UniformSampleCone, y

Percentage Accurate: 58.1% → 98.3%

Time: 7.2s

Alternatives: 20

Speedup: 4.6×

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: 58.1% 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

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 58.1%

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

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. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. +-commutative N/A

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

   5. associate-*r* N/A

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

   6. mul-1-neg N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-neg.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. lower--.f32 N/A

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

   12. lower--.f32 N/A

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

   13. count-2-rev N/A

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

   14. lower-+.f32 98.3

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

4. Applied rewrites 98.3%

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

5. Taylor expanded in maxCos around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. lower--.f32 98.3

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

7. Applied rewrites 98.3%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. count-2-rev N/A

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

   4. lift-+.f32 98.3

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

9. Applied rewrites 98.3%

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

    1. lift--.f32 N/A

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

    2. lift-fma.f32 N/A

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

    3. associate--l+ N/A

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

    4. lift-neg.f32 N/A

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

    5. mul-1-neg N/A

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

    6. lift-+.f32 N/A

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

    7. count-2-rev N/A

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

    8. lower-fma.f32 N/A

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

    9. mul-1-neg N/A

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

    10. lift-neg.f32 N/A

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

    11. metadata-eval N/A

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

    12. fp-cancel-sign-sub-inv N/A

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

    13. +-commutative N/A

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

    14. lift-fma.f32 98.3

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

11. Applied rewrites 98.3%

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

Alternative 2: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 58.1%

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

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. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. +-commutative N/A

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

   5. associate-*r* N/A

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

   6. mul-1-neg N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-neg.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. lower--.f32 N/A

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

   12. lower--.f32 N/A

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

   13. count-2-rev N/A

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

   14. lower-+.f32 98.3

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

4. Applied rewrites 98.3%

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

5. Taylor expanded in maxCos around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. lower--.f32 98.3

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

7. Applied rewrites 98.3%

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

   1. lift-*.f32 N/A

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

   2. lift-sin.f32 N/A

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

   3. lift-PI.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

9. Applied rewrites 98.3%

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

Alternative 3: 97.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 58.1%

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

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. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. +-commutative N/A

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

   5. associate-*r* N/A

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

   6. mul-1-neg N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-neg.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. lower--.f32 N/A

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

   12. lower--.f32 N/A

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

   13. count-2-rev N/A

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

   14. lower-+.f32 98.3

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

4. Applied rewrites 98.3%

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

5. Taylor expanded in maxCos around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. lower--.f32 98.3

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

7. Applied rewrites 98.3%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. count-2-rev N/A

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

   4. lift-+.f32 98.3

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

9. Applied rewrites 98.3%

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

10. Taylor expanded in maxCos around 0

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

    1. +-commutative N/A

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

    2. lower-+.f32 N/A

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

    3. +-commutative N/A

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

    4. *-commutative N/A

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

    5. lower-fma.f32 N/A

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

    6. lower--.f32 N/A

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

    7. count-2-rev N/A

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

    8. lower-+.f32 N/A

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

    9. mul-1-neg N/A

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

    10. lift-neg.f32 97.7

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

12. Applied rewrites 97.7%

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

Alternative 4: 97.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.014999999664723873:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), -1.3333333333333333, \pi + \pi\right) \cdot uy\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(-ux\right) + 2\right) \cdot ux} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= uy 0.014999999664723873)
   (*
    (* (fma (* (* uy uy) (* (* PI PI) PI)) -1.3333333333333333 (+ PI PI)) uy)
    (sqrt
     (*
      (- (fma (- ux) (* (- maxCos 1.0) (- maxCos 1.0)) 2.0) (+ maxCos maxCos))
      ux)))
   (* (sqrt (* (+ (- ux) 2.0) ux)) (sin (* PI (+ uy uy))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.014999999664723873f) {
		tmp = (fmaf(((uy * uy) * ((((float) M_PI) * ((float) M_PI)) * ((float) M_PI))), -1.3333333333333333f, (((float) M_PI) + ((float) M_PI))) * uy) * sqrtf(((fmaf(-ux, ((maxCos - 1.0f) * (maxCos - 1.0f)), 2.0f) - (maxCos + maxCos)) * ux));
	} else {
		tmp = sqrtf(((-ux + 2.0f) * ux)) * sinf((((float) M_PI) * (uy + uy)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.014999999664723873))
		tmp = Float32(Float32(fma(Float32(Float32(uy * uy) * Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi))), Float32(-1.3333333333333333), Float32(Float32(pi) + Float32(pi))) * uy) * sqrt(Float32(Float32(fma(Float32(-ux), Float32(Float32(maxCos - Float32(1.0)) * Float32(maxCos - Float32(1.0))), Float32(2.0)) - Float32(maxCos + maxCos)) * ux)));
	else
		tmp = Float32(sqrt(Float32(Float32(Float32(-ux) + Float32(2.0)) * ux)) * sin(Float32(Float32(pi) * Float32(uy + uy))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.0149999997

