UniformSampleCone, y

Percentage Accurate: 57.1% → 98.2%

Time: 14.1s

Alternatives: 22

Speedup: 4.2×

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: 57.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.2% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 55.2%

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

3. Taylor expanded in ux around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

5. Applied rewrites 98.4%

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

7. Applied rewrites 98.4%

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

8. Final simplification 98.4%

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

Alternative 2: 96.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 0.0149999997

   1. Initial program 55.1%

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

   3. Taylor expanded in ux around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   5. Applied rewrites 98.4%

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

   6. Taylor expanded in uy around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f32 N/A

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

      8. unpow3 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-*.f32 N/A

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

      11. lower-PI.f32 N/A

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

      12. lower-PI.f32 N/A

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

      13. lower-PI.f32 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f32 N/A

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

      16. lower-PI.f32 98.6

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

   8. Applied rewrites 98.6%

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

   if 0.0149999997 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 55.5%

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

   3. Taylor expanded in ux around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   5. Applied rewrites 98.1%

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

   7. Applied rewrites 98.1%

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

   8. Taylor expanded in maxCos around 0

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

   10. Applied rewrites 92.6%

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

4. Final simplification 97.2%

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

Alternative 3: 96.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 0.0149999997

   1. Initial program 55.1%

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

   3. Taylor expanded in ux around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   5. Applied rewrites 98.4%

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

   6. Taylor expanded in uy around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f32 N/A

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

      8. unpow3 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-*.f32 N/A

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

      11. lower-PI.f32 N/A

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

      12. lower-PI.f32 N/A

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

      13. lower-PI.f32 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f32 N/A

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

      16. lower-PI.f32 98.6

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

   8. Applied rewrites 98.6%

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

   if 0.0149999997 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 55.5%

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

   3. Taylor expanded in ux around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   5. Applied rewrites 98.1%

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

   6. Taylor expanded in maxCos around 0

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

   8. Applied rewrites 92.5%

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\frac{2}{ux} - 1\right) \cdot \color{blue}{\left(ux \cdot ux\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.2%

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

Alternative 4: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 55.2%

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

3. Taylor expanded in ux around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

5. Applied rewrites 98.4%

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

7. Applied rewrites 98.4%

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

8. Taylor expanded in maxCos around 0

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

10. Applied rewrites 98.3%

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

11. Final simplification 98.3%

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

Alternative 5: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 55.2%

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

3. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

5. Applied rewrites 98.3%

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

6. Final simplification 98.3%

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

Alternative 6: 94.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.0560000017285347:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(uy \cdot uy\right) \cdot -1.3333333333333333, \left(\pi \cdot \pi\right) \cdot \pi, \pi \cdot 2\right) \cdot uy\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, maxCos - 1, \frac{\mathsf{fma}\left(maxCos, -2, 2\right)}{ux}\right) \cdot \left(ux \cdot ux\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(4 - \left(\left(maxCos + 2\right) + maxCos\right)\right) \cdot ux} \cdot \sin \left(\pi \cdot \left(2 \cdot uy\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* 2.0 uy) 0.0560000017285347)
   (*
    (* (fma (* (* uy uy) -1.3333333333333333) (* (* PI PI) PI) (* PI 2.0)) uy)
    (sqrt
     (*
      (fma (- 1.0 maxCos) (- maxCos 1.0) (/ (fma maxCos -2.0 2.0) ux))
      (* ux ux))))
   (*
    (sqrt (* (- 4.0 (+ (+ maxCos 2.0) maxCos)) ux))
    (sin (* PI (* 2.0 uy))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if ((2.0f * uy) <= 0.0560000017285347f) {
		tmp = (fmaf(((uy * uy) * -1.3333333333333333f), ((((float) M_PI) * ((float) M_PI)) * ((float) M_PI)), (((float) M_PI) * 2.0f)) * uy) * sqrtf((fmaf((1.0f - maxCos), (maxCos - 1.0f), (fmaf(maxCos, -2.0f, 2.0f) / ux)) * (ux * ux)));
	} else {
		tmp = sqrtf(((4.0f - ((maxCos + 2.0f) + maxCos)) * ux)) * sinf((((float) M_PI) * (2.0f * uy)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (Float32(Float32(2.0) * uy) <= Float32(0.0560000017285347))
		tmp = Float32(Float32(fma(Float32(Float32(uy * uy) * Float32(-1.3333333333333333)), Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi)), Float32(Float32(pi) * Float32(2.0))) * uy) * sqrt(Float32(fma(Float32(Float32(1.0) - maxCos), Float32(maxCos - Float32(1.0)), Float32(fma(maxCos, Float32(-2.0), Float32(2.0)) / ux)) * Float32(ux * ux))));
	else
		tmp = Float32(sqrt(Float32(Float32(Float32(4.0) - Float32(Float32(maxCos + Float32(2.0)) + maxCos)) * ux)) * sin(Float32(Float32(pi) * Float32(Float32(2.0) * uy))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 0.0560000017

