UniformSampleCone, y

Percentage Accurate: 57.3% → 98.3%

Time: 8.1s

Alternatives: 21

Speedup: 4.0×

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: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.3%

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

2. Taylor expanded in ux around -inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

4. Applied rewrites 98.2%

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

5. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower-+.f32 N/A

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

   4. lower-fma.f32 N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f32 N/A

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

   7. mul-1-neg N/A

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

   8. lift-neg.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. lower--.f32 N/A

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

   12. lower--.f32 98.3

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

7. Applied rewrites 98.3%

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

Alternative 2: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.3%

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

2. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. +-commutative N/A

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

   5. associate-*r* N/A

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

   6. mul-1-neg N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-neg.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. lower--.f32 N/A

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

   12. lower--.f32 N/A

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

   13. count-2-rev N/A

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

   14. lower-+.f32 98.3

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

4. Applied rewrites 98.3%

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

Alternative 3: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.3%

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

2. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. +-commutative N/A

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

   5. associate-*r* N/A

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

   6. mul-1-neg N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-neg.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. lower--.f32 N/A

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

   12. lower--.f32 N/A

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

   13. count-2-rev N/A

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

   14. lower-+.f32 98.3

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

4. Applied rewrites 98.3%

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

6. Applied rewrites 98.3%

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

7. Taylor expanded in maxCos around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. lower--.f32 98.3

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

9. Applied rewrites 98.3%

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

Alternative 4: 97.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.3%

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

2. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. +-commutative N/A

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

   5. associate-*r* N/A

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

   6. mul-1-neg N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-neg.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. lower--.f32 N/A

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

   12. lower--.f32 N/A

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

   13. count-2-rev N/A

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

   14. lower-+.f32 98.3

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

4. Applied rewrites 98.3%

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

5. Taylor expanded in maxCos around 0

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

   1. associate-+r+ N/A

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

   2. lower-+.f32 N/A

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

   3. lower-+.f32 N/A

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

   4. mul-1-neg N/A

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

   5. lift-neg.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. lower--.f32 N/A

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

   9. count-2-rev N/A

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

   10. lower-+.f32 97.7

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

7. Applied rewrites 97.7%

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

8. Taylor expanded in maxCos around 0

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

   1. lower--.f32 N/A

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

   2. count-2-rev N/A

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

   3. *-commutative N/A

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

   4. lower-+.f32 N/A

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

   5. lift--.f32 N/A

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

   6. lift-+.f32 N/A

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

   7. lift-*.f32 97.7

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

10. Applied rewrites 97.7%

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

Alternative 5: 97.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.017999999225139618:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), -1.3333333333333333, \pi + \pi\right) \cdot uy\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(2 - ux\right) \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= uy 0.017999999225139618)
   (*
    (* (fma (* (* uy uy) (* (* PI PI) PI)) -1.3333333333333333 (+ PI PI)) uy)
    (sqrt
     (*
      (- (fma (- ux) (* (- maxCos 1.0) (- maxCos 1.0)) 2.0) (+ maxCos maxCos))
      ux)))
   (* (sin (* (* uy 2.0) PI)) (sqrt (* (- 2.0 ux) ux)))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.017999999225139618f) {
		tmp = (fmaf(((uy * uy) * ((((float) M_PI) * ((float) M_PI)) * ((float) M_PI))), -1.3333333333333333f, (((float) M_PI) + ((float) M_PI))) * uy) * sqrtf(((fmaf(-ux, ((maxCos - 1.0f) * (maxCos - 1.0f)), 2.0f) - (maxCos + maxCos)) * ux));
	} else {
		tmp = sinf(((uy * 2.0f) * ((float) M_PI))) * sqrtf(((2.0f - ux) * ux));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.017999999225139618))
		tmp = Float32(Float32(fma(Float32(Float32(uy * uy) * Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi))), Float32(-1.3333333333333333), Float32(Float32(pi) + Float32(pi))) * uy) * sqrt(Float32(Float32(fma(Float32(-ux), Float32(Float32(maxCos - Float32(1.0)) * Float32(maxCos - Float32(1.0))), Float32(2.0)) - Float32(maxCos + maxCos)) * ux)));
	else
		tmp = Float32(sin(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) * sqrt(Float32(Float32(Float32(2.0) - ux) * ux)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.0179999992

   1. Initial program 57.3%

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

   2. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f32 N/A

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

      8. lower-neg.f32 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f32 N/A

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

      11. lower--.f32 N/A

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

      12. lower--.f32 N/A

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

      13. count-2-rev N/A

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

      14. lower-+.f32 98.3

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

   4. Applied rewrites 98.3%

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

   5. Taylor expanded in uy around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   7. Applied rewrites 88.9%

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

   if 0.0179999992 < uy

   1. Initial program 57.3%

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

   2. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f32 N/A

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

      8. lower-neg.f32 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f32 N/A

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

      11. lower--.f32 N/A

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

      12. lower--.f32 N/A

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

      13. count-2-rev N/A

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

      14. lower-+.f32 98.3

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

   4. Applied rewrites 98.3%

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

   5. Taylor expanded in maxCos around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f32 N/A

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

      3. lower-+.f32 N/A

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

      4. mul-1-neg N/A

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

      5. lift-neg.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f32 N/A

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

      8. lower--.f32 N/A

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

      9. count-2-rev N/A

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

      10. lower-+.f32 97.7

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

   7. Applied rewrites 97.7%

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

   8. Taylor expanded in maxCos around 0

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

      1. lower--.f32 92.4

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

   10. Applied rewrites 92.4%

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

Alternative 6: 97.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.3%

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

2. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. +-commutative N/A

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

   5. associate-*r* N/A

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

   6. mul-1-neg N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-neg.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. lower--.f32 N/A

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

   12. lower--.f32 N/A

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

   13. count-2-rev N/A

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

   14. lower-+.f32 98.3

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

4. Applied rewrites 98.3%

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

5. Taylor expanded in maxCos around 0

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

   1. associate-+r+ N/A

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

   2. lower-+.f32 N/A

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

   3. lower-+.f32 N/A

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

   4. mul-1-neg N/A

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

   5. lift-neg.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. lower--.f32 N/A

