UniformSampleCone, x

Percentage Accurate: 57.4% → 99.0%

Time: 21.1s

Alternatives: 13

Speedup: 3.1×

Specification

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

Math

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

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: 57.4% 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, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.7%

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

   1. associate-*l* 57.7%

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

   2. +-commutative 57.7%

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

   3. associate-+r- 57.6%

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

   4. fma-def 57.6%

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

   5. +-commutative 57.6%

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

   6. associate-+r- 57.5%

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

   7. fma-def 57.5%

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

3. Simplified 57.5%

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

4. Taylor expanded in ux around -inf 99.1%

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

   1. metadata-eval 99.1%

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

   2. cancel-sign-sub-inv 99.1%

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

   3. *-commutative 99.1%

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

   4. fma-def 99.1%

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

   5. cancel-sign-sub-inv 99.1%

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

   6. metadata-eval 99.1%

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

   7. +-commutative 99.1%

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

   8. *-commutative 99.1%

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

   9. fma-def 99.1%

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

   10. mul-1-neg 99.1%

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

   11. distribute-rgt-neg-in 99.1%

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

   12. unpow2 99.1%

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

   13. mul-1-neg 99.1%

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

   14. unsub-neg 99.1%

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

6. Simplified 99.1%

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

   1. pow1 99.1%

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

   2. associate-*l* 99.1%

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

8. Applied egg-rr 99.1%

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

   1. unpow1 99.1%

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

   2. *-commutative 99.1%

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

   3. fma-udef 99.1%

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

   4. fma-udef 99.1%

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

   5. *-commutative 99.1%

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

   6. +-commutative 99.1%

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

   7. *-commutative 99.1%

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

   8. distribute-lft-out 99.1%

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

   9. +-commutative 99.1%

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

   10. fma-def 99.1%

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

   11. distribute-rgt-neg-out 99.1%

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

   12. distribute-lft-neg-in 99.1%

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

   13. associate-*r* 99.1%

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

10. Simplified 99.1%

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

11. Taylor expanded in maxCos around 0 99.1%

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

    1. associate-+r+ 99.0%

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

    2. mul-1-neg 99.0%

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

    3. unsub-neg 99.0%

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

    4. associate-*r* 99.1%

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

    5. distribute-rgt-out 99.1%

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

    6. neg-mul-1 99.1%

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

    7. +-commutative 99.1%

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

    8. unsub-neg 99.1%

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

    9. fma-neg 99.1%

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

    10. metadata-eval 99.1%

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

    11. *-commutative 99.1%

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

    12. unpow2 99.1%

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

    13. unpow2 99.1%

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

13. Simplified 99.1%

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

14. Final simplification 99.1%

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

Alternative 2: 96.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\left(2 \cdot uy\right) \cdot \pi\right)\\ \mathbf{if}\;t_0 \leq 0.9999998211860657:\\ \;\;\;\;t_0 \cdot \sqrt{ux \cdot 2 - ux \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{ux \cdot \left(2 + \left(maxCos \cdot -2 - ux \cdot {\left(1 - maxCos\right)}^{2}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (cos (* (* 2.0 uy) PI))))
   (if (<= t_0 0.9999998211860657)
     (* t_0 (sqrt (- (* ux 2.0) (* ux ux))))
     (sqrt
      (* ux (+ 2.0 (- (* maxCos -2.0) (* ux (pow (- 1.0 maxCos) 2.0)))))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = cosf(((2.0f * uy) * ((float) M_PI)));
	float tmp;
	if (t_0 <= 0.9999998211860657f) {
		tmp = t_0 * sqrtf(((ux * 2.0f) - (ux * ux)));
	} else {
		tmp = sqrtf((ux * (2.0f + ((maxCos * -2.0f) - (ux * powf((1.0f - maxCos), 2.0f))))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = cos(Float32(Float32(Float32(2.0) * uy) * Float32(pi)))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.9999998211860657))
		tmp = Float32(t_0 * sqrt(Float32(Float32(ux * Float32(2.0)) - Float32(ux * ux))));
	else
		tmp = sqrt(Float32(ux * Float32(Float32(2.0) + Float32(Float32(maxCos * Float32(-2.0)) - Float32(ux * (Float32(Float32(1.0) - maxCos) ^ Float32(2.0)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	t_0 = cos(((single(2.0) * uy) * single(pi)));
	tmp = single(0.0);
	if (t_0 <= single(0.9999998211860657))
		tmp = t_0 * sqrt(((ux * single(2.0)) - (ux * ux)));
	else
		tmp = sqrt((ux * (single(2.0) + ((maxCos * single(-2.0)) - (ux * ((single(1.0) - maxCos) ^ single(2.0)))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 (*.f32 uy 2) (PI.f32))) < 0.999999821