   1. Initial program 54.9%

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

   3. Taylor expanded in ux around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   5. Applied rewrites 98.5%

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

   6. Taylor expanded in uy around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f32 N/A

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

      8. unpow3 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-*.f32 N/A

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

      11. lower-PI.f32 N/A

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

      12. lower-PI.f32 N/A

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

      13. lower-PI.f32 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f32 N/A

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

      16. lower-PI.f32 98.0

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

   8. Applied rewrites 98.0%

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

   if 0.0560000017 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 56.2%

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

      1. lift--.f32 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f32 N/A

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

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

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

      6. lift-+.f32 N/A

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

      7. flip3-+ N/A

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

      8. associate-*r/ N/A

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

      9. div-inv N/A

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

      10. lower-fma.f32 N/A

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

   4. Applied rewrites 55.9%

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

   5. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\left(4 - \color{blue}{\left(maxCos - \left(-1 \cdot maxCos - 2\right)\right)}\right) \cdot ux} \]

      6. lower--.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\left(4 - \color{blue}{\left(maxCos - \left(-1 \cdot maxCos - 2\right)\right)}\right) \cdot ux} \]

      7. lower--.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\left(4 - \left(maxCos - \color{blue}{\left(-1 \cdot maxCos - 2\right)}\right)\right) \cdot ux} \]

      8. mul-1-neg N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\left(4 - \left(maxCos - \left(\color{blue}{\left(\mathsf{neg}\left(maxCos\right)\right)} - 2\right)\right)\right) \cdot ux} \]

      9. lower-neg.f32 77.1

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(4 - \left(maxCos - \left(\color{blue}{\left(-maxCos\right)} - 2\right)\right)\right) \cdot ux} \]

   7. Applied rewrites 77.1%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.0560000017285347:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(uy \cdot uy\right) \cdot -1.3333333333333333, \left(\pi \cdot \pi\right) \cdot \pi, \pi \cdot 2\right) \cdot uy\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, maxCos - 1, \frac{\mathsf{fma}\left(maxCos, -2, 2\right)}{ux}\right) \cdot \left(ux \cdot ux\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(4 - \left(\left(maxCos + 2\right) + maxCos\right)\right) \cdot ux} \cdot \sin \left(\pi \cdot \left(2 \cdot uy\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 94.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.0560000017285347:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(uy \cdot uy\right) \cdot -1.3333333333333333, \left(\pi \cdot \pi\right) \cdot \pi, \pi \cdot 2\right) \cdot uy\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, maxCos - 1, \frac{\mathsf{fma}\left(maxCos, -2, 2\right)}{ux}\right) \cdot \left(ux \cdot ux\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(maxCos, -2, 2\right) \cdot ux} \cdot \sin \left(\pi \cdot \left(2 \cdot uy\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* 2.0 uy) 0.0560000017285347)
   (*
    (* (fma (* (* uy uy) -1.3333333333333333) (* (* PI PI) PI) (* PI 2.0)) uy)
    (sqrt
     (*
      (fma (- 1.0 maxCos) (- maxCos 1.0) (/ (fma maxCos -2.0 2.0) ux))
      (* ux ux))))
   (* (sqrt (* (fma maxCos -2.0 2.0) ux)) (sin (* PI (* 2.0 uy))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if ((2.0f * uy) <= 0.0560000017285347f) {
		tmp = (fmaf(((uy * uy) * -1.3333333333333333f), ((((float) M_PI) * ((float) M_PI)) * ((float) M_PI)), (((float) M_PI) * 2.0f)) * uy) * sqrtf((fmaf((1.0f - maxCos), (maxCos - 1.0f), (fmaf(maxCos, -2.0f, 2.0f) / ux)) * (ux * ux)));
	} else {
		tmp = sqrtf((fmaf(maxCos, -2.0f, 2.0f) * ux)) * sinf((((float) M_PI) * (2.0f * uy)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (Float32(Float32(2.0) * uy) <= Float32(0.0560000017285347))
		tmp = Float32(Float32(fma(Float32(Float32(uy * uy) * Float32(-1.3333333333333333)), Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi)), Float32(Float32(pi) * Float32(2.0))) * uy) * sqrt(Float32(fma(Float32(Float32(1.0) - maxCos), Float32(maxCos - Float32(1.0)), Float32(fma(maxCos, Float32(-2.0), Float32(2.0)) / ux)) * Float32(ux * ux))));
	else
		tmp = Float32(sqrt(Float32(fma(maxCos, Float32(-2.0), Float32(2.0)) * ux)) * sin(Float32(Float32(pi) * Float32(Float32(2.0) * uy))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 0.0560000017

