UniformSampleCone, x

Percentage Accurate: 57.1% → 99.0%

Time: 14.8s

Alternatives: 22

Speedup: 9.8×

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

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

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))))
   (* (cos (* (* 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 cosf(((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(cos(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 = cos(((uy * single(2.0)) * single(pi))) * sqrt((single(1.0) - (t_0 * t_0)));
end

Initial Program: 57.1% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))))
   (* (cos (* (* 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 cosf(((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(cos(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 = cos(((uy * single(2.0)) * single(pi))) * sqrt((single(1.0) - (t_0 * t_0)));
end

Alternative 1: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.6%

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

3. Taylor expanded in ux around inf

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

   1. lower-*.f32 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f32 N/A

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

   4. sub-neg N/A

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

   5. +-commutative N/A

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

   6. +-commutative N/A

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

   7. distribute-neg-in N/A

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

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

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

   9. metadata-eval N/A

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

   10. associate-+l+ N/A

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

   11. unpow2 N/A

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

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

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

   13. neg-sub0 N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\left(ux \cdot ux\right) \cdot \left(\left(maxCos - 1\right) \cdot \color{blue}{\left(0 - \left(maxCos - 1\right)\right)} + \left(-2 \cdot \frac{maxCos}{ux} + 2 \cdot \frac{1}{ux}\right)\right)} \]

   14. associate-+l- N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\left(ux \cdot ux\right) \cdot \left(\left(maxCos - 1\right) \cdot \color{blue}{\left(\left(0 - maxCos\right) + 1\right)} + \left(-2 \cdot \frac{maxCos}{ux} + 2 \cdot \frac{1}{ux}\right)\right)} \]

   15. neg-sub0 N/A

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

   16. mul-1-neg N/A

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

   17. +-commutative N/A

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

   18. +-commutative N/A

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

5. Simplified 99.0%

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

7. Applied egg-rr 99.2%

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

8. Final simplification 99.2%

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

Alternative 2: 97.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\pi \cdot \left(2 \cdot uy\right)\right)\\ \mathbf{if}\;t\_0 \leq 0.999180018901825:\\ \;\;\;\;t\_0 \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \pi \cdot \pi, 1\right) \cdot \sqrt{\mathsf{fma}\left(ux \cdot ux, \left(maxCos + -1\right) \cdot \left(1 - maxCos\right), ux \cdot \mathsf{fma}\left(maxCos, -2, 2\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (cos (* PI (* 2.0 uy)))))
   (if (<= t_0 0.999180018901825)
     (* t_0 (sqrt (* ux (- 2.0 ux))))
     (*
      (fma (* -2.0 (* uy uy)) (* PI PI) 1.0)
      (sqrt
       (fma
        (* ux ux)
        (* (+ maxCos -1.0) (- 1.0 maxCos))
        (* ux (fma maxCos -2.0 2.0))))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = cosf((((float) M_PI) * (2.0f * uy)));
	float tmp;
	if (t_0 <= 0.999180018901825f) {
		tmp = t_0 * sqrtf((ux * (2.0f - ux)));
	} else {
		tmp = fmaf((-2.0f * (uy * uy)), (((float) M_PI) * ((float) M_PI)), 1.0f) * sqrtf(fmaf((ux * ux), ((maxCos + -1.0f) * (1.0f - maxCos)), (ux * fmaf(maxCos, -2.0f, 2.0f))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = cos(Float32(Float32(pi) * Float32(Float32(2.0) * uy)))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.999180018901825))
		tmp = Float32(t_0 * sqrt(Float32(ux * Float32(Float32(2.0) - ux))));
	else
		tmp = Float32(fma(Float32(Float32(-2.0) * Float32(uy * uy)), Float32(Float32(pi) * Float32(pi)), Float32(1.0)) * sqrt(fma(Float32(ux * ux), Float32(Float32(maxCos + Float32(-1.0)) * Float32(Float32(1.0) - maxCos)), Float32(ux * fma(maxCos, Float32(-2.0), Float32(2.0))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 (*.f32 uy #s(literal 2 binary32)) (PI.f32))) < 0.999180019

   1. Initial program 55.9%

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

   3. Taylor expanded in ux around 0

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

      1. lower-*.f32 N/A

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

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

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f32 N/A

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

      7. +-commutative N/A

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

      8. mul-1-neg N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

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

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

      12. neg-sub0 N/A

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

      13. associate-+l- N/A

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

      14. neg-sub0 N/A

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

      15. mul-1-neg N/A

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

      16. +-commutative N/A

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

      17. lower-fma.f32 N/A

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

   5. Simplified 97.9%

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

   6. Taylor expanded in maxCos around 0

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

      1. lower-*.f32 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f32 93.5

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

   8. Simplified 93.5%

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

   if 0.999180019 < (cos.f32 (*.f32 (*.f32 uy #s(literal 2 binary32)) (PI.f32)))

   1. Initial program 58.1%

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

   3. Taylor expanded in ux around inf

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

      1. lower-*.f32 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f32 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

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

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

      9. metadata-eval N/A

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

      10. associate-+l+ N/A

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

      11. unpow2 N/A

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

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

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

      13. neg-sub0 N/A

          \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\left(ux \cdot ux\right) \cdot \left(\left(maxCos - 1\right) \cdot \color{blue}{\left(0 - \left(maxCos - 1\right)\right)} + \left(-2 \cdot \frac{maxCos}{ux} + 2 \cdot \frac{1}{ux}\right)\right)} \]

      14. associate-+l- N/A

          \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\left(ux \cdot ux\right) \cdot \left(\left(maxCos - 1\right) \cdot \color{blue}{\left(\left(0 - maxCos\right) + 1\right)} + \left(-2 \cdot \frac{maxCos}{ux} + 2 \cdot \frac{1}{ux}\right)\right)} \]

      15. neg-sub0 N/A

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

      16. mul-1-neg N/A

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

      17. +-commutative N/A

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

      18. +-commutative N/A

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

   5. Simplified 99.2%

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

   7. Applied egg-rr 99.4%

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

      1. add-log-exp N/A

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

      2. *-un-lft-identity N/A

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

      3. lift-PI.f32 N/A

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

      4. exp-prod N/A

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

      5. log-pow N/A

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

      6. lower-*.f32 N/A

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

      7. lower-log.f32 N/A

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

      8. exp-1-e N/A

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

      9. lower-E.f32 99.4

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

   9. Applied egg-rr 99.4%

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

   10. Taylor expanded in uy around 0

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

       1. associate-*r* N/A

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

       2. distribute-rgt1-in N/A

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

   12. Simplified 99.4%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\pi \cdot \left(2 \cdot uy\right)\right) \leq 0.999180018901825:\\ \;\;\;\;\cos \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \pi \cdot \pi, 1\right) \cdot \sqrt{\mathsf{fma}\left(ux \cdot ux, \left(maxCos + -1\right) \cdot \left(1 - maxCos\right), ux \cdot \mathsf{fma}\left(maxCos, -2, 2\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\pi \cdot \left(2 \cdot uy\right)\right)\\ \mathbf{if}\;t\_0 \leq 0.9950000047683716:\\ \;\;\;\;t\_0 \cdot \sqrt{ux \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \pi \cdot \pi, 1\right) \cdot \sqrt{\mathsf{fma}\left(ux \cdot ux, \left(maxCos + -1\right) \cdot \left(1 - maxCos\right), ux \cdot \mathsf{fma}\left(maxCos, -2, 2\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (cos (* PI (* 2.0 uy)))))
   (if (<= t_0 0.9950000047683716)
     (* t_0 (sqrt (* ux 2.0)))
     (*
      (fma (* -2.0 (* uy uy)) (* PI PI) 1.0)
      (sqrt
       (fma
        (* ux ux)
        (* (+ maxCos -1.0) (- 1.0 maxCos))
        (* ux (fma maxCos -2.0 2.0))))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = cosf((((float) M_PI) * (2.0f * uy)));
	float tmp;
	if (t_0 <= 0.9950000047683716f) {
		tmp = t_0 * sqrtf((ux * 2.0f));
	} else {
		tmp = fmaf((-2.0f * (uy * uy)), (((float) M_PI) * ((float) M_PI)), 1.0f) * sqrtf(fmaf((ux * ux), ((maxCos + -1.0f) * (1.0f - maxCos)), (ux * fmaf(maxCos, -2.0f, 2.0f))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = cos(Float32(Float32(pi) * Float32(Float32(2.0) * uy)))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.9950000047683716))
		tmp = Float32(t_0 * sqrt(Float32(ux * Float32(2.0))));
	else
		tmp = Float32(fma(Float32(Float32(-2.0) * Float32(uy * uy)), Float32(Float32(pi) * Float32(pi)), Float32(1.0)) * sqrt(fma(Float32(ux * ux), Float32(Float32(maxCos + Float32(-1.0)) * Float32(Float32(1.0) - maxCos)), Float32(ux * fma(maxCos, Float32(-2.0), Float32(2.0))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 (*.f32 uy #s(literal 2 binary32)) (PI.f32))) < 0.995000005

   1. Initial program 54.6%

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

   3. Taylor expanded in ux around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f32 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f32 48.2

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

   5. Simplified 48.2%

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

   6. Taylor expanded in maxCos around 0

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

      1. lower-*.f32 76.6

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

   8. Simplified 76.6%

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

   if 0.995000005 < (cos.f32 (*.f32 (*.f32 uy #s(literal 2 binary32)) (PI.f32)))

   1. Initial program 58.2%

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

   3. Taylor expanded in ux around inf

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

      1. lower-*.f32 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f32 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

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

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

      9. metadata-eval N/A

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

      10. associate-+l+ N/A

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

      11. unpow2 N/A

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

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

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

      13. neg-sub0 N/A

          \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\left(ux \cdot ux\right) \cdot \left(\left(maxCos - 1\right) \cdot \color{blue}{\left(0 - \left(maxCos - 1\right)\right)} + \left(-2 \cdot \frac{maxCos}{ux} + 2 \cdot \frac{1}{ux}\right)\right)} \]

      14. associate-+l- N/A

          \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\left(ux \cdot ux\right) \cdot \left(\left(maxCos - 1\right) \cdot \color{blue}{\left(\left(0 - maxCos\right) + 1\right)} + \left(-2 \cdot \frac{maxCos}{ux} + 2 \cdot \frac{1}{ux}\right)\right)} \]

      15. neg-sub0 N/A

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

      16. mul-1-neg N/A

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

      17. +-commutative N/A

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

      18. +-commutative N/A

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

   5. Simplified 99.2%

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

   7. Applied egg-rr 99.4%

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

      1. add-log-exp N/A

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

      2. *-un-lft-identity N/A

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

      3. lift-PI.f32 N/A

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

      4. exp-prod N/A

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

      5. log-pow N/A

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

      6. lower-*.f32 N/A

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

      7. lower-log.f32 N/A

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

      8. exp-1-e N/A

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

      9. lower-E.f32 99.4

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

   9. Applied egg-rr 99.4%

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

   10. Taylor expanded in uy around 0

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

       1. associate-*r* N/A

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

       2. distribute-rgt1-in N/A

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

   12. Simplified 98.8%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\pi \cdot \left(2 \cdot uy\right)\right) \leq 0.9950000047683716:\\ \;\;\;\;\cos \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{ux \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \pi \cdot \pi, 1\right) \cdot \sqrt{\mathsf{fma}\left(ux \cdot ux, \left(maxCos + -1\right) \cdot \left(1 - maxCos\right), ux \cdot \mathsf{fma}\left(maxCos, -2, 2\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.012799999676644802:\\ \;\;\;\;\mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \pi \cdot \pi, 1\right) \cdot \sqrt{\mathsf{fma}\left(ux \cdot ux, \left(maxCos + -1\right) \cdot \left(1 - maxCos\right), ux \cdot \mathsf{fma}\left(maxCos, -2, 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{\left(ux \cdot ux\right) \cdot \left(-1 + \frac{2}{ux}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* 2.0 uy) 0.012799999676644802)
   (*
    (fma (* -2.0 (* uy uy)) (* PI PI) 1.0)
    (sqrt
     (fma
      (* ux ux)
      (* (+ maxCos -1.0) (- 1.0 maxCos))
      (* ux (fma maxCos -2.0 2.0)))))
   (* (cos (* PI (* 2.0 uy))) (sqrt (* (* ux ux) (+ -1.0 (/ 2.0 ux)))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if ((2.0f * uy) <= 0.012799999676644802f) {
		tmp = fmaf((-2.0f * (uy * uy)), (((float) M_PI) * ((float) M_PI)), 1.0f) * sqrtf(fmaf((ux * ux), ((maxCos + -1.0f) * (1.0f - maxCos)), (ux * fmaf(maxCos, -2.0f, 2.0f))));
	} else {
		tmp = cosf((((float) M_PI) * (2.0f * uy))) * sqrtf(((ux * ux) * (-1.0f + (2.0f / ux))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (Float32(Float32(2.0) * uy) <= Float32(0.012799999676644802))
		tmp = Float32(fma(Float32(Float32(-2.0) * Float32(uy * uy)), Float32(Float32(pi) * Float32(pi)), Float32(1.0)) * sqrt(fma(Float32(ux * ux), Float32(Float32(maxCos + Float32(-1.0)) * Float32(Float32(1.0) - maxCos)), Float32(ux * fma(maxCos, Float32(-2.0), Float32(2.0))))));
	else
		tmp = Float32(cos(Float32(Float32(pi) * Float32(Float32(2.0) * uy))) * 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)) < 0.0127999997

   1. Initial program 58.1%

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

   3. Taylor expanded in ux around inf

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

      1. lower-*.f32 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f32 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

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

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

      9. metadata-eval N/A

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

      10. associate-+l+ N/A

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

      11. unpow2 N/A

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

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

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

      13. neg-sub0 N/A

          \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\left(ux \cdot ux\right) \cdot \left(\left(maxCos - 1\right) \cdot \color{blue}{\left(0 - \left(maxCos - 1\right)\right)} + \left(-2 \cdot \frac{maxCos}{ux} + 2 \cdot \frac{1}{ux}\right)\right)} \]

      14. associate-+l- N/A

          \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\left(ux \cdot ux\right) \cdot \left(\left(maxCos - 1\right) \cdot \color{blue}{\left(\left(0 - maxCos\right) + 1\right)} + \left(-2 \cdot \frac{maxCos}{ux} + 2 \cdot \frac{1}{ux}\right)\right)} \]

      15. neg-sub0 N/A

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

      16. mul-1-neg N/A

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

      17. +-commutative N/A

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

      18. +-commutative N/A

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

   5. Simplified 99.2%

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

   7. Applied egg-rr 99.4%

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

      1. add-log-exp N/A

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

      2. *-un-lft-identity N/A

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

      3. lift-PI.f32 N/A

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

      4. exp-prod N/A

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

      5. log-pow N/A

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

      6. lower-*.f32 N/A

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

      7. lower-log.f32 N/A

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

      8. exp-1-e N/A

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

      9. lower-E.f32 99.4

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

   9. Applied egg-rr 99.4%

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

   10. Taylor expanded in uy around 0

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

       1. associate-*r* N/A

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

       2. distribute-rgt1-in N/A

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

   12. Simplified 99.4%

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

   if 0.0127999997 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 55.9%

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

   3. Taylor expanded in ux around inf

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

      1. lower-*.f32 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f32 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

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

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

      9. metadata-eval N/A

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

      10. associate-+l+ N/A

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

      11. unpow2 N/A

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

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

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

      13. neg-sub0 N/A

          \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\left(ux \cdot ux\right) \cdot \left(\left(maxCos - 1\right) \cdot \color{blue}{\left(0 - \left(maxCos - 1\right)\right)} + \left(-2 \cdot \frac{maxCos}{ux} + 2 \cdot \frac{1}{ux}\right)\right)} \]

      14. associate-+l- N/A

          \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\left(ux \cdot ux\right) \cdot \left(\left(maxCos - 1\right) \cdot \color{blue}{\left(\left(0 - maxCos\right) + 1\right)} + \left(-2 \cdot \frac{maxCos}{ux} + 2 \cdot \frac{1}{ux}\right)\right)} \]

      15. neg-sub0 N/A

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

      16. mul-1-neg N/A

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

      17. +-commutative N/A

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

      18. +-commutative N/A

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

   5. Simplified 98.3%

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

   6. Taylor expanded in maxCos around 0

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

      1. lower-*.f32 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f32 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-+.f32 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f32 93.8

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

   8. Simplified 93.8%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.012799999676644802:\\ \;\;\;\;\mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \pi \cdot \pi, 1\right) \cdot \sqrt{\mathsf{fma}\left(ux \cdot ux, \left(maxCos + -1\right) \cdot \left(1 - maxCos\right), ux \cdot \mathsf{fma}\left(maxCos, -2, 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{\left(ux \cdot ux\right) \cdot \left(-1 + \frac{2}{ux}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.012799999676644802:\\ \;\;\;\;\mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \pi \cdot \pi, 1\right) \cdot \sqrt{\mathsf{fma}\left(ux \cdot ux, \left(maxCos + -1\right) \cdot \left(1 - maxCos\right), ux \cdot \mathsf{fma}\left(maxCos, -2, 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(ux \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \cdot \sqrt{-1 + \frac{2}{ux}}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* 2.0 uy) 0.012799999676644802)
   (*
    (fma (* -2.0 (* uy uy)) (* PI PI) 1.0)
    (sqrt
     (fma
      (* ux ux)
      (* (+ maxCos -1.0) (- 1.0 maxCos))
      (* ux (fma maxCos -2.0 2.0)))))
   (* (* ux (cos (* 2.0 (* uy PI)))) (sqrt (+ -1.0 (/ 2.0 ux))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if ((2.0f * uy) <= 0.012799999676644802f) {
		tmp = fmaf((-2.0f * (uy * uy)), (((float) M_PI) * ((float) M_PI)), 1.0f) * sqrtf(fmaf((ux * ux), ((maxCos + -1.0f) * (1.0f - maxCos)), (ux * fmaf(maxCos, -2.0f, 2.0f))));
	} else {
		tmp = (ux * cosf((2.0f * (uy * ((float) M_PI))))) * sqrtf((-1.0f + (2.0f / ux)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (Float32(Float32(2.0) * uy) <= Float32(0.012799999676644802))
		tmp = Float32(fma(Float32(Float32(-2.0) * Float32(uy * uy)), Float32(Float32(pi) * Float32(pi)), Float32(1.0)) * sqrt(fma(Float32(ux * ux), Float32(Float32(maxCos + Float32(-1.0)) * Float32(Float32(1.0) - maxCos)), Float32(ux * fma(maxCos, Float32(-2.0), Float32(2.0))))));
	else
		tmp = Float32(Float32(ux * cos(Float32(Float32(2.0) * Float32(uy * Float32(pi))))) * sqrt(Float32(Float32(-1.0) + Float32(Float32(2.0) / ux))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 0.0127999997

