UniformSampleCone 2

Percentage Accurate: 98.9% → 98.9%

Time: 24.4s

Alternatives: 28

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := ux \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\\ t_1 := \left(uy \cdot 2\right) \cdot \pi\\ \left(\left(\cos t\_1 \cdot \sqrt{1 + t\_0 \cdot \left(ux \cdot \left(maxCos \cdot \left(ux + -1\right)\right)\right)}\right) \cdot xi + \left(\sin t\_1 \cdot \sqrt{{ux}^{4} \cdot \left(\frac{1}{{ux}^{4}} - \mathsf{fma}\left(maxCos, maxCos, \frac{\mathsf{fma}\left(maxCos, \frac{maxCos}{ux}, \left(maxCos \cdot maxCos\right) \cdot -2\right)}{ux}\right)\right)}\right) \cdot yi\right) + t\_0 \cdot zi \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* ux (* (- 1.0 ux) maxCos))) (t_1 (* (* uy 2.0) PI)))
   (+
    (+
     (* (* (cos t_1) (sqrt (+ 1.0 (* t_0 (* ux (* maxCos (+ ux -1.0))))))) xi)
     (*
      (*
       (sin t_1)
       (sqrt
        (*
         (pow ux 4.0)
         (-
          (/ 1.0 (pow ux 4.0))
          (fma
           maxCos
           maxCos
           (/ (fma maxCos (/ maxCos ux) (* (* maxCos maxCos) -2.0)) ux))))))
      yi))
    (* t_0 zi))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = ux * ((1.0f - ux) * maxCos);
	float t_1 = (uy * 2.0f) * ((float) M_PI);
	return (((cosf(t_1) * sqrtf((1.0f + (t_0 * (ux * (maxCos * (ux + -1.0f))))))) * xi) + ((sinf(t_1) * sqrtf((powf(ux, 4.0f) * ((1.0f / powf(ux, 4.0f)) - fmaf(maxCos, maxCos, (fmaf(maxCos, (maxCos / ux), ((maxCos * maxCos) * -2.0f)) / ux)))))) * yi)) + (t_0 * zi);
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(ux * Float32(Float32(Float32(1.0) - ux) * maxCos))
	t_1 = Float32(Float32(uy * Float32(2.0)) * Float32(pi))
	return Float32(Float32(Float32(Float32(cos(t_1) * sqrt(Float32(Float32(1.0) + Float32(t_0 * Float32(ux * Float32(maxCos * Float32(ux + Float32(-1.0)))))))) * xi) + Float32(Float32(sin(t_1) * sqrt(Float32((ux ^ Float32(4.0)) * Float32(Float32(Float32(1.0) / (ux ^ Float32(4.0))) - fma(maxCos, maxCos, Float32(fma(maxCos, Float32(maxCos / ux), Float32(Float32(maxCos * maxCos) * Float32(-2.0))) / ux)))))) * yi)) + Float32(t_0 * zi))
end

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in ux around inf

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

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

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

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

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

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

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

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

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

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

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

   6. associate-+r+ N/A

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

   7. +-commutative N/A

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

   8. unpow2 N/A

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

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

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

   10. +-commutative N/A

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

   11. unpow2 N/A

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

   12. associate-/r* N/A

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

   13. metadata-eval N/A

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

   14. cancel-sign-sub-inv N/A

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

   15. associate-*r/ N/A

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

5. Simplified 98.9%

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

6. Final simplification 98.9%

   \[\leadsto \left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 + \left(ux \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right) \cdot \left(ux \cdot \left(maxCos \cdot \left(ux + -1\right)\right)\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{{ux}^{4} \cdot \left(\frac{1}{{ux}^{4}} - \mathsf{fma}\left(maxCos, maxCos, \frac{\mathsf{fma}\left(maxCos, \frac{maxCos}{ux}, \left(maxCos \cdot maxCos\right) \cdot -2\right)}{ux}\right)\right)}\right) \cdot yi\right) + \left(ux \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right) \cdot zi \]
7. Add Preprocessing

Alternative 2: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

3. Final simplification 98.9%

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

Alternative 3: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

4. Applied egg-rr 98.8%

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

5. Final simplification 98.8%

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

Alternative 4: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in ux around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

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

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

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

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

   5. unpow2 N/A

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

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

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

   7. unpow2 N/A

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

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

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

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

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

   10. neg-lowering-neg.f32 98.7

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

5. Simplified 98.7%

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

6. Final simplification 98.7%

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

Alternative 5: 98.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in ux around 0

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

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

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

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

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

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

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

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

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

   5. PI-lowering-PI.f32 98.7

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

5. Simplified 98.7%

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

6. Final simplification 98.7%

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

Alternative 6: 98.2% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in 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 97.9%

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

6. Taylor expanded in yi around inf

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

8. Simplified 98.4%

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

9. Taylor expanded in ux around 0

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

    1. mul-1-neg N/A

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

    2. unpow2 N/A

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

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

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

    4. mul-1-neg N/A

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

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

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

    6. mul-1-neg N/A

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

    7. neg-lowering-neg.f32 98.1

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

11. Simplified 98.1%

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

    1. associate-*r* N/A

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

    2. *-commutative N/A

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

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

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

13. Applied egg-rr 98.2%

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

14. Final simplification 98.2%

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

Alternative 7: 98.2% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* 2.0 (* uy PI))))
   (+
    (* (* ux (* (- 1.0 ux) maxCos)) zi)
    (* xi (fma (sin t_0) (/ yi xi) (cos 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));
	return ((ux * ((1.0f - ux) * maxCos)) * zi) + (xi * fmaf(sinf(t_0), (yi / xi), cosf(t_0)));
}

Julia

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

Derivation

1. Initial program 98.9%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
2. 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 97.9%

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

6. Taylor expanded in maxCos around 0

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

8. Simplified 97.5%

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

9. Final simplification 97.5%

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

Alternative 8: 97.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \left(uy \cdot \pi\right)\\ t_1 := \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)}\\ \mathbf{if}\;uy \cdot 2 \leq 0.07999999821186066:\\ \;\;\;\;\mathsf{fma}\left(uy, \mathsf{fma}\left(uy, t\_1 \cdot \mathsf{fma}\left(-2, xi \cdot \left(\pi \cdot \pi\right), -1.3333333333333333 \cdot \left(uy \cdot \left(yi \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\right), t\_1 \cdot \left(2 \cdot \left(\pi \cdot yi\right)\right)\right), \mathsf{fma}\left(xi, t\_1, maxCos \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot zi\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(xi, \cos t\_0, yi \cdot \sin t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* 2.0 (* uy PI)))
        (t_1
         (sqrt
          (fma
           (* maxCos maxCos)
           (* (* ux ux) (* (- 1.0 ux) (+ ux -1.0)))
           1.0))))
   (if (<= (* uy 2.0) 0.07999999821186066)
     (fma
      uy
      (fma
       uy
       (*
        t_1
        (fma
         -2.0
         (* xi (* PI PI))
         (* -1.3333333333333333 (* uy (* yi (* PI (* PI PI)))))))
       (* t_1 (* 2.0 (* PI yi))))
      (fma xi t_1 (* maxCos (* (- 1.0 ux) (* ux zi)))))
     (fma xi (cos t_0) (* yi (sin t_0))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = 2.0f * (uy * ((float) M_PI));
	float t_1 = sqrtf(fmaf((maxCos * maxCos), ((ux * ux) * ((1.0f - ux) * (ux + -1.0f))), 1.0f));
	float tmp;
	if ((uy * 2.0f) <= 0.07999999821186066f) {
		tmp = fmaf(uy, fmaf(uy, (t_1 * fmaf(-2.0f, (xi * (((float) M_PI) * ((float) M_PI))), (-1.3333333333333333f * (uy * (yi * (((float) M_PI) * (((float) M_PI) * ((float) M_PI)))))))), (t_1 * (2.0f * (((float) M_PI) * yi)))), fmaf(xi, t_1, (maxCos * ((1.0f - ux) * (ux * zi)))));
	} else {
		tmp = fmaf(xi, cosf(t_0), (yi * sinf(t_0)));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(Float32(2.0) * Float32(uy * Float32(pi)))
	t_1 = sqrt(fma(Float32(maxCos * maxCos), Float32(Float32(ux * ux) * Float32(Float32(Float32(1.0) - ux) * Float32(ux + Float32(-1.0)))), Float32(1.0)))
	tmp = Float32(0.0)
	if (Float32(uy * Float32(2.0)) <= Float32(0.07999999821186066))
		tmp = fma(uy, fma(uy, Float32(t_1 * fma(Float32(-2.0), Float32(xi * Float32(Float32(pi) * Float32(pi))), Float32(Float32(-1.3333333333333333) * Float32(uy * Float32(yi * Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi)))))))), Float32(t_1 * Float32(Float32(2.0) * Float32(Float32(pi) * yi)))), fma(xi, t_1, Float32(maxCos * Float32(Float32(Float32(1.0) - ux) * Float32(ux * zi)))));
	else
		tmp = fma(xi, cos(t_0), Float32(yi * sin(t_0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 0.0799999982

