UniformSampleCone, x

Percentage Accurate: 57.1% → 98.9%

Time: 15.5s

Alternatives: 16

Speedup: 9.8×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 57.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(uy \cdot \left(\pi + \pi\right)\right)\\ \left(0.5 + \mathsf{fma}\left(0.5, t\_0, 0.5 \cdot t\_0 - 0.5\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(maxCos, -2, \mathsf{fma}\left(1 - maxCos, \mathsf{fma}\left(ux, maxCos, -ux\right), 2\right)\right)} \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (cos (* uy (+ PI PI)))))
   (*
    (+ 0.5 (fma 0.5 t_0 (- (* 0.5 t_0) 0.5)))
    (sqrt
     (*
      ux
      (fma maxCos -2.0 (fma (- 1.0 maxCos) (fma ux maxCos (- ux)) 2.0)))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = cosf((uy * (((float) M_PI) + ((float) M_PI))));
	return (0.5f + fmaf(0.5f, t_0, ((0.5f * t_0) - 0.5f))) * sqrtf((ux * fmaf(maxCos, -2.0f, fmaf((1.0f - maxCos), fmaf(ux, maxCos, -ux), 2.0f))));
}

Julia

function code(ux, uy, maxCos)
	t_0 = cos(Float32(uy * Float32(Float32(pi) + Float32(pi))))
	return Float32(Float32(Float32(0.5) + fma(Float32(0.5), t_0, Float32(Float32(Float32(0.5) * t_0) - Float32(0.5)))) * sqrt(Float32(ux * fma(maxCos, Float32(-2.0), fma(Float32(Float32(1.0) - maxCos), fma(ux, maxCos, Float32(-ux)), Float32(2.0))))))
end

Derivation

1. Initial program 60.2%

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

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f32 N/A

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

   7. +-commutative N/A

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

   8. mul-1-neg N/A

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

   9. *-commutative N/A

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

   10. unpow2 N/A

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

   11. associate-*l* N/A

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

   12. *-commutative N/A

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

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

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

   14. neg-sub0 N/A

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

   15. associate-+l- N/A

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

   16. neg-sub0 N/A

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

   17. mul-1-neg N/A

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

   18. +-commutative N/A

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

5. Applied rewrites 99.1%

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

   1. lift-cos.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-PI.f32 N/A

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

   5. *-commutative N/A

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

   6. lift-PI.f32 N/A

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

   7. associate-*r* N/A

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

   8. lift-*.f32 N/A

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

   9. cos-2 N/A

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

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

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

   11. lower-fma.f32 N/A

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

   12. lower-cos.f32 N/A

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

   13. lower-cos.f32 N/A

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

   14. lower-*.f32 N/A

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

   15. lower-neg.f32 N/A

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

   16. lower-sin.f32 N/A

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

   17. lower-sin.f32 99.0

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

7. Applied rewrites 99.0%

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

   1. lift-fma.f32 N/A

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

   2. lift-cos.f32 N/A

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

   3. lift-cos.f32 N/A

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

   4. sqr-cos-a N/A

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

   5. associate-+l+ N/A

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

   6. lower-+.f32 N/A

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

   7. count-2 N/A

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

   8. lower-fma.f32 N/A

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

9. Applied rewrites 99.1%

   \[\leadsto \color{blue}{\left(0.5 + \mathsf{fma}\left(0.5, \cos \left(uy \cdot \left(\pi + \pi\right)\right), -\left(0.5 - 0.5 \cdot \cos \left(uy \cdot \left(\pi + \pi\right)\right)\right)\right)\right)} \cdot \sqrt{ux \cdot \mathsf{fma}\left(maxCos, -2, \mathsf{fma}\left(1 - maxCos, \mathsf{fma}\left(ux, maxCos, -ux\right), 2\right)\right)} \]

10. Final simplification 99.1%

    \[\leadsto \left(0.5 + \mathsf{fma}\left(0.5, \cos \left(uy \cdot \left(\pi + \pi\right)\right), 0.5 \cdot \cos \left(uy \cdot \left(\pi + \pi\right)\right) - 0.5\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(maxCos, -2, \mathsf{fma}\left(1 - maxCos, \mathsf{fma}\left(ux, maxCos, -ux\right), 2\right)\right)} \]
11. Add Preprocessing

