UniformSampleCone, x

Percentage Accurate: 57.1% → 99.0%

Time: 17.5s

Alternatives: 19

Speedup: 9.8×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 57.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 54.8%

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

3. Taylor expanded in ux around 0

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

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

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

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

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

   3. +-commutative N/A

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

   4. metadata-eval N/A

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

   5. associate-+l+ N/A

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

   6. mul-1-neg N/A

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

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

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

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

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

   9. unpow2 N/A

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

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

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

   11. neg-sub0 N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos - 1\right) \cdot \color{blue}{\left(0 - \left(maxCos - 1\right)\right)}, 2 + -2 \cdot maxCos\right)} \]

   12. associate-+l- N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos - 1\right) \cdot \color{blue}{\left(\left(0 - maxCos\right) + 1\right)}, 2 + -2 \cdot maxCos\right)} \]

   13. neg-sub0 N/A

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

   14. mul-1-neg N/A

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

   15. +-commutative N/A

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

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

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

   17. sub-neg N/A

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

   18. metadata-eval N/A

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

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

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

   20. mul-1-neg N/A

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

   21. unsub-neg N/A

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

   22. --lowering--.f32 N/A

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

5. Simplified 98.9%

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

   1. associate-+r+ N/A

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

   2. distribute-rgt-in N/A

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

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

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

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

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

   5. sub-neg N/A

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

   6. metadata-eval N/A

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

   7. neg-mul-1 N/A

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

   8. distribute-lft-in N/A

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

   9. +-commutative N/A

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

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

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

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

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

   12. +-commutative N/A

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

   13. distribute-lft-in N/A

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

   14. metadata-eval N/A

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

   15. neg-mul-1 N/A

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

   16. sub-neg N/A

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

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

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

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

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

   19. *-lowering-*.f32 98.9

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

7. Applied egg-rr 98.9%

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

Alternative 2: 97.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{ux \cdot \left(\mathsf{fma}\left(-2, maxCos, 2\right) + \left(maxCos + -1\right) \cdot \mathsf{fma}\left(-maxCos, ux, ux\right)\right)}\\ \mathbf{if}\;\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \leq 0.9991999864578247:\\ \;\;\;\;\sqrt{ux} \cdot \left(\sqrt{2 - ux} \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2, t\_0 \cdot \left(\left(uy \cdot uy\right) \cdot \left(\pi \cdot \pi\right)\right), t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0
         (sqrt
          (*
           ux
           (+
            (fma -2.0 maxCos 2.0)
            (* (+ maxCos -1.0) (fma (- maxCos) ux ux)))))))
   (if (<= (cos (* (* uy 2.0) PI)) 0.9991999864578247)
     (* (sqrt ux) (* (sqrt (- 2.0 ux)) (cos (* 2.0 (* uy PI)))))
     (fma -2.0 (* t_0 (* (* uy uy) (* PI PI))) t_0))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = sqrtf((ux * (fmaf(-2.0f, maxCos, 2.0f) + ((maxCos + -1.0f) * fmaf(-maxCos, ux, ux)))));
	float tmp;
	if (cosf(((uy * 2.0f) * ((float) M_PI))) <= 0.9991999864578247f) {
		tmp = sqrtf(ux) * (sqrtf((2.0f - ux)) * cosf((2.0f * (uy * ((float) M_PI)))));
	} else {
		tmp = fmaf(-2.0f, (t_0 * ((uy * uy) * (((float) M_PI) * ((float) M_PI)))), t_0);
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = sqrt(Float32(ux * Float32(fma(Float32(-2.0), maxCos, Float32(2.0)) + Float32(Float32(maxCos + Float32(-1.0)) * fma(Float32(-maxCos), ux, ux)))))
	tmp = Float32(0.0)
	if (cos(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) <= Float32(0.9991999864578247))
		tmp = Float32(sqrt(ux) * Float32(sqrt(Float32(Float32(2.0) - ux)) * cos(Float32(Float32(2.0) * Float32(uy * Float32(pi))))));
	else
		tmp = fma(Float32(-2.0), Float32(t_0 * Float32(Float32(uy * uy) * Float32(Float32(pi) * Float32(pi)))), t_0);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 (*.f32 uy #s(literal 2 binary32)) (PI.f32))) < 0.999199986

   1. Initial program 56.3%

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

   3. Taylor expanded in ux around 0

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

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

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

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

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

      5. associate-+l+ N/A

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

      6. mul-1-neg N/A

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

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

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

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

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

      9. unpow2 N/A

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

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

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

      11. neg-sub0 N/A

          \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos - 1\right) \cdot \color{blue}{\left(0 - \left(maxCos - 1\right)\right)}, 2 + -2 \cdot maxCos\right)} \]

      12. associate-+l- N/A

          \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos - 1\right) \cdot \color{blue}{\left(\left(0 - maxCos\right) + 1\right)}, 2 + -2 \cdot maxCos\right)} \]

      13. neg-sub0 N/A

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

      14. mul-1-neg N/A

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

      15. +-commutative N/A

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

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

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

      17. sub-neg N/A

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

      18. metadata-eval N/A

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

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

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

      20. mul-1-neg N/A

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

      21. unsub-neg N/A

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

      22. --lowering--.f32 N/A

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

   5. Simplified 97.6%

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

   6. Taylor expanded in maxCos around 0

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

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

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

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

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

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

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

      4. mul-1-neg N/A

         \[\leadsto \sqrt{ux \cdot \left(2 + \color{blue}{\left(\mathsf{neg}\left(ux\right)\right)}\right)} \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \]

      5. unsub-neg N/A

         \[\leadsto \sqrt{ux \cdot \color{blue}{\left(2 - ux\right)}} \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \]

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

         \[\leadsto \sqrt{ux \cdot \color{blue}{\left(2 - ux\right)}} \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \]

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

         \[\leadsto \sqrt{ux \cdot \left(2 - ux\right)} \cdot \color{blue}{\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \]

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

         \[\leadsto \sqrt{ux \cdot \left(2 - ux\right)} \cdot \cos \color{blue}{\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \]

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

         \[\leadsto \sqrt{ux \cdot \left(2 - ux\right)} \cdot \cos \left(2 \cdot \color{blue}{\left(uy \cdot \mathsf{PI}\left(\right)\right)}\right) \]

      10. PI-lowering-PI.f32 91.6

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

   8. Simplified 91.6%

      \[\leadsto \color{blue}{\sqrt{ux \cdot \left(2 - ux\right)} \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right)} \]
   9. Step-by-step derivation

      1. pow1/2 N/A

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

      2. unpow-prod-down N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. add-sqr-sqrt N/A

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

      6. associate-*l* N/A

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

      7. associate-*l* N/A

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

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

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

      9. pow1/2 N/A

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

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

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

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

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

      12. pow1/2 N/A

          \[\leadsto \sqrt{ux} \cdot \left(\color{blue}{\sqrt{2 - ux}} \cdot \cos \left(\left(\left(uy \cdot 2\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right) \]

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

          \[\leadsto \sqrt{ux} \cdot \left(\color{blue}{\sqrt{2 - ux}} \cdot \cos \left(\left(\left(uy \cdot 2\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right) \]

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

          \[\leadsto \sqrt{ux} \cdot \left(\sqrt{\color{blue}{2 - ux}} \cdot \cos \left(\left(\left(uy \cdot 2\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right) \]

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

          \[\leadsto \sqrt{ux} \cdot \left(\sqrt{2 - ux} \cdot \color{blue}{\cos \left(\left(\left(uy \cdot 2\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right) \]

      16. associate-*l* N/A

          \[\leadsto \sqrt{ux} \cdot \left(\sqrt{2 - ux} \cdot \cos \color{blue}{\left(\left(uy \cdot 2\right) \cdot \left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right)}\right) \]

      17. add-sqr-sqrt N/A

          \[\leadsto \sqrt{ux} \cdot \left(\sqrt{2 - ux} \cdot \cos \left(\left(uy \cdot 2\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \]

   10. Applied egg-rr 91.9%

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

   if 0.999199986 < (cos.f32 (*.f32 (*.f32 uy #s(literal 2 binary32)) (PI.f32)))

   1. Initial program 54.4%

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

   3. Taylor expanded in ux around 0

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

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

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

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

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

      5. associate-+l+ N/A

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

      6. mul-1-neg N/A

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

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

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

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

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

      9. unpow2 N/A

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

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

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

      11. neg-sub0 N/A

          \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos - 1\right) \cdot \color{blue}{\left(0 - \left(maxCos - 1\right)\right)}, 2 + -2 \cdot maxCos\right)} \]

      12. associate-+l- N/A

          \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos - 1\right) \cdot \color{blue}{\left(\left(0 - maxCos\right) + 1\right)}, 2 + -2 \cdot maxCos\right)} \]

      13. neg-sub0 N/A

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

      14. mul-1-neg N/A

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

      15. +-commutative N/A

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

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

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

      17. sub-neg N/A

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

      18. metadata-eval N/A

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

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

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

      20. mul-1-neg N/A

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

      21. unsub-neg N/A

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

      22. --lowering--.f32 N/A

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

   5. Simplified 99.2%

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

   6. Taylor expanded in uy around 0

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

      1. +-commutative N/A

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

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

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

   8. Simplified 99.2%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \leq 0.9991999864578247:\\ \;\;\;\;\sqrt{ux} \cdot \left(\sqrt{2 - ux} \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2, \sqrt{ux \cdot \left(\mathsf{fma}\left(-2, maxCos, 2\right) + \left(maxCos + -1\right) \cdot \mathsf{fma}\left(-maxCos, ux, ux\right)\right)} \cdot \left(\left(uy \cdot uy\right) \cdot \left(\pi \cdot \pi\right)\right), \sqrt{ux \cdot \left(\mathsf{fma}\left(-2, maxCos, 2\right) + \left(maxCos + -1\right) \cdot \mathsf{fma}\left(-maxCos, ux, ux\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 54.8%

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

3. Taylor expanded in ux around 0

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

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

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

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

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

   3. +-commutative N/A

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

   4. metadata-eval N/A

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

   5. associate-+l+ N/A

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

   6. mul-1-neg N/A

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

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

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

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

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

   9. unpow2 N/A

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

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

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

   11. neg-sub0 N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos - 1\right) \cdot \color{blue}{\left(0 - \left(maxCos - 1\right)\right)}, 2 + -2 \cdot maxCos\right)} \]

   12. associate-+l- N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos - 1\right) \cdot \color{blue}{\left(\left(0 - maxCos\right) + 1\right)}, 2 + -2 \cdot maxCos\right)} \]

   13. neg-sub0 N/A

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

   14. mul-1-neg N/A

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

   15. +-commutative N/A

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

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

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

   17. sub-neg N/A

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

   18. metadata-eval N/A

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

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

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

   20. mul-1-neg N/A

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

   21. unsub-neg N/A

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

   22. --lowering--.f32 N/A

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

5. Simplified 98.9%

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

Alternative 4: 98.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (cos (* (* uy 2.0) PI))
  (sqrt (fma ux (- 2.0 ux) (* (* ux maxCos) (fma 2.0 ux -2.0))))))

C

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

Julia

function code(ux, uy, maxCos)
	return Float32(cos(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) * sqrt(fma(ux, Float32(Float32(2.0) - ux), Float32(Float32(ux * maxCos) * fma(Float32(2.0), ux, Float32(-2.0))))))
end

Derivation

1. Initial program 54.8%

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

3. Taylor expanded in ux around 0

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

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

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

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

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

   3. +-commutative N/A

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

   4. metadata-eval N/A

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

   5. associate-+l+ N/A

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

   6. mul-1-neg N/A

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

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

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

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

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

   9. unpow2 N/A

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

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

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

   11. neg-sub0 N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos - 1\right) \cdot \color{blue}{\left(0 - \left(maxCos - 1\right)\right)}, 2 + -2 \cdot maxCos\right)} \]

   12. associate-+l- N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos - 1\right) \cdot \color{blue}{\left(\left(0 - maxCos\right) + 1\right)}, 2 + -2 \cdot maxCos\right)} \]

   13. neg-sub0 N/A

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

   14. mul-1-neg N/A

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

   15. +-commutative N/A

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

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

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

   17. sub-neg N/A

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

   18. metadata-eval N/A

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

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

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

   20. mul-1-neg N/A

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

   21. unsub-neg N/A

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

   22. --lowering--.f32 N/A

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

5. Simplified 98.9%

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

6. Taylor expanded in maxCos around 0

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

   1. +-commutative N/A

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

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

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

   3. mul-1-neg N/A

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

   4. unsub-neg N/A

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

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

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

   6. associate-*r* N/A

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

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

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

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

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

   9. sub-neg N/A

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

   10. metadata-eval N/A

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

   11. accelerator-lowering-fma.f32 98.2

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

8. Simplified 98.2%

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

9. Final simplification 98.2%

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

Alternative 5: 97.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 54.8%

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

3. Taylor expanded in ux around 0

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

   1. +-commutative N/A

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

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

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

   3. +-commutative N/A

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

   4. associate--l+ N/A

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

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

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

   6. unpow2 N/A

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

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

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

   8. sub-neg N/A

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

   9. metadata-eval N/A

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

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

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

   11. sub-neg N/A

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

   12. metadata-eval N/A

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

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

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

   14. sub-neg N/A

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

   15. metadata-eval N/A

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

   16. accelerator-lowering-fma.f32 58.6

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

5. Simplified 58.6%

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

6. Taylor expanded in maxCos around 0

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

   1. sub-neg N/A

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

   2. associate-*r* N/A

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

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

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

   4. mul-1-neg N/A

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

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

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

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

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

   7. +-commutative N/A

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

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

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

   9. mul-1-neg N/A

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

   10. associate-*r* N/A

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

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

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

   12. mul-1-neg N/A

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

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

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

   14. sub-neg N/A

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

   15. metadata-eval N/A

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

   16. +-lowering-+.f32 98.2

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

8. Simplified 98.2%

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

9. Taylor expanded in ux around 0

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

    1. *-commutative N/A

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

    2. *-lowering-*.f32 97.3

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

11. Simplified 97.3%

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

12. Final simplification 97.3%

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

Alternative 6: 97.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{ux \cdot \left(\mathsf{fma}\left(-2, maxCos, 2\right) + \left(maxCos + -1\right) \cdot \mathsf{fma}\left(-maxCos, ux, ux\right)\right)}\\ \mathbf{if}\;uy \cdot 2 \leq 0.012000000104308128:\\ \;\;\;\;\mathsf{fma}\left(-2, t\_0 \cdot \left(\left(uy \cdot uy\right) \cdot \left(\pi \cdot \pi\right)\right), t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0
         (sqrt
          (*
           ux
           (+
            (fma -2.0 maxCos 2.0)
            (* (+ maxCos -1.0) (fma (- maxCos) ux ux)))))))
   (if (<= (* uy 2.0) 0.012000000104308128)
     (fma -2.0 (* t_0 (* (* uy uy) (* PI PI))) t_0)
     (* (cos (* 2.0 (* uy PI))) (sqrt (* ux (- 2.0 ux)))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = sqrtf((ux * (fmaf(-2.0f, maxCos, 2.0f) + ((maxCos + -1.0f) * fmaf(-maxCos, ux, ux)))));
	float tmp;
	if ((uy * 2.0f) <= 0.012000000104308128f) {
		tmp = fmaf(-2.0f, (t_0 * ((uy * uy) * (((float) M_PI) * ((float) M_PI)))), t_0);
	} else {
		tmp = cosf((2.0f * (uy * ((float) M_PI)))) * sqrtf((ux * (2.0f - ux)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = sqrt(Float32(ux * Float32(fma(Float32(-2.0), maxCos, Float32(2.0)) + Float32(Float32(maxCos + Float32(-1.0)) * fma(Float32(-maxCos), ux, ux)))))
	tmp = Float32(0.0)
	if (Float32(uy * Float32(2.0)) <= Float32(0.012000000104308128))
		tmp = fma(Float32(-2.0), Float32(t_0 * Float32(Float32(uy * uy) * Float32(Float32(pi) * Float32(pi)))), t_0);
	else
		tmp = Float32(cos(Float32(Float32(2.0) * Float32(uy * Float32(pi)))) * sqrt(Float32(ux * Float32(Float32(2.0) - ux))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 0.0120000001