   1. Initial program 55.5%

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

      1. associate-*l* 55.5%

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

      2. +-commutative 55.5%

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

      3. associate-+r- 55.5%

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

      4. fma-def 55.5%

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

      5. +-commutative 55.5%

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

      6. associate-+r- 55.4%

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

      7. fma-def 55.4%

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

   3. Simplified 55.4%

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

   4. Taylor expanded in ux around -inf 98.5%

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

      1. metadata-eval 98.5%

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

      2. cancel-sign-sub-inv 98.5%

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

      3. *-commutative 98.5%

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

      4. fma-def 98.5%

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

      5. cancel-sign-sub-inv 98.5%

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

      6. metadata-eval 98.5%

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

      7. +-commutative 98.5%

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

      8. *-commutative 98.5%

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

      9. fma-def 98.5%

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

      10. mul-1-neg 98.5%

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

      11. distribute-rgt-neg-in 98.5%

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

      12. unpow2 98.5%

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

      13. mul-1-neg 98.5%

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

      14. unsub-neg 98.5%

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

   6. Simplified 98.5%

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

   7. Taylor expanded in maxCos around 0 91.8%

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

      1. associate-*r* 91.8%

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

      2. +-commutative 91.8%

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

      3. mul-1-neg 91.8%

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

      4. unsub-neg 91.8%

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

      5. unpow2 91.8%

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

   9. Simplified 91.8%

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

   if 0.999999821 < (cos.f32 (*.f32 (*.f32 uy 2) (PI.f32)))

   1. Initial program 59.5%

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

      1. associate-*l* 59.5%

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

      2. +-commutative 59.5%

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

      3. associate-+r- 59.3%

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

      4. fma-def 59.3%

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

      5. +-commutative 59.3%

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

      6. associate-+r- 59.1%

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

      7. fma-def 59.1%

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

   3. Simplified 59.1%

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

   4. Taylor expanded in ux around -inf 99.5%

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

      1. metadata-eval 99.5%

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

      2. cancel-sign-sub-inv 99.5%

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

      3. *-commutative 99.5%

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

      4. fma-def 99.6%

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

      5. cancel-sign-sub-inv 99.6%

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

      6. metadata-eval 99.6%

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

      7. +-commutative 99.6%

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

      8. *-commutative 99.6%

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

      9. fma-def 99.6%

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

      10. mul-1-neg 99.6%

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

      11. distribute-rgt-neg-in 99.6%

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

      12. unpow2 99.6%

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

      13. mul-1-neg 99.6%

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

      14. unsub-neg 99.6%

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

   6. Simplified 99.6%

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

      1. pow1 99.6%

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

      2. associate-*l* 99.6%

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

   8. Applied egg-rr 99.6%

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

      1. unpow1 99.6%

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

      2. *-commutative 99.6%

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

      3. fma-udef 99.5%

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

      4. fma-udef 99.5%

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

      5. *-commutative 99.5%

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

      6. +-commutative 99.5%

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

      7. *-commutative 99.5%

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

      8. distribute-lft-out 99.6%

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

      9. +-commutative 99.6%

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

      10. fma-def 99.6%

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

      11. distribute-rgt-neg-out 99.6%

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

      12. distribute-lft-neg-in 99.6%

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

      13. associate-*r* 99.6%

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

   10. Simplified 99.6%

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

   11. Taylor expanded in uy around 0 99.5%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\left(2 \cdot uy\right) \cdot \pi\right) \leq 0.9999998211860657:\\ \;\;\;\;\cos \left(\left(2 \cdot uy\right) \cdot \pi\right) \cdot \sqrt{ux \cdot 2 - ux \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{ux \cdot \left(2 + \left(maxCos \cdot -2 - ux \cdot {\left(1 - maxCos\right)}^{2}\right)\right)}\\ \end{array} \]

