UniformSampleCone, y

Percentage Accurate: 58.0% → 98.3%

Time: 5.7s

Alternatives: 13

Speedup: 1.1×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 58.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 58.0%

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

2. 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. lower-pow.f32 N/A

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

   10. lower--.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 98.3

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

4. Applied rewrites 98.3%

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

6. Applied rewrites 98.3%

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

7. Taylor expanded in maxCos around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. lower--.f32 98.3

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

9. Applied rewrites 98.3%

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

Alternative 2: 82.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - ux\right) + ux \cdot maxCos\\ \mathbf{if}\;t\_0 \cdot t\_0 \leq 0.9996600151062012:\\ \;\;\;\;\left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{1 - \left(\frac{1 - ux \cdot ux}{1 + ux} + ux \cdot maxCos\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{2 \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))))
   (if (<= (* t_0 t_0) 0.9996600151062012)
     (*
      (* PI (* 2.0 uy))
      (sqrt
       (- 1.0 (* (+ (/ (- 1.0 (* ux ux)) (+ 1.0 ux)) (* ux maxCos)) t_0))))
     (* (sin (* (* uy 2.0) PI)) (sqrt (* 2.0 ux))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = (1.0f - ux) + (ux * maxCos);
	float tmp;
	if ((t_0 * t_0) <= 0.9996600151062012f) {
		tmp = (((float) M_PI) * (2.0f * uy)) * sqrtf((1.0f - ((((1.0f - (ux * ux)) / (1.0f + ux)) + (ux * maxCos)) * t_0)));
	} else {
		tmp = sinf(((uy * 2.0f) * ((float) M_PI))) * sqrtf((2.0f * ux));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(Float32(Float32(1.0) - ux) + Float32(ux * maxCos))
	tmp = Float32(0.0)
	if (Float32(t_0 * t_0) <= Float32(0.9996600151062012))
		tmp = Float32(Float32(Float32(pi) * Float32(Float32(2.0) * uy)) * sqrt(Float32(Float32(1.0) - Float32(Float32(Float32(Float32(Float32(1.0) - Float32(ux * ux)) / Float32(Float32(1.0) + ux)) + Float32(ux * maxCos)) * t_0))));
	else
		tmp = Float32(sin(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) * sqrt(Float32(Float32(2.0) * ux)));
	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(0.9996600151062012))
		tmp = (single(pi) * (single(2.0) * uy)) * sqrt((single(1.0) - ((((single(1.0) - (ux * ux)) / (single(1.0) + ux)) + (ux * maxCos)) * t_0)));
	else
		tmp = sin(((uy * single(2.0)) * single(pi))) * sqrt((single(2.0) * ux));
	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))) < 0.999660015

   1. Initial program 89.2%

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

   2. Taylor expanded in uy around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f32 75.4

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

   4. Applied rewrites 75.4%

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

      1. lift--.f32 N/A

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

      2. flip-- N/A

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

      3. lower-/.f32 N/A

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

      4. metadata-eval N/A

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

      5. unpow2 N/A

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

      6. lower--.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 N/A

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

      9. lower-+.f32 75.4

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

   6. Applied rewrites 75.4%

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

   if 0.999660015 < (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)))

   1. Initial program 37.2%

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

   2. Taylor expanded in 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. lower-pow.f32 N/A

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

      10. lower--.f32 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f32 98.3

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

   4. Applied rewrites 98.3%

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

   5. Taylor expanded in maxCos around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f32 92.0

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

   7. Applied rewrites 92.0%

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

   8. Taylor expanded in ux around 0

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

   10. Applied rewrites 86.8%

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

Alternative 3: 97.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 58.0%

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

2. 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. lower-pow.f32 N/A

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

   10. lower--.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 98.3

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

4. Applied rewrites 98.3%

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

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. +-commutative N/A

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

   2. lower-+.f32 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. lower--.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. mul-1-neg N/A

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

   10. lift-neg.f32 97.7

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

7. Applied rewrites 97.7%

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

Alternative 4: 96.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 58.0%

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

2. 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. lower-pow.f32 N/A

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

   10. lower--.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 98.3

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

4. Applied rewrites 98.3%

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

5. Taylor expanded in maxCos around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f32 96.9

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

7. Applied rewrites 96.9%

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

Alternative 5: 92.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.0%

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

2. 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. lower-pow.f32 N/A

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

   10. lower--.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 98.3

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

4. Applied rewrites 98.3%

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

5. Taylor expanded in maxCos around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f32 92.2

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

7. Applied rewrites 92.2%

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

   1. lift-fma.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. mul-1-neg N/A

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

   4. lift-neg.f32 92.2

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

9. Applied rewrites 92.2%

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

Alternative 6: 51.6% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.0%

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

2. Taylor expanded in uy around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 50.9

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

4. Applied rewrites 50.9%

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

5. Taylor expanded in ux around inf

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

   1. *-commutative N/A

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

   2. +-commutative N/A

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

   3. associate-+r- N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lower--.f32 N/A