   1. Initial program 58.1%

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

   3. Taylor expanded in ux around inf

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

      1. lower-*.f32 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f32 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

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

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

      9. metadata-eval N/A

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

      10. associate-+l+ N/A

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

      11. unpow2 N/A

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

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

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

      13. neg-sub0 N/A

          \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\left(ux \cdot ux\right) \cdot \left(\left(maxCos - 1\right) \cdot \color{blue}{\left(0 - \left(maxCos - 1\right)\right)} + \left(-2 \cdot \frac{maxCos}{ux} + 2 \cdot \frac{1}{ux}\right)\right)} \]

      14. associate-+l- N/A

          \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\left(ux \cdot ux\right) \cdot \left(\left(maxCos - 1\right) \cdot \color{blue}{\left(\left(0 - maxCos\right) + 1\right)} + \left(-2 \cdot \frac{maxCos}{ux} + 2 \cdot \frac{1}{ux}\right)\right)} \]

      15. neg-sub0 N/A

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

      16. mul-1-neg N/A

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

      17. +-commutative N/A

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

      18. +-commutative N/A

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

   5. Simplified 99.2%

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

   7. Applied egg-rr 99.4%

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

      1. add-log-exp N/A

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

      2. *-un-lft-identity N/A

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

      3. lift-PI.f32 N/A

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

      4. exp-prod N/A

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

      5. log-pow N/A

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

      6. lower-*.f32 N/A

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

      7. lower-log.f32 N/A

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

      8. exp-1-e N/A

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

      9. lower-E.f32 99.4

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

   9. Applied egg-rr 99.4%

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

   10. Taylor expanded in uy around 0

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

       1. associate-*r* N/A

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

       2. distribute-rgt1-in N/A

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

   12. Simplified 99.4%

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

   if 0.0127999997 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 55.9%

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

   3. Taylor expanded in ux around inf

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

      1. lower-*.f32 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f32 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

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

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

      9. metadata-eval N/A

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

      10. associate-+l+ N/A

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

      11. unpow2 N/A

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

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

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

      13. neg-sub0 N/A

          \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\left(ux \cdot ux\right) \cdot \left(\left(maxCos - 1\right) \cdot \color{blue}{\left(0 - \left(maxCos - 1\right)\right)} + \left(-2 \cdot \frac{maxCos}{ux} + 2 \cdot \frac{1}{ux}\right)\right)} \]

      14. associate-+l- N/A

          \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\left(ux \cdot ux\right) \cdot \left(\left(maxCos - 1\right) \cdot \color{blue}{\left(\left(0 - maxCos\right) + 1\right)} + \left(-2 \cdot \frac{maxCos}{ux} + 2 \cdot \frac{1}{ux}\right)\right)} \]

      15. neg-sub0 N/A

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

      16. mul-1-neg N/A

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

      17. +-commutative N/A

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

      18. +-commutative N/A

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

   5. Simplified 98.3%

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

   6. Taylor expanded in maxCos around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-cos.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-PI.f32 N/A

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

      7. lower-sqrt.f32 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. lower-+.f32 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f32 93.8

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

   8. Simplified 93.8%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.012799999676644802:\\ \;\;\;\;\mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \pi \cdot \pi, 1\right) \cdot \sqrt{\mathsf{fma}\left(ux \cdot ux, \left(maxCos + -1\right) \cdot \left(1 - maxCos\right), ux \cdot \mathsf{fma}\left(maxCos, -2, 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(ux \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right)\right) \cdot \sqrt{-1 + \frac{2}{ux}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.6%

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

3. Taylor expanded in ux around 0

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

   1. lower-*.f32 N/A

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

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

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

   3. metadata-eval N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f32 N/A

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

   7. +-commutative N/A

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

   8. mul-1-neg N/A

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

   9. unpow2 N/A

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

   10. associate-*r* N/A

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

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

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

   12. neg-sub0 N/A

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

   13. associate-+l- N/A

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

   14. neg-sub0 N/A

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

   15. mul-1-neg N/A

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

   16. +-commutative N/A

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

   17. lower-fma.f32 N/A

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

5. Simplified 99.1%

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

6. Final simplification 99.1%

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

Alternative 7: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.6%

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

3. Taylor expanded in ux around 0

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

   1. lower-*.f32 N/A

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

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

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

   3. metadata-eval N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f32 N/A

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

   7. +-commutative N/A

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

   8. mul-1-neg N/A

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

   9. unpow2 N/A

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

   10. associate-*r* N/A

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

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

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

   12. neg-sub0 N/A

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

   13. associate-+l- N/A

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

   14. neg-sub0 N/A

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

   15. mul-1-neg N/A

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

   16. +-commutative N/A

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

   17. lower-fma.f32 N/A

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

5. Simplified 99.1%

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

6. Taylor expanded in maxCos around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

   3. mul-1-neg N/A

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

   4. unsub-neg N/A

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

   5. lower--.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. sub-neg N/A

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

   9. metadata-eval N/A

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

   10. lower-fma.f32 98.4

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

8. Simplified 98.4%

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

9. Final simplification 98.4%

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

Alternative 8: 97.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.012799999676644802:\\ \;\;\;\;\mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \pi \cdot \pi, 1\right) \cdot \sqrt{\mathsf{fma}\left(ux \cdot ux, \left(maxCos + -1\right) \cdot \left(1 - maxCos\right), ux \cdot \mathsf{fma}\left(maxCos, -2, 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{\mathsf{fma}\left(-ux, ux, ux \cdot 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* 2.0 uy) 0.012799999676644802)
   (*
    (fma (* -2.0 (* uy uy)) (* PI PI) 1.0)
    (sqrt
     (fma
      (* ux ux)
      (* (+ maxCos -1.0) (- 1.0 maxCos))
      (* ux (fma maxCos -2.0 2.0)))))
   (* (cos (* PI (* 2.0 uy))) (sqrt (fma (- ux) ux (* ux 2.0))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if ((2.0f * uy) <= 0.012799999676644802f) {
		tmp = fmaf((-2.0f * (uy * uy)), (((float) M_PI) * ((float) M_PI)), 1.0f) * sqrtf(fmaf((ux * ux), ((maxCos + -1.0f) * (1.0f - maxCos)), (ux * fmaf(maxCos, -2.0f, 2.0f))));
	} else {
		tmp = cosf((((float) M_PI) * (2.0f * uy))) * sqrtf(fmaf(-ux, ux, (ux * 2.0f)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (Float32(Float32(2.0) * uy) <= Float32(0.012799999676644802))
		tmp = Float32(fma(Float32(Float32(-2.0) * Float32(uy * uy)), Float32(Float32(pi) * Float32(pi)), Float32(1.0)) * sqrt(fma(Float32(ux * ux), Float32(Float32(maxCos + Float32(-1.0)) * Float32(Float32(1.0) - maxCos)), Float32(ux * fma(maxCos, Float32(-2.0), Float32(2.0))))));
	else
		tmp = Float32(cos(Float32(Float32(pi) * Float32(Float32(2.0) * uy))) * sqrt(fma(Float32(-ux), ux, Float32(ux * Float32(2.0)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 0.0127999997

   1. Initial program 58.1%

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

   3. Taylor expanded in ux around inf

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

      1. lower-*.f32 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f32 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

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

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

      9. metadata-eval N/A

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

      10. associate-+l+ N/A

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

      11. unpow2 N/A

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

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

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

      13. neg-sub0 N/A

          \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\left(ux \cdot ux\right) \cdot \left(\left(maxCos - 1\right) \cdot \color{blue}{\left(0 - \left(maxCos - 1\right)\right)} + \left(-2 \cdot \frac{maxCos}{ux} + 2 \cdot \frac{1}{ux}\right)\right)} \]

      14. associate-+l- N/A

          \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\left(ux \cdot ux\right) \cdot \left(\left(maxCos - 1\right) \cdot \color{blue}{\left(\left(0 - maxCos\right) + 1\right)} + \left(-2 \cdot \frac{maxCos}{ux} + 2 \cdot \frac{1}{ux}\right)\right)} \]

      15. neg-sub0 N/A

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

      16. mul-1-neg N/A

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

      17. +-commutative N/A

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

      18. +-commutative N/A

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

   5. Simplified 99.2%

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

   7. Applied egg-rr 99.4%

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

      1. add-log-exp N/A

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

      2. *-un-lft-identity N/A

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

      3. lift-PI.f32 N/A

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

      4. exp-prod N/A

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

      5. log-pow N/A

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

      6. lower-*.f32 N/A

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

      7. lower-log.f32 N/A

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

      8. exp-1-e N/A

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

      9. lower-E.f32 99.4

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

   9. Applied egg-rr 99.4%

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

   10. Taylor expanded in uy around 0

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

       1. associate-*r* N/A

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

       2. distribute-rgt1-in N/A

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

   12. Simplified 99.4%

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

   if 0.0127999997 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 55.9%

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

   3. Taylor expanded in ux around inf

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

      1. lower-*.f32 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f32 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

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

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

      9. metadata-eval N/A

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

      10. associate-+l+ N/A

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

      11. unpow2 N/A

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

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

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

      13. neg-sub0 N/A

          \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\left(ux \cdot ux\right) \cdot \left(\left(maxCos - 1\right) \cdot \color{blue}{\left(0 - \left(maxCos - 1\right)\right)} + \left(-2 \cdot \frac{maxCos}{ux} + 2 \cdot \frac{1}{ux}\right)\right)} \]

      14. associate-+l- N/A

          \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\left(ux \cdot ux\right) \cdot \left(\left(maxCos - 1\right) \cdot \color{blue}{\left(\left(0 - maxCos\right) + 1\right)} + \left(-2 \cdot \frac{maxCos}{ux} + 2 \cdot \frac{1}{ux}\right)\right)} \]

      15. neg-sub0 N/A

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

      16. mul-1-neg N/A

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

      17. +-commutative N/A

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

      18. +-commutative N/A

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

   5. Simplified 98.3%

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

   6. Taylor expanded in maxCos around 0

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

      1. lower-*.f32 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f32 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-+.f32 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f32 93.8

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

   8. Simplified 93.8%

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

      1. lift-*.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. lift-*.f32 N/A

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

      6. associate-*r* N/A

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

      7. neg-mul-1 N/A

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

      8. lift-/.f32 N/A

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

      9. div-inv N/A

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

      10. associate-*l* N/A

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

      11. inv-pow N/A

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

      12. lift-*.f32 N/A

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

      13. pow2 N/A

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

      14. pow-prod-up N/A

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

      15. metadata-eval N/A

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

      16. unpow1 N/A

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

      17. lower-fma.f32 N/A

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

      18. lower-neg.f32 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f32 93.8

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

   10. Applied egg-rr 93.8%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.012799999676644802:\\ \;\;\;\;\mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \pi \cdot \pi, 1\right) \cdot \sqrt{\mathsf{fma}\left(ux \cdot ux, \left(maxCos + -1\right) \cdot \left(1 - maxCos\right), ux \cdot \mathsf{fma}\left(maxCos, -2, 2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{\mathsf{fma}\left(-ux, ux, ux \cdot 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 88.4% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.6%

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

3. Taylor expanded in ux around inf

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

   1. lower-*.f32 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f32 N/A

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

   4. sub-neg N/A

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

   5. +-commutative N/A

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

   6. +-commutative N/A

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

   7. distribute-neg-in N/A

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

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

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

   9. metadata-eval N/A

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

   10. associate-+l+ N/A

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

   11. unpow2 N/A

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

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

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

   13. neg-sub0 N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\left(ux \cdot ux\right) \cdot \left(\left(maxCos - 1\right) \cdot \color{blue}{\left(0 - \left(maxCos - 1\right)\right)} + \left(-2 \cdot \frac{maxCos}{ux} + 2 \cdot \frac{1}{ux}\right)\right)} \]

   14. associate-+l- N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\left(ux \cdot ux\right) \cdot \left(\left(maxCos - 1\right) \cdot \color{blue}{\left(\left(0 - maxCos\right) + 1\right)} + \left(-2 \cdot \frac{maxCos}{ux} + 2 \cdot \frac{1}{ux}\right)\right)} \]

   15. neg-sub0 N/A

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

   16. mul-1-neg N/A

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

   17. +-commutative N/A

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

   18. +-commutative N/A

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

5. Simplified 99.0%

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

7. Applied egg-rr 99.2%

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

   1. add-log-exp N/A

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

   2. *-un-lft-identity N/A

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

   3. lift-PI.f32 N/A

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

   4. exp-prod N/A

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

   5. log-pow N/A

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

   6. lower-*.f32 N/A

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

   7. lower-log.f32 N/A

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

   8. exp-1-e N/A

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

   9. lower-E.f32 99.1

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

9. Applied egg-rr 99.1%

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

10. Taylor expanded in uy around 0

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

    1. associate-*r* N/A

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

    2. distribute-rgt1-in N/A

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

12. Simplified 90.7%

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

13. Final simplification 90.7%

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

Alternative 10: 88.4% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.6%

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

3. Taylor expanded in ux around inf

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

   1. lower-*.f32 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f32 N/A

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

   4. sub-neg N/A

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

   5. +-commutative N/A

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

   6. +-commutative N/A

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

   7. distribute-neg-in N/A

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

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

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

   9. metadata-eval N/A

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

   10. associate-+l+ N/A

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

   11. unpow2 N/A

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

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

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

   13. neg-sub0 N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\left(ux \cdot ux\right) \cdot \left(\left(maxCos - 1\right) \cdot \color{blue}{\left(0 - \left(maxCos - 1\right)\right)} + \left(-2 \cdot \frac{maxCos}{ux} + 2 \cdot \frac{1}{ux}\right)\right)} \]

   14. associate-+l- N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\left(ux \cdot ux\right) \cdot \left(\left(maxCos - 1\right) \cdot \color{blue}{\left(\left(0 - maxCos\right) + 1\right)} + \left(-2 \cdot \frac{maxCos}{ux} + 2 \cdot \frac{1}{ux}\right)\right)} \]

   15. neg-sub0 N/A

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

   16. mul-1-neg N/A

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

   17. +-commutative N/A

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

   18. +-commutative N/A

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

5. Simplified 99.0%

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

7. Applied egg-rr 99.2%

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

8. Taylor expanded in uy around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 N/A

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

   9. lower-PI.f32 N/A

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

   10. lower-PI.f32 N/A

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

   11. lower-*.f32 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f32 90.7

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

10. Simplified 90.7%

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

11. Final simplification 90.7%

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

Alternative 11: 86.6% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.0008200000156648457:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(maxCos, -2, 2\right), ux, \left(maxCos + -1\right) \cdot \left(\left(1 - maxCos\right) \cdot \left(ux \cdot ux\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-1 + \frac{2}{ux}} \cdot \mathsf{fma}\left(-2, \left(\pi \cdot \pi\right) \cdot \left(ux \cdot \left(uy \cdot uy\right)\right), ux\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* 2.0 uy) 0.0008200000156648457)
   (sqrt
    (fma
     (fma maxCos -2.0 2.0)
     ux
     (* (+ maxCos -1.0) (* (- 1.0 maxCos) (* ux ux)))))
   (*
    (sqrt (+ -1.0 (/ 2.0 ux)))
    (fma -2.0 (* (* PI PI) (* ux (* uy uy))) ux))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if ((2.0f * uy) <= 0.0008200000156648457f) {
		tmp = sqrtf(fmaf(fmaf(maxCos, -2.0f, 2.0f), ux, ((maxCos + -1.0f) * ((1.0f - maxCos) * (ux * ux)))));
	} else {
		tmp = sqrtf((-1.0f + (2.0f / ux))) * fmaf(-2.0f, ((((float) M_PI) * ((float) M_PI)) * (ux * (uy * uy))), ux);
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (Float32(Float32(2.0) * uy) <= Float32(0.0008200000156648457))
		tmp = sqrt(fma(fma(maxCos, Float32(-2.0), Float32(2.0)), ux, Float32(Float32(maxCos + Float32(-1.0)) * Float32(Float32(Float32(1.0) - maxCos) * Float32(ux * ux)))));
	else
		tmp = Float32(sqrt(Float32(Float32(-1.0) + Float32(Float32(2.0) / ux))) * fma(Float32(-2.0), Float32(Float32(Float32(pi) * Float32(pi)) * Float32(ux * Float32(uy * uy))), ux));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 8.20000016e-4

   1. Initial program 58.5%

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

   3. Taylor expanded in uy around 0

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

      1. lower-sqrt.f32 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

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

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

      6. lower-fma.f32 N/A

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

   5. Simplified 58.6%

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

   6. Taylor expanded in ux around 0

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

      1. lower-*.f32 N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

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

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

      5. unpow2 N/A

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

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

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. distribute-neg-in N/A

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

      10. mul-1-neg N/A

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

      11. metadata-eval N/A

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

      12. +-commutative N/A

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

      13. mul-1-neg N/A

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

      14. sub-neg N/A

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

      15. lower-fma.f32 N/A

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

   8. Simplified 98.4%

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

      1. lift-+.f32 N/A

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

      2. lift--.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. lift-+.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f32 N/A

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

      10. lift-*.f32 N/A

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

      11. lift-+.f32 N/A

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

      12. distribute-rgt-in N/A

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

      13. metadata-eval N/A

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

      14. lift-fma.f32 N/A

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

      15. associate-*r* N/A

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

      16. lift-*.f32 N/A

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

      17. *-commutative N/A

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

      18. lift-*.f32 N/A

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

   10. Applied egg-rr 98.5%

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

   if 8.20000016e-4 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 55.9%

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

   3. Taylor expanded in ux around inf

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

      1. lower-*.f32 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f32 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

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

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

      9. metadata-eval N/A

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

      10. associate-+l+ N/A

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

      11. unpow2 N/A

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

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

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

      13. neg-sub0 N/A

          \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\left(ux \cdot ux\right) \cdot \left(\left(maxCos - 1\right) \cdot \color{blue}{\left(0 - \left(maxCos - 1\right)\right)} + \left(-2 \cdot \frac{maxCos}{ux} + 2 \cdot \frac{1}{ux}\right)\right)} \]