Alternative 2: 83.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\pi \cdot \left(uy \cdot 2\right)\right) \leq 0.9999949932098389:\\ \;\;\;\;\sqrt{ux \cdot 2} \cdot \mathsf{fma}\left(-2 \cdot \left(\pi \cdot \left(uy \cdot uy\right)\right), \pi, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(maxCos, -2, 2\right), ux, \left(\left(1 - maxCos\right) \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot ux\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (cos (* PI (* uy 2.0))) 0.9999949932098389)
   (* (sqrt (* ux 2.0)) (fma (* -2.0 (* PI (* uy uy))) PI 1.0))
   (sqrt
    (fma
     (fma maxCos -2.0 2.0)
     ux
     (* (* (- 1.0 maxCos) (+ maxCos -1.0)) (* ux ux))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (cosf((((float) M_PI) * (uy * 2.0f))) <= 0.9999949932098389f) {
		tmp = sqrtf((ux * 2.0f)) * fmaf((-2.0f * (((float) M_PI) * (uy * uy))), ((float) M_PI), 1.0f);
	} else {
		tmp = sqrtf(fmaf(fmaf(maxCos, -2.0f, 2.0f), ux, (((1.0f - maxCos) * (maxCos + -1.0f)) * (ux * ux))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (cos(Float32(Float32(pi) * Float32(uy * Float32(2.0)))) <= Float32(0.9999949932098389))
		tmp = Float32(sqrt(Float32(ux * Float32(2.0))) * fma(Float32(Float32(-2.0) * Float32(Float32(pi) * Float32(uy * uy))), Float32(pi), Float32(1.0)));
	else
		tmp = sqrt(fma(fma(maxCos, Float32(-2.0), Float32(2.0)), ux, Float32(Float32(Float32(Float32(1.0) - maxCos) * Float32(maxCos + Float32(-1.0))) * Float32(ux * ux))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 (*.f32 uy #s(literal 2 binary32)) (PI.f32))) < 0.999994993

   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 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. lower-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. lower--.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. lower-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. lower--.f32 55.0

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

   5. Applied rewrites 55.0%

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

   6. Taylor expanded in ux around 0

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

   8. Applied rewrites 72.8%

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

   9. Taylor expanded in uy around 0

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

       1. +-commutative N/A

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

       2. associate-*r* N/A

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

       3. lower-fma.f32 N/A

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

       4. lower-*.f32 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f32 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f32 N/A

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

       9. lower-PI.f32 N/A

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

       10. lower-PI.f32 58.0

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

   11. Applied rewrites 58.0%

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

   13. Applied rewrites 58.0%

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

   if 0.999994993 < (cos.f32 (*.f32 (*.f32 uy #s(literal 2 binary32)) (PI.f32)))

   1. Initial program 62.0%

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

   3. Taylor expanded in uy around 0

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

      1. lower-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. lower-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. Applied rewrites 62.3%

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

   6. Taylor expanded in ux around 0

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

   8. Applied rewrites 98.7%

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

   10. Applied rewrites 98.8%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\pi \cdot \left(uy \cdot 2\right)\right) \leq 0.9999949932098389:\\ \;\;\;\;\sqrt{ux \cdot 2} \cdot \mathsf{fma}\left(-2 \cdot \left(\pi \cdot \left(uy \cdot uy\right)\right), \pi, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(maxCos, -2, 2\right), ux, \left(\left(1 - maxCos\right) \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot ux\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\pi \cdot \left(uy \cdot 2\right)\right) \leq 0.9999949932098389:\\ \;\;\;\;\sqrt{ux \cdot 2} \cdot \mathsf{fma}\left(-2 \cdot \left(\pi \cdot \left(uy \cdot uy\right)\right), \pi, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{ux \cdot \mathsf{fma}\left(ux, \left(1 - maxCos\right) \cdot \left(maxCos + -1\right), \mathsf{fma}\left(maxCos, -2, 2\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (cos (* PI (* uy 2.0))) 0.9999949932098389)
   (* (sqrt (* ux 2.0)) (fma (* -2.0 (* PI (* uy uy))) PI 1.0))
   (sqrt
    (* ux (fma ux (* (- 1.0 maxCos) (+ maxCos -1.0)) (fma maxCos -2.0 2.0))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (cosf((((float) M_PI) * (uy * 2.0f))) <= 0.9999949932098389f) {
		tmp = sqrtf((ux * 2.0f)) * fmaf((-2.0f * (((float) M_PI) * (uy * uy))), ((float) M_PI), 1.0f);
	} else {
		tmp = sqrtf((ux * fmaf(ux, ((1.0f - maxCos) * (maxCos + -1.0f)), fmaf(maxCos, -2.0f, 2.0f))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (cos(Float32(Float32(pi) * Float32(uy * Float32(2.0)))) <= Float32(0.9999949932098389))
		tmp = Float32(sqrt(Float32(ux * Float32(2.0))) * fma(Float32(Float32(-2.0) * Float32(Float32(pi) * Float32(uy * uy))), Float32(pi), Float32(1.0)));
	else
		tmp = sqrt(Float32(ux * fma(ux, Float32(Float32(Float32(1.0) - maxCos) * Float32(maxCos + Float32(-1.0))), fma(maxCos, Float32(-2.0), Float32(2.0)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 (*.f32 uy #s(literal 2 binary32)) (PI.f32))) < 0.999994993

