UniformSampleCone, y

Percentage Accurate: 57.3% → 98.4%

Time: 15.8s

Alternatives: 15

Speedup: 5.0×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 57.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 58.2%

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

3. Taylor expanded in ux around inf

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

   1. lower-*.f32 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f32 N/A

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

   4. sub-neg N/A

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

   5. +-commutative N/A

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

   6. +-commutative N/A

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

   7. distribute-neg-in N/A

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

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

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

   9. metadata-eval N/A

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

   10. associate-+l+ N/A

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

5. Applied rewrites 98.4%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 98.4

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

7. Applied rewrites 98.5%

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

Alternative 2: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 58.2%

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

3. Taylor expanded in ux around 0

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

   1. lower-*.f32 N/A

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

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

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

   3. metadata-eval N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. +-commutative N/A

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

   7. mul-1-neg N/A

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

   8. unpow2 N/A

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

   9. associate-*r* N/A

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

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

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

   11. sub-neg N/A

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

   12. metadata-eval N/A

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

   13. distribute-neg-in N/A

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

   14. mul-1-neg N/A

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

   15. metadata-eval N/A

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

   16. +-commutative N/A

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

   17. lower-fma.f32 N/A

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

5. Applied rewrites 98.5%

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

6. Final simplification 98.5%

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

Alternative 3: 97.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 58.2%

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

3. Taylor expanded in ux around 0

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

   1. lower-*.f32 N/A

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

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

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

   3. metadata-eval N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. +-commutative N/A

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

   7. mul-1-neg N/A

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

   8. unpow2 N/A

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

   9. associate-*r* N/A

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

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

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

   11. sub-neg N/A

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

   12. metadata-eval N/A

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

   13. distribute-neg-in N/A

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

   14. mul-1-neg N/A

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

   15. metadata-eval N/A

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

   16. +-commutative N/A

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

   17. lower-fma.f32 N/A

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

5. Applied rewrites 98.5%

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

6. Taylor expanded in maxCos around 0

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

8. Applied rewrites 97.7%

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

9. Final simplification 97.7%

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

Alternative 4: 97.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.02199999988079071:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(ux, \mathsf{fma}\left(maxCos, -2, 2\right), \left(maxCos + -1\right) \cdot \left(\left(1 - maxCos\right) \cdot \left(ux \cdot ux\right)\right)\right)} \cdot \left(uy \cdot \mathsf{fma}\left(2, \pi, -1.3333333333333333 \cdot \left(\left(uy \cdot uy\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(ux, 2, -ux \cdot ux\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* 2.0 uy) 0.02199999988079071)
   (*
    (sqrt
     (fma
      ux
      (fma maxCos -2.0 2.0)
      (* (+ maxCos -1.0) (* (- 1.0 maxCos) (* ux ux)))))
    (* uy (fma 2.0 PI (* -1.3333333333333333 (* (* uy uy) (* PI (* PI PI)))))))
   (* (sin (* 2.0 (* uy PI))) (sqrt (fma ux 2.0 (- (* ux ux)))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if ((2.0f * uy) <= 0.02199999988079071f) {
		tmp = sqrtf(fmaf(ux, fmaf(maxCos, -2.0f, 2.0f), ((maxCos + -1.0f) * ((1.0f - maxCos) * (ux * ux))))) * (uy * fmaf(2.0f, ((float) M_PI), (-1.3333333333333333f * ((uy * uy) * (((float) M_PI) * (((float) M_PI) * ((float) M_PI)))))));
	} else {
		tmp = sinf((2.0f * (uy * ((float) M_PI)))) * sqrtf(fmaf(ux, 2.0f, -(ux * ux)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (Float32(Float32(2.0) * uy) <= Float32(0.02199999988079071))
		tmp = Float32(sqrt(fma(ux, fma(maxCos, Float32(-2.0), Float32(2.0)), Float32(Float32(maxCos + Float32(-1.0)) * Float32(Float32(Float32(1.0) - maxCos) * Float32(ux * ux))))) * Float32(uy * fma(Float32(2.0), Float32(pi), Float32(Float32(-1.3333333333333333) * Float32(Float32(uy * uy) * Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi))))))));
	else
		tmp = Float32(sin(Float32(Float32(2.0) * Float32(uy * Float32(pi)))) * sqrt(fma(ux, Float32(2.0), Float32(-Float32(ux * ux)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 0.0219999999

