UniformSampleCone, y

Percentage Accurate: 57.9% → 98.4%

Time: 20.9s

Alternatives: 23

Speedup: 5.0×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 57.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.4% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 56.6%

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

3. Taylor expanded in ux around inf

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

   1. *-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%

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

7. Applied egg-rr 98.4%

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

8. Final simplification 98.4%

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

Alternative 2: 97.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos + -1\right) \cdot \left(1 - maxCos\right), \mathsf{fma}\left(maxCos, -2, 2\right)\right)}\\ \mathbf{if}\;uy \cdot 2 \leq 0.02879999950528145:\\ \;\;\;\;\mathsf{fma}\left(t\_0 \cdot -1.3333333333333333, uy \cdot \left(uy \cdot \left(uy \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right)\right), uy \cdot \left(t\_0 \cdot \left(2 \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(ux \cdot ux\right) \cdot \left(-1 + \frac{2}{ux}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0
         (sqrt
          (*
           ux
           (fma
            ux
            (* (+ maxCos -1.0) (- 1.0 maxCos))
            (fma maxCos -2.0 2.0))))))
   (if (<= (* uy 2.0) 0.02879999950528145)
     (fma
      (* t_0 -1.3333333333333333)
      (* uy (* uy (* uy (* PI (* PI PI)))))
      (* uy (* t_0 (* 2.0 PI))))
     (* (sin (* (* uy 2.0) PI)) (sqrt (* (* ux ux) (+ -1.0 (/ 2.0 ux))))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = sqrtf((ux * fmaf(ux, ((maxCos + -1.0f) * (1.0f - maxCos)), fmaf(maxCos, -2.0f, 2.0f))));
	float tmp;
	if ((uy * 2.0f) <= 0.02879999950528145f) {
		tmp = fmaf((t_0 * -1.3333333333333333f), (uy * (uy * (uy * (((float) M_PI) * (((float) M_PI) * ((float) M_PI)))))), (uy * (t_0 * (2.0f * ((float) M_PI)))));
	} else {
		tmp = sinf(((uy * 2.0f) * ((float) M_PI))) * sqrtf(((ux * ux) * (-1.0f + (2.0f / ux))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = sqrt(Float32(ux * fma(ux, Float32(Float32(maxCos + Float32(-1.0)) * Float32(Float32(1.0) - maxCos)), fma(maxCos, Float32(-2.0), Float32(2.0)))))
	tmp = Float32(0.0)
	if (Float32(uy * Float32(2.0)) <= Float32(0.02879999950528145))
		tmp = fma(Float32(t_0 * Float32(-1.3333333333333333)), Float32(uy * Float32(uy * Float32(uy * Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi)))))), Float32(uy * Float32(t_0 * Float32(Float32(2.0) * Float32(pi)))));
	else
		tmp = Float32(sin(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) * sqrt(Float32(Float32(ux * ux) * 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)) < 0.0287999995

   1. Initial program 56.3%

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

   3. Taylor expanded in ux around 0

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

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

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

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

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

      5. associate-+l+ N/A

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

      6. mul-1-neg N/A

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

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

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

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

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

   5. Simplified 98.4%

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

   6. Taylor expanded in uy around 0

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

   8. Simplified 98.5%

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

   10. Applied egg-rr 98.6%

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

   if 0.0287999995 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 58.3%

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

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

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

      10. --lowering--.f32 55.4

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

   5. Simplified 55.4%

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

   6. Taylor expanded in ux around inf

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

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

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

      2. unpow2 N/A

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

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

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

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

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. /-lowering-/.f32 91.0

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

   8. Simplified 91.0%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.02879999950528145:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos + -1\right) \cdot \left(1 - maxCos\right), \mathsf{fma}\left(maxCos, -2, 2\right)\right)} \cdot -1.3333333333333333, uy \cdot \left(uy \cdot \left(uy \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right)\right), uy \cdot \left(\sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos + -1\right) \cdot \left(1 - maxCos\right), \mathsf{fma}\left(maxCos, -2, 2\right)\right)} \cdot \left(2 \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(ux \cdot ux\right) \cdot \left(-1 + \frac{2}{ux}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 56.6%

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

3. Taylor expanded in ux around inf

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

   1. *-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. distribute-lft-in N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. div-inv N/A

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

   5. associate-*l* N/A

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

   6. inv-pow N/A

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

   7. pow2 N/A

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

   8. pow-prod-up N/A

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

   9. metadata-eval N/A

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

   10. unpow1 N/A

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

   11. *-commutative N/A

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

   12. distribute-lft-in N/A

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

   13. associate-+r+ N/A

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

   14. distribute-rgt-in N/A

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

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

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

7. Applied egg-rr 98.4%

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

Alternative 4: 98.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (sin (* (* uy 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 sinf(((uy * 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(sin(Float32(Float32(uy * 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 56.6%

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

3. Taylor expanded in ux around 0

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

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

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

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

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

   3. +-commutative N/A

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

   4. metadata-eval N/A

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

   5. associate-+l+ N/A

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

   6. mul-1-neg N/A

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

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

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

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

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

5. Simplified 98.3%

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

Alternative 5: 97.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 56.6%

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

3. Taylor expanded in ux around inf

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

   1. *-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. distribute-lft-in N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. div-inv N/A

