UniformSampleCone, y

Percentage Accurate: 57.3% → 98.3%

Time: 6.3s

Alternatives: 10

Speedup: 2.3×

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

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

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))))
   (* (sin (* (* 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 sinf(((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(sin(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 = sin(((uy * single(2.0)) * single(pi))) * sqrt((single(1.0) - (t_0 * t_0)));
end

Initial Program: 57.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))))
   (* (sin (* (* 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 sinf(((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(sin(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 = sin(((uy * single(2.0)) * single(pi))) * sqrt((single(1.0) - (t_0 * t_0)));
end

Alternative 1: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.3%

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

2. Taylor expanded in ux around 0

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

   1. lower-*.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lower-+.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-pow.f32 N/A

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

   7. lower--.f32 N/A

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

   8. lower-*.f32 98.3

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

4. Applied rewrites 98.3%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 98.3

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

6. Applied rewrites 98.3%

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

   1. lift--.f32 N/A

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

   2. metadata-eval N/A

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

   3. associate--l+ N/A

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

   4. lower-+.f32 N/A

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

   5. lower--.f32 98.3

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

   6. lift-fma.f32 N/A

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

   7. add-flip N/A

      \[\leadsto \sqrt{\left(1 + \left(1 - \left(\left(ux \cdot \left(maxCos - 1\right)\right) \cdot \left(maxCos - 1\right) - \left(\mathsf{neg}\left(\left(maxCos + maxCos\right)\right)\right)\right)\right)\right) \cdot ux} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right) \]

   8. lift-+.f32 N/A

      \[\leadsto \sqrt{\left(1 + \left(1 - \left(\left(ux \cdot \left(maxCos - 1\right)\right) \cdot \left(maxCos - 1\right) - \left(\mathsf{neg}\left(\left(maxCos + maxCos\right)\right)\right)\right)\right)\right) \cdot ux} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right) \]

   9. count-2-rev N/A

      \[\leadsto \sqrt{\left(1 + \left(1 - \left(\left(ux \cdot \left(maxCos - 1\right)\right) \cdot \left(maxCos - 1\right) - \left(\mathsf{neg}\left(2 \cdot maxCos\right)\right)\right)\right)\right) \cdot ux} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right) \]

   10. lift-*.f32 N/A

       \[\leadsto \sqrt{\left(1 + \left(1 - \left(\left(ux \cdot \left(maxCos - 1\right)\right) \cdot \left(maxCos - 1\right) - \left(\mathsf{neg}\left(2 \cdot maxCos\right)\right)\right)\right)\right) \cdot ux} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right) \]

   11. sub-flip N/A

       \[\leadsto \sqrt{\left(1 + \left(1 - \left(\left(ux \cdot \left(maxCos - 1\right)\right) \cdot \left(maxCos - 1\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(2 \cdot maxCos\right)\right)\right)\right)\right)\right)\right) \cdot ux} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right) \]

   12. lift-*.f32 N/A

       \[\leadsto \sqrt{\left(1 + \left(1 - \left(\left(ux \cdot \left(maxCos - 1\right)\right) \cdot \left(maxCos - 1\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(2 \cdot maxCos\right)\right)\right)\right)\right)\right)\right) \cdot ux} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right) \]

   13. associate-*l* N/A

       \[\leadsto \sqrt{\left(1 + \left(1 - \left(ux \cdot \left(\left(maxCos - 1\right) \cdot \left(maxCos - 1\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(2 \cdot maxCos\right)\right)\right)\right)\right)\right)\right) \cdot ux} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right) \]

   14. lift--.f32 N/A

       \[\leadsto \sqrt{\left(1 + \left(1 - \left(ux \cdot \left(\left(maxCos - 1\right) \cdot \left(maxCos - 1\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(2 \cdot maxCos\right)\right)\right)\right)\right)\right)\right) \cdot ux} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right) \]

   15. sub-negate-rev N/A

       \[\leadsto \sqrt{\left(1 + \left(1 - \left(ux \cdot \left(\left(\mathsf{neg}\left(\left(1 - maxCos\right)\right)\right) \cdot \left(maxCos - 1\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(2 \cdot maxCos\right)\right)\right)\right)\right)\right)\right) \cdot ux} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right) \]