Alternative 3: 96.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\left(2 \cdot uy\right) \cdot \pi\right)\\ \mathbf{if}\;t_0 \leq 0.9999998211860657:\\ \;\;\;\;t_0 \cdot \sqrt{ux \cdot 2 - ux \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{ux \cdot \left(\mathsf{fma}\left(-2, maxCos, 2\right) - ux \cdot {\left(1 - maxCos\right)}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (cos (* (* 2.0 uy) PI))))
   (if (<= t_0 0.9999998211860657)
     (* t_0 (sqrt (- (* ux 2.0) (* ux ux))))
     (sqrt (* ux (- (fma -2.0 maxCos 2.0) (* ux (pow (- 1.0 maxCos) 2.0))))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = cosf(((2.0f * uy) * ((float) M_PI)));
	float tmp;
	if (t_0 <= 0.9999998211860657f) {
		tmp = t_0 * sqrtf(((ux * 2.0f) - (ux * ux)));
	} else {
		tmp = sqrtf((ux * (fmaf(-2.0f, maxCos, 2.0f) - (ux * powf((1.0f - maxCos), 2.0f)))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = cos(Float32(Float32(Float32(2.0) * uy) * Float32(pi)))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.9999998211860657))
		tmp = Float32(t_0 * sqrt(Float32(Float32(ux * Float32(2.0)) - Float32(ux * ux))));
	else
		tmp = sqrt(Float32(ux * Float32(fma(Float32(-2.0), maxCos, Float32(2.0)) - Float32(ux * (Float32(Float32(1.0) - maxCos) ^ Float32(2.0))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 (*.f32 uy 2) (PI.f32))) < 0.999999821

   1. Initial program 55.5%

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

      1. associate-*l* 55.5%

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

      2. +-commutative 55.5%

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

      3. associate-+r- 55.5%

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

      4. fma-def 55.5%

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

      5. +-commutative 55.5%

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

      6. associate-+r- 55.4%

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

      7. fma-def 55.4%

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

   3. Simplified 55.4%

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

   4. Taylor expanded in ux around -inf 98.5%

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

      1. metadata-eval 98.5%

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

      2. cancel-sign-sub-inv 98.5%

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

      3. *-commutative 98.5%

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

      4. fma-def 98.5%

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

      5. cancel-sign-sub-inv 98.5%

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

      6. metadata-eval 98.5%

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

      7. +-commutative 98.5%

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

      8. *-commutative 98.5%

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

      9. fma-def 98.5%

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

      10. mul-1-neg 98.5%

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

      11. distribute-rgt-neg-in 98.5%

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

      12. unpow2 98.5%

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

      13. mul-1-neg 98.5%

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

      14. unsub-neg 98.5%

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

   6. Simplified 98.5%

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

   7. Taylor expanded in maxCos around 0 91.8%

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

      1. associate-*r* 91.8%

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

      2. +-commutative 91.8%

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

      3. mul-1-neg 91.8%

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

      4. unsub-neg 91.8%

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

      5. unpow2 91.8%

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

   9. Simplified 91.8%

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

   if 0.999999821 < (cos.f32 (*.f32 (*.f32 uy 2) (PI.f32)))

   1. Initial program 59.5%

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

      1. associate-*l* 59.5%

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

      2. +-commutative 59.5%

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

      3. associate-+r- 59.3%

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

      4. fma-def 59.3%

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

      5. +-commutative 59.3%

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

      6. associate-+r- 59.1%

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

      7. fma-def 59.1%

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

   3. Simplified 59.1%

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

   4. Taylor expanded in ux around -inf 99.5%

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

      1. metadata-eval 99.5%

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

      2. cancel-sign-sub-inv 99.5%

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

      3. *-commutative 99.5%

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

      4. fma-def 99.6%

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

      5. cancel-sign-sub-inv 99.6%

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

      6. metadata-eval 99.6%

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

      7. +-commutative 99.6%

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

      8. *-commutative 99.6%

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

      9. fma-def 99.6%

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

      10. mul-1-neg 99.6%

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

      11. distribute-rgt-neg-in 99.6%

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

      12. unpow2 99.6%

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

      13. mul-1-neg 99.6%

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

      14. unsub-neg 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in uy around 0 99.4%

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

      1. fma-def 99.4%

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

      2. +-commutative 99.4%

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

      3. mul-1-neg 99.4%

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

      4. *-commutative 99.4%

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

      5. distribute-rgt-neg-in 99.4%

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

      6. unpow2 99.4%

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

      7. associate-*r* 99.4%

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

      8. fma-def 99.4%

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

      9. *-commutative 99.4%

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

      10. distribute-lft-out 99.5%

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

      11. +-commutative 99.5%

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

      12. fma-def 99.5%

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

      13. distribute-rgt-neg-out 99.5%

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

      14. distribute-lft-neg-in 99.5%

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

   9. Simplified 99.5%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\left(2 \cdot uy\right) \cdot \pi\right) \leq 0.9999998211860657:\\ \;\;\;\;\cos \left(\left(2 \cdot uy\right) \cdot \pi\right) \cdot \sqrt{ux \cdot 2 - ux \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{ux \cdot \left(\mathsf{fma}\left(-2, maxCos, 2\right) - ux \cdot {\left(1 - maxCos\right)}^{2}\right)}\\ \end{array} \]