   1. Initial program 99.2%

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

   3. Taylor expanded in uy around 0

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

   5. Simplified 98.3%

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

   if 0.0799999982 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 97.6%

      \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
   2. 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 90.5

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

   5. Simplified 90.5%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.07999999821186066:\\ \;\;\;\;\mathsf{fma}\left(uy, \mathsf{fma}\left(uy, \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)} \cdot \mathsf{fma}\left(-2, xi \cdot \left(\pi \cdot \pi\right), -1.3333333333333333 \cdot \left(uy \cdot \left(yi \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\right), \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)} \cdot \left(2 \cdot \left(\pi \cdot yi\right)\right)\right), \mathsf{fma}\left(xi, \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)}, maxCos \cdot \left(\left(1 - ux\right) \cdot \left(ux \cdot zi\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(xi, \cos \left(2 \cdot \left(uy \cdot \pi\right)\right), yi \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 96.1% accurate, 1.5× speedup

Math

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

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) (* zi (* ux maxCos))))))

C

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

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(zi * Float32(ux * maxCos))))
end

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in 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. associate-*r* N/A

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

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

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

   15. *-lowering-*.f32 95.5

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

5. Simplified 95.5%

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

6. Final simplification 95.5%

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

Alternative 10: 92.4% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
2. 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 97.9%

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

6. Taylor expanded in yi around inf

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

8. Simplified 98.4%

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

9. Taylor expanded in uy around 0

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

    1. +-commutative N/A

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

    2. associate-*r* N/A

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

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

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

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

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

    5. unpow2 N/A

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

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

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

    7. unpow2 N/A

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

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

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

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

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

    10. PI-lowering-PI.f32 91.4

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

11. Simplified 91.4%

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

12. Final simplification 91.4%

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

Alternative 11: 92.3% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
2. 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 97.9%

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

6. Taylor expanded in yi around inf

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

8. Simplified 98.4%

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

9. Taylor expanded in ux around 0

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

    1. mul-1-neg N/A

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

    2. unpow2 N/A

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

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

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

    4. mul-1-neg N/A

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

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

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

    6. mul-1-neg N/A

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

    7. neg-lowering-neg.f32 98.1

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

11. Simplified 98.1%

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

12. Taylor expanded in uy around 0

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

    1. +-commutative N/A

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

    2. associate-*r* N/A

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

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

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

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

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

    5. unpow2 N/A

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

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

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

    7. unpow2 N/A

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

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

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

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

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

    10. PI-lowering-PI.f32 91.0

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

14. Simplified 91.0%

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

15. Final simplification 91.0%

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

Alternative 12: 89.4% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
2. 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 97.9%

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

6. Taylor expanded in uy around 0

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

   1. +-commutative N/A

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

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

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

8. Simplified 87.0%

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

9. Taylor expanded in xi around 0

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

11. Simplified 88.1%

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

12. Final simplification 88.1%

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

Alternative 13: 88.4% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in 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 97.9%

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

6. Taylor expanded in yi around inf

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

8. Simplified 98.4%

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

9. Taylor expanded in ux around 0

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

    1. mul-1-neg N/A

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

    2. unpow2 N/A

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

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

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

    4. mul-1-neg N/A

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

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

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

    6. mul-1-neg N/A

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

    7. neg-lowering-neg.f32 98.1

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

11. Simplified 98.1%

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

12. Taylor expanded in uy around 0

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

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

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

14. Simplified 86.9%

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

15. Final simplification 86.9%

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

Alternative 14: 88.5% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
2. 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 97.9%

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

6. Taylor expanded in uy around 0

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

   1. +-commutative N/A

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

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

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

8. Simplified 87.0%

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

9. Taylor expanded in maxCos around 0

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

11. Simplified 86.6%

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

12. Final simplification 86.6%

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

Alternative 15: 84.8% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := ux \cdot \left(1 - ux\right)\\ \mathbf{if}\;ux \cdot \left(\left(1 - ux\right) \cdot maxCos\right) \leq 1.999999936531045 \cdot 10^{-21}:\\ \;\;\;\;xi \cdot \mathsf{fma}\left(uy, \mathsf{fma}\left(uy, \mathsf{fma}\left(-2, \pi \cdot \pi, \frac{-1.3333333333333333 \cdot \left(uy \cdot \left(yi \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right)\right)}{xi}\right), \frac{2 \cdot \left(\pi \cdot yi\right)}{xi}\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-maxCos \cdot maxCos, t\_0 \cdot t\_0, 1\right)}, \mathsf{fma}\left(2, uy \cdot \left(\pi \cdot yi\right), xi\right), \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) \cdot zi\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* ux (- 1.0 ux))))
   (if (<= (* ux (* (- 1.0 ux) maxCos)) 1.999999936531045e-21)
     (*
      xi
      (fma
       uy
       (fma
        uy
        (fma
         -2.0
         (* PI PI)
         (/ (* -1.3333333333333333 (* uy (* yi (* PI (* PI PI))))) xi))
        (/ (* 2.0 (* PI yi)) xi))
       1.0))
     (fma
      (sqrt (fma (- (* maxCos maxCos)) (* t_0 t_0) 1.0))
      (fma 2.0 (* uy (* PI yi)) xi)
      (* (* ux maxCos) (* (- 1.0 ux) zi))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = ux * (1.0f - ux);
	float tmp;
	if ((ux * ((1.0f - ux) * maxCos)) <= 1.999999936531045e-21f) {
		tmp = xi * fmaf(uy, fmaf(uy, fmaf(-2.0f, (((float) M_PI) * ((float) M_PI)), ((-1.3333333333333333f * (uy * (yi * (((float) M_PI) * (((float) M_PI) * ((float) M_PI)))))) / xi)), ((2.0f * (((float) M_PI) * yi)) / xi)), 1.0f);
	} else {
		tmp = fmaf(sqrtf(fmaf(-(maxCos * maxCos), (t_0 * t_0), 1.0f)), fmaf(2.0f, (uy * (((float) M_PI) * yi)), xi), ((ux * maxCos) * ((1.0f - ux) * zi)));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = Float32(ux * Float32(Float32(1.0) - ux))
	tmp = Float32(0.0)
	if (Float32(ux * Float32(Float32(Float32(1.0) - ux) * maxCos)) <= Float32(1.999999936531045e-21))
		tmp = Float32(xi * fma(uy, fma(uy, fma(Float32(-2.0), Float32(Float32(pi) * Float32(pi)), Float32(Float32(Float32(-1.3333333333333333) * Float32(uy * Float32(yi * Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi)))))) / xi)), Float32(Float32(Float32(2.0) * Float32(Float32(pi) * yi)) / xi)), Float32(1.0)));
	else
		tmp = fma(sqrt(fma(Float32(-Float32(maxCos * maxCos)), Float32(t_0 * t_0), Float32(1.0))), fma(Float32(2.0), Float32(uy * Float32(Float32(pi) * yi)), xi), Float32(Float32(ux * maxCos) * Float32(Float32(Float32(1.0) - ux) * zi)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 (-.f32 #s(literal 1 binary32) ux) maxCos) ux) < 1.9999999e-21

