UniformSampleCone, x

Percentage Accurate: 57.8% → 99.0%

Time: 17.0s

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: 57.8% 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 56.5%

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

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

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

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

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

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

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

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

   9. *-lowering-*.f32 99.0

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

7. Applied egg-rr 99.0%

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

Alternative 2: 83.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 + \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(ux + -1\right) - ux \cdot maxCos\right)} \leq 0.0018910999642685056:\\ \;\;\;\;\mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \pi \cdot \pi, 1\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(maxCos, -2, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1 - ux\right), \mathsf{fma}\left(maxCos, -ux, ux\right), ux \cdot \left(1 - maxCos\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<=
      (*
       (cos (* (* uy 2.0) PI))
       (sqrt
        (+
         1.0
         (* (+ (- 1.0 ux) (* ux maxCos)) (- (+ ux -1.0) (* ux maxCos))))))
      0.0018910999642685056)
   (*
    (fma (* -2.0 (* uy uy)) (* PI PI) 1.0)
    (sqrt (* ux (fma maxCos -2.0 2.0))))
   (sqrt
    (fma
     (fma ux maxCos (- 1.0 ux))
     (fma maxCos (- ux) ux)
     (* ux (- 1.0 maxCos))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if ((cosf(((uy * 2.0f) * ((float) M_PI))) * sqrtf((1.0f + (((1.0f - ux) + (ux * maxCos)) * ((ux + -1.0f) - (ux * maxCos)))))) <= 0.0018910999642685056f) {
		tmp = fmaf((-2.0f * (uy * uy)), (((float) M_PI) * ((float) M_PI)), 1.0f) * sqrtf((ux * fmaf(maxCos, -2.0f, 2.0f)));
	} else {
		tmp = sqrtf(fmaf(fmaf(ux, maxCos, (1.0f - ux)), fmaf(maxCos, -ux, ux), (ux * (1.0f - maxCos))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (Float32(cos(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) * sqrt(Float32(Float32(1.0) + Float32(Float32(Float32(Float32(1.0) - ux) + Float32(ux * maxCos)) * Float32(Float32(ux + Float32(-1.0)) - Float32(ux * maxCos)))))) <= Float32(0.0018910999642685056))
		tmp = Float32(fma(Float32(Float32(-2.0) * Float32(uy * uy)), Float32(Float32(pi) * Float32(pi)), Float32(1.0)) * sqrt(Float32(ux * fma(maxCos, Float32(-2.0), Float32(2.0)))));
	else
		tmp = sqrt(fma(fma(ux, maxCos, Float32(Float32(1.0) - ux)), fma(maxCos, Float32(-ux), ux), Float32(ux * Float32(Float32(1.0) - maxCos))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (cos.f32 (*.f32 (*.f32 uy #s(literal 2 binary32)) (PI.f32))) (sqrt.f32 (-.f32 #s(literal 1 binary32) (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)))))) < 0.00189109996

   1. Initial program 29.0%

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

   3. Taylor expanded in ux around 0

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

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

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

      2. metadata-eval N/A

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

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

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. accelerator-lowering-fma.f32 95.2

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

   5. Simplified 95.2%

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

   6. Taylor expanded in uy around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

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

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

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

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

      5. unpow2 N/A

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

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

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

      7. unpow2 N/A

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

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

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

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

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

      10. PI-lowering-PI.f32 83.7

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

   8. Simplified 83.7%

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

   if 0.00189109996 < (*.f32 (cos.f32 (*.f32 (*.f32 uy #s(literal 2 binary32)) (PI.f32))) (sqrt.f32 (-.f32 #s(literal 1 binary32) (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos))))))

   1. Initial program 81.5%

      \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   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 72.5%

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

      1. distribute-rgt-in N/A

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

      2. associate-+l+ N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

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

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

      6. +-commutative N/A

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

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

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

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

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

      9. distribute-rgt-in N/A

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

      10. neg-mul-1 N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

      13. *-lft-identity N/A

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

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

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

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

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

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

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

   7. Applied egg-rr 74.2%

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

   8. Taylor expanded in ux around -inf

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

      1. mul-1-neg N/A

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

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

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

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

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. +-lowering-+.f32 85.9

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

   10. Simplified 85.9%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 + \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(ux + -1\right) - ux \cdot maxCos\right)} \leq 0.0018910999642685056:\\ \;\;\;\;\mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \pi \cdot \pi, 1\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(maxCos, -2, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1 - ux\right), \mathsf{fma}\left(maxCos, -ux, ux\right), ux \cdot \left(1 - maxCos\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.0% 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 56.5%