   1. Initial program 54.9%

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

   3. Taylor expanded in ux around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   5. Applied rewrites 98.5%

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

   6. Taylor expanded in uy around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f32 N/A

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

      8. unpow3 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-*.f32 N/A

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

      11. lower-PI.f32 N/A

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

      12. lower-PI.f32 N/A

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

      13. lower-PI.f32 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f32 N/A

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

      16. lower-PI.f32 98.0

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

   8. Applied rewrites 98.0%

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

   if 0.0560000017 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 56.2%

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

   3. Taylor expanded in ux around 0

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

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

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f32 77.1

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

   5. Applied rewrites 77.1%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.0560000017285347:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(uy \cdot uy\right) \cdot -1.3333333333333333, \left(\pi \cdot \pi\right) \cdot \pi, \pi \cdot 2\right) \cdot uy\right) \cdot \sqrt{\mathsf{fma}\left(1 - maxCos, maxCos - 1, \frac{\mathsf{fma}\left(maxCos, -2, 2\right)}{ux}\right) \cdot \left(ux \cdot ux\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(maxCos, -2, 2\right) \cdot ux} \cdot \sin \left(\pi \cdot \left(2 \cdot uy\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 88.9% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 55.2%

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

3. Taylor expanded in ux around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

5. Applied rewrites 98.4%

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

6. Taylor expanded in uy around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. associate-*r* N/A

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

   4. lower-fma.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f32 N/A

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

   8. unpow3 N/A

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

   9. lower-*.f32 N/A

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

   10. lower-*.f32 N/A

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

   11. lower-PI.f32 N/A

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

   12. lower-PI.f32 N/A

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

   13. lower-PI.f32 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f32 N/A

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

   16. lower-PI.f32 88.3

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

8. Applied rewrites 88.3%

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

9. Final simplification 88.3%

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

Alternative 9: 76.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\pi \cdot 2\right) \cdot uy\\ t_1 := maxCos \cdot ux + \left(1 - ux\right)\\ \mathbf{if}\;1 - t\_1 \cdot t\_1 \leq 0.0003000000142492354:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux, ux - \mathsf{fma}\left(maxCos, ux, 1\right), 1\right)} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (* PI 2.0) uy)) (t_1 (+ (* maxCos ux) (- 1.0 ux))))
   (if (<= (- 1.0 (* t_1 t_1)) 0.0003000000142492354)
     (* (sqrt (* (fma -2.0 maxCos 2.0) ux)) t_0)
     (*
      (sqrt (fma (- (fma maxCos ux 1.0) ux) (- ux (fma maxCos ux 1.0)) 1.0))
      t_0))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = (((float) M_PI) * 2.0f) * uy;
	float t_1 = (maxCos * ux) + (1.0f - ux);
	float tmp;
	if ((1.0f - (t_1 * t_1)) <= 0.0003000000142492354f) {
		tmp = sqrtf((fmaf(-2.0f, maxCos, 2.0f) * ux)) * t_0;
	} else {
		tmp = sqrtf(fmaf((fmaf(maxCos, ux, 1.0f) - ux), (ux - fmaf(maxCos, ux, 1.0f)), 1.0f)) * t_0;
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(Float32(Float32(pi) * Float32(2.0)) * uy)
	t_1 = Float32(Float32(maxCos * ux) + Float32(Float32(1.0) - ux))
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - Float32(t_1 * t_1)) <= Float32(0.0003000000142492354))
		tmp = Float32(sqrt(Float32(fma(Float32(-2.0), maxCos, Float32(2.0)) * ux)) * t_0);
	else
		tmp = Float32(sqrt(fma(Float32(fma(maxCos, ux, Float32(1.0)) - ux), Float32(ux - fma(maxCos, ux, Float32(1.0))), Float32(1.0))) * t_0);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)))) < 3.00000014e-4

   1. Initial program 35.7%

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

   3. Taylor expanded in uy around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f32 N/A

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

      8. lower-PI.f32 N/A

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

      9. lower-sqrt.f32 N/A

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

      10. lower--.f32 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f32 N/A

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

      13. lower--.f32 N/A

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

      14. +-commutative N/A

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

      15. lower-fma.f32 N/A

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

      16. lower--.f32 N/A

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

      17. +-commutative N/A

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

      18. lower-fma.f32 33.2

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

   5. Applied rewrites 33.2%

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

   6. Taylor expanded in ux around 0

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

   8. Applied rewrites 78.5%

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

   if 3.00000014e-4 < (-.f32 #s(literal 1 binary32) (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos))))