   1. Initial program 57.2%

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

   3. Taylor expanded in ux around inf

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

      1. lower-*.f32 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f32 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

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

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

      9. metadata-eval N/A

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

      10. associate-+l+ N/A

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

   5. Applied rewrites 98.6%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f32 98.6

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

   7. Applied rewrites 98.7%

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

   8. Taylor expanded in uy around 0

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

      1. lower-*.f32 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-PI.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 N/A

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

      9. cube-mult N/A

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

      10. lower-*.f32 N/A

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

      11. lower-PI.f32 N/A

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

      12. lower-*.f32 N/A

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

      13. lower-PI.f32 N/A

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

      14. lower-PI.f32 98.7

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

   10. Applied rewrites 98.7%

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

   if 0.0219999999 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 61.7%

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

   3. Taylor expanded in ux around inf

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

      1. lower-*.f32 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f32 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

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

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

      9. metadata-eval N/A

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

      10. associate-+l+ N/A

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

   5. Applied rewrites 97.9%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f32 97.9

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

   7. Applied rewrites 98.1%

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

   8. Taylor expanded in maxCos around 0

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

   10. Applied rewrites 94.9%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.02199999988079071:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(ux, \mathsf{fma}\left(maxCos, -2, 2\right), \left(maxCos + -1\right) \cdot \left(\left(1 - maxCos\right) \cdot \left(ux \cdot ux\right)\right)\right)} \cdot \left(uy \cdot \mathsf{fma}\left(2, \pi, -1.3333333333333333 \cdot \left(\left(uy \cdot uy\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(ux, 2, -ux \cdot ux\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.02199999988079071:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(ux, \mathsf{fma}\left(maxCos, -2, 2\right), \left(maxCos + -1\right) \cdot \left(\left(1 - maxCos\right) \cdot \left(ux \cdot ux\right)\right)\right)} \cdot \left(uy \cdot \mathsf{fma}\left(2, \pi, -1.3333333333333333 \cdot \left(\left(uy \cdot uy\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* 2.0 uy) 0.02199999988079071)
   (*
    (sqrt
     (fma
      ux
      (fma maxCos -2.0 2.0)
      (* (+ maxCos -1.0) (* (- 1.0 maxCos) (* ux ux)))))
    (* uy (fma 2.0 PI (* -1.3333333333333333 (* (* uy uy) (* PI (* PI PI)))))))
   (* (sin (* PI (* 2.0 uy))) (sqrt (* ux (- 2.0 ux))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if ((2.0f * uy) <= 0.02199999988079071f) {
		tmp = sqrtf(fmaf(ux, fmaf(maxCos, -2.0f, 2.0f), ((maxCos + -1.0f) * ((1.0f - maxCos) * (ux * ux))))) * (uy * fmaf(2.0f, ((float) M_PI), (-1.3333333333333333f * ((uy * uy) * (((float) M_PI) * (((float) M_PI) * ((float) M_PI)))))));
	} else {
		tmp = sinf((((float) M_PI) * (2.0f * uy))) * sqrtf((ux * (2.0f - ux)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (Float32(Float32(2.0) * uy) <= Float32(0.02199999988079071))
		tmp = Float32(sqrt(fma(ux, fma(maxCos, Float32(-2.0), Float32(2.0)), Float32(Float32(maxCos + Float32(-1.0)) * Float32(Float32(Float32(1.0) - maxCos) * Float32(ux * ux))))) * Float32(uy * fma(Float32(2.0), Float32(pi), Float32(Float32(-1.3333333333333333) * Float32(Float32(uy * uy) * Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi))))))));
	else
		tmp = Float32(sin(Float32(Float32(pi) * Float32(Float32(2.0) * uy))) * sqrt(Float32(ux * Float32(Float32(2.0) - ux))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 0.0219999999