   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 98.5%

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

   6. Taylor expanded in uy around 0

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

      1. +-commutative N/A

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

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

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

   8. Simplified 85.7%

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

   9. Taylor expanded in maxCos around 0

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

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

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

       2. +-commutative N/A

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

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

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

   11. Simplified 84.9%

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

   if 1.9999999e-21 < (*.f32 (*.f32 (-.f32 #s(literal 1 binary32) ux) maxCos) ux)

   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 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 97.1%

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

   6. Taylor expanded in yi around inf

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

   8. Simplified 98.7%

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

   9. Taylor expanded in uy around 0

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

       1. +-commutative N/A

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

       2. associate-+r+ N/A

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

       3. associate-*r* N/A

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

       4. distribute-rgt-out N/A

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

   11. Simplified 85.0%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \cdot \left(\left(1 - ux\right) \cdot maxCos\right) \leq 1.999999936531045 \cdot 10^{-21}:\\ \;\;\;\;xi \cdot \mathsf{fma}\left(uy, \mathsf{fma}\left(uy, \mathsf{fma}\left(-2, \pi \cdot \pi, \frac{-1.3333333333333333 \cdot \left(uy \cdot \left(yi \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right)\right)}{xi}\right), \frac{2 \cdot \left(\pi \cdot yi\right)}{xi}\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(-maxCos \cdot maxCos, \left(ux \cdot \left(1 - ux\right)\right) \cdot \left(ux \cdot \left(1 - ux\right)\right), 1\right)}, \mathsf{fma}\left(2, uy \cdot \left(\pi \cdot yi\right), xi\right), \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) \cdot zi\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 86.0% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in 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 97.9%

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

6. Taylor expanded in uy around 0

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

   1. +-commutative N/A

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

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

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

8. Simplified 87.0%

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

9. Taylor expanded in ux around 0

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

    1. +-commutative N/A

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

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

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

11. Simplified 83.8%

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

Alternative 17: 60.7% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;xi \leq -9.999999998199587 \cdot 10^{-24}:\\ \;\;\;\;\left(ux \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right) \cdot zi + xi \cdot \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)}\\ \mathbf{elif}\;xi \leq 3.0000001167615996 \cdot 10^{-17}:\\ \;\;\;\;\left(uy \cdot yi\right) \cdot \mathsf{fma}\left(-1.3333333333333333 \cdot \left(uy \cdot uy\right), \pi \cdot \left(\pi \cdot \pi\right), 2 \cdot \pi\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(maxCos \cdot maxCos\right) \cdot \left(ux \cdot \left(-ux\right)\right), \left(1 - ux\right) \cdot \left(1 - ux\right), 1\right)} \cdot \left(xi \cdot \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \pi \cdot \pi, 1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (if (<= xi -9.999999998199587e-24)
   (+
    (* (* ux (* (- 1.0 ux) maxCos)) zi)
    (*
     xi
     (sqrt
      (fma (* maxCos maxCos) (* (* ux ux) (* (- 1.0 ux) (+ ux -1.0))) 1.0))))
   (if (<= xi 3.0000001167615996e-17)
     (*
      (* uy yi)
      (fma (* -1.3333333333333333 (* uy uy)) (* PI (* PI PI)) (* 2.0 PI)))
     (*
      (sqrt
       (fma (* (* maxCos maxCos) (* ux (- ux))) (* (- 1.0 ux) (- 1.0 ux)) 1.0))
      (* xi (fma (* -2.0 (* uy uy)) (* PI PI) 1.0))))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float tmp;
	if (xi <= -9.999999998199587e-24f) {
		tmp = ((ux * ((1.0f - ux) * maxCos)) * zi) + (xi * sqrtf(fmaf((maxCos * maxCos), ((ux * ux) * ((1.0f - ux) * (ux + -1.0f))), 1.0f)));
	} else if (xi <= 3.0000001167615996e-17f) {
		tmp = (uy * yi) * fmaf((-1.3333333333333333f * (uy * uy)), (((float) M_PI) * (((float) M_PI) * ((float) M_PI))), (2.0f * ((float) M_PI)));
	} else {
		tmp = sqrtf(fmaf(((maxCos * maxCos) * (ux * -ux)), ((1.0f - ux) * (1.0f - ux)), 1.0f)) * (xi * fmaf((-2.0f * (uy * uy)), (((float) M_PI) * ((float) M_PI)), 1.0f));
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	tmp = Float32(0.0)
	if (xi <= Float32(-9.999999998199587e-24))
		tmp = Float32(Float32(Float32(ux * Float32(Float32(Float32(1.0) - ux) * maxCos)) * zi) + Float32(xi * sqrt(fma(Float32(maxCos * maxCos), Float32(Float32(ux * ux) * Float32(Float32(Float32(1.0) - ux) * Float32(ux + Float32(-1.0)))), Float32(1.0)))));
	elseif (xi <= Float32(3.0000001167615996e-17))
		tmp = Float32(Float32(uy * yi) * fma(Float32(Float32(-1.3333333333333333) * Float32(uy * uy)), Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi))), Float32(Float32(2.0) * Float32(pi))));
	else
		tmp = Float32(sqrt(fma(Float32(Float32(maxCos * maxCos) * Float32(ux * Float32(-ux))), Float32(Float32(Float32(1.0) - ux) * Float32(Float32(1.0) - ux)), Float32(1.0))) * Float32(xi * fma(Float32(Float32(-2.0) * Float32(uy * uy)), Float32(Float32(pi) * Float32(pi)), Float32(1.0))));
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if xi < -1e-23

   1. Initial program 99.3%

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

   3. Taylor expanded in uy around 0

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

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

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

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

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. mul-1-neg N/A

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

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

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

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

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

   5. Simplified 66.7%

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

   if -1e-23 < xi < 3.0000001e-17

   1. Initial program 98.4%

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

   3. Taylor expanded in yi around inf

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

      1. *-commutative N/A

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

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

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

   5. Simplified 66.5%

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

   6. Taylor expanded in uy around 0

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

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

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

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

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

      3. associate-*r* N/A

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

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

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

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

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

      6. unpow2 N/A

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

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

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

      8. cube-mult N/A

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

      9. unpow2 N/A

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

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

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

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

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

      12. unpow2 N/A

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

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

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

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

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

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

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

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

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

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

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

      18. PI-lowering-PI.f32 62.0

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

   8. Simplified 62.0%

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

   9. Taylor expanded in yi around 0

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

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

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

       2. associate-*r* N/A

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

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

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

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

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

       5. unpow2 N/A

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

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

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

       7. cube-mult N/A

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

       8. unpow2 N/A

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

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

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

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

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

       11. unpow2 N/A

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

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

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

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

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

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

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

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

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

       16. PI-lowering-PI.f32 62.1

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

   11. Simplified 62.1%

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

   12. Taylor expanded in maxCos around 0

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

       1. associate-*r* N/A

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

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

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

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

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

       4. associate-*r* N/A

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

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

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

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

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

       7. unpow2 N/A

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

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

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

       9. cube-mult N/A

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

       10. unpow2 N/A

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

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

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

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

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

       13. unpow2 N/A

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

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

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

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

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

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

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

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

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

       18. PI-lowering-PI.f32 61.7

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

   14. Simplified 61.7%

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

   if 3.0000001e-17 < xi

   1. Initial program 99.2%

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

   3. Taylor expanded in 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 98.9%

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

   6. Taylor expanded in uy around 0

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

      1. +-commutative N/A

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

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

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

   8. Simplified 85.1%

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

   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. *-commutative N/A

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

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

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

   11. Simplified 73.0%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;xi \leq -9.999999998199587 \cdot 10^{-24}:\\ \;\;\;\;\left(ux \cdot \left(\left(1 - ux\right) \cdot maxCos\right)\right) \cdot zi + xi \cdot \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)}\\ \mathbf{elif}\;xi \leq 3.0000001167615996 \cdot 10^{-17}:\\ \;\;\;\;\left(uy \cdot yi\right) \cdot \mathsf{fma}\left(-1.3333333333333333 \cdot \left(uy \cdot uy\right), \pi \cdot \left(\pi \cdot \pi\right), 2 \cdot \pi\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(maxCos \cdot maxCos\right) \cdot \left(ux \cdot \left(-ux\right)\right), \left(1 - ux\right) \cdot \left(1 - ux\right), 1\right)} \cdot \left(xi \cdot \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \pi \cdot \pi, 1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 81.7% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in 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 97.9%