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

   5. associate-*l* N/A

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

   6. inv-pow N/A

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

   7. pow2 N/A

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

   8. pow-prod-up N/A

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

   9. metadata-eval N/A

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

   10. unpow1 N/A

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

   11. *-commutative N/A

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

   12. distribute-lft-in N/A

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

   13. associate-+r+ N/A

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

   14. distribute-rgt-in N/A

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

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

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

7. Applied egg-rr 98.4%

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

8. Taylor expanded in maxCos around 0

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

   1. associate-+r+ N/A

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

   2. +-commutative N/A

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

   3. mul-1-neg N/A

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

   4. unpow2 N/A

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

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

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

   6. mul-1-neg N/A

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

   7. distribute-rgt-in N/A

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

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

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

   9. mul-1-neg N/A

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

   10. unsub-neg N/A

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

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

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

   12. associate-*r* N/A

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

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

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

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

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

   15. sub-neg N/A

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

   16. metadata-eval N/A

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

   17. accelerator-lowering-fma.f32 97.8

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

10. Simplified 97.8%

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

11. Final simplification 97.8%

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

Alternative 6: 97.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos + -1\right) \cdot \left(1 - maxCos\right), \mathsf{fma}\left(maxCos, -2, 2\right)\right)}\\ \mathbf{if}\;uy \cdot 2 \leq 0.02879999950528145:\\ \;\;\;\;\mathsf{fma}\left(t\_0 \cdot -1.3333333333333333, uy \cdot \left(uy \cdot \left(uy \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right)\right), uy \cdot \left(t\_0 \cdot \left(2 \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0
         (sqrt
          (*
           ux
           (fma
            ux
            (* (+ maxCos -1.0) (- 1.0 maxCos))
            (fma maxCos -2.0 2.0))))))
   (if (<= (* uy 2.0) 0.02879999950528145)
     (fma
      (* t_0 -1.3333333333333333)
      (* uy (* uy (* uy (* PI (* PI PI)))))
      (* uy (* t_0 (* 2.0 PI))))
     (* (sin (* (* uy 2.0) PI)) (sqrt (* ux (- 2.0 ux)))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = sqrtf((ux * fmaf(ux, ((maxCos + -1.0f) * (1.0f - maxCos)), fmaf(maxCos, -2.0f, 2.0f))));
	float tmp;
	if ((uy * 2.0f) <= 0.02879999950528145f) {
		tmp = fmaf((t_0 * -1.3333333333333333f), (uy * (uy * (uy * (((float) M_PI) * (((float) M_PI) * ((float) M_PI)))))), (uy * (t_0 * (2.0f * ((float) M_PI)))));
	} else {
		tmp = sinf(((uy * 2.0f) * ((float) M_PI))) * sqrtf((ux * (2.0f - ux)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = sqrt(Float32(ux * fma(ux, Float32(Float32(maxCos + Float32(-1.0)) * Float32(Float32(1.0) - maxCos)), fma(maxCos, Float32(-2.0), Float32(2.0)))))
	tmp = Float32(0.0)
	if (Float32(uy * Float32(2.0)) <= Float32(0.02879999950528145))
		tmp = fma(Float32(t_0 * Float32(-1.3333333333333333)), Float32(uy * Float32(uy * Float32(uy * Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi)))))), Float32(uy * Float32(t_0 * Float32(Float32(2.0) * Float32(pi)))));
	else
		tmp = Float32(sin(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) * 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)) < 0.0287999995

