UniformSampleCone 2

Percentage Accurate: 98.9% → 98.5%

Time: 19.8s

Alternatives: 18

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: 98.5% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in maxCos around 0

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

   1. +-commutative N/A

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

   2. associate-+l+ N/A

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

   3. accelerator-lowering-fma.f32 N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   14. associate-*r* N/A

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

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

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

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

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

5. Simplified 99.0%

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

6. Taylor expanded in ux around inf

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

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

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

   2. unpow2 N/A

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

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

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

   4. mul-1-neg N/A

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

   5. distribute-rgt-neg-in N/A

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

   6. mul-1-neg N/A

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

   7. accelerator-lowering-fma.f32 N/A

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

   8. mul-1-neg N/A

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

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

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

   10. +-commutative N/A

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

   11. associate-/l* N/A

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

   12. accelerator-lowering-fma.f32 N/A

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

8. Simplified 99.1%

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

9. Final simplification 99.1%

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

Alternative 2: 98.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in maxCos around 0

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

   1. +-commutative N/A

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

   2. associate-+l+ N/A

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

   3. accelerator-lowering-fma.f32 N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   14. associate-*r* N/A

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

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

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

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

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

5. Simplified 99.0%

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

Alternative 3: 97.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \left(2 \cdot uy\right)\\ \mathbf{if}\;2 \cdot uy \leq 0.13500000536441803:\\ \;\;\;\;\mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), \mathsf{fma}\left(yi, uy \cdot \mathsf{fma}\left(-1.3333333333333333 \cdot \left(uy \cdot uy\right), \pi \cdot \left(\pi \cdot \pi\right), 2 \cdot \pi\right), maxCos \cdot \left(\left(ux \cdot zi\right) \cdot \left(1 - ux\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(yi, \sin t\_0, xi \cdot \cos t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* PI (* 2.0 uy))))
   (if (<= (* 2.0 uy) 0.13500000536441803)
     (fma
      xi
      (cos (* 2.0 (* uy PI)))
      (fma
       yi
       (*
        uy
        (fma (* -1.3333333333333333 (* uy uy)) (* PI (* PI PI)) (* 2.0 PI)))
       (* maxCos (* (* ux zi) (- 1.0 ux)))))
     (fma yi (sin t_0) (* xi (cos t_0))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = ((float) M_PI) * (2.0f * uy);
	float tmp;
	if ((2.0f * uy) <= 0.13500000536441803f) {
		tmp = fmaf(xi, cosf((2.0f * (uy * ((float) M_PI)))), fmaf(yi, (uy * fmaf((-1.3333333333333333f * (uy * uy)), (((float) M_PI) * (((float) M_PI) * ((float) M_PI))), (2.0f * ((float) M_PI)))), (maxCos * ((ux * zi) * (1.0f - ux)))));
	} else {
		tmp = fmaf(yi, sinf(t_0), (xi * cosf(t_0)));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(pi) * Float32(Float32(2.0) * uy))
	tmp = Float32(0.0)
	if (Float32(Float32(2.0) * uy) <= Float32(0.13500000536441803))
		tmp = fma(xi, cos(Float32(Float32(2.0) * Float32(uy * Float32(pi)))), fma(yi, Float32(uy * fma(Float32(Float32(-1.3333333333333333) * Float32(uy * uy)), Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi))), Float32(Float32(2.0) * Float32(pi)))), Float32(maxCos * Float32(Float32(ux * zi) * Float32(Float32(1.0) - ux)))));
	else
		tmp = fma(yi, sin(t_0), Float32(xi * cos(t_0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 0.135000005

   1. Initial program 99.3%

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

   3. Taylor expanded in maxCos around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. accelerator-lowering-fma.f32 N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      14. associate-*r* N/A

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

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

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

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

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

   5. Simplified 99.3%

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

   6. Taylor expanded in uy around 0

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

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

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

      2. associate-*r* N/A

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

      3. accelerator-lowering-fma.f32 N/A

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

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

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

      5. unpow2 N/A

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

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

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

      7. cube-mult N/A

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

      8. unpow2 N/A

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

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

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

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

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

      11. unpow2 N/A

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

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

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

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

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

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

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

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

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

      16. PI-lowering-PI.f32 98.5

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

   8. Simplified 98.5%

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

   if 0.135000005 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 96.8%

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

   3. Taylor expanded in maxCos around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. accelerator-lowering-fma.f32 N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      14. associate-*r* N/A

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

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

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

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

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

   5. Simplified 96.8%

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

   6. Taylor expanded in maxCos around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f32 N/A

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

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

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

      4. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

      10. associate-*r* N/A

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

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

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

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

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

      13. PI-lowering-PI.f32 96.3

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

   8. Simplified 96.3%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.13500000536441803:\\ \;\;\;\;\mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), \mathsf{fma}\left(yi, uy \cdot \mathsf{fma}\left(-1.3333333333333333 \cdot \left(uy \cdot uy\right), \pi \cdot \left(\pi \cdot \pi\right), 2 \cdot \pi\right), maxCos \cdot \left(\left(ux \cdot zi\right) \cdot \left(1 - ux\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(yi, \sin \left(\pi \cdot \left(2 \cdot uy\right)\right), xi \cdot \cos \left(\pi \cdot \left(2 \cdot uy\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \left(uy \cdot \pi\right)\\ t_1 := \cos t\_0\\ \mathbf{if}\;2 \cdot uy \leq 0.13500000536441803:\\ \;\;\;\;\mathsf{fma}\left(xi, t\_1, \mathsf{fma}\left(yi, uy \cdot \mathsf{fma}\left(-1.3333333333333333 \cdot \left(uy \cdot uy\right), \pi \cdot \left(\pi \cdot \pi\right), 2 \cdot \pi\right), maxCos \cdot \left(\left(ux \cdot zi\right) \cdot \left(1 - ux\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(xi, t\_1, yi \cdot \sin t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* 2.0 (* uy PI))) (t_1 (cos t_0)))
   (if (<= (* 2.0 uy) 0.13500000536441803)
     (fma
      xi
      t_1
      (fma
       yi
       (*
        uy
        (fma (* -1.3333333333333333 (* uy uy)) (* PI (* PI PI)) (* 2.0 PI)))
       (* maxCos (* (* ux zi) (- 1.0 ux)))))
     (fma xi t_1 (* yi (sin t_0))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = 2.0f * (uy * ((float) M_PI));
	float t_1 = cosf(t_0);
	float tmp;
	if ((2.0f * uy) <= 0.13500000536441803f) {
		tmp = fmaf(xi, t_1, fmaf(yi, (uy * fmaf((-1.3333333333333333f * (uy * uy)), (((float) M_PI) * (((float) M_PI) * ((float) M_PI))), (2.0f * ((float) M_PI)))), (maxCos * ((ux * zi) * (1.0f - ux)))));
	} else {
		tmp = fmaf(xi, t_1, (yi * sinf(t_0)));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(2.0) * Float32(uy * Float32(pi)))
	t_1 = cos(t_0)
	tmp = Float32(0.0)
	if (Float32(Float32(2.0) * uy) <= Float32(0.13500000536441803))
		tmp = fma(xi, t_1, fma(yi, Float32(uy * fma(Float32(Float32(-1.3333333333333333) * Float32(uy * uy)), Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi))), Float32(Float32(2.0) * Float32(pi)))), Float32(maxCos * Float32(Float32(ux * zi) * Float32(Float32(1.0) - ux)))));
	else
		tmp = fma(xi, t_1, Float32(yi * sin(t_0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 0.135000005