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

   9. count-2-rev N/A

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

   10. lower-+.f32 97.7

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

7. Applied rewrites 97.7%

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

8. Taylor expanded in ux around 0

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

10. Applied rewrites 97.0%

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

Alternative 7: 96.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.017999999225139618:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), -1.3333333333333333, \pi\right) + \pi\right) \cdot uy\right) \cdot \sqrt{\left(\left(2 + \left(-ux\right)\right) + \left(\left(ux + ux\right) - 2\right) \cdot maxCos\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(2 - ux\right) \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= uy 0.017999999225139618)
   (*
    (* (+ (fma (* (* uy uy) (* (* PI PI) PI)) -1.3333333333333333 PI) PI) uy)
    (sqrt (* (+ (+ 2.0 (- ux)) (* (- (+ ux ux) 2.0) maxCos)) ux)))
   (* (sin (* (* uy 2.0) PI)) (sqrt (* (- 2.0 ux) ux)))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.017999999225139618f) {
		tmp = ((fmaf(((uy * uy) * ((((float) M_PI) * ((float) M_PI)) * ((float) M_PI))), -1.3333333333333333f, ((float) M_PI)) + ((float) M_PI)) * uy) * sqrtf((((2.0f + -ux) + (((ux + ux) - 2.0f) * maxCos)) * ux));
	} else {
		tmp = sinf(((uy * 2.0f) * ((float) M_PI))) * sqrtf(((2.0f - ux) * ux));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.017999999225139618))
		tmp = Float32(Float32(Float32(fma(Float32(Float32(uy * uy) * Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi))), Float32(-1.3333333333333333), Float32(pi)) + Float32(pi)) * uy) * sqrt(Float32(Float32(Float32(Float32(2.0) + Float32(-ux)) + Float32(Float32(Float32(ux + ux) - Float32(2.0)) * maxCos)) * ux)));
	else
		tmp = Float32(sin(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) * sqrt(Float32(Float32(Float32(2.0) - ux) * ux)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.0179999992

   1. Initial program 57.3%

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

   2. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f32 N/A

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

      8. lower-neg.f32 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f32 N/A

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

      11. lower--.f32 N/A

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

      12. lower--.f32 N/A

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

      13. count-2-rev N/A

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

      14. lower-+.f32 98.3

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

   4. Applied rewrites 98.3%

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

   5. Taylor expanded in maxCos around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f32 N/A

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

      3. lower-+.f32 N/A

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

      4. mul-1-neg N/A

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

      5. lift-neg.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f32 N/A

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

      8. lower--.f32 N/A

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

      9. count-2-rev N/A

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

      10. lower-+.f32 97.7

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

   7. Applied rewrites 97.7%

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

   8. Taylor expanded in uy around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   10. Applied rewrites 88.4%

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

       1. lift-fma.f32 N/A

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

       2. lift-PI.f32 N/A

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

       3. lift-PI.f32 N/A

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

       4. lift-+.f32 N/A

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

       5. associate-+r+ N/A

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

       6. lower-+.f32 N/A

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

   12. Applied rewrites 88.4%

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

   if 0.0179999992 < uy

   1. Initial program 57.3%

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

   2. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f32 N/A

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

      8. lower-neg.f32 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f32 N/A

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

      11. lower--.f32 N/A

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

      12. lower--.f32 N/A

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

      13. count-2-rev N/A

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

      14. lower-+.f32 98.3

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

   4. Applied rewrites 98.3%

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

   5. Taylor expanded in maxCos around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f32 N/A

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

      3. lower-+.f32 N/A

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

      4. mul-1-neg N/A

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

      5. lift-neg.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f32 N/A

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

      8. lower--.f32 N/A

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

      9. count-2-rev N/A

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

      10. lower-+.f32 97.7

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

   7. Applied rewrites 97.7%

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

   8. Taylor expanded in maxCos around 0

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

      1. lower--.f32 92.4

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

   10. Applied rewrites 92.4%

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

Alternative 8: 93.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if uy < 0.0199999996

   1. Initial program 57.3%

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

   2. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f32 N/A

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

      8. lower-neg.f32 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f32 N/A

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

      11. lower--.f32 N/A

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

      12. lower--.f32 N/A

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

      13. count-2-rev N/A

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

      14. lower-+.f32 98.3

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

   4. Applied rewrites 98.3%

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

   5. Taylor expanded in maxCos around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f32 N/A

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

      3. lower-+.f32 N/A

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

      4. mul-1-neg N/A

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

      5. lift-neg.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f32 N/A

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

      8. lower--.f32 N/A

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

      9. count-2-rev N/A

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

      10. lower-+.f32 97.7

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

   7. Applied rewrites 97.7%

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

   8. Taylor expanded in uy around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   10. Applied rewrites 88.4%

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

       1. lift-fma.f32 N/A

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

       2. lift-PI.f32 N/A

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

       3. lift-PI.f32 N/A

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

       4. lift-+.f32 N/A

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

       5. associate-+r+ N/A

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

       6. lower-+.f32 N/A

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

   12. Applied rewrites 88.4%

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

   if 0.0199999996 < uy

   1. Initial program 57.3%

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

   2. Taylor expanded in maxCos around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower-sqrt.f32 N/A

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

      4. lower--.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. lift--.f32 N/A

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

      8. lift--.f32 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lift-*.f32 N/A

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

      12. lift-*.f32 N/A

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

      13. lift-PI.f32 N/A

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

      14. lift-sin.f32 55.4

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

      15. lift-*.f32 N/A

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

      16. lift-PI.f32 N/A

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

   4. Applied rewrites 55.4%

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

   5. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. lower-sqrt.f32 N/A

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

      6. lower-sin.f32 N/A

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

      7. associate-*r* N/A

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

      8. count-2-rev N/A

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

      9. lift-+.f32 N/A

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

      10. lower-*.f32 N/A

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

      11. lift-PI.f32 N/A

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

      12. lower-sqrt.f32 73.1

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

   7. Applied rewrites 73.1%

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

      1. lift-*.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-sqrt.f32 N/A

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

      4. *-commutative N/A

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

      5. lift-sin.f32 N/A

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

      6. lift-PI.f32 N/A

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

      7. lift-*.f32 N/A

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

      8. lift-+.f32 N/A

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

      9. count-2-rev N/A

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

      10. associate-*r* N/A

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

      11. lift-sqrt.f32 N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f32 N/A

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

   9. Applied rewrites 73.1%

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

Alternative 9: 93.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if uy < 0.0199999996

   1. Initial program 57.3%

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

   2. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f32 N/A

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

      8. lower-neg.f32 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f32 N/A

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

      11. lower--.f32 N/A

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

      12. lower--.f32 N/A

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

      13. count-2-rev N/A

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

      14. lower-+.f32 98.3

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

   4. Applied rewrites 98.3%

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

   5. Taylor expanded in maxCos around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f32 N/A

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

      3. lower-+.f32 N/A

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

      4. mul-1-neg N/A

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

      5. lift-neg.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f32 N/A

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

      8. lower--.f32 N/A

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

      9. count-2-rev N/A

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

      10. lower-+.f32 97.7

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

   7. Applied rewrites 97.7%

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

   8. Taylor expanded in uy around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   10. Applied rewrites 88.4%

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

       1. lift-fma.f32 N/A

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

       2. lift-PI.f32 N/A

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

       3. lift-PI.f32 N/A

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

       4. lift-+.f32 N/A

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

       5. associate-+r+ N/A

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

       6. lower-+.f32 N/A

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

   12. Applied rewrites 88.4%

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

   if 0.0199999996 < uy

   1. Initial program 57.3%

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

   2. Taylor expanded in maxCos around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower-sqrt.f32 N/A

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

      4. lower--.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. lift--.f32 N/A

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

      8. lift--.f32 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lift-*.f32 N/A