   1. Initial program 57.2%

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

   3. Taylor expanded in ux around inf

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

      1. lower-*.f32 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f32 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

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

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

      9. metadata-eval N/A

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

      10. associate-+l+ N/A

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

   5. Applied rewrites 98.6%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f32 98.6

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

   7. Applied rewrites 98.7%

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

   8. Taylor expanded in uy around 0

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

      1. lower-*.f32 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-PI.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 N/A

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

      9. cube-mult N/A

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

      10. lower-*.f32 N/A

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

      11. lower-PI.f32 N/A

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

      12. lower-*.f32 N/A

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

      13. lower-PI.f32 N/A

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

      14. lower-PI.f32 98.7

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

   10. Applied rewrites 98.7%

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

   if 0.0219999999 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 61.7%

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

   3. Taylor expanded in ux around 0

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

      1. lower-*.f32 N/A

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

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

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f32 N/A

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

      6. +-commutative N/A

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

      7. mul-1-neg N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

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

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

      11. sub-neg N/A

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

      12. metadata-eval N/A

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

      13. distribute-neg-in N/A

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

      14. mul-1-neg N/A

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

      15. metadata-eval N/A

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

      16. +-commutative N/A

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

      17. lower-fma.f32 N/A

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

   5. Applied rewrites 98.0%

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

   6. Taylor expanded in maxCos around 0

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

   8. Applied rewrites 94.9%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.02199999988079071:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(ux, \mathsf{fma}\left(maxCos, -2, 2\right), \left(maxCos + -1\right) \cdot \left(\left(1 - maxCos\right) \cdot \left(ux \cdot ux\right)\right)\right)} \cdot \left(uy \cdot \mathsf{fma}\left(2, \pi, -1.3333333333333333 \cdot \left(\left(uy \cdot uy\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 58.2%

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

3. Taylor expanded in ux around 0

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

   1. lower-*.f32 N/A

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

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

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

   3. metadata-eval N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. +-commutative N/A

