UniformSampleCone, y

Percentage Accurate: 57.2% → 98.3%

Time: 18.4s

Alternatives: 18

Speedup: 4.5×

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: 57.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 60.1%

   \[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
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. *-lowering-*.f32 N/A

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

   2. unpow2 N/A

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

   3. *-lowering-*.f32 N/A

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

   4. sub-neg N/A

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

   5. +-commutative N/A

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

   6. +-commutative N/A

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

   7. distribute-neg-in N/A

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

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

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

   9. metadata-eval N/A

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

   10. associate-+l+ N/A

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

5. Simplified 98.3%

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

   1. *-commutative N/A

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

   2. *-lowering-*.f32 N/A

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

7. Applied egg-rr 98.3%

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

8. Final simplification 98.3%

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

Alternative 2: 97.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.03999999910593033:\\ \;\;\;\;uy \cdot \left(\sqrt{\mathsf{fma}\left(ux, \mathsf{fma}\left(-2, maxCos, 2\right), \left(ux \cdot ux\right) \cdot \left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right)\right)} \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), 2 \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;ux \cdot \left(\sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{-1 + \frac{2}{ux}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* 2.0 uy) 0.03999999910593033)
   (*
    uy
    (*
     (sqrt
      (fma
       ux
       (fma -2.0 maxCos 2.0)
       (* (* ux ux) (* (+ maxCos -1.0) (- 1.0 maxCos)))))
     (fma -1.3333333333333333 (* (* uy uy) (* PI (* PI PI))) (* 2.0 PI))))
   (* ux (* (sin (* 2.0 (* uy PI))) (sqrt (+ -1.0 (/ 2.0 ux)))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if ((2.0f * uy) <= 0.03999999910593033f) {
		tmp = uy * (sqrtf(fmaf(ux, fmaf(-2.0f, maxCos, 2.0f), ((ux * ux) * ((maxCos + -1.0f) * (1.0f - maxCos))))) * fmaf(-1.3333333333333333f, ((uy * uy) * (((float) M_PI) * (((float) M_PI) * ((float) M_PI)))), (2.0f * ((float) M_PI))));
	} else {
		tmp = ux * (sinf((2.0f * (uy * ((float) M_PI)))) * sqrtf((-1.0f + (2.0f / ux))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (Float32(Float32(2.0) * uy) <= Float32(0.03999999910593033))
		tmp = Float32(uy * Float32(sqrt(fma(ux, fma(Float32(-2.0), maxCos, Float32(2.0)), Float32(Float32(ux * ux) * Float32(Float32(maxCos + Float32(-1.0)) * Float32(Float32(1.0) - maxCos))))) * fma(Float32(-1.3333333333333333), Float32(Float32(uy * uy) * Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi)))), Float32(Float32(2.0) * Float32(pi)))));
	else
		tmp = Float32(ux * Float32(sin(Float32(Float32(2.0) * Float32(uy * Float32(pi)))) * sqrt(Float32(Float32(-1.0) + Float32(Float32(2.0) / ux)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 0.0399999991

   1. Initial program 59.2%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   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. *-lowering-*.f32 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f32 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

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

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

      9. metadata-eval N/A

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

      10. associate-+l+ N/A

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

   5. Simplified 98.3%

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

      1. *-commutative N/A

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

      2. *-lowering-*.f32 N/A

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

   7. Applied egg-rr 98.4%

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

   8. Taylor expanded in uy around 0

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

   10. Simplified 98.3%

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

   if 0.0399999991 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 64.4%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. 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. *-lowering-*.f32 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f32 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

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

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

      9. metadata-eval N/A

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

      10. associate-+l+ N/A

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

   5. Simplified 98.2%

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

      1. *-commutative N/A

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

      2. pow1/2 N/A

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

      3. unpow-prod-down N/A

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

      4. associate-*l* N/A

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

      5. pow2 N/A

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

      6. pow-pow N/A

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

      7. metadata-eval N/A

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

      8. unpow1 N/A

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

      9. *-lowering-*.f32 N/A

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

      10. *-lowering-*.f32 N/A

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

   7. Applied egg-rr 98.0%

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

   8. Taylor expanded in maxCos around 0

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

      1. sqrt-lowering-sqrt.f32 N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. +-lowering-+.f32 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. /-lowering-/.f32 94.8