   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 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. lower-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. lower--.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. lower-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. lower--.f32 55.0

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

   5. Applied rewrites 55.0%

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

   6. Taylor expanded in ux around 0

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

   8. Applied rewrites 72.8%

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

   9. Taylor expanded in uy around 0

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

       1. +-commutative N/A

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

       2. associate-*r* N/A

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

       3. lower-fma.f32 N/A

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

       4. lower-*.f32 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f32 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f32 N/A

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

       9. lower-PI.f32 N/A

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

       10. lower-PI.f32 58.0

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

   11. Applied rewrites 58.0%

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

   13. Applied rewrites 58.0%

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

   if 0.999994993 < (cos.f32 (*.f32 (*.f32 uy #s(literal 2 binary32)) (PI.f32)))

   1. Initial program 62.0%

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

   3. Taylor expanded in uy around 0

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

      1. lower-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. lower-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. Applied rewrites 62.3%

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

   6. Taylor expanded in ux around 0

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

   8. Applied rewrites 98.7%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\pi \cdot \left(uy \cdot 2\right)\right) \leq 0.9999949932098389:\\ \;\;\;\;\sqrt{ux \cdot 2} \cdot \mathsf{fma}\left(-2 \cdot \left(\pi \cdot \left(uy \cdot uy\right)\right), \pi, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{ux \cdot \mathsf{fma}\left(ux, \left(1 - maxCos\right) \cdot \left(maxCos + -1\right), \mathsf{fma}\left(maxCos, -2, 2\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\pi \cdot \left(uy \cdot 2\right)\right) \leq 0.9999949932098389:\\ \;\;\;\;\sqrt{ux \cdot 2} \cdot \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \pi \cdot \pi, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{ux \cdot \mathsf{fma}\left(ux, \left(1 - maxCos\right) \cdot \left(maxCos + -1\right), \mathsf{fma}\left(maxCos, -2, 2\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (cos (* PI (* uy 2.0))) 0.9999949932098389)
   (* (sqrt (* ux 2.0)) (fma (* -2.0 (* uy uy)) (* PI PI) 1.0))
   (sqrt
    (* ux (fma ux (* (- 1.0 maxCos) (+ maxCos -1.0)) (fma maxCos -2.0 2.0))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (cosf((((float) M_PI) * (uy * 2.0f))) <= 0.9999949932098389f) {
		tmp = sqrtf((ux * 2.0f)) * fmaf((-2.0f * (uy * uy)), (((float) M_PI) * ((float) M_PI)), 1.0f);
	} else {
		tmp = sqrtf((ux * fmaf(ux, ((1.0f - maxCos) * (maxCos + -1.0f)), fmaf(maxCos, -2.0f, 2.0f))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (cos(Float32(Float32(pi) * Float32(uy * Float32(2.0)))) <= Float32(0.9999949932098389))
		tmp = Float32(sqrt(Float32(ux * Float32(2.0))) * fma(Float32(Float32(-2.0) * Float32(uy * uy)), Float32(Float32(pi) * Float32(pi)), Float32(1.0)));
	else
		tmp = sqrt(Float32(ux * fma(ux, Float32(Float32(Float32(1.0) - maxCos) * Float32(maxCos + Float32(-1.0))), fma(maxCos, Float32(-2.0), Float32(2.0)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 (*.f32 uy #s(literal 2 binary32)) (PI.f32))) < 0.999994993

   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 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. lower-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. lower--.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. lower-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. lower--.f32 55.0

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

   5. Applied rewrites 55.0%

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

   6. Taylor expanded in ux around 0

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

   8. Applied rewrites 72.8%

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

   9. Taylor expanded in uy around 0

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

       1. +-commutative N/A

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

       2. associate-*r* N/A

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

       3. lower-fma.f32 N/A

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

       4. lower-*.f32 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f32 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f32 N/A

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

       9. lower-PI.f32 N/A

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

       10. lower-PI.f32 58.0

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

   11. Applied rewrites 58.0%

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

   if 0.999994993 < (cos.f32 (*.f32 (*.f32 uy #s(literal 2 binary32)) (PI.f32)))

   1. Initial program 62.0%

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

   3. Taylor expanded in uy around 0

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

      1. lower-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. lower-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. Applied rewrites 62.3%

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

   6. Taylor expanded in ux around 0

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

   8. Applied rewrites 98.7%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\pi \cdot \left(uy \cdot 2\right)\right) \leq 0.9999949932098389:\\ \;\;\;\;\sqrt{ux \cdot 2} \cdot \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \pi \cdot \pi, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{ux \cdot \mathsf{fma}\left(ux, \left(1 - maxCos\right) \cdot \left(maxCos + -1\right), \mathsf{fma}\left(maxCos, -2, 2\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 60.2%