Alternative 4: 96.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\left(2 \cdot uy\right) \cdot \pi\right) \leq 0.9999998211860657:\\ \;\;\;\;\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{ux \cdot \left(2 + \left(maxCos \cdot -2 - ux \cdot {\left(1 - maxCos\right)}^{2}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (cos (* (* 2.0 uy) PI)) 0.9999998211860657)
   (* (cos (* uy (* 2.0 PI))) (sqrt (* ux (- 2.0 ux))))
   (sqrt (* ux (+ 2.0 (- (* maxCos -2.0) (* ux (pow (- 1.0 maxCos) 2.0))))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (cosf(((2.0f * uy) * ((float) M_PI))) <= 0.9999998211860657f) {
		tmp = cosf((uy * (2.0f * ((float) M_PI)))) * sqrtf((ux * (2.0f - ux)));
	} else {
		tmp = sqrtf((ux * (2.0f + ((maxCos * -2.0f) - (ux * powf((1.0f - maxCos), 2.0f))))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (cos(Float32(Float32(Float32(2.0) * uy) * Float32(pi))) <= Float32(0.9999998211860657))
		tmp = Float32(cos(Float32(uy * Float32(Float32(2.0) * Float32(pi)))) * sqrt(Float32(ux * Float32(Float32(2.0) - ux))));
	else
		tmp = sqrt(Float32(ux * Float32(Float32(2.0) + Float32(Float32(maxCos * Float32(-2.0)) - Float32(ux * (Float32(Float32(1.0) - maxCos) ^ Float32(2.0)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (cos(((single(2.0) * uy) * single(pi))) <= single(0.9999998211860657))
		tmp = cos((uy * (single(2.0) * single(pi)))) * sqrt((ux * (single(2.0) - ux)));
	else
		tmp = sqrt((ux * (single(2.0) + ((maxCos * single(-2.0)) - (ux * ((single(1.0) - maxCos) ^ single(2.0)))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 (*.f32 uy 2) (PI.f32))) < 0.999999821

   1. Initial program 55.5%

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

      1. associate-*l* 55.5%

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

      2. +-commutative 55.5%

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

      3. associate-+r- 55.5%

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

      4. fma-def 55.5%

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

      5. +-commutative 55.5%

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

      6. associate-+r- 55.4%

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

      7. fma-def 55.4%

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

   3. Simplified 55.4%

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

   4. Taylor expanded in ux around 0 98.5%

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

      1. cancel-sign-sub-inv 98.5%

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

      2. metadata-eval 98.5%

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

      3. +-commutative 98.5%

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

      4. mul-1-neg 98.5%

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

      5. unsub-neg 98.5%

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

      6. *-commutative 98.5%

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

      7. fma-def 98.5%

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

      8. sub-neg 98.5%

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

      9. metadata-eval 98.5%

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

      10. unpow2 98.5%

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

   6. Simplified 98.5%

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

   7. Taylor expanded in maxCos around 0 91.8%

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

      1. unpow2 91.8%

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

      2. distribute-rgt-out-- 91.8%

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

   9. Simplified 91.8%

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

   if 0.999999821 < (cos.f32 (*.f32 (*.f32 uy 2) (PI.f32)))