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

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

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (cos (* (* uy 2.0) PI))
  (sqrt (fma ux (- 2.0 ux) (* maxCos (* ux (fma ux 2.0 -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), (maxCos * (ux * fmaf(ux, 2.0f, -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(maxCos * Float32(ux * fma(ux, Float32(2.0), Float32(-2.0)))))))
end

Derivation

1. Initial program 56.5%

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

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

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

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

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

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

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

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

   9. *-lowering-*.f32 99.0

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

7. Applied egg-rr 99.0%

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

8. Taylor expanded in maxCos around 0

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

   1. associate-+r+ N/A

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

   2. +-commutative N/A

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

   3. mul-1-neg N/A

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

   4. unpow2 N/A

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

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

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

   6. mul-1-neg N/A

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

   7. distribute-rgt-in N/A

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

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

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

   9. mul-1-neg N/A

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

   10. unsub-neg N/A

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

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

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

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

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

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

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

   14. sub-neg N/A

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

   15. *-commutative N/A

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

   16. metadata-eval N/A

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

   17. accelerator-lowering-fma.f32 98.4

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

10. Simplified 98.4%

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

Alternative 5: 98.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (cos (* (* uy 2.0) PI))
  (sqrt (fma maxCos (* ux (fma ux 2.0 -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(maxCos, (ux * fmaf(ux, 2.0f, -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(maxCos, Float32(ux * fma(ux, Float32(2.0), Float32(-2.0))), Float32(ux * Float32(Float32(2.0) - ux)))))
end

Derivation

1. Initial program 56.5%

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

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

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

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

   3. sub-neg N/A

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

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

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

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

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

   8. mul-1-neg N/A

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

   9. unsub-neg N/A

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

   10. --lowering--.f32 98.4

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

8. Simplified 98.4%

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

Alternative 6: 97.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\\ \mathbf{if}\;uy \cdot 2 \leq 0.0154600003734231:\\ \;\;\;\;\mathsf{fma}\left(-2, \sqrt{ux \cdot \mathsf{fma}\left(ux, t\_0, \mathsf{fma}\left(-2, maxCos, 2\right)\right)} \cdot \left(\left(uy \cdot uy\right) \cdot \left(\pi \cdot \pi\right)\right), \sqrt{ux \cdot \left(2 + \mathsf{fma}\left(ux, t\_0, maxCos \cdot -2\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (+ maxCos -1.0) (- 1.0 maxCos))))
   (if (<= (* uy 2.0) 0.0154600003734231)
     (fma
      -2.0
      (*
       (sqrt (* ux (fma ux t_0 (fma -2.0 maxCos 2.0))))
       (* (* uy uy) (* PI PI)))
      (sqrt (* ux (+ 2.0 (fma ux t_0 (* maxCos -2.0))))))
     (* (cos (* (* uy 2.0) PI)) (sqrt (* ux (- 2.0 ux)))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = (maxCos + -1.0f) * (1.0f - maxCos);
	float tmp;
	if ((uy * 2.0f) <= 0.0154600003734231f) {
		tmp = fmaf(-2.0f, (sqrtf((ux * fmaf(ux, t_0, fmaf(-2.0f, maxCos, 2.0f)))) * ((uy * uy) * (((float) M_PI) * ((float) M_PI)))), sqrtf((ux * (2.0f + fmaf(ux, t_0, (maxCos * -2.0f))))));
	} else {
		tmp = cosf(((uy * 2.0f) * ((float) M_PI))) * sqrtf((ux * (2.0f - ux)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(Float32(maxCos + Float32(-1.0)) * Float32(Float32(1.0) - maxCos))
	tmp = Float32(0.0)
	if (Float32(uy * Float32(2.0)) <= Float32(0.0154600003734231))
		tmp = fma(Float32(-2.0), Float32(sqrt(Float32(ux * fma(ux, t_0, fma(Float32(-2.0), maxCos, Float32(2.0))))) * Float32(Float32(uy * uy) * Float32(Float32(pi) * Float32(pi)))), sqrt(Float32(ux * Float32(Float32(2.0) + fma(ux, t_0, Float32(maxCos * Float32(-2.0)))))));
	else
		tmp = Float32(cos(Float32(Float32(uy * Float32(2.0)) * 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)) < 0.0154600004

