UniformSampleCone 2

Percentage Accurate: 98.9% → 98.9%

Time: 24.6s

Alternatives: 20

Speedup: N/A×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

4. Applied rewrites 99.0%

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

5. Taylor expanded in ux around 0

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

   1. lower-sin.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 99.0

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

7. Applied rewrites 99.0%

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

8. Taylor expanded in ux around 0

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

   1. sub-neg N/A

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

   2. mul-1-neg N/A

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

   3. lower-fma.f32 N/A

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

   4. mul-1-neg N/A

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

   5. lower-neg.f32 99.0

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

10. Applied rewrites 99.0%

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

11. Final simplification 99.0%

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

Alternative 2: 94.9% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* 2.0 (* uy PI))))
   (fma yi (fma xi (/ (cos t_0) 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 = 2.0f * (uy * ((float) M_PI));
	return fmaf(yi, fmaf(xi, (cosf(t_0) / yi), sinf(t_0)), (maxCos * (ux * zi)));
}

Julia

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in yi around inf

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

5. Applied rewrites 98.4%

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

6. Taylor expanded in ux around 0

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

8. Applied rewrites 95.9%

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

Alternative 3: 97.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \left(2 \cdot uy\right)\\ t_1 := \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)}\\ \mathbf{if}\;2 \cdot uy \leq 0.02500000037252903:\\ \;\;\;\;\mathsf{fma}\left(uy, \mathsf{fma}\left(uy, t\_1 \cdot \mathsf{fma}\left(-1.3333333333333333, uy \cdot \left(yi \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right), -2 \cdot \left(xi \cdot \left(\pi \cdot \pi\right)\right)\right), t\_1 \cdot \left(2 \cdot \left(\pi \cdot yi\right)\right)\right), \mathsf{fma}\left(xi, t\_1, maxCos \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot zi\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(xi, \cos t\_0, yi \cdot \sin t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* PI (* 2.0 uy)))
        (t_1
         (sqrt
          (fma
           (* maxCos maxCos)
           (* (* ux ux) (* (- 1.0 ux) (+ ux -1.0)))
           1.0))))
   (if (<= (* 2.0 uy) 0.02500000037252903)
     (fma
      uy
      (fma
       uy
       (*
        t_1
        (fma
         -1.3333333333333333
         (* uy (* yi (* PI (* PI PI))))
         (* -2.0 (* xi (* PI PI)))))
       (* t_1 (* 2.0 (* PI yi))))
      (fma xi t_1 (* 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);
	float t_1 = sqrtf(fmaf((maxCos * maxCos), ((ux * ux) * ((1.0f - ux) * (ux + -1.0f))), 1.0f));
	float tmp;
	if ((2.0f * uy) <= 0.02500000037252903f) {
		tmp = fmaf(uy, fmaf(uy, (t_1 * fmaf(-1.3333333333333333f, (uy * (yi * (((float) M_PI) * (((float) M_PI) * ((float) M_PI))))), (-2.0f * (xi * (((float) M_PI) * ((float) M_PI)))))), (t_1 * (2.0f * (((float) M_PI) * yi)))), fmaf(xi, t_1, (maxCos * ((1.0f - ux) * (ux * zi)))));
	} else {
		tmp = fmaf(xi, cosf(t_0), (yi * sinf(t_0)));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(pi) * Float32(Float32(2.0) * uy))
	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(0.02500000037252903))
		tmp = fma(uy, fma(uy, Float32(t_1 * fma(Float32(-1.3333333333333333), Float32(uy * Float32(yi * Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi))))), Float32(Float32(-2.0) * Float32(xi * Float32(Float32(pi) * Float32(pi)))))), Float32(t_1 * Float32(Float32(2.0) * Float32(Float32(pi) * yi)))), fma(xi, t_1, Float32(maxCos * Float32(Float32(Float32(1.0) - ux) * Float32(ux * zi)))));
	else
		tmp = fma(xi, cos(t_0), Float32(yi * sin(t_0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 0.0250000004

   1. Initial program 99.1%

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

   3. Taylor expanded in uy around 0

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

   5. Applied rewrites 99.1%

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

   if 0.0250000004 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 97.9%

      \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
   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 90.2