      14. associate-+l- N/A

          \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\left(ux \cdot ux\right) \cdot \left(\left(maxCos - 1\right) \cdot \color{blue}{\left(\left(0 - maxCos\right) + 1\right)} + \left(-2 \cdot \frac{maxCos}{ux} + 2 \cdot \frac{1}{ux}\right)\right)} \]

      15. neg-sub0 N/A

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

      16. mul-1-neg N/A

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

      17. +-commutative N/A

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

      18. +-commutative N/A

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

   5. Simplified 98.7%

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

   6. Taylor expanded in maxCos around 0

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

      1. lower-*.f32 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f32 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-+.f32 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f32 94.2

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

   8. Simplified 94.2%

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

   9. Taylor expanded in uy around 0

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

       1. associate-*r* N/A

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

       2. distribute-rgt-out N/A

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

       3. lower-*.f32 N/A

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

       4. lower-sqrt.f32 N/A

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

       5. sub-neg N/A

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

       6. metadata-eval N/A

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

       7. lower-+.f32 N/A

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

       8. associate-*r/ N/A

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

       9. metadata-eval N/A

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

       10. lower-/.f32 N/A

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

       11. lower-fma.f32 N/A

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

   11. Simplified 71.7%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.0008200000156648457:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(maxCos, -2, 2\right), ux, \left(maxCos + -1\right) \cdot \left(\left(1 - maxCos\right) \cdot \left(ux \cdot ux\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-1 + \frac{2}{ux}} \cdot \mathsf{fma}\left(-2, \left(\pi \cdot \pi\right) \cdot \left(ux \cdot \left(uy \cdot uy\right)\right), ux\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 86.6% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.0008200000156648457:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(maxCos, -2, 2\right), ux, \left(maxCos + -1\right) \cdot \left(\left(1 - maxCos\right) \cdot \left(ux \cdot ux\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;ux \cdot \left(\mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \pi \cdot \pi, 1\right) \cdot \sqrt{-1 + \frac{2}{ux}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* 2.0 uy) 0.0008200000156648457)
   (sqrt
    (fma
     (fma maxCos -2.0 2.0)
     ux
     (* (+ maxCos -1.0) (* (- 1.0 maxCos) (* ux ux)))))
   (*
    ux
    (* (fma (* -2.0 (* uy uy)) (* PI PI) 1.0) (sqrt (+ -1.0 (/ 2.0 ux)))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if ((2.0f * uy) <= 0.0008200000156648457f) {
		tmp = sqrtf(fmaf(fmaf(maxCos, -2.0f, 2.0f), ux, ((maxCos + -1.0f) * ((1.0f - maxCos) * (ux * ux)))));
	} else {
		tmp = ux * (fmaf((-2.0f * (uy * uy)), (((float) M_PI) * ((float) M_PI)), 1.0f) * sqrtf((-1.0f + (2.0f / ux))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (Float32(Float32(2.0) * uy) <= Float32(0.0008200000156648457))
		tmp = sqrt(fma(fma(maxCos, Float32(-2.0), Float32(2.0)), ux, Float32(Float32(maxCos + Float32(-1.0)) * Float32(Float32(Float32(1.0) - maxCos) * Float32(ux * ux)))));
	else
		tmp = Float32(ux * Float32(fma(Float32(Float32(-2.0) * Float32(uy * uy)), Float32(Float32(pi) * Float32(pi)), Float32(1.0)) * sqrt(Float32(Float32(-1.0) + Float32(Float32(2.0) / ux)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 8.20000016e-4

   1. Initial program 58.5%

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

   3. Taylor expanded in uy around 0

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

      1. lower-sqrt.f32 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

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

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

      6. lower-fma.f32 N/A

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

   5. Simplified 58.6%

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

   6. Taylor expanded in ux around 0

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

      1. lower-*.f32 N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

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

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

      5. unpow2 N/A

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

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

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. distribute-neg-in N/A

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

      10. mul-1-neg N/A

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

      11. metadata-eval N/A

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

      12. +-commutative N/A

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

      13. mul-1-neg N/A

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

      14. sub-neg N/A

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

      15. lower-fma.f32 N/A

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

   8. Simplified 98.4%

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

      1. lift-+.f32 N/A

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

      2. lift--.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. lift-+.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f32 N/A

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

      10. lift-*.f32 N/A

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

      11. lift-+.f32 N/A

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

      12. distribute-rgt-in N/A

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

      13. metadata-eval N/A

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

      14. lift-fma.f32 N/A

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

      15. associate-*r* N/A

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

      16. lift-*.f32 N/A

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

      17. *-commutative N/A

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

      18. lift-*.f32 N/A

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

   10. Applied egg-rr 98.5%

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

   if 8.20000016e-4 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 55.9%

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

   3. Taylor expanded in ux around inf

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

      1. lower-*.f32 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f32 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

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

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

      9. metadata-eval N/A

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

      10. associate-+l+ N/A

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

      11. unpow2 N/A

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

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

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

      13. neg-sub0 N/A

          \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\left(ux \cdot ux\right) \cdot \left(\left(maxCos - 1\right) \cdot \color{blue}{\left(0 - \left(maxCos - 1\right)\right)} + \left(-2 \cdot \frac{maxCos}{ux} + 2 \cdot \frac{1}{ux}\right)\right)} \]

      14. associate-+l- N/A

          \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\left(ux \cdot ux\right) \cdot \left(\left(maxCos - 1\right) \cdot \color{blue}{\left(\left(0 - maxCos\right) + 1\right)} + \left(-2 \cdot \frac{maxCos}{ux} + 2 \cdot \frac{1}{ux}\right)\right)} \]

      15. neg-sub0 N/A

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

      16. mul-1-neg N/A

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

      17. +-commutative N/A

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

      18. +-commutative N/A

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

   5. Simplified 98.7%

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

   6. Taylor expanded in uy around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 N/A

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

      9. lower-PI.f32 N/A

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

      10. lower-PI.f32 73.5

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

   8. Simplified 73.5%

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

      1. lift-*.f32 N/A

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

      2. lift-+.f32 N/A

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

      3. lift--.f32 N/A

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

      4. lift-fma.f32 N/A

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

      5. lift-/.f32 N/A

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

      6. lift-fma.f32 N/A

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

      7. *-commutative N/A

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

      8. sqrt-prod N/A

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

      9. pow1/2 N/A

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

      10. lift-*.f32 N/A

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

      11. sqrt-prod N/A

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

      12. rem-square-sqrt N/A

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

      13. lower-*.f32 N/A

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

      14. pow1/2 N/A

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

      15. lower-sqrt.f32 73.3

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

   10. Applied egg-rr 73.3%

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

   11. Taylor expanded in maxCos around 0

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

       1. associate-*l* N/A

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

       2. lower-*.f32 N/A

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

       3. lower-*.f32 N/A

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

       4. +-commutative N/A

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

       5. associate-*r* N/A

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

       6. lower-fma.f32 N/A

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

       7. lower-*.f32 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f32 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f32 N/A

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

       12. lower-PI.f32 N/A

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

       13. lower-PI.f32 N/A

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

       14. lower-sqrt.f32 N/A

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

       15. sub-neg N/A

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

       16. metadata-eval N/A

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

       17. lower-+.f32 N/A

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

   13. Simplified 71.5%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.0008200000156648457:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(maxCos, -2, 2\right), ux, \left(maxCos + -1\right) \cdot \left(\left(1 - maxCos\right) \cdot \left(ux \cdot ux\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;ux \cdot \left(\mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \pi \cdot \pi, 1\right) \cdot \sqrt{-1 + \frac{2}{ux}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 88.4% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.6%

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

3. Taylor expanded in ux around 0

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

   1. lower-*.f32 N/A

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

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

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

   3. metadata-eval N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f32 N/A

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

   7. +-commutative N/A

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

   8. mul-1-neg N/A

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

   9. unpow2 N/A

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

   10. associate-*r* N/A

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

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

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

   12. neg-sub0 N/A

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

   13. associate-+l- N/A

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

   14. neg-sub0 N/A

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

   15. mul-1-neg N/A

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

   16. +-commutative N/A

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

   17. lower-fma.f32 N/A

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

5. Simplified 99.1%

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

6. Taylor expanded in uy around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 N/A

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

   9. lower-PI.f32 N/A

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

   10. lower-PI.f32 90.7

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

8. Simplified 90.7%

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

9. Final simplification 90.7%

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

Alternative 14: 83.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.004000000189989805:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(maxCos, -2, 2\right), ux, \left(maxCos + -1\right) \cdot \left(\left(1 - maxCos\right) \cdot \left(ux \cdot ux\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \pi \cdot \pi, 1\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(-2, maxCos, 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* 2.0 uy) 0.004000000189989805)
   (sqrt
    (fma
     (fma maxCos -2.0 2.0)
     ux
     (* (+ maxCos -1.0) (* (- 1.0 maxCos) (* ux ux)))))
   (*
    (fma (* -2.0 (* uy uy)) (* PI PI) 1.0)
    (sqrt (* ux (fma -2.0 maxCos 2.0))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if ((2.0f * uy) <= 0.004000000189989805f) {
		tmp = sqrtf(fmaf(fmaf(maxCos, -2.0f, 2.0f), ux, ((maxCos + -1.0f) * ((1.0f - maxCos) * (ux * ux)))));
	} else {
		tmp = fmaf((-2.0f * (uy * uy)), (((float) M_PI) * ((float) M_PI)), 1.0f) * sqrtf((ux * fmaf(-2.0f, maxCos, 2.0f)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (Float32(Float32(2.0) * uy) <= Float32(0.004000000189989805))
		tmp = sqrt(fma(fma(maxCos, Float32(-2.0), Float32(2.0)), ux, Float32(Float32(maxCos + Float32(-1.0)) * Float32(Float32(Float32(1.0) - maxCos) * Float32(ux * ux)))));
	else
		tmp = Float32(fma(Float32(Float32(-2.0) * Float32(uy * uy)), Float32(Float32(pi) * Float32(pi)), Float32(1.0)) * sqrt(Float32(ux * fma(Float32(-2.0), maxCos, Float32(2.0)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 0.00400000019

   1. Initial program 59.4%

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

   3. Taylor expanded in uy around 0

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

      1. lower-sqrt.f32 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

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

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

      6. lower-fma.f32 N/A

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

   5. Simplified 58.6%

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

   6. Taylor expanded in ux around 0

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

      1. lower-*.f32 N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

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

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

      5. unpow2 N/A

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

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

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. distribute-neg-in N/A

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

      10. mul-1-neg N/A

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

      11. metadata-eval N/A

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

      12. +-commutative N/A

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

      13. mul-1-neg N/A

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

      14. sub-neg N/A

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

      15. lower-fma.f32 N/A

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

   8. Simplified 96.7%

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

      1. lift-+.f32 N/A

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

      2. lift--.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. lift-+.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. +-commutative N/A

         \[\leadsto \sqrt{ux \cdot \color{blue}{\left(-2 \cdot \left(maxCos + -1\right) + ux \cdot \left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right)\right)}} \]

      7. distribute-lft-in N/A

         \[\leadsto \sqrt{\color{blue}{ux \cdot \left(-2 \cdot \left(maxCos + -1\right)\right) + ux \cdot \left(ux \cdot \left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right)\right)}} \]

      8. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\left(-2 \cdot \left(maxCos + -1\right)\right) \cdot ux} + ux \cdot \left(ux \cdot \left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right)\right)} \]

      9. lower-fma.f32 N/A

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(-2 \cdot \left(maxCos + -1\right), ux, ux \cdot \left(ux \cdot \left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right)\right)\right)}} \]

      10. lift-*.f32 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{-2 \cdot \left(maxCos + -1\right)}, ux, ux \cdot \left(ux \cdot \left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right)\right)\right)} \]

      11. lift-+.f32 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(-2 \cdot \color{blue}{\left(maxCos + -1\right)}, ux, ux \cdot \left(ux \cdot \left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right)\right)\right)} \]

      12. distribute-rgt-in N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{maxCos \cdot -2 + -1 \cdot -2}, ux, ux \cdot \left(ux \cdot \left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right)\right)\right)} \]

      13. metadata-eval N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(maxCos \cdot -2 + \color{blue}{2}, ux, ux \cdot \left(ux \cdot \left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right)\right)\right)} \]

      14. lift-fma.f32 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(maxCos, -2, 2\right)}, ux, ux \cdot \left(ux \cdot \left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right)\right)\right)} \]

      15. associate-*r* N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(maxCos, -2, 2\right), ux, \color{blue}{\left(ux \cdot ux\right) \cdot \left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right)}\right)} \]

      16. lift-*.f32 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(maxCos, -2, 2\right), ux, \color{blue}{\left(ux \cdot ux\right)} \cdot \left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right)\right)} \]

      17. *-commutative N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(maxCos, -2, 2\right), ux, \color{blue}{\left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right) \cdot \left(ux \cdot ux\right)}\right)} \]

      18. lift-*.f32 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(maxCos, -2, 2\right), ux, \color{blue}{\left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right)} \cdot \left(ux \cdot ux\right)\right)} \]

   10. Applied egg-rr 96.7%

       \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(maxCos, -2, 2\right), ux, \left(maxCos + -1\right) \cdot \left(\left(1 - maxCos\right) \cdot \left(ux \cdot ux\right)\right)\right)}} \]

   if 0.00400000019 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 53.2%

      \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in ux around inf

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{{ux}^{2} \cdot \left(2 \cdot \frac{1}{ux} - \left(2 \cdot \frac{maxCos}{ux} + {\left(maxCos - 1\right)}^{2}\right)\right)}} \]
   4. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{{ux}^{2} \cdot \left(2 \cdot \frac{1}{ux} - \left(2 \cdot \frac{maxCos}{ux} + {\left(maxCos - 1\right)}^{2}\right)\right)}} \]

      2. unpow2 N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{\left(ux \cdot ux\right)} \cdot \left(2 \cdot \frac{1}{ux} - \left(2 \cdot \frac{maxCos}{ux} + {\left(maxCos - 1\right)}^{2}\right)\right)} \]

      3. lower-*.f32 N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{\left(ux \cdot ux\right)} \cdot \left(2 \cdot \frac{1}{ux} - \left(2 \cdot \frac{maxCos}{ux} + {\left(maxCos - 1\right)}^{2}\right)\right)} \]

      4. sub-neg N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\left(ux \cdot ux\right) \cdot \color{blue}{\left(2 \cdot \frac{1}{ux} + \left(\mathsf{neg}\left(\left(2 \cdot \frac{maxCos}{ux} + {\left(maxCos - 1\right)}^{2}\right)\right)\right)\right)}} \]

      5. +-commutative N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\left(ux \cdot ux\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(2 \cdot \frac{maxCos}{ux} + {\left(maxCos - 1\right)}^{2}\right)\right)\right) + 2 \cdot \frac{1}{ux}\right)}} \]

      6. +-commutative N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\left(ux \cdot ux\right) \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\left({\left(maxCos - 1\right)}^{2} + 2 \cdot \frac{maxCos}{ux}\right)}\right)\right) + 2 \cdot \frac{1}{ux}\right)} \]

      7. distribute-neg-in N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\left(ux \cdot ux\right) \cdot \left(\color{blue}{\left(\left(\mathsf{neg}\left({\left(maxCos - 1\right)}^{2}\right)\right) + \left(\mathsf{neg}\left(2 \cdot \frac{maxCos}{ux}\right)\right)\right)} + 2 \cdot \frac{1}{ux}\right)} \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\left(ux \cdot ux\right) \cdot \left(\left(\left(\mathsf{neg}\left({\left(maxCos - 1\right)}^{2}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot \frac{maxCos}{ux}}\right) + 2 \cdot \frac{1}{ux}\right)} \]

      9. metadata-eval N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\left(ux \cdot ux\right) \cdot \left(\left(\left(\mathsf{neg}\left({\left(maxCos - 1\right)}^{2}\right)\right) + \color{blue}{-2} \cdot \frac{maxCos}{ux}\right) + 2 \cdot \frac{1}{ux}\right)} \]

      10. associate-+l+ N/A

          \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\left(ux \cdot ux\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left({\left(maxCos - 1\right)}^{2}\right)\right) + \left(-2 \cdot \frac{maxCos}{ux} + 2 \cdot \frac{1}{ux}\right)\right)}} \]

      11. unpow2 N/A

          \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\left(ux \cdot ux\right) \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\left(maxCos - 1\right) \cdot \left(maxCos - 1\right)}\right)\right) + \left(-2 \cdot \frac{maxCos}{ux} + 2 \cdot \frac{1}{ux}\right)\right)} \]

      12. distribute-rgt-neg-in N/A

          \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\left(ux \cdot ux\right) \cdot \left(\color{blue}{\left(maxCos - 1\right) \cdot \left(\mathsf{neg}\left(\left(maxCos - 1\right)\right)\right)} + \left(-2 \cdot \frac{maxCos}{ux} + 2 \cdot \frac{1}{ux}\right)\right)} \]

      13. neg-sub0 N/A

          \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\left(ux \cdot ux\right) \cdot \left(\left(maxCos - 1\right) \cdot \color{blue}{\left(0 - \left(maxCos - 1\right)\right)} + \left(-2 \cdot \frac{maxCos}{ux} + 2 \cdot \frac{1}{ux}\right)\right)} \]

      14. associate-+l- N/A

          \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\left(ux \cdot ux\right) \cdot \left(\left(maxCos - 1\right) \cdot \color{blue}{\left(\left(0 - maxCos\right) + 1\right)} + \left(-2 \cdot \frac{maxCos}{ux} + 2 \cdot \frac{1}{ux}\right)\right)} \]

      15. neg-sub0 N/A

          \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\left(ux \cdot ux\right) \cdot \left(\left(maxCos - 1\right) \cdot \left(\color{blue}{\left(\mathsf{neg}\left(maxCos\right)\right)} + 1\right) + \left(-2 \cdot \frac{maxCos}{ux} + 2 \cdot \frac{1}{ux}\right)\right)} \]

      16. mul-1-neg N/A

          \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\left(ux \cdot ux\right) \cdot \left(\left(maxCos - 1\right) \cdot \left(\color{blue}{-1 \cdot maxCos} + 1\right) + \left(-2 \cdot \frac{maxCos}{ux} + 2 \cdot \frac{1}{ux}\right)\right)} \]