   1. Initial program 56.3%

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

   3. Taylor expanded in ux around 0

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

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

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

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

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

      5. associate-+l+ N/A

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

      6. mul-1-neg N/A

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

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

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

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

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

   5. Simplified 98.4%

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

   6. Taylor expanded in uy around 0

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

   8. Simplified 98.5%

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

   10. Applied egg-rr 98.6%

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

   if 0.0287999995 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 58.3%

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

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

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

      10. --lowering--.f32 55.4

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

   5. Simplified 55.4%

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

   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 90.8

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

   8. Simplified 90.8%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.02879999950528145:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos + -1\right) \cdot \left(1 - maxCos\right), \mathsf{fma}\left(maxCos, -2, 2\right)\right)} \cdot -1.3333333333333333, uy \cdot \left(uy \cdot \left(uy \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right)\right), uy \cdot \left(\sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos + -1\right) \cdot \left(1 - maxCos\right), \mathsf{fma}\left(maxCos, -2, 2\right)\right)} \cdot \left(2 \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 94.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos + -1\right) \cdot \left(1 - maxCos\right), \mathsf{fma}\left(maxCos, -2, 2\right)\right)}\\ \mathbf{if}\;uy \cdot 2 \leq 0.05999999865889549:\\ \;\;\;\;\mathsf{fma}\left(t\_0 \cdot -1.3333333333333333, uy \cdot \left(uy \cdot \left(uy \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right)\right), uy \cdot \left(t\_0 \cdot \left(2 \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{2 \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0
         (sqrt
          (*
           ux
           (fma
            ux
            (* (+ maxCos -1.0) (- 1.0 maxCos))
            (fma maxCos -2.0 2.0))))))
   (if (<= (* uy 2.0) 0.05999999865889549)
     (fma
      (* t_0 -1.3333333333333333)
      (* uy (* uy (* uy (* PI (* PI PI)))))
      (* uy (* t_0 (* 2.0 PI))))
     (* (sin (* (* uy 2.0) PI)) (sqrt (* 2.0 ux))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = sqrtf((ux * fmaf(ux, ((maxCos + -1.0f) * (1.0f - maxCos)), fmaf(maxCos, -2.0f, 2.0f))));
	float tmp;
	if ((uy * 2.0f) <= 0.05999999865889549f) {
		tmp = fmaf((t_0 * -1.3333333333333333f), (uy * (uy * (uy * (((float) M_PI) * (((float) M_PI) * ((float) M_PI)))))), (uy * (t_0 * (2.0f * ((float) M_PI)))));
	} else {
		tmp = sinf(((uy * 2.0f) * ((float) M_PI))) * sqrtf((2.0f * ux));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = sqrt(Float32(ux * fma(ux, Float32(Float32(maxCos + Float32(-1.0)) * Float32(Float32(1.0) - maxCos)), fma(maxCos, Float32(-2.0), Float32(2.0)))))
	tmp = Float32(0.0)
	if (Float32(uy * Float32(2.0)) <= Float32(0.05999999865889549))
		tmp = fma(Float32(t_0 * Float32(-1.3333333333333333)), Float32(uy * Float32(uy * Float32(uy * Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi)))))), Float32(uy * Float32(t_0 * Float32(Float32(2.0) * Float32(pi)))));
	else
		tmp = Float32(sin(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) * sqrt(Float32(Float32(2.0) * ux)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 56.8%

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

   3. Taylor expanded in ux around 0

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

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

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

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

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

      5. associate-+l+ N/A

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

      6. mul-1-neg N/A

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

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

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

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

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

   5. Simplified 98.5%

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

   6. Taylor expanded in uy around 0

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

   8. Simplified 98.3%

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

   10. Applied egg-rr 98.4%

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

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

   1. Initial program 55.5%

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

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

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

      10. --lowering--.f32 52.2

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

   5. Simplified 52.2%

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

   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. *-lowering-*.f32 72.4

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

   8. Simplified 72.4%

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

4. Final simplification 94.3%

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

Alternative 8: 89.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos + -1\right) \cdot \left(1 - maxCos\right), \mathsf{fma}\left(maxCos, -2, 2\right)\right)}\\ \mathsf{fma}\left(t\_0 \cdot \left(-1.3333333333333333 \cdot \left(uy \cdot \left(uy \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\right), uy, uy \cdot \left(t\_0 \cdot \left(2 \cdot \pi\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0
         (sqrt
          (*
           ux
           (fma
            ux
            (* (+ maxCos -1.0) (- 1.0 maxCos))
            (fma maxCos -2.0 2.0))))))
   (fma
    (* t_0 (* -1.3333333333333333 (* uy (* uy (* PI (* PI PI))))))
    uy
    (* uy (* t_0 (* 2.0 PI))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = sqrtf((ux * fmaf(ux, ((maxCos + -1.0f) * (1.0f - maxCos)), fmaf(maxCos, -2.0f, 2.0f))));
	return fmaf((t_0 * (-1.3333333333333333f * (uy * (uy * (((float) M_PI) * (((float) M_PI) * ((float) M_PI))))))), uy, (uy * (t_0 * (2.0f * ((float) M_PI)))));
}

Julia

function code(ux, uy, maxCos)
	t_0 = sqrt(Float32(ux * fma(ux, Float32(Float32(maxCos + Float32(-1.0)) * Float32(Float32(1.0) - maxCos)), fma(maxCos, Float32(-2.0), Float32(2.0)))))
	return fma(Float32(t_0 * Float32(Float32(-1.3333333333333333) * Float32(uy * Float32(uy * Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi))))))), uy, Float32(uy * Float32(t_0 * Float32(Float32(2.0) * Float32(pi)))))
end

Derivation

1. Initial program 56.6%

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

3. Taylor expanded in ux around 0

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

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

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

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

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

   3. +-commutative N/A

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

   4. metadata-eval N/A

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

   5. associate-+l+ N/A

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

   6. mul-1-neg N/A

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

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

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

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

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

5. Simplified 98.3%

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

6. Taylor expanded in uy around 0

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

8. Simplified 88.8%

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-commutative N/A

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

10. Applied egg-rr 88.9%

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

11. Final simplification 88.9%

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

Alternative 9: 89.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos + -1\right) \cdot \left(1 - maxCos\right), \mathsf{fma}\left(maxCos, -2, 2\right)\right)}\\ \mathsf{fma}\left(t\_0 \cdot -1.3333333333333333, uy \cdot \left(uy \cdot \left(uy \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right)\right), uy \cdot \left(t\_0 \cdot \left(2 \cdot \pi\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0
         (sqrt
          (*
           ux
           (fma
            ux
            (* (+ maxCos -1.0) (- 1.0 maxCos))
            (fma maxCos -2.0 2.0))))))
   (fma
    (* t_0 -1.3333333333333333)
    (* uy (* uy (* uy (* PI (* PI PI)))))
    (* uy (* t_0 (* 2.0 PI))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = sqrtf((ux * fmaf(ux, ((maxCos + -1.0f) * (1.0f - maxCos)), fmaf(maxCos, -2.0f, 2.0f))));
	return fmaf((t_0 * -1.3333333333333333f), (uy * (uy * (uy * (((float) M_PI) * (((float) M_PI) * ((float) M_PI)))))), (uy * (t_0 * (2.0f * ((float) M_PI)))));
}