   1. Initial program 58.5%

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

   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. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f32 N/A

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

      8. lower-neg.f32 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f32 N/A

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

      11. lower--.f32 N/A

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

      12. lower--.f32 N/A

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

      13. count-2-rev N/A

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

      14. lower-+.f32 98.5

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

   4. Applied rewrites 98.5%

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

   5. Taylor expanded in uy around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. count-2-rev N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

   7. Applied rewrites 98.4%

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

   if 0.0149999997 < uy

   1. Initial program 56.2%

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

   2. Taylor expanded in maxCos around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower-sqrt.f32 N/A

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

      4. lower--.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. lift--.f32 N/A

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

      8. lift--.f32 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-sin.f32 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f32 N/A

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

      14. lift-PI.f32 N/A

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

      15. *-commutative N/A

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

      16. count-2-rev N/A

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

      17. lower-+.f32 54.0

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

   4. Applied rewrites 54.0%

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

   5. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f32 N/A

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

      5. mul-1-neg N/A

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

      6. lift-neg.f32 91.4

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

   7. Applied rewrites 91.4%

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

Alternative 5: 96.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.1%

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

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. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. +-commutative N/A

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

   5. associate-*r* N/A

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

   6. mul-1-neg N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-neg.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. lower--.f32 N/A

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

   12. lower--.f32 N/A

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

   13. count-2-rev N/A

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

   14. lower-+.f32 98.3

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

4. Applied rewrites 98.3%

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

5. Taylor expanded in maxCos around 0

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

   1. +-commutative N/A

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

   2. lower-+.f32 N/A

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

   3. mul-1-neg N/A

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

   4. lift-neg.f32 96.9

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

7. Applied rewrites 96.9%

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

Alternative 6: 96.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.1%

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

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. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. +-commutative N/A

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

   5. associate-*r* N/A

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

   6. mul-1-neg N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-neg.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. lower--.f32 N/A

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

   12. lower--.f32 N/A

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

   13. count-2-rev N/A

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

   14. lower-+.f32 98.3

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

4. Applied rewrites 98.3%

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

5. Taylor expanded in maxCos around 0

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

   1. +-commutative N/A

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

   2. lower-+.f32 N/A

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

   3. mul-1-neg N/A

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

   4. lift-neg.f32 96.9

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

7. Applied rewrites 96.9%

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

   1. lift-*.f32 N/A

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

   2. lift-sin.f32 N/A

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

   3. lift-PI.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

9. Applied rewrites 96.9%

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

Alternative 7: 93.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.019999999552965164:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), -1.3333333333333333, \pi + \pi\right) \cdot uy\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{2} \cdot \sin \left(\left(uy + uy\right) \cdot \pi\right)\right) \cdot \sqrt{ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= uy 0.019999999552965164)
   (*
    (* (fma (* (* uy uy) (* (* PI PI) PI)) -1.3333333333333333 (+ PI PI)) uy)
    (sqrt
     (*
      (- (fma (- ux) (* (- maxCos 1.0) (- maxCos 1.0)) 2.0) (+ maxCos maxCos))
      ux)))
   (* (* (sqrt 2.0) (sin (* (+ uy uy) PI))) (sqrt ux))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.019999999552965164f) {
		tmp = (fmaf(((uy * uy) * ((((float) M_PI) * ((float) M_PI)) * ((float) M_PI))), -1.3333333333333333f, (((float) M_PI) + ((float) M_PI))) * uy) * sqrtf(((fmaf(-ux, ((maxCos - 1.0f) * (maxCos - 1.0f)), 2.0f) - (maxCos + maxCos)) * ux));
	} else {
		tmp = (sqrtf(2.0f) * sinf(((uy + uy) * ((float) M_PI)))) * sqrtf(ux);
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.019999999552965164))
		tmp = Float32(Float32(fma(Float32(Float32(uy * uy) * Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi))), Float32(-1.3333333333333333), Float32(Float32(pi) + Float32(pi))) * uy) * sqrt(Float32(Float32(fma(Float32(-ux), Float32(Float32(maxCos - Float32(1.0)) * Float32(maxCos - Float32(1.0))), Float32(2.0)) - Float32(maxCos + maxCos)) * ux)));
	else
		tmp = Float32(Float32(sqrt(Float32(2.0)) * sin(Float32(Float32(uy + uy) * Float32(pi)))) * sqrt(ux));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.0199999996

   1. Initial program 58.5%

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

   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. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f32 N/A

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

      8. lower-neg.f32 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f32 N/A

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

      11. lower--.f32 N/A

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

      12. lower--.f32 N/A

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

      13. count-2-rev N/A

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

      14. lower-+.f32 98.5

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

   4. Applied rewrites 98.5%

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

   5. Taylor expanded in uy around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. count-2-rev N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