   1. Initial program 99.3%

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

   3. Taylor expanded in maxCos around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. accelerator-lowering-fma.f32 N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      14. associate-*r* N/A

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

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

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

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

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

   5. Simplified 99.3%

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

   6. Taylor expanded in uy around 0

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

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

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

      2. associate-*r* N/A

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

      3. accelerator-lowering-fma.f32 N/A

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

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

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

      5. unpow2 N/A

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

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

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

      7. cube-mult N/A

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

      8. unpow2 N/A

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

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

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

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

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

      11. unpow2 N/A

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

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

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

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

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

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

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

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

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

      16. PI-lowering-PI.f32 98.5

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

   8. Simplified 98.5%

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

   if 0.135000005 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 96.8%

      \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
   2. 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. accelerator-lowering-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. cos-lowering-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. *-lowering-*.f32 N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. PI-lowering-PI.f32 96.3

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

   5. Simplified 96.3%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.13500000536441803:\\ \;\;\;\;\mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), \mathsf{fma}\left(yi, uy \cdot \mathsf{fma}\left(-1.3333333333333333 \cdot \left(uy \cdot uy\right), \pi \cdot \left(\pi \cdot \pi\right), 2 \cdot \pi\right), maxCos \cdot \left(\left(ux \cdot zi\right) \cdot \left(1 - ux\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in ux around 0

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

   1. +-commutative N/A

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

   2. associate-+l+ N/A

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

   3. accelerator-lowering-fma.f32 N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   14. *-lowering-*.f32 97.5

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

5. Simplified 97.5%

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

Alternative 6: 92.6% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in maxCos around 0

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

   1. +-commutative N/A

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

   2. associate-+l+ N/A

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

   3. accelerator-lowering-fma.f32 N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   14. associate-*r* N/A

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

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

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

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

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

5. Simplified 99.0%

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

6. Taylor expanded in uy around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. accelerator-lowering-fma.f32 N/A

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

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

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

   5. unpow2 N/A

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

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

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

   7. unpow2 N/A

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

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

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

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

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

   10. PI-lowering-PI.f32 94.1

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

8. Simplified 94.1%

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

9. Final simplification 94.1%

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

Alternative 7: 91.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \left(uy \cdot \pi\right)\\ \mathbf{if}\;yi \leq -5.000000229068525 \cdot 10^{-19}:\\ \;\;\;\;xi \cdot \mathsf{fma}\left(yi, \frac{\sin t\_0}{xi}, \mathsf{fma}\left(\left(uy \cdot uy\right) \cdot -2, \pi \cdot \pi, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(xi, \cos t\_0, \mathsf{fma}\left(ux \cdot maxCos, zi \cdot \left(1 - ux\right), \left(2 \cdot uy\right) \cdot \left(\pi \cdot yi\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* 2.0 (* uy PI))))
   (if (<= yi -5.000000229068525e-19)
     (* xi (fma yi (/ (sin t_0) xi) (fma (* (* uy uy) -2.0) (* PI PI) 1.0)))
     (fma
      xi
      (cos t_0)
      (fma (* ux maxCos) (* zi (- 1.0 ux)) (* (* 2.0 uy) (* PI yi)))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = 2.0f * (uy * ((float) M_PI));
	float tmp;
	if (yi <= -5.000000229068525e-19f) {
		tmp = xi * fmaf(yi, (sinf(t_0) / xi), fmaf(((uy * uy) * -2.0f), (((float) M_PI) * ((float) M_PI)), 1.0f));
	} else {
		tmp = fmaf(xi, cosf(t_0), fmaf((ux * maxCos), (zi * (1.0f - ux)), ((2.0f * uy) * (((float) M_PI) * yi))));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(2.0) * Float32(uy * Float32(pi)))
	tmp = Float32(0.0)
	if (yi <= Float32(-5.000000229068525e-19))
		tmp = Float32(xi * fma(yi, Float32(sin(t_0) / xi), fma(Float32(Float32(uy * uy) * Float32(-2.0)), Float32(Float32(pi) * Float32(pi)), Float32(1.0))));
	else
		tmp = fma(xi, cos(t_0), fma(Float32(ux * maxCos), Float32(zi * Float32(Float32(1.0) - ux)), Float32(Float32(Float32(2.0) * uy) * Float32(Float32(pi) * yi))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if yi < -5.00000023e-19

   1. Initial program 98.7%

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

   3. Taylor expanded in xi around inf

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

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

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

      2. distribute-rgt-out N/A

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

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

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

   5. Simplified 98.4%

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

   6. Taylor expanded in uy around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. accelerator-lowering-fma.f32 N/A

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

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

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

      5. unpow2 N/A

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

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

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

      7. unpow2 N/A

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

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

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

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

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

      10. PI-lowering-PI.f32 98.0

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

   8. Simplified 98.0%

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

   9. Taylor expanded in maxCos around 0

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

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

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

       2. +-commutative N/A

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

       3. +-commutative N/A

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

       4. associate-+l+ N/A

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

       5. +-commutative N/A

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

       6. associate-/l* N/A

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

       7. accelerator-lowering-fma.f32 N/A

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

   11. Simplified 96.4%

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

   if -5.00000023e-19 < yi

   1. Initial program 99.1%

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

   3. Taylor expanded in maxCos around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. accelerator-lowering-fma.f32 N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      14. associate-*r* N/A