   16. lift--.f32 N/A

       \[\leadsto \sqrt{\left(1 + \left(1 - \left(ux \cdot \left(\left(\mathsf{neg}\left(\left(1 - maxCos\right)\right)\right) \cdot \left(maxCos - 1\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(2 \cdot maxCos\right)\right)\right)\right)\right)\right)\right) \cdot ux} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right) \]

   17. lift--.f32 N/A

       \[\leadsto \sqrt{\left(1 + \left(1 - \left(ux \cdot \left(\left(\mathsf{neg}\left(\left(1 - maxCos\right)\right)\right) \cdot \left(maxCos - 1\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(2 \cdot maxCos\right)\right)\right)\right)\right)\right)\right) \cdot ux} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right) \]

   18. sub-negate-rev N/A

       \[\leadsto \sqrt{\left(1 + \left(1 - \left(ux \cdot \left(\left(\mathsf{neg}\left(\left(1 - maxCos\right)\right)\right) \cdot \left(\mathsf{neg}\left(\left(1 - maxCos\right)\right)\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(2 \cdot maxCos\right)\right)\right)\right)\right)\right)\right) \cdot ux} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right) \]

   19. lift--.f32 N/A

       \[\leadsto \sqrt{\left(1 + \left(1 - \left(ux \cdot \left(\left(\mathsf{neg}\left(\left(1 - maxCos\right)\right)\right) \cdot \left(\mathsf{neg}\left(\left(1 - maxCos\right)\right)\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(2 \cdot maxCos\right)\right)\right)\right)\right)\right)\right) \cdot ux} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right) \]

   20. sqr-neg-rev N/A

       \[\leadsto \sqrt{\left(1 + \left(1 - \left(ux \cdot \left(\left(1 - maxCos\right) \cdot \left(1 - maxCos\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(2 \cdot maxCos\right)\right)\right)\right)\right)\right)\right) \cdot ux} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right) \]

   21. associate-*r* N/A

       \[\leadsto \sqrt{\left(1 + \left(1 - \left(\left(ux \cdot \left(1 - maxCos\right)\right) \cdot \left(1 - maxCos\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(2 \cdot maxCos\right)\right)\right)\right)\right)\right)\right) \cdot ux} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right) \]

8. Applied rewrites 98.3%

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

Alternative 2: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.3%

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

2. Taylor expanded in ux around 0

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

   1. lower-*.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lower-+.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-pow.f32 N/A

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

   7. lower--.f32 N/A

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

   8. lower-*.f32 98.3

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

4. Applied rewrites 98.3%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 98.3

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

6. Applied rewrites 98.3%

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

8. Applied rewrites 98.3%

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

Alternative 3: 98.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.3%

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

   1. lift--.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. metadata-eval N/A

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

   4. sqr-neg-rev N/A

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

   5. difference-of-squares N/A

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

   6. +-commutative N/A

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

   7. lift-+.f32 N/A

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

   8. lift--.f32 N/A

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

   9. associate-+l- N/A

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

   10. sub-negate N/A

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

   11. associate-+l- N/A

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

   12. associate-+l- N/A

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

   13. lift--.f32 N/A

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

   14. lift-+.f32 N/A

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

   15. lower-*.f32 N/A

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

3. Applied rewrites 98.3%

   \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\color{blue}{\left(\left(ux - maxCos \cdot ux\right) - 0\right) \cdot \left(\mathsf{fma}\left(maxCos, ux, 1 - ux\right) - -1\right)}} \]
4. Step-by-step derivation

   1. lift-sqrt.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. sqrt-prod N/A

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

   4. lower-unsound-*.f32 N/A

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

   5. lift--.f32 N/A

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

   6. --rgt-identity N/A

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

   7. lower-unsound-sqrt.f32 N/A

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

   8. lower-unsound-sqrt.f32 98.1

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

   9. lift--.f32 N/A

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

   10. sub-negate-rev N/A

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

   11. lower-neg.f32 N/A

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

   12. lift-fma.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. +-commutative N/A