   7. Applied rewrites 98.2%

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

   if 0.0199999996 < uy

   1. Initial program 56.4%

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

   2. Taylor expanded in maxCos around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower-sqrt.f32 N/A

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

      4. lower--.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. lift--.f32 N/A

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

      8. lift--.f32 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-sin.f32 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f32 N/A

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

      14. lift-PI.f32 N/A

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

      15. *-commutative N/A

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

      16. count-2-rev N/A

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

      17. lower-+.f32 54.2

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

   4. Applied rewrites 54.2%

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

   5. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. lower-sqrt.f32 N/A

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

      6. lower-sin.f32 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lift-*.f32 N/A

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

      10. lift-*.f32 N/A

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

      11. lift-PI.f32 N/A

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

      12. lift-*.f32 N/A

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

      13. *-commutative N/A

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

      14. count-2-rev N/A

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

      15. lift-+.f32 N/A

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

      16. lower-sqrt.f32 73.1

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

   7. Applied rewrites 73.1%

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

Alternative 8: 92.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.019999999552965164:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), -1.3333333333333333, \pi + \pi\right) \cdot uy\right) \cdot \sqrt{\left(\left(\frac{\mathsf{fma}\left(-2, maxCos, 2\right)}{ux} - 1\right) \cdot ux\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{2} \cdot \sin \left(\left(uy + uy\right) \cdot \pi\right)\right) \cdot \sqrt{ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= uy 0.019999999552965164)
   (*
    (* (fma (* (* uy uy) (* (* PI PI) PI)) -1.3333333333333333 (+ PI PI)) uy)
    (sqrt (* (* (- (/ (fma -2.0 maxCos 2.0) ux) 1.0) ux) ux)))
   (* (* (sqrt 2.0) (sin (* (+ uy uy) PI))) (sqrt ux))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.019999999552965164f) {
		tmp = (fmaf(((uy * uy) * ((((float) M_PI) * ((float) M_PI)) * ((float) M_PI))), -1.3333333333333333f, (((float) M_PI) + ((float) M_PI))) * uy) * sqrtf(((((fmaf(-2.0f, maxCos, 2.0f) / ux) - 1.0f) * ux) * ux));
	} else {
		tmp = (sqrtf(2.0f) * sinf(((uy + uy) * ((float) M_PI)))) * sqrtf(ux);
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.019999999552965164))
		tmp = Float32(Float32(fma(Float32(Float32(uy * uy) * Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi))), Float32(-1.3333333333333333), Float32(Float32(pi) + Float32(pi))) * uy) * sqrt(Float32(Float32(Float32(Float32(fma(Float32(-2.0), maxCos, Float32(2.0)) / ux) - Float32(1.0)) * ux) * ux)));
	else
		tmp = Float32(Float32(sqrt(Float32(2.0)) * sin(Float32(Float32(uy + uy) * Float32(pi)))) * sqrt(ux));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.0199999996

   1. Initial program 58.5%

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

   2. Taylor expanded in ux around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   4. Applied rewrites 98.4%

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

   5. Taylor expanded in uy around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. count-2-rev N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

   7. Applied rewrites 98.2%

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

   8. Taylor expanded in maxCos around 0

      \[\leadsto \left(\mathsf{fma}\left(\left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), \frac{-4}{3}, \pi + \pi\right) \cdot uy\right) \cdot \sqrt{\left(\frac{\mathsf{fma}\left(-2, maxCos, 2\right)}{ux} - 1\right) \cdot \left(ux \cdot ux\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 96.8%

       \[\leadsto \left(\mathsf{fma}\left(\left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), -1.3333333333333333, \pi + \pi\right) \cdot uy\right) \cdot \sqrt{\left(\frac{\mathsf{fma}\left(-2, maxCos, 2\right)}{ux} - 1\right) \cdot \left(ux \cdot ux\right)} \]
   11. Step-by-step derivation

       1. lift-*.f32 N/A

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

       2. lift-*.f32 N/A

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

       3. associate-*r* N/A

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

       4. lower-*.f32 N/A

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

       5. lower-*.f32 96.9

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

   12. Applied rewrites 96.9%

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

   if 0.0199999996 < uy

   1. Initial program 56.4%

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

   2. Taylor expanded in maxCos around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower-sqrt.f32 N/A

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

      4. lower--.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. lift--.f32 N/A

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

      8. lift--.f32 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-sin.f32 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f32 N/A

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

      14. lift-PI.f32 N/A

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

      15. *-commutative N/A

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

      16. count-2-rev N/A

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

      17. lower-+.f32 54.2

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

   4. Applied rewrites 54.2%

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

   5. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. lower-sqrt.f32 N/A

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

      6. lower-sin.f32 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lift-*.f32 N/A