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

   10. Simplified 94.8%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.03999999910593033:\\ \;\;\;\;uy \cdot \left(\sqrt{\mathsf{fma}\left(ux, \mathsf{fma}\left(-2, maxCos, 2\right), \left(ux \cdot ux\right) \cdot \left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right)\right)} \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), 2 \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;ux \cdot \left(\sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{-1 + \frac{2}{ux}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.03999999910593033:\\ \;\;\;\;uy \cdot \left(\sqrt{\mathsf{fma}\left(ux, \mathsf{fma}\left(-2, maxCos, 2\right), \left(ux \cdot ux\right) \cdot \left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right)\right)} \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), 2 \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(ux, 2, -ux \cdot ux\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* 2.0 uy) 0.03999999910593033)
   (*
    uy
    (*
     (sqrt
      (fma
       ux
       (fma -2.0 maxCos 2.0)
       (* (* ux ux) (* (+ maxCos -1.0) (- 1.0 maxCos)))))
     (fma -1.3333333333333333 (* (* uy uy) (* PI (* PI PI))) (* 2.0 PI))))
   (* (sin (* 2.0 (* uy PI))) (sqrt (fma ux 2.0 (- (* ux ux)))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if ((2.0f * uy) <= 0.03999999910593033f) {
		tmp = uy * (sqrtf(fmaf(ux, fmaf(-2.0f, maxCos, 2.0f), ((ux * ux) * ((maxCos + -1.0f) * (1.0f - maxCos))))) * fmaf(-1.3333333333333333f, ((uy * uy) * (((float) M_PI) * (((float) M_PI) * ((float) M_PI)))), (2.0f * ((float) M_PI))));
	} else {
		tmp = sinf((2.0f * (uy * ((float) M_PI)))) * sqrtf(fmaf(ux, 2.0f, -(ux * ux)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (Float32(Float32(2.0) * uy) <= Float32(0.03999999910593033))
		tmp = Float32(uy * Float32(sqrt(fma(ux, fma(Float32(-2.0), maxCos, Float32(2.0)), Float32(Float32(ux * ux) * Float32(Float32(maxCos + Float32(-1.0)) * Float32(Float32(1.0) - maxCos))))) * fma(Float32(-1.3333333333333333), Float32(Float32(uy * uy) * Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi)))), Float32(Float32(2.0) * Float32(pi)))));
	else
		tmp = Float32(sin(Float32(Float32(2.0) * Float32(uy * Float32(pi)))) * sqrt(fma(ux, Float32(2.0), Float32(-Float32(ux * ux)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 0.0399999991

   1. Initial program 59.2%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   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. *-lowering-*.f32 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f32 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

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

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

      9. metadata-eval N/A

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

      10. associate-+l+ N/A

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

   5. Simplified 98.3%

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

      1. *-commutative N/A

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

      2. *-lowering-*.f32 N/A

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

   7. Applied egg-rr 98.4%

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

   8. Taylor expanded in uy around 0

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

   10. Simplified 98.3%

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

   if 0.0399999991 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 64.4%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. 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. *-lowering-*.f32 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f32 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

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

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

      9. metadata-eval N/A

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

      10. associate-+l+ N/A

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

   5. Simplified 98.2%

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

      1. *-commutative N/A

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

      2. *-lowering-*.f32 N/A

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

   7. Applied egg-rr 97.9%

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

   8. Taylor expanded in maxCos around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f32 N/A

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

      4. mul-1-neg N/A

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

      5. unpow2 N/A

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

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

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

      7. mul-1-neg N/A

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

      8. *-lowering-*.f32 N/A

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

      9. mul-1-neg N/A

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

      10. neg-lowering-neg.f32 94.7

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

   10. Simplified 94.7%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.03999999910593033:\\ \;\;\;\;uy \cdot \left(\sqrt{\mathsf{fma}\left(ux, \mathsf{fma}\left(-2, maxCos, 2\right), \left(ux \cdot ux\right) \cdot \left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right)\right)} \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), 2 \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(ux, 2, -ux \cdot ux\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 60.1%

   \[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
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. *-lowering-*.f32 N/A