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

   5. Applied rewrites 90.2%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.02500000037252903:\\ \;\;\;\;\mathsf{fma}\left(uy, \mathsf{fma}\left(uy, \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)} \cdot \mathsf{fma}\left(-1.3333333333333333, uy \cdot \left(yi \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right), -2 \cdot \left(xi \cdot \left(\pi \cdot \pi\right)\right)\right), \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)} \cdot \left(2 \cdot \left(\pi \cdot yi\right)\right)\right), \mathsf{fma}\left(xi, \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)}, maxCos \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot zi\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(xi, \cos \left(\pi \cdot \left(2 \cdot uy\right)\right), yi \cdot \sin \left(\pi \cdot \left(2 \cdot uy\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.2% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in uy around 0

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

5. Applied rewrites 92.2%

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

6. Final simplification 92.2%

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

Alternative 5: 90.3% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

4. Applied rewrites 99.0%

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

5. Taylor expanded in ux around 0

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

   1. lower-sin.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 99.0

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

7. Applied rewrites 99.0%

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

8. Taylor expanded in uy around 0

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

10. Applied rewrites 91.6%

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

11. Final simplification 91.6%

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

Alternative 6: 88.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)}\\ t_1 := ux \cdot \left(maxCos \cdot \left(1 - ux\right)\right)\\ \mathbf{if}\;t\_1 \leq 1.000000045813705 \cdot 10^{-18}:\\ \;\;\;\;\mathsf{fma}\left(xi, \mathsf{fma}\left(-2, \left(uy \cdot \pi\right) \cdot \left(uy \cdot \pi\right), 1\right), yi \cdot \sin \left(\pi \cdot \left(2 \cdot uy\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(uy, t\_0 \cdot \mathsf{fma}\left(-2, uy \cdot \left(xi \cdot \left(\pi \cdot \pi\right)\right), 2 \cdot \left(\pi \cdot yi\right)\right), xi \cdot t\_0\right) + zi \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0
         (sqrt
          (fma
           (* maxCos maxCos)
           (* (* ux ux) (* (- 1.0 ux) (+ ux -1.0)))
           1.0)))
        (t_1 (* ux (* maxCos (- 1.0 ux)))))
   (if (<= t_1 1.000000045813705e-18)
     (fma
      xi
      (fma -2.0 (* (* uy PI) (* uy PI)) 1.0)
      (* yi (sin (* PI (* 2.0 uy)))))
     (+
      (fma
       uy
       (* t_0 (fma -2.0 (* uy (* xi (* PI PI))) (* 2.0 (* PI yi))))
       (* xi t_0))
      (* zi t_1)))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = sqrtf(fmaf((maxCos * maxCos), ((ux * ux) * ((1.0f - ux) * (ux + -1.0f))), 1.0f));
	float t_1 = ux * (maxCos * (1.0f - ux));
	float tmp;
	if (t_1 <= 1.000000045813705e-18f) {
		tmp = fmaf(xi, fmaf(-2.0f, ((uy * ((float) M_PI)) * (uy * ((float) M_PI))), 1.0f), (yi * sinf((((float) M_PI) * (2.0f * uy)))));
	} else {
		tmp = fmaf(uy, (t_0 * fmaf(-2.0f, (uy * (xi * (((float) M_PI) * ((float) M_PI)))), (2.0f * (((float) M_PI) * yi)))), (xi * t_0)) + (zi * t_1);
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = sqrt(fma(Float32(maxCos * maxCos), Float32(Float32(ux * ux) * Float32(Float32(Float32(1.0) - ux) * Float32(ux + Float32(-1.0)))), Float32(1.0)))
	t_1 = Float32(ux * Float32(maxCos * Float32(Float32(1.0) - ux)))
	tmp = Float32(0.0)
	if (t_1 <= Float32(1.000000045813705e-18))
		tmp = fma(xi, fma(Float32(-2.0), Float32(Float32(uy * Float32(pi)) * Float32(uy * Float32(pi))), Float32(1.0)), Float32(yi * sin(Float32(Float32(pi) * Float32(Float32(2.0) * uy)))));
	else
		tmp = Float32(fma(uy, Float32(t_0 * fma(Float32(-2.0), Float32(uy * Float32(xi * Float32(Float32(pi) * Float32(pi)))), Float32(Float32(2.0) * Float32(Float32(pi) * yi)))), Float32(xi * t_0)) + Float32(zi * t_1));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 (-.f32 #s(literal 1 binary32) ux) maxCos) ux) < 1.00000005e-18