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

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

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

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

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

   5. Simplified 99.0%

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

   6. Taylor expanded in uy around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. accelerator-lowering-fma.f32 N/A

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

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

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

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

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

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

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

      7. associate-*r* N/A

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

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

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

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

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

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

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

      11. PI-lowering-PI.f32 93.3

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

   8. Simplified 93.3%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;yi \leq -5.000000229068525 \cdot 10^{-19}:\\ \;\;\;\;xi \cdot \mathsf{fma}\left(yi, \frac{\sin \left(2 \cdot \left(uy \cdot \pi\right)\right)}{xi}, \mathsf{fma}\left(\left(uy \cdot uy\right) \cdot -2, \pi \cdot \pi, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), \mathsf{fma}\left(ux \cdot maxCos, zi \cdot \left(1 - ux\right), \left(2 \cdot uy\right) \cdot \left(\pi \cdot yi\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 91.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \left(uy \cdot \pi\right)\\ \mathbf{if}\;yi \leq -5.000000229068525 \cdot 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(xi, 1, \mathsf{fma}\left(yi, \sin t\_0, maxCos \cdot \left(\left(ux \cdot zi\right) \cdot \left(1 - ux\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(xi, \cos t\_0, \mathsf{fma}\left(ux \cdot maxCos, zi \cdot \left(1 - ux\right), \left(2 \cdot uy\right) \cdot \left(\pi \cdot yi\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* 2.0 (* uy PI))))
   (if (<= yi -5.000000229068525e-19)
     (fma xi 1.0 (fma yi (sin t_0) (* maxCos (* (* ux zi) (- 1.0 ux)))))
     (fma
      xi
      (cos t_0)
      (fma (* ux maxCos) (* zi (- 1.0 ux)) (* (* 2.0 uy) (* PI yi)))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = 2.0f * (uy * ((float) M_PI));
	float tmp;
	if (yi <= -5.000000229068525e-19f) {
		tmp = fmaf(xi, 1.0f, fmaf(yi, sinf(t_0), (maxCos * ((ux * zi) * (1.0f - ux)))));
	} else {
		tmp = fmaf(xi, cosf(t_0), fmaf((ux * maxCos), (zi * (1.0f - ux)), ((2.0f * uy) * (((float) M_PI) * yi))));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(2.0) * Float32(uy * Float32(pi)))
	tmp = Float32(0.0)
	if (yi <= Float32(-5.000000229068525e-19))
		tmp = fma(xi, Float32(1.0), fma(yi, sin(t_0), Float32(maxCos * Float32(Float32(ux * zi) * Float32(Float32(1.0) - ux)))));
	else
		tmp = fma(xi, cos(t_0), fma(Float32(ux * maxCos), Float32(zi * Float32(Float32(1.0) - ux)), Float32(Float32(Float32(2.0) * uy) * Float32(Float32(pi) * yi))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if yi < -5.00000023e-19

   1. Initial program 98.7%

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

   3. Taylor expanded in maxCos around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. accelerator-lowering-fma.f32 N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      14. associate-*r* N/A

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

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

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

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

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

   5. Simplified 98.8%

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

   6. Taylor expanded in uy around 0

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

   8. Simplified 94.5%

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

   if -5.00000023e-19 < yi

   1. Initial program 99.1%

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

   3. Taylor expanded in maxCos around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. accelerator-lowering-fma.f32 N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      14. associate-*r* N/A

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

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

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

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

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

   5. Simplified 99.0%

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

   6. Taylor expanded in uy around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. accelerator-lowering-fma.f32 N/A

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

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

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

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

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

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

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

      7. associate-*r* N/A

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

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

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

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

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

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

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

      11. PI-lowering-PI.f32 93.3

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

   8. Simplified 93.3%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;yi \leq -5.000000229068525 \cdot 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(xi, 1, \mathsf{fma}\left(yi, \sin \left(2 \cdot \left(uy \cdot \pi\right)\right), maxCos \cdot \left(\left(ux \cdot zi\right) \cdot \left(1 - ux\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), \mathsf{fma}\left(ux \cdot maxCos, zi \cdot \left(1 - ux\right), \left(2 \cdot uy\right) \cdot \left(\pi \cdot yi\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 89.1% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in maxCos around 0

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

   1. +-commutative N/A

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

   2. associate-+l+ N/A

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

   3. accelerator-lowering-fma.f32 N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   14. associate-*r* N/A

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

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

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

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

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

5. Simplified 99.0%

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

6. Taylor expanded in maxCos around inf

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

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

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

   2. +-commutative N/A

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

   3. associate-/l* N/A

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

   4. accelerator-lowering-fma.f32 N/A

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

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

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

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

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

   7. associate-*r* N/A

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

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

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

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

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

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

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

   11. associate-*r* N/A

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

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

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

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

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

   14. --lowering--.f32 99.0

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

8. Simplified 99.0%

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

9. Taylor expanded in uy around 0

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

11. Simplified 90.2%

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

12. Final simplification 90.2%

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

Alternative 10: 85.6% accurate, 6.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in maxCos around 0

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

   1. +-commutative N/A

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

   2. associate-+l+ N/A

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

   3. accelerator-lowering-fma.f32 N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   14. associate-*r* N/A

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

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

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

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

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

5. Simplified 99.0%

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

6. Taylor expanded in maxCos around inf

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

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

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

   2. +-commutative N/A

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

   3. associate-/l* N/A

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

   4. accelerator-lowering-fma.f32 N/A

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

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

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

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

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

   7. associate-*r* N/A

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

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

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

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

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

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

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

   11. associate-*r* N/A

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

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

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

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

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

   14. --lowering--.f32 99.0

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

8. Simplified 99.0%

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

9. Taylor expanded in uy around 0

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

    1. associate-+r+ N/A

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

    2. +-commutative N/A

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

    3. accelerator-lowering-fma.f32 N/A

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

    4. +-commutative N/A

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

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

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

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

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

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

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

    8. associate-*r* N/A

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

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

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

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

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

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

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

    12. unpow2 N/A

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

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

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

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

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

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

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

11. Simplified 86.9%

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

12. Final simplification 86.9%

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

Alternative 11: 62.9% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := xi \cdot \mathsf{fma}\left(\left(uy \cdot uy\right) \cdot -2, \pi \cdot \pi, 1\right)\\ \mathbf{if}\;xi \leq -1.200000048101681 \cdot 10^{-25}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;xi \leq 4.999999943633011 \cdot 10^{-27}:\\ \;\;\;\;\mathsf{fma}\left(maxCos, \left(ux \cdot zi\right) \cdot \left(1 - ux\right), 2 \cdot \left(uy \cdot \left(\pi \cdot yi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* xi (fma (* (* uy uy) -2.0) (* PI PI) 1.0))))
   (if (<= xi -1.200000048101681e-25)
     t_0
     (if (<= xi 4.999999943633011e-27)
       (fma maxCos (* (* ux zi) (- 1.0 ux)) (* 2.0 (* uy (* PI yi))))
       t_0))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = xi * fmaf(((uy * uy) * -2.0f), (((float) M_PI) * ((float) M_PI)), 1.0f);
	float tmp;
	if (xi <= -1.200000048101681e-25f) {
		tmp = t_0;
	} else if (xi <= 4.999999943633011e-27f) {
		tmp = fmaf(maxCos, ((ux * zi) * (1.0f - ux)), (2.0f * (uy * (((float) M_PI) * yi))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(xi * fma(Float32(Float32(uy * uy) * Float32(-2.0)), Float32(Float32(pi) * Float32(pi)), Float32(1.0)))
	tmp = Float32(0.0)
	if (xi <= Float32(-1.200000048101681e-25))
		tmp = t_0;
	elseif (xi <= Float32(4.999999943633011e-27))
		tmp = fma(maxCos, Float32(Float32(ux * zi) * Float32(Float32(1.0) - ux)), Float32(Float32(2.0) * Float32(uy * Float32(Float32(pi) * yi))));
	else
		tmp = t_0;
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if xi < -1.20000005e-25 or 4.99999994e-27 < xi