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

   2. unpow2 N/A

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

   3. *-lowering-*.f32 N/A

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

   4. sub-neg N/A

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

   5. +-commutative N/A

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

   6. +-commutative N/A

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

   7. distribute-neg-in N/A

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

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

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

   9. metadata-eval N/A

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

   10. associate-+l+ N/A

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

5. Simplified 98.3%

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

   1. *-commutative N/A

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

   2. *-lowering-*.f32 N/A

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

7. Applied egg-rr 98.3%

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

8. Taylor expanded in ux around 0

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

   1. *-lowering-*.f32 N/A

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

   2. +-lowering-+.f32 N/A

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

   3. +-commutative N/A

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

   4. sub-neg N/A

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

   5. mul-1-neg N/A

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

   6. accelerator-lowering-fma.f32 N/A

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

   7. *-commutative N/A

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

   8. *-lowering-*.f32 N/A

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

   9. sub-neg N/A

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

   10. metadata-eval N/A

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

   11. +-lowering-+.f32 N/A

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

   12. mul-1-neg N/A

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

   13. sub-neg N/A

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

   14. --lowering--.f32 N/A

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

   15. *-lowering-*.f32 98.3

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

10. Simplified 98.3%

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

11. Taylor expanded in maxCos around 0

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

    1. +-commutative N/A

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

    2. +-commutative N/A

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

    3. associate-+l+ N/A

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

    4. +-commutative N/A

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

    5. accelerator-lowering-fma.f32 N/A

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

    6. sub-neg N/A

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

    7. +-commutative N/A

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

    8. associate-*r* N/A

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

    9. mul-1-neg N/A

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

    10. distribute-rgt-out N/A

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

    11. mul-1-neg N/A

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

    12. metadata-eval N/A

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

    13. accelerator-lowering-fma.f32 N/A

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

    14. mul-1-neg N/A

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

    15. unsub-neg N/A

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

    16. --lowering--.f32 N/A

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

    17. mul-1-neg N/A

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

    18. unsub-neg N/A

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

    19. --lowering--.f32 98.3

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

13. Simplified 98.3%

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

14. Final simplification 98.3%

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

Alternative 5: 97.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 60.1%

   \[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
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. *-lowering-*.f32 N/A