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

   15. lift--.f32 N/A

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

   16. associate--r- N/A

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

   17. lift--.f32 N/A

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

   18. associate--r- N/A

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

   19. metadata-eval N/A

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

   20. metadata-eval N/A

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

   21. lower-+.f32 N/A

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

   22. metadata-eval 98.1

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

5. Applied rewrites 98.1%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 98.1

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

7. Applied rewrites 98.3%

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

Alternative 4: 97.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.3%

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

   1. lift--.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. metadata-eval N/A

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

   4. sqr-neg-rev N/A

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

   5. difference-of-squares N/A

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

   6. +-commutative N/A

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

   7. lift-+.f32 N/A

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

   8. lift--.f32 N/A

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

   9. associate-+l- N/A

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

   10. sub-negate N/A

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

   11. associate-+l- N/A

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

   12. associate-+l- N/A

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

   13. lift--.f32 N/A

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

   14. lift-+.f32 N/A

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

   15. lower-*.f32 N/A

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

3. Applied rewrites 98.3%

   \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\color{blue}{\left(\left(ux - maxCos \cdot ux\right) - 0\right) \cdot \left(\mathsf{fma}\left(maxCos, ux, 1 - ux\right) - -1\right)}} \]
4. Step-by-step derivation

   1. lift-sqrt.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. sqrt-prod N/A

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

   4. lower-unsound-*.f32 N/A

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

   5. lift--.f32 N/A

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

   6. --rgt-identity N/A

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

   7. lower-unsound-sqrt.f32 N/A

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

   8. lower-unsound-sqrt.f32 98.1

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

   9. lift--.f32 N/A

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

   10. sub-negate-rev N/A

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

   11. lower-neg.f32 N/A

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

   12. lift-fma.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. +-commutative N/A

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

   15. lift--.f32 N/A

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

   16. associate--r- N/A

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

   17. lift--.f32 N/A

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

   18. associate--r- N/A

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

   19. metadata-eval N/A

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

   20. metadata-eval N/A

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

   21. lower-+.f32 N/A

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

   22. metadata-eval 98.1

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

5. Applied rewrites 98.1%

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

6. Taylor expanded in maxCos around 0

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

   1. lower-sqrt.f32 N/A

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \left(\sqrt{ux - maxCos \cdot ux} \cdot \sqrt{2 - ux}\right) \]

   2. lower--.f32 97.0

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \left(\sqrt{ux - maxCos \cdot ux} \cdot \sqrt{2 - ux}\right) \]

8. Applied rewrites 97.0%

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

Alternative 5: 95.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 9.999999747378752 \cdot 10^{-5}:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{\left(ux - maxCos \cdot ux\right) \cdot \left(\left(2 + maxCos \cdot ux\right) - ux\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(1 + \left(1 - ux\right)\right) \cdot ux} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= uy 9.999999747378752e-5)
   (*
    2.0
    (* uy (* PI (sqrt (* (- ux (* maxCos ux)) (- (+ 2.0 (* maxCos ux)) ux))))))
   (* (sqrt (* (+ 1.0 (- 1.0 ux)) ux)) (sin (* PI (+ uy uy))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 9.999999747378752e-5f) {
		tmp = 2.0f * (uy * (((float) M_PI) * sqrtf(((ux - (maxCos * ux)) * ((2.0f + (maxCos * ux)) - ux)))));
	} else {
		tmp = sqrtf(((1.0f + (1.0f - ux)) * ux)) * sinf((((float) M_PI) * (uy + uy)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(9.999999747378752e-5))
		tmp = Float32(Float32(2.0) * Float32(uy * Float32(Float32(pi) * sqrt(Float32(Float32(ux - Float32(maxCos * ux)) * Float32(Float32(Float32(2.0) + Float32(maxCos * ux)) - ux))))));
	else
		tmp = Float32(sqrt(Float32(Float32(Float32(1.0) + Float32(Float32(1.0) - ux)) * ux)) * sin(Float32(Float32(pi) * Float32(uy + uy))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (uy <= single(9.999999747378752e-5))
		tmp = single(2.0) * (uy * (single(pi) * sqrt(((ux - (maxCos * ux)) * ((single(2.0) + (maxCos * ux)) - ux)))));
	else
		tmp = sqrt(((single(1.0) + (single(1.0) - ux)) * ux)) * sin((single(pi) * (uy + uy)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if uy < 9.99999975e-5