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

      10. lift-*.f32 N/A

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

      11. lift-PI.f32 N/A

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

      12. lift-*.f32 N/A

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

      13. *-commutative N/A

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

      14. count-2-rev N/A

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

      15. lift-+.f32 N/A

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

      16. lower-sqrt.f32 73.1

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

   7. Applied rewrites 73.1%

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

Alternative 9: 92.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.019999999552965164:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), -1.3333333333333333, \pi + \pi\right) \cdot uy\right) \cdot \sqrt{\left(\left(\frac{\mathsf{fma}\left(-2, maxCos, 2\right)}{ux} - 1\right) \cdot ux\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{ux + ux} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= uy 0.019999999552965164)
   (*
    (* (fma (* (* uy uy) (* (* PI PI) PI)) -1.3333333333333333 (+ PI PI)) uy)
    (sqrt (* (* (- (/ (fma -2.0 maxCos 2.0) ux) 1.0) ux) ux)))
   (* (sqrt (+ ux ux)) (sin (* PI (+ uy uy))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.019999999552965164f) {
		tmp = (fmaf(((uy * uy) * ((((float) M_PI) * ((float) M_PI)) * ((float) M_PI))), -1.3333333333333333f, (((float) M_PI) + ((float) M_PI))) * uy) * sqrtf(((((fmaf(-2.0f, maxCos, 2.0f) / ux) - 1.0f) * ux) * ux));
	} else {
		tmp = sqrtf((ux + ux)) * sinf((((float) M_PI) * (uy + uy)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.019999999552965164))
		tmp = Float32(Float32(fma(Float32(Float32(uy * uy) * Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi))), Float32(-1.3333333333333333), Float32(Float32(pi) + Float32(pi))) * uy) * sqrt(Float32(Float32(Float32(Float32(fma(Float32(-2.0), maxCos, Float32(2.0)) / ux) - Float32(1.0)) * ux) * ux)));
	else
		tmp = Float32(sqrt(Float32(ux + ux)) * sin(Float32(Float32(pi) * Float32(uy + uy))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.0199999996

   1. Initial program 58.5%

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

   2. Taylor expanded in ux around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   4. Applied rewrites 98.4%

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

   5. Taylor expanded in uy around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. count-2-rev N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

   7. Applied rewrites 98.2%

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

   8. Taylor expanded in maxCos around 0

      \[\leadsto \left(\mathsf{fma}\left(\left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), \frac{-4}{3}, \pi + \pi\right) \cdot uy\right) \cdot \sqrt{\left(\frac{\mathsf{fma}\left(-2, maxCos, 2\right)}{ux} - 1\right) \cdot \left(ux \cdot ux\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 96.8%

       \[\leadsto \left(\mathsf{fma}\left(\left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), -1.3333333333333333, \pi + \pi\right) \cdot uy\right) \cdot \sqrt{\left(\frac{\mathsf{fma}\left(-2, maxCos, 2\right)}{ux} - 1\right) \cdot \left(ux \cdot ux\right)} \]
   11. Step-by-step derivation

       1. lift-*.f32 N/A

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

       2. lift-*.f32 N/A

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

       3. associate-*r* N/A

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

       4. lower-*.f32 N/A

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

       5. lower-*.f32 96.9

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

   12. Applied rewrites 96.9%

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

   if 0.0199999996 < uy

   1. Initial program 56.4%

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

   2. Taylor expanded in maxCos around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower-sqrt.f32 N/A

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

      4. lower--.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. lift--.f32 N/A

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

      8. lift--.f32 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-sin.f32 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f32 N/A