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

      12. lift-*.f32 N/A

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

      13. lift-PI.f32 N/A

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

      14. lift-sin.f32 55.4

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

      15. lift-*.f32 N/A

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

      16. lift-PI.f32 N/A

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

   4. Applied rewrites 55.4%

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

   5. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. lower-sqrt.f32 N/A

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

      6. lower-sin.f32 N/A

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

      7. associate-*r* N/A

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

      8. count-2-rev N/A

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

      9. lift-+.f32 N/A

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

      10. lower-*.f32 N/A

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

      11. lift-PI.f32 N/A

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

      12. lower-sqrt.f32 73.1

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

   7. Applied rewrites 73.1%

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

Alternative 10: 93.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if uy < 0.0199999996

   1. Initial program 57.3%

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

   2. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f32 N/A

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

      8. lower-neg.f32 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f32 N/A

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

      11. lower--.f32 N/A

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

      12. lower--.f32 N/A

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

      13. count-2-rev N/A

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

      14. lower-+.f32 98.3

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

   4. Applied rewrites 98.3%

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

   5. Taylor expanded in maxCos around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f32 N/A

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

      3. lower-+.f32 N/A

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

      4. mul-1-neg N/A

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

      5. lift-neg.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f32 N/A

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

      8. lower--.f32 N/A

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

      9. count-2-rev N/A

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

      10. lower-+.f32 97.7

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

   7. Applied rewrites 97.7%

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

   8. Taylor expanded in uy around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   10. Applied rewrites 88.4%

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

       1. lift-fma.f32 N/A

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

       2. lift-PI.f32 N/A

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

       3. lift-PI.f32 N/A

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

       4. lift-+.f32 N/A

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

       5. associate-+r+ N/A

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

       6. lower-+.f32 N/A

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

   12. Applied rewrites 88.4%

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

   if 0.0199999996 < uy

   1. Initial program 57.3%

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

   2. Taylor expanded in maxCos around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower-sqrt.f32 N/A

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

      4. lower--.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. lift--.f32 N/A

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

      8. lift--.f32 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lift-*.f32 N/A

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

      12. lift-*.f32 N/A

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

      13. lift-PI.f32 N/A

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

      14. lift-sin.f32 55.4

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

      15. lift-*.f32 N/A

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

      16. lift-PI.f32 N/A

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

   4. Applied rewrites 55.4%

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

   5. Taylor expanded in ux around 0

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

      1. count-2-rev N/A

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

      2. lower-+.f32 73.0

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

   7. Applied rewrites 73.0%

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

Alternative 11: 88.4% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.3%

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

2. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. +-commutative N/A

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

   5. associate-*r* N/A

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

   6. mul-1-neg N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-neg.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. lower--.f32 N/A

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

   12. lower--.f32 N/A

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

   13. count-2-rev N/A

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

   14. lower-+.f32 98.3

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

4. Applied rewrites 98.3%

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

5. Taylor expanded in maxCos around 0

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

   1. associate-+r+ N/A

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

   2. lower-+.f32 N/A

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

   3. lower-+.f32 N/A

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

   4. mul-1-neg N/A

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

   5. lift-neg.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. lower--.f32 N/A

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

   9. count-2-rev N/A

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

   10. lower-+.f32 97.7

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

7. Applied rewrites 97.7%

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

8. Taylor expanded in uy around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

10. Applied rewrites 88.4%

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

    1. lift-fma.f32 N/A

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

    2. lift-PI.f32 N/A

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

    3. lift-PI.f32 N/A

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

    4. lift-+.f32 N/A

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

    5. associate-+r+ N/A

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

    6. lower-+.f32 N/A

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

12. Applied rewrites 88.4%

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

Alternative 12: 88.4% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.3%

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

2. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. +-commutative N/A

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

   5. associate-*r* N/A

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

   6. mul-1-neg N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-neg.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. lower--.f32 N/A

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

   12. lower--.f32 N/A

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

   13. count-2-rev N/A

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

   14. lower-+.f32 98.3

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

4. Applied rewrites 98.3%

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

5. Taylor expanded in maxCos around 0

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

   1. associate-+r+ N/A

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

   2. lower-+.f32 N/A

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

   3. lower-+.f32 N/A

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

   4. mul-1-neg N/A

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

   5. lift-neg.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. lower--.f32 N/A

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

   9. count-2-rev N/A

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

   10. lower-+.f32 97.7

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

7. Applied rewrites 97.7%

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

8. Taylor expanded in uy around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

10. Applied rewrites 88.4%

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

11. Taylor expanded in maxCos around 0

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

    1. lower--.f32 N/A

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

    2. +-commutative N/A

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

    3. *-commutative N/A

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

    4. lower-fma.f32 N/A

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

    5. lower--.f32 N/A

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

    6. count-2-rev N/A

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

    7. lift-+.f32 88.4

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

13. Applied rewrites 88.4%

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

Alternative 13: 87.8% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.3%

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

2. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. +-commutative N/A

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

   5. associate-*r* N/A

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

   6. mul-1-neg N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-neg.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. lower--.f32 N/A

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

   12. lower--.f32 N/A

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

   13. count-2-rev N/A

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

   14. lower-+.f32 98.3

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

4. Applied rewrites 98.3%

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

5. Taylor expanded in maxCos around 0

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

   1. associate-+r+ N/A

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

   2. lower-+.f32 N/A

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

   3. lower-+.f32 N/A

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

   4. mul-1-neg N/A

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

   5. lift-neg.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. lower--.f32 N/A