   1. Initial program 54.4%

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

   3. Taylor expanded in ux around 0

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

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

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

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

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

      5. associate-+l+ N/A

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

      6. mul-1-neg N/A

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

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

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

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

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

      9. unpow2 N/A

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

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

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

      11. neg-sub0 N/A

          \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos - 1\right) \cdot \color{blue}{\left(0 - \left(maxCos - 1\right)\right)}, 2 + -2 \cdot maxCos\right)} \]

      12. associate-+l- N/A

          \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos - 1\right) \cdot \color{blue}{\left(\left(0 - maxCos\right) + 1\right)}, 2 + -2 \cdot maxCos\right)} \]

      13. neg-sub0 N/A

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

      14. mul-1-neg N/A

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

      15. +-commutative N/A

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

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

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

      17. sub-neg N/A

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

      18. metadata-eval N/A

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

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

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

      20. mul-1-neg N/A

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

      21. unsub-neg N/A

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

      22. --lowering--.f32 N/A

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

   5. Simplified 99.2%

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

   6. Taylor expanded in uy around 0

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

      1. +-commutative N/A

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

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

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

   8. Simplified 99.2%

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

   if 0.0120000001 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 56.3%

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

   3. Taylor expanded in ux around 0

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

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

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

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

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

      5. associate-+l+ N/A

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

      6. mul-1-neg N/A

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

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

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

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

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

      9. unpow2 N/A

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

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

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

      11. neg-sub0 N/A

          \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos - 1\right) \cdot \color{blue}{\left(0 - \left(maxCos - 1\right)\right)}, 2 + -2 \cdot maxCos\right)} \]

      12. associate-+l- N/A

          \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos - 1\right) \cdot \color{blue}{\left(\left(0 - maxCos\right) + 1\right)}, 2 + -2 \cdot maxCos\right)} \]

      13. neg-sub0 N/A

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

      14. mul-1-neg N/A

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

      15. +-commutative N/A

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

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

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

      17. sub-neg N/A

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

      18. metadata-eval N/A

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

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

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

      20. mul-1-neg N/A

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

      21. unsub-neg N/A

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

      22. --lowering--.f32 N/A

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

   5. Simplified 97.6%

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

   6. Taylor expanded in maxCos around 0

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

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

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

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

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

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

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

      4. mul-1-neg N/A

         \[\leadsto \sqrt{ux \cdot \left(2 + \color{blue}{\left(\mathsf{neg}\left(ux\right)\right)}\right)} \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \]

      5. unsub-neg N/A

         \[\leadsto \sqrt{ux \cdot \color{blue}{\left(2 - ux\right)}} \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \]

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

         \[\leadsto \sqrt{ux \cdot \color{blue}{\left(2 - ux\right)}} \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \]

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

         \[\leadsto \sqrt{ux \cdot \left(2 - ux\right)} \cdot \color{blue}{\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \]

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

         \[\leadsto \sqrt{ux \cdot \left(2 - ux\right)} \cdot \cos \color{blue}{\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \]

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

         \[\leadsto \sqrt{ux \cdot \left(2 - ux\right)} \cdot \cos \left(2 \cdot \color{blue}{\left(uy \cdot \mathsf{PI}\left(\right)\right)}\right) \]

      10. PI-lowering-PI.f32 91.6

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

   8. Simplified 91.6%

      \[\leadsto \color{blue}{\sqrt{ux \cdot \left(2 - ux\right)} \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.012000000104308128:\\ \;\;\;\;\mathsf{fma}\left(-2, \sqrt{ux \cdot \left(\mathsf{fma}\left(-2, maxCos, 2\right) + \left(maxCos + -1\right) \cdot \mathsf{fma}\left(-maxCos, ux, ux\right)\right)} \cdot \left(\left(uy \cdot uy\right) \cdot \left(\pi \cdot \pi\right)\right), \sqrt{ux \cdot \left(\mathsf{fma}\left(-2, maxCos, 2\right) + \left(maxCos + -1\right) \cdot \mathsf{fma}\left(-maxCos, ux, ux\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 93.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{ux \cdot \left(\mathsf{fma}\left(-2, maxCos, 2\right) + \left(maxCos + -1\right) \cdot \mathsf{fma}\left(-maxCos, ux, ux\right)\right)}\\ \mathbf{if}\;uy \cdot 2 \leq 0.10000000149011612:\\ \;\;\;\;\mathsf{fma}\left(-2, t\_0 \cdot \left(\left(uy \cdot uy\right) \cdot \left(\pi \cdot \pi\right)\right), t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{2 \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0
         (sqrt
          (*
           ux
           (+
            (fma -2.0 maxCos 2.0)
            (* (+ maxCos -1.0) (fma (- maxCos) ux ux)))))))
   (if (<= (* uy 2.0) 0.10000000149011612)
     (fma -2.0 (* t_0 (* (* uy uy) (* PI PI))) t_0)
     (* (cos (* (* uy 2.0) PI)) (sqrt (* 2.0 ux))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = sqrtf((ux * (fmaf(-2.0f, maxCos, 2.0f) + ((maxCos + -1.0f) * fmaf(-maxCos, ux, ux)))));
	float tmp;
	if ((uy * 2.0f) <= 0.10000000149011612f) {
		tmp = fmaf(-2.0f, (t_0 * ((uy * uy) * (((float) M_PI) * ((float) M_PI)))), t_0);
	} else {
		tmp = cosf(((uy * 2.0f) * ((float) M_PI))) * sqrtf((2.0f * ux));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = sqrt(Float32(ux * Float32(fma(Float32(-2.0), maxCos, Float32(2.0)) + Float32(Float32(maxCos + Float32(-1.0)) * fma(Float32(-maxCos), ux, ux)))))
	tmp = Float32(0.0)
	if (Float32(uy * Float32(2.0)) <= Float32(0.10000000149011612))
		tmp = fma(Float32(-2.0), Float32(t_0 * Float32(Float32(uy * uy) * Float32(Float32(pi) * Float32(pi)))), t_0);
	else
		tmp = Float32(cos(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) * sqrt(Float32(Float32(2.0) * ux)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 0.100000001

   1. Initial program 54.8%

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

   3. Taylor expanded in ux around 0

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

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

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

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

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

      5. associate-+l+ N/A

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

      6. mul-1-neg N/A

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

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

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

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

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

      9. unpow2 N/A

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

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

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

      11. neg-sub0 N/A

          \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos - 1\right) \cdot \color{blue}{\left(0 - \left(maxCos - 1\right)\right)}, 2 + -2 \cdot maxCos\right)} \]

      12. associate-+l- N/A

          \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos - 1\right) \cdot \color{blue}{\left(\left(0 - maxCos\right) + 1\right)}, 2 + -2 \cdot maxCos\right)} \]

      13. neg-sub0 N/A

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

      14. mul-1-neg N/A

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

      15. +-commutative N/A

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

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

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

      17. sub-neg N/A

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

      18. metadata-eval N/A

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

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

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

      20. mul-1-neg N/A

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

      21. unsub-neg N/A

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

      22. --lowering--.f32 N/A

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

   5. Simplified 99.2%

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

   6. Taylor expanded in uy around 0

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

      1. +-commutative N/A

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

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

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

   8. Simplified 97.7%

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

   if 0.100000001 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 54.4%

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

   3. Taylor expanded in maxCos around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

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

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

      5. mul-1-neg N/A

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

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

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

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

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

      8. mul-1-neg N/A

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

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

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

      10. --lowering--.f32 53.4

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

   5. Simplified 53.4%

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

   6. Taylor expanded in ux around 0

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

      1. *-lowering-*.f32 73.8

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

   8. Simplified 73.8%

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\color{blue}{2 \cdot ux}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.10000000149011612:\\ \;\;\;\;\mathsf{fma}\left(-2, \sqrt{ux \cdot \left(\mathsf{fma}\left(-2, maxCos, 2\right) + \left(maxCos + -1\right) \cdot \mathsf{fma}\left(-maxCos, ux, ux\right)\right)} \cdot \left(\left(uy \cdot uy\right) \cdot \left(\pi \cdot \pi\right)\right), \sqrt{ux \cdot \left(\mathsf{fma}\left(-2, maxCos, 2\right) + \left(maxCos + -1\right) \cdot \mathsf{fma}\left(-maxCos, ux, ux\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{2 \cdot ux}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 88.2% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 54.8%

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

3. Taylor expanded in ux around 0

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

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

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

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

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

   3. +-commutative N/A

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

   4. metadata-eval N/A

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

   5. associate-+l+ N/A

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

   6. mul-1-neg N/A

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

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

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

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

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

   9. unpow2 N/A

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

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

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

   11. neg-sub0 N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos - 1\right) \cdot \color{blue}{\left(0 - \left(maxCos - 1\right)\right)}, 2 + -2 \cdot maxCos\right)} \]

   12. associate-+l- N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos - 1\right) \cdot \color{blue}{\left(\left(0 - maxCos\right) + 1\right)}, 2 + -2 \cdot maxCos\right)} \]

   13. neg-sub0 N/A

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

   14. mul-1-neg N/A

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

   15. +-commutative N/A

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

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

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

   17. sub-neg N/A

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

   18. metadata-eval N/A

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

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

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

   20. mul-1-neg N/A

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

   21. unsub-neg N/A

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

   22. --lowering--.f32 N/A

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

5. Simplified 98.9%

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

6. Taylor expanded in uy around 0

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

   1. +-commutative N/A

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

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

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

8. Simplified 87.4%

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

9. Final simplification 87.4%

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

Alternative 9: 88.2% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 54.8%

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

3. Taylor expanded in ux around 0

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

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

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

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

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

   3. +-commutative N/A

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

   4. metadata-eval N/A

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

   5. associate-+l+ N/A

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

   6. mul-1-neg N/A

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

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

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

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

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

   9. unpow2 N/A

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

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

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

   11. neg-sub0 N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos - 1\right) \cdot \color{blue}{\left(0 - \left(maxCos - 1\right)\right)}, 2 + -2 \cdot maxCos\right)} \]

   12. associate-+l- N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos - 1\right) \cdot \color{blue}{\left(\left(0 - maxCos\right) + 1\right)}, 2 + -2 \cdot maxCos\right)} \]

   13. neg-sub0 N/A

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

   14. mul-1-neg N/A

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

   15. +-commutative N/A

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

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

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

   17. sub-neg N/A

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

   18. metadata-eval N/A

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

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

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

   20. mul-1-neg N/A

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

   21. unsub-neg N/A

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

   22. --lowering--.f32 N/A

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

5. Simplified 98.9%

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

   1. associate-+r+ N/A

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

   2. distribute-rgt-in N/A

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

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

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

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

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

   5. sub-neg N/A

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

   6. metadata-eval N/A

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

   7. neg-mul-1 N/A

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

   8. distribute-lft-in N/A

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

   9. +-commutative N/A

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

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

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

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

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

   12. +-commutative N/A

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

   13. distribute-lft-in N/A

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

   14. metadata-eval N/A

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

   15. neg-mul-1 N/A

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

   16. sub-neg N/A

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

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

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

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

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

   19. *-lowering-*.f32 98.9

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

7. Applied egg-rr 98.9%

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

8. Taylor expanded in uy around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

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

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

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

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

   5. unpow2 N/A

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

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

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

   7. unpow2 N/A

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

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

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

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

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

   10. PI-lowering-PI.f32 87.3

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

10. Simplified 87.3%

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

11. Final simplification 87.3%

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

Alternative 10: 86.6% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{ux \cdot \left(2 - ux\right)}\\ \mathbf{if}\;maxCos \leq 4.999999987376214 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(-2, \left(\left(uy \cdot uy\right) \cdot \left(\pi \cdot \pi\right)\right) \cdot t\_0, t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{ux \cdot \left(\mathsf{fma}\left(-2, maxCos, 2\right) + \left(maxCos + -1\right) \cdot \mathsf{fma}\left(-maxCos, ux, ux\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (sqrt (* ux (- 2.0 ux)))))
   (if (<= maxCos 4.999999987376214e-7)
     (fma -2.0 (* (* (* uy uy) (* PI PI)) t_0) t_0)
     (sqrt
      (*
       ux
       (+
        (fma -2.0 maxCos 2.0)
        (* (+ maxCos -1.0) (fma (- maxCos) ux ux))))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = sqrtf((ux * (2.0f - ux)));
	float tmp;
	if (maxCos <= 4.999999987376214e-7f) {
		tmp = fmaf(-2.0f, (((uy * uy) * (((float) M_PI) * ((float) M_PI))) * t_0), t_0);
	} else {
		tmp = sqrtf((ux * (fmaf(-2.0f, maxCos, 2.0f) + ((maxCos + -1.0f) * fmaf(-maxCos, ux, ux)))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = sqrt(Float32(ux * Float32(Float32(2.0) - ux)))
	tmp = Float32(0.0)
	if (maxCos <= Float32(4.999999987376214e-7))
		tmp = fma(Float32(-2.0), Float32(Float32(Float32(uy * uy) * Float32(Float32(pi) * Float32(pi))) * t_0), t_0);
	else
		tmp = sqrt(Float32(ux * Float32(fma(Float32(-2.0), maxCos, Float32(2.0)) + Float32(Float32(maxCos + Float32(-1.0)) * fma(Float32(-maxCos), ux, ux)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if maxCos < 4.99999999e-7

   1. Initial program 53.7%

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

   3. Taylor expanded in ux around 0

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

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

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

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

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

      5. associate-+l+ N/A

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

      6. mul-1-neg N/A

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

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

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

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

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

      9. unpow2 N/A

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

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

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

      11. neg-sub0 N/A

          \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos - 1\right) \cdot \color{blue}{\left(0 - \left(maxCos - 1\right)\right)}, 2 + -2 \cdot maxCos\right)} \]

      12. associate-+l- N/A

          \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos - 1\right) \cdot \color{blue}{\left(\left(0 - maxCos\right) + 1\right)}, 2 + -2 \cdot maxCos\right)} \]

      13. neg-sub0 N/A

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

      14. mul-1-neg N/A

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

      15. +-commutative N/A

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

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

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

      17. sub-neg N/A

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

      18. metadata-eval N/A

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

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

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

      20. mul-1-neg N/A

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

      21. unsub-neg N/A

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

      22. --lowering--.f32 N/A

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

   5. Simplified 98.9%

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

   6. Taylor expanded in maxCos around 0

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

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

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

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

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

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

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

      4. mul-1-neg N/A

         \[\leadsto \sqrt{ux \cdot \left(2 + \color{blue}{\left(\mathsf{neg}\left(ux\right)\right)}\right)} \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \]

      5. unsub-neg N/A

         \[\leadsto \sqrt{ux \cdot \color{blue}{\left(2 - ux\right)}} \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \]

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

         \[\leadsto \sqrt{ux \cdot \color{blue}{\left(2 - ux\right)}} \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \]

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

         \[\leadsto \sqrt{ux \cdot \left(2 - ux\right)} \cdot \color{blue}{\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \]