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

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f32 N/A

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

   7. +-commutative N/A

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

   8. mul-1-neg N/A

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

   9. *-commutative N/A

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

   10. unpow2 N/A

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

   11. associate-*l* N/A

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

   12. *-commutative N/A

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

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

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

   14. neg-sub0 N/A

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

   15. associate-+l- N/A

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

   16. neg-sub0 N/A

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

   17. mul-1-neg N/A

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

   18. +-commutative N/A

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

5. Applied rewrites 99.1%

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

6. Final simplification 99.1%

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

Alternative 6: 98.2% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (cos (* PI (* uy 2.0)))
  (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((((float) M_PI) * (uy * 2.0f))) * 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(pi) * Float32(uy * Float32(2.0)))) * 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 60.2%

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

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f32 N/A

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

   7. +-commutative N/A

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

   8. mul-1-neg N/A

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

   9. *-commutative N/A

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

   10. unpow2 N/A

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

   11. associate-*l* N/A

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

   12. *-commutative N/A

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

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

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

   14. neg-sub0 N/A

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

   15. associate-+l- N/A

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

   16. neg-sub0 N/A

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

   17. mul-1-neg N/A

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

   18. +-commutative N/A

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

5. Applied rewrites 99.1%

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

6. Taylor expanded in maxCos around 0

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

8. Applied rewrites 98.6%

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

9. Final simplification 98.6%

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

Alternative 7: 97.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.017000000923871994:\\ \;\;\;\;\sqrt{ux \cdot \mathsf{fma}\left(maxCos, -2, \mathsf{fma}\left(1 - maxCos, \mathsf{fma}\left(ux, maxCos, -ux\right), 2\right)\right)} \cdot \left(0.5 + \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \pi \cdot \pi, 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\pi \cdot \left(uy \cdot 2\right)\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* uy 2.0) 0.017000000923871994)
   (*
    (sqrt
     (* ux (fma maxCos -2.0 (fma (- 1.0 maxCos) (fma ux maxCos (- ux)) 2.0))))
    (+ 0.5 (fma (* -2.0 (* uy uy)) (* PI PI) 0.5)))
   (* (cos (* PI (* uy 2.0))) (sqrt (* ux (- 2.0 ux))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if ((uy * 2.0f) <= 0.017000000923871994f) {
		tmp = sqrtf((ux * fmaf(maxCos, -2.0f, fmaf((1.0f - maxCos), fmaf(ux, maxCos, -ux), 2.0f)))) * (0.5f + fmaf((-2.0f * (uy * uy)), (((float) M_PI) * ((float) M_PI)), 0.5f));
	} else {
		tmp = cosf((((float) M_PI) * (uy * 2.0f))) * sqrtf((ux * (2.0f - ux)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (Float32(uy * Float32(2.0)) <= Float32(0.017000000923871994))
		tmp = Float32(sqrt(Float32(ux * fma(maxCos, Float32(-2.0), fma(Float32(Float32(1.0) - maxCos), fma(ux, maxCos, Float32(-ux)), Float32(2.0))))) * Float32(Float32(0.5) + fma(Float32(Float32(-2.0) * Float32(uy * uy)), Float32(Float32(pi) * Float32(pi)), Float32(0.5))));
	else
		tmp = Float32(cos(Float32(Float32(pi) * Float32(uy * Float32(2.0)))) * 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)) < 0.0170000009

   1. Initial program 60.9%

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

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f32 N/A

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

      7. +-commutative N/A

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

      8. mul-1-neg N/A

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

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

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

      14. neg-sub0 N/A

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

      15. associate-+l- N/A

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

      16. neg-sub0 N/A

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

      17. mul-1-neg N/A

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

      18. +-commutative N/A

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

   5. Applied rewrites 99.4%

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

      1. lift-cos.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. lift-PI.f32 N/A

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

      5. *-commutative N/A

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

      6. lift-PI.f32 N/A

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

      7. associate-*r* N/A

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

      8. lift-*.f32 N/A

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

      9. cos-2 N/A

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

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

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

      11. lower-fma.f32 N/A

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

      12. lower-cos.f32 N/A

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

      13. lower-cos.f32 N/A

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

      14. lower-*.f32 N/A

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

      15. lower-neg.f32 N/A

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

      16. lower-sin.f32 N/A

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

      17. lower-sin.f32 99.3

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

   7. Applied rewrites 99.3%

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

      1. lift-fma.f32 N/A

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

      2. lift-cos.f32 N/A

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

      3. lift-cos.f32 N/A

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

      4. sqr-cos-a N/A

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

      5. associate-+l+ N/A

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

      6. lower-+.f32 N/A

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

      7. count-2 N/A

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

      8. lower-fma.f32 N/A

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

   9. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\left(0.5 + \mathsf{fma}\left(0.5, \cos \left(uy \cdot \left(\pi + \pi\right)\right), -\left(0.5 - 0.5 \cdot \cos \left(uy \cdot \left(\pi + \pi\right)\right)\right)\right)\right)} \cdot \sqrt{ux \cdot \mathsf{fma}\left(maxCos, -2, \mathsf{fma}\left(1 - maxCos, \mathsf{fma}\left(ux, maxCos, -ux\right), 2\right)\right)} \]