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

   9. count-2-rev N/A

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

   10. lower-+.f32 97.7

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

7. Applied rewrites 97.7%

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

8. Taylor expanded in uy around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

10. Applied rewrites 88.4%

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

11. Taylor expanded in ux around 0

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

13. Applied rewrites 87.8%

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

Alternative 14: 87.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;maxCos \leq 7.599999776175537 \cdot 10^{-7}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), -1.3333333333333333, \pi + \pi\right) \cdot uy\right) \cdot \sqrt{\left(2 - ux\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - maxCos\right) - maxCos\right) \cdot ux} \cdot \left(\pi \cdot \left(uy + uy\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= maxCos 7.599999776175537e-7)
   (*
    (* (fma (* (* uy uy) (* (* PI PI) PI)) -1.3333333333333333 (+ PI PI)) uy)
    (sqrt (* (- 2.0 ux) ux)))
   (*
    (sqrt
     (*
      (- (- (fma (- ux) (* (- maxCos 1.0) (- maxCos 1.0)) 2.0) maxCos) maxCos)
      ux))
    (* PI (+ uy uy)))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (maxCos <= 7.599999776175537e-7f) {
		tmp = (fmaf(((uy * uy) * ((((float) M_PI) * ((float) M_PI)) * ((float) M_PI))), -1.3333333333333333f, (((float) M_PI) + ((float) M_PI))) * uy) * sqrtf(((2.0f - ux) * ux));
	} else {
		tmp = sqrtf((((fmaf(-ux, ((maxCos - 1.0f) * (maxCos - 1.0f)), 2.0f) - maxCos) - maxCos) * ux)) * (((float) M_PI) * (uy + uy));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (maxCos <= Float32(7.599999776175537e-7))
		tmp = Float32(Float32(fma(Float32(Float32(uy * uy) * Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi))), Float32(-1.3333333333333333), Float32(Float32(pi) + Float32(pi))) * uy) * sqrt(Float32(Float32(Float32(2.0) - ux) * ux)));
	else
		tmp = Float32(sqrt(Float32(Float32(Float32(fma(Float32(-ux), Float32(Float32(maxCos - Float32(1.0)) * Float32(maxCos - Float32(1.0))), Float32(2.0)) - maxCos) - maxCos) * ux)) * Float32(Float32(pi) * Float32(uy + uy)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if maxCos < 7.59999978e-7

   1. Initial program 57.3%

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

   2. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f32 N/A

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

      8. lower-neg.f32 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f32 N/A

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

      11. lower--.f32 N/A

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

      12. lower--.f32 N/A

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

      13. count-2-rev N/A

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

      14. lower-+.f32 98.3

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

   4. Applied rewrites 98.3%

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

   5. Taylor expanded in maxCos around 0

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

      1. associate-+r+ N/A

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

      2. lower-+.f32 N/A

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

      3. lower-+.f32 N/A

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

      4. mul-1-neg N/A

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

      5. lift-neg.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f32 N/A

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

      8. lower--.f32 N/A

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

      9. count-2-rev N/A

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

      10. lower-+.f32 97.7

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

   7. Applied rewrites 97.7%

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

   8. Taylor expanded in uy around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   10. Applied rewrites 88.4%

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

   11. Taylor expanded in maxCos around 0

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

       1. lower--.f32 83.9

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

   13. Applied rewrites 83.9%

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

   if 7.59999978e-7 < maxCos

   1. Initial program 57.3%

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

   2. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f32 N/A

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

      8. lower-neg.f32 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f32 N/A

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

      11. lower--.f32 N/A

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

      12. lower--.f32 N/A

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

      13. count-2-rev N/A

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

      14. lower-+.f32 98.3

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

   4. Applied rewrites 98.3%

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

   6. Applied rewrites 98.3%

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

   7. Taylor expanded in uy around 0

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

      1. count-2-rev N/A

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

      2. distribute-rgt-in N/A

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

      3. lift-+.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. lift-PI.f32 81.1

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

   9. Applied rewrites 81.1%

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

Alternative 15: 81.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.3%

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

2. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. +-commutative N/A

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

   5. associate-*r* N/A

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

   6. mul-1-neg N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-neg.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. lower--.f32 N/A

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

   12. lower--.f32 N/A

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

   13. count-2-rev N/A

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

   14. lower-+.f32 98.3

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

4. Applied rewrites 98.3%

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

6. Applied rewrites 98.3%

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

7. Taylor expanded in uy around 0

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

   1. count-2-rev N/A

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

   2. distribute-rgt-in N/A

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

   3. lift-+.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-PI.f32 81.1

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

9. Applied rewrites 81.1%

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

Alternative 16: 80.7% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.3%

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

2. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. +-commutative N/A

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

   5. associate-*r* N/A

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

   6. mul-1-neg N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-neg.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. lower--.f32 N/A

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

   12. lower--.f32 N/A

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

   13. count-2-rev N/A

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

   14. lower-+.f32 98.3

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

4. Applied rewrites 98.3%

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

5. Taylor expanded in maxCos around 0

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

   1. associate-+r+ N/A

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

   2. lower-+.f32 N/A

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

   3. lower-+.f32 N/A

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

   4. mul-1-neg N/A

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

   5. lift-neg.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. lower--.f32 N/A

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

   9. count-2-rev N/A

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

   10. lower-+.f32 97.7

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

7. Applied rewrites 97.7%

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

8. Taylor expanded in uy around 0

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

   1. associate-*r* N/A

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

   2. count-2-rev N/A

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

   3. lift-+.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-PI.f32 80.7

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

10. Applied rewrites 80.7%

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

Alternative 17: 75.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - ux\right) + ux \cdot maxCos\\ \mathbf{if}\;t\_0 \cdot t\_0 \leq 0.9995999932289124:\\ \;\;\;\;\left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\pi \cdot uy\right) \cdot \sqrt{\left(2 - \left(maxCos + maxCos\right)\right) \cdot ux}\right) \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))))
   (if (<= (* t_0 t_0) 0.9995999932289124)
     (* (* PI (+ uy uy)) (sqrt (- 1.0 (* (- 1.0 ux) (- 1.0 ux)))))
     (* (* (* PI uy) (sqrt (* (- 2.0 (+ maxCos maxCos)) ux))) 2.0))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = (1.0f - ux) + (ux * maxCos);
	float tmp;
	if ((t_0 * t_0) <= 0.9995999932289124f) {
		tmp = (((float) M_PI) * (uy + uy)) * sqrtf((1.0f - ((1.0f - ux) * (1.0f - ux))));
	} else {
		tmp = ((((float) M_PI) * uy) * sqrtf(((2.0f - (maxCos + maxCos)) * ux))) * 2.0f;
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(Float32(Float32(1.0) - ux) + Float32(ux * maxCos))
	tmp = Float32(0.0)
	if (Float32(t_0 * t_0) <= Float32(0.9995999932289124))
		tmp = Float32(Float32(Float32(pi) * Float32(uy + uy)) * sqrt(Float32(Float32(1.0) - Float32(Float32(Float32(1.0) - ux) * Float32(Float32(1.0) - ux)))));
	else
		tmp = Float32(Float32(Float32(Float32(pi) * uy) * sqrt(Float32(Float32(Float32(2.0) - Float32(maxCos + maxCos)) * ux))) * Float32(2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	t_0 = (single(1.0) - ux) + (ux * maxCos);
	tmp = single(0.0);
	if ((t_0 * t_0) <= single(0.9995999932289124))
		tmp = (single(pi) * (uy + uy)) * sqrt((single(1.0) - ((single(1.0) - ux) * (single(1.0) - ux))));
	else
		tmp = ((single(pi) * uy) * sqrt(((single(2.0) - (maxCos + maxCos)) * ux))) * single(2.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos))) < 0.999599993