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

         \[\leadsto \sqrt{ux \cdot \left(2 - ux\right)} \cdot \cos \color{blue}{\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \]

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

         \[\leadsto \sqrt{ux \cdot \left(2 - ux\right)} \cdot \cos \left(2 \cdot \color{blue}{\left(uy \cdot \mathsf{PI}\left(\right)\right)}\right) \]

      10. PI-lowering-PI.f32 98.8

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

   8. Simplified 98.8%

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

   9. Taylor expanded in uy around 0

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

       1. +-commutative N/A

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

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

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

   11. Simplified 87.8%

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

   if 4.99999999e-7 < maxCos

   1. Initial program 59.6%

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

   3. Taylor expanded in ux around 0

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

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

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

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

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

      5. associate-+l+ N/A

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

      6. mul-1-neg N/A

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

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

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

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

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

      9. unpow2 N/A

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

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

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

      11. neg-sub0 N/A

          \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos - 1\right) \cdot \color{blue}{\left(0 - \left(maxCos - 1\right)\right)}, 2 + -2 \cdot maxCos\right)} \]

      12. associate-+l- N/A

          \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos - 1\right) \cdot \color{blue}{\left(\left(0 - maxCos\right) + 1\right)}, 2 + -2 \cdot maxCos\right)} \]

      13. neg-sub0 N/A

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

      14. mul-1-neg N/A

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

      15. +-commutative N/A

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

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

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

      17. sub-neg N/A

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

      18. metadata-eval N/A

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

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

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

      20. mul-1-neg N/A

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

      21. unsub-neg N/A

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

      22. --lowering--.f32 N/A

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

   5. Simplified 98.9%

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

   6. Taylor expanded in uy around 0

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

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

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

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

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

      3. associate-+r+ N/A

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

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

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

      5. +-commutative N/A

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

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

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

      7. associate-*r* N/A

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

      8. sub-neg N/A

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

      9. mul-1-neg N/A

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

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

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

      11. distribute-rgt-in N/A

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

      12. *-lft-identity N/A

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

      13. associate-*r* N/A

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

      14. +-commutative N/A

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

      15. associate-*r* N/A

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

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

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

      17. mul-1-neg N/A

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

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

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

      19. sub-neg N/A

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

      20. metadata-eval N/A

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

      21. +-lowering-+.f32 77.2

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

   8. Simplified 77.2%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;maxCos \leq 4.999999987376214 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(-2, \left(\left(uy \cdot uy\right) \cdot \left(\pi \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}, \sqrt{ux \cdot \left(2 - ux\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{ux \cdot \left(\mathsf{fma}\left(-2, maxCos, 2\right) + \left(maxCos + -1\right) \cdot \mathsf{fma}\left(-maxCos, ux, ux\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 88.2% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 54.8%

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

3. Taylor expanded in ux around 0

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

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

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

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

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

   3. +-commutative N/A

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

   4. metadata-eval N/A

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

   5. associate-+l+ N/A

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

   6. mul-1-neg N/A

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

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

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

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

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

   9. unpow2 N/A

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

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

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

   11. neg-sub0 N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos - 1\right) \cdot \color{blue}{\left(0 - \left(maxCos - 1\right)\right)}, 2 + -2 \cdot maxCos\right)} \]

   12. associate-+l- N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos - 1\right) \cdot \color{blue}{\left(\left(0 - maxCos\right) + 1\right)}, 2 + -2 \cdot maxCos\right)} \]

   13. neg-sub0 N/A

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

   14. mul-1-neg N/A

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

   15. +-commutative N/A

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

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

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

   17. sub-neg N/A

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

   18. metadata-eval N/A

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

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

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

   20. mul-1-neg N/A

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

   21. unsub-neg N/A

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

   22. --lowering--.f32 N/A

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

5. Simplified 98.9%

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

6. Taylor expanded in uy around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

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

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

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

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

   5. unpow2 N/A

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

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

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

   7. unpow2 N/A

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

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

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

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

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

   10. PI-lowering-PI.f32 87.3

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

8. Simplified 87.3%

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

9. Final simplification 87.3%

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

Alternative 12: 86.5% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;maxCos \leq 4.999999987376214 \cdot 10^{-7}:\\ \;\;\;\;\sqrt{ux \cdot \left(2 - ux\right)} \cdot \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \pi \cdot \pi, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{ux \cdot \left(\mathsf{fma}\left(-2, maxCos, 2\right) + \left(maxCos + -1\right) \cdot \mathsf{fma}\left(-maxCos, ux, ux\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= maxCos 4.999999987376214e-7)
   (* (sqrt (* ux (- 2.0 ux))) (fma (* -2.0 (* uy uy)) (* PI PI) 1.0))
   (sqrt
    (*
     ux
     (+ (fma -2.0 maxCos 2.0) (* (+ maxCos -1.0) (fma (- maxCos) ux ux)))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (maxCos <= 4.999999987376214e-7f) {
		tmp = sqrtf((ux * (2.0f - ux))) * fmaf((-2.0f * (uy * uy)), (((float) M_PI) * ((float) M_PI)), 1.0f);
	} else {
		tmp = sqrtf((ux * (fmaf(-2.0f, maxCos, 2.0f) + ((maxCos + -1.0f) * fmaf(-maxCos, ux, ux)))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (maxCos <= Float32(4.999999987376214e-7))
		tmp = Float32(sqrt(Float32(ux * Float32(Float32(2.0) - ux))) * fma(Float32(Float32(-2.0) * Float32(uy * uy)), Float32(Float32(pi) * Float32(pi)), Float32(1.0)));
	else
		tmp = sqrt(Float32(ux * Float32(fma(Float32(-2.0), maxCos, Float32(2.0)) + Float32(Float32(maxCos + Float32(-1.0)) * fma(Float32(-maxCos), ux, ux)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if maxCos < 4.99999999e-7

   1. Initial program 53.7%

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

   3. Taylor expanded in ux around 0

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

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

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

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

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

      5. associate-+l+ N/A

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

      6. mul-1-neg N/A

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

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

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

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

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

      9. unpow2 N/A

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

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

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

      11. neg-sub0 N/A

          \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos - 1\right) \cdot \color{blue}{\left(0 - \left(maxCos - 1\right)\right)}, 2 + -2 \cdot maxCos\right)} \]

      12. associate-+l- N/A

          \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos - 1\right) \cdot \color{blue}{\left(\left(0 - maxCos\right) + 1\right)}, 2 + -2 \cdot maxCos\right)} \]

      13. neg-sub0 N/A

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

      14. mul-1-neg N/A

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

      15. +-commutative N/A

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

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

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

      17. sub-neg N/A

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

      18. metadata-eval N/A

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

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

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

      20. mul-1-neg N/A

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

      21. unsub-neg N/A

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

      22. --lowering--.f32 N/A

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

   5. Simplified 98.9%

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

   6. Taylor expanded in maxCos around 0

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

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

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

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

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

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

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

      4. mul-1-neg N/A

         \[\leadsto \sqrt{ux \cdot \left(2 + \color{blue}{\left(\mathsf{neg}\left(ux\right)\right)}\right)} \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \]

      5. unsub-neg N/A

         \[\leadsto \sqrt{ux \cdot \color{blue}{\left(2 - ux\right)}} \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \]

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

         \[\leadsto \sqrt{ux \cdot \color{blue}{\left(2 - ux\right)}} \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \]

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

         \[\leadsto \sqrt{ux \cdot \left(2 - ux\right)} \cdot \color{blue}{\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \]

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

         \[\leadsto \sqrt{ux \cdot \left(2 - ux\right)} \cdot \cos \color{blue}{\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \]

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

         \[\leadsto \sqrt{ux \cdot \left(2 - ux\right)} \cdot \cos \left(2 \cdot \color{blue}{\left(uy \cdot \mathsf{PI}\left(\right)\right)}\right) \]

      10. PI-lowering-PI.f32 98.8

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

   8. Simplified 98.8%

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

   9. Taylor expanded in uy around 0

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

       1. +-commutative N/A

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

       2. associate-*r* N/A

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

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

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

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

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

       5. unpow2 N/A

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

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

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

       7. unpow2 N/A

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

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

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

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

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

       10. PI-lowering-PI.f32 87.7

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

   11. Simplified 87.7%

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

   if 4.99999999e-7 < maxCos

   1. Initial program 59.6%

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

   3. Taylor expanded in ux around 0

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

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

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

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

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

      5. associate-+l+ N/A

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

      6. mul-1-neg N/A

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

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

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

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

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

      9. unpow2 N/A

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

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

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

      11. neg-sub0 N/A

          \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos - 1\right) \cdot \color{blue}{\left(0 - \left(maxCos - 1\right)\right)}, 2 + -2 \cdot maxCos\right)} \]

      12. associate-+l- N/A

          \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos - 1\right) \cdot \color{blue}{\left(\left(0 - maxCos\right) + 1\right)}, 2 + -2 \cdot maxCos\right)} \]

      13. neg-sub0 N/A

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

      14. mul-1-neg N/A

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

      15. +-commutative N/A

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

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

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

      17. sub-neg N/A

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

      18. metadata-eval N/A

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

      19. +-lowering-+.f32 N/A

          \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \color{blue}{\left(maxCos + -1\right)} \cdot \left(1 + -1 \cdot maxCos\right), 2 + -2 \cdot maxCos\right)} \]

      20. mul-1-neg N/A

          \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos + -1\right) \cdot \left(1 + \color{blue}{\left(\mathsf{neg}\left(maxCos\right)\right)}\right), 2 + -2 \cdot maxCos\right)} \]

      21. unsub-neg N/A

          \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos + -1\right) \cdot \color{blue}{\left(1 - maxCos\right)}, 2 + -2 \cdot maxCos\right)} \]

      22. --lowering--.f32 N/A

          \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos + -1\right) \cdot \color{blue}{\left(1 - maxCos\right)}, 2 + -2 \cdot maxCos\right)} \]

   5. Simplified 98.9%

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\color{blue}{ux \cdot \mathsf{fma}\left(ux, \left(maxCos + -1\right) \cdot \left(1 - maxCos\right), \mathsf{fma}\left(maxCos, -2, 2\right)\right)}} \]

   6. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{\sqrt{ux \cdot \left(2 + \left(-2 \cdot maxCos + ux \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos - 1\right)\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. sqrt-lowering-sqrt.f32 N/A

         \[\leadsto \color{blue}{\sqrt{ux \cdot \left(2 + \left(-2 \cdot maxCos + ux \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos - 1\right)\right)\right)\right)}} \]

      2. *-lowering-*.f32 N/A

         \[\leadsto \sqrt{\color{blue}{ux \cdot \left(2 + \left(-2 \cdot maxCos + ux \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos - 1\right)\right)\right)\right)}} \]

      3. associate-+r+ N/A

         \[\leadsto \sqrt{ux \cdot \color{blue}{\left(\left(2 + -2 \cdot maxCos\right) + ux \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos - 1\right)\right)\right)}} \]

      4. +-lowering-+.f32 N/A

         \[\leadsto \sqrt{ux \cdot \color{blue}{\left(\left(2 + -2 \cdot maxCos\right) + ux \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos - 1\right)\right)\right)}} \]

      5. +-commutative N/A

         \[\leadsto \sqrt{ux \cdot \left(\color{blue}{\left(-2 \cdot maxCos + 2\right)} + ux \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos - 1\right)\right)\right)} \]

      6. accelerator-lowering-fma.f32 N/A

         \[\leadsto \sqrt{ux \cdot \left(\color{blue}{\mathsf{fma}\left(-2, maxCos, 2\right)} + ux \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos - 1\right)\right)\right)} \]

      7. associate-*r* N/A

         \[\leadsto \sqrt{ux \cdot \left(\mathsf{fma}\left(-2, maxCos, 2\right) + \color{blue}{\left(ux \cdot \left(1 - maxCos\right)\right) \cdot \left(maxCos - 1\right)}\right)} \]

      8. sub-neg N/A

         \[\leadsto \sqrt{ux \cdot \left(\mathsf{fma}\left(-2, maxCos, 2\right) + \left(ux \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(maxCos\right)\right)\right)}\right) \cdot \left(maxCos - 1\right)\right)} \]

      9. mul-1-neg N/A

         \[\leadsto \sqrt{ux \cdot \left(\mathsf{fma}\left(-2, maxCos, 2\right) + \left(ux \cdot \left(1 + \color{blue}{-1 \cdot maxCos}\right)\right) \cdot \left(maxCos - 1\right)\right)} \]

      10. *-lowering-*.f32 N/A

          \[\leadsto \sqrt{ux \cdot \left(\mathsf{fma}\left(-2, maxCos, 2\right) + \color{blue}{\left(ux \cdot \left(1 + -1 \cdot maxCos\right)\right) \cdot \left(maxCos - 1\right)}\right)} \]

      11. distribute-rgt-in N/A

          \[\leadsto \sqrt{ux \cdot \left(\mathsf{fma}\left(-2, maxCos, 2\right) + \color{blue}{\left(1 \cdot ux + \left(-1 \cdot maxCos\right) \cdot ux\right)} \cdot \left(maxCos - 1\right)\right)} \]

      12. *-lft-identity N/A

          \[\leadsto \sqrt{ux \cdot \left(\mathsf{fma}\left(-2, maxCos, 2\right) + \left(\color{blue}{ux} + \left(-1 \cdot maxCos\right) \cdot ux\right) \cdot \left(maxCos - 1\right)\right)} \]

      13. associate-*r* N/A

          \[\leadsto \sqrt{ux \cdot \left(\mathsf{fma}\left(-2, maxCos, 2\right) + \left(ux + \color{blue}{-1 \cdot \left(maxCos \cdot ux\right)}\right) \cdot \left(maxCos - 1\right)\right)} \]

      14. +-commutative N/A

          \[\leadsto \sqrt{ux \cdot \left(\mathsf{fma}\left(-2, maxCos, 2\right) + \color{blue}{\left(-1 \cdot \left(maxCos \cdot ux\right) + ux\right)} \cdot \left(maxCos - 1\right)\right)} \]

      15. associate-*r* N/A

          \[\leadsto \sqrt{ux \cdot \left(\mathsf{fma}\left(-2, maxCos, 2\right) + \left(\color{blue}{\left(-1 \cdot maxCos\right) \cdot ux} + ux\right) \cdot \left(maxCos - 1\right)\right)} \]

      16. accelerator-lowering-fma.f32 N/A

          \[\leadsto \sqrt{ux \cdot \left(\mathsf{fma}\left(-2, maxCos, 2\right) + \color{blue}{\mathsf{fma}\left(-1 \cdot maxCos, ux, ux\right)} \cdot \left(maxCos - 1\right)\right)} \]

      17. mul-1-neg N/A

          \[\leadsto \sqrt{ux \cdot \left(\mathsf{fma}\left(-2, maxCos, 2\right) + \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(maxCos\right)}, ux, ux\right) \cdot \left(maxCos - 1\right)\right)} \]

      18. neg-lowering-neg.f32 N/A

          \[\leadsto \sqrt{ux \cdot \left(\mathsf{fma}\left(-2, maxCos, 2\right) + \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(maxCos\right)}, ux, ux\right) \cdot \left(maxCos - 1\right)\right)} \]

      19. sub-neg N/A

          \[\leadsto \sqrt{ux \cdot \left(\mathsf{fma}\left(-2, maxCos, 2\right) + \mathsf{fma}\left(\mathsf{neg}\left(maxCos\right), ux, ux\right) \cdot \color{blue}{\left(maxCos + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)} \]