   10. Taylor expanded in uy around 0

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

       1. +-commutative N/A

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

       2. associate-*r* N/A

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

       3. lower-fma.f32 N/A

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

       4. lower-*.f32 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f32 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f32 N/A

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

       9. lower-PI.f32 N/A

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

       10. lower-PI.f32 99.3

           \[\leadsto \left(0.5 + \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \pi \cdot \color{blue}{\pi}, 0.5\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(maxCos, -2, \mathsf{fma}\left(1 - maxCos, \mathsf{fma}\left(ux, maxCos, -ux\right), 2\right)\right)} \]

   12. Applied rewrites 99.3%

       \[\leadsto \left(0.5 + \color{blue}{\mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \pi \cdot \pi, 0.5\right)}\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(maxCos, -2, \mathsf{fma}\left(1 - maxCos, \mathsf{fma}\left(ux, maxCos, -ux\right), 2\right)\right)} \]

   if 0.0170000009 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 57.7%

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

   3. Taylor expanded in 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. lower-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. lower--.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. lower-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. lower--.f32 57.0

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

   5. Applied rewrites 57.0%

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

   6. Taylor expanded in ux around 0

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

   8. Applied rewrites 93.5%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.017000000923871994:\\ \;\;\;\;\sqrt{ux \cdot \mathsf{fma}\left(maxCos, -2, \mathsf{fma}\left(1 - maxCos, \mathsf{fma}\left(ux, maxCos, -ux\right), 2\right)\right)} \cdot \left(0.5 + \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \pi \cdot \pi, 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\pi \cdot \left(uy \cdot 2\right)\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 93.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.10000000149011612:\\ \;\;\;\;\sqrt{ux \cdot \mathsf{fma}\left(maxCos, -2, \mathsf{fma}\left(1 - maxCos, \mathsf{fma}\left(ux, maxCos, -ux\right), 2\right)\right)} \cdot \left(0.5 + \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \pi \cdot \pi, 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(uy \cdot \left(\pi + \pi\right)\right) \cdot \sqrt{ux \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* uy 2.0) 0.10000000149011612)
   (*
    (sqrt
     (* ux (fma maxCos -2.0 (fma (- 1.0 maxCos) (fma ux maxCos (- ux)) 2.0))))
    (+ 0.5 (fma (* -2.0 (* uy uy)) (* PI PI) 0.5)))
   (* (cos (* uy (+ PI PI))) (sqrt (* ux 2.0)))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if ((uy * 2.0f) <= 0.10000000149011612f) {
		tmp = sqrtf((ux * fmaf(maxCos, -2.0f, fmaf((1.0f - maxCos), fmaf(ux, maxCos, -ux), 2.0f)))) * (0.5f + fmaf((-2.0f * (uy * uy)), (((float) M_PI) * ((float) M_PI)), 0.5f));
	} else {
		tmp = cosf((uy * (((float) M_PI) + ((float) M_PI)))) * sqrtf((ux * 2.0f));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (Float32(uy * Float32(2.0)) <= Float32(0.10000000149011612))
		tmp = Float32(sqrt(Float32(ux * fma(maxCos, Float32(-2.0), fma(Float32(Float32(1.0) - maxCos), fma(ux, maxCos, Float32(-ux)), Float32(2.0))))) * Float32(Float32(0.5) + fma(Float32(Float32(-2.0) * Float32(uy * uy)), Float32(Float32(pi) * Float32(pi)), Float32(0.5))));
	else
		tmp = Float32(cos(Float32(uy * Float32(Float32(pi) + Float32(pi)))) * sqrt(Float32(ux * Float32(2.0))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 60.4%

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

   3. Taylor expanded in ux around 0

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

      1. lower-*.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. metadata-eval N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f32 N/A

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

      7. +-commutative N/A

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

      8. mul-1-neg N/A

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

      9. *-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

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

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

      14. neg-sub0 N/A

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

      15. associate-+l- N/A

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

      16. neg-sub0 N/A

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

      17. mul-1-neg N/A

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

      18. +-commutative N/A

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

   5. Applied rewrites 99.4%

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

      1. lift-cos.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. lift-PI.f32 N/A