   1. Initial program 57.3%

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

   2. Taylor expanded in uy around 0

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lift-*.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. lift-PI.f32 N/A

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

      7. lower-*.f32 N/A

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

      8. lift-*.f32 N/A

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

      9. lift-PI.f32 N/A

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

      10. lift-*.f32 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f32 N/A

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

      13. lift-PI.f32 N/A

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

      14. *-commutative N/A

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

      15. count-2-rev N/A

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

      16. lower-+.f32 N/A

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

      17. lower-sqrt.f32 N/A

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

      18. lower--.f32 N/A

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

   4. Applied rewrites 50.1%

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

   5. Taylor expanded in maxCos around 0

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

      1. lift--.f32 48.9

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

   7. Applied rewrites 48.9%

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

   8. Taylor expanded in maxCos around 0

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

      1. lift--.f32 48.7

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

   10. Applied rewrites 48.7%

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

   if 0.999599993 < (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)))

   1. Initial program 57.3%

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

   2. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{2 \cdot \left(\left(uy \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}\right)} \]
   3. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

      2. associate-*r* N/A

         \[\leadsto \left(\left(2 \cdot uy\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

      3. *-commutative N/A

         \[\leadsto \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{1} - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}} \]

      4. lift-*.f32 N/A

         \[\leadsto \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

      5. lift-*.f32 N/A

         \[\leadsto \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{1} - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}} \]

      6. lift-PI.f32 N/A

         \[\leadsto \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \color{blue}{{\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

      7. lower-*.f32 N/A

         \[\leadsto \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \color{blue}{\sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

      8. lift-*.f32 N/A

         \[\leadsto \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\color{blue}{1} - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}} \]

      9. lift-PI.f32 N/A

         \[\leadsto \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \color{blue}{{\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

      10. lift-*.f32 N/A

          \[\leadsto \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

      11. *-commutative N/A

          \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left(uy \cdot 2\right)\right) \cdot \sqrt{\color{blue}{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

      12. lower-*.f32 N/A

          \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left(uy \cdot 2\right)\right) \cdot \sqrt{\color{blue}{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

      13. lift-PI.f32 N/A

          \[\leadsto \left(\pi \cdot \left(uy \cdot 2\right)\right) \cdot \sqrt{\color{blue}{1} - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}} \]

      14. *-commutative N/A

          \[\leadsto \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{1 - \color{blue}{{\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

      15. count-2-rev N/A

          \[\leadsto \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{1 - \color{blue}{{\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

      16. lower-+.f32 N/A

          \[\leadsto \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{1 - \color{blue}{{\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

      17. lower-sqrt.f32 N/A

          \[\leadsto \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}} \]

      18. lower--.f32 N/A

          \[\leadsto \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}} \]

   4. Applied rewrites 50.1%

      \[\leadsto \color{blue}{\left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{1 - \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right) \cdot \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right)}} \]

   5. Taylor expanded in ux around 0

      \[\leadsto 2 \cdot \color{blue}{\left(\sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)} \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)} \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot 2 \]

      2. lower-*.f32 N/A

         \[\leadsto \left(\sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)} \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot 2 \]

      3. *-commutative N/A

         \[\leadsto \left(\left(uy \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\right) \cdot 2 \]

      4. lower-*.f32 N/A

         \[\leadsto \left(\left(uy \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\right) \cdot 2 \]

      5. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\right) \cdot 2 \]

      6. lower-*.f32 N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\right) \cdot 2 \]

      7. lift-PI.f32 N/A

         \[\leadsto \left(\left(\pi \cdot uy\right) \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\right) \cdot 2 \]

      8. lower-sqrt.f32 N/A

         \[\leadsto \left(\left(\pi \cdot uy\right) \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\right) \cdot 2 \]

      9. *-commutative N/A

         \[\leadsto \left(\left(\pi \cdot uy\right) \cdot \sqrt{\left(2 - 2 \cdot maxCos\right) \cdot ux}\right) \cdot 2 \]

      10. lower-*.f32 N/A

          \[\leadsto \left(\left(\pi \cdot uy\right) \cdot \sqrt{\left(2 - 2 \cdot maxCos\right) \cdot ux}\right) \cdot 2 \]

      11. lower--.f32 N/A

          \[\leadsto \left(\left(\pi \cdot uy\right) \cdot \sqrt{\left(2 - 2 \cdot maxCos\right) \cdot ux}\right) \cdot 2 \]

      12. count-2-rev N/A

          \[\leadsto \left(\left(\pi \cdot uy\right) \cdot \sqrt{\left(2 - \left(maxCos + maxCos\right)\right) \cdot ux}\right) \cdot 2 \]

      13. lift-+.f32 65.9

          \[\leadsto \left(\left(\pi \cdot uy\right) \cdot \sqrt{\left(2 - \left(maxCos + maxCos\right)\right) \cdot ux}\right) \cdot 2 \]

   7. Applied rewrites 65.9%

      \[\leadsto \left(\left(\pi \cdot uy\right) \cdot \sqrt{\left(2 - \left(maxCos + maxCos\right)\right) \cdot ux}\right) \cdot \color{blue}{2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 65.9% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \left(\left(\pi \cdot uy\right) \cdot \sqrt{\left(2 - \left(maxCos + maxCos\right)\right) \cdot ux}\right) \cdot 2 \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (* (* (* PI uy) (sqrt (* (- 2.0 (+ maxCos maxCos)) ux))) 2.0))

C

float code(float ux, float uy, float maxCos) {
	return ((((float) M_PI) * uy) * sqrtf(((2.0f - (maxCos + maxCos)) * ux))) * 2.0f;
}

Julia

function code(ux, uy, maxCos)
	return Float32(Float32(Float32(Float32(pi) * uy) * sqrt(Float32(Float32(Float32(2.0) - Float32(maxCos + maxCos)) * ux))) * Float32(2.0))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = ((single(pi) * uy) * sqrt(((single(2.0) - (maxCos + maxCos)) * ux))) * single(2.0);
end

Derivation

1. Initial program 57.3%

   \[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

2. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{2 \cdot \left(\left(uy \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}\right)} \]
3. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

   2. associate-*r* N/A

      \[\leadsto \left(\left(2 \cdot uy\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

   3. *-commutative N/A

      \[\leadsto \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{1} - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}} \]