   1. Initial program 99.3%

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

   3. Taylor expanded in xi around inf

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

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

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

      2. distribute-rgt-out N/A

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

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

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

   5. Simplified 99.1%

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

   6. Taylor expanded in uy around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. accelerator-lowering-fma.f32 N/A

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

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

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

      5. unpow2 N/A

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

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

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

      7. unpow2 N/A

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

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

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

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

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

      10. PI-lowering-PI.f32 93.2

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

   8. Simplified 93.2%

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

   9. Taylor expanded in xi around inf

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

       1. associate-*l* N/A

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

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

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

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

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

       4. +-commutative N/A

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

       5. associate-*r* N/A

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

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

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

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

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

       8. unpow2 N/A

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

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

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

       10. unpow2 N/A

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

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

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

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

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

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

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

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

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

   11. Simplified 66.9%

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

   12. Taylor expanded in maxCos around 0

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

       1. +-commutative N/A

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

       2. associate-*r* N/A

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

       3. accelerator-lowering-fma.f32 N/A

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

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

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

       5. unpow2 N/A

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

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

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

       7. unpow2 N/A

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

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

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

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

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

       10. PI-lowering-PI.f32 66.9

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

   14. Simplified 66.9%

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

   if -1.20000005e-25 < xi < 4.99999994e-27

   1. Initial program 98.3%

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

   3. Taylor expanded in maxCos around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. accelerator-lowering-fma.f32 N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      14. associate-*r* N/A

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

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

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

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

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

   5. Simplified 98.3%

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

   6. Taylor expanded in xi around 0

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

      1. accelerator-lowering-fma.f32 N/A

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

      2. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

      8. associate-*r* N/A

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

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

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

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

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

      11. PI-lowering-PI.f32 73.7

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

   8. Simplified 73.7%

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

   9. Taylor expanded in uy around 0

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

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

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

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

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

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

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

       4. PI-lowering-PI.f32 61.0

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

   11. Simplified 61.0%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;xi \leq -1.200000048101681 \cdot 10^{-25}:\\ \;\;\;\;xi \cdot \mathsf{fma}\left(\left(uy \cdot uy\right) \cdot -2, \pi \cdot \pi, 1\right)\\ \mathbf{elif}\;xi \leq 4.999999943633011 \cdot 10^{-27}:\\ \;\;\;\;\mathsf{fma}\left(maxCos, \left(ux \cdot zi\right) \cdot \left(1 - ux\right), 2 \cdot \left(uy \cdot \left(\pi \cdot yi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;xi \cdot \mathsf{fma}\left(\left(uy \cdot uy\right) \cdot -2, \pi \cdot \pi, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 81.4% accurate, 9.8× speedup

Math

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

FPCore

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

Julia

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

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in maxCos around 0

   \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + \left(xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{\left(xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)} \]

   2. associate-+l+ N/A

      \[\leadsto \color{blue}{xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + \left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right)} \]

   3. accelerator-lowering-fma.f32 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right)} \]

   4. cos-lowering-cos.f32 N/A

      \[\leadsto \mathsf{fma}\left(xi, \color{blue}{\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}, yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right) \]

   5. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(xi, \cos \color{blue}{\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}, yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right) \]

   6. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \color{blue}{\left(uy \cdot \mathsf{PI}\left(\right)\right)}\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right) \]

   7. PI-lowering-PI.f32 N/A

      \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right) \]

   8. accelerator-lowering-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{\mathsf{fma}\left(yi, \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right)}\right) \]

   9. sin-lowering-sin.f32 N/A

      \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), \mathsf{fma}\left(yi, \color{blue}{\sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}, maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right)\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), \mathsf{fma}\left(yi, \sin \color{blue}{\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}, maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right)\right) \]

   11. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), \mathsf{fma}\left(yi, \sin \left(2 \cdot \color{blue}{\left(uy \cdot \mathsf{PI}\left(\right)\right)}\right), maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right)\right) \]

   12. PI-lowering-PI.f32 N/A

       \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), \mathsf{fma}\left(yi, \sin \left(2 \cdot \left(uy \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right), maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right)\right) \]

   13. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), \mathsf{fma}\left(yi, \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)}\right)\right) \]

   14. associate-*r* N/A

       \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), \mathsf{fma}\left(yi, \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), maxCos \cdot \color{blue}{\left(\left(ux \cdot zi\right) \cdot \left(1 - ux\right)\right)}\right)\right) \]

   15. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), \mathsf{fma}\left(yi, \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), maxCos \cdot \color{blue}{\left(\left(ux \cdot zi\right) \cdot \left(1 - ux\right)\right)}\right)\right) \]

   16. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), \mathsf{fma}\left(yi, \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), maxCos \cdot \left(\color{blue}{\left(ux \cdot zi\right)} \cdot \left(1 - ux\right)\right)\right)\right) \]

5. Simplified 99.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), \mathsf{fma}\left(yi, \sin \left(2 \cdot \left(uy \cdot \pi\right)\right), maxCos \cdot \left(\left(ux \cdot zi\right) \cdot \left(1 - ux\right)\right)\right)\right)} \]

6. Taylor expanded in maxCos around inf

   \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right) + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}{maxCos}\right)}\right) \]
7. Step-by-step derivation

   1. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right) + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}{maxCos}\right)}\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), maxCos \cdot \color{blue}{\left(\frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}{maxCos} + ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)}\right) \]

   3. associate-/l* N/A

      \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), maxCos \cdot \left(\color{blue}{yi \cdot \frac{\sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}{maxCos}} + ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right) \]

   4. accelerator-lowering-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), maxCos \cdot \color{blue}{\mathsf{fma}\left(yi, \frac{\sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}{maxCos}, ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)}\right) \]

   5. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), maxCos \cdot \mathsf{fma}\left(yi, \color{blue}{\frac{\sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}{maxCos}}, ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right) \]

   6. sin-lowering-sin.f32 N/A

      \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), maxCos \cdot \mathsf{fma}\left(yi, \frac{\color{blue}{\sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}}{maxCos}, ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right) \]