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

      14. lift-PI.f32 N/A

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

      15. *-commutative N/A

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

      16. count-2-rev N/A

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

      17. lower-+.f32 54.2

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

   4. Applied rewrites 54.2%

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

   5. Taylor expanded in ux around 0

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

      1. count-2-rev N/A

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

      2. lower-+.f32 73.1

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

   7. Applied rewrites 73.1%

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

Alternative 10: 87.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.00011600000289035961:\\ \;\;\;\;\left(\left(uy + uy\right) \cdot \pi\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \mathsf{fma}\left(maxCos - 2, maxCos, 1\right), 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), -1.3333333333333333, \pi + \pi\right) \cdot uy\right) \cdot \sqrt{\left(\frac{2}{ux} - 1\right) \cdot \left(ux \cdot ux\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= uy 0.00011600000289035961)
   (*
    (* (+ uy uy) PI)
    (sqrt
     (*
      (- (fma (- ux) (fma (- maxCos 2.0) maxCos 1.0) 2.0) (+ maxCos maxCos))
      ux)))
   (*
    (* (fma (* (* uy uy) (* (* PI PI) PI)) -1.3333333333333333 (+ PI PI)) uy)
    (sqrt (* (- (/ 2.0 ux) 1.0) (* ux ux))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.00011600000289035961f) {
		tmp = ((uy + uy) * ((float) M_PI)) * sqrtf(((fmaf(-ux, fmaf((maxCos - 2.0f), maxCos, 1.0f), 2.0f) - (maxCos + maxCos)) * ux));
	} else {
		tmp = (fmaf(((uy * uy) * ((((float) M_PI) * ((float) M_PI)) * ((float) M_PI))), -1.3333333333333333f, (((float) M_PI) + ((float) M_PI))) * uy) * sqrtf((((2.0f / ux) - 1.0f) * (ux * ux)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.00011600000289035961))
		tmp = Float32(Float32(Float32(uy + uy) * Float32(pi)) * sqrt(Float32(Float32(fma(Float32(-ux), fma(Float32(maxCos - Float32(2.0)), maxCos, Float32(1.0)), Float32(2.0)) - Float32(maxCos + maxCos)) * ux)));
	else
		tmp = Float32(Float32(fma(Float32(Float32(uy * uy) * Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi))), Float32(-1.3333333333333333), Float32(Float32(pi) + Float32(pi))) * uy) * sqrt(Float32(Float32(Float32(Float32(2.0) / ux) - Float32(1.0)) * Float32(ux * ux))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 1.16000003e-4

   1. Initial program 58.6%

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

   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. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f32 N/A

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

      8. lower-neg.f32 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f32 N/A

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

      11. lower--.f32 N/A

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

      12. lower--.f32 N/A

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

      13. count-2-rev N/A

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

      14. lower-+.f32 98.5

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

   4. Applied rewrites 98.5%

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

   5. Taylor expanded in maxCos around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

      4. lower--.f32 98.5

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

   7. Applied rewrites 98.5%

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

   8. Taylor expanded in uy around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. count-2-rev N/A

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

      4. distribute-rgt-in N/A

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

      5. associate-*r* N/A

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

      6. count-2-rev N/A

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

      7. lift-+.f32 N/A

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

      8. lower-*.f32 N/A

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

      9. lift-PI.f32 98.5

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

   10. Applied rewrites 98.5%

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

   if 1.16000003e-4 < 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 -inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   4. Applied rewrites 97.9%

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

   5. Taylor expanded in uy around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. count-2-rev N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

   7. Applied rewrites 74.7%

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

   8. Taylor expanded in maxCos around 0

      \[\leadsto \left(\mathsf{fma}\left(\left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), \frac{-4}{3}, \pi + \pi\right) \cdot uy\right) \cdot \sqrt{\left(2 \cdot \frac{1}{ux} - 1\right) \cdot \left(\color{blue}{ux} \cdot ux\right)} \]
   9. Step-by-step derivation

      1. lower--.f32 N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), \frac{-4}{3}, \pi + \pi\right) \cdot uy\right) \cdot \sqrt{\left(2 \cdot \frac{1}{ux} - 1\right) \cdot \left(ux \cdot ux\right)} \]

      2. associate-*r/ N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), \frac{-4}{3}, \pi + \pi\right) \cdot uy\right) \cdot \sqrt{\left(\frac{2 \cdot 1}{ux} - 1\right) \cdot \left(ux \cdot ux\right)} \]

      3. metadata-eval N/A

         \[\leadsto \left(\mathsf{fma}\left(\left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), \frac{-4}{3}, \pi + \pi\right) \cdot uy\right) \cdot \sqrt{\left(\frac{2}{ux} - 1\right) \cdot \left(ux \cdot ux\right)} \]

      4. lift-/.f32 71.3

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

   10. Applied rewrites 71.3%

       \[\leadsto \left(\mathsf{fma}\left(\left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), -1.3333333333333333, \pi + \pi\right) \cdot uy\right) \cdot \sqrt{\left(\frac{2}{ux} - 1\right) \cdot \left(\color{blue}{ux} \cdot ux\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 87.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 58.1%

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

2. Taylor expanded in ux around -inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

4. Applied rewrites 98.2%

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

5. Taylor expanded in uy around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. count-2-rev N/A

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

   4. distribute-rgt-in N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

7. Applied rewrites 88.7%

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

8. Taylor expanded in maxCos around 0

   \[\leadsto \left(\mathsf{fma}\left(\left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), \frac{-4}{3}, \pi + \pi\right) \cdot uy\right) \cdot \sqrt{\left(\frac{\mathsf{fma}\left(-2, maxCos, 2\right)}{ux} - 1\right) \cdot \left(ux \cdot ux\right)} \]
9. Step-by-step derivation

10. Applied rewrites 87.5%

    \[\leadsto \left(\mathsf{fma}\left(\left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), -1.3333333333333333, \pi + \pi\right) \cdot uy\right) \cdot \sqrt{\left(\frac{\mathsf{fma}\left(-2, maxCos, 2\right)}{ux} - 1\right) \cdot \left(ux \cdot ux\right)} \]
11. Step-by-step derivation