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

      5. *-commutative N/A

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

      6. lift-PI.f32 N/A

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

      7. associate-*r* N/A

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

      8. lift-*.f32 N/A

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

      9. cos-2 N/A

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

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

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

      11. lower-fma.f32 N/A

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

      12. lower-cos.f32 N/A

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

      13. lower-cos.f32 N/A

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

      14. lower-*.f32 N/A

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

      15. lower-neg.f32 N/A

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

      16. lower-sin.f32 N/A

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

      17. lower-sin.f32 99.2

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

   7. Applied rewrites 99.2%

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

      1. lift-fma.f32 N/A

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

      2. lift-cos.f32 N/A

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

      3. lift-cos.f32 N/A

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

      4. sqr-cos-a N/A

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

      5. associate-+l+ N/A

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

      6. lower-+.f32 N/A

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

      7. count-2 N/A

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

      8. lower-fma.f32 N/A

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

   9. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\left(0.5 + \mathsf{fma}\left(0.5, \cos \left(uy \cdot \left(\pi + \pi\right)\right), -\left(0.5 - 0.5 \cdot \cos \left(uy \cdot \left(\pi + \pi\right)\right)\right)\right)\right)} \cdot \sqrt{ux \cdot \mathsf{fma}\left(maxCos, -2, \mathsf{fma}\left(1 - maxCos, \mathsf{fma}\left(ux, maxCos, -ux\right), 2\right)\right)} \]

   10. Taylor expanded in uy around 0

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

       1. +-commutative N/A

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

       2. associate-*r* N/A

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

       3. lower-fma.f32 N/A

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

       4. lower-*.f32 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f32 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f32 N/A

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

       9. lower-PI.f32 N/A

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

       10. lower-PI.f32 97.0

           \[\leadsto \left(0.5 + \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \pi \cdot \color{blue}{\pi}, 0.5\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(maxCos, -2, \mathsf{fma}\left(1 - maxCos, \mathsf{fma}\left(ux, maxCos, -ux\right), 2\right)\right)} \]

   12. Applied rewrites 97.0%

       \[\leadsto \left(0.5 + \color{blue}{\mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \pi \cdot \pi, 0.5\right)}\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(maxCos, -2, \mathsf{fma}\left(1 - maxCos, \mathsf{fma}\left(ux, maxCos, -ux\right), 2\right)\right)} \]

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

   1. Initial program 58.6%

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

   3. Taylor expanded in 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. lower-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. lower--.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. lower-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. lower--.f32 57.2

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

   5. Applied rewrites 57.2%

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

   6. Taylor expanded in ux around 0

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

   8. Applied rewrites 69.8%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lift-cos.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. lift-PI.f32 N/A

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

      7. *-commutative N/A

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

      8. lift-PI.f32 N/A

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

      9. associate-*r* N/A

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

      10. lift-*.f32 N/A

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

      11. count-2 N/A

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

   10. Applied rewrites 69.8%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.10000000149011612:\\ \;\;\;\;\sqrt{ux \cdot \mathsf{fma}\left(maxCos, -2, \mathsf{fma}\left(1 - maxCos, \mathsf{fma}\left(ux, maxCos, -ux\right), 2\right)\right)} \cdot \left(0.5 + \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \pi \cdot \pi, 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(uy \cdot \left(\pi + \pi\right)\right) \cdot \sqrt{ux \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 88.3% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \sqrt{ux \cdot \mathsf{fma}\left(maxCos, -2, \mathsf{fma}\left(1 - maxCos, \mathsf{fma}\left(ux, maxCos, -ux\right), 2\right)\right)} \cdot \left(0.5 + \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \pi \cdot \pi, 0.5\right)\right) \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (sqrt
   (* ux (fma maxCos -2.0 (fma (- 1.0 maxCos) (fma ux maxCos (- ux)) 2.0))))
  (+ 0.5 (fma (* -2.0 (* uy uy)) (* PI PI) 0.5))))

C

float code(float ux, float uy, float maxCos) {
	return sqrtf((ux * fmaf(maxCos, -2.0f, fmaf((1.0f - maxCos), fmaf(ux, maxCos, -ux), 2.0f)))) * (0.5f + fmaf((-2.0f * (uy * uy)), (((float) M_PI) * ((float) M_PI)), 0.5f));
}

Julia

function code(ux, uy, maxCos)
	return Float32(sqrt(Float32(ux * fma(maxCos, Float32(-2.0), fma(Float32(Float32(1.0) - maxCos), fma(ux, maxCos, Float32(-ux)), Float32(2.0))))) * Float32(Float32(0.5) + fma(Float32(Float32(-2.0) * Float32(uy * uy)), Float32(Float32(pi) * Float32(pi)), Float32(0.5))))
end