   7. associate-*r* N/A

      \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), maxCos \cdot \mathsf{fma}\left(yi, \frac{\sin \color{blue}{\left(\left(2 \cdot uy\right) \cdot \mathsf{PI}\left(\right)\right)}}{maxCos}, ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right) \]

   8. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), maxCos \cdot \mathsf{fma}\left(yi, \frac{\sin \color{blue}{\left(\left(2 \cdot uy\right) \cdot \mathsf{PI}\left(\right)\right)}}{maxCos}, ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right) \]

   9. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), maxCos \cdot \mathsf{fma}\left(yi, \frac{\sin \left(\color{blue}{\left(2 \cdot uy\right)} \cdot \mathsf{PI}\left(\right)\right)}{maxCos}, ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right) \]

   10. PI-lowering-PI.f32 N/A

       \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), maxCos \cdot \mathsf{fma}\left(yi, \frac{\sin \left(\left(2 \cdot uy\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)}{maxCos}, ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right) \]

   11. associate-*r* N/A

       \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), maxCos \cdot \mathsf{fma}\left(yi, \frac{\sin \left(\left(2 \cdot uy\right) \cdot \mathsf{PI}\left(\right)\right)}{maxCos}, \color{blue}{\left(ux \cdot zi\right) \cdot \left(1 - ux\right)}\right)\right) \]

   12. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), maxCos \cdot \mathsf{fma}\left(yi, \frac{\sin \left(\left(2 \cdot uy\right) \cdot \mathsf{PI}\left(\right)\right)}{maxCos}, \color{blue}{\left(ux \cdot zi\right) \cdot \left(1 - ux\right)}\right)\right) \]

   13. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), maxCos \cdot \mathsf{fma}\left(yi, \frac{\sin \left(\left(2 \cdot uy\right) \cdot \mathsf{PI}\left(\right)\right)}{maxCos}, \color{blue}{\left(ux \cdot zi\right)} \cdot \left(1 - ux\right)\right)\right) \]

   14. --lowering--.f32 99.0

       \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), maxCos \cdot \mathsf{fma}\left(yi, \frac{\sin \left(\left(2 \cdot uy\right) \cdot \pi\right)}{maxCos}, \left(ux \cdot zi\right) \cdot \color{blue}{\left(1 - ux\right)}\right)\right) \]

8. Simplified 99.0%

   \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), \color{blue}{maxCos \cdot \mathsf{fma}\left(yi, \frac{\sin \left(\left(2 \cdot uy\right) \cdot \pi\right)}{maxCos}, \left(ux \cdot zi\right) \cdot \left(1 - ux\right)\right)}\right) \]

9. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{xi + \left(2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right) + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right)} \]
10. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \color{blue}{\left(2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right) + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right) + xi} \]

    2. associate-+l+ N/A

       \[\leadsto \color{blue}{2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right) + \left(maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + xi\right)} \]

    3. +-commutative N/A

       \[\leadsto 2 \cdot \left(uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right) + \color{blue}{\left(xi + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right)} \]

    4. accelerator-lowering-fma.f32 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(2, uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right), xi + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right)} \]

    5. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(2, \color{blue}{uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)}, xi + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right) \]

    6. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(2, uy \cdot \color{blue}{\left(yi \cdot \mathsf{PI}\left(\right)\right)}, xi + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right) \]

    7. PI-lowering-PI.f32 N/A

       \[\leadsto \mathsf{fma}\left(2, uy \cdot \left(yi \cdot \color{blue}{\mathsf{PI}\left(\right)}\right), xi + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right) \]

    8. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(2, uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right), \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + xi}\right) \]

    9. associate-*r* N/A

       \[\leadsto \mathsf{fma}\left(2, uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right), \color{blue}{\left(maxCos \cdot ux\right) \cdot \left(zi \cdot \left(1 - ux\right)\right)} + xi\right) \]

    10. accelerator-lowering-fma.f32 N/A

        \[\leadsto \mathsf{fma}\left(2, uy \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right), \color{blue}{\mathsf{fma}\left(maxCos \cdot ux, zi \cdot \left(1 - ux\right), xi\right)}\right) \]

11. Simplified 82.7%

    \[\leadsto \color{blue}{\mathsf{fma}\left(2, uy \cdot \left(yi \cdot \pi\right), \mathsf{fma}\left(maxCos \cdot ux, \mathsf{fma}\left(ux, -zi, zi\right), xi\right)\right)} \]

12. Final simplification 82.7%

    \[\leadsto \mathsf{fma}\left(2, uy \cdot \left(\pi \cdot yi\right), \mathsf{fma}\left(ux \cdot maxCos, \mathsf{fma}\left(ux, -zi, zi\right), xi\right)\right) \]
13. Add Preprocessing

Alternative 13: 51.8% accurate, 17.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(ux \cdot maxCos, \mathsf{fma}\left(ux, -zi, zi\right), xi\right) \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (fma (* ux maxCos) (fma ux (- zi) zi) xi))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return fmaf((ux * maxCos), fmaf(ux, -zi, zi), xi);
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return fma(Float32(ux * maxCos), fma(ux, Float32(-zi), zi), xi)
end

Derivation

1. Initial program 99.0%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
2. Add Preprocessing

3. Taylor expanded in maxCos around 0

   \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + \left(xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{\left(xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)} \]

   2. associate-+l+ N/A

      \[\leadsto \color{blue}{xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + \left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right)} \]

   3. accelerator-lowering-fma.f32 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right)} \]

   4. cos-lowering-cos.f32 N/A

      \[\leadsto \mathsf{fma}\left(xi, \color{blue}{\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}, yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right) \]

   5. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(xi, \cos \color{blue}{\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}, yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right) \]

   6. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \color{blue}{\left(uy \cdot \mathsf{PI}\left(\right)\right)}\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right) \]

   7. PI-lowering-PI.f32 N/A

      \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right) \]

   8. accelerator-lowering-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{\mathsf{fma}\left(yi, \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right)}\right) \]

   9. sin-lowering-sin.f32 N/A

      \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), \mathsf{fma}\left(yi, \color{blue}{\sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}, maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right)\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), \mathsf{fma}\left(yi, \sin \color{blue}{\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}, maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right)\right) \]

   11. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), \mathsf{fma}\left(yi, \sin \left(2 \cdot \color{blue}{\left(uy \cdot \mathsf{PI}\left(\right)\right)}\right), maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right)\right) \]

   12. PI-lowering-PI.f32 N/A

       \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), \mathsf{fma}\left(yi, \sin \left(2 \cdot \left(uy \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right), maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right)\right) \]

   13. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), \mathsf{fma}\left(yi, \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)}\right)\right) \]