   1. Initial program 87.6%

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

   3. Taylor expanded in uy around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f32 N/A

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

      8. lower-PI.f32 N/A

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

      9. lower-sqrt.f32 N/A

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

      10. lower--.f32 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f32 N/A

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

      13. lower--.f32 N/A

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

      14. +-commutative N/A

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

      15. lower-fma.f32 N/A

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

      16. lower--.f32 N/A

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

      17. +-commutative N/A

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

      18. lower-fma.f32 73.2

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

   5. Applied rewrites 73.2%

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

   7. Applied rewrites 73.7%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - \left(maxCos \cdot ux + \left(1 - ux\right)\right) \cdot \left(maxCos \cdot ux + \left(1 - ux\right)\right) \leq 0.0003000000142492354:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux} \cdot \left(\left(\pi \cdot 2\right) \cdot uy\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux, ux - \mathsf{fma}\left(maxCos, ux, 1\right), 1\right)} \cdot \left(\left(\pi \cdot 2\right) \cdot uy\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 89.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 55.2%

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

3. Taylor expanded in uy around 0

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

   1. *-commutative N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f32 N/A

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

5. Applied rewrites 50.5%

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

6. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. mul-1-neg N/A

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

   5. unsub-neg N/A

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

   6. lower--.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. lower--.f32 N/A

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

   12. lower--.f32 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f32 88.2

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

8. Applied rewrites 88.2%

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

9. Taylor expanded in uy around 0

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

    1. *-commutative N/A

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

    2. lower-*.f32 N/A

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

    3. associate-*r* N/A

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

    4. lower-fma.f32 N/A

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

    5. *-commutative N/A

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

    6. lower-*.f32 N/A

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

    7. unpow2 N/A

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

    8. lower-*.f32 N/A

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

    9. unpow3 N/A

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

    10. lower-*.f32 N/A

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

    11. lower-*.f32 N/A

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

    12. lower-PI.f32 N/A

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

    13. lower-PI.f32 N/A

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

    14. lower-PI.f32 N/A

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

    15. *-commutative N/A

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

    16. lower-*.f32 N/A

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

    17. lower-PI.f32 88.3

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

11. Applied rewrites 88.3%

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

12. Final simplification 88.3%

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

Alternative 11: 89.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 55.2%

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

3. Taylor expanded in uy around 0

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

   1. *-commutative N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f32 N/A

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

5. Applied rewrites 50.5%

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

6. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. mul-1-neg N/A

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

   5. unsub-neg N/A

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

   6. lower--.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. lower--.f32 N/A

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

   12. lower--.f32 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f32 88.2

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

8. Applied rewrites 88.2%

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

9. Final simplification 88.2%

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

Alternative 12: 76.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(maxCos, ux, 1\right) - ux\\ t_1 := \left(\pi \cdot 2\right) \cdot uy\\ \mathbf{if}\;maxCos \cdot ux + \left(1 - ux\right) \leq 0.9998000264167786:\\ \;\;\;\;\sqrt{1 - t\_0 \cdot t\_0} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux} \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (- (fma maxCos ux 1.0) ux)) (t_1 (* (* PI 2.0) uy)))
   (if (<= (+ (* maxCos ux) (- 1.0 ux)) 0.9998000264167786)
     (* (sqrt (- 1.0 (* t_0 t_0))) t_1)
     (* (sqrt (* (fma -2.0 maxCos 2.0) ux)) t_1))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = fmaf(maxCos, ux, 1.0f) - ux;
	float t_1 = (((float) M_PI) * 2.0f) * uy;
	float tmp;
	if (((maxCos * ux) + (1.0f - ux)) <= 0.9998000264167786f) {
		tmp = sqrtf((1.0f - (t_0 * t_0))) * t_1;
	} else {
		tmp = sqrtf((fmaf(-2.0f, maxCos, 2.0f) * ux)) * t_1;
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(fma(maxCos, ux, Float32(1.0)) - ux)
	t_1 = Float32(Float32(Float32(pi) * Float32(2.0)) * uy)
	tmp = Float32(0.0)
	if (Float32(Float32(maxCos * ux) + Float32(Float32(1.0) - ux)) <= Float32(0.9998000264167786))
		tmp = Float32(sqrt(Float32(Float32(1.0) - Float32(t_0 * t_0))) * t_1);
	else
		tmp = Float32(sqrt(Float32(fma(Float32(-2.0), maxCos, Float32(2.0)) * ux)) * t_1);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) < 0.999800026

   1. Initial program 88.1%

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

   3. Taylor expanded in uy around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f32 N/A

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

      8. lower-PI.f32 N/A

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

      9. lower-sqrt.f32 N/A

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

      10. lower--.f32 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f32 N/A

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

      13. lower--.f32 N/A

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

      14. +-commutative N/A

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

      15. lower-fma.f32 N/A

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

      16. lower--.f32 N/A

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

      17. +-commutative N/A

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

      18. lower-fma.f32 73.2

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

   5. Applied rewrites 73.2%

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

   if 0.999800026 < (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos))

