UniformSampleCone, x

Percentage Accurate: 57.2% → 98.9%

Time: 15.2s

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: 57.2% 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: 98.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

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- 59.4%

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

   4. fma-def 59.4%

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

   5. +-commutative 59.4%

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

   6. associate-+r- 59.3%

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

   7. fma-def 59.3%

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

3. Simplified 59.3%

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

4. Taylor expanded in ux around -inf 99.0%

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

   1. metadata-eval 99.0%

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

   2. cancel-sign-sub-inv 99.0%

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

   3. *-commutative 99.0%

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

   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. *-commutative 99.1%

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

   12. distribute-rgt-neg-in 99.1%

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

   13. mul-1-neg 99.1%

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

   14. sub-neg 99.1%

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

   15. unpow2 99.1%

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

   16. distribute-rgt-neg-in 99.1%

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

6. Simplified 99.1%

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

   1. pow1 99.1%

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

8. Applied egg-rr 99.1%

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

   1. unpow1 99.1%

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

   2. *-commutative 99.1%

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

10. Simplified 99.1%

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

11. Taylor expanded in uy around inf 99.1%

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

12. Taylor expanded in maxCos around 0 99.1%

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

    1. fma-def 99.1%

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

    2. *-commutative 99.1%

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

    3. fma-def 99.1%

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

    4. mul-1-neg 99.1%

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

    5. *-commutative 99.1%

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

    6. unpow2 99.1%

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

    7. unpow2 99.1%

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

14. Simplified 99.1%

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

15. Final simplification 99.1%

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

Alternative 2: 99.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

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. sub-neg 59.4%

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

   3. +-commutative 59.4%

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

   4. distribute-rgt-neg-in 59.4%

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

   5. fma-def 59.6%

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

   6. +-commutative 59.6%

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

   7. associate-+r- 59.7%

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

   8. fma-def 59.7%

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

   9. neg-sub0 59.7%

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

   10. +-commutative 59.7%

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

   11. associate-+r- 59.6%

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

   12. associate--r- 59.6%

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

   13. neg-sub0 59.6%

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

   14. +-commutative 59.6%

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

   15. sub-neg 59.6%

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

   16. fma-def 59.6%

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

3. Simplified 59.6%

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

4. Taylor expanded in ux around 0 99.0%

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

   1. fma-def 99.0%

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

   2. sub-neg 99.0%

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

   3. metadata-eval 99.0%

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

   4. *-commutative 99.0%

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

   5. unpow2 99.0%

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

   6. associate--l+ 99.0%

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

   7. distribute-rgt-in 99.1%

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

   8. *-lft-identity 99.1%

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

   9. mul-1-neg 99.1%

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

   10. sub-neg 99.1%

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

   11. metadata-eval 99.1%

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

   12. distribute-neg-in 99.1%

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

   13. metadata-eval 99.1%

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

   14. +-commutative 99.1%

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

   15. sub-neg 99.1%

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

6. Simplified 99.1%