      20. metadata-eval N/A

          \[\leadsto \sqrt{ux \cdot \left(\mathsf{fma}\left(-2, maxCos, 2\right) + \mathsf{fma}\left(\mathsf{neg}\left(maxCos\right), ux, ux\right) \cdot \left(maxCos + \color{blue}{-1}\right)\right)} \]

      21. +-lowering-+.f32 77.2

          \[\leadsto \sqrt{ux \cdot \left(\mathsf{fma}\left(-2, maxCos, 2\right) + \mathsf{fma}\left(-maxCos, ux, ux\right) \cdot \color{blue}{\left(maxCos + -1\right)}\right)} \]

   8. Simplified 77.2%

      \[\leadsto \color{blue}{\sqrt{ux \cdot \left(\mathsf{fma}\left(-2, maxCos, 2\right) + \mathsf{fma}\left(-maxCos, ux, ux\right) \cdot \left(maxCos + -1\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;maxCos \leq 4.999999987376214 \cdot 10^{-7}:\\ \;\;\;\;\sqrt{ux \cdot \left(2 - ux\right)} \cdot \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \pi \cdot \pi, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{ux \cdot \left(\mathsf{fma}\left(-2, maxCos, 2\right) + \left(maxCos + -1\right) \cdot \mathsf{fma}\left(-maxCos, ux, ux\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 79.7% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \sqrt{ux \cdot \left(\mathsf{fma}\left(-2, maxCos, 2\right) + \left(maxCos + -1\right) \cdot \mathsf{fma}\left(-maxCos, ux, ux\right)\right)} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (sqrt
  (* ux (+ (fma -2.0 maxCos 2.0) (* (+ maxCos -1.0) (fma (- maxCos) ux ux))))))

C

float code(float ux, float uy, float maxCos) {
	return sqrtf((ux * (fmaf(-2.0f, maxCos, 2.0f) + ((maxCos + -1.0f) * fmaf(-maxCos, ux, ux)))));
}

Julia

function code(ux, uy, maxCos)
	return sqrt(Float32(ux * Float32(fma(Float32(-2.0), maxCos, Float32(2.0)) + Float32(Float32(maxCos + Float32(-1.0)) * fma(Float32(-maxCos), ux, ux)))))
end

Derivation

1. Initial program 54.8%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. Add Preprocessing

3. Taylor expanded in ux around 0

   \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right)}} \]
4. Step-by-step derivation

   1. *-lowering-*.f32 N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right)}} \]

   2. cancel-sign-sub-inv N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \color{blue}{\left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) + \left(\mathsf{neg}\left(2\right)\right) \cdot maxCos\right)}} \]

   3. +-commutative N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \left(\color{blue}{\left(-1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right) + 2\right)} + \left(\mathsf{neg}\left(2\right)\right) \cdot maxCos\right)} \]

   4. metadata-eval N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \left(\left(-1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right) + 2\right) + \color{blue}{-2} \cdot maxCos\right)} \]

   5. associate-+l+ N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \color{blue}{\left(-1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right) + \left(2 + -2 \cdot maxCos\right)\right)}} \]

   6. mul-1-neg N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \left(\color{blue}{\left(\mathsf{neg}\left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right)} + \left(2 + -2 \cdot maxCos\right)\right)} \]

   7. distribute-rgt-neg-in N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \left(\color{blue}{ux \cdot \left(\mathsf{neg}\left({\left(maxCos - 1\right)}^{2}\right)\right)} + \left(2 + -2 \cdot maxCos\right)\right)} \]

   8. accelerator-lowering-fma.f32 N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \color{blue}{\mathsf{fma}\left(ux, \mathsf{neg}\left({\left(maxCos - 1\right)}^{2}\right), 2 + -2 \cdot maxCos\right)}} \]

   9. unpow2 N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \mathsf{neg}\left(\color{blue}{\left(maxCos - 1\right) \cdot \left(maxCos - 1\right)}\right), 2 + -2 \cdot maxCos\right)} \]

   10. distribute-rgt-neg-in N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \color{blue}{\left(maxCos - 1\right) \cdot \left(\mathsf{neg}\left(\left(maxCos - 1\right)\right)\right)}, 2 + -2 \cdot maxCos\right)} \]

   11. neg-sub0 N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos - 1\right) \cdot \color{blue}{\left(0 - \left(maxCos - 1\right)\right)}, 2 + -2 \cdot maxCos\right)} \]

   12. associate-+l- N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos - 1\right) \cdot \color{blue}{\left(\left(0 - maxCos\right) + 1\right)}, 2 + -2 \cdot maxCos\right)} \]

   13. neg-sub0 N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos - 1\right) \cdot \left(\color{blue}{\left(\mathsf{neg}\left(maxCos\right)\right)} + 1\right), 2 + -2 \cdot maxCos\right)} \]

   14. mul-1-neg N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos - 1\right) \cdot \left(\color{blue}{-1 \cdot maxCos} + 1\right), 2 + -2 \cdot maxCos\right)} \]

   15. +-commutative N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos - 1\right) \cdot \color{blue}{\left(1 + -1 \cdot maxCos\right)}, 2 + -2 \cdot maxCos\right)} \]

   16. *-lowering-*.f32 N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \color{blue}{\left(maxCos - 1\right) \cdot \left(1 + -1 \cdot maxCos\right)}, 2 + -2 \cdot maxCos\right)} \]

   17. sub-neg N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \color{blue}{\left(maxCos + \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(1 + -1 \cdot maxCos\right), 2 + -2 \cdot maxCos\right)} \]

   18. metadata-eval N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos + \color{blue}{-1}\right) \cdot \left(1 + -1 \cdot maxCos\right), 2 + -2 \cdot maxCos\right)} \]

   19. +-lowering-+.f32 N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \color{blue}{\left(maxCos + -1\right)} \cdot \left(1 + -1 \cdot maxCos\right), 2 + -2 \cdot maxCos\right)} \]

   20. mul-1-neg N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos + -1\right) \cdot \left(1 + \color{blue}{\left(\mathsf{neg}\left(maxCos\right)\right)}\right), 2 + -2 \cdot maxCos\right)} \]

   21. unsub-neg N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos + -1\right) \cdot \color{blue}{\left(1 - maxCos\right)}, 2 + -2 \cdot maxCos\right)} \]

   22. --lowering--.f32 N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos + -1\right) \cdot \color{blue}{\left(1 - maxCos\right)}, 2 + -2 \cdot maxCos\right)} \]

5. Simplified 98.9%

   \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\color{blue}{ux \cdot \mathsf{fma}\left(ux, \left(maxCos + -1\right) \cdot \left(1 - maxCos\right), \mathsf{fma}\left(maxCos, -2, 2\right)\right)}} \]

6. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{\sqrt{ux \cdot \left(2 + \left(-2 \cdot maxCos + ux \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos - 1\right)\right)\right)\right)}} \]
7. Step-by-step derivation

   1. sqrt-lowering-sqrt.f32 N/A

      \[\leadsto \color{blue}{\sqrt{ux \cdot \left(2 + \left(-2 \cdot maxCos + ux \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos - 1\right)\right)\right)\right)}} \]

   2. *-lowering-*.f32 N/A

      \[\leadsto \sqrt{\color{blue}{ux \cdot \left(2 + \left(-2 \cdot maxCos + ux \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos - 1\right)\right)\right)\right)}} \]

   3. associate-+r+ N/A

      \[\leadsto \sqrt{ux \cdot \color{blue}{\left(\left(2 + -2 \cdot maxCos\right) + ux \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos - 1\right)\right)\right)}} \]

   4. +-lowering-+.f32 N/A

      \[\leadsto \sqrt{ux \cdot \color{blue}{\left(\left(2 + -2 \cdot maxCos\right) + ux \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos - 1\right)\right)\right)}} \]

   5. +-commutative N/A

      \[\leadsto \sqrt{ux \cdot \left(\color{blue}{\left(-2 \cdot maxCos + 2\right)} + ux \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos - 1\right)\right)\right)} \]

   6. accelerator-lowering-fma.f32 N/A

      \[\leadsto \sqrt{ux \cdot \left(\color{blue}{\mathsf{fma}\left(-2, maxCos, 2\right)} + ux \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos - 1\right)\right)\right)} \]

   7. associate-*r* N/A

      \[\leadsto \sqrt{ux \cdot \left(\mathsf{fma}\left(-2, maxCos, 2\right) + \color{blue}{\left(ux \cdot \left(1 - maxCos\right)\right) \cdot \left(maxCos - 1\right)}\right)} \]

   8. sub-neg N/A

      \[\leadsto \sqrt{ux \cdot \left(\mathsf{fma}\left(-2, maxCos, 2\right) + \left(ux \cdot \color{blue}{\left(1 + \left(\mathsf{neg}\left(maxCos\right)\right)\right)}\right) \cdot \left(maxCos - 1\right)\right)} \]

   9. mul-1-neg N/A

      \[\leadsto \sqrt{ux \cdot \left(\mathsf{fma}\left(-2, maxCos, 2\right) + \left(ux \cdot \left(1 + \color{blue}{-1 \cdot maxCos}\right)\right) \cdot \left(maxCos - 1\right)\right)} \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \sqrt{ux \cdot \left(\mathsf{fma}\left(-2, maxCos, 2\right) + \color{blue}{\left(ux \cdot \left(1 + -1 \cdot maxCos\right)\right) \cdot \left(maxCos - 1\right)}\right)} \]

   11. distribute-rgt-in N/A

       \[\leadsto \sqrt{ux \cdot \left(\mathsf{fma}\left(-2, maxCos, 2\right) + \color{blue}{\left(1 \cdot ux + \left(-1 \cdot maxCos\right) \cdot ux\right)} \cdot \left(maxCos - 1\right)\right)} \]

   12. *-lft-identity N/A

       \[\leadsto \sqrt{ux \cdot \left(\mathsf{fma}\left(-2, maxCos, 2\right) + \left(\color{blue}{ux} + \left(-1 \cdot maxCos\right) \cdot ux\right) \cdot \left(maxCos - 1\right)\right)} \]

   13. associate-*r* N/A

       \[\leadsto \sqrt{ux \cdot \left(\mathsf{fma}\left(-2, maxCos, 2\right) + \left(ux + \color{blue}{-1 \cdot \left(maxCos \cdot ux\right)}\right) \cdot \left(maxCos - 1\right)\right)} \]

   14. +-commutative N/A

       \[\leadsto \sqrt{ux \cdot \left(\mathsf{fma}\left(-2, maxCos, 2\right) + \color{blue}{\left(-1 \cdot \left(maxCos \cdot ux\right) + ux\right)} \cdot \left(maxCos - 1\right)\right)} \]

   15. associate-*r* N/A

       \[\leadsto \sqrt{ux \cdot \left(\mathsf{fma}\left(-2, maxCos, 2\right) + \left(\color{blue}{\left(-1 \cdot maxCos\right) \cdot ux} + ux\right) \cdot \left(maxCos - 1\right)\right)} \]

   16. accelerator-lowering-fma.f32 N/A

       \[\leadsto \sqrt{ux \cdot \left(\mathsf{fma}\left(-2, maxCos, 2\right) + \color{blue}{\mathsf{fma}\left(-1 \cdot maxCos, ux, ux\right)} \cdot \left(maxCos - 1\right)\right)} \]

   17. mul-1-neg N/A

       \[\leadsto \sqrt{ux \cdot \left(\mathsf{fma}\left(-2, maxCos, 2\right) + \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(maxCos\right)}, ux, ux\right) \cdot \left(maxCos - 1\right)\right)} \]

   18. neg-lowering-neg.f32 N/A

       \[\leadsto \sqrt{ux \cdot \left(\mathsf{fma}\left(-2, maxCos, 2\right) + \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(maxCos\right)}, ux, ux\right) \cdot \left(maxCos - 1\right)\right)} \]

   19. sub-neg N/A

       \[\leadsto \sqrt{ux \cdot \left(\mathsf{fma}\left(-2, maxCos, 2\right) + \mathsf{fma}\left(\mathsf{neg}\left(maxCos\right), ux, ux\right) \cdot \color{blue}{\left(maxCos + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)} \]

   20. metadata-eval N/A

       \[\leadsto \sqrt{ux \cdot \left(\mathsf{fma}\left(-2, maxCos, 2\right) + \mathsf{fma}\left(\mathsf{neg}\left(maxCos\right), ux, ux\right) \cdot \left(maxCos + \color{blue}{-1}\right)\right)} \]

   21. +-lowering-+.f32 79.2

       \[\leadsto \sqrt{ux \cdot \left(\mathsf{fma}\left(-2, maxCos, 2\right) + \mathsf{fma}\left(-maxCos, ux, ux\right) \cdot \color{blue}{\left(maxCos + -1\right)}\right)} \]

8. Simplified 79.2%

   \[\leadsto \color{blue}{\sqrt{ux \cdot \left(\mathsf{fma}\left(-2, maxCos, 2\right) + \mathsf{fma}\left(-maxCos, ux, ux\right) \cdot \left(maxCos + -1\right)\right)}} \]

9. Final simplification 79.2%

   \[\leadsto \sqrt{ux \cdot \left(\mathsf{fma}\left(-2, maxCos, 2\right) + \left(maxCos + -1\right) \cdot \mathsf{fma}\left(-maxCos, ux, ux\right)\right)} \]
10. Add Preprocessing

Alternative 14: 79.3% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \sqrt{\mathsf{fma}\left(-ux, ux + -2, maxCos \cdot \left(\left(-ux\right) \cdot \mathsf{fma}\left(ux, -2, 2\right)\right)\right)} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (sqrt (fma (- ux) (+ ux -2.0) (* maxCos (* (- ux) (fma ux -2.0 2.0))))))

C

float code(float ux, float uy, float maxCos) {
	return sqrtf(fmaf(-ux, (ux + -2.0f), (maxCos * (-ux * fmaf(ux, -2.0f, 2.0f)))));
}

Julia

function code(ux, uy, maxCos)
	return sqrt(fma(Float32(-ux), Float32(ux + Float32(-2.0)), Float32(maxCos * Float32(Float32(-ux) * fma(ux, Float32(-2.0), Float32(2.0))))))
end

Derivation

1. Initial program 54.8%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. Add Preprocessing

3. Taylor expanded in ux around 0

   \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \color{blue}{\left(1 + ux \cdot \left(\left(2 \cdot maxCos + ux \cdot {\left(maxCos - 1\right)}^{2}\right) - 2\right)\right)}} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \color{blue}{\left(ux \cdot \left(\left(2 \cdot maxCos + ux \cdot {\left(maxCos - 1\right)}^{2}\right) - 2\right) + 1\right)}} \]

   2. accelerator-lowering-fma.f32 N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \color{blue}{\mathsf{fma}\left(ux, \left(2 \cdot maxCos + ux \cdot {\left(maxCos - 1\right)}^{2}\right) - 2, 1\right)}} \]

   3. +-commutative N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \mathsf{fma}\left(ux, \color{blue}{\left(ux \cdot {\left(maxCos - 1\right)}^{2} + 2 \cdot maxCos\right)} - 2, 1\right)} \]

   4. associate--l+ N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \mathsf{fma}\left(ux, \color{blue}{ux \cdot {\left(maxCos - 1\right)}^{2} + \left(2 \cdot maxCos - 2\right)}, 1\right)} \]

   5. accelerator-lowering-fma.f32 N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \mathsf{fma}\left(ux, \color{blue}{\mathsf{fma}\left(ux, {\left(maxCos - 1\right)}^{2}, 2 \cdot maxCos - 2\right)}, 1\right)} \]