   1. Initial program 36.3%

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

   3. Taylor expanded in uy around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f32 N/A

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

      8. lower-PI.f32 N/A

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

      9. lower-sqrt.f32 N/A

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

      10. lower--.f32 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f32 N/A

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

      13. lower--.f32 N/A

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

      14. +-commutative N/A

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

      15. lower-fma.f32 N/A

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

      16. lower--.f32 N/A

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

      17. +-commutative N/A

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

      18. lower-fma.f32 33.9

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

   5. Applied rewrites 33.9%

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

   6. Taylor expanded in ux around 0

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

   8. Applied rewrites 78.4%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;maxCos \cdot ux + \left(1 - ux\right) \leq 0.9998000264167786:\\ \;\;\;\;\sqrt{1 - \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right) \cdot \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right)} \cdot \left(\left(\pi \cdot 2\right) \cdot uy\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux} \cdot \left(\left(\pi \cdot 2\right) \cdot uy\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 76.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(maxCos, ux, 1\right) - ux\\ \mathbf{if}\;maxCos \cdot ux + \left(1 - ux\right) \leq 0.9998000264167786:\\ \;\;\;\;\left(\sqrt{1 - t\_0 \cdot t\_0} \cdot \left(\pi \cdot 2\right)\right) \cdot uy\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux} \cdot \left(\left(\pi \cdot 2\right) \cdot uy\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (- (fma maxCos ux 1.0) ux)))
   (if (<= (+ (* maxCos ux) (- 1.0 ux)) 0.9998000264167786)
     (* (* (sqrt (- 1.0 (* t_0 t_0))) (* PI 2.0)) uy)
     (* (sqrt (* (fma -2.0 maxCos 2.0) ux)) (* (* PI 2.0) uy)))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = fmaf(maxCos, ux, 1.0f) - ux;
	float tmp;
	if (((maxCos * ux) + (1.0f - ux)) <= 0.9998000264167786f) {
		tmp = (sqrtf((1.0f - (t_0 * t_0))) * (((float) M_PI) * 2.0f)) * uy;
	} else {
		tmp = sqrtf((fmaf(-2.0f, maxCos, 2.0f) * ux)) * ((((float) M_PI) * 2.0f) * uy);
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(fma(maxCos, ux, Float32(1.0)) - ux)
	tmp = Float32(0.0)
	if (Float32(Float32(maxCos * ux) + Float32(Float32(1.0) - ux)) <= Float32(0.9998000264167786))
		tmp = Float32(Float32(sqrt(Float32(Float32(1.0) - Float32(t_0 * t_0))) * Float32(Float32(pi) * Float32(2.0))) * uy);
	else
		tmp = Float32(sqrt(Float32(fma(Float32(-2.0), maxCos, Float32(2.0)) * ux)) * Float32(Float32(Float32(pi) * Float32(2.0)) * uy));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) < 0.999800026

   1. Initial program 88.1%

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

   3. Taylor expanded in uy around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f32 N/A

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

      8. lower-PI.f32 N/A

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

      9. lower-sqrt.f32 N/A

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

      10. lower--.f32 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f32 N/A

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

      13. lower--.f32 N/A

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

      14. +-commutative N/A

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

      15. lower-fma.f32 N/A

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

      16. lower--.f32 N/A

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

      17. +-commutative N/A

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

      18. lower-fma.f32 73.2

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

   5. Applied rewrites 73.2%

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

   7. Applied rewrites 73.2%

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

   if 0.999800026 < (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos))

