UniformSampleCone, x

Percentage Accurate: 57.1% → 99.0%

Time: 19.0s

Alternatives: 11

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: 57.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.4%

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

   1. associate-*l* 57.4%

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

   2. +-commutative 57.4%

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

   3. associate-+r- 57.4%

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

   4. fma-def 57.4%

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

   5. +-commutative 57.4%

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

   6. associate-+r- 57.2%

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

   7. fma-def 57.2%

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

3. Simplified 57.2%

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

4. Taylor expanded in ux around -inf 98.9%

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

   1. +-commutative 98.9%

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

   2. metadata-eval 98.9%

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

   3. cancel-sign-sub-inv 98.9%

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

   4. fma-def 99.0%

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

   5. cancel-sign-sub-inv 99.0%

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

   6. metadata-eval 99.0%

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

   7. +-commutative 99.0%

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

   8. *-commutative 99.0%

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

   9. fma-def 99.0%

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

   10. mul-1-neg 99.0%

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

   11. *-commutative 99.0%

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

   12. distribute-rgt-neg-in 99.0%

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

   13. mul-1-neg 99.0%

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

   14. unsub-neg 99.0%

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

   15. unpow2 99.0%

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

   16. distribute-rgt-neg-in 99.0%

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

6. Simplified 99.0%

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

7. Final simplification 99.0%

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

Alternative 2: 99.0% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.4%

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

   1. associate-*l* 57.4%

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

   2. +-commutative 57.4%

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

   3. associate-+r- 57.4%

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

   4. fma-def 57.4%

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

   5. +-commutative 57.4%

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

   6. associate-+r- 57.2%

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

   7. fma-def 57.2%

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

3. Simplified 57.2%

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

4. Taylor expanded in ux around -inf 98.9%

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

   1. +-commutative 98.9%

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

   2. metadata-eval 98.9%

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

   3. cancel-sign-sub-inv 98.9%

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

   4. fma-def 99.0%

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

   5. cancel-sign-sub-inv 99.0%

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

   6. metadata-eval 99.0%

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

   7. +-commutative 99.0%

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

   8. *-commutative 99.0%

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

   9. fma-def 99.0%

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

   10. mul-1-neg 99.0%

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

   11. *-commutative 99.0%

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

   12. distribute-rgt-neg-in 99.0%

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

   13. mul-1-neg 99.0%

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

   14. unsub-neg 99.0%

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

   15. unpow2 99.0%

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

   16. distribute-rgt-neg-in 99.0%

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

6. Simplified 99.0%

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

   1. log1p-expm1-u 99.0%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(uy \cdot \left(2 \cdot \pi\right)\right)\right)\right)} \cdot \sqrt{\mathsf{fma}\left(ux, \mathsf{fma}\left(maxCos, -2, 2\right), {\left(1 - maxCos\right)}^{2} \cdot \left(ux \cdot \left(-ux\right)\right)\right)} \]

8. Applied egg-rr 99.0%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(uy \cdot \left(2 \cdot \pi\right)\right)\right)\right)} \cdot \sqrt{\mathsf{fma}\left(ux, \mathsf{fma}\left(maxCos, -2, 2\right), {\left(1 - maxCos\right)}^{2} \cdot \left(ux \cdot \left(-ux\right)\right)\right)} \]

9. Taylor expanded in ux around 0 98.9%

   \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(uy \cdot \left(2 \cdot \pi\right)\right)\right)\right) \cdot \sqrt{\color{blue}{-1 \cdot \left({ux}^{2} \cdot {\left(1 - maxCos\right)}^{2}\right) + ux \cdot \left(2 + -2 \cdot maxCos\right)}} \]
10. Step-by-step derivation

    1. +-commutative 98.9%

       \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(uy \cdot \left(2 \cdot \pi\right)\right)\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(2 + -2 \cdot maxCos\right) + -1 \cdot \left({ux}^{2} \cdot {\left(1 - maxCos\right)}^{2}\right)}} \]

    2. mul-1-neg 98.9%

       \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(uy \cdot \left(2 \cdot \pi\right)\right)\right)\right) \cdot \sqrt{ux \cdot \left(2 + -2 \cdot maxCos\right) + \color{blue}{\left(-{ux}^{2} \cdot {\left(1 - maxCos\right)}^{2}\right)}} \]

    3. unsub-neg 98.9%

       \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(uy \cdot \left(2 \cdot \pi\right)\right)\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(2 + -2 \cdot maxCos\right) - {ux}^{2} \cdot {\left(1 - maxCos\right)}^{2}}} \]

    4. +-commutative 98.9%

       \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(uy \cdot \left(2 \cdot \pi\right)\right)\right)\right) \cdot \sqrt{ux \cdot \color{blue}{\left(-2 \cdot maxCos + 2\right)} - {ux}^{2} \cdot {\left(1 - maxCos\right)}^{2}} \]

    5. fma-def 98.9%

       \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(uy \cdot \left(2 \cdot \pi\right)\right)\right)\right) \cdot \sqrt{ux \cdot \color{blue}{\mathsf{fma}\left(-2, maxCos, 2\right)} - {ux}^{2} \cdot {\left(1 - maxCos\right)}^{2}} \]

    6. unpow2 98.9%

       \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(uy \cdot \left(2 \cdot \pi\right)\right)\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(-2, maxCos, 2\right) - \color{blue}{\left(ux \cdot ux\right)} \cdot {\left(1 - maxCos\right)}^{2}} \]

    7. associate-*l* 98.9%

       \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(uy \cdot \left(2 \cdot \pi\right)\right)\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(-2, maxCos, 2\right) - \color{blue}{ux \cdot \left(ux \cdot {\left(1 - maxCos\right)}^{2}\right)}} \]

11. Simplified 98.9%

    \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(uy \cdot \left(2 \cdot \pi\right)\right)\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \mathsf{fma}\left(-2, maxCos, 2\right) - ux \cdot \left(ux \cdot {\left(1 - maxCos\right)}^{2}\right)}} \]
12. Step-by-step derivation