Julia

function code(ux, uy, maxCos)
	t_0 = sqrt(Float32(ux * fma(ux, Float32(Float32(maxCos + Float32(-1.0)) * Float32(Float32(1.0) - maxCos)), fma(maxCos, Float32(-2.0), Float32(2.0)))))
	return fma(Float32(t_0 * Float32(-1.3333333333333333)), Float32(uy * Float32(uy * Float32(uy * Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi)))))), Float32(uy * Float32(t_0 * Float32(Float32(2.0) * Float32(pi)))))
end

Derivation

1. Initial program 56.6%

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

3. Taylor expanded in ux around 0

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

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

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

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

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

   3. +-commutative N/A

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

   4. metadata-eval N/A

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

   5. associate-+l+ N/A

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

   6. mul-1-neg N/A

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

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

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

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

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

5. Simplified 98.3%

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

6. Taylor expanded in uy around 0

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

8. Simplified 88.8%

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

10. Applied egg-rr 88.9%

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

11. Final simplification 88.9%

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

Alternative 10: 89.3% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 56.6%

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

3. Taylor expanded in ux around 0

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

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

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

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

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

   3. +-commutative N/A

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

   4. metadata-eval N/A

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

   5. associate-+l+ N/A

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

   6. mul-1-neg N/A

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

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

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

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

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

5. Simplified 98.3%

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

6. Taylor expanded in uy around 0

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

8. Simplified 88.8%

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

   1. distribute-rgt-in N/A

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

   2. associate-*r* N/A

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

   3. distribute-rgt-in N/A

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

   4. *-commutative N/A

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

   5. neg-mul-1 N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. +-commutative N/A

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

   9. *-commutative N/A

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

   10. +-commutative N/A

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

   11. rem-square-sqrt N/A

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

   12. sqrt-prod N/A

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

   13. *-commutative N/A

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

   14. associate-*l* N/A

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

   15. *-commutative N/A

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

   16. sqrt-unprod N/A

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

10. Applied egg-rr 88.9%

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

11. Final simplification 88.9%

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

Alternative 11: 89.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos + -1\right) \cdot \left(1 - maxCos\right), \mathsf{fma}\left(maxCos, -2, 2\right)\right)}\\ uy \cdot \mathsf{fma}\left(2 \cdot t\_0, \pi, t\_0 \cdot \left(-1.3333333333333333 \cdot \left(uy \cdot \left(uy \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0
         (sqrt
          (*
           ux
           (fma
            ux
            (* (+ maxCos -1.0) (- 1.0 maxCos))
            (fma maxCos -2.0 2.0))))))
   (*
    uy
    (fma
     (* 2.0 t_0)
     PI
     (* t_0 (* -1.3333333333333333 (* uy (* uy (* PI (* PI PI))))))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = sqrtf((ux * fmaf(ux, ((maxCos + -1.0f) * (1.0f - maxCos)), fmaf(maxCos, -2.0f, 2.0f))));
	return uy * fmaf((2.0f * t_0), ((float) M_PI), (t_0 * (-1.3333333333333333f * (uy * (uy * (((float) M_PI) * (((float) M_PI) * ((float) M_PI))))))));
}

Julia

function code(ux, uy, maxCos)
	t_0 = sqrt(Float32(ux * fma(ux, Float32(Float32(maxCos + Float32(-1.0)) * Float32(Float32(1.0) - maxCos)), fma(maxCos, Float32(-2.0), Float32(2.0)))))
	return Float32(uy * fma(Float32(Float32(2.0) * t_0), Float32(pi), Float32(t_0 * Float32(Float32(-1.3333333333333333) * Float32(uy * Float32(uy * Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi)))))))))
end

Derivation

1. Initial program 56.6%

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

3. Taylor expanded in ux around 0

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

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

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

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

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

   3. +-commutative N/A

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

   4. metadata-eval N/A

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

   5. associate-+l+ N/A

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

   6. mul-1-neg N/A

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

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

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

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

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

5. Simplified 98.3%

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

6. Taylor expanded in uy around 0

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

8. Simplified 88.8%

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

   1. associate-*r* N/A

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

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

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

10. Applied egg-rr 88.8%

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

11. Final simplification 88.8%

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

Alternative 12: 89.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos + -1\right) \cdot \left(1 - maxCos\right), \mathsf{fma}\left(-2, maxCos, 2\right)\right)}\\ uy \cdot \mathsf{fma}\left(2, \pi \cdot t\_0, -1.3333333333333333 \cdot \left(t\_0 \cdot \left(\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(uy \cdot uy\right)\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0
         (sqrt
          (*
           ux
           (fma
            ux
            (* (+ maxCos -1.0) (- 1.0 maxCos))
            (fma -2.0 maxCos 2.0))))))
   (*
    uy
    (fma
     2.0
     (* PI t_0)
     (* -1.3333333333333333 (* t_0 (* (* PI (* PI PI)) (* uy uy))))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = sqrtf((ux * fmaf(ux, ((maxCos + -1.0f) * (1.0f - maxCos)), fmaf(-2.0f, maxCos, 2.0f))));
	return uy * fmaf(2.0f, (((float) M_PI) * t_0), (-1.3333333333333333f * (t_0 * ((((float) M_PI) * (((float) M_PI) * ((float) M_PI))) * (uy * uy)))));
}