   1. Initial program 36.3%

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

   3. Taylor expanded in uy around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f32 N/A

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

      8. lower-PI.f32 N/A

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

      9. lower-sqrt.f32 N/A

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

      10. lower--.f32 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f32 N/A

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

      13. lower--.f32 N/A

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

      14. +-commutative N/A

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

      15. lower-fma.f32 N/A

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

      16. lower--.f32 N/A

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

      17. +-commutative N/A

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

      18. lower-fma.f32 33.9

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

   5. Applied rewrites 33.9%

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

   6. Taylor expanded in ux around 0

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

   8. Applied rewrites 78.4%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;maxCos \cdot ux + \left(1 - ux\right) \leq 0.9998000264167786:\\ \;\;\;\;\left(\sqrt{1 - \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right) \cdot \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right)} \cdot \left(\pi \cdot 2\right)\right) \cdot uy\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux} \cdot \left(\left(\pi \cdot 2\right) \cdot uy\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 76.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(maxCos, ux, 1\right) - ux\\ \mathbf{if}\;maxCos \cdot ux + \left(1 - ux\right) \leq 0.9998000264167786:\\ \;\;\;\;\left(\sqrt{1 - t\_0 \cdot t\_0} \cdot \left(2 \cdot uy\right)\right) \cdot \pi\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux} \cdot \left(\left(\pi \cdot 2\right) \cdot uy\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (- (fma maxCos ux 1.0) ux)))
   (if (<= (+ (* maxCos ux) (- 1.0 ux)) 0.9998000264167786)
     (* (* (sqrt (- 1.0 (* t_0 t_0))) (* 2.0 uy)) PI)
     (* (sqrt (* (fma -2.0 maxCos 2.0) ux)) (* (* PI 2.0) uy)))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = fmaf(maxCos, ux, 1.0f) - ux;
	float tmp;
	if (((maxCos * ux) + (1.0f - ux)) <= 0.9998000264167786f) {
		tmp = (sqrtf((1.0f - (t_0 * t_0))) * (2.0f * uy)) * ((float) M_PI);
	} else {
		tmp = sqrtf((fmaf(-2.0f, maxCos, 2.0f) * ux)) * ((((float) M_PI) * 2.0f) * uy);
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(fma(maxCos, ux, Float32(1.0)) - ux)
	tmp = Float32(0.0)
	if (Float32(Float32(maxCos * ux) + Float32(Float32(1.0) - ux)) <= Float32(0.9998000264167786))
		tmp = Float32(Float32(sqrt(Float32(Float32(1.0) - Float32(t_0 * t_0))) * Float32(Float32(2.0) * uy)) * Float32(pi));
	else
		tmp = Float32(sqrt(Float32(fma(Float32(-2.0), maxCos, Float32(2.0)) * ux)) * Float32(Float32(Float32(pi) * Float32(2.0)) * uy));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) < 0.999800026

   1. Initial program 88.1%

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

   3. Taylor expanded in uy around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f32 N/A

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

      8. lower-PI.f32 N/A

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

      9. lower-sqrt.f32 N/A

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

      10. lower--.f32 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f32 N/A

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

      13. lower--.f32 N/A

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

      14. +-commutative N/A

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

      15. lower-fma.f32 N/A

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

      16. lower--.f32 N/A

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

      17. +-commutative N/A

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

      18. lower-fma.f32 73.2

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

   5. Applied rewrites 73.2%

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

   7. Applied rewrites 73.2%

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

   if 0.999800026 < (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos))

   1. Initial program 36.3%

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

   3. Taylor expanded in uy around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f32 N/A

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

      8. lower-PI.f32 N/A

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

      9. lower-sqrt.f32 N/A

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

      10. lower--.f32 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f32 N/A

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

      13. lower--.f32 N/A

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

      14. +-commutative N/A

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

      15. lower-fma.f32 N/A

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

      16. lower--.f32 N/A

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

      17. +-commutative N/A

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

      18. lower-fma.f32 33.9

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

   5. Applied rewrites 33.9%

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

   6. Taylor expanded in ux around 0

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

   8. Applied rewrites 78.4%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;maxCos \cdot ux + \left(1 - ux\right) \leq 0.9998000264167786:\\ \;\;\;\;\left(\sqrt{1 - \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right) \cdot \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right)} \cdot \left(2 \cdot uy\right)\right) \cdot \pi\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux} \cdot \left(\left(\pi \cdot 2\right) \cdot uy\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 88.5% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 55.2%

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

3. Taylor expanded in uy around 0

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

   1. *-commutative N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f32 N/A

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

5. Applied rewrites 50.5%

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

6. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. mul-1-neg N/A

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

   5. unsub-neg N/A

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

   6. lower--.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. lower--.f32 N/A