    1. pow1 98.9%

       \[\leadsto \color{blue}{{\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(uy \cdot \left(2 \cdot \pi\right)\right)\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(-2, maxCos, 2\right) - ux \cdot \left(ux \cdot {\left(1 - maxCos\right)}^{2}\right)}\right)}^{1}} \]

    2. log1p-expm1-u 98.9%

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

    3. distribute-lft-out-- 99.0%

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

13. Applied egg-rr 99.0%

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

    1. unpow1 99.0%

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

15. Simplified 99.0%

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

16. Final simplification 99.0%

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

Alternative 3: 96.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.00022000000171829015:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(ux, \mathsf{fma}\left(maxCos, -2, 2\right), {\left(1 - maxCos\right)}^{2} \cdot \left(ux \cdot \left(-ux\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux - ux \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* uy 2.0) 0.00022000000171829015)
   (sqrt
    (fma ux (fma maxCos -2.0 2.0) (* (pow (- 1.0 maxCos) 2.0) (* ux (- ux)))))
   (* (cos (* uy (* 2.0 PI))) (sqrt (- (* 2.0 ux) (* ux ux))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if ((uy * 2.0f) <= 0.00022000000171829015f) {
		tmp = sqrtf(fmaf(ux, fmaf(maxCos, -2.0f, 2.0f), (powf((1.0f - maxCos), 2.0f) * (ux * -ux))));
	} else {
		tmp = cosf((uy * (2.0f * ((float) M_PI)))) * sqrtf(((2.0f * ux) - (ux * ux)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (Float32(uy * Float32(2.0)) <= Float32(0.00022000000171829015))
		tmp = sqrt(fma(ux, fma(maxCos, Float32(-2.0), Float32(2.0)), Float32((Float32(Float32(1.0) - maxCos) ^ Float32(2.0)) * Float32(ux * Float32(-ux)))));
	else
		tmp = Float32(cos(Float32(uy * Float32(Float32(2.0) * Float32(pi)))) * sqrt(Float32(Float32(Float32(2.0) * ux) - Float32(ux * ux))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy 2) < 2.20000002e-4

   1. Initial program 57.6%

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

      1. associate-*l* 57.6%

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

      2. +-commutative 57.6%

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

      3. associate-+r- 57.7%

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

      4. fma-def 57.7%

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

      5. +-commutative 57.7%

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

      6. associate-+r- 57.5%

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

      7. fma-def 57.5%

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

   3. Simplified 57.5%

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

   4. Taylor expanded in ux around -inf 99.4%

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

      1. +-commutative 99.4%

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

      2. metadata-eval 99.4%

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

      3. cancel-sign-sub-inv 99.4%

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

      4. fma-def 99.5%

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

      5. cancel-sign-sub-inv 99.5%

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

      6. metadata-eval 99.5%

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

      7. +-commutative 99.5%

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

      8. *-commutative 99.5%

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

      9. fma-def 99.5%

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

      10. mul-1-neg 99.5%

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

      11. *-commutative 99.5%

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

      12. distribute-rgt-neg-in 99.5%

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

      13. mul-1-neg 99.5%

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

      14. unsub-neg 99.5%

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

      15. unpow2 99.5%

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

      16. distribute-rgt-neg-in 99.5%

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

   6. Simplified 99.5%

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

   7. Taylor expanded in uy around 0 99.3%

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

      1. +-commutative 99.3%

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

      2. +-commutative 99.3%

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

      3. *-commutative 99.3%

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

      4. fma-udef 99.3%

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

      5. mul-1-neg 99.3%

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

      6. *-commutative 99.3%

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

      7. distribute-rgt-neg-in 99.3%

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

      8. unpow2 99.3%

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

      9. distribute-rgt-neg-out 99.3%

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

      10. fma-udef 99.4%

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

   9. Simplified 99.4%

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

   if 2.20000002e-4 < (*.f32 uy 2)

   1. Initial program 57.0%

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

      1. associate-*l* 57.0%

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

      2. +-commutative 57.0%

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

      3. associate-+r- 57.0%

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

      4. fma-def 57.0%

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

      5. +-commutative 57.0%

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

      6. associate-+r- 56.8%

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

      7. fma-def 56.8%

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

   3. Simplified 56.8%

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

   4. Taylor expanded in ux around -inf 98.3%

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

      1. +-commutative 98.3%

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

      2. metadata-eval 98.3%

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

      3. cancel-sign-sub-inv 98.3%

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

      4. fma-def 98.4%

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

      5. cancel-sign-sub-inv 98.4%

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

      6. metadata-eval 98.4%

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

      7. +-commutative 98.4%

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

      8. *-commutative 98.4%

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

      9. fma-def 98.4%

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

      10. mul-1-neg 98.4%

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

      11. *-commutative 98.4%

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

      12. distribute-rgt-neg-in 98.4%

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

      13. mul-1-neg 98.4%

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

      14. unsub-neg 98.4%

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

      15. unpow2 98.4%

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

      16. distribute-rgt-neg-in 98.4%

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

   6. Simplified 98.4%

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

      1. log1p-expm1-u 98.3%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(uy \cdot \left(2 \cdot \pi\right)\right)\right)\right)} \cdot \sqrt{\mathsf{fma}\left(ux, \mathsf{fma}\left(maxCos, -2, 2\right), {\left(1 - maxCos\right)}^{2} \cdot \left(ux \cdot \left(-ux\right)\right)\right)} \]

   8. Applied egg-rr 98.3%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(uy \cdot \left(2 \cdot \pi\right)\right)\right)\right)} \cdot \sqrt{\mathsf{fma}\left(ux, \mathsf{fma}\left(maxCos, -2, 2\right), {\left(1 - maxCos\right)}^{2} \cdot \left(ux \cdot \left(-ux\right)\right)\right)} \]

   9. Taylor expanded in ux around 0 98.3%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(uy \cdot \left(2 \cdot \pi\right)\right)\right)\right) \cdot \sqrt{\color{blue}{-1 \cdot \left({ux}^{2} \cdot {\left(1 - maxCos\right)}^{2}\right) + ux \cdot \left(2 + -2 \cdot maxCos\right)}} \]
   10. Step-by-step derivation