   14. associate-*r* N/A

       \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), \mathsf{fma}\left(yi, \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), maxCos \cdot \color{blue}{\left(\left(ux \cdot zi\right) \cdot \left(1 - ux\right)\right)}\right)\right) \]

   15. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), \mathsf{fma}\left(yi, \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), maxCos \cdot \color{blue}{\left(\left(ux \cdot zi\right) \cdot \left(1 - ux\right)\right)}\right)\right) \]

   16. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), \mathsf{fma}\left(yi, \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), maxCos \cdot \left(\color{blue}{\left(ux \cdot zi\right)} \cdot \left(1 - ux\right)\right)\right)\right) \]

5. Simplified 99.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), \mathsf{fma}\left(yi, \sin \left(2 \cdot \left(uy \cdot \pi\right)\right), maxCos \cdot \left(\left(ux \cdot zi\right) \cdot \left(1 - ux\right)\right)\right)\right)} \]

6. Taylor expanded in maxCos around inf

   \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right) + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}{maxCos}\right)}\right) \]
7. Step-by-step derivation

   1. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right) + \frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}{maxCos}\right)}\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), maxCos \cdot \color{blue}{\left(\frac{yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}{maxCos} + ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)}\right) \]

   3. associate-/l* N/A

      \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), maxCos \cdot \left(\color{blue}{yi \cdot \frac{\sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}{maxCos}} + ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right) \]

   4. accelerator-lowering-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), maxCos \cdot \color{blue}{\mathsf{fma}\left(yi, \frac{\sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}{maxCos}, ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)}\right) \]

   5. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), maxCos \cdot \mathsf{fma}\left(yi, \color{blue}{\frac{\sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}{maxCos}}, ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right) \]

   6. sin-lowering-sin.f32 N/A

      \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), maxCos \cdot \mathsf{fma}\left(yi, \frac{\color{blue}{\sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}}{maxCos}, ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right) \]

   7. associate-*r* N/A

      \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), maxCos \cdot \mathsf{fma}\left(yi, \frac{\sin \color{blue}{\left(\left(2 \cdot uy\right) \cdot \mathsf{PI}\left(\right)\right)}}{maxCos}, ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right) \]

   8. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), maxCos \cdot \mathsf{fma}\left(yi, \frac{\sin \color{blue}{\left(\left(2 \cdot uy\right) \cdot \mathsf{PI}\left(\right)\right)}}{maxCos}, ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right) \]

   9. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), maxCos \cdot \mathsf{fma}\left(yi, \frac{\sin \left(\color{blue}{\left(2 \cdot uy\right)} \cdot \mathsf{PI}\left(\right)\right)}{maxCos}, ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right) \]

   10. PI-lowering-PI.f32 N/A

       \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), maxCos \cdot \mathsf{fma}\left(yi, \frac{\sin \left(\left(2 \cdot uy\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)}{maxCos}, ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right) \]

   11. associate-*r* N/A

       \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), maxCos \cdot \mathsf{fma}\left(yi, \frac{\sin \left(\left(2 \cdot uy\right) \cdot \mathsf{PI}\left(\right)\right)}{maxCos}, \color{blue}{\left(ux \cdot zi\right) \cdot \left(1 - ux\right)}\right)\right) \]

   12. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), maxCos \cdot \mathsf{fma}\left(yi, \frac{\sin \left(\left(2 \cdot uy\right) \cdot \mathsf{PI}\left(\right)\right)}{maxCos}, \color{blue}{\left(ux \cdot zi\right) \cdot \left(1 - ux\right)}\right)\right) \]

   13. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), maxCos \cdot \mathsf{fma}\left(yi, \frac{\sin \left(\left(2 \cdot uy\right) \cdot \mathsf{PI}\left(\right)\right)}{maxCos}, \color{blue}{\left(ux \cdot zi\right)} \cdot \left(1 - ux\right)\right)\right) \]

   14. --lowering--.f32 99.0

       \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), maxCos \cdot \mathsf{fma}\left(yi, \frac{\sin \left(\left(2 \cdot uy\right) \cdot \pi\right)}{maxCos}, \left(ux \cdot zi\right) \cdot \color{blue}{\left(1 - ux\right)}\right)\right) \]

8. Simplified 99.0%

   \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), \color{blue}{maxCos \cdot \mathsf{fma}\left(yi, \frac{\sin \left(\left(2 \cdot uy\right) \cdot \pi\right)}{maxCos}, \left(ux \cdot zi\right) \cdot \left(1 - ux\right)\right)}\right) \]

9. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{xi + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)} \]
10. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + xi} \]

    2. associate-*r* N/A

       \[\leadsto \color{blue}{\left(maxCos \cdot ux\right) \cdot \left(zi \cdot \left(1 - ux\right)\right)} + xi \]

    3. accelerator-lowering-fma.f32 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos \cdot ux, zi \cdot \left(1 - ux\right), xi\right)} \]

    4. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{maxCos \cdot ux}, zi \cdot \left(1 - ux\right), xi\right) \]

    5. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(maxCos \cdot ux, \color{blue}{\left(1 - ux\right) \cdot zi}, xi\right) \]

    6. sub-neg N/A

       \[\leadsto \mathsf{fma}\left(maxCos \cdot ux, \color{blue}{\left(1 + \left(\mathsf{neg}\left(ux\right)\right)\right)} \cdot zi, xi\right) \]

    7. mul-1-neg N/A

       \[\leadsto \mathsf{fma}\left(maxCos \cdot ux, \left(1 + \color{blue}{-1 \cdot ux}\right) \cdot zi, xi\right) \]

    8. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(maxCos \cdot ux, \color{blue}{\left(-1 \cdot ux + 1\right)} \cdot zi, xi\right) \]

    9. distribute-lft1-in N/A

       \[\leadsto \mathsf{fma}\left(maxCos \cdot ux, \color{blue}{\left(-1 \cdot ux\right) \cdot zi + zi}, xi\right) \]

    10. associate-*r* N/A

        \[\leadsto \mathsf{fma}\left(maxCos \cdot ux, \color{blue}{-1 \cdot \left(ux \cdot zi\right)} + zi, xi\right) \]

    11. mul-1-neg N/A

        \[\leadsto \mathsf{fma}\left(maxCos \cdot ux, \color{blue}{\left(\mathsf{neg}\left(ux \cdot zi\right)\right)} + zi, xi\right) \]

    12. distribute-rgt-neg-in N/A

        \[\leadsto \mathsf{fma}\left(maxCos \cdot ux, \color{blue}{ux \cdot \left(\mathsf{neg}\left(zi\right)\right)} + zi, xi\right) \]

    13. mul-1-neg N/A

        \[\leadsto \mathsf{fma}\left(maxCos \cdot ux, ux \cdot \color{blue}{\left(-1 \cdot zi\right)} + zi, xi\right) \]