   4. lift-*.f32 N/A

      \[\leadsto \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

   5. lift-*.f32 N/A

      \[\leadsto \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{1} - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}} \]

   6. lift-PI.f32 N/A

      \[\leadsto \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \color{blue}{{\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

   7. lower-*.f32 N/A

      \[\leadsto \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \color{blue}{\sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

   8. lift-*.f32 N/A

      \[\leadsto \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\color{blue}{1} - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}} \]

   9. lift-PI.f32 N/A

      \[\leadsto \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \color{blue}{{\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

   10. lift-*.f32 N/A

       \[\leadsto \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

   11. *-commutative N/A

       \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left(uy \cdot 2\right)\right) \cdot \sqrt{\color{blue}{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

   12. lower-*.f32 N/A

       \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left(uy \cdot 2\right)\right) \cdot \sqrt{\color{blue}{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

   13. lift-PI.f32 N/A

       \[\leadsto \left(\pi \cdot \left(uy \cdot 2\right)\right) \cdot \sqrt{\color{blue}{1} - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}} \]

   14. *-commutative N/A

       \[\leadsto \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{1 - \color{blue}{{\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

   15. count-2-rev N/A

       \[\leadsto \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{1 - \color{blue}{{\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

   16. lower-+.f32 N/A

       \[\leadsto \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{1 - \color{blue}{{\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

   17. lower-sqrt.f32 N/A

       \[\leadsto \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}} \]

   18. lower--.f32 N/A

       \[\leadsto \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}} \]

4. Applied rewrites 50.1%

   \[\leadsto \color{blue}{\left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{1 - \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right) \cdot \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right)}} \]

5. Taylor expanded in ux around 0

   \[\leadsto 2 \cdot \color{blue}{\left(\sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)} \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)} \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot 2 \]

   2. lower-*.f32 N/A

      \[\leadsto \left(\sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)} \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot 2 \]

   3. *-commutative N/A

      \[\leadsto \left(\left(uy \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\right) \cdot 2 \]

   4. lower-*.f32 N/A

      \[\leadsto \left(\left(uy \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\right) \cdot 2 \]

   5. *-commutative N/A

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\right) \cdot 2 \]

   6. lower-*.f32 N/A

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\right) \cdot 2 \]

   7. lift-PI.f32 N/A

      \[\leadsto \left(\left(\pi \cdot uy\right) \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\right) \cdot 2 \]

   8. lower-sqrt.f32 N/A

      \[\leadsto \left(\left(\pi \cdot uy\right) \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\right) \cdot 2 \]

   9. *-commutative N/A

      \[\leadsto \left(\left(\pi \cdot uy\right) \cdot \sqrt{\left(2 - 2 \cdot maxCos\right) \cdot ux}\right) \cdot 2 \]

   10. lower-*.f32 N/A

       \[\leadsto \left(\left(\pi \cdot uy\right) \cdot \sqrt{\left(2 - 2 \cdot maxCos\right) \cdot ux}\right) \cdot 2 \]

   11. lower--.f32 N/A

       \[\leadsto \left(\left(\pi \cdot uy\right) \cdot \sqrt{\left(2 - 2 \cdot maxCos\right) \cdot ux}\right) \cdot 2 \]

   12. count-2-rev N/A

       \[\leadsto \left(\left(\pi \cdot uy\right) \cdot \sqrt{\left(2 - \left(maxCos + maxCos\right)\right) \cdot ux}\right) \cdot 2 \]

   13. lift-+.f32 65.9

       \[\leadsto \left(\left(\pi \cdot uy\right) \cdot \sqrt{\left(2 - \left(maxCos + maxCos\right)\right) \cdot ux}\right) \cdot 2 \]

7. Applied rewrites 65.9%

   \[\leadsto \left(\left(\pi \cdot uy\right) \cdot \sqrt{\left(2 - \left(maxCos + maxCos\right)\right) \cdot ux}\right) \cdot \color{blue}{2} \]
8. Add Preprocessing

Alternative 19: 63.2% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(uy + uy\right) \cdot \left(\sqrt{2} \cdot \pi\right)\right) \cdot \sqrt{ux} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (* (* (+ uy uy) (* (sqrt 2.0) PI)) (sqrt ux)))

C

float code(float ux, float uy, float maxCos) {
	return ((uy + uy) * (sqrtf(2.0f) * ((float) M_PI))) * sqrtf(ux);
}

Julia

function code(ux, uy, maxCos)
	return Float32(Float32(Float32(uy + uy) * Float32(sqrt(Float32(2.0)) * Float32(pi))) * sqrt(ux))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = ((uy + uy) * (sqrt(single(2.0)) * single(pi))) * sqrt(ux);
end

Derivation

1. Initial program 57.3%

   \[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

2. Taylor expanded in maxCos around 0

   \[\leadsto \color{blue}{\sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{1 - {\left(1 - ux\right)}^{2}}} \]
3. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \sqrt{1 - {\left(1 - ux\right)}^{2}} \cdot \color{blue}{\sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \]

   2. lower-*.f32 N/A

      \[\leadsto \sqrt{1 - {\left(1 - ux\right)}^{2}} \cdot \color{blue}{\sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \]

   3. lower-sqrt.f32 N/A

      \[\leadsto \sqrt{1 - {\left(1 - ux\right)}^{2}} \cdot \sin \color{blue}{\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \]

   4. lower--.f32 N/A

      \[\leadsto \sqrt{1 - {\left(1 - ux\right)}^{2}} \cdot \sin \left(\color{blue}{2} \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \]

   5. unpow2 N/A

      \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \]

   6. lower-*.f32 N/A

      \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \]

   7. lift--.f32 N/A

      \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \]

   8. lift--.f32 N/A

      \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \]

   9. associate-*r* N/A

      \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \sin \left(\left(2 \cdot uy\right) \cdot \mathsf{PI}\left(\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \]

   11. lift-*.f32 N/A

       \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \]

   12. lift-*.f32 N/A

       \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \]

   13. lift-PI.f32 N/A

       \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \]

   14. lift-sin.f32 55.4

       \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \]

   15. lift-*.f32 N/A

       \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \]

   16. lift-PI.f32 N/A

       \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \]

4. Applied rewrites 55.4%

   \[\leadsto \color{blue}{\sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right)} \]

5. Taylor expanded in ux around 0

   \[\leadsto \sqrt{ux} \cdot \color{blue}{\left(\sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{2}\right)} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{2}\right) \cdot \sqrt{ux} \]

   2. lower-*.f32 N/A

      \[\leadsto \left(\sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{2}\right) \cdot \sqrt{ux} \]

   3. *-commutative N/A

      \[\leadsto \left(\sqrt{2} \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \sqrt{ux} \]