   1. Initial program 59.5%

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

      1. associate-*l* 59.5%

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

      2. +-commutative 59.5%

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

      3. associate-+r- 59.3%

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

      4. fma-def 59.3%

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

      5. +-commutative 59.3%

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

      6. associate-+r- 59.1%

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

      7. fma-def 59.1%

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

   3. Simplified 59.1%

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

   4. Taylor expanded in ux around -inf 99.5%

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

      1. metadata-eval 99.5%

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

      2. cancel-sign-sub-inv 99.5%

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

      3. *-commutative 99.5%

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

      4. fma-def 99.6%

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

      5. cancel-sign-sub-inv 99.6%

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

      6. metadata-eval 99.6%

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

      7. +-commutative 99.6%

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

      8. *-commutative 99.6%

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

      9. fma-def 99.6%

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

      10. mul-1-neg 99.6%

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

      11. distribute-rgt-neg-in 99.6%

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

      12. unpow2 99.6%

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

      13. mul-1-neg 99.6%

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

      14. unsub-neg 99.6%

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

   6. Simplified 99.6%

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

      1. pow1 99.6%

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

      2. associate-*l* 99.6%

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

   8. Applied egg-rr 99.6%

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

      1. unpow1 99.6%

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

      2. *-commutative 99.6%

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

      3. fma-udef 99.5%

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

      4. fma-udef 99.5%

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

      5. *-commutative 99.5%

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

      6. +-commutative 99.5%

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

      7. *-commutative 99.5%

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

      8. distribute-lft-out 99.6%

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

      9. +-commutative 99.6%

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

      10. fma-def 99.6%

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

      11. distribute-rgt-neg-out 99.6%

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

      12. distribute-lft-neg-in 99.6%

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

      13. associate-*r* 99.6%

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

   10. Simplified 99.6%

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

   11. Taylor expanded in uy around 0 99.5%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\left(2 \cdot uy\right) \cdot \pi\right) \leq 0.9999998211860657:\\ \;\;\;\;\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{ux \cdot \left(2 + \left(maxCos \cdot -2 - ux \cdot {\left(1 - maxCos\right)}^{2}\right)\right)}\\ \end{array} \]