   1. Initial program 57.3%

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

      1. lift--.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. metadata-eval N/A

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

      4. sqr-neg-rev N/A

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

      5. difference-of-squares N/A

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

      6. +-commutative N/A

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

      7. lift-+.f32 N/A

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

      8. lift--.f32 N/A

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

      9. associate-+l- N/A

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

      10. sub-negate N/A

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

      11. associate-+l- N/A

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

      12. associate-+l- N/A

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

      13. lift--.f32 N/A

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

      14. lift-+.f32 N/A

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

      15. lower-*.f32 N/A

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

   3. Applied rewrites 98.3%

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

   4. Taylor expanded in uy around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-PI.f32 N/A

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

      5. lower-sqrt.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lower--.f32 N/A

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

      8. lower-*.f32 N/A

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

      9. lower--.f32 N/A

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

      10. lower-+.f32 N/A

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

      11. lower-*.f32 81.1

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

   6. Applied rewrites 81.1%

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

   if 9.99999975e-5 < uy

   1. Initial program 57.3%

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

   2. Taylor expanded in ux around 0

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

      1. lower-*.f32 N/A

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

      2. lower--.f32 N/A

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

      3. lower-+.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower--.f32 N/A

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

      8. lower-*.f32 98.3

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

   4. Applied rewrites 98.3%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f32 98.3

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

   6. Applied rewrites 98.3%

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

      1. lift--.f32 N/A

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

      2. metadata-eval N/A

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

      3. associate--l+ N/A

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

      4. lower-+.f32 N/A

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

      5. lower--.f32 98.3

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

      6. lift-fma.f32 N/A

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

      7. add-flip N/A

         \[\leadsto \sqrt{\left(1 + \left(1 - \left(\left(ux \cdot \left(maxCos - 1\right)\right) \cdot \left(maxCos - 1\right) - \left(\mathsf{neg}\left(\left(maxCos + maxCos\right)\right)\right)\right)\right)\right) \cdot ux} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right) \]

      8. lift-+.f32 N/A

         \[\leadsto \sqrt{\left(1 + \left(1 - \left(\left(ux \cdot \left(maxCos - 1\right)\right) \cdot \left(maxCos - 1\right) - \left(\mathsf{neg}\left(\left(maxCos + maxCos\right)\right)\right)\right)\right)\right) \cdot ux} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right) \]

      9. count-2-rev N/A

         \[\leadsto \sqrt{\left(1 + \left(1 - \left(\left(ux \cdot \left(maxCos - 1\right)\right) \cdot \left(maxCos - 1\right) - \left(\mathsf{neg}\left(2 \cdot maxCos\right)\right)\right)\right)\right) \cdot ux} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right) \]

      10. lift-*.f32 N/A

          \[\leadsto \sqrt{\left(1 + \left(1 - \left(\left(ux \cdot \left(maxCos - 1\right)\right) \cdot \left(maxCos - 1\right) - \left(\mathsf{neg}\left(2 \cdot maxCos\right)\right)\right)\right)\right) \cdot ux} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right) \]

      11. sub-flip N/A

          \[\leadsto \sqrt{\left(1 + \left(1 - \left(\left(ux \cdot \left(maxCos - 1\right)\right) \cdot \left(maxCos - 1\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(2 \cdot maxCos\right)\right)\right)\right)\right)\right)\right) \cdot ux} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right) \]

      12. lift-*.f32 N/A

          \[\leadsto \sqrt{\left(1 + \left(1 - \left(\left(ux \cdot \left(maxCos - 1\right)\right) \cdot \left(maxCos - 1\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(2 \cdot maxCos\right)\right)\right)\right)\right)\right)\right) \cdot ux} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right) \]

      13. associate-*l* N/A

          \[\leadsto \sqrt{\left(1 + \left(1 - \left(ux \cdot \left(\left(maxCos - 1\right) \cdot \left(maxCos - 1\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(2 \cdot maxCos\right)\right)\right)\right)\right)\right)\right) \cdot ux} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right) \]