      17. +-commutative N/A

          \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\left(ux \cdot ux\right) \cdot \left(\left(maxCos - 1\right) \cdot \color{blue}{\left(1 + -1 \cdot maxCos\right)} + \left(-2 \cdot \frac{maxCos}{ux} + 2 \cdot \frac{1}{ux}\right)\right)} \]

      18. +-commutative N/A

          \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\left(ux \cdot ux\right) \cdot \left(\left(maxCos - 1\right) \cdot \left(1 + -1 \cdot maxCos\right) + \color{blue}{\left(2 \cdot \frac{1}{ux} + -2 \cdot \frac{maxCos}{ux}\right)}\right)} \]

   5. Simplified 98.5%

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\color{blue}{\left(ux \cdot ux\right) \cdot \mathsf{fma}\left(maxCos + -1, 1 - maxCos, \frac{\mathsf{fma}\left(maxCos, -2, 2\right)}{ux}\right)}} \]

   6. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{\left(1 + -2 \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \cdot \sqrt{\left(ux \cdot ux\right) \cdot \mathsf{fma}\left(maxCos + -1, 1 - maxCos, \frac{\mathsf{fma}\left(maxCos, -2, 2\right)}{ux}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(-2 \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 1\right)} \cdot \sqrt{\left(ux \cdot ux\right) \cdot \mathsf{fma}\left(maxCos + -1, 1 - maxCos, \frac{\mathsf{fma}\left(maxCos, -2, 2\right)}{ux}\right)} \]

      2. associate-*r* N/A

         \[\leadsto \left(\color{blue}{\left(-2 \cdot {uy}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2}} + 1\right) \cdot \sqrt{\left(ux \cdot ux\right) \cdot \mathsf{fma}\left(maxCos + -1, 1 - maxCos, \frac{\mathsf{fma}\left(maxCos, -2, 2\right)}{ux}\right)} \]

      3. lower-fma.f32 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-2 \cdot {uy}^{2}, {\mathsf{PI}\left(\right)}^{2}, 1\right)} \cdot \sqrt{\left(ux \cdot ux\right) \cdot \mathsf{fma}\left(maxCos + -1, 1 - maxCos, \frac{\mathsf{fma}\left(maxCos, -2, 2\right)}{ux}\right)} \]

      4. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{-2 \cdot {uy}^{2}}, {\mathsf{PI}\left(\right)}^{2}, 1\right) \cdot \sqrt{\left(ux \cdot ux\right) \cdot \mathsf{fma}\left(maxCos + -1, 1 - maxCos, \frac{\mathsf{fma}\left(maxCos, -2, 2\right)}{ux}\right)} \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(-2 \cdot \color{blue}{\left(uy \cdot uy\right)}, {\mathsf{PI}\left(\right)}^{2}, 1\right) \cdot \sqrt{\left(ux \cdot ux\right) \cdot \mathsf{fma}\left(maxCos + -1, 1 - maxCos, \frac{\mathsf{fma}\left(maxCos, -2, 2\right)}{ux}\right)} \]

      6. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(-2 \cdot \color{blue}{\left(uy \cdot uy\right)}, {\mathsf{PI}\left(\right)}^{2}, 1\right) \cdot \sqrt{\left(ux \cdot ux\right) \cdot \mathsf{fma}\left(maxCos + -1, 1 - maxCos, \frac{\mathsf{fma}\left(maxCos, -2, 2\right)}{ux}\right)} \]

      7. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \color{blue}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}, 1\right) \cdot \sqrt{\left(ux \cdot ux\right) \cdot \mathsf{fma}\left(maxCos + -1, 1 - maxCos, \frac{\mathsf{fma}\left(maxCos, -2, 2\right)}{ux}\right)} \]

      8. lower-*.f32 N/A

         \[\leadsto \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \color{blue}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}, 1\right) \cdot \sqrt{\left(ux \cdot ux\right) \cdot \mathsf{fma}\left(maxCos + -1, 1 - maxCos, \frac{\mathsf{fma}\left(maxCos, -2, 2\right)}{ux}\right)} \]

      9. lower-PI.f32 N/A

         \[\leadsto \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right), 1\right) \cdot \sqrt{\left(ux \cdot ux\right) \cdot \mathsf{fma}\left(maxCos + -1, 1 - maxCos, \frac{\mathsf{fma}\left(maxCos, -2, 2\right)}{ux}\right)} \]

      10. lower-PI.f32 68.4

          \[\leadsto \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \pi \cdot \color{blue}{\pi}, 1\right) \cdot \sqrt{\left(ux \cdot ux\right) \cdot \mathsf{fma}\left(maxCos + -1, 1 - maxCos, \frac{\mathsf{fma}\left(maxCos, -2, 2\right)}{ux}\right)} \]

   8. Simplified 68.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \pi \cdot \pi, 1\right)} \cdot \sqrt{\left(ux \cdot ux\right) \cdot \mathsf{fma}\left(maxCos + -1, 1 - maxCos, \frac{\mathsf{fma}\left(maxCos, -2, 2\right)}{ux}\right)} \]

   9. Taylor expanded in ux around 0

      \[\leadsto \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right) \cdot \color{blue}{\sqrt{ux \cdot \left(2 + -2 \cdot maxCos\right)}} \]
   10. Step-by-step derivation

       1. lower-sqrt.f32 N/A

          \[\leadsto \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right) \cdot \color{blue}{\sqrt{ux \cdot \left(2 + -2 \cdot maxCos\right)}} \]

       2. lower-*.f32 N/A

          \[\leadsto \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right) \cdot \sqrt{\color{blue}{ux \cdot \left(2 + -2 \cdot maxCos\right)}} \]

       3. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right) \cdot \sqrt{ux \cdot \color{blue}{\left(-2 \cdot maxCos + 2\right)}} \]

       4. lower-fma.f32 62.1

          \[\leadsto \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \pi \cdot \pi, 1\right) \cdot \sqrt{ux \cdot \color{blue}{\mathsf{fma}\left(-2, maxCos, 2\right)}} \]

   11. Simplified 62.1%

       \[\leadsto \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \pi \cdot \pi, 1\right) \cdot \color{blue}{\sqrt{ux \cdot \mathsf{fma}\left(-2, maxCos, 2\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.004000000189989805:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(maxCos, -2, 2\right), ux, \left(maxCos + -1\right) \cdot \left(\left(1 - maxCos\right) \cdot \left(ux \cdot ux\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \pi \cdot \pi, 1\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(-2, maxCos, 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 80.1% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(maxCos, -2, 2\right), ux, \left(maxCos + -1\right) \cdot \left(\left(1 - maxCos\right) \cdot \left(ux \cdot ux\right)\right)\right)} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (sqrt
  (fma
   (fma maxCos -2.0 2.0)
   ux
   (* (+ maxCos -1.0) (* (- 1.0 maxCos) (* ux ux))))))

C

float code(float ux, float uy, float maxCos) {
	return sqrtf(fmaf(fmaf(maxCos, -2.0f, 2.0f), ux, ((maxCos + -1.0f) * ((1.0f - maxCos) * (ux * ux)))));
}

Julia

function code(ux, uy, maxCos)
	return sqrt(fma(fma(maxCos, Float32(-2.0), Float32(2.0)), ux, Float32(Float32(maxCos + Float32(-1.0)) * Float32(Float32(Float32(1.0) - maxCos) * Float32(ux * ux)))))
end

Derivation

1. Initial program 57.6%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. Add Preprocessing

3. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{\sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]
4. Step-by-step derivation

   1. lower-sqrt.f32 N/A

      \[\leadsto \color{blue}{\sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

   2. sub-neg N/A

      \[\leadsto \sqrt{\color{blue}{1 + \left(\mathsf{neg}\left({\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}\right)\right)}} \]

   3. +-commutative N/A

      \[\leadsto \sqrt{\color{blue}{\left(\mathsf{neg}\left({\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}\right)\right) + 1}} \]

   4. unpow2 N/A

      \[\leadsto \sqrt{\left(\mathsf{neg}\left(\color{blue}{\left(\left(1 + maxCos \cdot ux\right) - ux\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)}\right)\right) + 1} \]

   5. distribute-rgt-neg-in N/A

      \[\leadsto \sqrt{\color{blue}{\left(\left(1 + maxCos \cdot ux\right) - ux\right) \cdot \left(\mathsf{neg}\left(\left(\left(1 + maxCos \cdot ux\right) - ux\right)\right)\right)} + 1} \]

   6. lower-fma.f32 N/A

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\left(1 + maxCos \cdot ux\right) - ux, \mathsf{neg}\left(\left(\left(1 + maxCos \cdot ux\right) - ux\right)\right), 1\right)}} \]

5. Simplified 51.4%

   \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1 - ux\right), \mathsf{fma}\left(-ux, maxCos + -1, -1\right), 1\right)}} \]

6. Taylor expanded in ux around 0

   \[\leadsto \sqrt{\color{blue}{ux \cdot \left(-2 \cdot \left(maxCos - 1\right) + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right)}} \]
7. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto \sqrt{\color{blue}{ux \cdot \left(-2 \cdot \left(maxCos - 1\right) + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right)}} \]

   2. +-commutative N/A

      \[\leadsto \sqrt{ux \cdot \color{blue}{\left(-1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right) + -2 \cdot \left(maxCos - 1\right)\right)}} \]

   3. mul-1-neg N/A

      \[\leadsto \sqrt{ux \cdot \left(\color{blue}{\left(\mathsf{neg}\left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right)} + -2 \cdot \left(maxCos - 1\right)\right)} \]

   4. distribute-rgt-neg-in N/A

      \[\leadsto \sqrt{ux \cdot \left(\color{blue}{ux \cdot \left(\mathsf{neg}\left({\left(maxCos - 1\right)}^{2}\right)\right)} + -2 \cdot \left(maxCos - 1\right)\right)} \]

   5. unpow2 N/A

      \[\leadsto \sqrt{ux \cdot \left(ux \cdot \left(\mathsf{neg}\left(\color{blue}{\left(maxCos - 1\right) \cdot \left(maxCos - 1\right)}\right)\right) + -2 \cdot \left(maxCos - 1\right)\right)} \]

   6. distribute-lft-neg-in N/A

      \[\leadsto \sqrt{ux \cdot \left(ux \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(maxCos - 1\right)\right)\right) \cdot \left(maxCos - 1\right)\right)} + -2 \cdot \left(maxCos - 1\right)\right)} \]

   7. sub-neg N/A

      \[\leadsto \sqrt{ux \cdot \left(ux \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\left(maxCos + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)\right) \cdot \left(maxCos - 1\right)\right) + -2 \cdot \left(maxCos - 1\right)\right)} \]

   8. metadata-eval N/A

      \[\leadsto \sqrt{ux \cdot \left(ux \cdot \left(\left(\mathsf{neg}\left(\left(maxCos + \color{blue}{-1}\right)\right)\right) \cdot \left(maxCos - 1\right)\right) + -2 \cdot \left(maxCos - 1\right)\right)} \]

   9. distribute-neg-in N/A

      \[\leadsto \sqrt{ux \cdot \left(ux \cdot \left(\color{blue}{\left(\left(\mathsf{neg}\left(maxCos\right)\right) + \left(\mathsf{neg}\left(-1\right)\right)\right)} \cdot \left(maxCos - 1\right)\right) + -2 \cdot \left(maxCos - 1\right)\right)} \]

   10. mul-1-neg N/A

       \[\leadsto \sqrt{ux \cdot \left(ux \cdot \left(\left(\color{blue}{-1 \cdot maxCos} + \left(\mathsf{neg}\left(-1\right)\right)\right) \cdot \left(maxCos - 1\right)\right) + -2 \cdot \left(maxCos - 1\right)\right)} \]

   11. metadata-eval N/A

       \[\leadsto \sqrt{ux \cdot \left(ux \cdot \left(\left(-1 \cdot maxCos + \color{blue}{1}\right) \cdot \left(maxCos - 1\right)\right) + -2 \cdot \left(maxCos - 1\right)\right)} \]

   12. +-commutative N/A

       \[\leadsto \sqrt{ux \cdot \left(ux \cdot \left(\color{blue}{\left(1 + -1 \cdot maxCos\right)} \cdot \left(maxCos - 1\right)\right) + -2 \cdot \left(maxCos - 1\right)\right)} \]

   13. mul-1-neg N/A

       \[\leadsto \sqrt{ux \cdot \left(ux \cdot \left(\left(1 + \color{blue}{\left(\mathsf{neg}\left(maxCos\right)\right)}\right) \cdot \left(maxCos - 1\right)\right) + -2 \cdot \left(maxCos - 1\right)\right)} \]

   14. sub-neg N/A

       \[\leadsto \sqrt{ux \cdot \left(ux \cdot \left(\color{blue}{\left(1 - maxCos\right)} \cdot \left(maxCos - 1\right)\right) + -2 \cdot \left(maxCos - 1\right)\right)} \]

   15. lower-fma.f32 N/A

       \[\leadsto \sqrt{ux \cdot \color{blue}{\mathsf{fma}\left(ux, \left(1 - maxCos\right) \cdot \left(maxCos - 1\right), -2 \cdot \left(maxCos - 1\right)\right)}} \]

8. Simplified 81.5%

   \[\leadsto \sqrt{\color{blue}{ux \cdot \mathsf{fma}\left(ux, \left(maxCos + -1\right) \cdot \left(1 - maxCos\right), -2 \cdot \left(maxCos + -1\right)\right)}} \]
9. Step-by-step derivation

   1. lift-+.f32 N/A

      \[\leadsto \sqrt{ux \cdot \left(ux \cdot \left(\color{blue}{\left(maxCos + -1\right)} \cdot \left(1 - maxCos\right)\right) + -2 \cdot \left(maxCos + -1\right)\right)} \]

   2. lift--.f32 N/A

      \[\leadsto \sqrt{ux \cdot \left(ux \cdot \left(\left(maxCos + -1\right) \cdot \color{blue}{\left(1 - maxCos\right)}\right) + -2 \cdot \left(maxCos + -1\right)\right)} \]

   3. lift-*.f32 N/A

      \[\leadsto \sqrt{ux \cdot \left(ux \cdot \color{blue}{\left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right)} + -2 \cdot \left(maxCos + -1\right)\right)} \]

   4. lift-+.f32 N/A

      \[\leadsto \sqrt{ux \cdot \left(ux \cdot \left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right) + -2 \cdot \color{blue}{\left(maxCos + -1\right)}\right)} \]

   5. lift-*.f32 N/A

      \[\leadsto \sqrt{ux \cdot \left(ux \cdot \left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right) + \color{blue}{-2 \cdot \left(maxCos + -1\right)}\right)} \]

   6. +-commutative N/A

      \[\leadsto \sqrt{ux \cdot \color{blue}{\left(-2 \cdot \left(maxCos + -1\right) + ux \cdot \left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right)\right)}} \]

   7. distribute-lft-in N/A

      \[\leadsto \sqrt{\color{blue}{ux \cdot \left(-2 \cdot \left(maxCos + -1\right)\right) + ux \cdot \left(ux \cdot \left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right)\right)}} \]

   8. *-commutative N/A

      \[\leadsto \sqrt{\color{blue}{\left(-2 \cdot \left(maxCos + -1\right)\right) \cdot ux} + ux \cdot \left(ux \cdot \left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right)\right)} \]

   9. lower-fma.f32 N/A

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(-2 \cdot \left(maxCos + -1\right), ux, ux \cdot \left(ux \cdot \left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right)\right)\right)}} \]

   10. lift-*.f32 N/A

       \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{-2 \cdot \left(maxCos + -1\right)}, ux, ux \cdot \left(ux \cdot \left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right)\right)\right)} \]

   11. lift-+.f32 N/A

       \[\leadsto \sqrt{\mathsf{fma}\left(-2 \cdot \color{blue}{\left(maxCos + -1\right)}, ux, ux \cdot \left(ux \cdot \left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right)\right)\right)} \]

   12. distribute-rgt-in N/A

       \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{maxCos \cdot -2 + -1 \cdot -2}, ux, ux \cdot \left(ux \cdot \left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right)\right)\right)} \]

   13. metadata-eval N/A

       \[\leadsto \sqrt{\mathsf{fma}\left(maxCos \cdot -2 + \color{blue}{2}, ux, ux \cdot \left(ux \cdot \left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right)\right)\right)} \]

   14. lift-fma.f32 N/A

       \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(maxCos, -2, 2\right)}, ux, ux \cdot \left(ux \cdot \left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right)\right)\right)} \]

   15. associate-*r* N/A

       \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(maxCos, -2, 2\right), ux, \color{blue}{\left(ux \cdot ux\right) \cdot \left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right)}\right)} \]

   16. lift-*.f32 N/A

       \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(maxCos, -2, 2\right), ux, \color{blue}{\left(ux \cdot ux\right)} \cdot \left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right)\right)} \]

   17. *-commutative N/A

       \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(maxCos, -2, 2\right), ux, \color{blue}{\left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right) \cdot \left(ux \cdot ux\right)}\right)} \]

   18. lift-*.f32 N/A

       \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(maxCos, -2, 2\right), ux, \color{blue}{\left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right)} \cdot \left(ux \cdot ux\right)\right)} \]

10. Applied egg-rr 81.6%

    \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(maxCos, -2, 2\right), ux, \left(maxCos + -1\right) \cdot \left(\left(1 - maxCos\right) \cdot \left(ux \cdot ux\right)\right)\right)}} \]
11. Add Preprocessing

Alternative 16: 80.1% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \sqrt{\mathsf{fma}\left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right), ux \cdot ux, ux \cdot \mathsf{fma}\left(maxCos, -2, 2\right)\right)} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (sqrt
  (fma
   (* (+ maxCos -1.0) (- 1.0 maxCos))
   (* ux ux)
   (* ux (fma maxCos -2.0 2.0)))))

C

float code(float ux, float uy, float maxCos) {
	return sqrtf(fmaf(((maxCos + -1.0f) * (1.0f - maxCos)), (ux * ux), (ux * fmaf(maxCos, -2.0f, 2.0f))));
}

Julia

function code(ux, uy, maxCos)
	return sqrt(fma(Float32(Float32(maxCos + Float32(-1.0)) * Float32(Float32(1.0) - maxCos)), Float32(ux * ux), Float32(ux * fma(maxCos, Float32(-2.0), Float32(2.0)))))
end

