (((float) M_PI) * yi)), xi), (maxCos * (ux * ((1.0f - ux) * zi))));
	} else {
		tmp = fmaf(uy, fmaf(uy, fmaf(-2.0f, (xi * (((float) M_PI) * ((float) M_PI))), (-1.3333333333333333f * ((uy * yi) * (((float) M_PI) * (((float) M_PI) * ((float) M_PI)))))), (2.0f * (((float) M_PI) * yi))), xi);
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	tmp = Float32(0.0)
	if (Float32(Float32(2.0) * uy) <= Float32(0.0006000000284984708))
		tmp = fma(sqrt(fma(Float32(maxCos * maxCos), Float32(Float32(ux * ux) * Float32(Float32(Float32(1.0) - ux) * Float32(ux + Float32(-1.0)))), Float32(1.0))), fma(Float32(2.0), Float32(uy * Float32(Float32(pi) * yi)), xi), Float32(maxCos * Float32(ux * Float32(Float32(Float32(1.0) - ux) * zi))));
	else
		tmp = fma(uy, fma(uy, fma(Float32(-2.0), Float32(xi * Float32(Float32(pi) * Float32(pi))), Float32(Float32(-1.3333333333333333) * Float32(Float32(uy * yi) * Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi)))))), Float32(Float32(2.0) * Float32(Float32(pi) * yi))), xi);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 6.00000028e-4

   1. Initial program 99.4%

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

   3. Taylor expanded in zi around inf

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

      1. lower-*.f32 N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lower--.f32 15.8

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

   5. Applied rewrites 15.8%

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

   6. Taylor expanded in uy around 0

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

   8. Applied rewrites 98.7%

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

   if 6.00000028e-4 < (*.f32 uy #s(literal 2 binary32))

   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. Add Preprocessing

   3. Taylor expanded in ux around 0

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

      1. lower-fma.f32 N/A

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

      2. lower-cos.f32 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-PI.f32 N/A

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

      7. lower-*.f32 N/A

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

      8. lower-sin.f32 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f32 N/A

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

      11. lower-*.f32 N/A

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

      12. lower-PI.f32 91.0

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

   5. Applied rewrites 91.0%

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

   6. Taylor expanded in uy around 0

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

   8. Applied rewrites 66.5%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.0006000000284984708:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)}, \mathsf{fma}\left(2, uy \cdot \left(\pi \cdot yi\right), xi\right), maxCos \cdot \left(ux \cdot \left(\left(1 - ux\right) \cdot zi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(uy, \mathsf{fma}\left(uy, \mathsf{fma}\left(-2, xi \cdot \left(\pi \cdot \pi\right), -1.3333333333333333 \cdot \left(\left(uy \cdot yi\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right)\right), 2 \cdot \left(\pi \cdot yi\right)\right), xi\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 89.2% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in uy around 0

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

5. Applied rewrites 89.8%

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

6. Taylor expanded in maxCos around 0

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

8. Applied rewrites 89.8%

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

9. Final simplification 89.8%

   \[\leadsto \mathsf{fma}\left(uy, \mathsf{fma}\left(uy, \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot yi\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), -2 \cdot \left(xi \cdot \left(\pi \cdot \pi\right)\right)\right), 2 \cdot \left(\pi \cdot yi\right)\right), xi\right) + zi \cdot \left(ux \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right) \]
10. Add Preprocessing

Alternative 11: 81.4% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(uy, \mathsf{fma}\left(uy, \mathsf{fma}\left(-2, xi \cdot \left(\pi \cdot \pi\right), -1.3333333333333333 \cdot \left(\left(uy \cdot yi\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right)\right), 2 \cdot \left(\pi \cdot yi\right)\right), xi\right) \end{array} \]

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in ux around 0

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

   1. lower-fma.f32 N/A

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

   2. lower-cos.f32 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-PI.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lower-sin.f32 N/A

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

   9. associate-*r* N/A

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

   10. lower-*.f32 N/A

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

   11. lower-*.f32 N/A

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

   12. lower-PI.f32 88.8

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

5. Applied rewrites 88.8%

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

6. Taylor expanded in uy around 0

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

8. Applied rewrites 80.2%

   \[\leadsto \mathsf{fma}\left(uy, \color{blue}{\mathsf{fma}\left(uy, \mathsf{fma}\left(-2, xi \cdot \left(\pi \cdot \pi\right), -1.3333333333333333 \cdot \left(\left(uy \cdot yi\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right)\right), 2 \cdot \left(\pi \cdot yi\right)\right)}, xi\right) \]
9. Add Preprocessing

Alternative 12: 78.0% accurate, 9.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in ux around 0

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

   1. lower-fma.f32 N/A

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

   2. lower-cos.f32 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-PI.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lower-sin.f32 N/A

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

   9. associate-*r* N/A

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

   10. lower-*.f32 N/A

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

   11. lower-*.f32 N/A

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

   12. lower-PI.f32 88.8

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

5. Applied rewrites 88.8%

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

6. Taylor expanded in uy around 0

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

8. Applied rewrites 77.3%

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

9. Final simplification 77.3%

   \[\leadsto \mathsf{fma}\left(uy, \mathsf{fma}\left(uy \cdot -2, xi \cdot \left(\pi \cdot \pi\right), 2 \cdot \left(\pi \cdot yi\right)\right), xi\right) \]
10. Add Preprocessing

Alternative 13: 74.1% accurate, 20.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in ux around 0

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

   1. lower-fma.f32 N/A

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

   2. lower-cos.f32 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-PI.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lower-sin.f32 N/A

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

   9. associate-*r* N/A

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

   10. lower-*.f32 N/A

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

   11. lower-*.f32 N/A

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

   12. lower-PI.f32 88.8

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

5. Applied rewrites 88.8%

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

6. Taylor expanded in uy around 0

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

8. Applied rewrites 73.2%

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

Alternative 14: 45.0% accurate, 58.8× speedup

Math

\[\begin{array}{l} \\ xi \cdot 1 \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in yi around 0

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

   1. +-commutative N/A

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

   2. associate-*l* N/A

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

   3. lower-fma.f32 N/A

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

5. Applied rewrites 61.9%

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

6. Taylor expanded in maxCos around 0

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

8. Applied rewrites 53.8%

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

9. Taylor expanded in uy around 0

   \[\leadsto xi \cdot 1 \]
10. Step-by-step derivation

11. Applied rewrites 46.7%

    \[\leadsto xi \cdot 1 \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024221 
(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)))