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

   7. mul-1-neg N/A

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

   8. unpow2 N/A

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

   9. associate-*r* N/A

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

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

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

   11. sub-neg N/A

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

   12. metadata-eval N/A

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

   13. distribute-neg-in N/A

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

   14. mul-1-neg N/A

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

   15. metadata-eval N/A

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

   16. +-commutative N/A

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

   17. lower-fma.f32 N/A

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

5. Applied rewrites 98.5%

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

6. Taylor expanded in maxCos around 0

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

8. Applied rewrites 98.5%

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

9. Taylor expanded in ux around 0

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

11. Applied rewrites 96.9%

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

12. Final simplification 96.9%

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

Alternative 7: 94.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.05999999865889549:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(ux, \mathsf{fma}\left(maxCos, -2, 2\right), \left(maxCos + -1\right) \cdot \left(\left(1 - maxCos\right) \cdot \left(ux \cdot ux\right)\right)\right)} \cdot \left(uy \cdot \mathsf{fma}\left(2, \pi, -1.3333333333333333 \cdot \left(\left(uy \cdot uy\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{ux \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* 2.0 uy) 0.05999999865889549)
   (*
    (sqrt
     (fma
      ux
      (fma maxCos -2.0 2.0)
      (* (+ maxCos -1.0) (* (- 1.0 maxCos) (* ux ux)))))
    (* uy (fma 2.0 PI (* -1.3333333333333333 (* (* uy uy) (* PI (* PI PI)))))))
   (* (sin (* 2.0 (* uy PI))) (sqrt (* ux 2.0)))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if ((2.0f * uy) <= 0.05999999865889549f) {
		tmp = sqrtf(fmaf(ux, fmaf(maxCos, -2.0f, 2.0f), ((maxCos + -1.0f) * ((1.0f - maxCos) * (ux * ux))))) * (uy * fmaf(2.0f, ((float) M_PI), (-1.3333333333333333f * ((uy * uy) * (((float) M_PI) * (((float) M_PI) * ((float) M_PI)))))));
	} else {
		tmp = sinf((2.0f * (uy * ((float) M_PI)))) * sqrtf((ux * 2.0f));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (Float32(Float32(2.0) * uy) <= Float32(0.05999999865889549))
		tmp = Float32(sqrt(fma(ux, fma(maxCos, Float32(-2.0), Float32(2.0)), Float32(Float32(maxCos + Float32(-1.0)) * Float32(Float32(Float32(1.0) - maxCos) * Float32(ux * ux))))) * Float32(uy * fma(Float32(2.0), Float32(pi), Float32(Float32(-1.3333333333333333) * Float32(Float32(uy * uy) * Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi))))))));
	else
		tmp = Float32(sin(Float32(Float32(2.0) * Float32(uy * Float32(pi)))) * sqrt(Float32(ux * Float32(2.0))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

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

   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. Add Preprocessing

   3. Taylor expanded in ux around inf

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

      1. lower-*.f32 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f32 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

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

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

      9. metadata-eval N/A

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

      10. associate-+l+ N/A

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

   5. Applied rewrites 98.6%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f32 98.6

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

   7. Applied rewrites 98.7%

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

   8. Taylor expanded in uy around 0

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

      1. lower-*.f32 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-PI.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 N/A

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

      9. cube-mult N/A

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

      10. lower-*.f32 N/A

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

      11. lower-PI.f32 N/A

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

      12. lower-*.f32 N/A

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

      13. lower-PI.f32 N/A

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

      14. lower-PI.f32 97.8

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

   10. Applied rewrites 97.8%

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

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

   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. Add Preprocessing

   3. Taylor expanded in ux around inf

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

      1. lower-*.f32 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f32 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

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

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

      9. metadata-eval N/A

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

      10. associate-+l+ N/A

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

   5. Applied rewrites 97.2%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f32 97.2

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

   7. Applied rewrites 97.6%

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

   8. Taylor expanded in ux around 0

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

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

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

      2. metadata-eval N/A

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

      3. lower-*.f32 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f32 76.2

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

   10. Applied rewrites 76.2%

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

   11. Taylor expanded in maxCos around 0

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

   13. Applied rewrites 76.2%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.05999999865889549:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(ux, \mathsf{fma}\left(maxCos, -2, 2\right), \left(maxCos + -1\right) \cdot \left(\left(1 - maxCos\right) \cdot \left(ux \cdot ux\right)\right)\right)} \cdot \left(uy \cdot \mathsf{fma}\left(2, \pi, -1.3333333333333333 \cdot \left(\left(uy \cdot uy\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{ux \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 89.3% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 58.2%

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

3. Taylor expanded in ux around inf

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

   1. lower-*.f32 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f32 N/A

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

   4. sub-neg N/A

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

   5. +-commutative N/A

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

   6. +-commutative N/A

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

   7. distribute-neg-in N/A

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

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

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

   9. metadata-eval N/A

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

   10. associate-+l+ N/A

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

5. Applied rewrites 98.4%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 98.4

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

7. Applied rewrites 98.5%

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

8. Taylor expanded in uy around 0

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

   1. lower-*.f32 N/A

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

   2. +-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-PI.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 N/A

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

   9. cube-mult N/A

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

   10. lower-*.f32 N/A

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

   11. lower-PI.f32 N/A

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

   12. lower-*.f32 N/A

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

   13. lower-PI.f32 N/A

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

   14. lower-PI.f32 88.2

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

10. Applied rewrites 88.2%

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

Alternative 9: 89.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 58.2%

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

3. Taylor expanded in ux around 0

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

   1. lower-*.f32 N/A

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

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

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

   3. metadata-eval N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. +-commutative N/A