Derivation

1. Initial program 57.6%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. Add Preprocessing

3. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{\sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]
4. Step-by-step derivation

   1. lower-sqrt.f32 N/A

      \[\leadsto \color{blue}{\sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

   2. sub-neg N/A

      \[\leadsto \sqrt{\color{blue}{1 + \left(\mathsf{neg}\left({\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}\right)\right)}} \]

   3. +-commutative N/A

      \[\leadsto \sqrt{\color{blue}{\left(\mathsf{neg}\left({\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}\right)\right) + 1}} \]

   4. unpow2 N/A

      \[\leadsto \sqrt{\left(\mathsf{neg}\left(\color{blue}{\left(\left(1 + maxCos \cdot ux\right) - ux\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)}\right)\right) + 1} \]

   5. distribute-rgt-neg-in N/A

      \[\leadsto \sqrt{\color{blue}{\left(\left(1 + maxCos \cdot ux\right) - ux\right) \cdot \left(\mathsf{neg}\left(\left(\left(1 + maxCos \cdot ux\right) - ux\right)\right)\right)} + 1} \]

   6. lower-fma.f32 N/A

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\left(1 + maxCos \cdot ux\right) - ux, \mathsf{neg}\left(\left(\left(1 + maxCos \cdot ux\right) - ux\right)\right), 1\right)}} \]

5. Simplified 51.4%

   \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1 - ux\right), \mathsf{fma}\left(-ux, maxCos + -1, -1\right), 1\right)}} \]

6. Taylor expanded in ux around 0

   \[\leadsto \sqrt{\color{blue}{ux \cdot \left(-2 \cdot \left(maxCos - 1\right) + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right)}} \]
7. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto \sqrt{\color{blue}{ux \cdot \left(-2 \cdot \left(maxCos - 1\right) + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right)}} \]

   2. +-commutative N/A

      \[\leadsto \sqrt{ux \cdot \color{blue}{\left(-1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right) + -2 \cdot \left(maxCos - 1\right)\right)}} \]

   3. mul-1-neg N/A

      \[\leadsto \sqrt{ux \cdot \left(\color{blue}{\left(\mathsf{neg}\left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right)} + -2 \cdot \left(maxCos - 1\right)\right)} \]

   4. distribute-rgt-neg-in N/A

      \[\leadsto \sqrt{ux \cdot \left(\color{blue}{ux \cdot \left(\mathsf{neg}\left({\left(maxCos - 1\right)}^{2}\right)\right)} + -2 \cdot \left(maxCos - 1\right)\right)} \]

   5. unpow2 N/A

      \[\leadsto \sqrt{ux \cdot \left(ux \cdot \left(\mathsf{neg}\left(\color{blue}{\left(maxCos - 1\right) \cdot \left(maxCos - 1\right)}\right)\right) + -2 \cdot \left(maxCos - 1\right)\right)} \]

   6. distribute-lft-neg-in N/A

      \[\leadsto \sqrt{ux \cdot \left(ux \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(maxCos - 1\right)\right)\right) \cdot \left(maxCos - 1\right)\right)} + -2 \cdot \left(maxCos - 1\right)\right)} \]

   7. sub-neg N/A

      \[\leadsto \sqrt{ux \cdot \left(ux \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\left(maxCos + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)\right) \cdot \left(maxCos - 1\right)\right) + -2 \cdot \left(maxCos - 1\right)\right)} \]

   8. metadata-eval N/A

      \[\leadsto \sqrt{ux \cdot \left(ux \cdot \left(\left(\mathsf{neg}\left(\left(maxCos + \color{blue}{-1}\right)\right)\right) \cdot \left(maxCos - 1\right)\right) + -2 \cdot \left(maxCos - 1\right)\right)} \]

   9. distribute-neg-in N/A

      \[\leadsto \sqrt{ux \cdot \left(ux \cdot \left(\color{blue}{\left(\left(\mathsf{neg}\left(maxCos\right)\right) + \left(\mathsf{neg}\left(-1\right)\right)\right)} \cdot \left(maxCos - 1\right)\right) + -2 \cdot \left(maxCos - 1\right)\right)} \]

   10. mul-1-neg N/A

       \[\leadsto \sqrt{ux \cdot \left(ux \cdot \left(\left(\color{blue}{-1 \cdot maxCos} + \left(\mathsf{neg}\left(-1\right)\right)\right) \cdot \left(maxCos - 1\right)\right) + -2 \cdot \left(maxCos - 1\right)\right)} \]

   11. metadata-eval N/A

       \[\leadsto \sqrt{ux \cdot \left(ux \cdot \left(\left(-1 \cdot maxCos + \color{blue}{1}\right) \cdot \left(maxCos - 1\right)\right) + -2 \cdot \left(maxCos - 1\right)\right)} \]

   12. +-commutative N/A

       \[\leadsto \sqrt{ux \cdot \left(ux \cdot \left(\color{blue}{\left(1 + -1 \cdot maxCos\right)} \cdot \left(maxCos - 1\right)\right) + -2 \cdot \left(maxCos - 1\right)\right)} \]

   13. mul-1-neg N/A

       \[\leadsto \sqrt{ux \cdot \left(ux \cdot \left(\left(1 + \color{blue}{\left(\mathsf{neg}\left(maxCos\right)\right)}\right) \cdot \left(maxCos - 1\right)\right) + -2 \cdot \left(maxCos - 1\right)\right)} \]

   14. sub-neg N/A

       \[\leadsto \sqrt{ux \cdot \left(ux \cdot \left(\color{blue}{\left(1 - maxCos\right)} \cdot \left(maxCos - 1\right)\right) + -2 \cdot \left(maxCos - 1\right)\right)} \]

   15. lower-fma.f32 N/A

       \[\leadsto \sqrt{ux \cdot \color{blue}{\mathsf{fma}\left(ux, \left(1 - maxCos\right) \cdot \left(maxCos - 1\right), -2 \cdot \left(maxCos - 1\right)\right)}} \]

8. Simplified 81.5%

   \[\leadsto \sqrt{\color{blue}{ux \cdot \mathsf{fma}\left(ux, \left(maxCos + -1\right) \cdot \left(1 - maxCos\right), -2 \cdot \left(maxCos + -1\right)\right)}} \]
9. Step-by-step derivation

   1. lift-+.f32 N/A

      \[\leadsto \sqrt{ux \cdot \left(ux \cdot \left(\color{blue}{\left(maxCos + -1\right)} \cdot \left(1 - maxCos\right)\right) + -2 \cdot \left(maxCos + -1\right)\right)} \]

   2. lift--.f32 N/A

      \[\leadsto \sqrt{ux \cdot \left(ux \cdot \left(\left(maxCos + -1\right) \cdot \color{blue}{\left(1 - maxCos\right)}\right) + -2 \cdot \left(maxCos + -1\right)\right)} \]

   3. lift-*.f32 N/A

      \[\leadsto \sqrt{ux \cdot \left(ux \cdot \color{blue}{\left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right)} + -2 \cdot \left(maxCos + -1\right)\right)} \]

   4. lift-+.f32 N/A

      \[\leadsto \sqrt{ux \cdot \left(ux \cdot \left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right) + -2 \cdot \color{blue}{\left(maxCos + -1\right)}\right)} \]

   5. lift-*.f32 N/A

      \[\leadsto \sqrt{ux \cdot \left(ux \cdot \left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right) + \color{blue}{-2 \cdot \left(maxCos + -1\right)}\right)} \]

   6. distribute-rgt-in N/A

      \[\leadsto \sqrt{\color{blue}{\left(ux \cdot \left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right)\right) \cdot ux + \left(-2 \cdot \left(maxCos + -1\right)\right) \cdot ux}} \]

   7. *-commutative N/A

      \[\leadsto \sqrt{\color{blue}{\left(\left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right) \cdot ux\right)} \cdot ux + \left(-2 \cdot \left(maxCos + -1\right)\right) \cdot ux} \]

   8. associate-*r* N/A

      \[\leadsto \sqrt{\color{blue}{\left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right) \cdot \left(ux \cdot ux\right)} + \left(-2 \cdot \left(maxCos + -1\right)\right) \cdot ux} \]

   9. lift-*.f32 N/A

      \[\leadsto \sqrt{\left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right) \cdot \color{blue}{\left(ux \cdot ux\right)} + \left(-2 \cdot \left(maxCos + -1\right)\right) \cdot ux} \]

   10. lift-*.f32 N/A

       \[\leadsto \sqrt{\left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right) \cdot \left(ux \cdot ux\right) + \color{blue}{\left(-2 \cdot \left(maxCos + -1\right)\right)} \cdot ux} \]

   11. lift-+.f32 N/A

       \[\leadsto \sqrt{\left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right) \cdot \left(ux \cdot ux\right) + \left(-2 \cdot \color{blue}{\left(maxCos + -1\right)}\right) \cdot ux} \]

   12. distribute-rgt-in N/A

       \[\leadsto \sqrt{\left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right) \cdot \left(ux \cdot ux\right) + \color{blue}{\left(maxCos \cdot -2 + -1 \cdot -2\right)} \cdot ux} \]

   13. metadata-eval N/A

       \[\leadsto \sqrt{\left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right) \cdot \left(ux \cdot ux\right) + \left(maxCos \cdot -2 + \color{blue}{2}\right) \cdot ux} \]

   14. lift-fma.f32 N/A

       \[\leadsto \sqrt{\left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right) \cdot \left(ux \cdot ux\right) + \color{blue}{\mathsf{fma}\left(maxCos, -2, 2\right)} \cdot ux} \]

   15. lift-*.f32 N/A

       \[\leadsto \sqrt{\left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right) \cdot \left(ux \cdot ux\right) + \color{blue}{\mathsf{fma}\left(maxCos, -2, 2\right) \cdot ux}} \]

   16. lower-fma.f32 81.6

       \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right), ux \cdot ux, \mathsf{fma}\left(maxCos, -2, 2\right) \cdot ux\right)}} \]

10. Applied egg-rr 81.6%

    \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right), ux \cdot ux, ux \cdot \mathsf{fma}\left(maxCos, -2, 2\right)\right)}} \]
11. Add Preprocessing

Alternative 17: 80.0% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \sqrt{ux \cdot \left(\left(maxCos + -1\right) \cdot \mathsf{fma}\left(ux, 1 - maxCos, -2\right)\right)} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (sqrt (* ux (* (+ maxCos -1.0) (fma ux (- 1.0 maxCos) -2.0)))))

C

float code(float ux, float uy, float maxCos) {
	return sqrtf((ux * ((maxCos + -1.0f) * fmaf(ux, (1.0f - maxCos), -2.0f))));
}

Julia

function code(ux, uy, maxCos)
	return sqrt(Float32(ux * Float32(Float32(maxCos + Float32(-1.0)) * fma(ux, Float32(Float32(1.0) - maxCos), Float32(-2.0)))))
end

Derivation

1. Initial program 57.6%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. Add Preprocessing

3. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{\sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]
4. Step-by-step derivation

   1. lower-sqrt.f32 N/A

      \[\leadsto \color{blue}{\sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

   2. sub-neg N/A

      \[\leadsto \sqrt{\color{blue}{1 + \left(\mathsf{neg}\left({\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}\right)\right)}} \]

   3. +-commutative N/A

      \[\leadsto \sqrt{\color{blue}{\left(\mathsf{neg}\left({\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}\right)\right) + 1}} \]

   4. unpow2 N/A

      \[\leadsto \sqrt{\left(\mathsf{neg}\left(\color{blue}{\left(\left(1 + maxCos \cdot ux\right) - ux\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)}\right)\right) + 1} \]

   5. distribute-rgt-neg-in N/A

      \[\leadsto \sqrt{\color{blue}{\left(\left(1 + maxCos \cdot ux\right) - ux\right) \cdot \left(\mathsf{neg}\left(\left(\left(1 + maxCos \cdot ux\right) - ux\right)\right)\right)} + 1} \]

   6. lower-fma.f32 N/A

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\left(1 + maxCos \cdot ux\right) - ux, \mathsf{neg}\left(\left(\left(1 + maxCos \cdot ux\right) - ux\right)\right), 1\right)}} \]

5. Simplified 51.4%

   \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1 - ux\right), \mathsf{fma}\left(-ux, maxCos + -1, -1\right), 1\right)}} \]

6. Taylor expanded in ux around 0

   \[\leadsto \sqrt{\color{blue}{ux \cdot \left(-2 \cdot \left(maxCos - 1\right) + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right)}} \]
7. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto \sqrt{\color{blue}{ux \cdot \left(-2 \cdot \left(maxCos - 1\right) + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right)}} \]

   2. +-commutative N/A

      \[\leadsto \sqrt{ux \cdot \color{blue}{\left(-1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right) + -2 \cdot \left(maxCos - 1\right)\right)}} \]

   3. mul-1-neg N/A

      \[\leadsto \sqrt{ux \cdot \left(\color{blue}{\left(\mathsf{neg}\left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right)} + -2 \cdot \left(maxCos - 1\right)\right)} \]

   4. distribute-rgt-neg-in N/A

      \[\leadsto \sqrt{ux \cdot \left(\color{blue}{ux \cdot \left(\mathsf{neg}\left({\left(maxCos - 1\right)}^{2}\right)\right)} + -2 \cdot \left(maxCos - 1\right)\right)} \]

   5. unpow2 N/A

      \[\leadsto \sqrt{ux \cdot \left(ux \cdot \left(\mathsf{neg}\left(\color{blue}{\left(maxCos - 1\right) \cdot \left(maxCos - 1\right)}\right)\right) + -2 \cdot \left(maxCos - 1\right)\right)} \]

   6. distribute-lft-neg-in N/A

      \[\leadsto \sqrt{ux \cdot \left(ux \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(maxCos - 1\right)\right)\right) \cdot \left(maxCos - 1\right)\right)} + -2 \cdot \left(maxCos - 1\right)\right)} \]

   7. sub-neg N/A

      \[\leadsto \sqrt{ux \cdot \left(ux \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\left(maxCos + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)\right) \cdot \left(maxCos - 1\right)\right) + -2 \cdot \left(maxCos - 1\right)\right)} \]

   8. metadata-eval N/A

      \[\leadsto \sqrt{ux \cdot \left(ux \cdot \left(\left(\mathsf{neg}\left(\left(maxCos + \color{blue}{-1}\right)\right)\right) \cdot \left(maxCos - 1\right)\right) + -2 \cdot \left(maxCos - 1\right)\right)} \]

   9. distribute-neg-in N/A

      \[\leadsto \sqrt{ux \cdot \left(ux \cdot \left(\color{blue}{\left(\left(\mathsf{neg}\left(maxCos\right)\right) + \left(\mathsf{neg}\left(-1\right)\right)\right)} \cdot \left(maxCos - 1\right)\right) + -2 \cdot \left(maxCos - 1\right)\right)} \]

   10. mul-1-neg N/A

       \[\leadsto \sqrt{ux \cdot \left(ux \cdot \left(\left(\color{blue}{-1 \cdot maxCos} + \left(\mathsf{neg}\left(-1\right)\right)\right) \cdot \left(maxCos - 1\right)\right) + -2 \cdot \left(maxCos - 1\right)\right)} \]

   11. metadata-eval N/A

       \[\leadsto \sqrt{ux \cdot \left(ux \cdot \left(\left(-1 \cdot maxCos + \color{blue}{1}\right) \cdot \left(maxCos - 1\right)\right) + -2 \cdot \left(maxCos - 1\right)\right)} \]

   12. +-commutative N/A

       \[\leadsto \sqrt{ux \cdot \left(ux \cdot \left(\color{blue}{\left(1 + -1 \cdot maxCos\right)} \cdot \left(maxCos - 1\right)\right) + -2 \cdot \left(maxCos - 1\right)\right)} \]

   13. mul-1-neg N/A

       \[\leadsto \sqrt{ux \cdot \left(ux \cdot \left(\left(1 + \color{blue}{\left(\mathsf{neg}\left(maxCos\right)\right)}\right) \cdot \left(maxCos - 1\right)\right) + -2 \cdot \left(maxCos - 1\right)\right)} \]

   14. sub-neg N/A

       \[\leadsto \sqrt{ux \cdot \left(ux \cdot \left(\color{blue}{\left(1 - maxCos\right)} \cdot \left(maxCos - 1\right)\right) + -2 \cdot \left(maxCos - 1\right)\right)} \]

   15. lower-fma.f32 N/A

       \[\leadsto \sqrt{ux \cdot \color{blue}{\mathsf{fma}\left(ux, \left(1 - maxCos\right) \cdot \left(maxCos - 1\right), -2 \cdot \left(maxCos - 1\right)\right)}} \]

8. Simplified 81.5%

   \[\leadsto \sqrt{\color{blue}{ux \cdot \mathsf{fma}\left(ux, \left(maxCos + -1\right) \cdot \left(1 - maxCos\right), -2 \cdot \left(maxCos + -1\right)\right)}} \]

9. Taylor expanded in ux around 0

   \[\leadsto \sqrt{\color{blue}{ux \cdot \left(-2 \cdot \left(maxCos - 1\right) + ux \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos - 1\right)\right)\right)}} \]
10. Step-by-step derivation

    1. lower-*.f32 N/A

       \[\leadsto \sqrt{\color{blue}{ux \cdot \left(-2 \cdot \left(maxCos - 1\right) + ux \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos - 1\right)\right)\right)}} \]

    2. +-commutative N/A

       \[\leadsto \sqrt{ux \cdot \color{blue}{\left(ux \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos - 1\right)\right) + -2 \cdot \left(maxCos - 1\right)\right)}} \]

    3. associate-*r* N/A

       \[\leadsto \sqrt{ux \cdot \left(\color{blue}{\left(ux \cdot \left(1 - maxCos\right)\right) \cdot \left(maxCos - 1\right)} + -2 \cdot \left(maxCos - 1\right)\right)} \]

    4. distribute-rgt-out N/A

       \[\leadsto \sqrt{ux \cdot \color{blue}{\left(\left(maxCos - 1\right) \cdot \left(ux \cdot \left(1 - maxCos\right) + -2\right)\right)}} \]

    5. lower-*.f32 N/A

       \[\leadsto \sqrt{ux \cdot \color{blue}{\left(\left(maxCos - 1\right) \cdot \left(ux \cdot \left(1 - maxCos\right) + -2\right)\right)}} \]