Julia

function code(ux, uy, maxCos)
	t_0 = sqrt(Float32(ux * fma(ux, Float32(Float32(maxCos + Float32(-1.0)) * Float32(Float32(1.0) - maxCos)), fma(Float32(-2.0), maxCos, Float32(2.0)))))
	return Float32(uy * fma(Float32(2.0), Float32(Float32(pi) * t_0), Float32(Float32(-1.3333333333333333) * Float32(t_0 * Float32(Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi))) * Float32(uy * uy))))))
end

Derivation

1. Initial program 56.6%

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

3. Taylor expanded in ux around 0

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

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

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

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

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

   3. +-commutative N/A

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

   4. metadata-eval N/A

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

   5. associate-+l+ N/A

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

   6. mul-1-neg N/A

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

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

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

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

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

5. Simplified 98.3%

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

6. Taylor expanded in uy around 0

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

8. Simplified 88.8%

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

9. Final simplification 88.8%

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

Alternative 13: 89.4% accurate, 1.3× speedup

Math

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

FPCore

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

C

float code(float ux, float uy, float maxCos) {
	return sqrtf(fmaf(sqrtf((ux * (ux * fmaf(maxCos, -2.0f, 2.0f)))), sqrtf(fmaf(maxCos, -2.0f, 2.0f)), (fmaf(ux, maxCos, -ux) * fmaf(maxCos, -ux, ux)))) * (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(fma(sqrt(Float32(ux * Float32(ux * fma(maxCos, Float32(-2.0), Float32(2.0))))), sqrt(fma(maxCos, Float32(-2.0), Float32(2.0))), Float32(fma(ux, maxCos, Float32(-ux)) * fma(maxCos, Float32(-ux), ux)))) * Float32(uy * fma(Float32(Float32(-1.3333333333333333) * Float32(uy * uy)), Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi))), Float32(Float32(2.0) * Float32(pi)))))
end

Derivation

1. Initial program 56.6%

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

3. Taylor expanded in ux around inf

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

   1. *-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%

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

7. Applied egg-rr 98.4%

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

8. Taylor expanded in uy around 0

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

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

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

   2. associate-*r* N/A

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

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

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

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

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

   5. unpow2 N/A

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

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

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

   7. cube-mult N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   14. PI-lowering-PI.f32 88.8

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

10. Simplified 88.8%

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

11. Final simplification 88.8%

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

Alternative 14: 89.3% accurate, 1.8× speedup

Math

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

FPCore

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

C

float code(float ux, float uy, float maxCos) {
	return sqrtf(fmaf(fmaf(ux, ((maxCos + -1.0f) * (1.0f - maxCos)), (maxCos * -2.0f)), ux, (2.0f * ux))) * (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(fma(fma(ux, Float32(Float32(maxCos + Float32(-1.0)) * Float32(Float32(1.0) - maxCos)), Float32(maxCos * Float32(-2.0))), ux, Float32(Float32(2.0) * ux))) * Float32(uy * fma(Float32(Float32(-1.3333333333333333) * Float32(uy * uy)), Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi))), Float32(Float32(2.0) * Float32(pi)))))
end

Derivation

1. Initial program 56.6%

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

3. Taylor expanded in ux around inf

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

   1. *-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. distribute-lft-in N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. div-inv N/A

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

   5. associate-*l* N/A

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

   6. inv-pow N/A

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

   7. pow2 N/A

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

   8. pow-prod-up N/A

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

   9. metadata-eval N/A

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

   10. unpow1 N/A

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

   11. *-commutative N/A

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

   12. distribute-lft-in N/A

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

   13. associate-+r+ N/A

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

   14. distribute-rgt-in N/A

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

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

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

7. Applied egg-rr 98.4%

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

8. Taylor expanded in uy around 0

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

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

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

   2. associate-*r* N/A

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

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

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

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

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

   5. unpow2 N/A

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

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

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

   7. cube-mult N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   14. PI-lowering-PI.f32 88.8

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

10. Simplified 88.8%

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

11. Final simplification 88.8%

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

Alternative 15: 89.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 56.6%

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

3. Taylor expanded in ux around 0

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

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

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

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

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

   3. +-commutative N/A

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

   4. metadata-eval N/A

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

   5. associate-+l+ N/A

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

   6. mul-1-neg N/A

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

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

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

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

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

5. Simplified 98.3%

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

6. Taylor expanded in uy around 0

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

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

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

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

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

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

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

   4. unpow2 N/A

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

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

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

   6. cube-mult N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   13. PI-lowering-PI.f32 88.7