       1. +-commutative 98.3%

          \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(uy \cdot \left(2 \cdot \pi\right)\right)\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(2 + -2 \cdot maxCos\right) + -1 \cdot \left({ux}^{2} \cdot {\left(1 - maxCos\right)}^{2}\right)}} \]

       2. mul-1-neg 98.3%

          \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(uy \cdot \left(2 \cdot \pi\right)\right)\right)\right) \cdot \sqrt{ux \cdot \left(2 + -2 \cdot maxCos\right) + \color{blue}{\left(-{ux}^{2} \cdot {\left(1 - maxCos\right)}^{2}\right)}} \]

       3. unsub-neg 98.3%

          \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(uy \cdot \left(2 \cdot \pi\right)\right)\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(2 + -2 \cdot maxCos\right) - {ux}^{2} \cdot {\left(1 - maxCos\right)}^{2}}} \]

       4. +-commutative 98.3%

          \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(uy \cdot \left(2 \cdot \pi\right)\right)\right)\right) \cdot \sqrt{ux \cdot \color{blue}{\left(-2 \cdot maxCos + 2\right)} - {ux}^{2} \cdot {\left(1 - maxCos\right)}^{2}} \]

       5. fma-def 98.3%

          \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(uy \cdot \left(2 \cdot \pi\right)\right)\right)\right) \cdot \sqrt{ux \cdot \color{blue}{\mathsf{fma}\left(-2, maxCos, 2\right)} - {ux}^{2} \cdot {\left(1 - maxCos\right)}^{2}} \]

       6. unpow2 98.3%

          \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(uy \cdot \left(2 \cdot \pi\right)\right)\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(-2, maxCos, 2\right) - \color{blue}{\left(ux \cdot ux\right)} \cdot {\left(1 - maxCos\right)}^{2}} \]

       7. associate-*l* 98.3%

          \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(uy \cdot \left(2 \cdot \pi\right)\right)\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(-2, maxCos, 2\right) - \color{blue}{ux \cdot \left(ux \cdot {\left(1 - maxCos\right)}^{2}\right)}} \]

   11. Simplified 98.3%

       \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(uy \cdot \left(2 \cdot \pi\right)\right)\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \mathsf{fma}\left(-2, maxCos, 2\right) - ux \cdot \left(ux \cdot {\left(1 - maxCos\right)}^{2}\right)}} \]

   12. Taylor expanded in maxCos around 0 92.5%

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

       1. associate-*r* 92.5%

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

       2. *-commutative 92.5%

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

       3. associate-*r* 92.5%

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

       4. *-commutative 92.5%

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

       5. unpow2 92.5%

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

   14. Simplified 92.5%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.00022000000171829015:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(ux, \mathsf{fma}\left(maxCos, -2, 2\right), {\left(1 - maxCos\right)}^{2} \cdot \left(ux \cdot \left(-ux\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux - ux \cdot ux}\\ \end{array} \]

Alternative 4: 96.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.00022000000171829015:\\ \;\;\;\;\sqrt{ux \cdot \mathsf{fma}\left(maxCos, -2, 2\right) - \left(ux \cdot ux\right) \cdot {\left(maxCos + -1\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux - ux \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* uy 2.0) 0.00022000000171829015)
   (sqrt
    (- (* ux (fma maxCos -2.0 2.0)) (* (* ux ux) (pow (+ maxCos -1.0) 2.0))))
   (* (cos (* uy (* 2.0 PI))) (sqrt (- (* 2.0 ux) (* ux ux))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if ((uy * 2.0f) <= 0.00022000000171829015f) {
		tmp = sqrtf(((ux * fmaf(maxCos, -2.0f, 2.0f)) - ((ux * ux) * powf((maxCos + -1.0f), 2.0f))));
	} else {
		tmp = cosf((uy * (2.0f * ((float) M_PI)))) * sqrtf(((2.0f * ux) - (ux * ux)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (Float32(uy * Float32(2.0)) <= Float32(0.00022000000171829015))
		tmp = sqrt(Float32(Float32(ux * fma(maxCos, Float32(-2.0), Float32(2.0))) - Float32(Float32(ux * ux) * (Float32(maxCos + Float32(-1.0)) ^ Float32(2.0)))));
	else
		tmp = Float32(cos(Float32(uy * Float32(Float32(2.0) * Float32(pi)))) * sqrt(Float32(Float32(Float32(2.0) * ux) - Float32(ux * ux))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy 2) < 2.20000002e-4

   1. Initial program 57.6%

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

      1. associate-*l* 57.6%

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

      2. +-commutative 57.6%

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

      3. associate-+r- 57.7%

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

      4. fma-def 57.7%

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

      5. +-commutative 57.7%

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

      6. associate-+r- 57.5%

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

      7. fma-def 57.5%

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

   3. Simplified 57.5%

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

   4. Taylor expanded in uy around 0 57.5%

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

   5. Taylor expanded in ux around 0 99.3%

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

      1. +-commutative 99.3%

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

      2. cancel-sign-sub-inv 99.3%

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

      3. metadata-eval 99.3%

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

      4. mul-1-neg 99.3%

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

      5. unsub-neg 99.3%

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

      6. +-commutative 99.3%

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

      7. *-commutative 99.3%

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

      8. fma-udef 99.3%

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

      9. unpow2 99.3%

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

   7. Simplified 99.3%

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

   if 2.20000002e-4 < (*.f32 uy 2)

   1. Initial program 57.0%

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

      1. associate-*l* 57.0%

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

      2. +-commutative 57.0%

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

      3. associate-+r- 57.0%

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

      4. fma-def 57.0%

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

      5. +-commutative 57.0%

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

      6. associate-+r- 56.8%

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

      7. fma-def 56.8%

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

   3. Simplified 56.8%

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

   4. Taylor expanded in ux around -inf 98.3%

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

      1. +-commutative 98.3%

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

      2. metadata-eval 98.3%

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

      3. cancel-sign-sub-inv 98.3%

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

      4. fma-def 98.4%

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

      5. cancel-sign-sub-inv 98.4%

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

      6. metadata-eval 98.4%

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

      7. +-commutative 98.4%

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

      8. *-commutative 98.4%

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

      9. fma-def 98.4%

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

      10. mul-1-neg 98.4%

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

      11. *-commutative 98.4%

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

      12. distribute-rgt-neg-in 98.4%

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

      13. mul-1-neg 98.4%

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

      14. unsub-neg 98.4%

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

      15. unpow2 98.4%

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

      16. distribute-rgt-neg-in 98.4%

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

   6. Simplified 98.4%

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

      1. log1p-expm1-u 98.3%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(uy \cdot \left(2 \cdot \pi\right)\right)\right)\right)} \cdot \sqrt{\mathsf{fma}\left(ux, \mathsf{fma}\left(maxCos, -2, 2\right), {\left(1 - maxCos\right)}^{2} \cdot \left(ux \cdot \left(-ux\right)\right)\right)} \]