   6. unpow2 N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \mathsf{fma}\left(ux, \mathsf{fma}\left(ux, \color{blue}{\left(maxCos - 1\right) \cdot \left(maxCos - 1\right)}, 2 \cdot maxCos - 2\right), 1\right)} \]

   7. *-lowering-*.f32 N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \mathsf{fma}\left(ux, \mathsf{fma}\left(ux, \color{blue}{\left(maxCos - 1\right) \cdot \left(maxCos - 1\right)}, 2 \cdot maxCos - 2\right), 1\right)} \]

   8. sub-neg N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \mathsf{fma}\left(ux, \mathsf{fma}\left(ux, \color{blue}{\left(maxCos + \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(maxCos - 1\right), 2 \cdot maxCos - 2\right), 1\right)} \]

   9. metadata-eval N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \mathsf{fma}\left(ux, \mathsf{fma}\left(ux, \left(maxCos + \color{blue}{-1}\right) \cdot \left(maxCos - 1\right), 2 \cdot maxCos - 2\right), 1\right)} \]

   10. +-lowering-+.f32 N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \mathsf{fma}\left(ux, \mathsf{fma}\left(ux, \color{blue}{\left(maxCos + -1\right)} \cdot \left(maxCos - 1\right), 2 \cdot maxCos - 2\right), 1\right)} \]

   11. sub-neg N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \mathsf{fma}\left(ux, \mathsf{fma}\left(ux, \left(maxCos + -1\right) \cdot \color{blue}{\left(maxCos + \left(\mathsf{neg}\left(1\right)\right)\right)}, 2 \cdot maxCos - 2\right), 1\right)} \]

   12. metadata-eval N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \mathsf{fma}\left(ux, \mathsf{fma}\left(ux, \left(maxCos + -1\right) \cdot \left(maxCos + \color{blue}{-1}\right), 2 \cdot maxCos - 2\right), 1\right)} \]

   13. +-lowering-+.f32 N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \mathsf{fma}\left(ux, \mathsf{fma}\left(ux, \left(maxCos + -1\right) \cdot \color{blue}{\left(maxCos + -1\right)}, 2 \cdot maxCos - 2\right), 1\right)} \]

   14. sub-neg N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \mathsf{fma}\left(ux, \mathsf{fma}\left(ux, \left(maxCos + -1\right) \cdot \left(maxCos + -1\right), \color{blue}{2 \cdot maxCos + \left(\mathsf{neg}\left(2\right)\right)}\right), 1\right)} \]

   15. metadata-eval N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \mathsf{fma}\left(ux, \mathsf{fma}\left(ux, \left(maxCos + -1\right) \cdot \left(maxCos + -1\right), 2 \cdot maxCos + \color{blue}{-2}\right), 1\right)} \]

   16. accelerator-lowering-fma.f32 58.6

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \mathsf{fma}\left(ux, \mathsf{fma}\left(ux, \left(maxCos + -1\right) \cdot \left(maxCos + -1\right), \color{blue}{\mathsf{fma}\left(2, maxCos, -2\right)}\right), 1\right)} \]

5. Simplified 58.6%

   \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \color{blue}{\mathsf{fma}\left(ux, \mathsf{fma}\left(ux, \left(maxCos + -1\right) \cdot \left(maxCos + -1\right), \mathsf{fma}\left(2, maxCos, -2\right)\right), 1\right)}} \]

6. Taylor expanded in maxCos around 0

   \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{-1 \cdot \left(maxCos \cdot \left(ux \cdot \left(2 + -2 \cdot ux\right)\right)\right) - ux \cdot \left(ux - 2\right)}} \]
7. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{-1 \cdot \left(maxCos \cdot \left(ux \cdot \left(2 + -2 \cdot ux\right)\right)\right) + \left(\mathsf{neg}\left(ux \cdot \left(ux - 2\right)\right)\right)}} \]

   2. associate-*r* N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{\left(-1 \cdot maxCos\right) \cdot \left(ux \cdot \left(2 + -2 \cdot ux\right)\right)} + \left(\mathsf{neg}\left(ux \cdot \left(ux - 2\right)\right)\right)} \]

   3. accelerator-lowering-fma.f32 N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(-1 \cdot maxCos, ux \cdot \left(2 + -2 \cdot ux\right), \mathsf{neg}\left(ux \cdot \left(ux - 2\right)\right)\right)}} \]

   4. mul-1-neg N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(maxCos\right)}, ux \cdot \left(2 + -2 \cdot ux\right), \mathsf{neg}\left(ux \cdot \left(ux - 2\right)\right)\right)} \]

   5. neg-lowering-neg.f32 N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(maxCos\right)}, ux \cdot \left(2 + -2 \cdot ux\right), \mathsf{neg}\left(ux \cdot \left(ux - 2\right)\right)\right)} \]

   6. *-lowering-*.f32 N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{neg}\left(maxCos\right), \color{blue}{ux \cdot \left(2 + -2 \cdot ux\right)}, \mathsf{neg}\left(ux \cdot \left(ux - 2\right)\right)\right)} \]

   7. +-commutative N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{neg}\left(maxCos\right), ux \cdot \color{blue}{\left(-2 \cdot ux + 2\right)}, \mathsf{neg}\left(ux \cdot \left(ux - 2\right)\right)\right)} \]

   8. accelerator-lowering-fma.f32 N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{neg}\left(maxCos\right), ux \cdot \color{blue}{\mathsf{fma}\left(-2, ux, 2\right)}, \mathsf{neg}\left(ux \cdot \left(ux - 2\right)\right)\right)} \]

   9. mul-1-neg N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{neg}\left(maxCos\right), ux \cdot \mathsf{fma}\left(-2, ux, 2\right), \color{blue}{-1 \cdot \left(ux \cdot \left(ux - 2\right)\right)}\right)} \]

   10. associate-*r* N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{neg}\left(maxCos\right), ux \cdot \mathsf{fma}\left(-2, ux, 2\right), \color{blue}{\left(-1 \cdot ux\right) \cdot \left(ux - 2\right)}\right)} \]

   11. *-lowering-*.f32 N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{neg}\left(maxCos\right), ux \cdot \mathsf{fma}\left(-2, ux, 2\right), \color{blue}{\left(-1 \cdot ux\right) \cdot \left(ux - 2\right)}\right)} \]

   12. mul-1-neg N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{neg}\left(maxCos\right), ux \cdot \mathsf{fma}\left(-2, ux, 2\right), \color{blue}{\left(\mathsf{neg}\left(ux\right)\right)} \cdot \left(ux - 2\right)\right)} \]

   13. neg-lowering-neg.f32 N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{neg}\left(maxCos\right), ux \cdot \mathsf{fma}\left(-2, ux, 2\right), \color{blue}{\left(\mathsf{neg}\left(ux\right)\right)} \cdot \left(ux - 2\right)\right)} \]

   14. sub-neg N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{neg}\left(maxCos\right), ux \cdot \mathsf{fma}\left(-2, ux, 2\right), \left(\mathsf{neg}\left(ux\right)\right) \cdot \color{blue}{\left(ux + \left(\mathsf{neg}\left(2\right)\right)\right)}\right)} \]

   15. metadata-eval N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{neg}\left(maxCos\right), ux \cdot \mathsf{fma}\left(-2, ux, 2\right), \left(\mathsf{neg}\left(ux\right)\right) \cdot \left(ux + \color{blue}{-2}\right)\right)} \]

   16. +-lowering-+.f32 98.2

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\mathsf{fma}\left(-maxCos, ux \cdot \mathsf{fma}\left(-2, ux, 2\right), \left(-ux\right) \cdot \color{blue}{\left(ux + -2\right)}\right)} \]

8. Simplified 98.2%

   \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(-maxCos, ux \cdot \mathsf{fma}\left(-2, ux, 2\right), \left(-ux\right) \cdot \left(ux + -2\right)\right)}} \]

9. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{\sqrt{-1 \cdot \left(maxCos \cdot \left(ux \cdot \left(2 + -2 \cdot ux\right)\right)\right) + -1 \cdot \left(ux \cdot \left(ux - 2\right)\right)}} \]
10. Step-by-step derivation

    1. sqrt-lowering-sqrt.f32 N/A

       \[\leadsto \color{blue}{\sqrt{-1 \cdot \left(maxCos \cdot \left(ux \cdot \left(2 + -2 \cdot ux\right)\right)\right) + -1 \cdot \left(ux \cdot \left(ux - 2\right)\right)}} \]

    2. +-commutative N/A

       \[\leadsto \sqrt{\color{blue}{-1 \cdot \left(ux \cdot \left(ux - 2\right)\right) + -1 \cdot \left(maxCos \cdot \left(ux \cdot \left(2 + -2 \cdot ux\right)\right)\right)}} \]

    3. associate-*r* N/A

       \[\leadsto \sqrt{\color{blue}{\left(-1 \cdot ux\right) \cdot \left(ux - 2\right)} + -1 \cdot \left(maxCos \cdot \left(ux \cdot \left(2 + -2 \cdot ux\right)\right)\right)} \]

    4. accelerator-lowering-fma.f32 N/A

       \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(-1 \cdot ux, ux - 2, -1 \cdot \left(maxCos \cdot \left(ux \cdot \left(2 + -2 \cdot ux\right)\right)\right)\right)}} \]

    5. mul-1-neg N/A

       \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(ux\right)}, ux - 2, -1 \cdot \left(maxCos \cdot \left(ux \cdot \left(2 + -2 \cdot ux\right)\right)\right)\right)} \]

    6. neg-lowering-neg.f32 N/A

       \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(ux\right)}, ux - 2, -1 \cdot \left(maxCos \cdot \left(ux \cdot \left(2 + -2 \cdot ux\right)\right)\right)\right)} \]

    7. sub-neg N/A

       \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{neg}\left(ux\right), \color{blue}{ux + \left(\mathsf{neg}\left(2\right)\right)}, -1 \cdot \left(maxCos \cdot \left(ux \cdot \left(2 + -2 \cdot ux\right)\right)\right)\right)} \]

    8. metadata-eval N/A

       \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{neg}\left(ux\right), ux + \color{blue}{-2}, -1 \cdot \left(maxCos \cdot \left(ux \cdot \left(2 + -2 \cdot ux\right)\right)\right)\right)} \]

    9. +-lowering-+.f32 N/A

       \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{neg}\left(ux\right), \color{blue}{ux + -2}, -1 \cdot \left(maxCos \cdot \left(ux \cdot \left(2 + -2 \cdot ux\right)\right)\right)\right)} \]

    10. associate-*r* N/A

        \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{neg}\left(ux\right), ux + -2, \color{blue}{\left(-1 \cdot maxCos\right) \cdot \left(ux \cdot \left(2 + -2 \cdot ux\right)\right)}\right)} \]

    11. *-lowering-*.f32 N/A

        \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{neg}\left(ux\right), ux + -2, \color{blue}{\left(-1 \cdot maxCos\right) \cdot \left(ux \cdot \left(2 + -2 \cdot ux\right)\right)}\right)} \]

    12. mul-1-neg N/A

        \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{neg}\left(ux\right), ux + -2, \color{blue}{\left(\mathsf{neg}\left(maxCos\right)\right)} \cdot \left(ux \cdot \left(2 + -2 \cdot ux\right)\right)\right)} \]

    13. neg-lowering-neg.f32 N/A

        \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{neg}\left(ux\right), ux + -2, \color{blue}{\left(\mathsf{neg}\left(maxCos\right)\right)} \cdot \left(ux \cdot \left(2 + -2 \cdot ux\right)\right)\right)} \]

    14. *-lowering-*.f32 N/A

        \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{neg}\left(ux\right), ux + -2, \left(\mathsf{neg}\left(maxCos\right)\right) \cdot \color{blue}{\left(ux \cdot \left(2 + -2 \cdot ux\right)\right)}\right)} \]

    15. +-commutative N/A

        \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{neg}\left(ux\right), ux + -2, \left(\mathsf{neg}\left(maxCos\right)\right) \cdot \left(ux \cdot \color{blue}{\left(-2 \cdot ux + 2\right)}\right)\right)} \]

    16. *-commutative N/A

        \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{neg}\left(ux\right), ux + -2, \left(\mathsf{neg}\left(maxCos\right)\right) \cdot \left(ux \cdot \left(\color{blue}{ux \cdot -2} + 2\right)\right)\right)} \]

    17. accelerator-lowering-fma.f32 78.6

        \[\leadsto \sqrt{\mathsf{fma}\left(-ux, ux + -2, \left(-maxCos\right) \cdot \left(ux \cdot \color{blue}{\mathsf{fma}\left(ux, -2, 2\right)}\right)\right)} \]

11. Simplified 78.6%

    \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(-ux, ux + -2, \left(-maxCos\right) \cdot \left(ux \cdot \mathsf{fma}\left(ux, -2, 2\right)\right)\right)}} \]

12. Final simplification 78.6%

    \[\leadsto \sqrt{\mathsf{fma}\left(-ux, ux + -2, maxCos \cdot \left(\left(-ux\right) \cdot \mathsf{fma}\left(ux, -2, 2\right)\right)\right)} \]
13. Add Preprocessing

Alternative 15: 75.8% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{ux \cdot \left(2 + \mathsf{fma}\left(ux, \left(maxCos + -1\right) \cdot \left(1 - maxCos\right), -maxCos\right)\right)} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (sqrt (* ux (+ 2.0 (fma ux (* (+ maxCos -1.0) (- 1.0 maxCos)) (- maxCos))))))

C

float code(float ux, float uy, float maxCos) {
	return sqrtf((ux * (2.0f + fmaf(ux, ((maxCos + -1.0f) * (1.0f - maxCos)), -maxCos))));
}

Julia

function code(ux, uy, maxCos)
	return sqrt(Float32(ux * Float32(Float32(2.0) + fma(ux, Float32(Float32(maxCos + Float32(-1.0)) * Float32(Float32(1.0) - maxCos)), Float32(-maxCos)))))
end

Derivation

1. Initial program 54.8%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. Add Preprocessing

3. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{\sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]
4. Step-by-step derivation

   1. sqrt-lowering-sqrt.f32 N/A

      \[\leadsto \color{blue}{\sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

   2. sub-neg N/A

      \[\leadsto \sqrt{\color{blue}{1 + \left(\mathsf{neg}\left({\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}\right)\right)}} \]

   3. +-commutative N/A

      \[\leadsto \sqrt{\color{blue}{\left(\mathsf{neg}\left({\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}\right)\right) + 1}} \]

   4. unpow2 N/A

      \[\leadsto \sqrt{\left(\mathsf{neg}\left(\color{blue}{\left(\left(1 + maxCos \cdot ux\right) - ux\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)}\right)\right) + 1} \]

   5. distribute-rgt-neg-in N/A

      \[\leadsto \sqrt{\color{blue}{\left(\left(1 + maxCos \cdot ux\right) - ux\right) \cdot \left(\mathsf{neg}\left(\left(\left(1 + maxCos \cdot ux\right) - ux\right)\right)\right)} + 1} \]

   6. accelerator-lowering-fma.f32 N/A

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\left(1 + maxCos \cdot ux\right) - ux, \mathsf{neg}\left(\left(\left(1 + maxCos \cdot ux\right) - ux\right)\right), 1\right)}} \]

5. Simplified 46.7%

   \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1 - ux\right), \mathsf{fma}\left(-ux, maxCos + -1, -1\right), 1\right)}} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \sqrt{\color{blue}{1 + \left(ux \cdot maxCos + \left(1 - ux\right)\right) \cdot \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(maxCos + -1\right) + -1\right)}} \]