   4. lower-*.f32 N/A

      \[\leadsto \left(\sqrt{2} \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \sqrt{ux} \]

   5. lower-sqrt.f32 N/A

      \[\leadsto \left(\sqrt{2} \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \sqrt{ux} \]

   6. lower-sin.f32 N/A

      \[\leadsto \left(\sqrt{2} \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \sqrt{ux} \]

   7. associate-*r* N/A

      \[\leadsto \left(\sqrt{2} \cdot \sin \left(\left(2 \cdot uy\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{ux} \]

   8. count-2-rev N/A

      \[\leadsto \left(\sqrt{2} \cdot \sin \left(\left(uy + uy\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{ux} \]

   9. lift-+.f32 N/A

      \[\leadsto \left(\sqrt{2} \cdot \sin \left(\left(uy + uy\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{ux} \]

   10. lower-*.f32 N/A

       \[\leadsto \left(\sqrt{2} \cdot \sin \left(\left(uy + uy\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{ux} \]

   11. lift-PI.f32 N/A

       \[\leadsto \left(\sqrt{2} \cdot \sin \left(\left(uy + uy\right) \cdot \pi\right)\right) \cdot \sqrt{ux} \]

   12. lower-sqrt.f32 73.1

       \[\leadsto \left(\sqrt{2} \cdot \sin \left(\left(uy + uy\right) \cdot \pi\right)\right) \cdot \sqrt{ux} \]

7. Applied rewrites 73.1%

   \[\leadsto \left(\sqrt{2} \cdot \sin \left(\left(uy + uy\right) \cdot \pi\right)\right) \cdot \color{blue}{\sqrt{ux}} \]

8. Taylor expanded in uy around 0

   \[\leadsto \left(2 \cdot \left(uy \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)\right)\right) \cdot \sqrt{ux} \]
9. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \left(\left(2 \cdot uy\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)\right) \cdot \sqrt{ux} \]

   2. count-2-rev N/A

      \[\leadsto \left(\left(uy + uy\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)\right) \cdot \sqrt{ux} \]

   3. lift-+.f32 N/A

      \[\leadsto \left(\left(uy + uy\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)\right) \cdot \sqrt{ux} \]

   4. lower-*.f32 N/A

      \[\leadsto \left(\left(uy + uy\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)\right) \cdot \sqrt{ux} \]

   5. *-commutative N/A

      \[\leadsto \left(\left(uy + uy\right) \cdot \left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{ux} \]

   6. lower-*.f32 N/A

      \[\leadsto \left(\left(uy + uy\right) \cdot \left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{ux} \]

   7. lift-sqrt.f32 N/A

      \[\leadsto \left(\left(uy + uy\right) \cdot \left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{ux} \]

   8. lift-PI.f32 63.2

      \[\leadsto \left(\left(uy + uy\right) \cdot \left(\sqrt{2} \cdot \pi\right)\right) \cdot \sqrt{ux} \]

10. Applied rewrites 63.2%

    \[\leadsto \left(\left(uy + uy\right) \cdot \left(\sqrt{2} \cdot \pi\right)\right) \cdot \sqrt{ux} \]
11. Add Preprocessing

Alternative 20: 63.2% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(\left(uy + uy\right) \cdot \sqrt{2}\right) \cdot \pi\right) \cdot \sqrt{ux} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (* (* (* (+ uy uy) (sqrt 2.0)) PI) (sqrt ux)))

C

float code(float ux, float uy, float maxCos) {
	return (((uy + uy) * sqrtf(2.0f)) * ((float) M_PI)) * sqrtf(ux);
}

Julia

function code(ux, uy, maxCos)
	return Float32(Float32(Float32(Float32(uy + uy) * sqrt(Float32(2.0))) * Float32(pi)) * sqrt(ux))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = (((uy + uy) * sqrt(single(2.0))) * single(pi)) * sqrt(ux);
end

Derivation

1. Initial program 57.3%

   \[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

2. Taylor expanded in maxCos around 0

   \[\leadsto \color{blue}{\sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{1 - {\left(1 - ux\right)}^{2}}} \]
3. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \sqrt{1 - {\left(1 - ux\right)}^{2}} \cdot \color{blue}{\sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \]

   2. lower-*.f32 N/A

      \[\leadsto \sqrt{1 - {\left(1 - ux\right)}^{2}} \cdot \color{blue}{\sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \]

   3. lower-sqrt.f32 N/A

      \[\leadsto \sqrt{1 - {\left(1 - ux\right)}^{2}} \cdot \sin \color{blue}{\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \]

   4. lower--.f32 N/A

      \[\leadsto \sqrt{1 - {\left(1 - ux\right)}^{2}} \cdot \sin \left(\color{blue}{2} \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \]

   5. unpow2 N/A

      \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \]

   6. lower-*.f32 N/A

      \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \]

   7. lift--.f32 N/A

      \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \]

   8. lift--.f32 N/A

      \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \]

   9. associate-*r* N/A

      \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \sin \left(\left(2 \cdot uy\right) \cdot \mathsf{PI}\left(\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \]

   11. lift-*.f32 N/A

       \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \]

   12. lift-*.f32 N/A

       \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \]

   13. lift-PI.f32 N/A

       \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \]

   14. lift-sin.f32 55.4

       \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \]

   15. lift-*.f32 N/A

       \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \]

   16. lift-PI.f32 N/A

       \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \]

4. Applied rewrites 55.4%

   \[\leadsto \color{blue}{\sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right)} \]

5. Taylor expanded in ux around 0

   \[\leadsto \sqrt{ux} \cdot \color{blue}{\left(\sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{2}\right)} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{2}\right) \cdot \sqrt{ux} \]

   2. lower-*.f32 N/A

      \[\leadsto \left(\sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{2}\right) \cdot \sqrt{ux} \]

   3. *-commutative N/A

      \[\leadsto \left(\sqrt{2} \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \sqrt{ux} \]

   4. lower-*.f32 N/A

      \[\leadsto \left(\sqrt{2} \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \sqrt{ux} \]

   5. lower-sqrt.f32 N/A

      \[\leadsto \left(\sqrt{2} \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \sqrt{ux} \]

   6. lower-sin.f32 N/A

      \[\leadsto \left(\sqrt{2} \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \sqrt{ux} \]

   7. associate-*r* N/A

      \[\leadsto \left(\sqrt{2} \cdot \sin \left(\left(2 \cdot uy\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{ux} \]

   8. count-2-rev N/A

      \[\leadsto \left(\sqrt{2} \cdot \sin \left(\left(uy + uy\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{ux} \]

   9. lift-+.f32 N/A

      \[\leadsto \left(\sqrt{2} \cdot \sin \left(\left(uy + uy\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{ux} \]