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

7. Final simplification 99.1%

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

Alternative 3: 99.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

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- 59.4%

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

   4. fma-def 59.4%

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

   5. +-commutative 59.4%

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

   6. associate-+r- 59.3%

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

   7. fma-def 59.3%

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

3. Simplified 59.3%

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

4. Taylor expanded in ux around -inf 99.0%

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

   1. metadata-eval 99.0%

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

   2. cancel-sign-sub-inv 99.0%

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

   3. *-commutative 99.0%

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

   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. *-commutative 99.1%

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

   12. distribute-rgt-neg-in 99.1%

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

   13. mul-1-neg 99.1%

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

   14. sub-neg 99.1%

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

   15. unpow2 99.1%

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

   16. distribute-rgt-neg-in 99.1%

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

6. Simplified 99.1%

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

   1. pow1 99.1%

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

8. Applied egg-rr 99.1%

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

   1. unpow1 99.1%

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

   2. *-commutative 99.1%

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

10. Simplified 99.1%

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

11. Taylor expanded in uy around inf 99.1%

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

12. Final simplification 99.1%

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

Alternative 4: 98.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

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- 59.4%

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

   4. fma-def 59.4%

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

   5. +-commutative 59.4%

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

   6. associate-+r- 59.3%

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

   7. fma-def 59.3%

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

3. Simplified 59.3%

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

4. Taylor expanded in ux around -inf 99.0%

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

   1. metadata-eval 99.0%

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

   2. cancel-sign-sub-inv 99.0%

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

   3. *-commutative 99.0%

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

   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. *-commutative 99.1%

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

   12. distribute-rgt-neg-in 99.1%

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

   13. mul-1-neg 99.1%

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

   14. sub-neg 99.1%

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

   15. unpow2 99.1%

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

   16. distribute-rgt-neg-in 99.1%

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

6. Simplified 99.1%

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

   1. pow1 99.1%

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

8. Applied egg-rr 99.1%

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

   1. unpow1 99.1%

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

   2. *-commutative 99.1%

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

10. Simplified 99.1%

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

11. Taylor expanded in maxCos around 0 98.6%

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

12. Final simplification 98.6%

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

Alternative 5: 95.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(uy \cdot \left(2 \cdot \pi\right)\right)\\ \mathbf{if}\;maxCos \leq 9.999999747378752 \cdot 10^{-5}:\\ \;\;\;\;t_0 \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (cos (* uy (* 2.0 PI)))))
   (if (<= maxCos 9.999999747378752e-5)
     (* t_0 (sqrt (* ux (- 2.0 ux))))
     (* t_0 (sqrt (* ux (- 2.0 (* 2.0 maxCos))))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = cosf((uy * (2.0f * ((float) M_PI))));
	float tmp;
	if (maxCos <= 9.999999747378752e-5f) {
		tmp = t_0 * sqrtf((ux * (2.0f - ux)));
	} else {
		tmp = t_0 * sqrtf((ux * (2.0f - (2.0f * maxCos))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = cos(Float32(uy * Float32(Float32(2.0) * Float32(pi))))
	tmp = Float32(0.0)
	if (maxCos <= Float32(9.999999747378752e-5))
		tmp = Float32(t_0 * sqrt(Float32(ux * Float32(Float32(2.0) - ux))));
	else
		tmp = Float32(t_0 * sqrt(Float32(ux * Float32(Float32(2.0) - Float32(Float32(2.0) * maxCos)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	t_0 = cos((uy * (single(2.0) * single(pi))));
	tmp = single(0.0);
	if (maxCos <= single(9.999999747378752e-5))
		tmp = t_0 * sqrt((ux * (single(2.0) - ux)));
	else
		tmp = t_0 * sqrt((ux * (single(2.0) - (single(2.0) * maxCos))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if maxCos < 9.99999975e-5