   1. Initial program 98.9%

      \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
   2. 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 97.0

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

   5. Applied rewrites 97.0%

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

   6. Taylor expanded in uy around 0

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

   8. Applied rewrites 94.2%

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

   if 1.00000005e-18 < (*.f32 (*.f32 (-.f32 #s(literal 1 binary32) ux) maxCos) ux)

   1. Initial program 98.9%

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

   3. Taylor expanded in uy around 0

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

      1. lower-fma.f32 N/A

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

   5. Applied rewrites 91.1%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \cdot \left(maxCos \cdot \left(1 - ux\right)\right) \leq 1.000000045813705 \cdot 10^{-18}:\\ \;\;\;\;\mathsf{fma}\left(xi, \mathsf{fma}\left(-2, \left(uy \cdot \pi\right) \cdot \left(uy \cdot \pi\right), 1\right), yi \cdot \sin \left(\pi \cdot \left(2 \cdot uy\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(uy, \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)} \cdot \mathsf{fma}\left(-2, uy \cdot \left(xi \cdot \left(\pi \cdot \pi\right)\right), 2 \cdot \left(\pi \cdot yi\right)\right), xi \cdot \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)}\right) + zi \cdot \left(ux \cdot \left(maxCos \cdot \left(1 - ux\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 88.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)}\\ \mathbf{if}\;ux \cdot \left(maxCos \cdot \left(1 - ux\right)\right) \leq 1.000000045813705 \cdot 10^{-18}:\\ \;\;\;\;\mathsf{fma}\left(xi, \mathsf{fma}\left(-2, \left(uy \cdot \pi\right) \cdot \left(uy \cdot \pi\right), 1\right), yi \cdot \sin \left(\pi \cdot \left(2 \cdot uy\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(xi, t\_0, \mathsf{fma}\left(uy, t\_0 \cdot \mathsf{fma}\left(-2, uy \cdot \left(xi \cdot \left(\pi \cdot \pi\right)\right), 2 \cdot \left(\pi \cdot yi\right)\right), maxCos \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot zi\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0
         (sqrt
          (fma
           (* maxCos maxCos)
           (* (* ux ux) (* (- 1.0 ux) (+ ux -1.0)))
           1.0))))
   (if (<= (* ux (* maxCos (- 1.0 ux))) 1.000000045813705e-18)
     (fma
      xi
      (fma -2.0 (* (* uy PI) (* uy PI)) 1.0)
      (* yi (sin (* PI (* 2.0 uy)))))
     (fma
      xi
      t_0
      (fma
       uy
       (* t_0 (fma -2.0 (* uy (* xi (* PI PI))) (* 2.0 (* PI yi))))
       (* maxCos (* (- 1.0 ux) (* ux zi))))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = sqrtf(fmaf((maxCos * maxCos), ((ux * ux) * ((1.0f - ux) * (ux + -1.0f))), 1.0f));
	float tmp;
	if ((ux * (maxCos * (1.0f - ux))) <= 1.000000045813705e-18f) {
		tmp = fmaf(xi, fmaf(-2.0f, ((uy * ((float) M_PI)) * (uy * ((float) M_PI))), 1.0f), (yi * sinf((((float) M_PI) * (2.0f * uy)))));
	} else {
		tmp = fmaf(xi, t_0, fmaf(uy, (t_0 * fmaf(-2.0f, (uy * (xi * (((float) M_PI) * ((float) M_PI)))), (2.0f * (((float) M_PI) * yi)))), (maxCos * ((1.0f - ux) * (ux * zi)))));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = 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(ux * Float32(maxCos * Float32(Float32(1.0) - ux))) <= Float32(1.000000045813705e-18))
		tmp = fma(xi, fma(Float32(-2.0), Float32(Float32(uy * Float32(pi)) * Float32(uy * Float32(pi))), Float32(1.0)), Float32(yi * sin(Float32(Float32(pi) * Float32(Float32(2.0) * uy)))));
	else
		tmp = fma(xi, t_0, fma(uy, Float32(t_0 * fma(Float32(-2.0), Float32(uy * Float32(xi * Float32(Float32(pi) * Float32(pi)))), Float32(Float32(2.0) * Float32(Float32(pi) * yi)))), Float32(maxCos * Float32(Float32(Float32(1.0) - ux) * Float32(ux * zi)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 (-.f32 #s(literal 1 binary32) ux) maxCos) ux) < 1.00000005e-18