   8. Applied egg-rr 98.3%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(uy \cdot \left(2 \cdot \pi\right)\right)\right)\right)} \cdot \sqrt{\mathsf{fma}\left(ux, \mathsf{fma}\left(maxCos, -2, 2\right), {\left(1 - maxCos\right)}^{2} \cdot \left(ux \cdot \left(-ux\right)\right)\right)} \]

   9. Taylor expanded in ux around 0 98.3%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(uy \cdot \left(2 \cdot \pi\right)\right)\right)\right) \cdot \sqrt{\color{blue}{-1 \cdot \left({ux}^{2} \cdot {\left(1 - maxCos\right)}^{2}\right) + ux \cdot \left(2 + -2 \cdot maxCos\right)}} \]
   10. Step-by-step derivation

       1. +-commutative 98.3%

          \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(uy \cdot \left(2 \cdot \pi\right)\right)\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(2 + -2 \cdot maxCos\right) + -1 \cdot \left({ux}^{2} \cdot {\left(1 - maxCos\right)}^{2}\right)}} \]

       2. mul-1-neg 98.3%

          \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(uy \cdot \left(2 \cdot \pi\right)\right)\right)\right) \cdot \sqrt{ux \cdot \left(2 + -2 \cdot maxCos\right) + \color{blue}{\left(-{ux}^{2} \cdot {\left(1 - maxCos\right)}^{2}\right)}} \]

       3. unsub-neg 98.3%

          \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(uy \cdot \left(2 \cdot \pi\right)\right)\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(2 + -2 \cdot maxCos\right) - {ux}^{2} \cdot {\left(1 - maxCos\right)}^{2}}} \]

       4. +-commutative 98.3%

          \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(uy \cdot \left(2 \cdot \pi\right)\right)\right)\right) \cdot \sqrt{ux \cdot \color{blue}{\left(-2 \cdot maxCos + 2\right)} - {ux}^{2} \cdot {\left(1 - maxCos\right)}^{2}} \]

       5. fma-def 98.3%

          \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(uy \cdot \left(2 \cdot \pi\right)\right)\right)\right) \cdot \sqrt{ux \cdot \color{blue}{\mathsf{fma}\left(-2, maxCos, 2\right)} - {ux}^{2} \cdot {\left(1 - maxCos\right)}^{2}} \]

       6. unpow2 98.3%

          \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(uy \cdot \left(2 \cdot \pi\right)\right)\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(-2, maxCos, 2\right) - \color{blue}{\left(ux \cdot ux\right)} \cdot {\left(1 - maxCos\right)}^{2}} \]

       7. associate-*l* 98.3%

          \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(uy \cdot \left(2 \cdot \pi\right)\right)\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(-2, maxCos, 2\right) - \color{blue}{ux \cdot \left(ux \cdot {\left(1 - maxCos\right)}^{2}\right)}} \]

   11. Simplified 98.3%

       \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(uy \cdot \left(2 \cdot \pi\right)\right)\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \mathsf{fma}\left(-2, maxCos, 2\right) - ux \cdot \left(ux \cdot {\left(1 - maxCos\right)}^{2}\right)}} \]

   12. Taylor expanded in maxCos around 0 92.5%

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

       1. associate-*r* 92.5%

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

       2. *-commutative 92.5%

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

       3. associate-*r* 92.5%

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

       4. *-commutative 92.5%

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

       5. unpow2 92.5%

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

   14. Simplified 92.5%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.00022000000171829015:\\ \;\;\;\;\sqrt{ux \cdot \mathsf{fma}\left(maxCos, -2, 2\right) - \left(ux \cdot ux\right) \cdot {\left(maxCos + -1\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux - ux \cdot ux}\\ \end{array} \]

Alternative 5: 95.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.00022000000171829015:\\ \;\;\;\;\sqrt{ux \cdot \left(2 - ux\right) + maxCos \cdot \left(-2 \cdot \left(ux - ux \cdot ux\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux - ux \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* uy 2.0) 0.00022000000171829015)
   (sqrt (+ (* ux (- 2.0 ux)) (* maxCos (* -2.0 (- ux (* ux ux))))))
   (* (cos (* uy (* 2.0 PI))) (sqrt (- (* 2.0 ux) (* ux ux))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if ((uy * 2.0f) <= 0.00022000000171829015f) {
		tmp = sqrtf(((ux * (2.0f - ux)) + (maxCos * (-2.0f * (ux - (ux * ux))))));
	} else {
		tmp = cosf((uy * (2.0f * ((float) M_PI)))) * sqrtf(((2.0f * ux) - (ux * ux)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (Float32(uy * Float32(2.0)) <= Float32(0.00022000000171829015))
		tmp = sqrt(Float32(Float32(ux * Float32(Float32(2.0) - ux)) + Float32(maxCos * Float32(Float32(-2.0) * Float32(ux - Float32(ux * ux))))));
	else
		tmp = Float32(cos(Float32(uy * Float32(Float32(2.0) * Float32(pi)))) * sqrt(Float32(Float32(Float32(2.0) * ux) - Float32(ux * ux))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if ((uy * single(2.0)) <= single(0.00022000000171829015))
		tmp = sqrt(((ux * (single(2.0) - ux)) + (maxCos * (single(-2.0) * (ux - (ux * ux))))));
	else
		tmp = cos((uy * (single(2.0) * single(pi)))) * sqrt(((single(2.0) * ux) - (ux * ux)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy 2) < 2.20000002e-4