   1. Initial program 60.5%

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

      1. associate-*l* 60.5%

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

      2. +-commutative 60.5%

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

      3. associate-+r- 60.6%

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

      4. fma-def 60.6%

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

      5. +-commutative 60.6%

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

      6. associate-+r- 60.5%

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

      7. fma-def 60.5%

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

   3. Simplified 60.5%

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

   4. Taylor expanded in ux around -inf 99.2%

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

      1. metadata-eval 99.2%

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

      2. cancel-sign-sub-inv 99.2%

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

      3. *-commutative 99.2%

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

      4. fma-def 99.2%

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

      5. cancel-sign-sub-inv 99.2%

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

      6. metadata-eval 99.2%

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

      7. +-commutative 99.2%

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

      8. *-commutative 99.2%

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

      9. fma-def 99.2%

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

      10. mul-1-neg 99.2%

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

      11. *-commutative 99.2%

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

      12. distribute-rgt-neg-in 99.2%

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

      13. mul-1-neg 99.2%

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

      14. sub-neg 99.2%

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

      15. unpow2 99.2%

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

      16. distribute-rgt-neg-in 99.2%

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

   6. Simplified 99.2%

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

   7. Taylor expanded in maxCos around 0 98.2%

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

      1. mul-1-neg 98.2%

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

      2. +-commutative 98.2%

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

      3. sub-neg 98.2%

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

      4. unpow2 98.2%

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

      5. distribute-rgt-out-- 98.2%

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

   9. Simplified 98.2%

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

   if 9.99999975e-5 < maxCos

   1. Initial program 50.3%

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

      1. associate-*l* 50.3%

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

      2. +-commutative 50.3%

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

      3. associate-+r- 50.4%

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

      4. fma-def 50.4%

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

      5. +-commutative 50.4%

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

      6. associate-+r- 49.4%

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

      7. fma-def 49.4%

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

   3. Simplified 49.4%

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

   4. Taylor expanded in ux around 0 82.3%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;maxCos \leq 9.999999747378752 \cdot 10^{-5}:\\ \;\;\;\;\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\\ \end{array} \]

Alternative 6: 95.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(uy \cdot \left(2 \cdot \pi\right)\right)\\ \mathbf{if}\;maxCos \leq 9.999999747378752 \cdot 10^{-5}:\\ \;\;\;\;t_0 \cdot \sqrt{2 \cdot ux - ux \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (cos (* uy (* 2.0 PI)))))
   (if (<= maxCos 9.999999747378752e-5)
     (* t_0 (sqrt (- (* 2.0 ux) (* ux ux))))
     (* t_0 (sqrt (* ux (- 2.0 (* 2.0 maxCos))))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = cosf((uy * (2.0f * ((float) M_PI))));
	float tmp;
	if (maxCos <= 9.999999747378752e-5f) {
		tmp = t_0 * sqrtf(((2.0f * ux) - (ux * ux)));
	} else {
		tmp = t_0 * sqrtf((ux * (2.0f - (2.0f * maxCos))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = cos(Float32(uy * Float32(Float32(2.0) * Float32(pi))))
	tmp = Float32(0.0)
	if (maxCos <= Float32(9.999999747378752e-5))
		tmp = Float32(t_0 * sqrt(Float32(Float32(Float32(2.0) * ux) - Float32(ux * ux))));
	else
		tmp = Float32(t_0 * sqrt(Float32(ux * Float32(Float32(2.0) - Float32(Float32(2.0) * maxCos)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	t_0 = cos((uy * (single(2.0) * single(pi))));
	tmp = single(0.0);
	if (maxCos <= single(9.999999747378752e-5))
		tmp = t_0 * sqrt(((single(2.0) * ux) - (ux * ux)));
	else
		tmp = t_0 * sqrt((ux * (single(2.0) - (single(2.0) * maxCos))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if maxCos < 9.99999975e-5

   1. Initial program 60.5%

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

      1. associate-*l* 60.5%

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

      2. +-commutative 60.5%

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

      3. associate-+r- 60.6%

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

      4. fma-def 60.6%

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

      5. +-commutative 60.6%

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

      6. associate-+r- 60.5%

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

      7. fma-def 60.5%

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

   3. Simplified 60.5%

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

   4. Taylor expanded in ux around -inf 99.2%

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

      1. metadata-eval 99.2%

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

      2. cancel-sign-sub-inv 99.2%

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

      3. *-commutative 99.2%

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

      4. fma-def 99.2%

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

      5. cancel-sign-sub-inv 99.2%

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

      6. metadata-eval 99.2%

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

      7. +-commutative 99.2%

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

      8. *-commutative 99.2%

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

      9. fma-def 99.2%

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

      10. mul-1-neg 99.2%

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

      11. *-commutative 99.2%

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

      12. distribute-rgt-neg-in 99.2%

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

      13. mul-1-neg 99.2%

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

      14. sub-neg 99.2%

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

      15. unpow2 99.2%

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

      16. distribute-rgt-neg-in 99.2%

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

   6. Simplified 99.2%

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

   7. Taylor expanded in maxCos around 0 98.2%

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

      1. mul-1-neg 98.2%

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

      2. +-commutative 98.2%

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

      3. sub-neg 98.2%

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

      4. unpow2 98.2%

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

   9. Simplified 98.2%

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

   if 9.99999975e-5 < maxCos

   1. Initial program 50.3%

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

      1. associate-*l* 50.3%

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

      2. +-commutative 50.3%

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

      3. associate-+r- 50.4%

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

      4. fma-def 50.4%

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

      5. +-commutative 50.4%

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

      6. associate-+r- 49.4%

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

      7. fma-def 49.4%

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

   3. Simplified 49.4%

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

   4. Taylor expanded in ux around 0 82.3%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;maxCos \leq 9.999999747378752 \cdot 10^{-5}:\\ \;\;\;\;\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux - ux \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\\ \end{array} \]