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

   2. unpow2 N/A

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

   3. *-lowering-*.f32 N/A

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

   4. sub-neg N/A

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

   5. +-commutative N/A

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

   6. +-commutative N/A

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

   7. distribute-neg-in N/A

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

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

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

   9. metadata-eval N/A

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

   10. associate-+l+ N/A

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

5. Simplified 98.3%

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

   1. *-commutative N/A

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

   2. *-lowering-*.f32 N/A

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

7. Applied egg-rr 98.3%

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

8. Taylor expanded in ux around 0

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

   1. *-lowering-*.f32 N/A

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

   2. +-lowering-+.f32 N/A

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

   3. +-commutative N/A

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

   4. sub-neg N/A

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

   5. mul-1-neg N/A

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

   6. accelerator-lowering-fma.f32 N/A

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

   7. *-commutative N/A

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

   8. *-lowering-*.f32 N/A

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

   9. sub-neg N/A

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

   10. metadata-eval N/A

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

   11. +-lowering-+.f32 N/A

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

   12. mul-1-neg N/A

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

   13. sub-neg N/A

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

   14. --lowering--.f32 N/A

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

   15. *-lowering-*.f32 98.3

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

10. Simplified 98.3%

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

11. Taylor expanded in maxCos around 0

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

    1. +-commutative N/A

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

    2. +-commutative N/A

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

    3. associate-+l+ N/A

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

    4. +-commutative N/A

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

    5. accelerator-lowering-fma.f32 N/A

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

    6. sub-neg N/A

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

    7. metadata-eval N/A

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

    8. accelerator-lowering-fma.f32 N/A

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

    9. mul-1-neg N/A

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

    10. unsub-neg N/A

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

    11. --lowering--.f32 97.4

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

13. Simplified 97.4%

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

14. Final simplification 97.4%

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

Alternative 6: 97.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.03999999910593033:\\ \;\;\;\;uy \cdot \left(\sqrt{\mathsf{fma}\left(ux, \mathsf{fma}\left(-2, maxCos, 2\right), \left(ux \cdot ux\right) \cdot \left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right)\right)} \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), 2 \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* 2.0 uy) 0.03999999910593033)
   (*
    uy
    (*
     (sqrt
      (fma
       ux
       (fma -2.0 maxCos 2.0)
       (* (* ux ux) (* (+ maxCos -1.0) (- 1.0 maxCos)))))
     (fma -1.3333333333333333 (* (* uy uy) (* PI (* PI PI))) (* 2.0 PI))))
   (* (sin (* PI (* 2.0 uy))) (sqrt (* ux (- 2.0 ux))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if ((2.0f * uy) <= 0.03999999910593033f) {
		tmp = uy * (sqrtf(fmaf(ux, fmaf(-2.0f, maxCos, 2.0f), ((ux * ux) * ((maxCos + -1.0f) * (1.0f - maxCos))))) * fmaf(-1.3333333333333333f, ((uy * uy) * (((float) M_PI) * (((float) M_PI) * ((float) M_PI)))), (2.0f * ((float) M_PI))));
	} else {
		tmp = sinf((((float) M_PI) * (2.0f * uy))) * sqrtf((ux * (2.0f - ux)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (Float32(Float32(2.0) * uy) <= Float32(0.03999999910593033))
		tmp = Float32(uy * Float32(sqrt(fma(ux, fma(Float32(-2.0), maxCos, Float32(2.0)), Float32(Float32(ux * ux) * Float32(Float32(maxCos + Float32(-1.0)) * Float32(Float32(1.0) - maxCos))))) * fma(Float32(-1.3333333333333333), Float32(Float32(uy * uy) * Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi)))), Float32(Float32(2.0) * Float32(pi)))));
	else
		tmp = Float32(sin(Float32(Float32(pi) * Float32(Float32(2.0) * uy))) * sqrt(Float32(ux * Float32(Float32(2.0) - ux))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 0.0399999991

   1. Initial program 59.2%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   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. *-lowering-*.f32 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f32 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

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

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

      9. metadata-eval N/A

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

      10. associate-+l+ N/A

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

   5. Simplified 98.3%

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

      1. *-commutative N/A

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

      2. *-lowering-*.f32 N/A

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

   7. Applied egg-rr 98.4%

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

   8. Taylor expanded in uy around 0

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

   10. Simplified 98.3%

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

   if 0.0399999991 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 64.4%

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

   3. Taylor expanded in maxCos around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

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

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

      5. mul-1-neg N/A

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

      6. accelerator-lowering-fma.f32 N/A

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

      7. --lowering--.f32 N/A

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

      8. mul-1-neg N/A

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

      9. neg-sub0 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{fma}\left(1 - ux, \color{blue}{0 - \left(1 - ux\right)}, 1\right)} \]

      10. --lowering--.f32 N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{fma}\left(1 - ux, \color{blue}{0 - \left(1 - ux\right)}, 1\right)} \]

      11. --lowering--.f32 62.4

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\mathsf{fma}\left(1 - ux, 0 - \color{blue}{\left(1 - ux\right)}, 1\right)} \]

   5. Simplified 62.4%

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(1 - ux, 0 - \left(1 - ux\right), 1\right)}} \]

   6. Taylor expanded in ux around 0

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

      1. *-lowering-*.f32 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. --lowering--.f32 94.7

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

   8. Simplified 94.7%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.03999999910593033:\\ \;\;\;\;uy \cdot \left(\sqrt{\mathsf{fma}\left(ux, \mathsf{fma}\left(-2, maxCos, 2\right), \left(ux \cdot ux\right) \cdot \left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right)\right)} \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), 2 \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 94.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.05950000137090683:\\ \;\;\;\;uy \cdot \left(\sqrt{\mathsf{fma}\left(ux, \mathsf{fma}\left(-2, maxCos, 2\right), \left(ux \cdot ux\right) \cdot \left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right)\right)} \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), 2 \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{ux \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* 2.0 uy) 0.05950000137090683)
   (*
    uy
    (*
     (sqrt
      (fma
       ux
       (fma -2.0 maxCos 2.0)
       (* (* ux ux) (* (+ maxCos -1.0) (- 1.0 maxCos)))))
     (fma -1.3333333333333333 (* (* uy uy) (* PI (* PI PI))) (* 2.0 PI))))
   (* (sin (* PI (* 2.0 uy))) (sqrt (* ux 2.0)))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if ((2.0f * uy) <= 0.05950000137090683f) {
		tmp = uy * (sqrtf(fmaf(ux, fmaf(-2.0f, maxCos, 2.0f), ((ux * ux) * ((maxCos + -1.0f) * (1.0f - maxCos))))) * fmaf(-1.3333333333333333f, ((uy * uy) * (((float) M_PI) * (((float) M_PI) * ((float) M_PI)))), (2.0f * ((float) M_PI))));
	} else {
		tmp = sinf((((float) M_PI) * (2.0f * uy))) * sqrtf((ux * 2.0f));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (Float32(Float32(2.0) * uy) <= Float32(0.05950000137090683))
		tmp = Float32(uy * Float32(sqrt(fma(ux, fma(Float32(-2.0), maxCos, Float32(2.0)), Float32(Float32(ux * ux) * Float32(Float32(maxCos + Float32(-1.0)) * Float32(Float32(1.0) - maxCos))))) * fma(Float32(-1.3333333333333333), Float32(Float32(uy * uy) * Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi)))), Float32(Float32(2.0) * Float32(pi)))));
	else
		tmp = Float32(sin(Float32(Float32(pi) * Float32(Float32(2.0) * uy))) * sqrt(Float32(ux * Float32(2.0))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 0.0595000014