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

   7. mul-1-neg N/A

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

   8. unpow2 N/A

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

   9. associate-*r* N/A

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

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

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

   11. sub-neg N/A

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

   12. metadata-eval N/A

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

   13. distribute-neg-in N/A

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

   14. mul-1-neg N/A

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

   15. metadata-eval N/A

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

   16. +-commutative N/A

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

   17. lower-fma.f32 N/A

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

5. Applied rewrites 98.5%

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

6. Taylor expanded in uy around 0

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

   1. lower-*.f32 N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f32 N/A

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

   6. cube-mult N/A

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

   7. lower-*.f32 N/A

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

   8. lower-PI.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lower-PI.f32 N/A

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

   11. lower-PI.f32 N/A

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

   12. lower-*.f32 N/A

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

   13. lower-PI.f32 88.1

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

8. Applied rewrites 88.1%

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

9. Final simplification 88.1%

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

Alternative 10: 84.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.003000000026077032:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(ux, \mathsf{fma}\left(maxCos, -2, 2\right), \left(maxCos + -1\right) \cdot \left(\left(1 - maxCos\right) \cdot \left(ux \cdot ux\right)\right)\right)} \cdot \left(2 \cdot \left(uy \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(uy \cdot \mathsf{fma}\left(2, \pi, -1.3333333333333333 \cdot \left(\left(uy \cdot uy\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(maxCos, -2, 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* 2.0 uy) 0.003000000026077032)
   (*
    (sqrt
     (fma
      ux
      (fma maxCos -2.0 2.0)
      (* (+ maxCos -1.0) (* (- 1.0 maxCos) (* ux ux)))))
    (* 2.0 (* uy PI)))
   (*
    (* uy (fma 2.0 PI (* -1.3333333333333333 (* (* uy uy) (* PI (* PI PI))))))
    (sqrt (* ux (fma maxCos -2.0 2.0))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if ((2.0f * uy) <= 0.003000000026077032f) {
		tmp = sqrtf(fmaf(ux, fmaf(maxCos, -2.0f, 2.0f), ((maxCos + -1.0f) * ((1.0f - maxCos) * (ux * ux))))) * (2.0f * (uy * ((float) M_PI)));
	} else {
		tmp = (uy * fmaf(2.0f, ((float) M_PI), (-1.3333333333333333f * ((uy * uy) * (((float) M_PI) * (((float) M_PI) * ((float) M_PI))))))) * sqrtf((ux * fmaf(maxCos, -2.0f, 2.0f)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (Float32(Float32(2.0) * uy) <= Float32(0.003000000026077032))
		tmp = Float32(sqrt(fma(ux, fma(maxCos, Float32(-2.0), Float32(2.0)), Float32(Float32(maxCos + Float32(-1.0)) * Float32(Float32(Float32(1.0) - maxCos) * Float32(ux * ux))))) * Float32(Float32(2.0) * Float32(uy * Float32(pi))));
	else
		tmp = Float32(Float32(uy * fma(Float32(2.0), Float32(pi), Float32(Float32(-1.3333333333333333) * Float32(Float32(uy * uy) * Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi))))))) * sqrt(Float32(ux * fma(maxCos, Float32(-2.0), Float32(2.0)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 0.00300000003

   1. Initial program 58.0%

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

   3. Taylor expanded in ux around inf

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

      1. lower-*.f32 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f32 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

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

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

      9. metadata-eval N/A

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

      10. associate-+l+ N/A

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

   5. Applied rewrites 98.5%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f32 98.5

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

   7. Applied rewrites 98.6%

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

   8. Taylor expanded in uy around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-PI.f32 97.0

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

   10. Applied rewrites 97.0%

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

   if 0.00300000003 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 58.7%

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

   3. Taylor expanded in maxCos around 0

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

      1. unpow2 N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. lower--.f32 56.8