   1. Initial program 57.6%

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

      1. associate-*l* 57.6%

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

      2. +-commutative 57.6%

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

      3. associate-+r- 57.7%

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

      4. fma-def 57.7%

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

      5. +-commutative 57.7%

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

      6. associate-+r- 57.5%

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

      7. fma-def 57.5%

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

   3. Simplified 57.5%

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

   4. Taylor expanded in ux around -inf 99.4%

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

      1. +-commutative 99.4%

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

      2. metadata-eval 99.4%

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

      3. cancel-sign-sub-inv 99.4%

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

      4. fma-def 99.5%

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

      5. cancel-sign-sub-inv 99.5%

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

      6. metadata-eval 99.5%

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

      7. +-commutative 99.5%

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

      8. *-commutative 99.5%

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

      9. fma-def 99.5%

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

      10. mul-1-neg 99.5%

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

      11. *-commutative 99.5%

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

      12. distribute-rgt-neg-in 99.5%

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

      13. mul-1-neg 99.5%

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

      14. unsub-neg 99.5%

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

      15. unpow2 99.5%

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

      16. distribute-rgt-neg-in 99.5%

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

   6. Simplified 99.5%

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

   7. Taylor expanded in uy around 0 99.3%

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

      1. +-commutative 99.3%

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

      2. +-commutative 99.3%

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

      3. *-commutative 99.3%

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

      4. fma-udef 99.3%

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

      5. mul-1-neg 99.3%

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

      6. *-commutative 99.3%

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

      7. distribute-rgt-neg-in 99.3%

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

      8. unpow2 99.3%

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

      9. distribute-rgt-neg-out 99.3%

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

      10. fma-udef 99.4%

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

   9. Simplified 99.4%

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

   10. Taylor expanded in maxCos around 0 99.0%

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

       1. associate-+r+ 99.0%

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

       2. +-commutative 99.0%

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

       3. *-commutative 99.0%

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

       4. mul-1-neg 99.0%

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

       5. unpow2 99.0%

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

       6. distribute-rgt-neg-out 99.0%

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

       7. distribute-lft-in 99.1%

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

       8. sub-neg 99.1%

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

       9. fma-def 99.1%

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

       10. unpow2 99.1%

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

   12. Simplified 99.1%

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

   13. Taylor expanded in maxCos around 0 99.1%

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

       1. metadata-eval 99.1%

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

       2. cancel-sign-sub-inv 99.1%

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

       3. distribute-lft-out-- 99.1%

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

       4. unpow2 99.1%

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

   15. Simplified 99.1%

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

   if 2.20000002e-4 < (*.f32 uy 2)

   1. Initial program 57.0%

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

      1. associate-*l* 57.0%

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

      2. +-commutative 57.0%

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

      3. associate-+r- 57.0%

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

      4. fma-def 57.0%

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

      5. +-commutative 57.0%

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

      6. associate-+r- 56.8%

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

      7. fma-def 56.8%

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

   3. Simplified 56.8%

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

   4. Taylor expanded in ux around -inf 98.3%

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

      1. +-commutative 98.3%

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

      2. metadata-eval 98.3%

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

      3. cancel-sign-sub-inv 98.3%

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

      4. fma-def 98.4%

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

      5. cancel-sign-sub-inv 98.4%

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

      6. metadata-eval 98.4%

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

      7. +-commutative 98.4%

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

      8. *-commutative 98.4%

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

      9. fma-def 98.4%

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

      10. mul-1-neg 98.4%

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

      11. *-commutative 98.4%

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

      12. distribute-rgt-neg-in 98.4%

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

      13. mul-1-neg 98.4%

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

      14. unsub-neg 98.4%

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

      15. unpow2 98.4%

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

      16. distribute-rgt-neg-in 98.4%

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

   6. Simplified 98.4%

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

      1. log1p-expm1-u 98.3%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(uy \cdot \left(2 \cdot \pi\right)\right)\right)\right)} \cdot \sqrt{\mathsf{fma}\left(ux, \mathsf{fma}\left(maxCos, -2, 2\right), {\left(1 - maxCos\right)}^{2} \cdot \left(ux \cdot \left(-ux\right)\right)\right)} \]

   8. Applied egg-rr 98.3%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(uy \cdot \left(2 \cdot \pi\right)\right)\right)\right)} \cdot \sqrt{\mathsf{fma}\left(ux, \mathsf{fma}\left(maxCos, -2, 2\right), {\left(1 - maxCos\right)}^{2} \cdot \left(ux \cdot \left(-ux\right)\right)\right)} \]

   9. Taylor expanded in ux around 0 98.3%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(uy \cdot \left(2 \cdot \pi\right)\right)\right)\right) \cdot \sqrt{\color{blue}{-1 \cdot \left({ux}^{2} \cdot {\left(1 - maxCos\right)}^{2}\right) + ux \cdot \left(2 + -2 \cdot maxCos\right)}} \]
   10. Step-by-step derivation

       1. +-commutative 98.3%

          \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(uy \cdot \left(2 \cdot \pi\right)\right)\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(2 + -2 \cdot maxCos\right) + -1 \cdot \left({ux}^{2} \cdot {\left(1 - maxCos\right)}^{2}\right)}} \]

       2. mul-1-neg 98.3%

          \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(uy \cdot \left(2 \cdot \pi\right)\right)\right)\right) \cdot \sqrt{ux \cdot \left(2 + -2 \cdot maxCos\right) + \color{blue}{\left(-{ux}^{2} \cdot {\left(1 - maxCos\right)}^{2}\right)}} \]

       3. unsub-neg 98.3%

          \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(uy \cdot \left(2 \cdot \pi\right)\right)\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(2 + -2 \cdot maxCos\right) - {ux}^{2} \cdot {\left(1 - maxCos\right)}^{2}}} \]

       4. +-commutative 98.3%

          \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(uy \cdot \left(2 \cdot \pi\right)\right)\right)\right) \cdot \sqrt{ux \cdot \color{blue}{\left(-2 \cdot maxCos + 2\right)} - {ux}^{2} \cdot {\left(1 - maxCos\right)}^{2}} \]