    6. sub-neg N/A

       \[\leadsto \sqrt{ux \cdot \left(\color{blue}{\left(maxCos + \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(ux \cdot \left(1 - maxCos\right) + -2\right)\right)} \]

    7. metadata-eval N/A

       \[\leadsto \sqrt{ux \cdot \left(\left(maxCos + \color{blue}{-1}\right) \cdot \left(ux \cdot \left(1 - maxCos\right) + -2\right)\right)} \]

    8. lower-+.f32 N/A

       \[\leadsto \sqrt{ux \cdot \left(\color{blue}{\left(maxCos + -1\right)} \cdot \left(ux \cdot \left(1 - maxCos\right) + -2\right)\right)} \]

    9. lower-fma.f32 N/A

       \[\leadsto \sqrt{ux \cdot \left(\left(maxCos + -1\right) \cdot \color{blue}{\mathsf{fma}\left(ux, 1 - maxCos, -2\right)}\right)} \]

    10. lower--.f32 81.5

        \[\leadsto \sqrt{ux \cdot \left(\left(maxCos + -1\right) \cdot \mathsf{fma}\left(ux, \color{blue}{1 - maxCos}, -2\right)\right)} \]

11. Simplified 81.5%

    \[\leadsto \sqrt{\color{blue}{ux \cdot \left(\left(maxCos + -1\right) \cdot \mathsf{fma}\left(ux, 1 - maxCos, -2\right)\right)}} \]
12. Add Preprocessing

Alternative 18: 79.1% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(maxCos, -2, 2\right), ux, -ux \cdot ux\right)} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (sqrt (fma (fma maxCos -2.0 2.0) ux (- (* ux ux)))))

C

float code(float ux, float uy, float maxCos) {
	return sqrtf(fmaf(fmaf(maxCos, -2.0f, 2.0f), ux, -(ux * ux)));
}

Julia

function code(ux, uy, maxCos)
	return sqrt(fma(fma(maxCos, Float32(-2.0), Float32(2.0)), ux, Float32(-Float32(ux * ux))))
end

Derivation

1. Initial program 57.6%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. Add Preprocessing

3. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{\sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]
4. Step-by-step derivation

   1. lower-sqrt.f32 N/A

      \[\leadsto \color{blue}{\sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

   2. sub-neg N/A

      \[\leadsto \sqrt{\color{blue}{1 + \left(\mathsf{neg}\left({\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}\right)\right)}} \]

   3. +-commutative N/A

      \[\leadsto \sqrt{\color{blue}{\left(\mathsf{neg}\left({\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}\right)\right) + 1}} \]

   4. unpow2 N/A

      \[\leadsto \sqrt{\left(\mathsf{neg}\left(\color{blue}{\left(\left(1 + maxCos \cdot ux\right) - ux\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)}\right)\right) + 1} \]

   5. distribute-rgt-neg-in N/A

      \[\leadsto \sqrt{\color{blue}{\left(\left(1 + maxCos \cdot ux\right) - ux\right) \cdot \left(\mathsf{neg}\left(\left(\left(1 + maxCos \cdot ux\right) - ux\right)\right)\right)} + 1} \]

   6. lower-fma.f32 N/A

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\left(1 + maxCos \cdot ux\right) - ux, \mathsf{neg}\left(\left(\left(1 + maxCos \cdot ux\right) - ux\right)\right), 1\right)}} \]

5. Simplified 51.4%

   \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1 - ux\right), \mathsf{fma}\left(-ux, maxCos + -1, -1\right), 1\right)}} \]

6. Taylor expanded in ux around 0

   \[\leadsto \sqrt{\color{blue}{ux \cdot \left(-2 \cdot \left(maxCos - 1\right) + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right)}} \]
7. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto \sqrt{\color{blue}{ux \cdot \left(-2 \cdot \left(maxCos - 1\right) + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right)}} \]

   2. +-commutative N/A

      \[\leadsto \sqrt{ux \cdot \color{blue}{\left(-1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right) + -2 \cdot \left(maxCos - 1\right)\right)}} \]

   3. mul-1-neg N/A

      \[\leadsto \sqrt{ux \cdot \left(\color{blue}{\left(\mathsf{neg}\left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right)} + -2 \cdot \left(maxCos - 1\right)\right)} \]

   4. distribute-rgt-neg-in N/A

      \[\leadsto \sqrt{ux \cdot \left(\color{blue}{ux \cdot \left(\mathsf{neg}\left({\left(maxCos - 1\right)}^{2}\right)\right)} + -2 \cdot \left(maxCos - 1\right)\right)} \]

   5. unpow2 N/A

      \[\leadsto \sqrt{ux \cdot \left(ux \cdot \left(\mathsf{neg}\left(\color{blue}{\left(maxCos - 1\right) \cdot \left(maxCos - 1\right)}\right)\right) + -2 \cdot \left(maxCos - 1\right)\right)} \]

   6. distribute-lft-neg-in N/A

      \[\leadsto \sqrt{ux \cdot \left(ux \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(maxCos - 1\right)\right)\right) \cdot \left(maxCos - 1\right)\right)} + -2 \cdot \left(maxCos - 1\right)\right)} \]

   7. sub-neg N/A

      \[\leadsto \sqrt{ux \cdot \left(ux \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\left(maxCos + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)\right) \cdot \left(maxCos - 1\right)\right) + -2 \cdot \left(maxCos - 1\right)\right)} \]

   8. metadata-eval N/A

      \[\leadsto \sqrt{ux \cdot \left(ux \cdot \left(\left(\mathsf{neg}\left(\left(maxCos + \color{blue}{-1}\right)\right)\right) \cdot \left(maxCos - 1\right)\right) + -2 \cdot \left(maxCos - 1\right)\right)} \]

   9. distribute-neg-in N/A

      \[\leadsto \sqrt{ux \cdot \left(ux \cdot \left(\color{blue}{\left(\left(\mathsf{neg}\left(maxCos\right)\right) + \left(\mathsf{neg}\left(-1\right)\right)\right)} \cdot \left(maxCos - 1\right)\right) + -2 \cdot \left(maxCos - 1\right)\right)} \]

   10. mul-1-neg N/A

       \[\leadsto \sqrt{ux \cdot \left(ux \cdot \left(\left(\color{blue}{-1 \cdot maxCos} + \left(\mathsf{neg}\left(-1\right)\right)\right) \cdot \left(maxCos - 1\right)\right) + -2 \cdot \left(maxCos - 1\right)\right)} \]

   11. metadata-eval N/A

       \[\leadsto \sqrt{ux \cdot \left(ux \cdot \left(\left(-1 \cdot maxCos + \color{blue}{1}\right) \cdot \left(maxCos - 1\right)\right) + -2 \cdot \left(maxCos - 1\right)\right)} \]

   12. +-commutative N/A

       \[\leadsto \sqrt{ux \cdot \left(ux \cdot \left(\color{blue}{\left(1 + -1 \cdot maxCos\right)} \cdot \left(maxCos - 1\right)\right) + -2 \cdot \left(maxCos - 1\right)\right)} \]

   13. mul-1-neg N/A

       \[\leadsto \sqrt{ux \cdot \left(ux \cdot \left(\left(1 + \color{blue}{\left(\mathsf{neg}\left(maxCos\right)\right)}\right) \cdot \left(maxCos - 1\right)\right) + -2 \cdot \left(maxCos - 1\right)\right)} \]

   14. sub-neg N/A

       \[\leadsto \sqrt{ux \cdot \left(ux \cdot \left(\color{blue}{\left(1 - maxCos\right)} \cdot \left(maxCos - 1\right)\right) + -2 \cdot \left(maxCos - 1\right)\right)} \]

   15. lower-fma.f32 N/A

       \[\leadsto \sqrt{ux \cdot \color{blue}{\mathsf{fma}\left(ux, \left(1 - maxCos\right) \cdot \left(maxCos - 1\right), -2 \cdot \left(maxCos - 1\right)\right)}} \]

8. Simplified 81.5%

   \[\leadsto \sqrt{\color{blue}{ux \cdot \mathsf{fma}\left(ux, \left(maxCos + -1\right) \cdot \left(1 - maxCos\right), -2 \cdot \left(maxCos + -1\right)\right)}} \]
9. Step-by-step derivation

   1. lift-+.f32 N/A

      \[\leadsto \sqrt{ux \cdot \left(ux \cdot \left(\color{blue}{\left(maxCos + -1\right)} \cdot \left(1 - maxCos\right)\right) + -2 \cdot \left(maxCos + -1\right)\right)} \]

   2. lift--.f32 N/A

      \[\leadsto \sqrt{ux \cdot \left(ux \cdot \left(\left(maxCos + -1\right) \cdot \color{blue}{\left(1 - maxCos\right)}\right) + -2 \cdot \left(maxCos + -1\right)\right)} \]

   3. lift-*.f32 N/A

      \[\leadsto \sqrt{ux \cdot \left(ux \cdot \color{blue}{\left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right)} + -2 \cdot \left(maxCos + -1\right)\right)} \]

   4. lift-+.f32 N/A

      \[\leadsto \sqrt{ux \cdot \left(ux \cdot \left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right) + -2 \cdot \color{blue}{\left(maxCos + -1\right)}\right)} \]

   5. lift-*.f32 N/A

      \[\leadsto \sqrt{ux \cdot \left(ux \cdot \left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right) + \color{blue}{-2 \cdot \left(maxCos + -1\right)}\right)} \]

   6. +-commutative N/A

      \[\leadsto \sqrt{ux \cdot \color{blue}{\left(-2 \cdot \left(maxCos + -1\right) + ux \cdot \left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right)\right)}} \]

   7. distribute-lft-in N/A

      \[\leadsto \sqrt{\color{blue}{ux \cdot \left(-2 \cdot \left(maxCos + -1\right)\right) + ux \cdot \left(ux \cdot \left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right)\right)}} \]

   8. *-commutative N/A

      \[\leadsto \sqrt{\color{blue}{\left(-2 \cdot \left(maxCos + -1\right)\right) \cdot ux} + ux \cdot \left(ux \cdot \left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right)\right)} \]

   9. lower-fma.f32 N/A

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(-2 \cdot \left(maxCos + -1\right), ux, ux \cdot \left(ux \cdot \left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right)\right)\right)}} \]

   10. lift-*.f32 N/A

       \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{-2 \cdot \left(maxCos + -1\right)}, ux, ux \cdot \left(ux \cdot \left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right)\right)\right)} \]

   11. lift-+.f32 N/A

       \[\leadsto \sqrt{\mathsf{fma}\left(-2 \cdot \color{blue}{\left(maxCos + -1\right)}, ux, ux \cdot \left(ux \cdot \left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right)\right)\right)} \]

   12. distribute-rgt-in N/A

       \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{maxCos \cdot -2 + -1 \cdot -2}, ux, ux \cdot \left(ux \cdot \left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right)\right)\right)} \]

   13. metadata-eval N/A

       \[\leadsto \sqrt{\mathsf{fma}\left(maxCos \cdot -2 + \color{blue}{2}, ux, ux \cdot \left(ux \cdot \left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right)\right)\right)} \]

   14. lift-fma.f32 N/A

       \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(maxCos, -2, 2\right)}, ux, ux \cdot \left(ux \cdot \left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right)\right)\right)} \]

   15. associate-*r* N/A

       \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(maxCos, -2, 2\right), ux, \color{blue}{\left(ux \cdot ux\right) \cdot \left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right)}\right)} \]

   16. lift-*.f32 N/A

       \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(maxCos, -2, 2\right), ux, \color{blue}{\left(ux \cdot ux\right)} \cdot \left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right)\right)} \]

   17. *-commutative N/A

       \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(maxCos, -2, 2\right), ux, \color{blue}{\left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right) \cdot \left(ux \cdot ux\right)}\right)} \]

   18. lift-*.f32 N/A

       \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(maxCos, -2, 2\right), ux, \color{blue}{\left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right)} \cdot \left(ux \cdot ux\right)\right)} \]

10. Applied egg-rr 81.6%

    \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(maxCos, -2, 2\right), ux, \left(maxCos + -1\right) \cdot \left(\left(1 - maxCos\right) \cdot \left(ux \cdot ux\right)\right)\right)}} \]

11. Taylor expanded in maxCos around 0

    \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(maxCos, -2, 2\right), ux, \color{blue}{-1 \cdot {ux}^{2}}\right)} \]
12. Step-by-step derivation

    1. mul-1-neg N/A

       \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(maxCos, -2, 2\right), ux, \color{blue}{\mathsf{neg}\left({ux}^{2}\right)}\right)} \]

    2. lower-neg.f32 N/A

       \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(maxCos, -2, 2\right), ux, \color{blue}{\mathsf{neg}\left({ux}^{2}\right)}\right)} \]

    3. unpow2 N/A

       \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(maxCos, -2, 2\right), ux, \mathsf{neg}\left(\color{blue}{ux \cdot ux}\right)\right)} \]

    4. lower-*.f32 80.1

       \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(maxCos, -2, 2\right), ux, -\color{blue}{ux \cdot ux}\right)} \]

13. Simplified 80.1%

    \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(maxCos, -2, 2\right), ux, \color{blue}{-ux \cdot ux}\right)} \]
14. Add Preprocessing

Alternative 19: 79.0% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \sqrt{ux \cdot \mathsf{fma}\left(ux, -1, \left(maxCos + -1\right) \cdot -2\right)} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (sqrt (* ux (fma ux -1.0 (* (+ maxCos -1.0) -2.0)))))

C

float code(float ux, float uy, float maxCos) {
	return sqrtf((ux * fmaf(ux, -1.0f, ((maxCos + -1.0f) * -2.0f))));
}

Julia

function code(ux, uy, maxCos)
	return sqrt(Float32(ux * fma(ux, Float32(-1.0), Float32(Float32(maxCos + Float32(-1.0)) * Float32(-2.0)))))
end

Derivation

1. Initial program 57.6%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. Add Preprocessing

3. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{\sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]
4. Step-by-step derivation

   1. lower-sqrt.f32 N/A

      \[\leadsto \color{blue}{\sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

   2. sub-neg N/A

      \[\leadsto \sqrt{\color{blue}{1 + \left(\mathsf{neg}\left({\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}\right)\right)}} \]

   3. +-commutative N/A

      \[\leadsto \sqrt{\color{blue}{\left(\mathsf{neg}\left({\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}\right)\right) + 1}} \]

   4. unpow2 N/A

      \[\leadsto \sqrt{\left(\mathsf{neg}\left(\color{blue}{\left(\left(1 + maxCos \cdot ux\right) - ux\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)}\right)\right) + 1} \]

   5. distribute-rgt-neg-in N/A

      \[\leadsto \sqrt{\color{blue}{\left(\left(1 + maxCos \cdot ux\right) - ux\right) \cdot \left(\mathsf{neg}\left(\left(\left(1 + maxCos \cdot ux\right) - ux\right)\right)\right)} + 1} \]

   6. lower-fma.f32 N/A

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\left(1 + maxCos \cdot ux\right) - ux, \mathsf{neg}\left(\left(\left(1 + maxCos \cdot ux\right) - ux\right)\right), 1\right)}} \]

5. Simplified 51.4%

   \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1 - ux\right), \mathsf{fma}\left(-ux, maxCos + -1, -1\right), 1\right)}} \]

6. Taylor expanded in ux around 0

   \[\leadsto \sqrt{\color{blue}{ux \cdot \left(-2 \cdot \left(maxCos - 1\right) + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right)}} \]
7. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto \sqrt{\color{blue}{ux \cdot \left(-2 \cdot \left(maxCos - 1\right) + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right)}} \]

   2. +-commutative N/A

      \[\leadsto \sqrt{ux \cdot \color{blue}{\left(-1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right) + -2 \cdot \left(maxCos - 1\right)\right)}} \]

   3. mul-1-neg N/A

      \[\leadsto \sqrt{ux \cdot \left(\color{blue}{\left(\mathsf{neg}\left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right)} + -2 \cdot \left(maxCos - 1\right)\right)} \]

   4. distribute-rgt-neg-in N/A

      \[\leadsto \sqrt{ux \cdot \left(\color{blue}{ux \cdot \left(\mathsf{neg}\left({\left(maxCos - 1\right)}^{2}\right)\right)} + -2 \cdot \left(maxCos - 1\right)\right)} \]

   5. unpow2 N/A

      \[\leadsto \sqrt{ux \cdot \left(ux \cdot \left(\mathsf{neg}\left(\color{blue}{\left(maxCos - 1\right) \cdot \left(maxCos - 1\right)}\right)\right) + -2 \cdot \left(maxCos - 1\right)\right)} \]

   6. distribute-lft-neg-in N/A

      \[\leadsto \sqrt{ux \cdot \left(ux \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(maxCos - 1\right)\right)\right) \cdot \left(maxCos - 1\right)\right)} + -2 \cdot \left(maxCos - 1\right)\right)} \]

   7. sub-neg N/A

      \[\leadsto \sqrt{ux \cdot \left(ux \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\left(maxCos + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)\right) \cdot \left(maxCos - 1\right)\right) + -2 \cdot \left(maxCos - 1\right)\right)} \]

   8. metadata-eval N/A

      \[\leadsto \sqrt{ux \cdot \left(ux \cdot \left(\left(\mathsf{neg}\left(\left(maxCos + \color{blue}{-1}\right)\right)\right) \cdot \left(maxCos - 1\right)\right) + -2 \cdot \left(maxCos - 1\right)\right)} \]

   9. distribute-neg-in N/A

      \[\leadsto \sqrt{ux \cdot \left(ux \cdot \left(\color{blue}{\left(\left(\mathsf{neg}\left(maxCos\right)\right) + \left(\mathsf{neg}\left(-1\right)\right)\right)} \cdot \left(maxCos - 1\right)\right) + -2 \cdot \left(maxCos - 1\right)\right)} \]

   10. mul-1-neg N/A

       \[\leadsto \sqrt{ux \cdot \left(ux \cdot \left(\left(\color{blue}{-1 \cdot maxCos} + \left(\mathsf{neg}\left(-1\right)\right)\right) \cdot \left(maxCos - 1\right)\right) + -2 \cdot \left(maxCos - 1\right)\right)} \]

   11. metadata-eval N/A

       \[\leadsto \sqrt{ux \cdot \left(ux \cdot \left(\left(-1 \cdot maxCos + \color{blue}{1}\right) \cdot \left(maxCos - 1\right)\right) + -2 \cdot \left(maxCos - 1\right)\right)} \]