   10. lower-*.f32 N/A

       \[\leadsto \left(\sqrt{2} \cdot \sin \left(\left(uy + uy\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{ux} \]

   11. lift-PI.f32 N/A

       \[\leadsto \left(\sqrt{2} \cdot \sin \left(\left(uy + uy\right) \cdot \pi\right)\right) \cdot \sqrt{ux} \]

   12. lower-sqrt.f32 73.1

       \[\leadsto \left(\sqrt{2} \cdot \sin \left(\left(uy + uy\right) \cdot \pi\right)\right) \cdot \sqrt{ux} \]

7. Applied rewrites 73.1%

   \[\leadsto \left(\sqrt{2} \cdot \sin \left(\left(uy + uy\right) \cdot \pi\right)\right) \cdot \color{blue}{\sqrt{ux}} \]

8. Taylor expanded in uy around 0

   \[\leadsto \left(2 \cdot \left(uy \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)\right)\right) \cdot \sqrt{ux} \]
9. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \left(\left(2 \cdot uy\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)\right) \cdot \sqrt{ux} \]

   2. count-2-rev N/A

      \[\leadsto \left(\left(uy + uy\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)\right) \cdot \sqrt{ux} \]

   3. lift-+.f32 N/A

      \[\leadsto \left(\left(uy + uy\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)\right) \cdot \sqrt{ux} \]

   4. lower-*.f32 N/A

      \[\leadsto \left(\left(uy + uy\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)\right) \cdot \sqrt{ux} \]

   5. *-commutative N/A

      \[\leadsto \left(\left(uy + uy\right) \cdot \left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{ux} \]

   6. lower-*.f32 N/A

      \[\leadsto \left(\left(uy + uy\right) \cdot \left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{ux} \]

   7. lift-sqrt.f32 N/A

      \[\leadsto \left(\left(uy + uy\right) \cdot \left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{ux} \]

   8. lift-PI.f32 63.2

      \[\leadsto \left(\left(uy + uy\right) \cdot \left(\sqrt{2} \cdot \pi\right)\right) \cdot \sqrt{ux} \]

10. Applied rewrites 63.2%

    \[\leadsto \left(\left(uy + uy\right) \cdot \left(\sqrt{2} \cdot \pi\right)\right) \cdot \sqrt{ux} \]
11. Step-by-step derivation

    1. lift-*.f32 N/A

       \[\leadsto \left(\left(uy + uy\right) \cdot \left(\sqrt{2} \cdot \pi\right)\right) \cdot \sqrt{ux} \]

    2. lift-PI.f32 N/A

       \[\leadsto \left(\left(uy + uy\right) \cdot \left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{ux} \]

    3. lift-*.f32 N/A

       \[\leadsto \left(\left(uy + uy\right) \cdot \left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{ux} \]

    4. lift-sqrt.f32 N/A

       \[\leadsto \left(\left(uy + uy\right) \cdot \left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{ux} \]

    5. associate-*r* N/A

       \[\leadsto \left(\left(\left(uy + uy\right) \cdot \sqrt{2}\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux} \]

    6. lower-*.f32 N/A

       \[\leadsto \left(\left(\left(uy + uy\right) \cdot \sqrt{2}\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux} \]

    7. lower-*.f32 N/A

       \[\leadsto \left(\left(\left(uy + uy\right) \cdot \sqrt{2}\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux} \]

    8. lift-sqrt.f32 N/A

       \[\leadsto \left(\left(\left(uy + uy\right) \cdot \sqrt{2}\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux} \]

    9. lift-PI.f32 63.2

       \[\leadsto \left(\left(\left(uy + uy\right) \cdot \sqrt{2}\right) \cdot \pi\right) \cdot \sqrt{ux} \]

12. Applied rewrites 63.2%

    \[\leadsto \left(\left(\left(uy + uy\right) \cdot \sqrt{2}\right) \cdot \pi\right) \cdot \sqrt{ux} \]
13. Add Preprocessing

Alternative 21: 7.1% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{1 - 1} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (* (* PI (+ uy uy)) (sqrt (- 1.0 1.0))))

C

float code(float ux, float uy, float maxCos) {
	return (((float) M_PI) * (uy + uy)) * sqrtf((1.0f - 1.0f));
}

Julia

function code(ux, uy, maxCos)
	return Float32(Float32(Float32(pi) * Float32(uy + uy)) * sqrt(Float32(Float32(1.0) - Float32(1.0))))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = (single(pi) * (uy + uy)) * sqrt((single(1.0) - single(1.0)));
end

Derivation

1. Initial program 57.3%

   \[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

2. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{2 \cdot \left(\left(uy \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}\right)} \]
3. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

   2. associate-*r* N/A

      \[\leadsto \left(\left(2 \cdot uy\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

   3. *-commutative N/A

      \[\leadsto \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{1} - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}} \]

   4. lift-*.f32 N/A

      \[\leadsto \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

   5. lift-*.f32 N/A

      \[\leadsto \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{1} - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}} \]

   6. lift-PI.f32 N/A

      \[\leadsto \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \color{blue}{{\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

   7. lower-*.f32 N/A

      \[\leadsto \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \color{blue}{\sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

   8. lift-*.f32 N/A

      \[\leadsto \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\color{blue}{1} - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}} \]

   9. lift-PI.f32 N/A

      \[\leadsto \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \color{blue}{{\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

   10. lift-*.f32 N/A

       \[\leadsto \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

   11. *-commutative N/A

       \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left(uy \cdot 2\right)\right) \cdot \sqrt{\color{blue}{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

   12. lower-*.f32 N/A

       \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left(uy \cdot 2\right)\right) \cdot \sqrt{\color{blue}{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

   13. lift-PI.f32 N/A

       \[\leadsto \left(\pi \cdot \left(uy \cdot 2\right)\right) \cdot \sqrt{\color{blue}{1} - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}} \]

   14. *-commutative N/A

       \[\leadsto \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{1 - \color{blue}{{\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

   15. count-2-rev N/A

       \[\leadsto \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{1 - \color{blue}{{\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

   16. lower-+.f32 N/A

       \[\leadsto \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{1 - \color{blue}{{\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

   17. lower-sqrt.f32 N/A

       \[\leadsto \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}} \]

   18. lower--.f32 N/A

       \[\leadsto \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}} \]

4. Applied rewrites 50.1%

   \[\leadsto \color{blue}{\left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{1 - \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right) \cdot \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right)}} \]

5. Taylor expanded in ux around 0

   \[\leadsto \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{1 - 1} \]
6. Step-by-step derivation

7. Applied rewrites 7.1%

   \[\leadsto \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{1 - 1} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025131 
(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)))))))