   1. Initial program 98.9%

      \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
   2. 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 97.0

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

   5. Applied rewrites 97.0%

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

   6. Taylor expanded in uy around 0

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

   8. Applied rewrites 94.2%

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

   if 1.00000005e-18 < (*.f32 (*.f32 (-.f32 #s(literal 1 binary32) ux) maxCos) ux)

   1. Initial program 98.9%

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

   3. Taylor expanded in uy around 0

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

   5. Applied rewrites 91.0%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \cdot \left(maxCos \cdot \left(1 - ux\right)\right) \leq 1.000000045813705 \cdot 10^{-18}:\\ \;\;\;\;\mathsf{fma}\left(xi, \mathsf{fma}\left(-2, \left(uy \cdot \pi\right) \cdot \left(uy \cdot \pi\right), 1\right), yi \cdot \sin \left(\pi \cdot \left(2 \cdot uy\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(xi, \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)}, \mathsf{fma}\left(uy, \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)} \cdot \mathsf{fma}\left(-2, uy \cdot \left(xi \cdot \left(\pi \cdot \pi\right)\right), 2 \cdot \left(\pi \cdot yi\right)\right), maxCos \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot zi\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 90.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.00044999999227002263:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)} \cdot \mathsf{fma}\left(2, uy \cdot \left(\pi \cdot yi\right), xi\right) - zi \cdot \left(ux \cdot \left(maxCos \cdot \left(ux + -1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(xi, \mathsf{fma}\left(-2, \left(uy \cdot \pi\right) \cdot \left(uy \cdot \pi\right), 1\right), yi \cdot \sin \left(\pi \cdot \left(2 \cdot uy\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (if (<= (* 2.0 uy) 0.00044999999227002263)
   (-
    (*
     (sqrt
      (fma (* maxCos maxCos) (* (* ux ux) (* (- 1.0 ux) (+ ux -1.0))) 1.0))
     (fma 2.0 (* uy (* PI yi)) xi))
    (* zi (* ux (* maxCos (+ ux -1.0)))))
   (fma
    xi
    (fma -2.0 (* (* uy PI) (* uy PI)) 1.0)
    (* yi (sin (* PI (* 2.0 uy)))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float tmp;
	if ((2.0f * uy) <= 0.00044999999227002263f) {
		tmp = (sqrtf(fmaf((maxCos * maxCos), ((ux * ux) * ((1.0f - ux) * (ux + -1.0f))), 1.0f)) * fmaf(2.0f, (uy * (((float) M_PI) * yi)), xi)) - (zi * (ux * (maxCos * (ux + -1.0f))));
	} else {
		tmp = fmaf(xi, fmaf(-2.0f, ((uy * ((float) M_PI)) * (uy * ((float) M_PI))), 1.0f), (yi * sinf((((float) M_PI) * (2.0f * uy)))));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	tmp = Float32(0.0)
	if (Float32(Float32(2.0) * uy) <= Float32(0.00044999999227002263))
		tmp = Float32(Float32(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(zi * Float32(ux * Float32(maxCos * Float32(ux + Float32(-1.0))))));
	else
		tmp = fma(xi, fma(Float32(-2.0), Float32(Float32(uy * Float32(pi)) * Float32(uy * Float32(pi))), Float32(1.0)), Float32(yi * sin(Float32(Float32(pi) * Float32(Float32(2.0) * uy)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 4.49999992e-4