   12. +-commutative N/A

       \[\leadsto \sqrt{ux \cdot \left(ux \cdot \left(\color{blue}{\left(1 + -1 \cdot maxCos\right)} \cdot \left(maxCos - 1\right)\right) + -2 \cdot \left(maxCos - 1\right)\right)} \]

   13. mul-1-neg N/A

       \[\leadsto \sqrt{ux \cdot \left(ux \cdot \left(\left(1 + \color{blue}{\left(\mathsf{neg}\left(maxCos\right)\right)}\right) \cdot \left(maxCos - 1\right)\right) + -2 \cdot \left(maxCos - 1\right)\right)} \]

   14. sub-neg N/A

       \[\leadsto \sqrt{ux \cdot \left(ux \cdot \left(\color{blue}{\left(1 - maxCos\right)} \cdot \left(maxCos - 1\right)\right) + -2 \cdot \left(maxCos - 1\right)\right)} \]

   15. lower-fma.f32 N/A

       \[\leadsto \sqrt{ux \cdot \color{blue}{\mathsf{fma}\left(ux, \left(1 - maxCos\right) \cdot \left(maxCos - 1\right), -2 \cdot \left(maxCos - 1\right)\right)}} \]

8. Simplified 81.5%

   \[\leadsto \sqrt{\color{blue}{ux \cdot \mathsf{fma}\left(ux, \left(maxCos + -1\right) \cdot \left(1 - maxCos\right), -2 \cdot \left(maxCos + -1\right)\right)}} \]

9. Taylor expanded in maxCos around 0

   \[\leadsto \sqrt{ux \cdot \mathsf{fma}\left(ux, \color{blue}{-1}, -2 \cdot \left(maxCos + -1\right)\right)} \]
10. Step-by-step derivation

11. Simplified 80.1%

    \[\leadsto \sqrt{ux \cdot \mathsf{fma}\left(ux, \color{blue}{-1}, -2 \cdot \left(maxCos + -1\right)\right)} \]

12. Final simplification 80.1%

    \[\leadsto \sqrt{ux \cdot \mathsf{fma}\left(ux, -1, \left(maxCos + -1\right) \cdot -2\right)} \]
13. Add Preprocessing

Alternative 20: 75.8% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \sqrt{\mathsf{fma}\left(-ux, ux, ux \cdot 2\right)} \end{array} \]

FPCore

(FPCore (ux uy maxCos) :precision binary32 (sqrt (fma (- ux) ux (* ux 2.0))))

C

float code(float ux, float uy, float maxCos) {
	return sqrtf(fmaf(-ux, ux, (ux * 2.0f)));
}

Julia

function code(ux, uy, maxCos)
	return sqrt(fma(Float32(-ux), ux, Float32(ux * Float32(2.0))))
end

Derivation

1. Initial program 57.6%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. Add Preprocessing

3. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{\sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]
4. Step-by-step derivation

   1. lower-sqrt.f32 N/A

      \[\leadsto \color{blue}{\sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

   2. sub-neg N/A

      \[\leadsto \sqrt{\color{blue}{1 + \left(\mathsf{neg}\left({\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}\right)\right)}} \]

   3. +-commutative N/A

      \[\leadsto \sqrt{\color{blue}{\left(\mathsf{neg}\left({\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}\right)\right) + 1}} \]

   4. unpow2 N/A

      \[\leadsto \sqrt{\left(\mathsf{neg}\left(\color{blue}{\left(\left(1 + maxCos \cdot ux\right) - ux\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)}\right)\right) + 1} \]

   5. distribute-rgt-neg-in N/A

      \[\leadsto \sqrt{\color{blue}{\left(\left(1 + maxCos \cdot ux\right) - ux\right) \cdot \left(\mathsf{neg}\left(\left(\left(1 + maxCos \cdot ux\right) - ux\right)\right)\right)} + 1} \]

   6. lower-fma.f32 N/A

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\left(1 + maxCos \cdot ux\right) - ux, \mathsf{neg}\left(\left(\left(1 + maxCos \cdot ux\right) - ux\right)\right), 1\right)}} \]

5. Simplified 51.4%

   \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1 - ux\right), \mathsf{fma}\left(-ux, maxCos + -1, -1\right), 1\right)}} \]

6. Taylor expanded in ux around 0

   \[\leadsto \sqrt{\color{blue}{ux \cdot \left(-2 \cdot \left(maxCos - 1\right) + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right)}} \]
7. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto \sqrt{\color{blue}{ux \cdot \left(-2 \cdot \left(maxCos - 1\right) + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right)}} \]

   2. +-commutative N/A

      \[\leadsto \sqrt{ux \cdot \color{blue}{\left(-1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right) + -2 \cdot \left(maxCos - 1\right)\right)}} \]

   3. mul-1-neg N/A

      \[\leadsto \sqrt{ux \cdot \left(\color{blue}{\left(\mathsf{neg}\left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right)} + -2 \cdot \left(maxCos - 1\right)\right)} \]

   4. distribute-rgt-neg-in N/A

      \[\leadsto \sqrt{ux \cdot \left(\color{blue}{ux \cdot \left(\mathsf{neg}\left({\left(maxCos - 1\right)}^{2}\right)\right)} + -2 \cdot \left(maxCos - 1\right)\right)} \]

   5. unpow2 N/A

      \[\leadsto \sqrt{ux \cdot \left(ux \cdot \left(\mathsf{neg}\left(\color{blue}{\left(maxCos - 1\right) \cdot \left(maxCos - 1\right)}\right)\right) + -2 \cdot \left(maxCos - 1\right)\right)} \]

   6. distribute-lft-neg-in N/A

      \[\leadsto \sqrt{ux \cdot \left(ux \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(maxCos - 1\right)\right)\right) \cdot \left(maxCos - 1\right)\right)} + -2 \cdot \left(maxCos - 1\right)\right)} \]

   7. sub-neg N/A

      \[\leadsto \sqrt{ux \cdot \left(ux \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\left(maxCos + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)\right) \cdot \left(maxCos - 1\right)\right) + -2 \cdot \left(maxCos - 1\right)\right)} \]

   8. metadata-eval N/A

      \[\leadsto \sqrt{ux \cdot \left(ux \cdot \left(\left(\mathsf{neg}\left(\left(maxCos + \color{blue}{-1}\right)\right)\right) \cdot \left(maxCos - 1\right)\right) + -2 \cdot \left(maxCos - 1\right)\right)} \]

   9. distribute-neg-in N/A

      \[\leadsto \sqrt{ux \cdot \left(ux \cdot \left(\color{blue}{\left(\left(\mathsf{neg}\left(maxCos\right)\right) + \left(\mathsf{neg}\left(-1\right)\right)\right)} \cdot \left(maxCos - 1\right)\right) + -2 \cdot \left(maxCos - 1\right)\right)} \]

   10. mul-1-neg N/A

       \[\leadsto \sqrt{ux \cdot \left(ux \cdot \left(\left(\color{blue}{-1 \cdot maxCos} + \left(\mathsf{neg}\left(-1\right)\right)\right) \cdot \left(maxCos - 1\right)\right) + -2 \cdot \left(maxCos - 1\right)\right)} \]

   11. metadata-eval N/A

       \[\leadsto \sqrt{ux \cdot \left(ux \cdot \left(\left(-1 \cdot maxCos + \color{blue}{1}\right) \cdot \left(maxCos - 1\right)\right) + -2 \cdot \left(maxCos - 1\right)\right)} \]

   12. +-commutative N/A

       \[\leadsto \sqrt{ux \cdot \left(ux \cdot \left(\color{blue}{\left(1 + -1 \cdot maxCos\right)} \cdot \left(maxCos - 1\right)\right) + -2 \cdot \left(maxCos - 1\right)\right)} \]

   13. mul-1-neg N/A

       \[\leadsto \sqrt{ux \cdot \left(ux \cdot \left(\left(1 + \color{blue}{\left(\mathsf{neg}\left(maxCos\right)\right)}\right) \cdot \left(maxCos - 1\right)\right) + -2 \cdot \left(maxCos - 1\right)\right)} \]

   14. sub-neg N/A

       \[\leadsto \sqrt{ux \cdot \left(ux \cdot \left(\color{blue}{\left(1 - maxCos\right)} \cdot \left(maxCos - 1\right)\right) + -2 \cdot \left(maxCos - 1\right)\right)} \]

   15. lower-fma.f32 N/A

       \[\leadsto \sqrt{ux \cdot \color{blue}{\mathsf{fma}\left(ux, \left(1 - maxCos\right) \cdot \left(maxCos - 1\right), -2 \cdot \left(maxCos - 1\right)\right)}} \]

8. Simplified 81.5%

   \[\leadsto \sqrt{\color{blue}{ux \cdot \mathsf{fma}\left(ux, \left(maxCos + -1\right) \cdot \left(1 - maxCos\right), -2 \cdot \left(maxCos + -1\right)\right)}} \]

9. Taylor expanded in maxCos around 0

   \[\leadsto \color{blue}{\sqrt{ux \cdot \left(2 + -1 \cdot ux\right)}} \]
10. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \sqrt{ux \cdot \color{blue}{\left(-1 \cdot ux + 2\right)}} \]

    2. distribute-rgt-in N/A

       \[\leadsto \sqrt{\color{blue}{\left(-1 \cdot ux\right) \cdot ux + 2 \cdot ux}} \]

    3. associate-*r* N/A

       \[\leadsto \sqrt{\color{blue}{-1 \cdot \left(ux \cdot ux\right)} + 2 \cdot ux} \]

    4. unpow2 N/A

       \[\leadsto \sqrt{-1 \cdot \color{blue}{{ux}^{2}} + 2 \cdot ux} \]

    5. lower-sqrt.f32 N/A

       \[\leadsto \color{blue}{\sqrt{-1 \cdot {ux}^{2} + 2 \cdot ux}} \]

    6. +-commutative N/A

       \[\leadsto \sqrt{\color{blue}{2 \cdot ux + -1 \cdot {ux}^{2}}} \]

    7. mul-1-neg N/A

       \[\leadsto \sqrt{2 \cdot ux + \color{blue}{\left(\mathsf{neg}\left({ux}^{2}\right)\right)}} \]

    8. unpow2 N/A

       \[\leadsto \sqrt{2 \cdot ux + \left(\mathsf{neg}\left(\color{blue}{ux \cdot ux}\right)\right)} \]

    9. distribute-lft-neg-in N/A

       \[\leadsto \sqrt{2 \cdot ux + \color{blue}{\left(\mathsf{neg}\left(ux\right)\right) \cdot ux}} \]

    10. mul-1-neg N/A

        \[\leadsto \sqrt{2 \cdot ux + \color{blue}{\left(-1 \cdot ux\right)} \cdot ux} \]

    11. distribute-rgt-in N/A

        \[\leadsto \sqrt{\color{blue}{ux \cdot \left(2 + -1 \cdot ux\right)}} \]

    12. lower-*.f32 N/A

        \[\leadsto \sqrt{\color{blue}{ux \cdot \left(2 + -1 \cdot ux\right)}} \]

    13. mul-1-neg N/A

        \[\leadsto \sqrt{ux \cdot \left(2 + \color{blue}{\left(\mathsf{neg}\left(ux\right)\right)}\right)} \]

    14. unsub-neg N/A

        \[\leadsto \sqrt{ux \cdot \color{blue}{\left(2 - ux\right)}} \]

    15. lower--.f32 76.9

        \[\leadsto \sqrt{ux \cdot \color{blue}{\left(2 - ux\right)}} \]

11. Simplified 76.9%

    \[\leadsto \color{blue}{\sqrt{ux \cdot \left(2 - ux\right)}} \]
12. Step-by-step derivation

    1. sub-neg N/A

       \[\leadsto \sqrt{ux \cdot \color{blue}{\left(2 + \left(\mathsf{neg}\left(ux\right)\right)\right)}} \]

    2. distribute-rgt-in N/A

       \[\leadsto \sqrt{\color{blue}{2 \cdot ux + \left(\mathsf{neg}\left(ux\right)\right) \cdot ux}} \]

    3. unpow1 N/A

       \[\leadsto \sqrt{2 \cdot \color{blue}{{ux}^{1}} + \left(\mathsf{neg}\left(ux\right)\right) \cdot ux} \]

    4. metadata-eval N/A

       \[\leadsto \sqrt{2 \cdot {ux}^{\color{blue}{\left(-1 + 2\right)}} + \left(\mathsf{neg}\left(ux\right)\right) \cdot ux} \]

    5. pow-prod-up N/A

       \[\leadsto \sqrt{2 \cdot \color{blue}{\left({ux}^{-1} \cdot {ux}^{2}\right)} + \left(\mathsf{neg}\left(ux\right)\right) \cdot ux} \]

    6. inv-pow N/A

       \[\leadsto \sqrt{2 \cdot \left(\color{blue}{\frac{1}{ux}} \cdot {ux}^{2}\right) + \left(\mathsf{neg}\left(ux\right)\right) \cdot ux} \]

    7. pow2 N/A

       \[\leadsto \sqrt{2 \cdot \left(\frac{1}{ux} \cdot \color{blue}{\left(ux \cdot ux\right)}\right) + \left(\mathsf{neg}\left(ux\right)\right) \cdot ux} \]

    8. lift-*.f32 N/A

       \[\leadsto \sqrt{2 \cdot \left(\frac{1}{ux} \cdot \color{blue}{\left(ux \cdot ux\right)}\right) + \left(\mathsf{neg}\left(ux\right)\right) \cdot ux} \]

    9. associate-*l* N/A

       \[\leadsto \sqrt{\color{blue}{\left(2 \cdot \frac{1}{ux}\right) \cdot \left(ux \cdot ux\right)} + \left(\mathsf{neg}\left(ux\right)\right) \cdot ux} \]

    10. div-inv N/A

        \[\leadsto \sqrt{\color{blue}{\frac{2}{ux}} \cdot \left(ux \cdot ux\right) + \left(\mathsf{neg}\left(ux\right)\right) \cdot ux} \]

    11. lift-/.f32 N/A

        \[\leadsto \sqrt{\color{blue}{\frac{2}{ux}} \cdot \left(ux \cdot ux\right) + \left(\mathsf{neg}\left(ux\right)\right) \cdot ux} \]

    12. neg-mul-1 N/A

        \[\leadsto \sqrt{\frac{2}{ux} \cdot \left(ux \cdot ux\right) + \color{blue}{\left(-1 \cdot ux\right)} \cdot ux} \]

    13. associate-*r* N/A

        \[\leadsto \sqrt{\frac{2}{ux} \cdot \left(ux \cdot ux\right) + \color{blue}{-1 \cdot \left(ux \cdot ux\right)}} \]

    14. lift-*.f32 N/A

        \[\leadsto \sqrt{\frac{2}{ux} \cdot \left(ux \cdot ux\right) + -1 \cdot \color{blue}{\left(ux \cdot ux\right)}} \]

    15. +-commutative N/A

        \[\leadsto \sqrt{\color{blue}{-1 \cdot \left(ux \cdot ux\right) + \frac{2}{ux} \cdot \left(ux \cdot ux\right)}} \]

    16. lift-/.f32 N/A

        \[\leadsto \sqrt{-1 \cdot \left(ux \cdot ux\right) + \color{blue}{\frac{2}{ux}} \cdot \left(ux \cdot ux\right)} \]

    17. div-inv N/A

        \[\leadsto \sqrt{-1 \cdot \left(ux \cdot ux\right) + \color{blue}{\left(2 \cdot \frac{1}{ux}\right)} \cdot \left(ux \cdot ux\right)} \]

    18. associate-*l* N/A

        \[\leadsto \sqrt{-1 \cdot \left(ux \cdot ux\right) + \color{blue}{2 \cdot \left(\frac{1}{ux} \cdot \left(ux \cdot ux\right)\right)}} \]

    19. inv-pow N/A

        \[\leadsto \sqrt{-1 \cdot \left(ux \cdot ux\right) + 2 \cdot \left(\color{blue}{{ux}^{-1}} \cdot \left(ux \cdot ux\right)\right)} \]

    20. lift-*.f32 N/A

        \[\leadsto \sqrt{-1 \cdot \left(ux \cdot ux\right) + 2 \cdot \left({ux}^{-1} \cdot \color{blue}{\left(ux \cdot ux\right)}\right)} \]

    21. pow2 N/A

        \[\leadsto \sqrt{-1 \cdot \left(ux \cdot ux\right) + 2 \cdot \left({ux}^{-1} \cdot \color{blue}{{ux}^{2}}\right)} \]

    22. pow-prod-up N/A

        \[\leadsto \sqrt{-1 \cdot \left(ux \cdot ux\right) + 2 \cdot \color{blue}{{ux}^{\left(-1 + 2\right)}}} \]

    23. metadata-eval N/A

        \[\leadsto \sqrt{-1 \cdot \left(ux \cdot ux\right) + 2 \cdot {ux}^{\color{blue}{1}}} \]

    24. unpow1 N/A

        \[\leadsto \sqrt{-1 \cdot \left(ux \cdot ux\right) + 2 \cdot \color{blue}{ux}} \]

13. Applied egg-rr 77.0%

    \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(-ux, ux, ux \cdot 2\right)}} \]
14. Add Preprocessing

Alternative 21: 75.8% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \sqrt{ux \cdot \left(2 - ux\right)} \end{array} \]

FPCore

(FPCore (ux uy maxCos) :precision binary32 (sqrt (* ux (- 2.0 ux))))

C

float code(float ux, float uy, float maxCos) {
	return sqrtf((ux * (2.0f - ux)));
}

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = sqrt((ux * (2.0e0 - ux)))
end function

Julia

function code(ux, uy, maxCos)
	return sqrt(Float32(ux * Float32(Float32(2.0) - ux)))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = sqrt((ux * (single(2.0) - ux)));
end

Derivation

1. Initial program 57.6%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. Add Preprocessing

3. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{\sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]
4. Step-by-step derivation

   1. lower-sqrt.f32 N/A

      \[\leadsto \color{blue}{\sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

   2. sub-neg N/A

      \[\leadsto \sqrt{\color{blue}{1 + \left(\mathsf{neg}\left({\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}\right)\right)}} \]

   3. +-commutative N/A

      \[\leadsto \sqrt{\color{blue}{\left(\mathsf{neg}\left({\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}\right)\right) + 1}} \]