   2. distribute-rgt-in N/A

      \[\leadsto \sqrt{1 + \color{blue}{\left(\left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot maxCos + \left(1 - ux\right)\right) + -1 \cdot \left(ux \cdot maxCos + \left(1 - ux\right)\right)\right)}} \]

   3. associate-+r+ N/A

      \[\leadsto \sqrt{\color{blue}{\left(1 + \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot maxCos + \left(1 - ux\right)\right)\right) + -1 \cdot \left(ux \cdot maxCos + \left(1 - ux\right)\right)}} \]

   4. +-lowering-+.f32 N/A

      \[\leadsto \sqrt{\color{blue}{\left(1 + \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot maxCos + \left(1 - ux\right)\right)\right) + -1 \cdot \left(ux \cdot maxCos + \left(1 - ux\right)\right)}} \]

7. Applied egg-rr 46.9%

   \[\leadsto \sqrt{\color{blue}{\left(1 + \mathsf{fma}\left(maxCos, -ux, ux\right) \cdot \mathsf{fma}\left(ux, maxCos, 1 - ux\right)\right) + \left(-\mathsf{fma}\left(ux, maxCos, 1 - ux\right)\right)}} \]

8. Taylor expanded in maxCos around 0

   \[\leadsto \sqrt{\left(1 + \mathsf{fma}\left(maxCos, \mathsf{neg}\left(ux\right), ux\right) \cdot \mathsf{fma}\left(ux, maxCos, 1 - ux\right)\right) + \color{blue}{\left(ux - 1\right)}} \]
9. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \sqrt{\left(1 + \mathsf{fma}\left(maxCos, \mathsf{neg}\left(ux\right), ux\right) \cdot \mathsf{fma}\left(ux, maxCos, 1 - ux\right)\right) + \color{blue}{\left(ux + \left(\mathsf{neg}\left(1\right)\right)\right)}} \]

   2. metadata-eval N/A

      \[\leadsto \sqrt{\left(1 + \mathsf{fma}\left(maxCos, \mathsf{neg}\left(ux\right), ux\right) \cdot \mathsf{fma}\left(ux, maxCos, 1 - ux\right)\right) + \left(ux + \color{blue}{-1}\right)} \]

   3. +-lowering-+.f32 45.3

      \[\leadsto \sqrt{\left(1 + \mathsf{fma}\left(maxCos, -ux, ux\right) \cdot \mathsf{fma}\left(ux, maxCos, 1 - ux\right)\right) + \color{blue}{\left(ux + -1\right)}} \]

10. Simplified 45.3%

    \[\leadsto \sqrt{\left(1 + \mathsf{fma}\left(maxCos, -ux, ux\right) \cdot \mathsf{fma}\left(ux, maxCos, 1 - ux\right)\right) + \color{blue}{\left(ux + -1\right)}} \]

11. Taylor expanded in ux around 0

    \[\leadsto \sqrt{\color{blue}{ux \cdot \left(2 + \left(-1 \cdot maxCos + ux \cdot \left(\left(1 + -1 \cdot maxCos\right) \cdot \left(maxCos - 1\right)\right)\right)\right)}} \]
12. Step-by-step derivation

    1. *-lowering-*.f32 N/A

       \[\leadsto \sqrt{\color{blue}{ux \cdot \left(2 + \left(-1 \cdot maxCos + ux \cdot \left(\left(1 + -1 \cdot maxCos\right) \cdot \left(maxCos - 1\right)\right)\right)\right)}} \]

    2. +-lowering-+.f32 N/A

       \[\leadsto \sqrt{ux \cdot \color{blue}{\left(2 + \left(-1 \cdot maxCos + ux \cdot \left(\left(1 + -1 \cdot maxCos\right) \cdot \left(maxCos - 1\right)\right)\right)\right)}} \]

    3. +-commutative N/A

       \[\leadsto \sqrt{ux \cdot \left(2 + \color{blue}{\left(ux \cdot \left(\left(1 + -1 \cdot maxCos\right) \cdot \left(maxCos - 1\right)\right) + -1 \cdot maxCos\right)}\right)} \]

    4. associate-*r* N/A

       \[\leadsto \sqrt{ux \cdot \left(2 + \left(\color{blue}{\left(ux \cdot \left(1 + -1 \cdot maxCos\right)\right) \cdot \left(maxCos - 1\right)} + -1 \cdot maxCos\right)\right)} \]

    5. mul-1-neg N/A

       \[\leadsto \sqrt{ux \cdot \left(2 + \left(\left(ux \cdot \left(1 + \color{blue}{\left(\mathsf{neg}\left(maxCos\right)\right)}\right)\right) \cdot \left(maxCos - 1\right) + -1 \cdot maxCos\right)\right)} \]

    6. sub-neg N/A

       \[\leadsto \sqrt{ux \cdot \left(2 + \left(\left(ux \cdot \color{blue}{\left(1 - maxCos\right)}\right) \cdot \left(maxCos - 1\right) + -1 \cdot maxCos\right)\right)} \]

    7. associate-*r* N/A

       \[\leadsto \sqrt{ux \cdot \left(2 + \left(\color{blue}{ux \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos - 1\right)\right)} + -1 \cdot maxCos\right)\right)} \]

    8. accelerator-lowering-fma.f32 N/A

       \[\leadsto \sqrt{ux \cdot \left(2 + \color{blue}{\mathsf{fma}\left(ux, \left(1 - maxCos\right) \cdot \left(maxCos - 1\right), -1 \cdot maxCos\right)}\right)} \]

    9. *-lowering-*.f32 N/A

       \[\leadsto \sqrt{ux \cdot \left(2 + \mathsf{fma}\left(ux, \color{blue}{\left(1 - maxCos\right) \cdot \left(maxCos - 1\right)}, -1 \cdot maxCos\right)\right)} \]

    10. --lowering--.f32 N/A

        \[\leadsto \sqrt{ux \cdot \left(2 + \mathsf{fma}\left(ux, \color{blue}{\left(1 - maxCos\right)} \cdot \left(maxCos - 1\right), -1 \cdot maxCos\right)\right)} \]

    11. sub-neg N/A

        \[\leadsto \sqrt{ux \cdot \left(2 + \mathsf{fma}\left(ux, \left(1 - maxCos\right) \cdot \color{blue}{\left(maxCos + \left(\mathsf{neg}\left(1\right)\right)\right)}, -1 \cdot maxCos\right)\right)} \]

    12. metadata-eval N/A

        \[\leadsto \sqrt{ux \cdot \left(2 + \mathsf{fma}\left(ux, \left(1 - maxCos\right) \cdot \left(maxCos + \color{blue}{-1}\right), -1 \cdot maxCos\right)\right)} \]

    13. +-lowering-+.f32 N/A

        \[\leadsto \sqrt{ux \cdot \left(2 + \mathsf{fma}\left(ux, \left(1 - maxCos\right) \cdot \color{blue}{\left(maxCos + -1\right)}, -1 \cdot maxCos\right)\right)} \]

    14. mul-1-neg N/A

        \[\leadsto \sqrt{ux \cdot \left(2 + \mathsf{fma}\left(ux, \left(1 - maxCos\right) \cdot \left(maxCos + -1\right), \color{blue}{\mathsf{neg}\left(maxCos\right)}\right)\right)} \]

    15. neg-lowering-neg.f32 75.7

        \[\leadsto \sqrt{ux \cdot \left(2 + \mathsf{fma}\left(ux, \left(1 - maxCos\right) \cdot \left(maxCos + -1\right), \color{blue}{-maxCos}\right)\right)} \]

13. Simplified 75.7%

    \[\leadsto \sqrt{\color{blue}{ux \cdot \left(2 + \mathsf{fma}\left(ux, \left(1 - maxCos\right) \cdot \left(maxCos + -1\right), -maxCos\right)\right)}} \]

14. Final simplification 75.7%

    \[\leadsto \sqrt{ux \cdot \left(2 + \mathsf{fma}\left(ux, \left(maxCos + -1\right) \cdot \left(1 - maxCos\right), -maxCos\right)\right)} \]
15. Add Preprocessing

Alternative 16: 75.5% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \sqrt{\mathsf{fma}\left(ux, 2, ux \cdot \left(-ux\right)\right)} \end{array} \]

FPCore

(FPCore (ux uy maxCos) :precision binary32 (sqrt (fma ux 2.0 (* ux (- ux)))))

C

float code(float ux, float uy, float maxCos) {
	return sqrtf(fmaf(ux, 2.0f, (ux * -ux)));
}

Julia

function code(ux, uy, maxCos)
	return sqrt(fma(ux, Float32(2.0), Float32(ux * Float32(-ux))))
end

Derivation

1. Initial program 54.8%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. Add Preprocessing

3. Taylor expanded in ux around 0

   \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right)}} \]
4. Step-by-step derivation

   1. *-lowering-*.f32 N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right)}} \]

   2. cancel-sign-sub-inv N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \color{blue}{\left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) + \left(\mathsf{neg}\left(2\right)\right) \cdot maxCos\right)}} \]

   3. +-commutative N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \left(\color{blue}{\left(-1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right) + 2\right)} + \left(\mathsf{neg}\left(2\right)\right) \cdot maxCos\right)} \]

   4. metadata-eval N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \left(\left(-1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right) + 2\right) + \color{blue}{-2} \cdot maxCos\right)} \]

   5. associate-+l+ N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \color{blue}{\left(-1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right) + \left(2 + -2 \cdot maxCos\right)\right)}} \]

   6. mul-1-neg N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \left(\color{blue}{\left(\mathsf{neg}\left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right)} + \left(2 + -2 \cdot maxCos\right)\right)} \]

   7. distribute-rgt-neg-in N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \left(\color{blue}{ux \cdot \left(\mathsf{neg}\left({\left(maxCos - 1\right)}^{2}\right)\right)} + \left(2 + -2 \cdot maxCos\right)\right)} \]

   8. accelerator-lowering-fma.f32 N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \color{blue}{\mathsf{fma}\left(ux, \mathsf{neg}\left({\left(maxCos - 1\right)}^{2}\right), 2 + -2 \cdot maxCos\right)}} \]

   9. unpow2 N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \mathsf{neg}\left(\color{blue}{\left(maxCos - 1\right) \cdot \left(maxCos - 1\right)}\right), 2 + -2 \cdot maxCos\right)} \]

   10. distribute-rgt-neg-in N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \color{blue}{\left(maxCos - 1\right) \cdot \left(\mathsf{neg}\left(\left(maxCos - 1\right)\right)\right)}, 2 + -2 \cdot maxCos\right)} \]

   11. neg-sub0 N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos - 1\right) \cdot \color{blue}{\left(0 - \left(maxCos - 1\right)\right)}, 2 + -2 \cdot maxCos\right)} \]

   12. associate-+l- N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos - 1\right) \cdot \color{blue}{\left(\left(0 - maxCos\right) + 1\right)}, 2 + -2 \cdot maxCos\right)} \]

   13. neg-sub0 N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos - 1\right) \cdot \left(\color{blue}{\left(\mathsf{neg}\left(maxCos\right)\right)} + 1\right), 2 + -2 \cdot maxCos\right)} \]

   14. mul-1-neg N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos - 1\right) \cdot \left(\color{blue}{-1 \cdot maxCos} + 1\right), 2 + -2 \cdot maxCos\right)} \]

   15. +-commutative N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos - 1\right) \cdot \color{blue}{\left(1 + -1 \cdot maxCos\right)}, 2 + -2 \cdot maxCos\right)} \]

   16. *-lowering-*.f32 N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \color{blue}{\left(maxCos - 1\right) \cdot \left(1 + -1 \cdot maxCos\right)}, 2 + -2 \cdot maxCos\right)} \]

   17. sub-neg N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \color{blue}{\left(maxCos + \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(1 + -1 \cdot maxCos\right), 2 + -2 \cdot maxCos\right)} \]

   18. metadata-eval N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos + \color{blue}{-1}\right) \cdot \left(1 + -1 \cdot maxCos\right), 2 + -2 \cdot maxCos\right)} \]

   19. +-lowering-+.f32 N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \color{blue}{\left(maxCos + -1\right)} \cdot \left(1 + -1 \cdot maxCos\right), 2 + -2 \cdot maxCos\right)} \]

   20. mul-1-neg N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos + -1\right) \cdot \left(1 + \color{blue}{\left(\mathsf{neg}\left(maxCos\right)\right)}\right), 2 + -2 \cdot maxCos\right)} \]

   21. unsub-neg N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos + -1\right) \cdot \color{blue}{\left(1 - maxCos\right)}, 2 + -2 \cdot maxCos\right)} \]

   22. --lowering--.f32 N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos + -1\right) \cdot \color{blue}{\left(1 - maxCos\right)}, 2 + -2 \cdot maxCos\right)} \]

5. Simplified 98.9%

   \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\color{blue}{ux \cdot \mathsf{fma}\left(ux, \left(maxCos + -1\right) \cdot \left(1 - maxCos\right), \mathsf{fma}\left(maxCos, -2, 2\right)\right)}} \]

6. Taylor expanded in maxCos around 0

   \[\leadsto \color{blue}{\sqrt{ux \cdot \left(2 + -1 \cdot ux\right)} \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \]
7. Step-by-step derivation

   1. *-lowering-*.f32 N/A

      \[\leadsto \color{blue}{\sqrt{ux \cdot \left(2 + -1 \cdot ux\right)} \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \]

   2. sqrt-lowering-sqrt.f32 N/A

      \[\leadsto \color{blue}{\sqrt{ux \cdot \left(2 + -1 \cdot ux\right)}} \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \]

   3. *-lowering-*.f32 N/A

      \[\leadsto \sqrt{\color{blue}{ux \cdot \left(2 + -1 \cdot ux\right)}} \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \]

   4. mul-1-neg N/A

      \[\leadsto \sqrt{ux \cdot \left(2 + \color{blue}{\left(\mathsf{neg}\left(ux\right)\right)}\right)} \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \]

   5. unsub-neg N/A

      \[\leadsto \sqrt{ux \cdot \color{blue}{\left(2 - ux\right)}} \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \]

   6. --lowering--.f32 N/A

      \[\leadsto \sqrt{ux \cdot \color{blue}{\left(2 - ux\right)}} \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \]

   7. cos-lowering-cos.f32 N/A

      \[\leadsto \sqrt{ux \cdot \left(2 - ux\right)} \cdot \color{blue}{\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \]

   8. *-lowering-*.f32 N/A

      \[\leadsto \sqrt{ux \cdot \left(2 - ux\right)} \cdot \cos \color{blue}{\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \]

   9. *-lowering-*.f32 N/A

      \[\leadsto \sqrt{ux \cdot \left(2 - ux\right)} \cdot \cos \left(2 \cdot \color{blue}{\left(uy \cdot \mathsf{PI}\left(\right)\right)}\right) \]

   10. PI-lowering-PI.f32 93.0

       \[\leadsto \sqrt{ux \cdot \left(2 - ux\right)} \cdot \cos \left(2 \cdot \left(uy \cdot \color{blue}{\pi}\right)\right) \]

8. Simplified 93.0%

   \[\leadsto \color{blue}{\sqrt{ux \cdot \left(2 - ux\right)} \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right)} \]

9. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{\sqrt{ux \cdot \left(2 - ux\right)}} \]
10. Step-by-step derivation

    1. sub-neg N/A

       \[\leadsto \sqrt{ux \cdot \color{blue}{\left(2 + \left(\mathsf{neg}\left(ux\right)\right)\right)}} \]

    2. mul-1-neg N/A

       \[\leadsto \sqrt{ux \cdot \left(2 + \color{blue}{-1 \cdot ux}\right)} \]