   1. Initial program 59.3%

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

   3. Taylor expanded in 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. *-lowering-*.f32 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f32 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

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

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

      9. metadata-eval N/A

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

      10. associate-+l+ N/A

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

   5. Simplified 98.3%

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

      1. *-commutative N/A

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

      2. *-lowering-*.f32 N/A

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

   7. Applied egg-rr 98.4%

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

   8. Taylor expanded in uy around 0

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

   10. Simplified 97.9%

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

   if 0.0595000014 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 64.4%

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

   3. Taylor expanded in maxCos around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

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

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

      5. mul-1-neg N/A

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

      6. accelerator-lowering-fma.f32 N/A

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

      7. --lowering--.f32 N/A

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

      8. mul-1-neg N/A

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

      9. neg-sub0 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{fma}\left(1 - ux, \color{blue}{0 - \left(1 - ux\right)}, 1\right)} \]

      10. --lowering--.f32 N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{fma}\left(1 - ux, \color{blue}{0 - \left(1 - ux\right)}, 1\right)} \]

      11. --lowering--.f32 62.1

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\mathsf{fma}\left(1 - ux, 0 - \color{blue}{\left(1 - ux\right)}, 1\right)} \]

   5. Simplified 62.1%

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(1 - ux, 0 - \left(1 - ux\right), 1\right)}} \]

   6. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f32 70.5

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

   8. Simplified 70.5%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.05950000137090683:\\ \;\;\;\;uy \cdot \left(\sqrt{\mathsf{fma}\left(ux, \mathsf{fma}\left(-2, maxCos, 2\right), \left(ux \cdot ux\right) \cdot \left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right)\right)} \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), 2 \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{ux \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 89.2% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 60.1%

   \[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
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. *-lowering-*.f32 N/A

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

   2. unpow2 N/A

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

   3. *-lowering-*.f32 N/A

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

   4. sub-neg N/A

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

   5. +-commutative N/A

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

   6. +-commutative N/A

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

   7. distribute-neg-in N/A

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

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

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

   9. metadata-eval N/A

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

   10. associate-+l+ N/A

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

5. Simplified 98.3%

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

   1. *-commutative N/A

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

   2. *-lowering-*.f32 N/A

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

7. Applied egg-rr 98.3%

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

8. Taylor expanded in uy around 0

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

10. Simplified 90.1%

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

Alternative 9: 89.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 60.1%

   \[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
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. *-lowering-*.f32 N/A