   1. Initial program 56.8%

      \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   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 99.3%

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

   6. Taylor expanded in uy around 0

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

   8. Simplified 99.3%

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

   9. Taylor expanded in ux around 0

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

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

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

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

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

       3. +-commutative N/A

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

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

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

       5. *-commutative N/A

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

       6. sub-neg N/A

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

       7. mul-1-neg N/A

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

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

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

       9. sub-neg N/A

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

       10. metadata-eval N/A

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

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

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

       12. mul-1-neg N/A

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

       13. sub-neg N/A

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

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

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

       15. *-lowering-*.f32 99.4

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

   11. Simplified 99.4%

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

   if 0.0154600004 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 55.1%

      \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   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 52.4

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

   5. Simplified 52.4%

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

   6. Taylor expanded in ux around 0

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

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

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. --lowering--.f32 89.8

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

   8. Simplified 89.8%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.0154600003734231:\\ \;\;\;\;\mathsf{fma}\left(-2, \sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos + -1\right) \cdot \left(1 - maxCos\right), \mathsf{fma}\left(-2, maxCos, 2\right)\right)} \cdot \left(\left(uy \cdot uy\right) \cdot \left(\pi \cdot \pi\right)\right), \sqrt{ux \cdot \left(2 + \mathsf{fma}\left(ux, \left(maxCos + -1\right) \cdot \left(1 - maxCos\right), maxCos \cdot -2\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 94.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\\ \mathbf{if}\;uy \cdot 2 \leq 0.05999999865889549:\\ \;\;\;\;\mathsf{fma}\left(-2, \sqrt{ux \cdot \mathsf{fma}\left(ux, t\_0, \mathsf{fma}\left(-2, maxCos, 2\right)\right)} \cdot \left(\left(uy \cdot uy\right) \cdot \left(\pi \cdot \pi\right)\right), \sqrt{ux \cdot \left(2 + \mathsf{fma}\left(ux, t\_0, maxCos \cdot -2\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{2 \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (+ maxCos -1.0) (- 1.0 maxCos))))
   (if (<= (* uy 2.0) 0.05999999865889549)
     (fma
      -2.0
      (*
       (sqrt (* ux (fma ux t_0 (fma -2.0 maxCos 2.0))))
       (* (* uy uy) (* PI PI)))
      (sqrt (* ux (+ 2.0 (fma ux t_0 (* maxCos -2.0))))))
     (* (cos (* (* uy 2.0) PI)) (sqrt (* 2.0 ux))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = (maxCos + -1.0f) * (1.0f - maxCos);
	float tmp;
	if ((uy * 2.0f) <= 0.05999999865889549f) {
		tmp = fmaf(-2.0f, (sqrtf((ux * fmaf(ux, t_0, fmaf(-2.0f, maxCos, 2.0f)))) * ((uy * uy) * (((float) M_PI) * ((float) M_PI)))), sqrtf((ux * (2.0f + fmaf(ux, t_0, (maxCos * -2.0f))))));
	} else {
		tmp = cosf(((uy * 2.0f) * ((float) M_PI))) * sqrtf((2.0f * ux));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(Float32(maxCos + Float32(-1.0)) * Float32(Float32(1.0) - maxCos))
	tmp = Float32(0.0)
	if (Float32(uy * Float32(2.0)) <= Float32(0.05999999865889549))
		tmp = fma(Float32(-2.0), Float32(sqrt(Float32(ux * fma(ux, t_0, fma(Float32(-2.0), maxCos, Float32(2.0))))) * Float32(Float32(uy * uy) * Float32(Float32(pi) * Float32(pi)))), sqrt(Float32(ux * Float32(Float32(2.0) + fma(ux, t_0, Float32(maxCos * Float32(-2.0)))))));
	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)) < 0.0599999987

   1. Initial program 56.8%

      \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   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 99.3%