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

   12. lower--.f32 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f32 88.2

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

8. Applied rewrites 88.2%

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

9. Taylor expanded in maxCos around 0

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

11. Applied rewrites 88.0%

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

12. Final simplification 88.0%

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

Alternative 16: 87.6% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;maxCos \leq 9.999999747378752 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{\left(2 - ux\right) \cdot ux} \cdot \left(\left(\mathsf{fma}\left(\left(uy \cdot uy\right) \cdot -1.3333333333333333, \pi \cdot \pi, 2\right) \cdot \pi\right) \cdot uy\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 - \left(\left(1 - maxCos\right) \cdot \left(1 - maxCos\right)\right) \cdot ux\right) - maxCos \cdot 2\right) \cdot ux} \cdot \left(\left(\pi \cdot 2\right) \cdot uy\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= maxCos 9.999999747378752e-6)
   (*
    (sqrt (* (- 2.0 ux) ux))
    (* (* (fma (* (* uy uy) -1.3333333333333333) (* PI PI) 2.0) PI) uy))
   (*
    (sqrt
     (*
      (- (- 2.0 (* (* (- 1.0 maxCos) (- 1.0 maxCos)) ux)) (* maxCos 2.0))
      ux))
    (* (* PI 2.0) uy))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (maxCos <= 9.999999747378752e-6f) {
		tmp = sqrtf(((2.0f - ux) * ux)) * ((fmaf(((uy * uy) * -1.3333333333333333f), (((float) M_PI) * ((float) M_PI)), 2.0f) * ((float) M_PI)) * uy);
	} else {
		tmp = sqrtf((((2.0f - (((1.0f - maxCos) * (1.0f - maxCos)) * ux)) - (maxCos * 2.0f)) * ux)) * ((((float) M_PI) * 2.0f) * uy);
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (maxCos <= Float32(9.999999747378752e-6))
		tmp = Float32(sqrt(Float32(Float32(Float32(2.0) - ux) * ux)) * Float32(Float32(fma(Float32(Float32(uy * uy) * Float32(-1.3333333333333333)), Float32(Float32(pi) * Float32(pi)), Float32(2.0)) * Float32(pi)) * uy));
	else
		tmp = Float32(sqrt(Float32(Float32(Float32(Float32(2.0) - Float32(Float32(Float32(Float32(1.0) - maxCos) * Float32(Float32(1.0) - maxCos)) * ux)) - Float32(maxCos * Float32(2.0))) * ux)) * Float32(Float32(Float32(pi) * Float32(2.0)) * uy));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if maxCos < 9.99999975e-6

   1. Initial program 56.1%

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

   3. Taylor expanded in uy around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f32 N/A

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

   5. Applied rewrites 51.3%

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

   6. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

      6. lower--.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f32 N/A

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

      11. lower--.f32 N/A

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

      12. lower--.f32 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f32 88.0

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

   8. Applied rewrites 88.0%

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

   9. Taylor expanded in maxCos around 0

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

   11. Applied rewrites 87.8%

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

   if 9.99999975e-6 < maxCos

   1. Initial program 48.7%

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

   3. Taylor expanded in uy around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f32 N/A

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

      8. lower-PI.f32 N/A

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

      9. lower-sqrt.f32 N/A

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

      10. lower--.f32 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f32 N/A

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

      13. lower--.f32 N/A

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

      14. +-commutative N/A

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

      15. lower-fma.f32 N/A

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

      16. lower--.f32 N/A

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

      17. +-commutative N/A

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

      18. lower-fma.f32 42.8

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

   5. Applied rewrites 42.8%

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

   6. Taylor expanded in ux around 0

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

   8. Applied rewrites 83.9%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;maxCos \leq 9.999999747378752 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{\left(2 - ux\right) \cdot ux} \cdot \left(\left(\mathsf{fma}\left(\left(uy \cdot uy\right) \cdot -1.3333333333333333, \pi \cdot \pi, 2\right) \cdot \pi\right) \cdot uy\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 - \left(\left(1 - maxCos\right) \cdot \left(1 - maxCos\right)\right) \cdot ux\right) - maxCos \cdot 2\right) \cdot ux} \cdot \left(\left(\pi \cdot 2\right) \cdot uy\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 87.9% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 55.2%

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

3. Taylor expanded in uy around 0

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

   1. *-commutative N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f32 N/A

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

5. Applied rewrites 50.5%

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

6. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. mul-1-neg N/A

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

   5. unsub-neg N/A

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

   6. lower--.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. lower--.f32 N/A

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

   12. lower--.f32 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f32 88.2

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

8. Applied rewrites 88.2%

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

9. Taylor expanded in maxCos around 0

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

11. Applied rewrites 87.4%

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

12. Final simplification 87.4%

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

Alternative 18: 75.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\pi \cdot 2\right) \cdot uy\\ \mathbf{if}\;maxCos \cdot ux + \left(1 - ux\right) \leq 0.9998000264167786:\\ \;\;\;\;\sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (* PI 2.0) uy)))
   (if (<= (+ (* maxCos ux) (- 1.0 ux)) 0.9998000264167786)
     (* (sqrt (- 1.0 (* (- 1.0 ux) (- 1.0 ux)))) t_0)
     (* (sqrt (* (fma -2.0 maxCos 2.0) ux)) t_0))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = (((float) M_PI) * 2.0f) * uy;
	float tmp;
	if (((maxCos * ux) + (1.0f - ux)) <= 0.9998000264167786f) {
		tmp = sqrtf((1.0f - ((1.0f - ux) * (1.0f - ux)))) * t_0;
	} else {
		tmp = sqrtf((fmaf(-2.0f, maxCos, 2.0f) * ux)) * t_0;
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(Float32(Float32(pi) * Float32(2.0)) * uy)
	tmp = Float32(0.0)
	if (Float32(Float32(maxCos * ux) + Float32(Float32(1.0) - ux)) <= Float32(0.9998000264167786))
		tmp = Float32(sqrt(Float32(Float32(1.0) - Float32(Float32(Float32(1.0) - ux) * Float32(Float32(1.0) - ux)))) * t_0);
	else
		tmp = Float32(sqrt(Float32(fma(Float32(-2.0), maxCos, Float32(2.0)) * ux)) * t_0);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) < 0.999800026