    3. +-commutative N/A

       \[\leadsto \sqrt{ux \cdot \color{blue}{\left(-1 \cdot ux + 2\right)}} \]

    4. distribute-rgt-in N/A

       \[\leadsto \sqrt{\color{blue}{\left(-1 \cdot ux\right) \cdot ux + 2 \cdot ux}} \]

    5. mul-1-neg N/A

       \[\leadsto \sqrt{\color{blue}{\left(\mathsf{neg}\left(ux\right)\right)} \cdot ux + 2 \cdot ux} \]

    6. distribute-lft-neg-in N/A

       \[\leadsto \sqrt{\color{blue}{\left(\mathsf{neg}\left(ux \cdot ux\right)\right)} + 2 \cdot ux} \]

    7. unpow2 N/A

       \[\leadsto \sqrt{\left(\mathsf{neg}\left(\color{blue}{{ux}^{2}}\right)\right) + 2 \cdot ux} \]

    8. mul-1-neg N/A

       \[\leadsto \sqrt{\color{blue}{-1 \cdot {ux}^{2}} + 2 \cdot ux} \]

    9. sqrt-lowering-sqrt.f32 N/A

       \[\leadsto \color{blue}{\sqrt{-1 \cdot {ux}^{2} + 2 \cdot ux}} \]

    10. +-commutative N/A

        \[\leadsto \sqrt{\color{blue}{2 \cdot ux + -1 \cdot {ux}^{2}}} \]

    11. mul-1-neg N/A

        \[\leadsto \sqrt{2 \cdot ux + \color{blue}{\left(\mathsf{neg}\left({ux}^{2}\right)\right)}} \]

    12. unpow2 N/A

        \[\leadsto \sqrt{2 \cdot ux + \left(\mathsf{neg}\left(\color{blue}{ux \cdot ux}\right)\right)} \]

    13. distribute-lft-neg-in N/A

        \[\leadsto \sqrt{2 \cdot ux + \color{blue}{\left(\mathsf{neg}\left(ux\right)\right) \cdot ux}} \]

    14. mul-1-neg N/A

        \[\leadsto \sqrt{2 \cdot ux + \color{blue}{\left(-1 \cdot ux\right)} \cdot ux} \]

    15. distribute-rgt-in N/A

        \[\leadsto \sqrt{\color{blue}{ux \cdot \left(2 + -1 \cdot ux\right)}} \]

    16. *-lowering-*.f32 N/A

        \[\leadsto \sqrt{\color{blue}{ux \cdot \left(2 + -1 \cdot ux\right)}} \]

    17. mul-1-neg N/A

        \[\leadsto \sqrt{ux \cdot \left(2 + \color{blue}{\left(\mathsf{neg}\left(ux\right)\right)}\right)} \]

    18. sub-neg N/A

        \[\leadsto \sqrt{ux \cdot \color{blue}{\left(2 - ux\right)}} \]

    19. --lowering--.f32 75.4

        \[\leadsto \sqrt{ux \cdot \color{blue}{\left(2 - ux\right)}} \]

11. Simplified 75.4%

    \[\leadsto \color{blue}{\sqrt{ux \cdot \left(2 - ux\right)}} \]
12. Step-by-step derivation

    1. sub-neg N/A

       \[\leadsto \sqrt{ux \cdot \color{blue}{\left(2 + \left(\mathsf{neg}\left(ux\right)\right)\right)}} \]

    2. distribute-rgt-in N/A

       \[\leadsto \sqrt{\color{blue}{2 \cdot ux + \left(\mathsf{neg}\left(ux\right)\right) \cdot ux}} \]

    3. *-commutative N/A

       \[\leadsto \sqrt{\color{blue}{ux \cdot 2} + \left(\mathsf{neg}\left(ux\right)\right) \cdot ux} \]

    4. accelerator-lowering-fma.f32 N/A

       \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(ux, 2, \left(\mathsf{neg}\left(ux\right)\right) \cdot ux\right)}} \]

    5. distribute-lft-neg-out N/A

       \[\leadsto \sqrt{\mathsf{fma}\left(ux, 2, \color{blue}{\mathsf{neg}\left(ux \cdot ux\right)}\right)} \]

    6. neg-lowering-neg.f32 N/A

       \[\leadsto \sqrt{\mathsf{fma}\left(ux, 2, \color{blue}{\mathsf{neg}\left(ux \cdot ux\right)}\right)} \]

    7. *-lowering-*.f32 75.5

       \[\leadsto \sqrt{\mathsf{fma}\left(ux, 2, -\color{blue}{ux \cdot ux}\right)} \]

13. Applied egg-rr 75.5%

    \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(ux, 2, -ux \cdot ux\right)}} \]

14. Final simplification 75.5%

    \[\leadsto \sqrt{\mathsf{fma}\left(ux, 2, ux \cdot \left(-ux\right)\right)} \]
15. Add Preprocessing

Alternative 17: 75.5% accurate, 7.8× speedup

Math

\[\begin{array}{l} \\ \sqrt{\mathsf{fma}\left(ux, 1 - ux, ux\right)} \end{array} \]

FPCore

(FPCore (ux uy maxCos) :precision binary32 (sqrt (fma ux (- 1.0 ux) ux)))

C

float code(float ux, float uy, float maxCos) {
	return sqrtf(fmaf(ux, (1.0f - ux), ux));
}

Julia

function code(ux, uy, maxCos)
	return sqrt(fma(ux, Float32(Float32(1.0) - ux), ux))
end

Derivation

1. Initial program 54.8%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. Add Preprocessing

3. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{\sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]
4. Step-by-step derivation

   1. sqrt-lowering-sqrt.f32 N/A

      \[\leadsto \color{blue}{\sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

   2. sub-neg N/A

      \[\leadsto \sqrt{\color{blue}{1 + \left(\mathsf{neg}\left({\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}\right)\right)}} \]

   3. +-commutative N/A

      \[\leadsto \sqrt{\color{blue}{\left(\mathsf{neg}\left({\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}\right)\right) + 1}} \]

   4. unpow2 N/A

      \[\leadsto \sqrt{\left(\mathsf{neg}\left(\color{blue}{\left(\left(1 + maxCos \cdot ux\right) - ux\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)}\right)\right) + 1} \]

   5. distribute-rgt-neg-in N/A

      \[\leadsto \sqrt{\color{blue}{\left(\left(1 + maxCos \cdot ux\right) - ux\right) \cdot \left(\mathsf{neg}\left(\left(\left(1 + maxCos \cdot ux\right) - ux\right)\right)\right)} + 1} \]

   6. accelerator-lowering-fma.f32 N/A

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\left(1 + maxCos \cdot ux\right) - ux, \mathsf{neg}\left(\left(\left(1 + maxCos \cdot ux\right) - ux\right)\right), 1\right)}} \]

5. Simplified 46.7%

   \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1 - ux\right), \mathsf{fma}\left(-ux, maxCos + -1, -1\right), 1\right)}} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \sqrt{\color{blue}{1 + \left(ux \cdot maxCos + \left(1 - ux\right)\right) \cdot \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(maxCos + -1\right) + -1\right)}} \]

   2. distribute-rgt-in N/A

      \[\leadsto \sqrt{1 + \color{blue}{\left(\left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot maxCos + \left(1 - ux\right)\right) + -1 \cdot \left(ux \cdot maxCos + \left(1 - ux\right)\right)\right)}} \]

   3. associate-+r+ N/A

      \[\leadsto \sqrt{\color{blue}{\left(1 + \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot maxCos + \left(1 - ux\right)\right)\right) + -1 \cdot \left(ux \cdot maxCos + \left(1 - ux\right)\right)}} \]

   4. +-lowering-+.f32 N/A

      \[\leadsto \sqrt{\color{blue}{\left(1 + \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot maxCos + \left(1 - ux\right)\right)\right) + -1 \cdot \left(ux \cdot maxCos + \left(1 - ux\right)\right)}} \]

7. Applied egg-rr 46.9%

   \[\leadsto \sqrt{\color{blue}{\left(1 + \mathsf{fma}\left(maxCos, -ux, ux\right) \cdot \mathsf{fma}\left(ux, maxCos, 1 - ux\right)\right) + \left(-\mathsf{fma}\left(ux, maxCos, 1 - ux\right)\right)}} \]

8. Taylor expanded in maxCos around 0

   \[\leadsto \sqrt{\color{blue}{ux + ux \cdot \left(1 - ux\right)}} \]
9. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \sqrt{\color{blue}{ux \cdot \left(1 - ux\right) + ux}} \]

   2. accelerator-lowering-fma.f32 N/A

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(ux, 1 - ux, ux\right)}} \]

   3. --lowering--.f32 75.4

      \[\leadsto \sqrt{\mathsf{fma}\left(ux, \color{blue}{1 - ux}, ux\right)} \]

10. Simplified 75.4%

    \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(ux, 1 - ux, ux\right)}} \]
11. Add Preprocessing

Alternative 18: 75.5% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \sqrt{ux \cdot \left(2 - ux\right)} \end{array} \]

FPCore

(FPCore (ux uy maxCos) :precision binary32 (sqrt (* ux (- 2.0 ux))))

C

float code(float ux, float uy, float maxCos) {
	return sqrtf((ux * (2.0f - ux)));
}

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = sqrt((ux * (2.0e0 - ux)))
end function

Julia

function code(ux, uy, maxCos)
	return sqrt(Float32(ux * Float32(Float32(2.0) - ux)))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = sqrt((ux * (single(2.0) - ux)));
end

Derivation

1. Initial program 54.8%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. Add Preprocessing

3. Taylor expanded in ux around 0

   \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right)}} \]
4. Step-by-step derivation

   1. *-lowering-*.f32 N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right)}} \]

   2. cancel-sign-sub-inv N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \color{blue}{\left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) + \left(\mathsf{neg}\left(2\right)\right) \cdot maxCos\right)}} \]

   3. +-commutative N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \left(\color{blue}{\left(-1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right) + 2\right)} + \left(\mathsf{neg}\left(2\right)\right) \cdot maxCos\right)} \]

   4. metadata-eval N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \left(\left(-1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right) + 2\right) + \color{blue}{-2} \cdot maxCos\right)} \]

   5. associate-+l+ N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \color{blue}{\left(-1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right) + \left(2 + -2 \cdot maxCos\right)\right)}} \]

   6. mul-1-neg N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \left(\color{blue}{\left(\mathsf{neg}\left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right)} + \left(2 + -2 \cdot maxCos\right)\right)} \]

   7. distribute-rgt-neg-in N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \left(\color{blue}{ux \cdot \left(\mathsf{neg}\left({\left(maxCos - 1\right)}^{2}\right)\right)} + \left(2 + -2 \cdot maxCos\right)\right)} \]

   8. accelerator-lowering-fma.f32 N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \color{blue}{\mathsf{fma}\left(ux, \mathsf{neg}\left({\left(maxCos - 1\right)}^{2}\right), 2 + -2 \cdot maxCos\right)}} \]

   9. unpow2 N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \mathsf{neg}\left(\color{blue}{\left(maxCos - 1\right) \cdot \left(maxCos - 1\right)}\right), 2 + -2 \cdot maxCos\right)} \]

   10. distribute-rgt-neg-in N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \color{blue}{\left(maxCos - 1\right) \cdot \left(\mathsf{neg}\left(\left(maxCos - 1\right)\right)\right)}, 2 + -2 \cdot maxCos\right)} \]

   11. neg-sub0 N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos - 1\right) \cdot \color{blue}{\left(0 - \left(maxCos - 1\right)\right)}, 2 + -2 \cdot maxCos\right)} \]

   12. associate-+l- N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos - 1\right) \cdot \color{blue}{\left(\left(0 - maxCos\right) + 1\right)}, 2 + -2 \cdot maxCos\right)} \]

   13. neg-sub0 N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos - 1\right) \cdot \left(\color{blue}{\left(\mathsf{neg}\left(maxCos\right)\right)} + 1\right), 2 + -2 \cdot maxCos\right)} \]

   14. mul-1-neg N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos - 1\right) \cdot \left(\color{blue}{-1 \cdot maxCos} + 1\right), 2 + -2 \cdot maxCos\right)} \]

   15. +-commutative N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos - 1\right) \cdot \color{blue}{\left(1 + -1 \cdot maxCos\right)}, 2 + -2 \cdot maxCos\right)} \]

   16. *-lowering-*.f32 N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \color{blue}{\left(maxCos - 1\right) \cdot \left(1 + -1 \cdot maxCos\right)}, 2 + -2 \cdot maxCos\right)} \]

   17. sub-neg N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \color{blue}{\left(maxCos + \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(1 + -1 \cdot maxCos\right), 2 + -2 \cdot maxCos\right)} \]

   18. metadata-eval N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos + \color{blue}{-1}\right) \cdot \left(1 + -1 \cdot maxCos\right), 2 + -2 \cdot maxCos\right)} \]

   19. +-lowering-+.f32 N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \color{blue}{\left(maxCos + -1\right)} \cdot \left(1 + -1 \cdot maxCos\right), 2 + -2 \cdot maxCos\right)} \]

   20. mul-1-neg N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos + -1\right) \cdot \left(1 + \color{blue}{\left(\mathsf{neg}\left(maxCos\right)\right)}\right), 2 + -2 \cdot maxCos\right)} \]

   21. unsub-neg N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos + -1\right) \cdot \color{blue}{\left(1 - maxCos\right)}, 2 + -2 \cdot maxCos\right)} \]

   22. --lowering--.f32 N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos + -1\right) \cdot \color{blue}{\left(1 - maxCos\right)}, 2 + -2 \cdot maxCos\right)} \]

5. Simplified 98.9%

   \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\color{blue}{ux \cdot \mathsf{fma}\left(ux, \left(maxCos + -1\right) \cdot \left(1 - maxCos\right), \mathsf{fma}\left(maxCos, -2, 2\right)\right)}} \]

6. Taylor expanded in maxCos around 0

   \[\leadsto \color{blue}{\sqrt{ux \cdot \left(2 + -1 \cdot ux\right)} \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \]
7. Step-by-step derivation

   1. *-lowering-*.f32 N/A

      \[\leadsto \color{blue}{\sqrt{ux \cdot \left(2 + -1 \cdot ux\right)} \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \]

   2. sqrt-lowering-sqrt.f32 N/A

      \[\leadsto \color{blue}{\sqrt{ux \cdot \left(2 + -1 \cdot ux\right)}} \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \]

   3. *-lowering-*.f32 N/A

      \[\leadsto \sqrt{\color{blue}{ux \cdot \left(2 + -1 \cdot ux\right)}} \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \]

   4. mul-1-neg N/A

      \[\leadsto \sqrt{ux \cdot \left(2 + \color{blue}{\left(\mathsf{neg}\left(ux\right)\right)}\right)} \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \]

   5. unsub-neg N/A

      \[\leadsto \sqrt{ux \cdot \color{blue}{\left(2 - ux\right)}} \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \]

   6. --lowering--.f32 N/A

      \[\leadsto \sqrt{ux \cdot \color{blue}{\left(2 - ux\right)}} \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \]

   7. cos-lowering-cos.f32 N/A

      \[\leadsto \sqrt{ux \cdot \left(2 - ux\right)} \cdot \color{blue}{\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \]

   8. *-lowering-*.f32 N/A

      \[\leadsto \sqrt{ux \cdot \left(2 - ux\right)} \cdot \cos \color{blue}{\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \]

   9. *-lowering-*.f32 N/A

      \[\leadsto \sqrt{ux \cdot \left(2 - ux\right)} \cdot \cos \left(2 \cdot \color{blue}{\left(uy \cdot \mathsf{PI}\left(\right)\right)}\right) \]