Alternative 7: 95.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(uy \cdot \left(2 \cdot \pi\right)\right)\\ \mathbf{if}\;maxCos \leq 9.999999747378752 \cdot 10^{-5}:\\ \;\;\;\;t_0 \cdot \sqrt{2 \cdot ux - ux \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \sqrt{ux \cdot \left(\left(2 - maxCos\right) - maxCos\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (cos (* uy (* 2.0 PI)))))
   (if (<= maxCos 9.999999747378752e-5)
     (* t_0 (sqrt (- (* 2.0 ux) (* ux ux))))
     (* t_0 (sqrt (* ux (- (- 2.0 maxCos) maxCos)))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = cosf((uy * (2.0f * ((float) M_PI))));
	float tmp;
	if (maxCos <= 9.999999747378752e-5f) {
		tmp = t_0 * sqrtf(((2.0f * ux) - (ux * ux)));
	} else {
		tmp = t_0 * sqrtf((ux * ((2.0f - maxCos) - maxCos)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = cos(Float32(uy * Float32(Float32(2.0) * Float32(pi))))
	tmp = Float32(0.0)
	if (maxCos <= Float32(9.999999747378752e-5))
		tmp = Float32(t_0 * sqrt(Float32(Float32(Float32(2.0) * ux) - Float32(ux * ux))));
	else
		tmp = Float32(t_0 * sqrt(Float32(ux * Float32(Float32(Float32(2.0) - maxCos) - maxCos))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	t_0 = cos((uy * (single(2.0) * single(pi))));
	tmp = single(0.0);
	if (maxCos <= single(9.999999747378752e-5))
		tmp = t_0 * sqrt(((single(2.0) * ux) - (ux * ux)));
	else
		tmp = t_0 * sqrt((ux * ((single(2.0) - maxCos) - maxCos)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if maxCos < 9.99999975e-5