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

8. Simplified 88.7%

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

9. Final simplification 88.7%

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

Alternative 16: 81.2% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 56.6%

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

3. Taylor expanded in ux around 0

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

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

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

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

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

   3. +-commutative N/A

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

   4. metadata-eval N/A

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

   5. associate-+l+ N/A

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

   6. mul-1-neg N/A

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

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

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

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

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

5. Simplified 98.3%

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

6. Taylor expanded in uy around 0

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

8. Simplified 88.8%

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

9. Taylor expanded in uy around 0

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

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

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

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

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

11. Simplified 82.8%

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

12. Final simplification 82.8%

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

Alternative 17: 81.1% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 56.6%

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

3. Taylor expanded in ux around 0

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

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

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

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

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

   3. +-commutative N/A

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

   4. metadata-eval N/A

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

   5. associate-+l+ N/A

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

   6. mul-1-neg N/A

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

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

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

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

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

5. Simplified 98.3%

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

6. Taylor expanded in uy around 0

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

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

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

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

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

   3. PI-lowering-PI.f32 82.7

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

8. Simplified 82.7%

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

9. Final simplification 82.7%

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

Alternative 18: 77.2% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 56.6%

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

3. Taylor expanded in uy around 0

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

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

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

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

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

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

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

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

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

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

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

   6. sub-neg N/A

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

   7. +-commutative N/A

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

   8. unpow2 N/A

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

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

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

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

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

5. Simplified 49.9%

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

   1. distribute-lft-in N/A

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

   2. associate-+l+ N/A

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

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

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

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

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

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

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

   6. distribute-rgt-in N/A

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

   7. neg-mul-1 N/A

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

   8. associate-*r* N/A

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

   9. metadata-eval N/A

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

   10. *-lft-identity N/A

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

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

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

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

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

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

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

7. Applied egg-rr 56.3%

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

8. Taylor expanded in maxCos around 0

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

10. Simplified 78.1%

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

Alternative 19: 76.9% accurate, 4.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 56.6%

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

3. Taylor expanded in ux around 0

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

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

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

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

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

   3. +-commutative N/A

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

   4. metadata-eval N/A

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

   5. associate-+l+ N/A

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

   6. mul-1-neg N/A

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

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

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

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

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

5. Simplified 98.3%

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

6. Taylor expanded in uy around 0

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

8. Simplified 88.8%

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

9. Taylor expanded in maxCos around 0

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

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

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

    2. distribute-rgt-in N/A

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

    3. mul-1-neg N/A

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

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

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

    5. unpow2 N/A

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

    6. mul-1-neg N/A

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

    7. +-commutative N/A

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

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

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

    9. +-commutative N/A

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

    10. mul-1-neg N/A

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

    11. unpow2 N/A

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

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

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

    13. mul-1-neg N/A

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

    14. distribute-rgt-in N/A

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

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

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

    16. mul-1-neg N/A

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

    17. unsub-neg N/A

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

    18. --lowering--.f32 N/A

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

    19. PI-lowering-PI.f32 82.9

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

11. Simplified 82.9%

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

12. Taylor expanded in uy around 0

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

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

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

    2. *-commutative N/A

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

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

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

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

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

    5. sub-neg N/A

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

    6. mul-1-neg N/A

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

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

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

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

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

    9. mul-1-neg N/A

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

    10. sub-neg N/A

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

    11. --lowering--.f32 77.8

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

14. Simplified 77.8%

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

Alternative 20: 76.9% accurate, 4.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 56.6%

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

3. Taylor expanded in uy around 0

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

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

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

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

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

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

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

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

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

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

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

   6. sub-neg N/A

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

   7. +-commutative N/A

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

   8. unpow2 N/A

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

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

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

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

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

5. Simplified 49.9%

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

   1. distribute-lft-in N/A

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

   2. associate-+l+ N/A

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

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

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

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

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

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

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

   6. distribute-rgt-in N/A

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

   7. neg-mul-1 N/A

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

   8. associate-*r* N/A

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

   9. metadata-eval N/A

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

   10. *-lft-identity N/A

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

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

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

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

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

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

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

7. Applied egg-rr 56.3%

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

8. Taylor expanded in maxCos around 0

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

   1. +-commutative N/A

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

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

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

   3. --lowering--.f32 77.7

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

10. Simplified 77.7%

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

11. Taylor expanded in ux around 0

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

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

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

    2. mul-1-neg N/A

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

    3. sub-neg N/A

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

    4. --lowering--.f32 77.7

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

13. Simplified 77.7%

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

14. Final simplification 77.7%

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

Alternative 21: 62.9% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 56.6%

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

3. Taylor expanded in uy around 0

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

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

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

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

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

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

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

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

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

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

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

   6. sub-neg N/A

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

   7. +-commutative N/A

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

   8. unpow2 N/A

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

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

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

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

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

5. Simplified 49.9%

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

   1. distribute-lft-in N/A

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

   2. associate-+l+ N/A

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

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

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

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

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

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

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

   6. distribute-rgt-in N/A

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

   7. neg-mul-1 N/A

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

   8. associate-*r* N/A

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

   9. metadata-eval N/A

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

   10. *-lft-identity N/A

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

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

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

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

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

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

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

7. Applied egg-rr 56.3%

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

8. Taylor expanded in maxCos around 0

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

   1. +-commutative N/A

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

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

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

   3. --lowering--.f32 77.7

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

10. Simplified 77.7%

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

11. Taylor expanded in ux around 0

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

    1. *-lowering-*.f32 64.2

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

13. Simplified 64.2%

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

14. Final simplification 64.2%

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

Alternative 22: 9.8% accurate, 7.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 56.6%