Alternative 5: 99.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.7%

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

   1. associate-*l* 57.7%

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

   2. +-commutative 57.7%

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

   3. associate-+r- 57.6%

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

   4. fma-def 57.6%

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

   5. +-commutative 57.6%

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

   6. associate-+r- 57.5%

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

   7. fma-def 57.5%

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

3. Simplified 57.5%

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

4. Taylor expanded in ux around -inf 99.1%

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

   1. metadata-eval 99.1%

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

   2. cancel-sign-sub-inv 99.1%

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

   3. *-commutative 99.1%

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

   4. fma-def 99.1%

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

   5. cancel-sign-sub-inv 99.1%

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

   6. metadata-eval 99.1%

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

   7. +-commutative 99.1%

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

   8. *-commutative 99.1%

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

   9. fma-def 99.1%

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

   10. mul-1-neg 99.1%

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

   11. distribute-rgt-neg-in 99.1%

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

   12. unpow2 99.1%

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

   13. mul-1-neg 99.1%

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

   14. unsub-neg 99.1%

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

6. Simplified 99.1%

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

   1. pow1 99.1%

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

   2. associate-*l* 99.1%

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

8. Applied egg-rr 99.1%

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

   1. unpow1 99.1%

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

   2. *-commutative 99.1%

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

   3. fma-udef 99.1%

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

   4. fma-udef 99.1%

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

   5. *-commutative 99.1%

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

   6. +-commutative 99.1%

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

   7. *-commutative 99.1%

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

   8. distribute-lft-out 99.1%

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

   9. +-commutative 99.1%

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

   10. fma-def 99.1%

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

   11. distribute-rgt-neg-out 99.1%

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

   12. distribute-lft-neg-in 99.1%

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

   13. associate-*r* 99.1%

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

10. Simplified 99.1%

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

    1. unpow2 99.1%

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

12. Applied egg-rr 99.1%

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

13. Final simplification 99.1%

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

Alternative 6: 98.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.7%

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

   1. associate-*l* 57.7%

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

   2. +-commutative 57.7%

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

   3. associate-+r- 57.6%

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

   4. fma-def 57.6%

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

   5. +-commutative 57.6%

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

   6. associate-+r- 57.5%

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

   7. fma-def 57.5%

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

3. Simplified 57.5%

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

4. Taylor expanded in ux around -inf 99.1%

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

   1. metadata-eval 99.1%

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

   2. cancel-sign-sub-inv 99.1%

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

   3. *-commutative 99.1%

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

   4. fma-def 99.1%

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

   5. cancel-sign-sub-inv 99.1%

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

   6. metadata-eval 99.1%

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

   7. +-commutative 99.1%

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

   8. *-commutative 99.1%

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

   9. fma-def 99.1%

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

   10. mul-1-neg 99.1%

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

   11. distribute-rgt-neg-in 99.1%

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

   12. unpow2 99.1%

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

   13. mul-1-neg 99.1%

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

   14. unsub-neg 99.1%

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

6. Simplified 99.1%

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

   1. pow1 99.1%

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

   2. associate-*l* 99.1%

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

8. Applied egg-rr 99.1%

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

   1. unpow1 99.1%

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

   2. *-commutative 99.1%

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

   3. fma-udef 99.1%

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

   4. fma-udef 99.1%

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

   5. *-commutative 99.1%

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

   6. +-commutative 99.1%

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

   7. *-commutative 99.1%

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

   8. distribute-lft-out 99.1%

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

   9. +-commutative 99.1%

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

   10. fma-def 99.1%

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

   11. distribute-rgt-neg-out 99.1%

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

   12. distribute-lft-neg-in 99.1%

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

   13. associate-*r* 99.1%

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

10. Simplified 99.1%

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

11. Taylor expanded in maxCos around 0 97.9%

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

12. Final simplification 97.9%

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

Alternative 7: 89.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.0015999999595806003:\\ \;\;\;\;\sqrt{ux \cdot \left(2 + \left(maxCos \cdot -2 - ux \cdot {\left(1 - maxCos\right)}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= uy 0.0015999999595806003)
   (sqrt (* ux (+ 2.0 (- (* maxCos -2.0) (* ux (pow (- 1.0 maxCos) 2.0))))))
   (* (cos (* uy (* 2.0 PI))) (sqrt (* ux 2.0)))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.0015999999595806003f) {
		tmp = sqrtf((ux * (2.0f + ((maxCos * -2.0f) - (ux * powf((1.0f - maxCos), 2.0f))))));
	} else {
		tmp = cosf((uy * (2.0f * ((float) M_PI)))) * sqrtf((ux * 2.0f));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.0015999999595806003))
		tmp = sqrt(Float32(ux * Float32(Float32(2.0) + Float32(Float32(maxCos * Float32(-2.0)) - Float32(ux * (Float32(Float32(1.0) - maxCos) ^ Float32(2.0)))))));
	else
		tmp = Float32(cos(Float32(uy * Float32(Float32(2.0) * Float32(pi)))) * sqrt(Float32(ux * Float32(2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (uy <= single(0.0015999999595806003))
		tmp = sqrt((ux * (single(2.0) + ((maxCos * single(-2.0)) - (ux * ((single(1.0) - maxCos) ^ single(2.0)))))));
	else
		tmp = cos((uy * (single(2.0) * single(pi)))) * sqrt((ux * single(2.0)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.00159999996

   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. Step-by-step derivation

      1. associate-*l* 59.4%

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

      2. +-commutative 59.4%

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

      3. associate-+r- 59.2%

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

      4. fma-def 59.2%

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

      5. +-commutative 59.2%

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

      6. associate-+r- 59.1%

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

      7. fma-def 59.1%

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

   3. Simplified 59.1%

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

   4. Taylor expanded in ux around -inf 99.5%

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

      1. metadata-eval 99.5%

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

      2. cancel-sign-sub-inv 99.5%

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

      3. *-commutative 99.5%

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

      4. fma-def 99.5%

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

      5. cancel-sign-sub-inv 99.5%

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

      6. metadata-eval 99.5%

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

      7. +-commutative 99.5%

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

      8. *-commutative 99.5%

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

      9. fma-def 99.5%

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

      10. mul-1-neg 99.5%

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

      11. distribute-rgt-neg-in 99.5%

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

      12. unpow2 99.5%

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

      13. mul-1-neg 99.5%

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

      14. unsub-neg 99.5%

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

   6. Simplified 99.5%

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

      1. pow1 99.5%

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

      2. associate-*l* 99.5%

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

   8. Applied egg-rr 99.5%

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

      1. unpow1 99.5%

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

      2. *-commutative 99.5%

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

      3. fma-udef 99.5%

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

      4. fma-udef 99.5%

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

      5. *-commutative 99.5%

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

      6. +-commutative 99.5%

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

      7. *-commutative 99.5%

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

      8. distribute-lft-out 99.5%

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

      9. +-commutative 99.5%

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

      10. fma-def 99.5%

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

      11. distribute-rgt-neg-out 99.5%

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

      12. distribute-lft-neg-in 99.5%

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

      13. associate-*r* 99.5%

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

   10. Simplified 99.5%

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

   11. Taylor expanded in uy around 0 96.0%

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

   if 0.00159999996 < uy

   1. Initial program 53.6%

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

      1. associate-*l* 53.6%

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

      2. +-commutative 53.6%

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

      3. associate-+r- 53.6%

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

      4. fma-def 53.6%

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

      5. +-commutative 53.6%

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

      6. associate-+r- 53.5%

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

      7. fma-def 53.5%

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

   3. Simplified 53.5%

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

   4. Taylor expanded in ux around 0 43.0%

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

   5. Taylor expanded in maxCos around 0 75.4%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \leq 0.0015999999595806003:\\ \;\;\;\;\sqrt{ux \cdot \left(2 + \left(maxCos \cdot -2 - ux \cdot {\left(1 - maxCos\right)}^{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot 2}\\ \end{array} \]