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

6. Taylor expanded in yi around inf

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

8. Simplified 98.4%

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

9. Taylor expanded in uy around 0

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

    1. +-commutative N/A

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

    2. associate-+r+ N/A

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

    3. associate-*r* N/A

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

    4. distribute-rgt-out N/A

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

11. Simplified 80.8%

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

12. Final simplification 80.8%

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

Alternative 19: 81.7% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in uy around 0

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

   1. +-commutative N/A

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

   2. associate-+r+ N/A

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

   3. associate-*r* N/A

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

   4. distribute-rgt-out N/A

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

5. Simplified 80.8%

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

6. Final simplification 80.8%

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

Alternative 20: 81.2% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in 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 97.9%

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

6. Taylor expanded in yi around inf

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

8. Simplified 98.4%

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

9. Taylor expanded in ux around 0

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

    1. mul-1-neg N/A

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

    2. unpow2 N/A

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

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

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

    4. mul-1-neg N/A

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

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

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

    6. mul-1-neg N/A

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

    7. neg-lowering-neg.f32 98.1

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

11. Simplified 98.1%

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

12. Taylor expanded in uy around 0

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

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

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

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

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

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

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

    4. /-lowering-/.f32 80.3

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

14. Simplified 80.3%

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

15. Final simplification 80.3%

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

Alternative 21: 63.0% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-1.3333333333333333 \cdot \left(uy \cdot uy\right), \pi \cdot \left(\pi \cdot \pi\right), 2 \cdot \pi\right)\\ \mathbf{if}\;yi \leq -1.999999987845058 \cdot 10^{-8}:\\ \;\;\;\;uy \cdot \left(yi \cdot t\_0\right)\\ \mathbf{elif}\;yi \leq 9.999999717180685 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(xi, \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)}, \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) \cdot zi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(uy \cdot yi\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0
         (fma (* -1.3333333333333333 (* uy uy)) (* PI (* PI PI)) (* 2.0 PI))))
   (if (<= yi -1.999999987845058e-8)
     (* uy (* yi t_0))
     (if (<= yi 9.999999717180685e-10)
       (fma
        xi
        (sqrt
         (fma (* maxCos maxCos) (* (* ux ux) (* (- 1.0 ux) (+ ux -1.0))) 1.0))
        (* (* ux maxCos) (* (- 1.0 ux) zi)))
       (* (* uy yi) t_0)))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = fmaf((-1.3333333333333333f * (uy * uy)), (((float) M_PI) * (((float) M_PI) * ((float) M_PI))), (2.0f * ((float) M_PI)));
	float tmp;
	if (yi <= -1.999999987845058e-8f) {
		tmp = uy * (yi * t_0);
	} else if (yi <= 9.999999717180685e-10f) {
		tmp = fmaf(xi, sqrtf(fmaf((maxCos * maxCos), ((ux * ux) * ((1.0f - ux) * (ux + -1.0f))), 1.0f)), ((ux * maxCos) * ((1.0f - ux) * zi)));
	} else {
		tmp = (uy * yi) * t_0;
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = fma(Float32(Float32(-1.3333333333333333) * Float32(uy * uy)), Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi))), Float32(Float32(2.0) * Float32(pi)))
	tmp = Float32(0.0)
	if (yi <= Float32(-1.999999987845058e-8))
		tmp = Float32(uy * Float32(yi * t_0));
	elseif (yi <= Float32(9.999999717180685e-10))
		tmp = fma(xi, sqrt(fma(Float32(maxCos * maxCos), Float32(Float32(ux * ux) * Float32(Float32(Float32(1.0) - ux) * Float32(ux + Float32(-1.0)))), Float32(1.0))), Float32(Float32(ux * maxCos) * Float32(Float32(Float32(1.0) - ux) * zi)));
	else
		tmp = Float32(Float32(uy * yi) * t_0);
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if yi < -1.99999999e-8

   1. Initial program 98.9%

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

   3. Taylor expanded in yi around inf

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

      1. *-commutative N/A

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

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

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

   5. Simplified 67.0%

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

   6. Taylor expanded in uy around 0

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

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

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

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

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

      3. associate-*r* N/A

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

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

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

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

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

      6. unpow2 N/A

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

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

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

      8. cube-mult N/A

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

      9. unpow2 N/A

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

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

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

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

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

      12. unpow2 N/A

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

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

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

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

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

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

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

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

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

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

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

      18. PI-lowering-PI.f32 64.5

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

   8. Simplified 64.5%

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

   9. Taylor expanded in yi around 0

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

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

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

       2. associate-*r* N/A

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

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

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

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

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

       5. unpow2 N/A

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

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

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

       7. cube-mult N/A

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

       8. unpow2 N/A

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

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

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

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

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

       11. unpow2 N/A

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

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

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

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

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

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

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

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

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

       16. PI-lowering-PI.f32 64.5

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

   11. Simplified 64.5%

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

   12. Taylor expanded in maxCos around 0

       \[\leadsto \color{blue}{1} \cdot \left(uy \cdot \left(yi \cdot \mathsf{fma}\left(\frac{-4}{3} \cdot \left(uy \cdot uy\right), \mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), 2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]
   13. Step-by-step derivation