   1. Initial program 99.2%

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

   3. Taylor expanded in uy around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt-out N/A

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

      3. lower-*.f32 N/A

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

   5. Applied rewrites 99.0%

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

   if 4.49999992e-4 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 98.5%

      \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
   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 89.5

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

   5. Applied rewrites 89.5%

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

   6. Taylor expanded in uy around 0

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

   8. Applied rewrites 81.0%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.00044999999227002263:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)} \cdot \mathsf{fma}\left(2, uy \cdot \left(\pi \cdot yi\right), xi\right) - zi \cdot \left(ux \cdot \left(maxCos \cdot \left(ux + -1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(xi, \mathsf{fma}\left(-2, \left(uy \cdot \pi\right) \cdot \left(uy \cdot \pi\right), 1\right), yi \cdot \sin \left(\pi \cdot \left(2 \cdot uy\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 86.6% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.00044999999227002263:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)} \cdot \mathsf{fma}\left(2, uy \cdot \left(\pi \cdot yi\right), xi\right) - zi \cdot \left(ux \cdot \left(maxCos \cdot \left(ux + -1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(uy, \mathsf{fma}\left(uy, \mathsf{fma}\left(-1.3333333333333333, \left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(uy \cdot yi\right), -2 \cdot \left(xi \cdot \left(\pi \cdot \pi\right)\right)\right), 2 \cdot \left(\pi \cdot yi\right)\right), xi\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (if (<= (* 2.0 uy) 0.00044999999227002263)
   (-
    (*
     (sqrt
      (fma (* maxCos maxCos) (* (* ux ux) (* (- 1.0 ux) (+ ux -1.0))) 1.0))
     (fma 2.0 (* uy (* PI yi)) xi))
    (* zi (* ux (* maxCos (+ ux -1.0)))))
   (fma
    uy
    (fma
     uy
     (fma
      -1.3333333333333333
      (* (* PI (* PI PI)) (* uy yi))
      (* -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) {
	float tmp;
	if ((2.0f * uy) <= 0.00044999999227002263f) {
		tmp = (sqrtf(fmaf((maxCos * maxCos), ((ux * ux) * ((1.0f - ux) * (ux + -1.0f))), 1.0f)) * fmaf(2.0f, (uy * (((float) M_PI) * yi)), xi)) - (zi * (ux * (maxCos * (ux + -1.0f))));
	} else {
		tmp = fmaf(uy, fmaf(uy, fmaf(-1.3333333333333333f, ((((float) M_PI) * (((float) M_PI) * ((float) M_PI))) * (uy * yi)), (-2.0f * (xi * (((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(0.00044999999227002263))
		tmp = Float32(Float32(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(zi * Float32(ux * Float32(maxCos * Float32(ux + Float32(-1.0))))));
	else
		tmp = fma(uy, fma(uy, fma(Float32(-1.3333333333333333), Float32(Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi))) * Float32(uy * yi)), Float32(Float32(-2.0) * Float32(xi * 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)) < 4.49999992e-4

   1. Initial program 99.2%

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

   3. Taylor expanded in uy around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt-out N/A

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

      3. lower-*.f32 N/A

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

   5. Applied rewrites 99.0%

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

   if 4.49999992e-4 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 98.5%

      \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
   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 89.5