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

   2. unpow2 N/A

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

   3. *-lowering-*.f32 N/A

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

   4. sub-neg N/A

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

   5. +-commutative N/A

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

   6. +-commutative N/A

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

   7. distribute-neg-in N/A

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

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

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

   9. metadata-eval N/A

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

   10. associate-+l+ N/A

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

5. Simplified 98.3%

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

   1. *-commutative N/A

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

   2. *-lowering-*.f32 N/A

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

7. Applied egg-rr 98.3%

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

8. Taylor expanded in ux around 0

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

   1. *-lowering-*.f32 N/A

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

   2. +-lowering-+.f32 N/A

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

   3. +-commutative N/A

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

   4. sub-neg N/A

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

   5. mul-1-neg N/A

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

   6. accelerator-lowering-fma.f32 N/A

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

   7. *-commutative N/A

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

   8. *-lowering-*.f32 N/A

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

   9. sub-neg N/A

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

   10. metadata-eval N/A

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

   11. +-lowering-+.f32 N/A

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

   12. mul-1-neg N/A

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

   13. sub-neg N/A

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

   14. --lowering--.f32 N/A

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

   15. *-lowering-*.f32 98.3

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

10. Simplified 98.3%

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

11. Taylor expanded in uy around 0

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

    1. *-lowering-*.f32 N/A

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

    2. accelerator-lowering-fma.f32 N/A

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

    3. *-lowering-*.f32 N/A

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

    4. unpow2 N/A

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

    5. *-lowering-*.f32 N/A

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

    6. cube-mult N/A

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

    7. *-lowering-*.f32 N/A

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

    8. PI-lowering-PI.f32 N/A

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

    9. *-lowering-*.f32 N/A

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

    10. PI-lowering-PI.f32 N/A

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

    11. PI-lowering-PI.f32 N/A

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

    12. *-lowering-*.f32 N/A

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

    13. PI-lowering-PI.f32 90.0

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

13. Simplified 90.0%

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

14. Final simplification 90.0%

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

Alternative 10: 89.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 60.1%

   \[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
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. *-lowering-*.f32 N/A

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

   2. unpow2 N/A

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

   3. *-lowering-*.f32 N/A

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

   4. sub-neg N/A

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

   5. +-commutative N/A

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

   6. +-commutative N/A

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

   7. distribute-neg-in N/A

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

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

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

   9. metadata-eval N/A

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

   10. associate-+l+ N/A

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

5. Simplified 98.3%

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

   1. *-commutative N/A

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

   2. *-lowering-*.f32 N/A

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

7. Applied egg-rr 98.3%

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

   1. associate-*l* N/A

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

   2. *-commutative N/A

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

   3. distribute-lft-out N/A

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

   4. *-lowering-*.f32 N/A

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

   5. accelerator-lowering-fma.f32 N/A

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

   6. *-lowering-*.f32 N/A

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

   7. +-lowering-+.f32 N/A

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

   8. --lowering--.f32 N/A

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

   9. accelerator-lowering-fma.f32 98.2

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

9. Applied egg-rr 98.2%

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

10. Taylor expanded in uy around 0

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

    1. *-lowering-*.f32 N/A

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

    2. accelerator-lowering-fma.f32 N/A

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

    3. *-lowering-*.f32 N/A

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

    4. unpow2 N/A

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

    5. *-lowering-*.f32 N/A

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

    6. cube-mult N/A

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

    7. *-lowering-*.f32 N/A

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

    8. PI-lowering-PI.f32 N/A

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

    9. *-lowering-*.f32 N/A

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

    10. PI-lowering-PI.f32 N/A

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

    11. PI-lowering-PI.f32 N/A

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

    12. *-lowering-*.f32 N/A

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

    13. PI-lowering-PI.f32 89.9

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

12. Simplified 89.9%

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

13. Final simplification 89.9%

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

Alternative 11: 81.6% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 60.1%

   \[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
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. *-lowering-*.f32 N/A

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

   2. unpow2 N/A

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

   3. *-lowering-*.f32 N/A

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

   4. sub-neg N/A

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

   5. +-commutative N/A

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

   6. +-commutative N/A

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

   7. distribute-neg-in N/A

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

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

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

   9. metadata-eval N/A

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

   10. associate-+l+ N/A

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

5. Simplified 98.3%

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

   1. *-commutative N/A

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

   2. *-lowering-*.f32 N/A

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

7. Applied egg-rr 98.3%

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

8. Taylor expanded in uy around 0

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

10. Simplified 82.4%

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

Alternative 12: 81.6% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 60.1%

   \[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
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. *-lowering-*.f32 N/A