   14. Simplified 63.6%

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

   if -1.99999999e-8 < yi < 9.99999972e-10

   1. Initial program 98.9%

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

   4. Applied egg-rr 98.8%

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

   5. Taylor expanded in uy around 0

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

      1. +-commutative N/A

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

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

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

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

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

      4. +-commutative N/A

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

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

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

      6. unpow2 N/A

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

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

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

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

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

      9. unpow2 N/A

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

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

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

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

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

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

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

      13. sub-neg N/A

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

      14. metadata-eval N/A

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

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

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

   7. Simplified 64.2%

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

   if 9.99999972e-10 < yi

   1. Initial program 98.8%

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

   3. Taylor expanded in yi around inf

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

      1. *-commutative N/A

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

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

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

   5. Simplified 79.5%

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

   6. Taylor expanded in uy around 0

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

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

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

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

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

      3. associate-*r* N/A

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

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

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

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

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

      6. unpow2 N/A

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

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

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

      8. cube-mult N/A

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

      9. unpow2 N/A

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

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

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

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

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

      12. unpow2 N/A

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

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

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

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

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

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

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

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

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

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

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

      18. PI-lowering-PI.f32 70.6

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

   8. Simplified 70.6%

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

   9. Taylor expanded in yi around 0

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

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

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

       2. associate-*r* N/A

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

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

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

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

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

       5. unpow2 N/A

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

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

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

       7. cube-mult N/A

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

       8. unpow2 N/A

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

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

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

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

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

       11. unpow2 N/A

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

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

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

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

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

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

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

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

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

       16. PI-lowering-PI.f32 70.7

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

   11. Simplified 70.7%

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

   12. Taylor expanded in maxCos around 0

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

       1. associate-*r* N/A

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

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

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

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

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

       4. associate-*r* N/A

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

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

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

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

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

       7. unpow2 N/A

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

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

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

       9. cube-mult N/A

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

       10. unpow2 N/A

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

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

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

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

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

       13. unpow2 N/A

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

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

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

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

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

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

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

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

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

       18. PI-lowering-PI.f32 70.9

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

   14. Simplified 70.9%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;yi \leq -1.999999987845058 \cdot 10^{-8}:\\ \;\;\;\;uy \cdot \left(yi \cdot \mathsf{fma}\left(-1.3333333333333333 \cdot \left(uy \cdot uy\right), \pi \cdot \left(\pi \cdot \pi\right), 2 \cdot \pi\right)\right)\\ \mathbf{elif}\;yi \leq 9.999999717180685 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(xi, \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(ux \cdot ux\right) \cdot \left(\left(1 - ux\right) \cdot \left(ux + -1\right)\right), 1\right)}, \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) \cdot zi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(uy \cdot yi\right) \cdot \mathsf{fma}\left(-1.3333333333333333 \cdot \left(uy \cdot uy\right), \pi \cdot \left(\pi \cdot \pi\right), 2 \cdot \pi\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 60.0% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(xi, \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, ux \cdot \left(-ux\right), 1\right)}, \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) \cdot zi\right)\right)\\ \mathbf{if}\;xi \leq -9.999999998199587 \cdot 10^{-24}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;xi \leq 4.9999998413276127 \cdot 10^{-20}:\\ \;\;\;\;uy \cdot \mathsf{fma}\left(-1.3333333333333333 \cdot \left(uy \cdot uy\right), yi \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), \pi \cdot \left(2 \cdot yi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0
         (fma
          xi
          (sqrt (fma (* maxCos maxCos) (* ux (- ux)) 1.0))
          (* (* ux maxCos) (* (- 1.0 ux) zi)))))
   (if (<= xi -9.999999998199587e-24)
     t_0
     (if (<= xi 4.9999998413276127e-20)
       (*
        uy
        (fma
         (* -1.3333333333333333 (* uy uy))
         (* yi (* PI (* PI PI)))
         (* PI (* 2.0 yi))))
       t_0))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	float t_0 = fmaf(xi, sqrtf(fmaf((maxCos * maxCos), (ux * -ux), 1.0f)), ((ux * maxCos) * ((1.0f - ux) * zi)));
	float tmp;
	if (xi <= -9.999999998199587e-24f) {
		tmp = t_0;
	} else if (xi <= 4.9999998413276127e-20f) {
		tmp = uy * fmaf((-1.3333333333333333f * (uy * uy)), (yi * (((float) M_PI) * (((float) M_PI) * ((float) M_PI)))), (((float) M_PI) * (2.0f * yi)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	t_0 = fma(xi, sqrt(fma(Float32(maxCos * maxCos), Float32(ux * Float32(-ux)), Float32(1.0))), Float32(Float32(ux * maxCos) * Float32(Float32(Float32(1.0) - ux) * zi)))
	tmp = Float32(0.0)
	if (xi <= Float32(-9.999999998199587e-24))
		tmp = t_0;
	elseif (xi <= Float32(4.9999998413276127e-20))
		tmp = Float32(uy * fma(Float32(Float32(-1.3333333333333333) * Float32(uy * uy)), Float32(yi * Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi)))), Float32(Float32(pi) * Float32(Float32(2.0) * yi))));
	else
		tmp = t_0;
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if xi < -1e-23 or 4.99999984e-20 < xi

   1. Initial program 99.2%

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

   3. Taylor expanded in 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 98.9%

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

   6. Taylor expanded in yi around inf

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

   8. Simplified 98.5%

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

   9. Taylor expanded in ux around 0

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

       1. mul-1-neg N/A

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

       2. unpow2 N/A

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

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

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

       4. mul-1-neg N/A

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

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

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

       6. mul-1-neg N/A

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

       7. neg-lowering-neg.f32 98.6

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

   11. Simplified 98.6%

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

   12. Taylor expanded in uy around 0

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

       1. +-commutative N/A

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

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

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

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

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

       4. +-commutative N/A

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

       5. mul-1-neg N/A

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

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

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

       7. mul-1-neg N/A

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

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

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

       9. unpow2 N/A

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

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

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

       11. mul-1-neg N/A

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

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

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

       13. unpow2 N/A

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

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

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

       15. associate-*r* N/A

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

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

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

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

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

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

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

       19. --lowering--.f32 66.2

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

   14. Simplified 66.2%

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

   if -1e-23 < xi < 4.99999984e-20

   1. Initial program 98.5%

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

   3. Taylor expanded in yi around inf

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

      1. *-commutative N/A

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

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

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

   5. Simplified 68.8%

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

   6. Taylor expanded in uy around 0

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

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

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

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

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

      3. associate-*r* N/A

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

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

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

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

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

      6. unpow2 N/A

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

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

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

      8. cube-mult N/A

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

      9. unpow2 N/A

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

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

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

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

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

      12. unpow2 N/A

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

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

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

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

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

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

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

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

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

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

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

      18. PI-lowering-PI.f32 63.9

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

   8. Simplified 63.9%

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

   9. Taylor expanded in maxCos around 0

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

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

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

       2. associate-*r* N/A

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

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

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

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

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

       5. unpow2 N/A

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

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

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

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

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

       8. cube-mult N/A

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

       9. unpow2 N/A

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

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

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

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

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

       12. unpow2 N/A

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

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

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

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

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

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

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

       16. associate-*r* N/A

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

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

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

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

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

       19. PI-lowering-PI.f32 63.6

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

   11. Simplified 63.6%

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

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;xi \leq -9.999999998199587 \cdot 10^{-24}:\\ \;\;\;\;\mathsf{fma}\left(xi, \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, ux \cdot \left(-ux\right), 1\right)}, \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) \cdot zi\right)\right)\\ \mathbf{elif}\;xi \leq 4.9999998413276127 \cdot 10^{-20}:\\ \;\;\;\;uy \cdot \mathsf{fma}\left(-1.3333333333333333 \cdot \left(uy \cdot uy\right), yi \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), \pi \cdot \left(2 \cdot yi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(xi, \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, ux \cdot \left(-ux\right), 1\right)}, \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) \cdot zi\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 35.7% accurate, 8.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in yi around inf

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

   1. *-commutative N/A

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

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

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

5. Simplified 41.1%

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

6. Taylor expanded in uy around 0

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

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

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

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

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

   3. associate-*r* N/A

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

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

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

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

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

   6. unpow2 N/A

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

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

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

   8. cube-mult N/A

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

   9. unpow2 N/A

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

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

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

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

       \[\leadsto \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(ux \cdot \left(\mathsf{neg}\left(ux\right)\right)\right), 1\right)} \cdot \left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(\left(uy \cdot uy\right) \cdot yi\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot {\mathsf{PI}\left(\right)}^{2}\right), 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   12. unpow2 N/A

       \[\leadsto \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(ux \cdot \left(\mathsf{neg}\left(ux\right)\right)\right), 1\right)} \cdot \left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(\left(uy \cdot uy\right) \cdot yi\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}\right), 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   13. *-lowering-*.f32 N/A

       \[\leadsto \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(ux \cdot \left(\mathsf{neg}\left(ux\right)\right)\right), 1\right)} \cdot \left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(\left(uy \cdot uy\right) \cdot yi\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}\right), 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   14. PI-lowering-PI.f32 N/A

       \[\leadsto \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(ux \cdot \left(\mathsf{neg}\left(ux\right)\right)\right), 1\right)} \cdot \left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(\left(uy \cdot uy\right) \cdot yi\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right)\right), 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   15. PI-lowering-PI.f32 N/A

       \[\leadsto \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(ux \cdot \left(\mathsf{neg}\left(ux\right)\right)\right), 1\right)} \cdot \left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(\left(uy \cdot uy\right) \cdot yi\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right), 2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

   16. *-lowering-*.f32 N/A

       \[\leadsto \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(ux \cdot \left(\mathsf{neg}\left(ux\right)\right)\right), 1\right)} \cdot \left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(\left(uy \cdot uy\right) \cdot yi\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \]

   17. *-lowering-*.f32 N/A

       \[\leadsto \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(ux \cdot \left(\mathsf{neg}\left(ux\right)\right)\right), 1\right)} \cdot \left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(\left(uy \cdot uy\right) \cdot yi\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right), 2 \cdot \color{blue}{\left(yi \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \]

   18. PI-lowering-PI.f32 38.0

       \[\leadsto \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(ux \cdot \left(-ux\right)\right), 1\right)} \cdot \left(uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(\left(uy \cdot uy\right) \cdot yi\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), 2 \cdot \left(yi \cdot \color{blue}{\pi}\right)\right)\right) \]

8. Simplified 38.0%

   \[\leadsto \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(ux \cdot \left(-ux\right)\right), 1\right)} \cdot \color{blue}{\left(uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(\left(uy \cdot uy\right) \cdot yi\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), 2 \cdot \left(yi \cdot \pi\right)\right)\right)} \]

9. Taylor expanded in yi around 0

   \[\leadsto \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(ux \cdot \left(\mathsf{neg}\left(ux\right)\right)\right), 1\right)} \cdot \left(uy \cdot \color{blue}{\left(yi \cdot \left(\frac{-4}{3} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]
10. Step-by-step derivation