Derivation

1. Initial program 60.2%

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

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f32 N/A

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

   7. +-commutative N/A

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

   8. mul-1-neg N/A

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

   9. *-commutative N/A

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

   10. unpow2 N/A

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

   11. associate-*l* N/A

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

   12. *-commutative N/A

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

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

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

   14. neg-sub0 N/A

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

   15. associate-+l- N/A

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

   16. neg-sub0 N/A

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

   17. mul-1-neg N/A

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

   18. +-commutative N/A

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

5. Applied rewrites 99.1%

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

   1. lift-cos.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-PI.f32 N/A

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

   5. *-commutative N/A

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

   6. lift-PI.f32 N/A

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

   7. associate-*r* N/A

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

   8. lift-*.f32 N/A

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

   9. cos-2 N/A

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

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

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

   11. lower-fma.f32 N/A

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

   12. lower-cos.f32 N/A

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

   13. lower-cos.f32 N/A

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

   14. lower-*.f32 N/A

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

   15. lower-neg.f32 N/A

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

   16. lower-sin.f32 N/A

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

   17. lower-sin.f32 99.0

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

7. Applied rewrites 99.0%

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

   1. lift-fma.f32 N/A

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

   2. lift-cos.f32 N/A

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

   3. lift-cos.f32 N/A

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

   4. sqr-cos-a N/A

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

   5. associate-+l+ N/A

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

   6. lower-+.f32 N/A

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

   7. count-2 N/A

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

   8. lower-fma.f32 N/A

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

9. Applied rewrites 99.1%

   \[\leadsto \color{blue}{\left(0.5 + \mathsf{fma}\left(0.5, \cos \left(uy \cdot \left(\pi + \pi\right)\right), -\left(0.5 - 0.5 \cdot \cos \left(uy \cdot \left(\pi + \pi\right)\right)\right)\right)\right)} \cdot \sqrt{ux \cdot \mathsf{fma}\left(maxCos, -2, \mathsf{fma}\left(1 - maxCos, \mathsf{fma}\left(ux, maxCos, -ux\right), 2\right)\right)} \]

10. Taylor expanded in uy around 0

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

    1. +-commutative N/A

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

    2. associate-*r* N/A

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

    3. lower-fma.f32 N/A

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

    4. lower-*.f32 N/A

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

    5. unpow2 N/A

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

    6. lower-*.f32 N/A

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

    7. unpow2 N/A

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

    8. lower-*.f32 N/A

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

    9. lower-PI.f32 N/A

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

    10. lower-PI.f32 89.7

        \[\leadsto \left(0.5 + \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \pi \cdot \color{blue}{\pi}, 0.5\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(maxCos, -2, \mathsf{fma}\left(1 - maxCos, \mathsf{fma}\left(ux, maxCos, -ux\right), 2\right)\right)} \]

12. Applied rewrites 89.7%

    \[\leadsto \left(0.5 + \color{blue}{\mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \pi \cdot \pi, 0.5\right)}\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(maxCos, -2, \mathsf{fma}\left(1 - maxCos, \mathsf{fma}\left(ux, maxCos, -ux\right), 2\right)\right)} \]

13. Final simplification 89.7%

    \[\leadsto \sqrt{ux \cdot \mathsf{fma}\left(maxCos, -2, \mathsf{fma}\left(1 - maxCos, \mathsf{fma}\left(ux, maxCos, -ux\right), 2\right)\right)} \cdot \left(0.5 + \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \pi \cdot \pi, 0.5\right)\right) \]
14. Add Preprocessing

Alternative 10: 88.3% accurate, 2.4× speedup

Math

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

FPCore

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

C

float code(float ux, float uy, float maxCos) {
	return sqrtf((ux * fmaf(maxCos, -2.0f, fmaf((1.0f - maxCos), fmaf(ux, maxCos, -ux), 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(maxCos, Float32(-2.0), fma(Float32(Float32(1.0) - maxCos), fma(ux, maxCos, Float32(-ux)), Float32(2.0))))) * fma(Float32(Float32(-2.0) * Float32(uy * uy)), Float32(Float32(pi) * Float32(pi)), Float32(1.0)))
end