       5. fma-def 98.3%

          \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(uy \cdot \left(2 \cdot \pi\right)\right)\right)\right) \cdot \sqrt{ux \cdot \color{blue}{\mathsf{fma}\left(-2, maxCos, 2\right)} - {ux}^{2} \cdot {\left(1 - maxCos\right)}^{2}} \]

       6. unpow2 98.3%

          \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(uy \cdot \left(2 \cdot \pi\right)\right)\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(-2, maxCos, 2\right) - \color{blue}{\left(ux \cdot ux\right)} \cdot {\left(1 - maxCos\right)}^{2}} \]

       7. associate-*l* 98.3%

          \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(uy \cdot \left(2 \cdot \pi\right)\right)\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(-2, maxCos, 2\right) - \color{blue}{ux \cdot \left(ux \cdot {\left(1 - maxCos\right)}^{2}\right)}} \]

   11. Simplified 98.3%

       \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(uy \cdot \left(2 \cdot \pi\right)\right)\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \mathsf{fma}\left(-2, maxCos, 2\right) - ux \cdot \left(ux \cdot {\left(1 - maxCos\right)}^{2}\right)}} \]

   12. Taylor expanded in maxCos around 0 92.5%

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

       1. associate-*r* 92.5%

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

       2. *-commutative 92.5%

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

       3. associate-*r* 92.5%

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

       4. *-commutative 92.5%

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

       5. unpow2 92.5%

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

   14. Simplified 92.5%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.00022000000171829015:\\ \;\;\;\;\sqrt{ux \cdot \left(2 - ux\right) + maxCos \cdot \left(-2 \cdot \left(ux - ux \cdot ux\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux - ux \cdot ux}\\ \end{array} \]

Alternative 6: 95.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := ux \cdot \left(2 - ux\right)\\ \mathbf{if}\;uy \cdot 2 \leq 0.00022000000171829015:\\ \;\;\;\;\sqrt{t_0 + maxCos \cdot \left(-2 \cdot \left(ux - ux \cdot ux\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* ux (- 2.0 ux))))
   (if (<= (* uy 2.0) 0.00022000000171829015)
     (sqrt (+ t_0 (* maxCos (* -2.0 (- ux (* ux ux))))))
     (* (cos (* uy (* 2.0 PI))) (sqrt t_0)))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = ux * (2.0f - ux);
	float tmp;
	if ((uy * 2.0f) <= 0.00022000000171829015f) {
		tmp = sqrtf((t_0 + (maxCos * (-2.0f * (ux - (ux * ux))))));
	} else {
		tmp = cosf((uy * (2.0f * ((float) M_PI)))) * sqrtf(t_0);
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(ux * Float32(Float32(2.0) - ux))
	tmp = Float32(0.0)
	if (Float32(uy * Float32(2.0)) <= Float32(0.00022000000171829015))
		tmp = sqrt(Float32(t_0 + Float32(maxCos * Float32(Float32(-2.0) * Float32(ux - Float32(ux * ux))))));
	else
		tmp = Float32(cos(Float32(uy * Float32(Float32(2.0) * Float32(pi)))) * sqrt(t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	t_0 = ux * (single(2.0) - ux);
	tmp = single(0.0);
	if ((uy * single(2.0)) <= single(0.00022000000171829015))
		tmp = sqrt((t_0 + (maxCos * (single(-2.0) * (ux - (ux * ux))))));
	else
		tmp = cos((uy * (single(2.0) * single(pi)))) * sqrt(t_0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy 2) < 2.20000002e-4

   1. Initial program 57.6%

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

      1. associate-*l* 57.6%

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

      2. +-commutative 57.6%

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

      3. associate-+r- 57.7%

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

      4. fma-def 57.7%

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

      5. +-commutative 57.7%

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

      6. associate-+r- 57.5%

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

      7. fma-def 57.5%

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

   3. Simplified 57.5%

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

   4. Taylor expanded in ux around -inf 99.4%

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

      1. +-commutative 99.4%

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

      2. metadata-eval 99.4%

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

      3. cancel-sign-sub-inv 99.4%

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

      4. fma-def 99.5%

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

      5. cancel-sign-sub-inv 99.5%

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

      6. metadata-eval 99.5%

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

      7. +-commutative 99.5%

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

      8. *-commutative 99.5%

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

      9. fma-def 99.5%

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

      10. mul-1-neg 99.5%

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

      11. *-commutative 99.5%

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

      12. distribute-rgt-neg-in 99.5%

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

      13. mul-1-neg 99.5%

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

      14. unsub-neg 99.5%

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

      15. unpow2 99.5%

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

      16. distribute-rgt-neg-in 99.5%

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

   6. Simplified 99.5%

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

   7. Taylor expanded in uy around 0 99.3%

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

      1. +-commutative 99.3%

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

      2. +-commutative 99.3%

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

      3. *-commutative 99.3%

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

      4. fma-udef 99.3%

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

      5. mul-1-neg 99.3%

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

      6. *-commutative 99.3%

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

      7. distribute-rgt-neg-in 99.3%

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

      8. unpow2 99.3%

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

      9. distribute-rgt-neg-out 99.3%

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

      10. fma-udef 99.4%

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

   9. Simplified 99.4%

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

   10. Taylor expanded in maxCos around 0 99.0%

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

       1. associate-+r+ 99.0%

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

       2. +-commutative 99.0%

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

       3. *-commutative 99.0%

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

       4. mul-1-neg 99.0%

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

       5. unpow2 99.0%

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

       6. distribute-rgt-neg-out 99.0%

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

       7. distribute-lft-in 99.1%

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

       8. sub-neg 99.1%

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

       9. fma-def 99.1%

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

       10. unpow2 99.1%

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

   12. Simplified 99.1%

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

   13. Taylor expanded in maxCos around 0 99.1%

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

       1. metadata-eval 99.1%

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

       2. cancel-sign-sub-inv 99.1%

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

       3. distribute-lft-out-- 99.1%

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

       4. unpow2 99.1%

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

   15. Simplified 99.1%

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

   if 2.20000002e-4 < (*.f32 uy 2)