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

   5. Applied rewrites 89.5%

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

   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 72.7%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.00044999999227002263:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)} \cdot \mathsf{fma}\left(2, uy \cdot \left(\pi \cdot yi\right), xi\right) - zi \cdot \left(ux \cdot \left(maxCos \cdot \left(ux + -1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(uy, \mathsf{fma}\left(uy, \mathsf{fma}\left(-1.3333333333333333, \left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(uy \cdot yi\right), -2 \cdot \left(xi \cdot \left(\pi \cdot \pi\right)\right)\right), 2 \cdot \left(\pi \cdot yi\right)\right), xi\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 86.6% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.00044999999227002263:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)}, \mathsf{fma}\left(2, uy \cdot \left(\pi \cdot yi\right), xi\right), maxCos \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot zi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(uy, \mathsf{fma}\left(uy, \mathsf{fma}\left(-1.3333333333333333, \left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(uy \cdot yi\right), -2 \cdot \left(xi \cdot \left(\pi \cdot \pi\right)\right)\right), 2 \cdot \left(\pi \cdot yi\right)\right), xi\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (if (<= (* 2.0 uy) 0.00044999999227002263)
   (fma
    (sqrt (fma (* maxCos maxCos) (* (* ux ux) (* (- 1.0 ux) (+ ux -1.0))) 1.0))
    (fma 2.0 (* uy (* PI yi)) xi)
    (* maxCos (* (- 1.0 ux) (* ux zi))))
   (fma
    uy
    (fma
     uy
     (fma
      -1.3333333333333333
      (* (* PI (* PI PI)) (* uy yi))
      (* -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) {
	float tmp;
	if ((2.0f * uy) <= 0.00044999999227002263f) {
		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 * ((1.0f - ux) * (ux * zi))));
	} else {
		tmp = fmaf(uy, fmaf(uy, fmaf(-1.3333333333333333f, ((((float) M_PI) * (((float) M_PI) * ((float) M_PI))) * (uy * yi)), (-2.0f * (xi * (((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(0.00044999999227002263))
		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(Float32(Float32(1.0) - ux) * Float32(ux * zi))));
	else
		tmp = fma(uy, fma(uy, fma(Float32(-1.3333333333333333), Float32(Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi))) * Float32(uy * yi)), Float32(Float32(-2.0) * Float32(xi * 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)) < 4.49999992e-4

   1. Initial program 99.2%

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

   3. Taylor expanded in uy around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

   5. Applied rewrites 99.0%

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

   if 4.49999992e-4 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 98.5%

      \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
   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 89.5

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

   5. Applied rewrites 89.5%

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

   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 72.7%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.00044999999227002263:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)}, \mathsf{fma}\left(2, uy \cdot \left(\pi \cdot yi\right), xi\right), maxCos \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot zi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(uy, \mathsf{fma}\left(uy, \mathsf{fma}\left(-1.3333333333333333, \left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(uy \cdot yi\right), -2 \cdot \left(xi \cdot \left(\pi \cdot \pi\right)\right)\right), 2 \cdot \left(\pi \cdot yi\right)\right), xi\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 86.5% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.00044999999227002263:\\ \;\;\;\;xi + \mathsf{fma}\left(ux \cdot maxCos, \left(1 - ux\right) \cdot zi, \left(\pi \cdot yi\right) \cdot \left(2 \cdot uy\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(uy, \mathsf{fma}\left(uy, \mathsf{fma}\left(-1.3333333333333333, \left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(uy \cdot yi\right), -2 \cdot \left(xi \cdot \left(\pi \cdot \pi\right)\right)\right), 2 \cdot \left(\pi \cdot yi\right)\right), xi\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (if (<= (* 2.0 uy) 0.00044999999227002263)
   (+ xi (fma (* ux maxCos) (* (- 1.0 ux) zi) (* (* PI yi) (* 2.0 uy))))
   (fma
    uy
    (fma
     uy
     (fma
      -1.3333333333333333
      (* (* PI (* PI PI)) (* uy yi))
      (* -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) {
	float tmp;
	if ((2.0f * uy) <= 0.00044999999227002263f) {
		tmp = xi + fmaf((ux * maxCos), ((1.0f - ux) * zi), ((((float) M_PI) * yi) * (2.0f * uy)));
	} else {
		tmp = fmaf(uy, fmaf(uy, fmaf(-1.3333333333333333f, ((((float) M_PI) * (((float) M_PI) * ((float) M_PI))) * (uy * yi)), (-2.0f * (xi * (((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(0.00044999999227002263))
		tmp = Float32(xi + fma(Float32(ux * maxCos), Float32(Float32(Float32(1.0) - ux) * zi), Float32(Float32(Float32(pi) * yi) * Float32(Float32(2.0) * uy))));
	else
		tmp = fma(uy, fma(uy, fma(Float32(-1.3333333333333333), Float32(Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi))) * Float32(uy * yi)), Float32(Float32(-2.0) * Float32(xi * 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)) < 4.49999992e-4

   1. Initial program 99.2%

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

   3. Taylor expanded in uy around 0

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

   5. Applied rewrites 99.0%

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

   6. Taylor expanded in maxCos around 0

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

   8. Applied rewrites 98.7%

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

   if 4.49999992e-4 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 98.5%

      \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
   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 89.5