      14. lift--.f32 N/A

          \[\leadsto \sqrt{\left(1 + \left(1 - \left(ux \cdot \left(\left(maxCos - 1\right) \cdot \left(maxCos - 1\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(2 \cdot maxCos\right)\right)\right)\right)\right)\right)\right) \cdot ux} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right) \]

      15. sub-negate-rev N/A

          \[\leadsto \sqrt{\left(1 + \left(1 - \left(ux \cdot \left(\left(\mathsf{neg}\left(\left(1 - maxCos\right)\right)\right) \cdot \left(maxCos - 1\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(2 \cdot maxCos\right)\right)\right)\right)\right)\right)\right) \cdot ux} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right) \]

      16. lift--.f32 N/A

          \[\leadsto \sqrt{\left(1 + \left(1 - \left(ux \cdot \left(\left(\mathsf{neg}\left(\left(1 - maxCos\right)\right)\right) \cdot \left(maxCos - 1\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(2 \cdot maxCos\right)\right)\right)\right)\right)\right)\right) \cdot ux} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right) \]

      17. lift--.f32 N/A

          \[\leadsto \sqrt{\left(1 + \left(1 - \left(ux \cdot \left(\left(\mathsf{neg}\left(\left(1 - maxCos\right)\right)\right) \cdot \left(maxCos - 1\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(2 \cdot maxCos\right)\right)\right)\right)\right)\right)\right) \cdot ux} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right) \]

      18. sub-negate-rev N/A

          \[\leadsto \sqrt{\left(1 + \left(1 - \left(ux \cdot \left(\left(\mathsf{neg}\left(\left(1 - maxCos\right)\right)\right) \cdot \left(\mathsf{neg}\left(\left(1 - maxCos\right)\right)\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(2 \cdot maxCos\right)\right)\right)\right)\right)\right)\right) \cdot ux} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right) \]

      19. lift--.f32 N/A

          \[\leadsto \sqrt{\left(1 + \left(1 - \left(ux \cdot \left(\left(\mathsf{neg}\left(\left(1 - maxCos\right)\right)\right) \cdot \left(\mathsf{neg}\left(\left(1 - maxCos\right)\right)\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(2 \cdot maxCos\right)\right)\right)\right)\right)\right)\right) \cdot ux} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right) \]

      20. sqr-neg-rev N/A

          \[\leadsto \sqrt{\left(1 + \left(1 - \left(ux \cdot \left(\left(1 - maxCos\right) \cdot \left(1 - maxCos\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(2 \cdot maxCos\right)\right)\right)\right)\right)\right)\right) \cdot ux} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right) \]

      21. associate-*r* N/A

          \[\leadsto \sqrt{\left(1 + \left(1 - \left(\left(ux \cdot \left(1 - maxCos\right)\right) \cdot \left(1 - maxCos\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(2 \cdot maxCos\right)\right)\right)\right)\right)\right)\right) \cdot ux} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right) \]

   8. Applied rewrites 98.3%

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

   9. Taylor expanded in maxCos around 0

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

       1. lower--.f32 92.1

          \[\leadsto \sqrt{\left(1 + \left(1 - ux\right)\right) \cdot ux} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right) \]