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

   5. Applied rewrites 56.8%

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

   6. Taylor expanded in ux around 0

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

   8. Applied rewrites 59.5%

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

   9. Taylor expanded in uy around 0

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

       1. lower-*.f32 N/A

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

       2. +-commutative N/A

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

       3. lower-fma.f32 N/A

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

       4. lower-PI.f32 N/A

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

       5. lower-*.f32 N/A

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

       6. lower-*.f32 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f32 N/A

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

       9. cube-mult N/A

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

       10. lower-*.f32 N/A

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

       11. lower-PI.f32 N/A

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

       12. lower-*.f32 N/A

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

       13. lower-PI.f32 N/A

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

       14. lower-PI.f32 44.1

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

   11. Applied rewrites 44.1%

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

   12. Taylor expanded in ux around 0

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

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

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

       2. metadata-eval N/A

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

       3. lower-*.f32 N/A

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f32 55.1

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

   14. Applied rewrites 55.1%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.003000000026077032:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(ux, \mathsf{fma}\left(maxCos, -2, 2\right), \left(maxCos + -1\right) \cdot \left(\left(1 - maxCos\right) \cdot \left(ux \cdot ux\right)\right)\right)} \cdot \left(2 \cdot \left(uy \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(uy \cdot \mathsf{fma}\left(2, \pi, -1.3333333333333333 \cdot \left(\left(uy \cdot uy\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(maxCos, -2, 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 81.5% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 58.2%

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

3. Taylor expanded in ux around inf

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

   1. lower-*.f32 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f32 N/A

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

   4. sub-neg N/A

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

   5. +-commutative N/A

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

   6. +-commutative N/A

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

   7. distribute-neg-in N/A

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

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

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

   9. metadata-eval N/A

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

   10. associate-+l+ N/A

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

5. Applied rewrites 98.4%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 98.4

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

7. Applied rewrites 98.5%

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

8. Taylor expanded in uy around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-PI.f32 80.8

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

10. Applied rewrites 80.8%

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

Alternative 12: 81.5% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 58.2%

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

3. Taylor expanded in ux around 0

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

   1. lower-*.f32 N/A

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

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

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

   3. metadata-eval N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. +-commutative N/A

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

   7. mul-1-neg N/A

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

   8. unpow2 N/A

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

   9. associate-*r* N/A

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

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

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

   11. sub-neg N/A

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

   12. metadata-eval N/A

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

   13. distribute-neg-in N/A

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

   14. mul-1-neg N/A

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

   15. metadata-eval N/A

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

   16. +-commutative N/A

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

   17. lower-fma.f32 N/A

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

5. Applied rewrites 98.5%

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

6. Taylor expanded in uy around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-PI.f32 80.8

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

8. Applied rewrites 80.8%

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

9. Final simplification 80.8%

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

Alternative 13: 77.3% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.2%

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

3. Taylor expanded in uy around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

5. Applied rewrites 51.1%

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

6. Taylor expanded in maxCos around 0

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

8. Applied rewrites 49.5%

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

9. Taylor expanded in ux around inf

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

11. Applied rewrites 76.9%

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

12. Final simplification 76.9%

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

Alternative 14: 77.3% accurate, 4.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.2%

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

3. Taylor expanded in uy around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

5. Applied rewrites 51.1%

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

6. Taylor expanded in maxCos around 0

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

8. Applied rewrites 49.5%

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

9. Taylor expanded in ux around 0

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

11. Applied rewrites 76.8%

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

12. Final simplification 76.8%

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

Alternative 15: 63.7% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.2%

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

3. Taylor expanded in uy around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

5. Applied rewrites 51.1%

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

6. Taylor expanded in maxCos around 0

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

8. Applied rewrites 49.5%

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

9. Taylor expanded in ux around 0

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

11. Applied rewrites 63.7%

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

12. Final simplification 63.7%

    \[\leadsto \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{ux \cdot 2} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024227 
(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)))))))