Derivation

1. Initial program 60.2%

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

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f32 N/A

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

   7. +-commutative N/A

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

   8. mul-1-neg N/A

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

   9. *-commutative N/A

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

   10. unpow2 N/A

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

   11. associate-*l* N/A

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

   12. *-commutative N/A

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

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

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

   14. neg-sub0 N/A

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

   15. associate-+l- N/A

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

   16. neg-sub0 N/A

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

   17. mul-1-neg N/A

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

   18. +-commutative N/A

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

5. Applied rewrites 99.1%

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

6. Taylor expanded in uy around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 N/A

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

   9. lower-PI.f32 N/A

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

   10. lower-PI.f32 89.7

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

8. Applied rewrites 89.7%

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

9. Final simplification 89.7%

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

Alternative 11: 86.9% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;maxCos \leq 1.9999999949504854 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{ux \cdot \left(2 - ux\right)} \cdot \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \pi \cdot \pi, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(maxCos, -2, 2\right), ux, \left(\left(1 - maxCos\right) \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot ux\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= maxCos 1.9999999949504854e-6)
   (* (sqrt (* ux (- 2.0 ux))) (fma (* -2.0 (* uy uy)) (* PI PI) 1.0))
   (sqrt
    (fma
     (fma maxCos -2.0 2.0)
     ux
     (* (* (- 1.0 maxCos) (+ maxCos -1.0)) (* ux ux))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (maxCos <= 1.9999999949504854e-6f) {
		tmp = sqrtf((ux * (2.0f - ux))) * fmaf((-2.0f * (uy * uy)), (((float) M_PI) * ((float) M_PI)), 1.0f);
	} else {
		tmp = sqrtf(fmaf(fmaf(maxCos, -2.0f, 2.0f), ux, (((1.0f - maxCos) * (maxCos + -1.0f)) * (ux * ux))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (maxCos <= Float32(1.9999999949504854e-6))
		tmp = Float32(sqrt(Float32(ux * Float32(Float32(2.0) - ux))) * fma(Float32(Float32(-2.0) * Float32(uy * uy)), Float32(Float32(pi) * Float32(pi)), Float32(1.0)));
	else
		tmp = sqrt(fma(fma(maxCos, Float32(-2.0), Float32(2.0)), ux, Float32(Float32(Float32(Float32(1.0) - maxCos) * Float32(maxCos + Float32(-1.0))) * Float32(ux * ux))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if maxCos < 1.99999999e-6

   1. Initial program 60.9%

      \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   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. lower-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. lower--.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. lower-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. lower--.f32 61.1

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

   5. Applied rewrites 61.1%

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

   6. Taylor expanded in ux around 0

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

   8. Applied rewrites 75.0%

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

   9. Taylor expanded in uy around 0

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

       1. +-commutative N/A

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

       2. associate-*r* N/A

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

       3. lower-fma.f32 N/A

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

       4. lower-*.f32 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f32 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f32 N/A

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

       9. lower-PI.f32 N/A

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

       10. lower-PI.f32 70.0

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

   11. Applied rewrites 70.0%

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

   12. Taylor expanded in ux around 0

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

   14. Applied rewrites 89.5%

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

   if 1.99999999e-6 < maxCos

   1. Initial program 56.9%

      \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   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. lower-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. lower-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. Applied rewrites 54.2%

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

   6. Taylor expanded in ux around 0

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

   8. Applied rewrites 83.1%

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

   10. Applied rewrites 83.2%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;maxCos \leq 1.9999999949504854 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{ux \cdot \left(2 - ux\right)} \cdot \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \pi \cdot \pi, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(maxCos, -2, 2\right), ux, \left(\left(1 - maxCos\right) \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot ux\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 79.7% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 60.2%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
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. lower-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. lower-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. Applied rewrites 53.1%

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

6. Taylor expanded in ux around 0

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

8. Applied rewrites 81.6%

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

9. Final simplification 81.6%

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

Alternative 13: 79.2% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 60.2%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
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. lower-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. lower-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. Applied rewrites 53.1%

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

6. Taylor expanded in ux around 0

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

8. Applied rewrites 81.6%

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

9. Taylor expanded in maxCos around 0

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

11. Applied rewrites 81.2%

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

12. Final simplification 81.2%

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

Alternative 14: 75.4% accurate, 6.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 60.2%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
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. lower-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. lower-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. Applied rewrites 53.1%

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

6. Taylor expanded in ux around 0

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

8. Applied rewrites 81.6%

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

9. Taylor expanded in maxCos around 0

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

11. Applied rewrites 76.9%

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

13. Applied rewrites 76.9%

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

14. Final simplification 76.9%

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

Alternative 15: 75.4% 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 60.2%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
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. lower-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. lower-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. Applied rewrites 53.1%

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

6. Taylor expanded in ux around 0

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

8. Applied rewrites 81.6%

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

9. Taylor expanded in maxCos around 0

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

11. Applied rewrites 76.9%

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

Alternative 16: 61.9% accurate, 9.8× speedup

Math

\[\begin{array}{l} \\ \sqrt{ux \cdot 2} \end{array} \]

FPCore

(FPCore (ux uy maxCos) :precision binary32 (sqrt (* ux 2.0)))

C

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

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))
end function

Julia

function code(ux, uy, maxCos)
	return sqrt(Float32(ux * Float32(2.0)))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = sqrt((ux * single(2.0)));
end

Derivation

1. Initial program 60.2%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
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. lower-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. lower-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. Applied rewrites 53.1%

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

6. Taylor expanded in ux around 0

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

8. Applied rewrites 81.6%

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

9. Taylor expanded in maxCos around 0

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

11. Applied rewrites 76.9%

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

12. Taylor expanded in ux around 0

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

14. Applied rewrites 62.1%

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

15. Final simplification 62.1%

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

Reproduce

herbie shell --seed 2024237 
(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)))))))