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

   7. +-commutative N/A

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

   8. lower-+.f32 N/A

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

   9. lift-/.f32 51.6

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

7. Applied rewrites 51.6%

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

Alternative 7: 51.0% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.0%

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

2. Taylor expanded in uy around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 50.9

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

4. Applied rewrites 50.9%

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

5. Taylor expanded in maxCos around inf

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

   1. *-commutative N/A

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

   2. +-commutative N/A

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

   3. associate-+r- N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. associate--l+ N/A

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

   7. div-sub N/A

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

   8. lower-+.f32 N/A

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

   9. lower-/.f32 N/A

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

   10. lift--.f32 51.0

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

7. Applied rewrites 51.0%

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

Alternative 8: 50.9% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 58.0%

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

2. Taylor expanded in uy around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 50.9

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

4. Applied rewrites 50.9%

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

   1. lift-*.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. lift--.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-+.f32 N/A

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

   6. lift--.f32 N/A

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

   7. distribute-lft-in N/A

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

   8. lower-fma.f32 N/A

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

   9. *-commutative N/A

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

   10. +-commutative N/A

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

   11. lift-fma.f32 N/A

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

   12. lift--.f32 N/A

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

   13. lift--.f32 N/A

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

   14. lift-*.f32 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f32 N/A

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

   17. *-commutative N/A

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

   18. +-commutative N/A

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

   19. lift-fma.f32 N/A

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

   20. lift--.f32 N/A

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

   21. lower-*.f32 50.9

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

6. Applied rewrites 50.9%

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

Alternative 9: 51.0% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.0%

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

2. Taylor expanded in uy around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 50.9

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

4. Applied rewrites 50.9%

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

   1. lift-+.f32 N/A

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

   2. lift--.f32 N/A

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

   3. associate-+l- N/A

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

   4. lower--.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower--.f32 N/A

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

   8. lower-*.f32 51.0

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

6. Applied rewrites 51.0%

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

Alternative 10: 50.9% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 58.0%

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

2. Taylor expanded in uy around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 50.9

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

4. Applied rewrites 50.9%

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

   1. lift-+.f32 N/A

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

   2. lift--.f32 N/A

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

   3. +-commutative N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. lift-fma.f32 N/A

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

   7. lift--.f32 50.9

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

   8. lift-+.f32 N/A

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

   9. lift--.f32 N/A

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

   10. +-commutative N/A

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

   11. lift-*.f32 N/A

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

   12. *-commutative N/A

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

   13. lift-fma.f32 N/A

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

   14. lift--.f32 50.9

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

6. Applied rewrites 50.9%

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

Alternative 11: 49.5% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.0%

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

2. Taylor expanded in uy around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 50.9

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

4. Applied rewrites 50.9%

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

5. Taylor expanded in maxCos around 0

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

   1. *-commutative N/A

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

   2. +-commutative N/A

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

   3. associate-+r- N/A

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

   4. lift--.f32 49.5

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

7. Applied rewrites 49.5%

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

Alternative 12: 41.9% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 58.0%

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

2. Taylor expanded in uy around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 50.9

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

4. Applied rewrites 50.9%

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

5. Taylor expanded in ux around 0

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

7. Applied rewrites 7.1%

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

   1. lift-*.f32 N/A

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

   2. count-2-rev N/A

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

   3. lower-+.f32 7.1

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

9. Applied rewrites 7.1%

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

10. Taylor expanded in ux around 0

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

    1. +-commutative N/A

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

    2. *-commutative N/A

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

    3. associate--l+ N/A

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

    4. +-commutative N/A

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

    5. *-commutative N/A

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

    6. lower-fma.f32 N/A

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

    7. lower--.f32 N/A

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

    8. lower-*.f32 41.9

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

12. Applied rewrites 41.9%

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

Alternative 13: 7.1% accurate, 5.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.0%

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

2. Taylor expanded in uy around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 50.9

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

4. Applied rewrites 50.9%

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

5. Taylor expanded in ux around 0

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

7. Applied rewrites 7.1%

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

   1. lift-*.f32 N/A

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

   2. count-2-rev N/A

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

   3. lower-+.f32 7.1

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

9. Applied rewrites 7.1%

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

Reproduce

herbie shell --seed 2025105 
(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)))))))