   10. PI-lowering-PI.f32 93.0

       \[\leadsto \sqrt{ux \cdot \left(2 - ux\right)} \cdot \cos \left(2 \cdot \left(uy \cdot \color{blue}{\pi}\right)\right) \]

8. Simplified 93.0%

   \[\leadsto \color{blue}{\sqrt{ux \cdot \left(2 - ux\right)} \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right)} \]

9. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{\sqrt{ux \cdot \left(2 - ux\right)}} \]
10. Step-by-step derivation

    1. sub-neg N/A

       \[\leadsto \sqrt{ux \cdot \color{blue}{\left(2 + \left(\mathsf{neg}\left(ux\right)\right)\right)}} \]

    2. mul-1-neg N/A

       \[\leadsto \sqrt{ux \cdot \left(2 + \color{blue}{-1 \cdot ux}\right)} \]

    3. +-commutative N/A

       \[\leadsto \sqrt{ux \cdot \color{blue}{\left(-1 \cdot ux + 2\right)}} \]

    4. distribute-rgt-in N/A

       \[\leadsto \sqrt{\color{blue}{\left(-1 \cdot ux\right) \cdot ux + 2 \cdot ux}} \]

    5. mul-1-neg N/A

       \[\leadsto \sqrt{\color{blue}{\left(\mathsf{neg}\left(ux\right)\right)} \cdot ux + 2 \cdot ux} \]

    6. distribute-lft-neg-in N/A

       \[\leadsto \sqrt{\color{blue}{\left(\mathsf{neg}\left(ux \cdot ux\right)\right)} + 2 \cdot ux} \]

    7. unpow2 N/A

       \[\leadsto \sqrt{\left(\mathsf{neg}\left(\color{blue}{{ux}^{2}}\right)\right) + 2 \cdot ux} \]

    8. mul-1-neg N/A

       \[\leadsto \sqrt{\color{blue}{-1 \cdot {ux}^{2}} + 2 \cdot ux} \]

    9. sqrt-lowering-sqrt.f32 N/A

       \[\leadsto \color{blue}{\sqrt{-1 \cdot {ux}^{2} + 2 \cdot ux}} \]

    10. +-commutative N/A

        \[\leadsto \sqrt{\color{blue}{2 \cdot ux + -1 \cdot {ux}^{2}}} \]

    11. mul-1-neg N/A

        \[\leadsto \sqrt{2 \cdot ux + \color{blue}{\left(\mathsf{neg}\left({ux}^{2}\right)\right)}} \]

    12. unpow2 N/A

        \[\leadsto \sqrt{2 \cdot ux + \left(\mathsf{neg}\left(\color{blue}{ux \cdot ux}\right)\right)} \]

    13. distribute-lft-neg-in N/A

        \[\leadsto \sqrt{2 \cdot ux + \color{blue}{\left(\mathsf{neg}\left(ux\right)\right) \cdot ux}} \]

    14. mul-1-neg N/A

        \[\leadsto \sqrt{2 \cdot ux + \color{blue}{\left(-1 \cdot ux\right)} \cdot ux} \]

    15. distribute-rgt-in N/A

        \[\leadsto \sqrt{\color{blue}{ux \cdot \left(2 + -1 \cdot ux\right)}} \]

    16. *-lowering-*.f32 N/A

        \[\leadsto \sqrt{\color{blue}{ux \cdot \left(2 + -1 \cdot ux\right)}} \]

    17. mul-1-neg N/A

        \[\leadsto \sqrt{ux \cdot \left(2 + \color{blue}{\left(\mathsf{neg}\left(ux\right)\right)}\right)} \]

    18. sub-neg N/A

        \[\leadsto \sqrt{ux \cdot \color{blue}{\left(2 - ux\right)}} \]

    19. --lowering--.f32 75.4

        \[\leadsto \sqrt{ux \cdot \color{blue}{\left(2 - ux\right)}} \]

11. Simplified 75.4%

    \[\leadsto \color{blue}{\sqrt{ux \cdot \left(2 - ux\right)}} \]
12. Add Preprocessing

Alternative 19: 61.8% accurate, 9.8× speedup

Math

\[\begin{array}{l} \\ \sqrt{2 \cdot ux} \end{array} \]

FPCore

(FPCore (ux uy maxCos) :precision binary32 (sqrt (* 2.0 ux)))

C

float code(float ux, float uy, float maxCos) {
	return sqrtf((2.0f * ux));
}

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = sqrt((2.0e0 * ux))
end function

Julia

function code(ux, uy, maxCos)
	return sqrt(Float32(Float32(2.0) * ux))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = sqrt((single(2.0) * ux));
end

Derivation

1. Initial program 54.8%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. Add Preprocessing

3. Taylor expanded in ux around 0

   \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right)}} \]
4. Step-by-step derivation

   1. *-lowering-*.f32 N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right)}} \]

   2. cancel-sign-sub-inv N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \color{blue}{\left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) + \left(\mathsf{neg}\left(2\right)\right) \cdot maxCos\right)}} \]

   3. +-commutative N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \left(\color{blue}{\left(-1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right) + 2\right)} + \left(\mathsf{neg}\left(2\right)\right) \cdot maxCos\right)} \]

   4. metadata-eval N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \left(\left(-1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right) + 2\right) + \color{blue}{-2} \cdot maxCos\right)} \]

   5. associate-+l+ N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \color{blue}{\left(-1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right) + \left(2 + -2 \cdot maxCos\right)\right)}} \]

   6. mul-1-neg N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \left(\color{blue}{\left(\mathsf{neg}\left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right)} + \left(2 + -2 \cdot maxCos\right)\right)} \]

   7. distribute-rgt-neg-in N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \left(\color{blue}{ux \cdot \left(\mathsf{neg}\left({\left(maxCos - 1\right)}^{2}\right)\right)} + \left(2 + -2 \cdot maxCos\right)\right)} \]

   8. accelerator-lowering-fma.f32 N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \color{blue}{\mathsf{fma}\left(ux, \mathsf{neg}\left({\left(maxCos - 1\right)}^{2}\right), 2 + -2 \cdot maxCos\right)}} \]

   9. unpow2 N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \mathsf{neg}\left(\color{blue}{\left(maxCos - 1\right) \cdot \left(maxCos - 1\right)}\right), 2 + -2 \cdot maxCos\right)} \]

   10. distribute-rgt-neg-in N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \color{blue}{\left(maxCos - 1\right) \cdot \left(\mathsf{neg}\left(\left(maxCos - 1\right)\right)\right)}, 2 + -2 \cdot maxCos\right)} \]

   11. neg-sub0 N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos - 1\right) \cdot \color{blue}{\left(0 - \left(maxCos - 1\right)\right)}, 2 + -2 \cdot maxCos\right)} \]

   12. associate-+l- N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos - 1\right) \cdot \color{blue}{\left(\left(0 - maxCos\right) + 1\right)}, 2 + -2 \cdot maxCos\right)} \]

   13. neg-sub0 N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos - 1\right) \cdot \left(\color{blue}{\left(\mathsf{neg}\left(maxCos\right)\right)} + 1\right), 2 + -2 \cdot maxCos\right)} \]

   14. mul-1-neg N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos - 1\right) \cdot \left(\color{blue}{-1 \cdot maxCos} + 1\right), 2 + -2 \cdot maxCos\right)} \]

   15. +-commutative N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos - 1\right) \cdot \color{blue}{\left(1 + -1 \cdot maxCos\right)}, 2 + -2 \cdot maxCos\right)} \]

   16. *-lowering-*.f32 N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \color{blue}{\left(maxCos - 1\right) \cdot \left(1 + -1 \cdot maxCos\right)}, 2 + -2 \cdot maxCos\right)} \]

   17. sub-neg N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \color{blue}{\left(maxCos + \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(1 + -1 \cdot maxCos\right), 2 + -2 \cdot maxCos\right)} \]

   18. metadata-eval N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos + \color{blue}{-1}\right) \cdot \left(1 + -1 \cdot maxCos\right), 2 + -2 \cdot maxCos\right)} \]

   19. +-lowering-+.f32 N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \color{blue}{\left(maxCos + -1\right)} \cdot \left(1 + -1 \cdot maxCos\right), 2 + -2 \cdot maxCos\right)} \]

   20. mul-1-neg N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos + -1\right) \cdot \left(1 + \color{blue}{\left(\mathsf{neg}\left(maxCos\right)\right)}\right), 2 + -2 \cdot maxCos\right)} \]

   21. unsub-neg N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos + -1\right) \cdot \color{blue}{\left(1 - maxCos\right)}, 2 + -2 \cdot maxCos\right)} \]

   22. --lowering--.f32 N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos + -1\right) \cdot \color{blue}{\left(1 - maxCos\right)}, 2 + -2 \cdot maxCos\right)} \]

5. Simplified 98.9%

   \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\color{blue}{ux \cdot \mathsf{fma}\left(ux, \left(maxCos + -1\right) \cdot \left(1 - maxCos\right), \mathsf{fma}\left(maxCos, -2, 2\right)\right)}} \]

6. Taylor expanded in maxCos around 0

   \[\leadsto \color{blue}{\sqrt{ux \cdot \left(2 + -1 \cdot ux\right)} \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \]
7. Step-by-step derivation

   1. *-lowering-*.f32 N/A

      \[\leadsto \color{blue}{\sqrt{ux \cdot \left(2 + -1 \cdot ux\right)} \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \]

   2. sqrt-lowering-sqrt.f32 N/A

      \[\leadsto \color{blue}{\sqrt{ux \cdot \left(2 + -1 \cdot ux\right)}} \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \]

   3. *-lowering-*.f32 N/A

      \[\leadsto \sqrt{\color{blue}{ux \cdot \left(2 + -1 \cdot ux\right)}} \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \]

   4. mul-1-neg N/A

      \[\leadsto \sqrt{ux \cdot \left(2 + \color{blue}{\left(\mathsf{neg}\left(ux\right)\right)}\right)} \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \]

   5. unsub-neg N/A

      \[\leadsto \sqrt{ux \cdot \color{blue}{\left(2 - ux\right)}} \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \]

   6. --lowering--.f32 N/A

      \[\leadsto \sqrt{ux \cdot \color{blue}{\left(2 - ux\right)}} \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \]

   7. cos-lowering-cos.f32 N/A

      \[\leadsto \sqrt{ux \cdot \left(2 - ux\right)} \cdot \color{blue}{\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \]

   8. *-lowering-*.f32 N/A

      \[\leadsto \sqrt{ux \cdot \left(2 - ux\right)} \cdot \cos \color{blue}{\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \]

   9. *-lowering-*.f32 N/A

      \[\leadsto \sqrt{ux \cdot \left(2 - ux\right)} \cdot \cos \left(2 \cdot \color{blue}{\left(uy \cdot \mathsf{PI}\left(\right)\right)}\right) \]

   10. PI-lowering-PI.f32 93.0

       \[\leadsto \sqrt{ux \cdot \left(2 - ux\right)} \cdot \cos \left(2 \cdot \left(uy \cdot \color{blue}{\pi}\right)\right) \]

8. Simplified 93.0%

   \[\leadsto \color{blue}{\sqrt{ux \cdot \left(2 - ux\right)} \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right)} \]

9. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{\sqrt{ux \cdot \left(2 - ux\right)}} \]
10. Step-by-step derivation

    1. sub-neg N/A

       \[\leadsto \sqrt{ux \cdot \color{blue}{\left(2 + \left(\mathsf{neg}\left(ux\right)\right)\right)}} \]

    2. mul-1-neg N/A

       \[\leadsto \sqrt{ux \cdot \left(2 + \color{blue}{-1 \cdot ux}\right)} \]

    3. +-commutative N/A

       \[\leadsto \sqrt{ux \cdot \color{blue}{\left(-1 \cdot ux + 2\right)}} \]

    4. distribute-rgt-in N/A

       \[\leadsto \sqrt{\color{blue}{\left(-1 \cdot ux\right) \cdot ux + 2 \cdot ux}} \]

    5. mul-1-neg N/A

       \[\leadsto \sqrt{\color{blue}{\left(\mathsf{neg}\left(ux\right)\right)} \cdot ux + 2 \cdot ux} \]

    6. distribute-lft-neg-in N/A

       \[\leadsto \sqrt{\color{blue}{\left(\mathsf{neg}\left(ux \cdot ux\right)\right)} + 2 \cdot ux} \]

    7. unpow2 N/A

       \[\leadsto \sqrt{\left(\mathsf{neg}\left(\color{blue}{{ux}^{2}}\right)\right) + 2 \cdot ux} \]

    8. mul-1-neg N/A

       \[\leadsto \sqrt{\color{blue}{-1 \cdot {ux}^{2}} + 2 \cdot ux} \]

    9. sqrt-lowering-sqrt.f32 N/A

       \[\leadsto \color{blue}{\sqrt{-1 \cdot {ux}^{2} + 2 \cdot ux}} \]

    10. +-commutative N/A

        \[\leadsto \sqrt{\color{blue}{2 \cdot ux + -1 \cdot {ux}^{2}}} \]

    11. mul-1-neg N/A

        \[\leadsto \sqrt{2 \cdot ux + \color{blue}{\left(\mathsf{neg}\left({ux}^{2}\right)\right)}} \]

    12. unpow2 N/A

        \[\leadsto \sqrt{2 \cdot ux + \left(\mathsf{neg}\left(\color{blue}{ux \cdot ux}\right)\right)} \]

    13. distribute-lft-neg-in N/A

        \[\leadsto \sqrt{2 \cdot ux + \color{blue}{\left(\mathsf{neg}\left(ux\right)\right) \cdot ux}} \]

    14. mul-1-neg N/A

        \[\leadsto \sqrt{2 \cdot ux + \color{blue}{\left(-1 \cdot ux\right)} \cdot ux} \]

    15. distribute-rgt-in N/A

        \[\leadsto \sqrt{\color{blue}{ux \cdot \left(2 + -1 \cdot ux\right)}} \]

    16. *-lowering-*.f32 N/A

        \[\leadsto \sqrt{\color{blue}{ux \cdot \left(2 + -1 \cdot ux\right)}} \]

    17. mul-1-neg N/A

        \[\leadsto \sqrt{ux \cdot \left(2 + \color{blue}{\left(\mathsf{neg}\left(ux\right)\right)}\right)} \]

    18. sub-neg N/A

        \[\leadsto \sqrt{ux \cdot \color{blue}{\left(2 - ux\right)}} \]

    19. --lowering--.f32 75.4

        \[\leadsto \sqrt{ux \cdot \color{blue}{\left(2 - ux\right)}} \]

11. Simplified 75.4%

    \[\leadsto \color{blue}{\sqrt{ux \cdot \left(2 - ux\right)}} \]

12. Taylor expanded in ux around 0

    \[\leadsto \sqrt{\color{blue}{2 \cdot ux}} \]
13. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \sqrt{\color{blue}{ux \cdot 2}} \]

    2. *-lowering-*.f32 63.8

       \[\leadsto \sqrt{\color{blue}{ux \cdot 2}} \]

14. Simplified 63.8%

    \[\leadsto \sqrt{\color{blue}{ux \cdot 2}} \]

15. Final simplification 63.8%

    \[\leadsto \sqrt{2 \cdot ux} \]
16. Add Preprocessing

Reproduce

herbie shell --seed 2024199 
(FPCore (ux uy maxCos)
  :name "UniformSampleCone, x"
  :precision binary32
  :pre (and (and (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) (* ux maxCos)) (+ (- 1.0 ux) (* ux maxCos)))))))