    14. accelerator-lowering-fma.f32 N/A

        \[\leadsto \mathsf{fma}\left(maxCos \cdot ux, \color{blue}{\mathsf{fma}\left(ux, -1 \cdot zi, zi\right)}, xi\right) \]

    15. mul-1-neg N/A

        \[\leadsto \mathsf{fma}\left(maxCos \cdot ux, \mathsf{fma}\left(ux, \color{blue}{\mathsf{neg}\left(zi\right)}, zi\right), xi\right) \]

    16. neg-lowering-neg.f32 52.9

        \[\leadsto \mathsf{fma}\left(maxCos \cdot ux, \mathsf{fma}\left(ux, \color{blue}{-zi}, zi\right), xi\right) \]

11. Simplified 52.9%

    \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos \cdot ux, \mathsf{fma}\left(ux, -zi, zi\right), xi\right)} \]

12. Final simplification 52.9%

    \[\leadsto \mathsf{fma}\left(ux \cdot maxCos, \mathsf{fma}\left(ux, -zi, zi\right), xi\right) \]
13. Add Preprocessing

Alternative 14: 51.8% accurate, 17.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(ux \cdot maxCos, zi \cdot \left(1 - ux\right), xi\right) \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (fma (* ux maxCos) (* zi (- 1.0 ux)) xi))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return fmaf((ux * maxCos), (zi * (1.0f - ux)), xi);
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return fma(Float32(ux * maxCos), Float32(zi * Float32(Float32(1.0) - ux)), xi)
end

Derivation

1. Initial program 99.0%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
2. Add Preprocessing

3. Taylor expanded in maxCos around 0

   \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + \left(xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{\left(xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)} \]

   2. associate-+l+ N/A

      \[\leadsto \color{blue}{xi \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + \left(yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right)} \]

   3. accelerator-lowering-fma.f32 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right)} \]

   4. cos-lowering-cos.f32 N/A

      \[\leadsto \mathsf{fma}\left(xi, \color{blue}{\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}, yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right) \]

   5. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(xi, \cos \color{blue}{\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}, yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right) \]

   6. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \color{blue}{\left(uy \cdot \mathsf{PI}\left(\right)\right)}\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right) \]

   7. PI-lowering-PI.f32 N/A

      \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right) \]

   8. accelerator-lowering-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{\mathsf{fma}\left(yi, \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right)}\right) \]

   9. sin-lowering-sin.f32 N/A

      \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), \mathsf{fma}\left(yi, \color{blue}{\sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}, maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right)\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), \mathsf{fma}\left(yi, \sin \color{blue}{\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}, maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right)\right) \]

   11. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), \mathsf{fma}\left(yi, \sin \left(2 \cdot \color{blue}{\left(uy \cdot \mathsf{PI}\left(\right)\right)}\right), maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right)\right) \]

   12. PI-lowering-PI.f32 N/A

       \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), \mathsf{fma}\left(yi, \sin \left(2 \cdot \left(uy \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right), maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)\right)\right) \]

   13. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), \mathsf{fma}\left(yi, \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)}\right)\right) \]

   14. associate-*r* N/A

       \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), \mathsf{fma}\left(yi, \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), maxCos \cdot \color{blue}{\left(\left(ux \cdot zi\right) \cdot \left(1 - ux\right)\right)}\right)\right) \]

   15. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), \mathsf{fma}\left(yi, \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), maxCos \cdot \color{blue}{\left(\left(ux \cdot zi\right) \cdot \left(1 - ux\right)\right)}\right)\right) \]

   16. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), \mathsf{fma}\left(yi, \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right), maxCos \cdot \left(\color{blue}{\left(ux \cdot zi\right)} \cdot \left(1 - ux\right)\right)\right)\right) \]

5. Simplified 99.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), \mathsf{fma}\left(yi, \sin \left(2 \cdot \left(uy \cdot \pi\right)\right), maxCos \cdot \left(\left(ux \cdot zi\right) \cdot \left(1 - ux\right)\right)\right)\right)} \]

6. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{xi + maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + xi} \]

   2. associate-*r* N/A

      \[\leadsto \color{blue}{\left(maxCos \cdot ux\right) \cdot \left(zi \cdot \left(1 - ux\right)\right)} + xi \]

   3. accelerator-lowering-fma.f32 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos \cdot ux, zi \cdot \left(1 - ux\right), xi\right)} \]

   4. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{maxCos \cdot ux}, zi \cdot \left(1 - ux\right), xi\right) \]

   5. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(maxCos \cdot ux, \color{blue}{zi \cdot \left(1 - ux\right)}, xi\right) \]

   6. --lowering--.f32 52.8

      \[\leadsto \mathsf{fma}\left(maxCos \cdot ux, zi \cdot \color{blue}{\left(1 - ux\right)}, xi\right) \]

8. Simplified 52.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(maxCos \cdot ux, zi \cdot \left(1 - ux\right), xi\right)} \]

9. Final simplification 52.8%

   \[\leadsto \mathsf{fma}\left(ux \cdot maxCos, zi \cdot \left(1 - ux\right), xi\right) \]
10. Add Preprocessing

Alternative 15: 13.6% accurate, 18.6× speedup

Math

\[\begin{array}{l} \\ maxCos \cdot \left(zi \cdot \left(ux \cdot \left(1 - ux\right)\right)\right) \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (* maxCos (* zi (* ux (- 1.0 ux)))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return maxCos * (zi * (ux * (1.0f - ux)));
}

Fortran

real(4) function code(xi, yi, zi, ux, uy, maxcos)
    real(4), intent (in) :: xi
    real(4), intent (in) :: yi
    real(4), intent (in) :: zi
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = maxcos * (zi * (ux * (1.0e0 - ux)))
end function

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return Float32(maxCos * Float32(zi * Float32(ux * Float32(Float32(1.0) - ux))))
end

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	tmp = maxCos * (zi * (ux * (single(1.0) - ux)));
end

Derivation

1. Initial program 99.0%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
2. Add Preprocessing

3. Taylor expanded in zi around inf

   \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)} \]
4. Step-by-step derivation

   1. *-lowering-*.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. *-lowering-*.f32 N/A

      \[\leadsto maxCos \cdot \color{blue}{\left(\left(ux \cdot zi\right) \cdot \left(1 - ux\right)\right)} \]

   4. *-lowering-*.f32 N/A

      \[\leadsto maxCos \cdot \left(\color{blue}{\left(ux \cdot zi\right)} \cdot \left(1 - ux\right)\right) \]

   5. --lowering--.f32 12.1

      \[\leadsto maxCos \cdot \left(\left(ux \cdot zi\right) \cdot \color{blue}{\left(1 - ux\right)}\right) \]