    1. *-lowering-*.f32 N/A

       \[\leadsto \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(ux \cdot \left(\mathsf{neg}\left(ux\right)\right)\right), 1\right)} \cdot \left(uy \cdot \color{blue}{\left(yi \cdot \left(\frac{-4}{3} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

    2. associate-*r* N/A

       \[\leadsto \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(ux \cdot \left(\mathsf{neg}\left(ux\right)\right)\right), 1\right)} \cdot \left(uy \cdot \left(yi \cdot \left(\color{blue}{\left(\frac{-4}{3} \cdot {uy}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{3}} + 2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

    3. accelerator-lowering-fma.f32 N/A

       \[\leadsto \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(ux \cdot \left(\mathsf{neg}\left(ux\right)\right)\right), 1\right)} \cdot \left(uy \cdot \left(yi \cdot \color{blue}{\mathsf{fma}\left(\frac{-4}{3} \cdot {uy}^{2}, {\mathsf{PI}\left(\right)}^{3}, 2 \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \]

    4. *-lowering-*.f32 N/A

       \[\leadsto \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(ux \cdot \left(\mathsf{neg}\left(ux\right)\right)\right), 1\right)} \cdot \left(uy \cdot \left(yi \cdot \mathsf{fma}\left(\color{blue}{\frac{-4}{3} \cdot {uy}^{2}}, {\mathsf{PI}\left(\right)}^{3}, 2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

    5. unpow2 N/A

       \[\leadsto \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(ux \cdot \left(\mathsf{neg}\left(ux\right)\right)\right), 1\right)} \cdot \left(uy \cdot \left(yi \cdot \mathsf{fma}\left(\frac{-4}{3} \cdot \color{blue}{\left(uy \cdot uy\right)}, {\mathsf{PI}\left(\right)}^{3}, 2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

    6. *-lowering-*.f32 N/A

       \[\leadsto \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(ux \cdot \left(\mathsf{neg}\left(ux\right)\right)\right), 1\right)} \cdot \left(uy \cdot \left(yi \cdot \mathsf{fma}\left(\frac{-4}{3} \cdot \color{blue}{\left(uy \cdot uy\right)}, {\mathsf{PI}\left(\right)}^{3}, 2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

    7. cube-mult N/A

       \[\leadsto \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(ux \cdot \left(\mathsf{neg}\left(ux\right)\right)\right), 1\right)} \cdot \left(uy \cdot \left(yi \cdot \mathsf{fma}\left(\frac{-4}{3} \cdot \left(uy \cdot uy\right), \color{blue}{\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}, 2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

    8. unpow2 N/A

       \[\leadsto \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(ux \cdot \left(\mathsf{neg}\left(ux\right)\right)\right), 1\right)} \cdot \left(uy \cdot \left(yi \cdot \mathsf{fma}\left(\frac{-4}{3} \cdot \left(uy \cdot uy\right), \mathsf{PI}\left(\right) \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{2}}, 2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

    9. *-lowering-*.f32 N/A

       \[\leadsto \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(ux \cdot \left(\mathsf{neg}\left(ux\right)\right)\right), 1\right)} \cdot \left(uy \cdot \left(yi \cdot \mathsf{fma}\left(\frac{-4}{3} \cdot \left(uy \cdot uy\right), \color{blue}{\mathsf{PI}\left(\right) \cdot {\mathsf{PI}\left(\right)}^{2}}, 2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

    10. PI-lowering-PI.f32 N/A

        \[\leadsto \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(ux \cdot \left(\mathsf{neg}\left(ux\right)\right)\right), 1\right)} \cdot \left(uy \cdot \left(yi \cdot \mathsf{fma}\left(\frac{-4}{3} \cdot \left(uy \cdot uy\right), \color{blue}{\mathsf{PI}\left(\right)} \cdot {\mathsf{PI}\left(\right)}^{2}, 2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

    11. unpow2 N/A

        \[\leadsto \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(ux \cdot \left(\mathsf{neg}\left(ux\right)\right)\right), 1\right)} \cdot \left(uy \cdot \left(yi \cdot \mathsf{fma}\left(\frac{-4}{3} \cdot \left(uy \cdot uy\right), \mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}, 2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

    12. *-lowering-*.f32 N/A

        \[\leadsto \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(ux \cdot \left(\mathsf{neg}\left(ux\right)\right)\right), 1\right)} \cdot \left(uy \cdot \left(yi \cdot \mathsf{fma}\left(\frac{-4}{3} \cdot \left(uy \cdot uy\right), \mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}, 2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

    13. PI-lowering-PI.f32 N/A

        \[\leadsto \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(ux \cdot \left(\mathsf{neg}\left(ux\right)\right)\right), 1\right)} \cdot \left(uy \cdot \left(yi \cdot \mathsf{fma}\left(\frac{-4}{3} \cdot \left(uy \cdot uy\right), \mathsf{PI}\left(\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right), 2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

    14. PI-lowering-PI.f32 N/A

        \[\leadsto \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(ux \cdot \left(\mathsf{neg}\left(ux\right)\right)\right), 1\right)} \cdot \left(uy \cdot \left(yi \cdot \mathsf{fma}\left(\frac{-4}{3} \cdot \left(uy \cdot uy\right), \mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right), 2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

    15. *-lowering-*.f32 N/A

        \[\leadsto \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(ux \cdot \left(\mathsf{neg}\left(ux\right)\right)\right), 1\right)} \cdot \left(uy \cdot \left(yi \cdot \mathsf{fma}\left(\frac{-4}{3} \cdot \left(uy \cdot uy\right), \mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), \color{blue}{2 \cdot \mathsf{PI}\left(\right)}\right)\right)\right) \]

    16. PI-lowering-PI.f32 38.0

        \[\leadsto \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(ux \cdot \left(-ux\right)\right), 1\right)} \cdot \left(uy \cdot \left(yi \cdot \mathsf{fma}\left(-1.3333333333333333 \cdot \left(uy \cdot uy\right), \pi \cdot \left(\pi \cdot \pi\right), 2 \cdot \color{blue}{\pi}\right)\right)\right) \]

11. Simplified 38.0%

    \[\leadsto \sqrt{\mathsf{fma}\left(maxCos \cdot maxCos, \left(\left(1 - ux\right) \cdot \left(1 - ux\right)\right) \cdot \left(ux \cdot \left(-ux\right)\right), 1\right)} \cdot \left(uy \cdot \color{blue}{\left(yi \cdot \mathsf{fma}\left(-1.3333333333333333 \cdot \left(uy \cdot uy\right), \pi \cdot \left(\pi \cdot \pi\right), 2 \cdot \pi\right)\right)}\right) \]

12. Taylor expanded in maxCos around 0

    \[\leadsto \color{blue}{uy \cdot \left(yi \cdot \left(\frac{-4}{3} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]
13. Step-by-step derivation

    1. associate-*r* N/A

       \[\leadsto \color{blue}{\left(uy \cdot yi\right) \cdot \left(\frac{-4}{3} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)} \]

    2. *-lowering-*.f32 N/A

       \[\leadsto \color{blue}{\left(uy \cdot yi\right) \cdot \left(\frac{-4}{3} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)} \]

    3. *-lowering-*.f32 N/A

       \[\leadsto \color{blue}{\left(uy \cdot yi\right)} \cdot \left(\frac{-4}{3} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right) \]

    4. associate-*r* N/A

       \[\leadsto \left(uy \cdot yi\right) \cdot \left(\color{blue}{\left(\frac{-4}{3} \cdot {uy}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{3}} + 2 \cdot \mathsf{PI}\left(\right)\right) \]

    5. accelerator-lowering-fma.f32 N/A

       \[\leadsto \left(uy \cdot yi\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{-4}{3} \cdot {uy}^{2}, {\mathsf{PI}\left(\right)}^{3}, 2 \cdot \mathsf{PI}\left(\right)\right)} \]

    6. *-lowering-*.f32 N/A

       \[\leadsto \left(uy \cdot yi\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{-4}{3} \cdot {uy}^{2}}, {\mathsf{PI}\left(\right)}^{3}, 2 \cdot \mathsf{PI}\left(\right)\right) \]