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

   2. unpow2 N/A

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

   3. *-lowering-*.f32 N/A

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

   4. sub-neg N/A

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

   5. +-commutative N/A

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

   6. +-commutative N/A

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

   7. distribute-neg-in N/A

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

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

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

   9. metadata-eval N/A

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

   10. associate-+l+ N/A

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

5. Simplified 98.3%

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

   1. *-commutative N/A

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

   2. *-lowering-*.f32 N/A

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

7. Applied egg-rr 98.3%

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

8. Taylor expanded in uy around 0

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

   1. *-lowering-*.f32 N/A

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

   2. *-lowering-*.f32 N/A

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

   3. PI-lowering-PI.f32 82.4

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

10. Simplified 82.4%

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

11. Final simplification 82.4%

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

Alternative 13: 81.6% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 60.1%

   \[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
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. *-commutative N/A

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

   5. *-lowering-*.f32 N/A

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

5. Simplified 52.9%

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. associate-+r+ N/A

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

   4. +-lowering-+.f32 N/A

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

7. Applied egg-rr 52.6%

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

8. Taylor expanded in uy around 0

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

   1. associate-*r* N/A

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

   2. *-lowering-*.f32 N/A

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

   3. *-lowering-*.f32 N/A

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

   4. *-lowering-*.f32 N/A

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

   5. PI-lowering-PI.f32 N/A

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

   6. sqrt-lowering-sqrt.f32 N/A

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

   7. +-commutative N/A

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

   8. *-commutative N/A

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

   9. *-lft-identity N/A

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

   10. associate-*r* N/A

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

   11. distribute-rgt-in N/A

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

   12. mul-1-neg N/A

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

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

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

   14. sub-neg N/A

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

   15. metadata-eval N/A

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

10. Simplified 82.4%

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

Alternative 14: 81.6% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 60.1%

   \[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
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. *-lowering-*.f32 N/A

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

   2. unpow2 N/A

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

   3. *-lowering-*.f32 N/A

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

   4. sub-neg N/A

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

   5. +-commutative N/A

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

   6. +-commutative N/A

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

   7. distribute-neg-in N/A

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

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

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

   9. metadata-eval N/A

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

   10. associate-+l+ N/A

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

5. Simplified 98.3%

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

   1. *-commutative N/A

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

   2. *-lowering-*.f32 N/A

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

7. Applied egg-rr 98.3%

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

8. Taylor expanded in ux around 0

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

   1. *-lowering-*.f32 N/A

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

   2. +-lowering-+.f32 N/A

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

   3. +-commutative N/A

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

   4. sub-neg N/A

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

   5. mul-1-neg N/A

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

   6. accelerator-lowering-fma.f32 N/A

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

   7. *-commutative N/A

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

   8. *-lowering-*.f32 N/A

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

   9. sub-neg N/A

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

   10. metadata-eval N/A

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

   11. +-lowering-+.f32 N/A

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

   12. mul-1-neg N/A

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

   13. sub-neg N/A

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

   14. --lowering--.f32 N/A

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

   15. *-lowering-*.f32 98.3

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

10. Simplified 98.3%

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

11. Taylor expanded in uy around 0

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

    1. *-lowering-*.f32 N/A

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

    2. *-lowering-*.f32 N/A

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

    3. PI-lowering-PI.f32 82.4

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

13. Simplified 82.4%

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

14. Final simplification 82.4%

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

Alternative 15: 81.6% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 60.1%

   \[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
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. *-lowering-*.f32 N/A

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

   2. unpow2 N/A

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

   3. *-lowering-*.f32 N/A

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

   4. sub-neg N/A

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

   5. +-commutative N/A

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

   6. +-commutative N/A

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

   7. distribute-neg-in N/A

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

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

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

   9. metadata-eval N/A

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

   10. associate-+l+ N/A

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

5. Simplified 98.3%

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

   1. *-commutative N/A

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

   2. *-lowering-*.f32 N/A

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

7. Applied egg-rr 98.3%

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

   1. associate-*l* N/A

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

   2. *-commutative N/A

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

   3. distribute-lft-out N/A

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

   4. *-lowering-*.f32 N/A

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

   5. accelerator-lowering-fma.f32 N/A

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

   6. *-lowering-*.f32 N/A

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

   7. +-lowering-+.f32 N/A

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

   8. --lowering--.f32 N/A

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

   9. accelerator-lowering-fma.f32 98.2

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

9. Applied egg-rr 98.2%

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

10. Taylor expanded in uy around 0

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

    1. *-lowering-*.f32 N/A

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

    2. *-lowering-*.f32 N/A

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

    3. PI-lowering-PI.f32 82.3

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

12. Simplified 82.3%

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

13. Final simplification 82.3%

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

Alternative 16: 81.5% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 60.1%

   \[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
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. *-commutative N/A