   4. unpow2 N/A

      \[\leadsto \sqrt{\left(\mathsf{neg}\left(\color{blue}{\left(\left(1 + maxCos \cdot ux\right) - ux\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)}\right)\right) + 1} \]

   5. distribute-rgt-neg-in N/A

      \[\leadsto \sqrt{\color{blue}{\left(\left(1 + maxCos \cdot ux\right) - ux\right) \cdot \left(\mathsf{neg}\left(\left(\left(1 + maxCos \cdot ux\right) - ux\right)\right)\right)} + 1} \]

   6. lower-fma.f32 N/A

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\left(1 + maxCos \cdot ux\right) - ux, \mathsf{neg}\left(\left(\left(1 + maxCos \cdot ux\right) - ux\right)\right), 1\right)}} \]

5. Simplified 51.4%

   \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1 - ux\right), \mathsf{fma}\left(-ux, maxCos + -1, -1\right), 1\right)}} \]

6. Taylor expanded in ux around 0

   \[\leadsto \sqrt{\color{blue}{ux \cdot \left(-2 \cdot \left(maxCos - 1\right) + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right)}} \]
7. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto \sqrt{\color{blue}{ux \cdot \left(-2 \cdot \left(maxCos - 1\right) + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right)}} \]

   2. +-commutative N/A

      \[\leadsto \sqrt{ux \cdot \color{blue}{\left(-1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right) + -2 \cdot \left(maxCos - 1\right)\right)}} \]

   3. mul-1-neg N/A

      \[\leadsto \sqrt{ux \cdot \left(\color{blue}{\left(\mathsf{neg}\left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right)} + -2 \cdot \left(maxCos - 1\right)\right)} \]

   4. distribute-rgt-neg-in N/A

      \[\leadsto \sqrt{ux \cdot \left(\color{blue}{ux \cdot \left(\mathsf{neg}\left({\left(maxCos - 1\right)}^{2}\right)\right)} + -2 \cdot \left(maxCos - 1\right)\right)} \]

   5. unpow2 N/A

      \[\leadsto \sqrt{ux \cdot \left(ux \cdot \left(\mathsf{neg}\left(\color{blue}{\left(maxCos - 1\right) \cdot \left(maxCos - 1\right)}\right)\right) + -2 \cdot \left(maxCos - 1\right)\right)} \]

   6. distribute-lft-neg-in N/A

      \[\leadsto \sqrt{ux \cdot \left(ux \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(maxCos - 1\right)\right)\right) \cdot \left(maxCos - 1\right)\right)} + -2 \cdot \left(maxCos - 1\right)\right)} \]

   7. sub-neg N/A

      \[\leadsto \sqrt{ux \cdot \left(ux \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\left(maxCos + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)\right) \cdot \left(maxCos - 1\right)\right) + -2 \cdot \left(maxCos - 1\right)\right)} \]

   8. metadata-eval N/A

      \[\leadsto \sqrt{ux \cdot \left(ux \cdot \left(\left(\mathsf{neg}\left(\left(maxCos + \color{blue}{-1}\right)\right)\right) \cdot \left(maxCos - 1\right)\right) + -2 \cdot \left(maxCos - 1\right)\right)} \]

   9. distribute-neg-in N/A

      \[\leadsto \sqrt{ux \cdot \left(ux \cdot \left(\color{blue}{\left(\left(\mathsf{neg}\left(maxCos\right)\right) + \left(\mathsf{neg}\left(-1\right)\right)\right)} \cdot \left(maxCos - 1\right)\right) + -2 \cdot \left(maxCos - 1\right)\right)} \]

   10. mul-1-neg N/A

       \[\leadsto \sqrt{ux \cdot \left(ux \cdot \left(\left(\color{blue}{-1 \cdot maxCos} + \left(\mathsf{neg}\left(-1\right)\right)\right) \cdot \left(maxCos - 1\right)\right) + -2 \cdot \left(maxCos - 1\right)\right)} \]

   11. metadata-eval N/A

       \[\leadsto \sqrt{ux \cdot \left(ux \cdot \left(\left(-1 \cdot maxCos + \color{blue}{1}\right) \cdot \left(maxCos - 1\right)\right) + -2 \cdot \left(maxCos - 1\right)\right)} \]

   12. +-commutative N/A

       \[\leadsto \sqrt{ux \cdot \left(ux \cdot \left(\color{blue}{\left(1 + -1 \cdot maxCos\right)} \cdot \left(maxCos - 1\right)\right) + -2 \cdot \left(maxCos - 1\right)\right)} \]

   13. mul-1-neg N/A

       \[\leadsto \sqrt{ux \cdot \left(ux \cdot \left(\left(1 + \color{blue}{\left(\mathsf{neg}\left(maxCos\right)\right)}\right) \cdot \left(maxCos - 1\right)\right) + -2 \cdot \left(maxCos - 1\right)\right)} \]

   14. sub-neg N/A

       \[\leadsto \sqrt{ux \cdot \left(ux \cdot \left(\color{blue}{\left(1 - maxCos\right)} \cdot \left(maxCos - 1\right)\right) + -2 \cdot \left(maxCos - 1\right)\right)} \]

   15. lower-fma.f32 N/A

       \[\leadsto \sqrt{ux \cdot \color{blue}{\mathsf{fma}\left(ux, \left(1 - maxCos\right) \cdot \left(maxCos - 1\right), -2 \cdot \left(maxCos - 1\right)\right)}} \]

8. Simplified 81.5%

   \[\leadsto \sqrt{\color{blue}{ux \cdot \mathsf{fma}\left(ux, \left(maxCos + -1\right) \cdot \left(1 - maxCos\right), -2 \cdot \left(maxCos + -1\right)\right)}} \]

9. Taylor expanded in maxCos around 0

   \[\leadsto \color{blue}{\sqrt{ux \cdot \left(2 + -1 \cdot ux\right)}} \]
10. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \sqrt{ux \cdot \color{blue}{\left(-1 \cdot ux + 2\right)}} \]

    2. distribute-rgt-in N/A

       \[\leadsto \sqrt{\color{blue}{\left(-1 \cdot ux\right) \cdot ux + 2 \cdot ux}} \]

    3. associate-*r* N/A

       \[\leadsto \sqrt{\color{blue}{-1 \cdot \left(ux \cdot ux\right)} + 2 \cdot ux} \]

    4. unpow2 N/A

       \[\leadsto \sqrt{-1 \cdot \color{blue}{{ux}^{2}} + 2 \cdot ux} \]

    5. lower-sqrt.f32 N/A

       \[\leadsto \color{blue}{\sqrt{-1 \cdot {ux}^{2} + 2 \cdot ux}} \]

    6. +-commutative N/A

       \[\leadsto \sqrt{\color{blue}{2 \cdot ux + -1 \cdot {ux}^{2}}} \]

    7. mul-1-neg N/A

       \[\leadsto \sqrt{2 \cdot ux + \color{blue}{\left(\mathsf{neg}\left({ux}^{2}\right)\right)}} \]

    8. unpow2 N/A

       \[\leadsto \sqrt{2 \cdot ux + \left(\mathsf{neg}\left(\color{blue}{ux \cdot ux}\right)\right)} \]

    9. distribute-lft-neg-in N/A

       \[\leadsto \sqrt{2 \cdot ux + \color{blue}{\left(\mathsf{neg}\left(ux\right)\right) \cdot ux}} \]

    10. mul-1-neg N/A

        \[\leadsto \sqrt{2 \cdot ux + \color{blue}{\left(-1 \cdot ux\right)} \cdot ux} \]

    11. distribute-rgt-in N/A

        \[\leadsto \sqrt{\color{blue}{ux \cdot \left(2 + -1 \cdot ux\right)}} \]

    12. lower-*.f32 N/A

        \[\leadsto \sqrt{\color{blue}{ux \cdot \left(2 + -1 \cdot ux\right)}} \]

    13. mul-1-neg N/A

        \[\leadsto \sqrt{ux \cdot \left(2 + \color{blue}{\left(\mathsf{neg}\left(ux\right)\right)}\right)} \]

    14. unsub-neg N/A

        \[\leadsto \sqrt{ux \cdot \color{blue}{\left(2 - ux\right)}} \]

    15. lower--.f32 76.9

        \[\leadsto \sqrt{ux \cdot \color{blue}{\left(2 - ux\right)}} \]

11. Simplified 76.9%

    \[\leadsto \color{blue}{\sqrt{ux \cdot \left(2 - ux\right)}} \]
12. Add Preprocessing

Alternative 22: 62.4% accurate, 9.8× speedup

Math

\[\begin{array}{l} \\ \sqrt{ux \cdot 2} \end{array} \]

FPCore

(FPCore (ux uy maxCos) :precision binary32 (sqrt (* ux 2.0)))

C

float code(float ux, float uy, float maxCos) {
	return sqrtf((ux * 2.0f));
}

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = sqrt((ux * 2.0e0))
end function

Julia

function code(ux, uy, maxCos)
	return sqrt(Float32(ux * Float32(2.0)))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = sqrt((ux * single(2.0)));
end

Derivation

1. Initial program 57.6%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. Add Preprocessing

3. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{\sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]
4. Step-by-step derivation

   1. lower-sqrt.f32 N/A

      \[\leadsto \color{blue}{\sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

   2. sub-neg N/A

      \[\leadsto \sqrt{\color{blue}{1 + \left(\mathsf{neg}\left({\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}\right)\right)}} \]

   3. +-commutative N/A

      \[\leadsto \sqrt{\color{blue}{\left(\mathsf{neg}\left({\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}\right)\right) + 1}} \]

   4. unpow2 N/A

      \[\leadsto \sqrt{\left(\mathsf{neg}\left(\color{blue}{\left(\left(1 + maxCos \cdot ux\right) - ux\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)}\right)\right) + 1} \]

   5. distribute-rgt-neg-in N/A

      \[\leadsto \sqrt{\color{blue}{\left(\left(1 + maxCos \cdot ux\right) - ux\right) \cdot \left(\mathsf{neg}\left(\left(\left(1 + maxCos \cdot ux\right) - ux\right)\right)\right)} + 1} \]

   6. lower-fma.f32 N/A

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\left(1 + maxCos \cdot ux\right) - ux, \mathsf{neg}\left(\left(\left(1 + maxCos \cdot ux\right) - ux\right)\right), 1\right)}} \]

5. Simplified 51.4%

   \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1 - ux\right), \mathsf{fma}\left(-ux, maxCos + -1, -1\right), 1\right)}} \]

6. Taylor expanded in ux around 0

   \[\leadsto \sqrt{\color{blue}{ux \cdot \left(-2 \cdot \left(maxCos - 1\right) + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right)}} \]
7. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto \sqrt{\color{blue}{ux \cdot \left(-2 \cdot \left(maxCos - 1\right) + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right)}} \]

   2. +-commutative N/A

      \[\leadsto \sqrt{ux \cdot \color{blue}{\left(-1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right) + -2 \cdot \left(maxCos - 1\right)\right)}} \]

   3. mul-1-neg N/A

      \[\leadsto \sqrt{ux \cdot \left(\color{blue}{\left(\mathsf{neg}\left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right)} + -2 \cdot \left(maxCos - 1\right)\right)} \]

   4. distribute-rgt-neg-in N/A

      \[\leadsto \sqrt{ux \cdot \left(\color{blue}{ux \cdot \left(\mathsf{neg}\left({\left(maxCos - 1\right)}^{2}\right)\right)} + -2 \cdot \left(maxCos - 1\right)\right)} \]

   5. unpow2 N/A

      \[\leadsto \sqrt{ux \cdot \left(ux \cdot \left(\mathsf{neg}\left(\color{blue}{\left(maxCos - 1\right) \cdot \left(maxCos - 1\right)}\right)\right) + -2 \cdot \left(maxCos - 1\right)\right)} \]

   6. distribute-lft-neg-in N/A

      \[\leadsto \sqrt{ux \cdot \left(ux \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(maxCos - 1\right)\right)\right) \cdot \left(maxCos - 1\right)\right)} + -2 \cdot \left(maxCos - 1\right)\right)} \]

   7. sub-neg N/A

      \[\leadsto \sqrt{ux \cdot \left(ux \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\left(maxCos + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)\right) \cdot \left(maxCos - 1\right)\right) + -2 \cdot \left(maxCos - 1\right)\right)} \]

   8. metadata-eval N/A

      \[\leadsto \sqrt{ux \cdot \left(ux \cdot \left(\left(\mathsf{neg}\left(\left(maxCos + \color{blue}{-1}\right)\right)\right) \cdot \left(maxCos - 1\right)\right) + -2 \cdot \left(maxCos - 1\right)\right)} \]

   9. distribute-neg-in N/A

      \[\leadsto \sqrt{ux \cdot \left(ux \cdot \left(\color{blue}{\left(\left(\mathsf{neg}\left(maxCos\right)\right) + \left(\mathsf{neg}\left(-1\right)\right)\right)} \cdot \left(maxCos - 1\right)\right) + -2 \cdot \left(maxCos - 1\right)\right)} \]

   10. mul-1-neg N/A

       \[\leadsto \sqrt{ux \cdot \left(ux \cdot \left(\left(\color{blue}{-1 \cdot maxCos} + \left(\mathsf{neg}\left(-1\right)\right)\right) \cdot \left(maxCos - 1\right)\right) + -2 \cdot \left(maxCos - 1\right)\right)} \]

   11. metadata-eval N/A

       \[\leadsto \sqrt{ux \cdot \left(ux \cdot \left(\left(-1 \cdot maxCos + \color{blue}{1}\right) \cdot \left(maxCos - 1\right)\right) + -2 \cdot \left(maxCos - 1\right)\right)} \]

   12. +-commutative N/A

       \[\leadsto \sqrt{ux \cdot \left(ux \cdot \left(\color{blue}{\left(1 + -1 \cdot maxCos\right)} \cdot \left(maxCos - 1\right)\right) + -2 \cdot \left(maxCos - 1\right)\right)} \]

   13. mul-1-neg N/A

       \[\leadsto \sqrt{ux \cdot \left(ux \cdot \left(\left(1 + \color{blue}{\left(\mathsf{neg}\left(maxCos\right)\right)}\right) \cdot \left(maxCos - 1\right)\right) + -2 \cdot \left(maxCos - 1\right)\right)} \]

   14. sub-neg N/A

       \[\leadsto \sqrt{ux \cdot \left(ux \cdot \left(\color{blue}{\left(1 - maxCos\right)} \cdot \left(maxCos - 1\right)\right) + -2 \cdot \left(maxCos - 1\right)\right)} \]

   15. lower-fma.f32 N/A

       \[\leadsto \sqrt{ux \cdot \color{blue}{\mathsf{fma}\left(ux, \left(1 - maxCos\right) \cdot \left(maxCos - 1\right), -2 \cdot \left(maxCos - 1\right)\right)}} \]

8. Simplified 81.5%

   \[\leadsto \sqrt{\color{blue}{ux \cdot \mathsf{fma}\left(ux, \left(maxCos + -1\right) \cdot \left(1 - maxCos\right), -2 \cdot \left(maxCos + -1\right)\right)}} \]

9. Taylor expanded in maxCos around 0

   \[\leadsto \color{blue}{\sqrt{ux \cdot \left(2 + -1 \cdot ux\right)}} \]
10. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \sqrt{ux \cdot \color{blue}{\left(-1 \cdot ux + 2\right)}} \]

    2. distribute-rgt-in N/A

       \[\leadsto \sqrt{\color{blue}{\left(-1 \cdot ux\right) \cdot ux + 2 \cdot ux}} \]

    3. associate-*r* N/A

       \[\leadsto \sqrt{\color{blue}{-1 \cdot \left(ux \cdot ux\right)} + 2 \cdot ux} \]

    4. unpow2 N/A

       \[\leadsto \sqrt{-1 \cdot \color{blue}{{ux}^{2}} + 2 \cdot ux} \]

    5. lower-sqrt.f32 N/A

       \[\leadsto \color{blue}{\sqrt{-1 \cdot {ux}^{2} + 2 \cdot ux}} \]

    6. +-commutative N/A

       \[\leadsto \sqrt{\color{blue}{2 \cdot ux + -1 \cdot {ux}^{2}}} \]

    7. mul-1-neg N/A

       \[\leadsto \sqrt{2 \cdot ux + \color{blue}{\left(\mathsf{neg}\left({ux}^{2}\right)\right)}} \]

    8. unpow2 N/A

       \[\leadsto \sqrt{2 \cdot ux + \left(\mathsf{neg}\left(\color{blue}{ux \cdot ux}\right)\right)} \]

    9. distribute-lft-neg-in N/A

       \[\leadsto \sqrt{2 \cdot ux + \color{blue}{\left(\mathsf{neg}\left(ux\right)\right) \cdot ux}} \]

    10. mul-1-neg N/A

        \[\leadsto \sqrt{2 \cdot ux + \color{blue}{\left(-1 \cdot ux\right)} \cdot ux} \]

    11. distribute-rgt-in N/A

        \[\leadsto \sqrt{\color{blue}{ux \cdot \left(2 + -1 \cdot ux\right)}} \]

    12. lower-*.f32 N/A

        \[\leadsto \sqrt{\color{blue}{ux \cdot \left(2 + -1 \cdot ux\right)}} \]

    13. mul-1-neg N/A

        \[\leadsto \sqrt{ux \cdot \left(2 + \color{blue}{\left(\mathsf{neg}\left(ux\right)\right)}\right)} \]

    14. unsub-neg N/A

        \[\leadsto \sqrt{ux \cdot \color{blue}{\left(2 - ux\right)}} \]

    15. lower--.f32 76.9

        \[\leadsto \sqrt{ux \cdot \color{blue}{\left(2 - ux\right)}} \]

11. Simplified 76.9%

    \[\leadsto \color{blue}{\sqrt{ux \cdot \left(2 - ux\right)}} \]

12. Taylor expanded in ux around 0

    \[\leadsto \sqrt{ux \cdot \color{blue}{2}} \]
13. Step-by-step derivation

14. Simplified 62.6%

    \[\leadsto \sqrt{ux \cdot \color{blue}{2}} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024219 
(FPCore (ux uy maxCos)
  :name "UniformSampleCone, x"
  :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)))
  (* (cos (* (* uy 2.0) PI)) (sqrt (- 1.0 (* (+ (- 1.0 ux) (* ux maxCos)) (+ (- 1.0 ux) (* ux maxCos)))))))