Alternative 8: 80.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

float code(float ux, float uy, float maxCos) {
	return sqrtf((ux * (2.0f + ((maxCos * -2.0f) - (ux * powf((1.0f - maxCos), 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 + ((maxcos * (-2.0e0)) - (ux * ((1.0e0 - maxcos) ** 2.0e0))))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 57.7%

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

   1. associate-*l* 57.7%

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

   2. +-commutative 57.7%

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

   3. associate-+r- 57.6%

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

   4. fma-def 57.6%

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

   5. +-commutative 57.6%

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

   6. associate-+r- 57.5%

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

   7. fma-def 57.5%

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

3. Simplified 57.5%

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

4. Taylor expanded in ux around -inf 99.1%

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

   1. metadata-eval 99.1%

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

   2. cancel-sign-sub-inv 99.1%

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

   3. *-commutative 99.1%

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

   4. fma-def 99.1%

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

   5. cancel-sign-sub-inv 99.1%

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

   6. metadata-eval 99.1%

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

   7. +-commutative 99.1%

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

   8. *-commutative 99.1%

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

   9. fma-def 99.1%

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

   10. mul-1-neg 99.1%

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

   11. distribute-rgt-neg-in 99.1%

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

   12. unpow2 99.1%

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

   13. mul-1-neg 99.1%

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

   14. unsub-neg 99.1%

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

6. Simplified 99.1%

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

   1. pow1 99.1%

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

   2. associate-*l* 99.1%

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

8. Applied egg-rr 99.1%

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

   1. unpow1 99.1%

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

   2. *-commutative 99.1%

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

   3. fma-udef 99.1%

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

   4. fma-udef 99.1%

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

   5. *-commutative 99.1%

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

   6. +-commutative 99.1%

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

   7. *-commutative 99.1%

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

   8. distribute-lft-out 99.1%

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

   9. +-commutative 99.1%

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

   10. fma-def 99.1%

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

   11. distribute-rgt-neg-out 99.1%

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

   12. distribute-lft-neg-in 99.1%

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

   13. associate-*r* 99.1%

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

10. Simplified 99.1%

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

11. Taylor expanded in uy around 0 79.8%

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

12. Final simplification 79.8%

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

Alternative 9: 73.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ux \leq 0.00015999999595806003:\\ \;\;\;\;\sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - {\left(1 - ux\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= ux 0.00015999999595806003)
   (sqrt (* ux (- 2.0 (* 2.0 maxCos))))
   (sqrt (- 1.0 (pow (- 1.0 ux) 2.0)))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (ux <= 0.00015999999595806003f) {
		tmp = sqrtf((ux * (2.0f - (2.0f * maxCos))));
	} else {
		tmp = sqrtf((1.0f - powf((1.0f - ux), 2.0f)));
	}
	return tmp;
}

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    real(4) :: tmp
    if (ux <= 0.00015999999595806003e0) then
        tmp = sqrt((ux * (2.0e0 - (2.0e0 * maxcos))))
    else
        tmp = sqrt((1.0e0 - ((1.0e0 - ux) ** 2.0e0)))
    end if
    code = tmp
end function

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (ux <= Float32(0.00015999999595806003))
		tmp = sqrt(Float32(ux * Float32(Float32(2.0) - Float32(Float32(2.0) * maxCos))));
	else
		tmp = sqrt(Float32(Float32(1.0) - (Float32(Float32(1.0) - ux) ^ Float32(2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (ux <= single(0.00015999999595806003))
		tmp = sqrt((ux * (single(2.0) - (single(2.0) * maxCos))));
	else
		tmp = sqrt((single(1.0) - ((single(1.0) - ux) ^ single(2.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if ux < 1.59999996e-4

   1. Initial program 35.4%

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

      1. associate-*l* 35.4%

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

      2. +-commutative 35.4%

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

      3. associate-+r- 35.4%

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

      4. fma-def 35.4%

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

      5. +-commutative 35.4%

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

      6. associate-+r- 35.3%

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

      7. fma-def 35.3%

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

   3. Simplified 35.3%

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

   4. Taylor expanded in uy around 0 32.2%

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

   5. Taylor expanded in ux around 0 73.6%

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

   if 1.59999996e-4 < ux

   1. Initial program 89.3%

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

      1. associate-*l* 89.3%

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

      2. +-commutative 89.3%

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

      3. associate-+r- 89.1%

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

      4. fma-def 89.1%

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

      5. +-commutative 89.1%

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

      6. associate-+r- 88.9%

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

      7. fma-def 88.9%

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

   3. Simplified 88.9%

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

   4. Taylor expanded in uy around 0 76.1%

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

   5. Taylor expanded in maxCos around 0 72.6%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \leq 0.00015999999595806003:\\ \;\;\;\;\sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - {\left(1 - ux\right)}^{2}}\\ \end{array} \]