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

   6. Taylor expanded in uy around 0

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

   8. Simplified 98.6%

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

   9. Taylor expanded in ux around 0

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

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

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

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

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

       3. +-commutative N/A

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

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

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

       5. *-commutative N/A

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

       6. sub-neg N/A

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

       7. mul-1-neg N/A

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

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

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

       9. sub-neg N/A

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

       10. metadata-eval N/A

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

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

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

       12. mul-1-neg N/A

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

       13. sub-neg N/A

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

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

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

       15. *-lowering-*.f32 98.7

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

   11. Simplified 98.7%

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

   if 0.0599999987 < (*.f32 uy #s(literal 2 binary32))

   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 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 51.4

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

   5. Simplified 51.4%

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

   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. *-commutative N/A

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

      2. *-lowering-*.f32 71.1

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

   8. Simplified 71.1%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.05999999865889549:\\ \;\;\;\;\mathsf{fma}\left(-2, \sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos + -1\right) \cdot \left(1 - maxCos\right), \mathsf{fma}\left(-2, maxCos, 2\right)\right)} \cdot \left(\left(uy \cdot uy\right) \cdot \left(\pi \cdot \pi\right)\right), \sqrt{ux \cdot \left(2 + \mathsf{fma}\left(ux, \left(maxCos + -1\right) \cdot \left(1 - maxCos\right), maxCos \cdot -2\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{2 \cdot ux}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 88.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 56.5%

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

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

8. Simplified 88.3%

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

9. Taylor expanded in ux around 0

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

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

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

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

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

    3. +-commutative N/A

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

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

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

    5. *-commutative N/A

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

    6. sub-neg N/A

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

    7. mul-1-neg N/A

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

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

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

    9. sub-neg N/A

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

    10. metadata-eval N/A

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

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

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

    12. mul-1-neg N/A

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

    13. sub-neg N/A

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

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

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

    15. *-lowering-*.f32 88.4

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

11. Simplified 88.4%

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

12. Final simplification 88.4%

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

Alternative 9: 86.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{ux \cdot \left(2 - ux\right)}\\ \mathbf{if}\;uy \cdot 2 \leq 0.00013499999477062374:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1 - ux\right), \mathsf{fma}\left(maxCos, -ux, ux\right), \mathsf{fma}\left(ux, -maxCos, ux\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2, \left(\left(uy \cdot uy\right) \cdot \left(\pi \cdot \pi\right)\right) \cdot t\_0, t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (sqrt (* ux (- 2.0 ux)))))
   (if (<= (* uy 2.0) 0.00013499999477062374)
     (sqrt
      (fma
       (fma ux maxCos (- 1.0 ux))
       (fma maxCos (- ux) ux)
       (fma ux (- maxCos) ux)))
     (fma -2.0 (* (* (* uy uy) (* PI PI)) t_0) t_0))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = sqrtf((ux * (2.0f - ux)));
	float tmp;
	if ((uy * 2.0f) <= 0.00013499999477062374f) {
		tmp = sqrtf(fmaf(fmaf(ux, maxCos, (1.0f - ux)), fmaf(maxCos, -ux, ux), fmaf(ux, -maxCos, ux)));
	} else {
		tmp = fmaf(-2.0f, (((uy * uy) * (((float) M_PI) * ((float) M_PI))) * t_0), t_0);
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = sqrt(Float32(ux * Float32(Float32(2.0) - ux)))
	tmp = Float32(0.0)
	if (Float32(uy * Float32(2.0)) <= Float32(0.00013499999477062374))
		tmp = sqrt(fma(fma(ux, maxCos, Float32(Float32(1.0) - ux)), fma(maxCos, Float32(-ux), ux), fma(ux, Float32(-maxCos), ux)));
	else
		tmp = fma(Float32(-2.0), Float32(Float32(Float32(uy * uy) * Float32(Float32(pi) * Float32(pi))) * t_0), t_0);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 1.34999995e-4

   1. Initial program 57.4%

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

   3. Taylor expanded in 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 57.5%

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

      1. distribute-rgt-in N/A

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

      2. associate-+l+ N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

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

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

      6. +-commutative N/A

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

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

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

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

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

      9. distribute-rgt-in N/A

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

      10. neg-mul-1 N/A

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

      11. associate-*r* N/A

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

      12. metadata-eval N/A

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

      13. *-lft-identity N/A

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

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

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

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

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

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

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

   7. Applied egg-rr 63.9%

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

   8. Taylor expanded in maxCos around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

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

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

      5. mul-1-neg N/A

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

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

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

      7. mul-1-neg N/A

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

      8. neg-lowering-neg.f32 99.5

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

   10. Simplified 99.5%

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

   if 1.34999995e-4 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 55.1%

      \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   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 98.2%