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

3. Taylor expanded in uy around 0

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

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

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

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

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

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

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

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

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

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

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

   6. sub-neg N/A

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

   7. +-commutative N/A

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

   8. unpow2 N/A

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

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

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

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

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

5. Simplified 49.9%

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

   1. distribute-lft-in N/A

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

   2. associate-+l+ N/A

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

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

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

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

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

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

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

   6. distribute-rgt-in N/A

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

   7. neg-mul-1 N/A

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

   8. associate-*r* N/A

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

   9. metadata-eval N/A

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

   10. *-lft-identity N/A

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

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

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

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

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

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

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

7. Applied egg-rr 56.3%

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

9. Applied egg-rr 16.4%

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

10. Taylor expanded in maxCos around inf

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

    1. associate-*r* N/A

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

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

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

    3. *-commutative N/A

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

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

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

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

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

    6. PI-lowering-PI.f32 10.0

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

12. Simplified 10.0%

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

13. Final simplification 10.0%

    \[\leadsto 2 \cdot \left(\left(uy \cdot \pi\right) \cdot \left(ux \cdot maxCos\right)\right) \]
14. Add Preprocessing

Alternative 23: 6.0% accurate, 7.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 56.6%

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

3. Taylor expanded in uy around 0

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

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

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

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

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

   3. *-lowering-*.f32 N/A

      \[\leadsto 2 \cdot \left(\color{blue}{\left(uy \cdot \mathsf{PI}\left(\right)\right)} \cdot \sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}\right) \]

   4. PI-lowering-PI.f32 N/A

      \[\leadsto 2 \cdot \left(\left(uy \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}\right) \]

   5. sqrt-lowering-sqrt.f32 N/A

      \[\leadsto 2 \cdot \left(\left(uy \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}}\right) \]

   6. sub-neg N/A

      \[\leadsto 2 \cdot \left(\left(uy \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{1 + \left(\mathsf{neg}\left({\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}\right)\right)}}\right) \]

   7. +-commutative N/A

      \[\leadsto 2 \cdot \left(\left(uy \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{\left(\mathsf{neg}\left({\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}\right)\right) + 1}}\right) \]

   8. unpow2 N/A

      \[\leadsto 2 \cdot \left(\left(uy \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\left(\mathsf{neg}\left(\color{blue}{\left(\left(1 + maxCos \cdot ux\right) - ux\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)}\right)\right) + 1}\right) \]

   9. distribute-rgt-neg-in N/A

      \[\leadsto 2 \cdot \left(\left(uy \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{\left(\left(1 + maxCos \cdot ux\right) - ux\right) \cdot \left(\mathsf{neg}\left(\left(\left(1 + maxCos \cdot ux\right) - ux\right)\right)\right)} + 1}\right) \]

   10. accelerator-lowering-fma.f32 N/A

       \[\leadsto 2 \cdot \left(\left(uy \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(\left(1 + maxCos \cdot ux\right) - ux, \mathsf{neg}\left(\left(\left(1 + maxCos \cdot ux\right) - ux\right)\right), 1\right)}}\right) \]

5. Simplified 49.9%

   \[\leadsto \color{blue}{2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1 - ux\right), \mathsf{fma}\left(-ux, maxCos + -1, -1\right), 1\right)}\right)} \]
6. Step-by-step derivation

   1. distribute-lft-in N/A

      \[\leadsto 2 \cdot \left(\left(uy \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{\left(\left(ux \cdot maxCos + \left(1 - ux\right)\right) \cdot \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(maxCos + -1\right)\right) + \left(ux \cdot maxCos + \left(1 - ux\right)\right) \cdot -1\right)} + 1}\right) \]

   2. associate-+l+ N/A

      \[\leadsto 2 \cdot \left(\left(uy \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{\left(ux \cdot maxCos + \left(1 - ux\right)\right) \cdot \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(maxCos + -1\right)\right) + \left(\left(ux \cdot maxCos + \left(1 - ux\right)\right) \cdot -1 + 1\right)}}\right) \]

   3. accelerator-lowering-fma.f32 N/A

      \[\leadsto 2 \cdot \left(\left(uy \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(ux \cdot maxCos + \left(1 - ux\right), \left(\mathsf{neg}\left(ux\right)\right) \cdot \left(maxCos + -1\right), \left(ux \cdot maxCos + \left(1 - ux\right)\right) \cdot -1 + 1\right)}}\right) \]

   4. accelerator-lowering-fma.f32 N/A

      \[\leadsto 2 \cdot \left(\left(uy \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(ux, maxCos, 1 - ux\right)}, \left(\mathsf{neg}\left(ux\right)\right) \cdot \left(maxCos + -1\right), \left(ux \cdot maxCos + \left(1 - ux\right)\right) \cdot -1 + 1\right)}\right) \]

   5. --lowering--.f32 N/A

      \[\leadsto 2 \cdot \left(\left(uy \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, \color{blue}{1 - ux}\right), \left(\mathsf{neg}\left(ux\right)\right) \cdot \left(maxCos + -1\right), \left(ux \cdot maxCos + \left(1 - ux\right)\right) \cdot -1 + 1\right)}\right) \]