   11. Applied rewrites 92.1%

       \[\leadsto \sqrt{\left(1 + \left(1 - ux\right)\right) \cdot ux} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 95.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 9.999999747378752 \cdot 10^{-5}:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{\left(ux - maxCos \cdot ux\right) \cdot \left(\left(2 + maxCos \cdot ux\right) - ux\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 - ux\right) \cdot ux} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= uy 9.999999747378752e-5)
   (*
    2.0
    (* uy (* PI (sqrt (* (- ux (* maxCos ux)) (- (+ 2.0 (* maxCos ux)) ux))))))
   (* (sqrt (* (- 2.0 ux) ux)) (sin (* PI (+ uy uy))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 9.999999747378752e-5f) {
		tmp = 2.0f * (uy * (((float) M_PI) * sqrtf(((ux - (maxCos * ux)) * ((2.0f + (maxCos * ux)) - ux)))));
	} else {
		tmp = sqrtf(((2.0f - ux) * ux)) * sinf((((float) M_PI) * (uy + uy)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(9.999999747378752e-5))
		tmp = Float32(Float32(2.0) * Float32(uy * Float32(Float32(pi) * sqrt(Float32(Float32(ux - Float32(maxCos * ux)) * Float32(Float32(Float32(2.0) + Float32(maxCos * ux)) - ux))))));
	else
		tmp = Float32(sqrt(Float32(Float32(Float32(2.0) - ux) * ux)) * sin(Float32(Float32(pi) * Float32(uy + uy))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (uy <= single(9.999999747378752e-5))
		tmp = single(2.0) * (uy * (single(pi) * sqrt(((ux - (maxCos * ux)) * ((single(2.0) + (maxCos * ux)) - ux)))));
	else
		tmp = sqrt(((single(2.0) - ux) * ux)) * sin((single(pi) * (uy + uy)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if uy < 9.99999975e-5

   1. Initial program 57.3%

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

      1. lift--.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. metadata-eval N/A

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

      4. sqr-neg-rev N/A

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

      5. difference-of-squares N/A

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

      6. +-commutative N/A

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

      7. lift-+.f32 N/A

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

      8. lift--.f32 N/A

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

      9. associate-+l- N/A

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

      10. sub-negate N/A

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

      11. associate-+l- N/A

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

      12. associate-+l- N/A

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

      13. lift--.f32 N/A

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

      14. lift-+.f32 N/A

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

      15. lower-*.f32 N/A

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

   3. Applied rewrites 98.3%

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

   4. Taylor expanded in uy around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-PI.f32 N/A

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

      5. lower-sqrt.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lower--.f32 N/A

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

      8. lower-*.f32 N/A

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

      9. lower--.f32 N/A

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

      10. lower-+.f32 N/A

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

      11. lower-*.f32 81.1

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

   6. Applied rewrites 81.1%

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

   if 9.99999975e-5 < uy

   1. Initial program 57.3%

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

   2. Taylor expanded in ux around 0

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

      1. lower-*.f32 N/A

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

      2. lower--.f32 N/A

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

      3. lower-+.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower--.f32 N/A

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

      8. lower-*.f32 98.3

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

   4. Applied rewrites 98.3%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f32 98.3

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

   6. Applied rewrites 98.3%

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

   7. Taylor expanded in maxCos around 0

      \[\leadsto \sqrt{\left(2 - ux\right) \cdot ux} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right) \]
   8. Step-by-step derivation

      1. lower--.f32 92.1

         \[\leadsto \sqrt{\left(2 - ux\right) \cdot ux} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right) \]

   9. Applied rewrites 92.1%

      \[\leadsto \sqrt{\left(2 - ux\right) \cdot ux} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 81.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.3%

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

2. Taylor expanded in ux around 0

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

   1. lower-*.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lower-+.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-pow.f32 N/A

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

   7. lower--.f32 N/A

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

   8. lower-*.f32 98.3

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

4. Applied rewrites 98.3%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 98.3

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

6. Applied rewrites 98.3%

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

   1. lift--.f32 N/A

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

   2. metadata-eval N/A

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

   3. associate--l+ N/A

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

   4. lower-+.f32 N/A

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

   5. lower--.f32 98.3

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

   6. lift-fma.f32 N/A

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

   7. add-flip N/A

      \[\leadsto \sqrt{\left(1 + \left(1 - \left(\left(ux \cdot \left(maxCos - 1\right)\right) \cdot \left(maxCos - 1\right) - \left(\mathsf{neg}\left(\left(maxCos + maxCos\right)\right)\right)\right)\right)\right) \cdot ux} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right) \]

   8. lift-+.f32 N/A

      \[\leadsto \sqrt{\left(1 + \left(1 - \left(\left(ux \cdot \left(maxCos - 1\right)\right) \cdot \left(maxCos - 1\right) - \left(\mathsf{neg}\left(\left(maxCos + maxCos\right)\right)\right)\right)\right)\right) \cdot ux} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right) \]