    1. lift-*.f32 N/A

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

    2. lift-*.f32 N/A

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

    3. associate-*r* N/A

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

    4. lower-*.f32 N/A

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

    5. lower-*.f32 87.6

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

12. Applied rewrites 87.6%

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

Alternative 12: 87.3% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 58.1%

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

2. Taylor expanded in ux around -inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

4. Applied rewrites 98.2%

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

5. Taylor expanded in uy around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. count-2-rev N/A

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

   4. distribute-rgt-in N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

7. Applied rewrites 88.7%

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

8. Taylor expanded in maxCos around 0

   \[\leadsto \left(\mathsf{fma}\left(\left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), \frac{-4}{3}, \pi + \pi\right) \cdot uy\right) \cdot \sqrt{\left(\frac{\mathsf{fma}\left(-2, maxCos, 2\right)}{ux} - 1\right) \cdot \left(ux \cdot ux\right)} \]
9. Step-by-step derivation

10. Applied rewrites 87.5%

    \[\leadsto \left(\mathsf{fma}\left(\left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), -1.3333333333333333, \pi + \pi\right) \cdot uy\right) \cdot \sqrt{\left(\frac{\mathsf{fma}\left(-2, maxCos, 2\right)}{ux} - 1\right) \cdot \left(ux \cdot ux\right)} \]
11. Step-by-step derivation

    1. lift-fma.f32 N/A

       \[\leadsto \left(\left(\left(\left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)\right) \cdot \frac{-4}{3} + \left(\pi + \pi\right)\right) \cdot uy\right) \cdot \sqrt{\left(\frac{\mathsf{fma}\left(-2, maxCos, 2\right)}{ux} - 1\right) \cdot \left(ux \cdot ux\right)} \]

    2. lift-PI.f32 N/A

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

    3. lift-PI.f32 N/A

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

    4. lift-+.f32 N/A

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

    5. associate-+r+ N/A

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

    6. lower-+.f32 N/A

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

12. Applied rewrites 87.5%

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

Alternative 13: 81.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 58.1%

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

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. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. +-commutative N/A

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

   5. associate-*r* N/A

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

   6. mul-1-neg N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-neg.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. lower--.f32 N/A

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

   12. lower--.f32 N/A

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

   13. count-2-rev N/A

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

   14. lower-+.f32 98.3

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

4. Applied rewrites 98.3%

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

5. Taylor expanded in maxCos around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. lower--.f32 98.3

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

7. Applied rewrites 98.3%

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

8. Taylor expanded in uy around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. count-2-rev N/A

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

   4. distribute-rgt-in N/A

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

   5. associate-*r* N/A

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

   6. count-2-rev N/A

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

   7. lift-+.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lift-PI.f32 81.0

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

10. Applied rewrites 81.0%

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

Alternative 14: 80.1% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 58.1%

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

2. Taylor expanded in ux around -inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

4. Applied rewrites 98.2%

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

5. Taylor expanded in uy around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. count-2-rev N/A

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

   4. distribute-rgt-in N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

7. Applied rewrites 88.7%

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

8. Taylor expanded in maxCos around 0

   \[\leadsto \left(\mathsf{fma}\left(\left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), \frac{-4}{3}, \pi + \pi\right) \cdot uy\right) \cdot \sqrt{\left(\frac{\mathsf{fma}\left(-2, maxCos, 2\right)}{ux} - 1\right) \cdot \left(ux \cdot ux\right)} \]
9. Step-by-step derivation