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

   5. Applied rewrites 89.5%

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

   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 72.7%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.00044999999227002263:\\ \;\;\;\;xi + \mathsf{fma}\left(ux \cdot maxCos, \left(1 - ux\right) \cdot zi, \left(\pi \cdot yi\right) \cdot \left(2 \cdot uy\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(uy, \mathsf{fma}\left(uy, \mathsf{fma}\left(-1.3333333333333333, \left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(uy \cdot yi\right), -2 \cdot \left(xi \cdot \left(\pi \cdot \pi\right)\right)\right), 2 \cdot \left(\pi \cdot yi\right)\right), xi\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 81.5% accurate, 8.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in uy around 0

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

   1. +-commutative N/A

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

   2. associate-+r+ N/A

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

   3. associate-*r* N/A

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

   4. distribute-rgt-out N/A

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

5. Applied rewrites 84.3%

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

6. Taylor expanded in maxCos around 0

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

8. Applied rewrites 84.1%

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

9. Final simplification 84.1%

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

Alternative 13: 81.5% accurate, 9.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in uy around 0

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

   1. +-commutative N/A

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

   2. associate-+r+ N/A

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

   3. associate-*r* N/A

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

   4. distribute-rgt-out N/A

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

5. Applied rewrites 84.3%

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

6. Taylor expanded in maxCos around 0

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

8. Applied rewrites 84.1%

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

9. Final simplification 84.1%

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

Alternative 14: 78.0% accurate, 9.3× speedup

Math

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

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (fma uy (fma -2.0 (* uy (* 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(-2.0f, (uy * (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(-2.0), Float32(uy * Float32(xi * Float32(Float32(pi) * Float32(pi)))), Float32(Float32(2.0) * Float32(Float32(pi) * yi))), xi)
end

Derivation

1. Initial program 98.9%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
2. 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 91.6

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

5. Applied rewrites 91.6%

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

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 82.1%

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

Alternative 15: 78.8% accurate, 11.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in uy around 0

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

   1. +-commutative N/A

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

   2. associate-+r+ N/A

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

   3. associate-*r* N/A

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

   4. distribute-rgt-out N/A

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

5. Applied rewrites 84.3%

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

6. Taylor expanded in ux around 0

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

8. Applied rewrites 82.0%

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

9. Final simplification 82.0%

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

Alternative 16: 74.0% accurate, 20.8× speedup

Math

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

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(Float32(2.0) * uy), Float32(Float32(pi) * yi), xi)
end

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in uy around 0

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

   1. +-commutative N/A

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

   2. associate-+r+ N/A

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

   3. associate-*r* N/A

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

   4. distribute-rgt-out N/A

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

5. Applied rewrites 84.3%

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

6. Taylor expanded in maxCos around 0

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

8. Applied rewrites 77.6%

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

9. Final simplification 77.6%

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

Alternative 17: 74.0% 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 98.9%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
2. 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 91.6

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

5. Applied rewrites 91.6%

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

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 77.6%

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

Alternative 18: 11.8% accurate, 32.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
2. 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 13.1

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

5. Applied rewrites 13.1%

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

6. Taylor expanded in ux around 0

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

8. Applied rewrites 11.6%

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

Alternative 19: 11.8% accurate, 32.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
2. 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 13.1

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

5. Applied rewrites 13.1%

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

6. Taylor expanded in ux around 0

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

8. Applied rewrites 11.6%

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

9. Taylor expanded in ux around 0

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

11. Applied rewrites 11.6%

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

13. Applied rewrites 11.6%

    \[\leadsto \left(zi \cdot maxCos\right) \cdot ux \]

14. Final simplification 11.6%

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

Alternative 20: 11.8% accurate, 32.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
2. 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 13.1

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

5. Applied rewrites 13.1%

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

6. Taylor expanded in ux around 0

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

8. Applied rewrites 11.6%

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

9. Taylor expanded in ux around 0

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

11. Applied rewrites 11.6%

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

12. Final simplification 11.6%

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

Reproduce

herbie shell --seed 2024233 
(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)))