   1. Initial program 88.1%

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

   3. Taylor expanded in uy around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f32 N/A

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

      8. lower-PI.f32 N/A

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

      9. lower-sqrt.f32 N/A

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

      10. lower--.f32 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f32 N/A

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

      13. lower--.f32 N/A

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

      14. +-commutative N/A

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

      15. lower-fma.f32 N/A

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

      16. lower--.f32 N/A

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

      17. +-commutative N/A

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

      18. lower-fma.f32 73.2

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

   5. Applied rewrites 73.2%

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

   6. Taylor expanded in maxCos around 0

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

   8. Applied rewrites 70.9%

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

   if 0.999800026 < (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos))

   1. Initial program 36.3%

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

   3. Taylor expanded in uy around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f32 N/A

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

      8. lower-PI.f32 N/A

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

      9. lower-sqrt.f32 N/A

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

      10. lower--.f32 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f32 N/A

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

      13. lower--.f32 N/A

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

      14. +-commutative N/A

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

      15. lower-fma.f32 N/A

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

      16. lower--.f32 N/A

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

      17. +-commutative N/A

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

      18. lower-fma.f32 33.9

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

   5. Applied rewrites 33.9%

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

   6. Taylor expanded in ux around 0

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

   8. Applied rewrites 78.4%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;maxCos \cdot ux + \left(1 - ux\right) \leq 0.9998000264167786:\\ \;\;\;\;\sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \left(\left(\pi \cdot 2\right) \cdot uy\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux} \cdot \left(\left(\pi \cdot 2\right) \cdot uy\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 83.9% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 55.2%

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

3. Taylor expanded in uy around 0

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

   1. *-commutative N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f32 N/A

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

5. Applied rewrites 50.5%

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

6. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. mul-1-neg N/A

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

   5. unsub-neg N/A

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

   6. lower--.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. lower--.f32 N/A

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

   12. lower--.f32 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f32 88.2

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

8. Applied rewrites 88.2%

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

9. Taylor expanded in maxCos around 0

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

11. Applied rewrites 83.1%

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

12. Final simplification 83.1%

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

Alternative 20: 66.1% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 55.2%

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

3. Taylor expanded in uy around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. lower-PI.f32 N/A

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

   9. lower-sqrt.f32 N/A

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

   10. lower--.f32 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f32 N/A

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

   13. lower--.f32 N/A

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

   14. +-commutative N/A

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

   15. lower-fma.f32 N/A

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

   16. lower--.f32 N/A

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

   17. +-commutative N/A

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

   18. lower-fma.f32 48.2

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

5. Applied rewrites 48.2%

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

6. Taylor expanded in ux around 0

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

8. Applied rewrites 67.0%

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

9. Final simplification 67.0%

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

Alternative 21: 66.1% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 55.2%

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

3. Taylor expanded in uy around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. lower-PI.f32 N/A

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

   9. lower-sqrt.f32 N/A

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

   10. lower--.f32 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f32 N/A

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

   13. lower--.f32 N/A

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

   14. +-commutative N/A

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

   15. lower-fma.f32 N/A

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

   16. lower--.f32 N/A

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

   17. +-commutative N/A

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

   18. lower-fma.f32 48.2

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

5. Applied rewrites 48.2%

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

6. Taylor expanded in ux around 0

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

8. Applied rewrites 67.0%

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

10. Applied rewrites 67.0%

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

11. Final simplification 67.0%

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

Alternative 22: 7.1% accurate, 5.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 55.2%

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

3. Taylor expanded in uy around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. lower-PI.f32 N/A

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

   9. lower-sqrt.f32 N/A

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

   10. lower--.f32 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f32 N/A

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

   13. lower--.f32 N/A

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

   14. +-commutative N/A

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

   15. lower-fma.f32 N/A

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

   16. lower--.f32 N/A

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

   17. +-commutative N/A

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

   18. lower-fma.f32 48.2

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

5. Applied rewrites 48.2%

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

6. Taylor expanded in ux around 0

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

8. Applied rewrites 7.1%

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

9. Final simplification 7.1%

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

Reproduce

herbie shell --seed 2024235 
(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)))))))