   9. count-2-rev N/A

      \[\leadsto \sqrt{\left(1 + \left(1 - \left(\left(ux \cdot \left(maxCos - 1\right)\right) \cdot \left(maxCos - 1\right) - \left(\mathsf{neg}\left(2 \cdot maxCos\right)\right)\right)\right)\right) \cdot ux} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right) \]

   10. lift-*.f32 N/A

       \[\leadsto \sqrt{\left(1 + \left(1 - \left(\left(ux \cdot \left(maxCos - 1\right)\right) \cdot \left(maxCos - 1\right) - \left(\mathsf{neg}\left(2 \cdot maxCos\right)\right)\right)\right)\right) \cdot ux} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right) \]

   11. sub-flip N/A

       \[\leadsto \sqrt{\left(1 + \left(1 - \left(\left(ux \cdot \left(maxCos - 1\right)\right) \cdot \left(maxCos - 1\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(2 \cdot maxCos\right)\right)\right)\right)\right)\right)\right) \cdot ux} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right) \]

   12. lift-*.f32 N/A

       \[\leadsto \sqrt{\left(1 + \left(1 - \left(\left(ux \cdot \left(maxCos - 1\right)\right) \cdot \left(maxCos - 1\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(2 \cdot maxCos\right)\right)\right)\right)\right)\right)\right) \cdot ux} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right) \]

   13. associate-*l* N/A

       \[\leadsto \sqrt{\left(1 + \left(1 - \left(ux \cdot \left(\left(maxCos - 1\right) \cdot \left(maxCos - 1\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(2 \cdot maxCos\right)\right)\right)\right)\right)\right)\right) \cdot ux} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right) \]

   14. lift--.f32 N/A

       \[\leadsto \sqrt{\left(1 + \left(1 - \left(ux \cdot \left(\left(maxCos - 1\right) \cdot \left(maxCos - 1\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(2 \cdot maxCos\right)\right)\right)\right)\right)\right)\right) \cdot ux} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right) \]

   15. sub-negate-rev N/A

       \[\leadsto \sqrt{\left(1 + \left(1 - \left(ux \cdot \left(\left(\mathsf{neg}\left(\left(1 - maxCos\right)\right)\right) \cdot \left(maxCos - 1\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(2 \cdot maxCos\right)\right)\right)\right)\right)\right)\right) \cdot ux} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right) \]

   16. lift--.f32 N/A

       \[\leadsto \sqrt{\left(1 + \left(1 - \left(ux \cdot \left(\left(\mathsf{neg}\left(\left(1 - maxCos\right)\right)\right) \cdot \left(maxCos - 1\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(2 \cdot maxCos\right)\right)\right)\right)\right)\right)\right) \cdot ux} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right) \]

   17. lift--.f32 N/A

       \[\leadsto \sqrt{\left(1 + \left(1 - \left(ux \cdot \left(\left(\mathsf{neg}\left(\left(1 - maxCos\right)\right)\right) \cdot \left(maxCos - 1\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(2 \cdot maxCos\right)\right)\right)\right)\right)\right)\right) \cdot ux} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right) \]

   18. sub-negate-rev N/A

       \[\leadsto \sqrt{\left(1 + \left(1 - \left(ux \cdot \left(\left(\mathsf{neg}\left(\left(1 - maxCos\right)\right)\right) \cdot \left(\mathsf{neg}\left(\left(1 - maxCos\right)\right)\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(2 \cdot maxCos\right)\right)\right)\right)\right)\right)\right) \cdot ux} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right) \]

   19. lift--.f32 N/A

       \[\leadsto \sqrt{\left(1 + \left(1 - \left(ux \cdot \left(\left(\mathsf{neg}\left(\left(1 - maxCos\right)\right)\right) \cdot \left(\mathsf{neg}\left(\left(1 - maxCos\right)\right)\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(2 \cdot maxCos\right)\right)\right)\right)\right)\right)\right) \cdot ux} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right) \]