   1. Initial program 60.5%

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

      1. associate-*l* 60.5%

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

      2. +-commutative 60.5%

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

      3. associate-+r- 60.6%

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

      4. fma-def 60.6%

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

      5. +-commutative 60.6%

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

      6. associate-+r- 60.5%

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

      7. fma-def 60.5%

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

   3. Simplified 60.5%

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

   4. Taylor expanded in ux around -inf 99.2%

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

      1. metadata-eval 99.2%

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

      2. cancel-sign-sub-inv 99.2%

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

      3. *-commutative 99.2%

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

      4. fma-def 99.2%

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

      5. cancel-sign-sub-inv 99.2%

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

      6. metadata-eval 99.2%

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

      7. +-commutative 99.2%

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

      8. *-commutative 99.2%

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

      9. fma-def 99.2%

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

      10. mul-1-neg 99.2%

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

      11. *-commutative 99.2%

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

      12. distribute-rgt-neg-in 99.2%

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

      13. mul-1-neg 99.2%

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

      14. sub-neg 99.2%

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

      15. unpow2 99.2%

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

      16. distribute-rgt-neg-in 99.2%

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

   6. Simplified 99.2%

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

   7. Taylor expanded in maxCos around 0 98.2%

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

      1. mul-1-neg 98.2%

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

      2. +-commutative 98.2%

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

      3. sub-neg 98.2%

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

      4. unpow2 98.2%

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

   9. Simplified 98.2%

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

   if 9.99999975e-5 < maxCos

   1. Initial program 50.3%

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

      1. associate-*l* 50.3%

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

      2. sub-neg 50.3%

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

      3. +-commutative 50.3%

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

      4. distribute-rgt-neg-in 50.3%

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

      5. fma-def 50.4%

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

      6. +-commutative 50.4%

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

      7. associate-+r- 50.5%

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

      8. fma-def 50.5%

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

      9. neg-sub0 50.5%

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

      10. +-commutative 50.5%

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

      11. associate-+r- 49.5%

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

      12. associate--r- 49.5%

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

      13. neg-sub0 49.5%

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

      14. +-commutative 49.5%

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

      15. sub-neg 49.5%

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

      16. fma-def 49.5%

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

   3. Simplified 49.5%

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

   4. Taylor expanded in ux around 0 82.4%

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

      1. mul-1-neg 82.4%

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

      2. sub-neg 82.4%

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

      3. metadata-eval 82.4%

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

   6. Simplified 82.4%

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

      1. pow1 82.4%

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

      2. unsub-neg 82.4%

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

   8. Applied egg-rr 82.4%

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

      1. unpow1 82.4%

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

      2. associate--r+ 82.4%

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

      3. sub-neg 82.4%

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

      4. sub-neg 82.4%

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

      5. metadata-eval 82.4%

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

      6. associate-+r+ 82.4%

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

      7. +-commutative 82.4%

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

      8. associate-+r+ 82.5%

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

      9. metadata-eval 82.5%

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

   10. Simplified 82.5%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;maxCos \leq 9.999999747378752 \cdot 10^{-5}:\\ \;\;\;\;\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux - ux \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(\left(2 - maxCos\right) - maxCos\right)}\\ \end{array} \]

Alternative 8: 89.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.003000000026077032:\\ \;\;\;\;\sqrt{ux \cdot \left(\left(2 + maxCos \cdot -2\right) - ux \cdot {\left(1 - maxCos\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= uy 0.003000000026077032)
   (sqrt (* ux (- (+ 2.0 (* maxCos -2.0)) (* ux (pow (- 1.0 maxCos) 2.0)))))
   (* (cos (* uy (* 2.0 PI))) (sqrt (* 2.0 ux)))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.003000000026077032f) {
		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((2.0f * ux));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.003000000026077032))
		tmp = sqrt(Float32(ux * Float32(Float32(Float32(2.0) + 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(Float32(2.0) * ux)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (uy <= single(0.003000000026077032))
		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((single(2.0) * ux));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.00300000003

   1. Initial program 61.2%

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

      1. associate-*l* 61.2%

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

      2. +-commutative 61.2%

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

      3. associate-+r- 61.2%

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

      4. fma-def 61.2%

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

      5. +-commutative 61.2%

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

      6. associate-+r- 61.0%

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

      7. fma-def 61.0%

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

   3. Simplified 61.0%

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

   4. Taylor expanded in ux around -inf 99.4%

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

      1. metadata-eval 99.4%

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

      2. cancel-sign-sub-inv 99.4%

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

      3. *-commutative 99.4%

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

      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. *-commutative 99.5%

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

      12. distribute-rgt-neg-in 99.5%

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

      13. mul-1-neg 99.5%

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

      14. sub-neg 99.5%

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

      15. unpow2 99.5%

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

      16. distribute-rgt-neg-in 99.5%

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

   6. Simplified 99.5%

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

      1. pow1 99.5%

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

   8. Applied egg-rr 99.5%

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

      1. unpow1 99.5%

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

      2. *-commutative 99.5%

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

   10. Simplified 99.4%

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

   11. Taylor expanded in uy around 0 95.1%

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

   if 0.00300000003 < uy

   1. Initial program 54.9%

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

      1. associate-*l* 54.9%

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

      2. sub-neg 54.9%

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

      3. +-commutative 54.9%

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

      4. distribute-rgt-neg-in 54.9%

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

      5. fma-def 55.3%

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

      6. +-commutative 55.3%

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

      7. associate-+r- 55.5%

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

      8. fma-def 55.5%

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

      9. neg-sub0 55.5%

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

      10. +-commutative 55.5%

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

      11. associate-+r- 55.3%

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

      12. associate--r- 55.3%

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

      13. neg-sub0 55.3%

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

      14. +-commutative 55.3%

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

      15. sub-neg 55.3%

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

      16. fma-def 55.3%

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

   3. Simplified 55.3%

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

   4. Taylor expanded in ux around 0 78.0%

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

      1. mul-1-neg 78.0%

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

      2. sub-neg 78.0%

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

      3. metadata-eval 78.0%

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

   6. Simplified 78.0%

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

   7. Taylor expanded in maxCos around 0 72.9%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \leq 0.003000000026077032:\\ \;\;\;\;\sqrt{ux \cdot \left(\left(2 + maxCos \cdot -2\right) - ux \cdot {\left(1 - maxCos\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux}\\ \end{array} \]