5. Simplified 12.1%

   \[\leadsto \color{blue}{maxCos \cdot \left(\left(ux \cdot zi\right) \cdot \left(1 - ux\right)\right)} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto maxCos \cdot \color{blue}{\left(\left(1 - ux\right) \cdot \left(ux \cdot zi\right)\right)} \]

   2. associate-*r* N/A

      \[\leadsto maxCos \cdot \color{blue}{\left(\left(\left(1 - ux\right) \cdot ux\right) \cdot zi\right)} \]

   3. *-lowering-*.f32 N/A

      \[\leadsto maxCos \cdot \color{blue}{\left(\left(\left(1 - ux\right) \cdot ux\right) \cdot zi\right)} \]

   4. *-lowering-*.f32 N/A

      \[\leadsto maxCos \cdot \left(\color{blue}{\left(\left(1 - ux\right) \cdot ux\right)} \cdot zi\right) \]

   5. --lowering--.f32 12.1

      \[\leadsto maxCos \cdot \left(\left(\color{blue}{\left(1 - ux\right)} \cdot ux\right) \cdot zi\right) \]

7. Applied egg-rr 12.1%

   \[\leadsto maxCos \cdot \color{blue}{\left(\left(\left(1 - ux\right) \cdot ux\right) \cdot zi\right)} \]

8. Final simplification 12.1%

   \[\leadsto maxCos \cdot \left(zi \cdot \left(ux \cdot \left(1 - ux\right)\right)\right) \]
9. Add Preprocessing

Alternative 16: 13.6% accurate, 18.6× speedup

Math

\[\begin{array}{l} \\ maxCos \cdot \left(\left(ux \cdot zi\right) \cdot \left(1 - ux\right)\right) \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (* maxCos (* (* ux zi) (- 1.0 ux))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return maxCos * ((ux * zi) * (1.0f - ux));
}

Fortran

real(4) function code(xi, yi, zi, ux, uy, maxcos)
    real(4), intent (in) :: xi
    real(4), intent (in) :: yi
    real(4), intent (in) :: zi
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = maxcos * ((ux * zi) * (1.0e0 - ux))
end function

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return Float32(maxCos * Float32(Float32(ux * zi) * Float32(Float32(1.0) - ux)))
end

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	tmp = maxCos * ((ux * zi) * (single(1.0) - ux));
end

Derivation

1. Initial program 99.0%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
2. Add Preprocessing

3. Taylor expanded in zi around inf

   \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)} \]
4. Step-by-step derivation

   1. *-lowering-*.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. *-lowering-*.f32 N/A

      \[\leadsto maxCos \cdot \color{blue}{\left(\left(ux \cdot zi\right) \cdot \left(1 - ux\right)\right)} \]

   4. *-lowering-*.f32 N/A

      \[\leadsto maxCos \cdot \left(\color{blue}{\left(ux \cdot zi\right)} \cdot \left(1 - ux\right)\right) \]

   5. --lowering--.f32 12.1

      \[\leadsto maxCos \cdot \left(\left(ux \cdot zi\right) \cdot \color{blue}{\left(1 - ux\right)}\right) \]

5. Simplified 12.1%

   \[\leadsto \color{blue}{maxCos \cdot \left(\left(ux \cdot zi\right) \cdot \left(1 - ux\right)\right)} \]
6. Add Preprocessing

Alternative 17: 12.1% 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 99.0%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
2. Add Preprocessing

3. Taylor expanded in zi around inf

   \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)} \]
4. Step-by-step derivation

   1. *-lowering-*.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. *-lowering-*.f32 N/A

      \[\leadsto maxCos \cdot \color{blue}{\left(\left(ux \cdot zi\right) \cdot \left(1 - ux\right)\right)} \]

   4. *-lowering-*.f32 N/A

      \[\leadsto maxCos \cdot \left(\color{blue}{\left(ux \cdot zi\right)} \cdot \left(1 - ux\right)\right) \]

   5. --lowering--.f32 12.1

      \[\leadsto maxCos \cdot \left(\left(ux \cdot zi\right) \cdot \color{blue}{\left(1 - ux\right)}\right) \]

5. Simplified 12.1%

   \[\leadsto \color{blue}{maxCos \cdot \left(\left(ux \cdot zi\right) \cdot \left(1 - ux\right)\right)} \]

6. Taylor expanded in ux around 0

   \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot zi\right)} \]
7. Step-by-step derivation

   1. *-lowering-*.f32 N/A

      \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot zi\right)} \]

   2. *-lowering-*.f32 11.2

      \[\leadsto maxCos \cdot \color{blue}{\left(ux \cdot zi\right)} \]

8. Simplified 11.2%

   \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot zi\right)} \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto maxCos \cdot \color{blue}{\left(zi \cdot ux\right)} \]

   2. associate-*r* N/A

      \[\leadsto \color{blue}{\left(maxCos \cdot zi\right) \cdot ux} \]

   3. *-lowering-*.f32 N/A

      \[\leadsto \color{blue}{\left(maxCos \cdot zi\right) \cdot ux} \]

   4. *-lowering-*.f32 11.2

      \[\leadsto \color{blue}{\left(maxCos \cdot zi\right)} \cdot ux \]

10. Applied egg-rr 11.2%

    \[\leadsto \color{blue}{\left(maxCos \cdot zi\right) \cdot ux} \]

11. Final simplification 11.2%

    \[\leadsto ux \cdot \left(maxCos \cdot zi\right) \]
12. Add Preprocessing

Alternative 18: 12.1% 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 99.0%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
2. Add Preprocessing

3. Taylor expanded in zi around inf

   \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)} \]
4. Step-by-step derivation

   1. *-lowering-*.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. *-lowering-*.f32 N/A

      \[\leadsto maxCos \cdot \color{blue}{\left(\left(ux \cdot zi\right) \cdot \left(1 - ux\right)\right)} \]

   4. *-lowering-*.f32 N/A

      \[\leadsto maxCos \cdot \left(\color{blue}{\left(ux \cdot zi\right)} \cdot \left(1 - ux\right)\right) \]

   5. --lowering--.f32 12.1

      \[\leadsto maxCos \cdot \left(\left(ux \cdot zi\right) \cdot \color{blue}{\left(1 - ux\right)}\right) \]

5. Simplified 12.1%

   \[\leadsto \color{blue}{maxCos \cdot \left(\left(ux \cdot zi\right) \cdot \left(1 - ux\right)\right)} \]

6. Taylor expanded in ux around 0

   \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot zi\right)} \]
7. Step-by-step derivation

   1. *-lowering-*.f32 N/A

      \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot zi\right)} \]

   2. *-lowering-*.f32 11.2

      \[\leadsto maxCos \cdot \color{blue}{\left(ux \cdot zi\right)} \]

8. Simplified 11.2%

   \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot zi\right)} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024199 
(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)))