   1. Initial program 57.0%

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

      1. associate-*l* 57.0%

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

      2. +-commutative 57.0%

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

      3. associate-+r- 57.0%

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

      4. fma-def 57.0%

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

      5. +-commutative 57.0%

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

      6. associate-+r- 56.8%

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

      7. fma-def 56.8%

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

   3. Simplified 56.8%

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

   4. Taylor expanded in ux around -inf 98.3%

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

      1. +-commutative 98.3%

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

      2. metadata-eval 98.3%

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

      3. cancel-sign-sub-inv 98.3%

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

      4. fma-def 98.4%

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

      5. cancel-sign-sub-inv 98.4%

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

      6. metadata-eval 98.4%

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

      7. +-commutative 98.4%

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

      8. *-commutative 98.4%

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

      9. fma-def 98.4%

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

      10. mul-1-neg 98.4%

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

      11. *-commutative 98.4%

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

      12. distribute-rgt-neg-in 98.4%

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

      13. mul-1-neg 98.4%

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

      14. unsub-neg 98.4%

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

      15. unpow2 98.4%

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

      16. distribute-rgt-neg-in 98.4%

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

   6. Simplified 98.4%

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

      1. log1p-expm1-u 98.3%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(uy \cdot \left(2 \cdot \pi\right)\right)\right)\right)} \cdot \sqrt{\mathsf{fma}\left(ux, \mathsf{fma}\left(maxCos, -2, 2\right), {\left(1 - maxCos\right)}^{2} \cdot \left(ux \cdot \left(-ux\right)\right)\right)} \]

   8. Applied egg-rr 98.3%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(uy \cdot \left(2 \cdot \pi\right)\right)\right)\right)} \cdot \sqrt{\mathsf{fma}\left(ux, \mathsf{fma}\left(maxCos, -2, 2\right), {\left(1 - maxCos\right)}^{2} \cdot \left(ux \cdot \left(-ux\right)\right)\right)} \]

   9. Taylor expanded in maxCos around 0 92.5%

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

       1. *-commutative 92.5%

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

       2. associate-*r* 92.5%

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

       3. *-commutative 92.5%

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

       4. +-commutative 92.5%

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

       5. *-commutative 92.5%

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

       6. mul-1-neg 92.5%

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

       7. unpow2 92.5%

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

       8. distribute-rgt-neg-out 92.5%

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

       9. distribute-lft-in 92.5%

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

       10. sub-neg 92.5%

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

   11. Simplified 92.5%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.00022000000171829015:\\ \;\;\;\;\sqrt{ux \cdot \left(2 - ux\right) + maxCos \cdot \left(-2 \cdot \left(ux - ux \cdot ux\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \end{array} \]

Alternative 7: 88.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.0002300000051036477:\\ \;\;\;\;\sqrt{ux \cdot \left(2 - ux\right) + maxCos \cdot \left(-2 \cdot \left(ux - ux \cdot ux\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= uy 0.0002300000051036477)
   (sqrt (+ (* ux (- 2.0 ux)) (* maxCos (* -2.0 (- ux (* ux ux))))))
   (* (cos (* uy (* 2.0 PI))) (sqrt (* 2.0 ux)))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.0002300000051036477f) {
		tmp = sqrtf(((ux * (2.0f - ux)) + (maxCos * (-2.0f * (ux - (ux * ux))))));
	} 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(0.0002300000051036477))
		tmp = sqrt(Float32(Float32(ux * Float32(Float32(2.0) - ux)) + Float32(maxCos * Float32(Float32(-2.0) * Float32(ux - Float32(ux * ux))))));
	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(0.0002300000051036477))
		tmp = sqrt(((ux * (single(2.0) - ux)) + (maxCos * (single(-2.0) * (ux - (ux * ux))))));
	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 < 2.30000005e-4

   1. Initial program 58.6%

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

      1. associate-*l* 58.6%

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

      2. +-commutative 58.6%

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

      3. associate-+r- 58.7%

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

      4. fma-def 58.7%

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

      5. +-commutative 58.7%

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

      6. associate-+r- 58.6%

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

      7. fma-def 58.6%

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

   3. Simplified 58.6%

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

   4. Taylor expanded in ux around -inf 99.3%

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

      1. +-commutative 99.3%

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

      2. metadata-eval 99.3%

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

      3. cancel-sign-sub-inv 99.3%

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

      4. fma-def 99.4%

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

      5. cancel-sign-sub-inv 99.4%

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

      6. metadata-eval 99.4%

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

      7. +-commutative 99.4%

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

      8. *-commutative 99.4%

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

      9. fma-def 99.4%

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

      10. mul-1-neg 99.4%

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

      11. *-commutative 99.4%

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

      12. distribute-rgt-neg-in 99.4%

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

      13. mul-1-neg 99.4%

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

      14. unsub-neg 99.4%

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

      15. unpow2 99.4%

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

      16. distribute-rgt-neg-in 99.4%

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

   6. Simplified 99.4%

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

   7. Taylor expanded in uy around 0 98.8%

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

      1. +-commutative 98.8%

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

      2. +-commutative 98.8%

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

      3. *-commutative 98.8%

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

      4. fma-udef 98.8%

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

      5. mul-1-neg 98.8%

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

      6. *-commutative 98.8%

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

      7. distribute-rgt-neg-in 98.8%

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

      8. unpow2 98.8%

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

      9. distribute-rgt-neg-out 98.8%

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

      10. fma-udef 98.9%

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

   9. Simplified 98.9%

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

   10. Taylor expanded in maxCos around 0 98.5%

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

       1. associate-+r+ 98.5%

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

       2. +-commutative 98.5%

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

       3. *-commutative 98.5%

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

       4. mul-1-neg 98.5%

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

       5. unpow2 98.5%

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

       6. distribute-rgt-neg-out 98.5%

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

       7. distribute-lft-in 98.6%

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

       8. sub-neg 98.6%

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

       9. fma-def 98.6%

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

       10. unpow2 98.6%

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

   12. Simplified 98.6%

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

   13. Taylor expanded in maxCos around 0 98.6%

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

       1. metadata-eval 98.6%

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

       2. cancel-sign-sub-inv 98.6%

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

       3. distribute-lft-out-- 98.6%

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

       4. unpow2 98.6%

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

   15. Simplified 98.6%

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

   if 2.30000005e-4 < uy

   1. Initial program 55.4%

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

      1. associate-*l* 55.4%

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

      2. +-commutative 55.4%

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

      3. associate-+r- 55.3%

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

      4. fma-def 55.3%

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

      5. +-commutative 55.3%

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

      6. associate-+r- 55.2%

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

      7. fma-def 55.2%

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

   3. Simplified 55.2%

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

   4. Taylor expanded in ux around 0 46.1%

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

   5. Taylor expanded in maxCos around 0 75.7%

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

      1. *-commutative 75.7%

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

   7. Simplified 75.7%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \leq 0.0002300000051036477:\\ \;\;\;\;\sqrt{ux \cdot \left(2 - ux\right) + maxCos \cdot \left(-2 \cdot \left(ux - ux \cdot ux\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux}\\ \end{array} \]