Alternative 9: 92.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

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- 59.4%

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

   4. fma-def 59.4%

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

   5. +-commutative 59.4%

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

   6. associate-+r- 59.3%

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

   7. fma-def 59.3%

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

3. Simplified 59.3%

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

4. Taylor expanded in ux around -inf 99.0%

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

   1. metadata-eval 99.0%

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

   2. cancel-sign-sub-inv 99.0%

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

   3. *-commutative 99.0%

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

   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. *-commutative 99.1%

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

   12. distribute-rgt-neg-in 99.1%

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

   13. mul-1-neg 99.1%

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

   14. sub-neg 99.1%

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

   15. unpow2 99.1%

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

   16. distribute-rgt-neg-in 99.1%

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

6. Simplified 99.1%

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

7. Taylor expanded in maxCos around 0 92.7%

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

   1. mul-1-neg 92.7%

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

   2. +-commutative 92.7%

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

   3. sub-neg 92.7%

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

   4. unpow2 92.7%

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

   5. distribute-rgt-out-- 92.6%

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

9. Simplified 92.6%

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

10. Final simplification 92.6%

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

Alternative 10: 80.2% accurate, 1.5× speedup

Math

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

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(Float32(2.0) + 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 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- 59.4%

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

   4. fma-def 59.4%

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

   5. +-commutative 59.4%

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

   6. associate-+r- 59.3%

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

   7. fma-def 59.3%

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

3. Simplified 59.3%

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

4. Taylor expanded in ux around -inf 99.0%

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

   1. metadata-eval 99.0%

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

   2. cancel-sign-sub-inv 99.0%

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

   3. *-commutative 99.0%

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

   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. *-commutative 99.1%

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

   12. distribute-rgt-neg-in 99.1%

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

   13. mul-1-neg 99.1%

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

   14. sub-neg 99.1%

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

   15. unpow2 99.1%

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

   16. distribute-rgt-neg-in 99.1%

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

6. Simplified 99.1%

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

   1. pow1 99.1%

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

8. Applied egg-rr 99.1%

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

   1. unpow1 99.1%

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

   2. *-commutative 99.1%

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

10. Simplified 99.1%

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

11. Taylor expanded in uy around 0 78.7%

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

12. Final simplification 78.7%

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

Alternative 11: 80.2% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

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. sub-neg 59.4%

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

   3. +-commutative 59.4%

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

   4. distribute-rgt-neg-in 59.4%

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

   5. fma-def 59.6%

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

   6. +-commutative 59.6%

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

   7. associate-+r- 59.7%

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

   8. fma-def 59.7%

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

   9. neg-sub0 59.7%

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

   10. +-commutative 59.7%

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

   11. associate-+r- 59.6%

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

   12. associate--r- 59.6%

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

   13. neg-sub0 59.6%

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

   14. +-commutative 59.6%

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

   15. sub-neg 59.6%

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

   16. fma-def 59.6%

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

3. Simplified 59.6%

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

4. Taylor expanded in ux around 0 99.0%

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

   1. fma-def 99.0%

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

   2. sub-neg 99.0%

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

   3. metadata-eval 99.0%

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

   4. *-commutative 99.0%

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

   5. unpow2 99.0%

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

   6. associate--l+ 99.0%

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

   7. distribute-rgt-in 99.1%

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

   8. *-lft-identity 99.1%

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

   9. mul-1-neg 99.1%

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

   10. sub-neg 99.1%

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

   11. metadata-eval 99.1%

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

   12. distribute-neg-in 99.1%

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

   13. metadata-eval 99.1%

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

   14. +-commutative 99.1%

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

   15. sub-neg 99.1%

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

6. Simplified 99.1%

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

7. Taylor expanded in uy around 0 78.7%

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

   1. fma-def 78.7%