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

   6. Taylor expanded in uy around 0

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

   8. Simplified 72.6%

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

   9. Taylor expanded in maxCos around 0

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

       1. +-commutative N/A

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

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

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

   11. Simplified 69.5%

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

Alternative 10: 88.5% 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 56.5%

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

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

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

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

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

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

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

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

   9. *-lowering-*.f32 99.0

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

7. Applied egg-rr 99.0%

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

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 88.4

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

10. Simplified 88.4%

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

11. Final simplification 88.4%

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

Alternative 11: 88.4% 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 56.5%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
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 88.3

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

8. Simplified 88.3%

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

9. Final simplification 88.3%

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

Alternative 12: 79.7% accurate, 3.7× speedup

Math

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

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (sqrt
  (fma
   (fma ux maxCos (- 1.0 ux))
   (fma maxCos (- ux) ux)
   (fma ux (- maxCos) ux))))

C

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

Julia

function code(ux, uy, maxCos)
	return sqrt(fma(fma(ux, maxCos, Float32(Float32(1.0) - ux)), fma(maxCos, Float32(-ux), ux), fma(ux, Float32(-maxCos), ux)))
end

Derivation

1. Initial program 56.5%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
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 49.0%

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

   1. distribute-rgt-in N/A

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

   2. associate-+l+ N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

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

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

   6. +-commutative N/A

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

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

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

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

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

   9. distribute-rgt-in N/A

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

   10. neg-mul-1 N/A

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

   11. associate-*r* N/A

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

   12. metadata-eval N/A

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

   13. *-lft-identity N/A

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

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

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

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

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

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

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

7. Applied egg-rr 55.2%

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

8. Taylor expanded in maxCos around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. *-commutative N/A

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

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

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

   5. mul-1-neg N/A

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

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

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

   7. mul-1-neg N/A

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

   8. neg-lowering-neg.f32 81.2

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

10. Simplified 81.2%

    \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1 - ux\right), \mathsf{fma}\left(maxCos, -ux, ux\right), \color{blue}{\mathsf{fma}\left(ux, -maxCos, ux\right)}\right)} \]
11. Add Preprocessing

Alternative 13: 79.7% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

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 * Float32(-2.0))))))
end

Derivation

1. Initial program 56.5%

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

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

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

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

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

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

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

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

   9. *-lowering-*.f32 99.0

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

7. Applied egg-rr 99.0%

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

8. Taylor expanded in uy around 0

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

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

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

   2. *-commutative N/A

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

   3. distribute-lft-in N/A

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

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

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

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

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

   6. +-commutative N/A

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

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

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

   8. *-commutative N/A

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

   9. sub-neg N/A

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

   10. mul-1-neg N/A

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

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

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

   12. sub-neg N/A

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

   13. metadata-eval N/A

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

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

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

   15. mul-1-neg N/A

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

   16. sub-neg N/A

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

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

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

   18. *-lowering-*.f32 81.2

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

10. Simplified 81.2%

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

11. Final simplification 81.2%

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

Alternative 14: 79.7% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 56.5%

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

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

8. Simplified 81.2%

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

Alternative 15: 78.8% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 56.5%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
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 49.0%

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

   1. distribute-rgt-in N/A

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

   2. associate-+l+ N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

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

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

   6. +-commutative N/A

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

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

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

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

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

   9. distribute-rgt-in N/A

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

   10. neg-mul-1 N/A

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

   11. associate-*r* N/A

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

   12. metadata-eval N/A

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

   13. *-lft-identity N/A

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

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

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

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

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

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

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

7. Applied egg-rr 55.2%

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

8. Taylor expanded in ux around -inf

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

   1. mul-1-neg N/A

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

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

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

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

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

   4. sub-neg N/A

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

   5. metadata-eval N/A

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

   6. +-lowering-+.f32 81.2

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

10. Simplified 81.2%

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

11. Taylor expanded in maxCos around 0

    \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{1 - ux}, \mathsf{fma}\left(maxCos, \mathsf{neg}\left(ux\right), ux\right), \mathsf{neg}\left(ux \cdot \left(maxCos + -1\right)\right)\right)} \]
12. Step-by-step derivation