   20. sqr-neg-rev N/A

       \[\leadsto \sqrt{\left(1 + \left(1 - \left(ux \cdot \left(\left(1 - maxCos\right) \cdot \left(1 - maxCos\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(2 \cdot maxCos\right)\right)\right)\right)\right)\right)\right) \cdot ux} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right) \]

   21. associate-*r* N/A

       \[\leadsto \sqrt{\left(1 + \left(1 - \left(\left(ux \cdot \left(1 - maxCos\right)\right) \cdot \left(1 - maxCos\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(2 \cdot maxCos\right)\right)\right)\right)\right)\right)\right) \cdot ux} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right) \]

8. Applied rewrites 98.3%

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

9. Taylor expanded in uy around 0

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

    1. lower-*.f32 N/A

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

    2. lower-*.f32 N/A

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

    3. lower-PI.f32 81.1

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

11. Applied rewrites 81.1%

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

Alternative 8: 81.1% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.3%

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

   1. lift--.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. metadata-eval N/A

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

   4. sqr-neg-rev N/A

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

   5. difference-of-squares N/A

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

   6. +-commutative N/A

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

   7. lift-+.f32 N/A

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

   8. lift--.f32 N/A

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

   9. associate-+l- N/A

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

   10. sub-negate N/A

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

   11. associate-+l- N/A

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

   12. associate-+l- N/A

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

   13. lift--.f32 N/A

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

   14. lift-+.f32 N/A

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

   15. lower-*.f32 N/A

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

3. Applied rewrites 98.3%

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

4. Taylor expanded in uy around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower--.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lower--.f32 N/A

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

   10. lower-+.f32 N/A

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

   11. lower-*.f32 81.1

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

6. Applied rewrites 81.1%

   \[\leadsto \color{blue}{2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{\left(ux - maxCos \cdot ux\right) \cdot \left(\left(2 + maxCos \cdot ux\right) - ux\right)}\right)\right)} \]
7. Add Preprocessing

Alternative 9: 48.8% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.3%

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

2. Taylor expanded in uy around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-PI.f32 50.3

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

4. Applied rewrites 50.3%

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

5. Taylor expanded in maxCos around 0

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

   1. lower--.f32 49.0

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

7. Applied rewrites 49.0%

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

8. Taylor expanded in maxCos around 0

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

   1. lower--.f32 48.8

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

10. Applied rewrites 48.8%

    \[\leadsto \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{1 - \left(1 - ux\right) \cdot \color{blue}{\left(1 - ux\right)}} \]
11. Add Preprocessing

Alternative 10: 7.1% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \left(\left(uy + uy\right) \cdot \pi\right) \cdot \sqrt{1 - 1} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.3%

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

2. Taylor expanded in uy around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-PI.f32 50.3

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

4. Applied rewrites 50.3%

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

5. Taylor expanded in ux around 0

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

7. Applied rewrites 7.1%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

      \[\leadsto \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - 1} \]

   5. lift-PI.f32 N/A

      \[\leadsto \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - 1} \]

   6. lift-*.f32 N/A

      \[\leadsto \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - 1} \]

   7. lift-PI.f32 N/A

      \[\leadsto \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - 1} \]

   8. lift-*.f32 7.1

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

   9. lift-*.f32 N/A

      \[\leadsto \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - 1} \]

   10. *-commutative N/A

       \[\leadsto \left(\left(2 \cdot uy\right) \cdot \pi\right) \cdot \sqrt{1 - 1} \]

   11. count-2 N/A

       \[\leadsto \left(\left(uy + uy\right) \cdot \pi\right) \cdot \sqrt{1 - 1} \]

   12. lift-+.f32 7.1

       \[\leadsto \left(\left(uy + uy\right) \cdot \pi\right) \cdot \sqrt{1 - 1} \]

9. Applied rewrites 7.1%

   \[\leadsto \color{blue}{\left(\left(uy + uy\right) \cdot \pi\right) \cdot \sqrt{1 - 1}} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025162 
(FPCore (ux uy maxCos)
  :name "UniformSampleCone, y"
  :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)))
  (* (sin (* (* uy 2.0) PI)) (sqrt (- 1.0 (* (+ (- 1.0 ux) (* ux maxCos)) (+ (- 1.0 ux) (* ux maxCos)))))))