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

   2. count-2 78.7%

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

   3. sub-neg 78.7%

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

   4. metadata-eval 78.7%

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

   5. unpow2 78.7%

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

9. Simplified 78.7%

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

    1. fma-udef 78.7%

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

11. Applied egg-rr 78.7%

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

12. Final simplification 78.7%

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

Alternative 12: 75.7% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = sqrt((ux * (2.0e0 - ux)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 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. sub-neg 59.4%

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

   3. +-commutative 59.4%

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

   4. distribute-rgt-neg-in 59.4%

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

   5. fma-def 59.6%

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

   6. +-commutative 59.6%

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

   7. associate-+r- 59.7%

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

   8. fma-def 59.7%

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

   9. neg-sub0 59.7%

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

   10. +-commutative 59.7%

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

   11. associate-+r- 59.6%

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

   12. associate--r- 59.6%

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

   13. neg-sub0 59.6%

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

   14. +-commutative 59.6%

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

   15. sub-neg 59.6%

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

   16. fma-def 59.6%

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

3. Simplified 59.6%

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

4. Taylor expanded in ux around 0 99.0%

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

   1. fma-def 99.0%

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

   2. sub-neg 99.0%

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

   3. metadata-eval 99.0%

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

   4. *-commutative 99.0%

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

   5. unpow2 99.0%

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

   6. associate--l+ 99.0%

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

   7. distribute-rgt-in 99.1%

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

   8. *-lft-identity 99.1%

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

   9. mul-1-neg 99.1%

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

   10. sub-neg 99.1%

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

   11. metadata-eval 99.1%

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

   12. distribute-neg-in 99.1%

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

   13. metadata-eval 99.1%

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

   14. +-commutative 99.1%

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

   15. sub-neg 99.1%

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

6. Simplified 99.1%

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

7. Taylor expanded in uy around 0 78.7%

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

   1. fma-def 78.7%

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

   2. count-2 78.7%

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

   3. sub-neg 78.7%

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

   4. metadata-eval 78.7%

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

   5. unpow2 78.7%

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

9. Simplified 78.7%

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

10. Taylor expanded in maxCos around 0 74.8%

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

    1. +-commutative 74.8%

       \[\leadsto \sqrt{\color{blue}{2 \cdot ux + -1 \cdot {ux}^{2}}} \]

    2. mul-1-neg 74.8%

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

    3. unpow2 74.8%

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

    4. distribute-lft-neg-in 74.8%

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

    5. distribute-rgt-in 74.8%

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

    6. sub-neg 74.8%

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

12. Simplified 74.8%

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

13. Final simplification 74.8%

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

Alternative 13: 62.3% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{2 \cdot ux} \end{array} \]

FPCore

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

C

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

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = sqrt((2.0e0 * ux))
end function

Julia

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

MATLAB

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

Derivation

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. sub-neg 59.4%

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

   3. +-commutative 59.4%

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

   4. distribute-rgt-neg-in 59.4%

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

   5. fma-def 59.6%

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

   6. +-commutative 59.6%

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

   7. associate-+r- 59.7%

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

   8. fma-def 59.7%

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

   9. neg-sub0 59.7%

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

   10. +-commutative 59.7%

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

   11. associate-+r- 59.6%

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

   12. associate--r- 59.6%

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

   13. neg-sub0 59.6%

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

   14. +-commutative 59.6%

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

   15. sub-neg 59.6%

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

   16. fma-def 59.6%

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

3. Simplified 59.6%

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

4. Taylor expanded in ux around 0 75.9%

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

   1. mul-1-neg 75.9%

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

   2. sub-neg 75.9%

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

   3. metadata-eval 75.9%

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

6. Simplified 75.9%

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

7. Taylor expanded in uy around 0 62.9%

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

8. Taylor expanded in maxCos around 0 60.3%

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

9. Final simplification 60.3%

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

Reproduce

herbie shell --seed 2023228 
(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)))))))