    7. unpow2 N/A

       \[\leadsto \left(uy \cdot yi\right) \cdot \mathsf{fma}\left(\frac{-4}{3} \cdot \color{blue}{\left(uy \cdot uy\right)}, {\mathsf{PI}\left(\right)}^{3}, 2 \cdot \mathsf{PI}\left(\right)\right) \]

    8. *-lowering-*.f32 N/A

       \[\leadsto \left(uy \cdot yi\right) \cdot \mathsf{fma}\left(\frac{-4}{3} \cdot \color{blue}{\left(uy \cdot uy\right)}, {\mathsf{PI}\left(\right)}^{3}, 2 \cdot \mathsf{PI}\left(\right)\right) \]

    9. cube-mult N/A

       \[\leadsto \left(uy \cdot yi\right) \cdot \mathsf{fma}\left(\frac{-4}{3} \cdot \left(uy \cdot uy\right), \color{blue}{\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}, 2 \cdot \mathsf{PI}\left(\right)\right) \]

    10. unpow2 N/A

        \[\leadsto \left(uy \cdot yi\right) \cdot \mathsf{fma}\left(\frac{-4}{3} \cdot \left(uy \cdot uy\right), \mathsf{PI}\left(\right) \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{2}}, 2 \cdot \mathsf{PI}\left(\right)\right) \]

    11. *-lowering-*.f32 N/A

        \[\leadsto \left(uy \cdot yi\right) \cdot \mathsf{fma}\left(\frac{-4}{3} \cdot \left(uy \cdot uy\right), \color{blue}{\mathsf{PI}\left(\right) \cdot {\mathsf{PI}\left(\right)}^{2}}, 2 \cdot \mathsf{PI}\left(\right)\right) \]

    12. PI-lowering-PI.f32 N/A

        \[\leadsto \left(uy \cdot yi\right) \cdot \mathsf{fma}\left(\frac{-4}{3} \cdot \left(uy \cdot uy\right), \color{blue}{\mathsf{PI}\left(\right)} \cdot {\mathsf{PI}\left(\right)}^{2}, 2 \cdot \mathsf{PI}\left(\right)\right) \]

    13. unpow2 N/A

        \[\leadsto \left(uy \cdot yi\right) \cdot \mathsf{fma}\left(\frac{-4}{3} \cdot \left(uy \cdot uy\right), \mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}, 2 \cdot \mathsf{PI}\left(\right)\right) \]

    14. *-lowering-*.f32 N/A

        \[\leadsto \left(uy \cdot yi\right) \cdot \mathsf{fma}\left(\frac{-4}{3} \cdot \left(uy \cdot uy\right), \mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}, 2 \cdot \mathsf{PI}\left(\right)\right) \]

    15. PI-lowering-PI.f32 N/A

        \[\leadsto \left(uy \cdot yi\right) \cdot \mathsf{fma}\left(\frac{-4}{3} \cdot \left(uy \cdot uy\right), \mathsf{PI}\left(\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right), 2 \cdot \mathsf{PI}\left(\right)\right) \]

    16. PI-lowering-PI.f32 N/A

        \[\leadsto \left(uy \cdot yi\right) \cdot \mathsf{fma}\left(\frac{-4}{3} \cdot \left(uy \cdot uy\right), \mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right), 2 \cdot \mathsf{PI}\left(\right)\right) \]

    17. *-lowering-*.f32 N/A

        \[\leadsto \left(uy \cdot yi\right) \cdot \mathsf{fma}\left(\frac{-4}{3} \cdot \left(uy \cdot uy\right), \mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), \color{blue}{2 \cdot \mathsf{PI}\left(\right)}\right) \]

    18. PI-lowering-PI.f32 37.9

        \[\leadsto \left(uy \cdot yi\right) \cdot \mathsf{fma}\left(-1.3333333333333333 \cdot \left(uy \cdot uy\right), \pi \cdot \left(\pi \cdot \pi\right), 2 \cdot \color{blue}{\pi}\right) \]

14. Simplified 37.9%

    \[\leadsto \color{blue}{\left(uy \cdot yi\right) \cdot \mathsf{fma}\left(-1.3333333333333333 \cdot \left(uy \cdot uy\right), \pi \cdot \left(\pi \cdot \pi\right), 2 \cdot \pi\right)} \]
15. Add Preprocessing

Alternative 24: 13.0% accurate, 18.6× speedup

Math

\[\begin{array}{l} \\ \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) \cdot zi\right) \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (* (* ux maxCos) (* (- 1.0 ux) zi)))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return (ux * maxCos) * ((1.0f - ux) * zi);
}

Fortran

real(4) function code(xi, yi, zi, ux, uy, maxcos)
    real(4), intent (in) :: xi
    real(4), intent (in) :: yi
    real(4), intent (in) :: zi
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = (ux * maxcos) * ((1.0e0 - ux) * zi)
end function

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return Float32(Float32(ux * maxCos) * Float32(Float32(Float32(1.0) - ux) * zi))
end

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	tmp = (ux * maxCos) * ((single(1.0) - ux) * zi);
end

Derivation

1. Initial program 98.9%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
2. Add Preprocessing

3. Taylor expanded in zi around inf

   \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)} \]
4. Step-by-step derivation

   1. *-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 13.7

      \[\leadsto maxCos \cdot \left(\left(ux \cdot zi\right) \cdot \color{blue}{\left(1 - ux\right)}\right) \]

5. Simplified 13.7%

   \[\leadsto \color{blue}{maxCos \cdot \left(\left(ux \cdot zi\right) \cdot \left(1 - ux\right)\right)} \]
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

      \[\leadsto \color{blue}{\left(maxCos \cdot \left(1 - ux\right)\right) \cdot \left(ux \cdot zi\right)} \]

   3. *-commutative N/A

      \[\leadsto \color{blue}{\left(\left(1 - ux\right) \cdot maxCos\right)} \cdot \left(ux \cdot zi\right) \]

   4. associate-*l* N/A

      \[\leadsto \color{blue}{\left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi} \]

   5. *-commutative N/A

      \[\leadsto \color{blue}{zi \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)} \]

   6. associate-*l* N/A

      \[\leadsto zi \cdot \color{blue}{\left(\left(1 - ux\right) \cdot \left(maxCos \cdot ux\right)\right)} \]

   7. associate-*r* N/A

      \[\leadsto \color{blue}{\left(zi \cdot \left(1 - ux\right)\right) \cdot \left(maxCos \cdot ux\right)} \]

   8. *-lowering-*.f32 N/A

      \[\leadsto \color{blue}{\left(zi \cdot \left(1 - ux\right)\right) \cdot \left(maxCos \cdot ux\right)} \]

   9. *-commutative N/A

      \[\leadsto \color{blue}{\left(\left(1 - ux\right) \cdot zi\right)} \cdot \left(maxCos \cdot ux\right) \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \color{blue}{\left(\left(1 - ux\right) \cdot zi\right)} \cdot \left(maxCos \cdot ux\right) \]

   11. --lowering--.f32 N/A

       \[\leadsto \left(\color{blue}{\left(1 - ux\right)} \cdot zi\right) \cdot \left(maxCos \cdot ux\right) \]

   12. *-commutative N/A

       \[\leadsto \left(\left(1 - ux\right) \cdot zi\right) \cdot \color{blue}{\left(ux \cdot maxCos\right)} \]

   13. *-lowering-*.f32 13.7

       \[\leadsto \left(\left(1 - ux\right) \cdot zi\right) \cdot \color{blue}{\left(ux \cdot maxCos\right)} \]

7. Applied egg-rr 13.7%

   \[\leadsto \color{blue}{\left(\left(1 - ux\right) \cdot zi\right) \cdot \left(ux \cdot maxCos\right)} \]

8. Final simplification 13.7%

   \[\leadsto \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) \cdot zi\right) \]
9. Add Preprocessing

Alternative 25: 13.0% accurate, 18.6× speedup

Math

\[\begin{array}{l} \\ maxCos \cdot \left(ux \cdot \mathsf{fma}\left(ux, -zi, zi\right)\right) \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (* maxCos (* ux (fma ux (- zi) zi))))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return maxCos * (ux * fmaf(ux, -zi, zi));
}