10. Applied rewrites 87.5%

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

11. Taylor expanded in uy around 0

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

    1. count-2-rev N/A

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

    2. lift-+.f32 N/A

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

    3. lift-PI.f32 N/A

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

    4. lift-PI.f32 80.1

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

13. Applied rewrites 80.1%

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

Alternative 15: 75.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - ux\right) + ux \cdot maxCos\\ \mathbf{if}\;\sqrt{1 - t\_0 \cdot t\_0} \leq 0.02199999988079071:\\ \;\;\;\;\left(2 \cdot \left(\sqrt{\mathsf{fma}\left(-2, maxCos, 2\right)} \cdot \sqrt{ux}\right)\right) \cdot \left(\pi \cdot uy\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))))
   (if (<= (sqrt (- 1.0 (* t_0 t_0))) 0.02199999988079071)
     (* (* 2.0 (* (sqrt (fma -2.0 maxCos 2.0)) (sqrt ux))) (* PI uy))
     (* (* PI (+ uy uy)) (sqrt (- 1.0 (* (- 1.0 ux) (- 1.0 ux))))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = (1.0f - ux) + (ux * maxCos);
	float tmp;
	if (sqrtf((1.0f - (t_0 * t_0))) <= 0.02199999988079071f) {
		tmp = (2.0f * (sqrtf(fmaf(-2.0f, maxCos, 2.0f)) * sqrtf(ux))) * (((float) M_PI) * uy);
	} else {
		tmp = (((float) M_PI) * (uy + uy)) * sqrtf((1.0f - ((1.0f - ux) * (1.0f - ux))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(Float32(Float32(1.0) - ux) + Float32(ux * maxCos))
	tmp = Float32(0.0)
	if (sqrt(Float32(Float32(1.0) - Float32(t_0 * t_0))) <= Float32(0.02199999988079071))
		tmp = Float32(Float32(Float32(2.0) * Float32(sqrt(fma(Float32(-2.0), maxCos, Float32(2.0))) * sqrt(ux))) * Float32(Float32(pi) * uy));
	else
		tmp = Float32(Float32(Float32(pi) * Float32(uy + uy)) * sqrt(Float32(Float32(1.0) - Float32(Float32(Float32(1.0) - ux) * Float32(Float32(1.0) - ux)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (sqrt.f32 (-.f32 #s(literal 1 binary32) (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos))))) < 0.0219999999

   1. Initial program 38.3%

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

   2. Taylor expanded in uy around 0

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. lift-PI.f32 N/A

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

      8. *-commutative N/A

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

      9. count-2-rev N/A

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

      10. lower-+.f32 N/A

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

      11. lower-sqrt.f32 N/A

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

      12. lower--.f32 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f32 N/A

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

   4. Applied rewrites 35.3%

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

   5. Taylor expanded in ux around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-sqrt.f32 N/A

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

      5. metadata-eval N/A

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

      6. fp-cancel-sign-sub-inv N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 N/A

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

      9. +-commutative N/A

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

      10. lift-fma.f32 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f32 N/A

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

      13. lift-PI.f32 76.3

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

   7. Applied rewrites 76.3%

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

      1. lift-sqrt.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-fma.f32 N/A

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

      4. sqrt-prod N/A

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

      5. +-commutative N/A

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

      6. fp-cancel-sign-sub-inv N/A

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

      7. metadata-eval N/A

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

      8. lower-*.f32 N/A

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

      9. lower-sqrt.f32 N/A

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

      10. metadata-eval N/A

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

      11. fp-cancel-sign-sub-inv N/A

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

      12. +-commutative N/A

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

      13. lift-fma.f32 N/A

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

      14. lower-sqrt.f32 76.3

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

   9. Applied rewrites 76.3%

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

   if 0.0219999999 < (sqrt.f32 (-.f32 #s(literal 1 binary32) (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)))))

   1. Initial program 90.0%

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

   2. Taylor expanded in uy around 0

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. lift-PI.f32 N/A

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

      8. *-commutative N/A

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

      9. count-2-rev N/A

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

      10. lower-+.f32 N/A

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

      11. lower-sqrt.f32 N/A

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

      12. lower--.f32 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f32 N/A

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

   4. Applied rewrites 76.0%

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

   5. Taylor expanded in ux around 0

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

   7. Applied rewrites 73.4%

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

   8. Taylor expanded in ux around 0

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

   10. Applied rewrites 73.1%

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

Alternative 16: 65.2% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 58.1%

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

2. Taylor expanded in uy around 0

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

   1. associate-*r* N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. lift-PI.f32 N/A

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

   8. *-commutative N/A

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

   9. count-2-rev N/A

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

   10. lower-+.f32 N/A

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

   11. lower-sqrt.f32 N/A

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

   12. lower--.f32 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f32 N/A

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

4. Applied rewrites 50.9%

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

5. Taylor expanded in ux around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-sqrt.f32 N/A

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

   5. metadata-eval N/A

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

   6. fp-cancel-sign-sub-inv N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. +-commutative N/A

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

   10. lift-fma.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 N/A

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

   13. lift-PI.f32 65.1

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

7. Applied rewrites 65.1%

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

   1. lift-sqrt.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-fma.f32 N/A

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

   4. sqrt-prod N/A

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

   5. +-commutative N/A

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

   6. fp-cancel-sign-sub-inv N/A

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

   7. metadata-eval N/A

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

   8. lower-*.f32 N/A

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

   9. lower-sqrt.f32 N/A

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

   10. metadata-eval N/A

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

   11. fp-cancel-sign-sub-inv N/A

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

   12. +-commutative N/A

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

   13. lift-fma.f32 N/A

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

   14. lower-sqrt.f32 65.2

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

9. Applied rewrites 65.2%

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

Alternative 17: 65.1% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 58.1%

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

2. Taylor expanded in uy around 0

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

   1. associate-*r* N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. lift-PI.f32 N/A