    1. --lowering--.f32 80.4

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

13. Simplified 80.4%

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

14. Final simplification 80.4%

    \[\leadsto \sqrt{\mathsf{fma}\left(1 - ux, \mathsf{fma}\left(maxCos, -ux, ux\right), ux \cdot \left(1 - maxCos\right)\right)} \]
15. Add Preprocessing

Alternative 16: 75.9% accurate, 4.6× speedup

Math

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

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (sqrt (fma (fma ux maxCos (- 1.0 ux)) (fma maxCos (- ux) ux) ux)))

C

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

Julia

function code(ux, uy, maxCos)
	return sqrt(fma(fma(ux, maxCos, Float32(Float32(1.0) - ux)), fma(maxCos, Float32(-ux), ux), ux))
end

Derivation

1. Initial program 56.5%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
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 49.0%

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

   1. distribute-rgt-in N/A

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

   2. associate-+l+ N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

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

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

   6. +-commutative N/A

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

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

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

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

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

   9. distribute-rgt-in N/A

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

   10. neg-mul-1 N/A

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

   11. associate-*r* N/A

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

   12. metadata-eval N/A

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

   13. *-lft-identity N/A

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

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

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

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

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

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

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

7. Applied egg-rr 55.2%

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

8. Taylor expanded in maxCos around 0

   \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1 - ux\right), \mathsf{fma}\left(maxCos, \mathsf{neg}\left(ux\right), ux\right), \color{blue}{ux}\right)} \]
9. Step-by-step derivation

10. Simplified 76.6%

    \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1 - ux\right), \mathsf{fma}\left(maxCos, -ux, ux\right), \color{blue}{ux}\right)} \]
11. 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 56.5%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
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 49.0%

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

   1. distribute-rgt-in N/A

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

   2. associate-+l+ N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

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

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

   6. +-commutative N/A

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

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

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

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

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

   9. distribute-rgt-in N/A

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

   10. neg-mul-1 N/A

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

   11. associate-*r* N/A

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

   12. metadata-eval N/A

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

   13. *-lft-identity N/A

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

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

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

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

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

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

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

7. Applied egg-rr 55.2%

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

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 76.2

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

10. Simplified 76.2%

    \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(ux, 1 - ux, ux\right)}} \]
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 56.5%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
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 49.0%

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

   1. distribute-rgt-in N/A

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

   2. associate-+l+ N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

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

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

   6. +-commutative N/A

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

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

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

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

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

   9. distribute-rgt-in N/A

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

   10. neg-mul-1 N/A

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

   11. associate-*r* N/A

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

   12. metadata-eval N/A

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

   13. *-lft-identity N/A

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

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

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

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

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

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

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

7. Applied egg-rr 55.2%

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

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 76.2

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

10. Simplified 76.2%

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

11. Taylor expanded in ux around 0

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

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

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

    2. mul-1-neg N/A

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

    3. unsub-neg N/A

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

    4. --lowering--.f32 76.2

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

13. Simplified 76.2%

    \[\leadsto \sqrt{\color{blue}{ux \cdot \left(2 - ux\right)}} \]
14. Add Preprocessing

Alternative 19: 61.7% 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 56.5%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
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 49.0%

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

   1. distribute-rgt-in N/A

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

   2. associate-+l+ N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

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

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

   6. +-commutative N/A

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

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

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

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

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

   9. distribute-rgt-in N/A

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

   10. neg-mul-1 N/A

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

   11. associate-*r* N/A

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

   12. metadata-eval N/A

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

   13. *-lft-identity N/A

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

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

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

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

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

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

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

7. Applied egg-rr 55.2%

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

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 76.2

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

10. Simplified 76.2%

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

11. Taylor expanded in ux around 0

    \[\leadsto \sqrt{\color{blue}{2 \cdot ux}} \]
12. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \sqrt{\color{blue}{ux \cdot 2}} \]

    2. *-lowering-*.f32 63.0

       \[\leadsto \sqrt{\color{blue}{ux \cdot 2}} \]

13. Simplified 63.0%

    \[\leadsto \sqrt{\color{blue}{ux \cdot 2}} \]

14. Final simplification 63.0%

    \[\leadsto \sqrt{2 \cdot ux} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024204 
(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)))))))