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

   5. *-lowering-*.f32 N/A

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

5. Simplified 52.9%

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. associate-+r+ N/A

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

   4. +-lowering-+.f32 N/A

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

7. Applied egg-rr 52.6%

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

8. Taylor expanded in ux around 0

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

   1. *-lowering-*.f32 N/A

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

   2. associate-+r+ N/A

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

   3. +-commutative N/A

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

   4. mul-1-neg N/A

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

   5. sub-neg N/A

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

   6. metadata-eval N/A

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

   7. distribute-neg-in N/A

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

   8. mul-1-neg N/A

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

   9. metadata-eval N/A

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

   10. +-commutative N/A

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

   11. associate-+r+ N/A

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

10. Simplified 82.3%

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

11. Final simplification 82.3%

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

Alternative 17: 77.3% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 60.1%

   \[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
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. *-commutative N/A

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

   5. *-lowering-*.f32 N/A

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

5. Simplified 52.9%

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. associate-+r+ N/A

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

   4. +-lowering-+.f32 N/A

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

7. Applied egg-rr 52.6%

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

8. Taylor expanded in maxCos around 0

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

   1. associate-*r* N/A

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

   2. *-lowering-*.f32 N/A

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

   3. *-lowering-*.f32 N/A

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

   4. *-lowering-*.f32 N/A

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

   5. PI-lowering-PI.f32 N/A

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

   6. sqrt-lowering-sqrt.f32 N/A

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

   7. +-commutative N/A

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

   8. accelerator-lowering-fma.f32 N/A

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

   9. mul-1-neg N/A

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

   10. sub-neg N/A

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

   11. --lowering--.f32 77.1

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

10. Simplified 77.1%

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

Alternative 18: 30.8% accurate, 6.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 60.1%

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

3. Taylor expanded in maxCos around 0

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. unpow2 N/A

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

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

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

   5. mul-1-neg N/A

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

   6. accelerator-lowering-fma.f32 N/A

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

   7. --lowering--.f32 N/A

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

   8. mul-1-neg N/A

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

   9. neg-sub0 N/A

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{fma}\left(1 - ux, \color{blue}{0 - \left(1 - ux\right)}, 1\right)} \]

   10. --lowering--.f32 N/A

       \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{fma}\left(1 - ux, \color{blue}{0 - \left(1 - ux\right)}, 1\right)} \]

   11. --lowering--.f32 57.4

       \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\mathsf{fma}\left(1 - ux, 0 - \color{blue}{\left(1 - ux\right)}, 1\right)} \]

5. Simplified 57.4%

   \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(1 - ux, 0 - \left(1 - ux\right), 1\right)}} \]

6. Taylor expanded in ux around 0

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

8. Simplified 27.0%

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

9. Taylor expanded in uy around 0

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

    1. *-lowering-*.f32 N/A

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

    2. *-lowering-*.f32 N/A

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

    3. *-lowering-*.f32 N/A

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

    4. PI-lowering-PI.f32 N/A

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

    5. sqrt-lowering-sqrt.f32 N/A

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

    6. mul-1-neg N/A

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

    7. sub-neg N/A

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

    8. mul-1-neg N/A

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

    9. unsub-neg N/A

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

    10. --lowering--.f32 N/A

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

    11. mul-1-neg N/A

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

    12. sub-neg N/A

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

    13. --lowering--.f32 26.2

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

11. Simplified 26.2%

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

    1. *-commutative N/A

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

    2. associate-*r* N/A

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

    3. *-lowering-*.f32 N/A

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

    4. *-lowering-*.f32 N/A

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

    5. associate--r- N/A

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

    6. metadata-eval N/A

       \[\leadsto 2 \cdot \left(\left(\sqrt{\color{blue}{0} + ux} \cdot uy\right) \cdot \mathsf{PI}\left(\right)\right) \]

    7. +-lft-identity N/A

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

    8. sqrt-lowering-sqrt.f32 N/A

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

    9. PI-lowering-PI.f32 30.7

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

13. Applied egg-rr 30.7%

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

14. Final simplification 30.7%

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

Reproduce

herbie shell --seed 2024195 
(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)))))))