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return Float32(maxCos * Float32(ux * fma(ux, Float32(-zi), zi)))
end

Derivation

1. Initial program 98.9%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
2. Add Preprocessing

3. Taylor expanded in zi around inf

   \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)} \]
4. Step-by-step derivation

   1. *-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 13.7

      \[\leadsto maxCos \cdot \left(\left(ux \cdot zi\right) \cdot \color{blue}{\left(1 - ux\right)}\right) \]

5. Simplified 13.7%

   \[\leadsto \color{blue}{maxCos \cdot \left(\left(ux \cdot zi\right) \cdot \left(1 - ux\right)\right)} \]

6. Taylor expanded in ux around 0

   \[\leadsto maxCos \cdot \color{blue}{\left(ux \cdot \left(zi + -1 \cdot \left(ux \cdot zi\right)\right)\right)} \]
7. Step-by-step derivation

   1. *-lowering-*.f32 N/A

      \[\leadsto maxCos \cdot \color{blue}{\left(ux \cdot \left(zi + -1 \cdot \left(ux \cdot zi\right)\right)\right)} \]

   2. +-commutative N/A

      \[\leadsto maxCos \cdot \left(ux \cdot \color{blue}{\left(-1 \cdot \left(ux \cdot zi\right) + zi\right)}\right) \]

   3. mul-1-neg N/A

      \[\leadsto maxCos \cdot \left(ux \cdot \left(\color{blue}{\left(\mathsf{neg}\left(ux \cdot zi\right)\right)} + zi\right)\right) \]

   4. distribute-rgt-neg-in N/A

      \[\leadsto maxCos \cdot \left(ux \cdot \left(\color{blue}{ux \cdot \left(\mathsf{neg}\left(zi\right)\right)} + zi\right)\right) \]

   5. mul-1-neg N/A

      \[\leadsto maxCos \cdot \left(ux \cdot \left(ux \cdot \color{blue}{\left(-1 \cdot zi\right)} + zi\right)\right) \]

   6. accelerator-lowering-fma.f32 N/A

      \[\leadsto maxCos \cdot \left(ux \cdot \color{blue}{\mathsf{fma}\left(ux, -1 \cdot zi, zi\right)}\right) \]

   7. mul-1-neg N/A

      \[\leadsto maxCos \cdot \left(ux \cdot \mathsf{fma}\left(ux, \color{blue}{\mathsf{neg}\left(zi\right)}, zi\right)\right) \]

   8. neg-lowering-neg.f32 13.7

      \[\leadsto maxCos \cdot \left(ux \cdot \mathsf{fma}\left(ux, \color{blue}{-zi}, zi\right)\right) \]

8. Simplified 13.7%

   \[\leadsto maxCos \cdot \color{blue}{\left(ux \cdot \mathsf{fma}\left(ux, -zi, zi\right)\right)} \]
9. Add Preprocessing

Alternative 26: 11.7% accurate, 32.1× speedup

Math

\[\begin{array}{l} \\ zi \cdot \left(ux \cdot maxCos\right) \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos) :precision binary32 (* zi (* ux maxCos)))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return zi * (ux * maxCos);
}

Fortran

real(4) function code(xi, yi, zi, ux, uy, maxcos)
    real(4), intent (in) :: xi
    real(4), intent (in) :: yi
    real(4), intent (in) :: zi
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = zi * (ux * maxcos)
end function

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return Float32(zi * Float32(ux * maxCos))
end

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	tmp = zi * (ux * maxCos);
end

Derivation

1. Initial program 98.9%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
2. Add Preprocessing

3. Taylor expanded in zi around inf

   \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)} \]
4. Step-by-step derivation

   1. *-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 13.7

      \[\leadsto maxCos \cdot \left(\left(ux \cdot zi\right) \cdot \color{blue}{\left(1 - ux\right)}\right) \]

5. Simplified 13.7%

   \[\leadsto \color{blue}{maxCos \cdot \left(\left(ux \cdot zi\right) \cdot \left(1 - ux\right)\right)} \]

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 12.1

      \[\leadsto maxCos \cdot \color{blue}{\left(ux \cdot zi\right)} \]

8. Simplified 12.1%

   \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot zi\right)} \]
9. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \color{blue}{\left(maxCos \cdot ux\right) \cdot zi} \]

   2. *-lowering-*.f32 N/A

      \[\leadsto \color{blue}{\left(maxCos \cdot ux\right) \cdot zi} \]

   3. *-commutative N/A

      \[\leadsto \color{blue}{\left(ux \cdot maxCos\right)} \cdot zi \]

   4. *-lowering-*.f32 12.1

      \[\leadsto \color{blue}{\left(ux \cdot maxCos\right)} \cdot zi \]

10. Applied egg-rr 12.1%

    \[\leadsto \color{blue}{\left(ux \cdot maxCos\right) \cdot zi} \]

11. Final simplification 12.1%

    \[\leadsto zi \cdot \left(ux \cdot maxCos\right) \]
12. Add Preprocessing

Alternative 27: 11.7% accurate, 32.1× speedup

Math

\[\begin{array}{l} \\ ux \cdot \left(maxCos \cdot zi\right) \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos) :precision binary32 (* ux (* maxCos zi)))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return ux * (maxCos * zi);
}

Fortran

real(4) function code(xi, yi, zi, ux, uy, maxcos)
    real(4), intent (in) :: xi
    real(4), intent (in) :: yi
    real(4), intent (in) :: zi
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = ux * (maxcos * zi)
end function

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return Float32(ux * Float32(maxCos * zi))
end

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	tmp = ux * (maxCos * zi);
end

Derivation

1. Initial program 98.9%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
2. Add Preprocessing

3. Taylor expanded in zi around inf

   \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)} \]
4. Step-by-step derivation

   1. *-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 13.7

      \[\leadsto maxCos \cdot \left(\left(ux \cdot zi\right) \cdot \color{blue}{\left(1 - ux\right)}\right) \]

5. Simplified 13.7%

   \[\leadsto \color{blue}{maxCos \cdot \left(\left(ux \cdot zi\right) \cdot \left(1 - ux\right)\right)} \]

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 12.1

      \[\leadsto maxCos \cdot \color{blue}{\left(ux \cdot zi\right)} \]

8. Simplified 12.1%

   \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot zi\right)} \]
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 12.1

      \[\leadsto \color{blue}{\left(maxCos \cdot zi\right)} \cdot ux \]

10. Applied egg-rr 12.1%

    \[\leadsto \color{blue}{\left(maxCos \cdot zi\right) \cdot ux} \]

11. Final simplification 12.1%

    \[\leadsto ux \cdot \left(maxCos \cdot zi\right) \]
12. Add Preprocessing

Alternative 28: 11.7% accurate, 32.1× speedup

Math

\[\begin{array}{l} \\ maxCos \cdot \left(ux \cdot zi\right) \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos) :precision binary32 (* maxCos (* ux zi)))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return maxCos * (ux * zi);
}

Fortran

real(4) function code(xi, yi, zi, ux, uy, maxcos)
    real(4), intent (in) :: xi
    real(4), intent (in) :: yi
    real(4), intent (in) :: zi
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = maxcos * (ux * zi)
end function

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return Float32(maxCos * Float32(ux * zi))
end

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	tmp = maxCos * (ux * zi);
end

Derivation

1. Initial program 98.9%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
2. Add Preprocessing

3. Taylor expanded in zi around inf

   \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right)} \]
4. Step-by-step derivation

   1. *-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 13.7

      \[\leadsto maxCos \cdot \left(\left(ux \cdot zi\right) \cdot \color{blue}{\left(1 - ux\right)}\right) \]

5. Simplified 13.7%

   \[\leadsto \color{blue}{maxCos \cdot \left(\left(ux \cdot zi\right) \cdot \left(1 - ux\right)\right)} \]

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 12.1

      \[\leadsto maxCos \cdot \color{blue}{\left(ux \cdot zi\right)} \]

8. Simplified 12.1%

   \[\leadsto \color{blue}{maxCos \cdot \left(ux \cdot zi\right)} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024198 
(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)))