Alternative 10: 78.9% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.7%

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

   1. associate-*l* 57.7%

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

   2. +-commutative 57.7%

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

   3. associate-+r- 57.6%

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

   4. fma-def 57.6%

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

   5. +-commutative 57.6%

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

   6. associate-+r- 57.5%

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

   7. fma-def 57.5%

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

3. Simplified 57.5%

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

4. Taylor expanded in ux around -inf 99.1%

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

   1. metadata-eval 99.1%

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

   2. cancel-sign-sub-inv 99.1%

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

   3. *-commutative 99.1%

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

   4. fma-def 99.1%

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

   5. cancel-sign-sub-inv 99.1%

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

   6. metadata-eval 99.1%

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

   7. +-commutative 99.1%

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

   8. *-commutative 99.1%

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

   9. fma-def 99.1%

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

   10. mul-1-neg 99.1%

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

   11. distribute-rgt-neg-in 99.1%

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

   12. unpow2 99.1%

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

   13. mul-1-neg 99.1%

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

   14. unsub-neg 99.1%

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

6. Simplified 99.1%

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

   1. add-log-exp 97.6%

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

8. Applied egg-rr 97.6%

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

9. Taylor expanded in maxCos around 0 95.7%

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

10. Taylor expanded in uy around 0 78.5%

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

    1. +-commutative 78.5%

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

    2. mul-1-neg 78.5%

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

    3. unpow2 78.5%

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

    4. distribute-rgt-neg-in 78.5%

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

    5. *-commutative 78.5%

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

    6. distribute-lft-out 78.5%

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

    7. fma-def 78.5%

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

12. Simplified 78.5%

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

13. Final simplification 78.5%

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

Alternative 11: 64.8% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

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 - (2.0e0 * maxcos))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 57.7%

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

   1. associate-*l* 57.7%

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

   2. +-commutative 57.7%

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

   3. associate-+r- 57.6%

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

   4. fma-def 57.6%

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

   5. +-commutative 57.6%

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

   6. associate-+r- 57.5%

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

   7. fma-def 57.5%

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

3. Simplified 57.5%

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

4. Taylor expanded in uy around 0 50.4%

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

5. Taylor expanded in ux around 0 63.9%

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

6. Final simplification 63.9%

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

Alternative 12: 61.9% accurate, 3.1× 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 57.7%

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

   1. associate-*l* 57.7%

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

   2. +-commutative 57.7%

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

   3. associate-+r- 57.6%

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

   4. fma-def 57.6%

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

   5. +-commutative 57.6%

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

   6. associate-+r- 57.5%

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

   7. fma-def 57.5%

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

3. Simplified 57.5%

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

4. Taylor expanded in uy around 0 50.4%

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

5. Taylor expanded in ux around 0 63.9%

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

6. Taylor expanded in maxCos around 0 60.9%

   \[\leadsto \sqrt{\color{blue}{2 \cdot ux}} \]

7. Final simplification 60.9%

   \[\leadsto \sqrt{ux \cdot 2} \]

Alternative 13: 6.6% accurate, 322.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (ux uy maxCos) :precision binary32 0.0)

C

float code(float ux, float uy, float maxCos) {
	return 0.0f;
}

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = 0.0e0
end function

Julia

function code(ux, uy, maxCos)
	return Float32(0.0)
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = single(0.0);
end

Derivation

1. Initial program 57.7%

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

   1. associate-*l* 57.7%

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

   2. +-commutative 57.7%

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

   3. associate-+r- 57.6%

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

   4. fma-def 57.6%

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

   5. +-commutative 57.6%

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

   6. associate-+r- 57.5%

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

   7. fma-def 57.5%

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

3. Simplified 57.5%

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

4. Taylor expanded in uy around 0 50.4%

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

5. Taylor expanded in ux around 0 63.9%

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

   1. add-log-exp 55.9%

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

   2. *-commutative 55.9%

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

   3. cancel-sign-sub-inv 55.9%

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

   4. metadata-eval 55.9%

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

7. Applied egg-rr 55.9%

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

8. Taylor expanded in ux around 0 6.6%

   \[\leadsto \log \color{blue}{1} \]

9. Final simplification 6.6%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023274 
(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)))))))