   6. distribute-rgt-in N/A

      \[\leadsto 2 \cdot \left(\left(uy \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1 - ux\right), \color{blue}{maxCos \cdot \left(\mathsf{neg}\left(ux\right)\right) + -1 \cdot \left(\mathsf{neg}\left(ux\right)\right)}, \left(ux \cdot maxCos + \left(1 - ux\right)\right) \cdot -1 + 1\right)}\right) \]

   7. neg-mul-1 N/A

      \[\leadsto 2 \cdot \left(\left(uy \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1 - ux\right), maxCos \cdot \left(\mathsf{neg}\left(ux\right)\right) + -1 \cdot \color{blue}{\left(-1 \cdot ux\right)}, \left(ux \cdot maxCos + \left(1 - ux\right)\right) \cdot -1 + 1\right)}\right) \]

   8. associate-*r* N/A

      \[\leadsto 2 \cdot \left(\left(uy \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1 - ux\right), maxCos \cdot \left(\mathsf{neg}\left(ux\right)\right) + \color{blue}{\left(-1 \cdot -1\right) \cdot ux}, \left(ux \cdot maxCos + \left(1 - ux\right)\right) \cdot -1 + 1\right)}\right) \]

   9. metadata-eval N/A

      \[\leadsto 2 \cdot \left(\left(uy \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1 - ux\right), maxCos \cdot \left(\mathsf{neg}\left(ux\right)\right) + \color{blue}{1} \cdot ux, \left(ux \cdot maxCos + \left(1 - ux\right)\right) \cdot -1 + 1\right)}\right) \]

   10. *-lft-identity N/A

       \[\leadsto 2 \cdot \left(\left(uy \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1 - ux\right), maxCos \cdot \left(\mathsf{neg}\left(ux\right)\right) + \color{blue}{ux}, \left(ux \cdot maxCos + \left(1 - ux\right)\right) \cdot -1 + 1\right)}\right) \]

   11. accelerator-lowering-fma.f32 N/A

       \[\leadsto 2 \cdot \left(\left(uy \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1 - ux\right), \color{blue}{\mathsf{fma}\left(maxCos, \mathsf{neg}\left(ux\right), ux\right)}, \left(ux \cdot maxCos + \left(1 - ux\right)\right) \cdot -1 + 1\right)}\right) \]

   12. neg-lowering-neg.f32 N/A

       \[\leadsto 2 \cdot \left(\left(uy \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1 - ux\right), \mathsf{fma}\left(maxCos, \color{blue}{\mathsf{neg}\left(ux\right)}, ux\right), \left(ux \cdot maxCos + \left(1 - ux\right)\right) \cdot -1 + 1\right)}\right) \]

   13. +-lowering-+.f32 N/A

       \[\leadsto 2 \cdot \left(\left(uy \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1 - ux\right), \mathsf{fma}\left(maxCos, \mathsf{neg}\left(ux\right), ux\right), \color{blue}{\left(ux \cdot maxCos + \left(1 - ux\right)\right) \cdot -1 + 1}\right)}\right) \]

7. Applied egg-rr 56.3%

   \[\leadsto 2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1 - ux\right), \mathsf{fma}\left(maxCos, -ux, ux\right), \mathsf{fma}\left(maxCos, -ux, -1 + ux\right) + 1\right)}}\right) \]

9. Applied egg-rr 16.4%

   \[\leadsto 2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(-ux, maxCos, \left(ux + 0\right) + \mathsf{fma}\left(ux, maxCos, -ux\right) \cdot \mathsf{fma}\left(ux, maxCos, 1 - ux\right)\right)}}\right) \]

10. Taylor expanded in maxCos around -inf

    \[\leadsto \color{blue}{-2 \cdot \left(maxCos \cdot \left(ux \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]
11. Step-by-step derivation

    1. *-lowering-*.f32 N/A

       \[\leadsto \color{blue}{-2 \cdot \left(maxCos \cdot \left(ux \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

    2. associate-*r* N/A

       \[\leadsto -2 \cdot \color{blue}{\left(\left(maxCos \cdot ux\right) \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \]

    3. *-lowering-*.f32 N/A

       \[\leadsto -2 \cdot \color{blue}{\left(\left(maxCos \cdot ux\right) \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \]

    4. *-commutative N/A

       \[\leadsto -2 \cdot \left(\color{blue}{\left(ux \cdot maxCos\right)} \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \]

    5. *-lowering-*.f32 N/A

       \[\leadsto -2 \cdot \left(\color{blue}{\left(ux \cdot maxCos\right)} \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \]

    6. *-lowering-*.f32 N/A

       \[\leadsto -2 \cdot \left(\left(ux \cdot maxCos\right) \cdot \color{blue}{\left(uy \cdot \mathsf{PI}\left(\right)\right)}\right) \]

    7. PI-lowering-PI.f32 6.0

       \[\leadsto -2 \cdot \left(\left(ux \cdot maxCos\right) \cdot \left(uy \cdot \color{blue}{\pi}\right)\right) \]

12. Simplified 6.0%

    \[\leadsto \color{blue}{-2 \cdot \left(\left(ux \cdot maxCos\right) \cdot \left(uy \cdot \pi\right)\right)} \]

13. Final simplification 6.0%

    \[\leadsto -2 \cdot \left(\left(uy \cdot \pi\right) \cdot \left(ux \cdot maxCos\right)\right) \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024204 
(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)))))))