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

   8. *-commutative N/A

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

   9. count-2-rev N/A

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

   10. lower-+.f32 N/A

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

   11. lower-sqrt.f32 N/A

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

   12. lower--.f32 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f32 N/A

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

4. Applied rewrites 50.9%

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

5. Taylor expanded in ux around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-sqrt.f32 N/A

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

   5. metadata-eval N/A

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

   6. fp-cancel-sign-sub-inv N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. +-commutative N/A

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

   10. lift-fma.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 N/A

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

   13. lift-PI.f32 65.1

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

7. Applied rewrites 65.1%

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

Alternative 18: 62.8% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.1%

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

2. Taylor expanded in uy around 0

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

   1. associate-*r* N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. lift-PI.f32 N/A

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

   8. *-commutative N/A

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

   9. count-2-rev N/A

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

   10. lower-+.f32 N/A

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

   11. lower-sqrt.f32 N/A

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

   12. lower--.f32 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f32 N/A

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

4. Applied rewrites 50.9%

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

5. Taylor expanded in ux around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-sqrt.f32 N/A

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

   5. metadata-eval N/A

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

   6. fp-cancel-sign-sub-inv N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. +-commutative N/A

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

   10. lift-fma.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 N/A

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

   13. lift-PI.f32 65.1

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

7. Applied rewrites 65.1%

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

8. Taylor expanded in maxCos around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-sqrt.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. lower-sqrt.f32 N/A

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

   10. lift-PI.f32 62.8

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

10. Applied rewrites 62.8%

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

Alternative 19: 62.7% accurate, 4.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.1%

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

2. Taylor expanded in uy around 0

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

   1. associate-*r* N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. lift-PI.f32 N/A

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

   8. *-commutative N/A

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

   9. count-2-rev N/A

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

   10. lower-+.f32 N/A

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

   11. lower-sqrt.f32 N/A

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

   12. lower--.f32 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f32 N/A

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

4. Applied rewrites 50.9%

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

5. Taylor expanded in ux around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-sqrt.f32 N/A

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

   5. metadata-eval N/A

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

   6. fp-cancel-sign-sub-inv N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. +-commutative N/A

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

   10. lift-fma.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 N/A

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

   13. lift-PI.f32 65.1

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

7. Applied rewrites 65.1%

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

8. Taylor expanded in maxCos around 0

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

   1. count-2-rev N/A

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

   2. lower-+.f32 62.7

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

10. Applied rewrites 62.7%

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

Alternative 20: 7.1% accurate, 4.7× speedup

Math

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

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (* (* PI (+ uy uy)) (sqrt (- 1.0 1.0))))

C

float code(float ux, float uy, float maxCos) {
	return (((float) M_PI) * (uy + uy)) * sqrtf((1.0f - 1.0f));
}

Julia

function code(ux, uy, maxCos)
	return Float32(Float32(Float32(pi) * Float32(uy + uy)) * sqrt(Float32(Float32(1.0) - Float32(1.0))))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = (single(pi) * (uy + uy)) * sqrt((single(1.0) - single(1.0)));
end

Derivation

1. Initial program 58.1%

   \[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

2. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{2 \cdot \left(\left(uy \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}\right)} \]
3. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

   2. associate-*r* N/A

      \[\leadsto \left(\left(2 \cdot uy\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

   3. *-commutative N/A

      \[\leadsto \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{1} - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}} \]

   4. lower-*.f32 N/A

      \[\leadsto \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

   5. *-commutative N/A

      \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left(uy \cdot 2\right)\right) \cdot \sqrt{\color{blue}{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

   6. lower-*.f32 N/A

      \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left(uy \cdot 2\right)\right) \cdot \sqrt{\color{blue}{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

   7. lift-PI.f32 N/A

      \[\leadsto \left(\pi \cdot \left(uy \cdot 2\right)\right) \cdot \sqrt{\color{blue}{1} - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}} \]

   8. *-commutative N/A

      \[\leadsto \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{1 - \color{blue}{{\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

   9. count-2-rev N/A

      \[\leadsto \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{1 - \color{blue}{{\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

   10. lower-+.f32 N/A

       \[\leadsto \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{1 - \color{blue}{{\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

   11. lower-sqrt.f32 N/A

       \[\leadsto \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}} \]

   12. lower--.f32 N/A

       \[\leadsto \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}} \]

   13. unpow2 N/A

       \[\leadsto \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{1 - \left(\left(1 + maxCos \cdot ux\right) - ux\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)} \]

   14. lower-*.f32 N/A

       \[\leadsto \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{1 - \left(\left(1 + maxCos \cdot ux\right) - ux\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)} \]

4. Applied rewrites 50.9%

   \[\leadsto \color{blue}{\left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{1 - \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right) \cdot \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right)}} \]

5. Taylor expanded in ux around 0

   \[\leadsto \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{1 - 1} \]
6. Step-by-step derivation

7. Applied rewrites 7.1%

   \[\leadsto \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{1 - 1} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025116 
(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)))))))