Alternative 8: 79.5% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.4%

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

   1. associate-*l* 57.4%

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

   2. +-commutative 57.4%

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

   3. associate-+r- 57.4%

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

   4. fma-def 57.4%

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

   5. +-commutative 57.4%

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

   6. associate-+r- 57.2%

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

   7. fma-def 57.2%

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

3. Simplified 57.2%

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

4. Taylor expanded in ux around -inf 98.9%

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

   1. +-commutative 98.9%

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

   2. metadata-eval 98.9%

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

   3. cancel-sign-sub-inv 98.9%

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

   4. fma-def 99.0%

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

   5. cancel-sign-sub-inv 99.0%

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

   6. metadata-eval 99.0%

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

   7. +-commutative 99.0%

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

   8. *-commutative 99.0%

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

   9. fma-def 99.0%

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

   10. mul-1-neg 99.0%

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

   11. *-commutative 99.0%

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

   12. distribute-rgt-neg-in 99.0%

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

   13. mul-1-neg 99.0%

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

   14. unsub-neg 99.0%

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

   15. unpow2 99.0%

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

   16. distribute-rgt-neg-in 99.0%

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

6. Simplified 99.0%

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

7. Taylor expanded in uy around 0 78.5%

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

   1. +-commutative 78.5%

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

   2. +-commutative 78.5%

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

   3. *-commutative 78.5%

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

   4. fma-udef 78.5%

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

   5. mul-1-neg 78.5%

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

   6. *-commutative 78.5%

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

   7. distribute-rgt-neg-in 78.5%

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

   8. unpow2 78.5%

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

   9. distribute-rgt-neg-out 78.5%

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

   10. fma-udef 78.6%

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

9. Simplified 78.6%

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

10. Taylor expanded in maxCos around 0 78.3%

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

    1. associate-+r+ 78.3%

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

    2. +-commutative 78.3%

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

    3. *-commutative 78.3%

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

    4. mul-1-neg 78.3%

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

    5. unpow2 78.3%

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

    6. distribute-rgt-neg-out 78.3%

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

    7. distribute-lft-in 78.4%

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

    8. sub-neg 78.4%

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

    9. fma-def 78.4%

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

    10. unpow2 78.4%

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

12. Simplified 78.4%

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

13. Taylor expanded in maxCos around 0 78.4%

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

    1. metadata-eval 78.4%

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

    2. cancel-sign-sub-inv 78.4%

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

    3. distribute-lft-out-- 78.4%

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

    4. unpow2 78.4%

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

15. Simplified 78.4%

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

16. Final simplification 78.4%

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

Alternative 9: 78.9% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.4%

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

   1. associate-*l* 57.4%

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

   2. +-commutative 57.4%

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

   3. associate-+r- 57.4%

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

   4. fma-def 57.4%

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

   5. +-commutative 57.4%

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

   6. associate-+r- 57.2%

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

   7. fma-def 57.2%

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

3. Simplified 57.2%

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

4. Taylor expanded in ux around -inf 98.9%

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

   1. +-commutative 98.9%

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

   2. metadata-eval 98.9%

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

   3. cancel-sign-sub-inv 98.9%

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

   4. fma-def 99.0%

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

   5. cancel-sign-sub-inv 99.0%

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

   6. metadata-eval 99.0%

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

   7. +-commutative 99.0%

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

   8. *-commutative 99.0%

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

   9. fma-def 99.0%

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

   10. mul-1-neg 99.0%

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

   11. *-commutative 99.0%

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

   12. distribute-rgt-neg-in 99.0%

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

   13. mul-1-neg 99.0%

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

   14. unsub-neg 99.0%

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

   15. unpow2 99.0%

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

   16. distribute-rgt-neg-in 99.0%

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

6. Simplified 99.0%

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

7. Taylor expanded in uy around 0 78.5%

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

   1. +-commutative 78.5%

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

   2. +-commutative 78.5%

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

   3. *-commutative 78.5%

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

   4. fma-udef 78.5%

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

   5. mul-1-neg 78.5%

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

   6. *-commutative 78.5%

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

   7. distribute-rgt-neg-in 78.5%

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

   8. unpow2 78.5%

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

   9. distribute-rgt-neg-out 78.5%

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

   10. fma-udef 78.6%

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

9. Simplified 78.6%

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

10. Taylor expanded in maxCos around 0 78.3%

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

    1. associate-+r+ 78.3%

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

    2. +-commutative 78.3%

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

    3. *-commutative 78.3%

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

    4. mul-1-neg 78.3%

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

    5. unpow2 78.3%

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

    6. distribute-rgt-neg-out 78.3%

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

    7. distribute-lft-in 78.4%

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

    8. sub-neg 78.4%

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

    9. fma-def 78.4%

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

    10. unpow2 78.4%

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

12. Simplified 78.4%

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

13. Taylor expanded in ux around 0 78.0%

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

    1. associate-*r* 78.0%

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

    2. *-commutative 78.0%

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

15. Simplified 78.0%

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

16. Final simplification 78.0%

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

Alternative 10: 75.6% 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 57.4%

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

   1. associate-*l* 57.4%

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

   2. +-commutative 57.4%

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

   3. associate-+r- 57.4%

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

   4. fma-def 57.4%

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

   5. +-commutative 57.4%

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

   6. associate-+r- 57.2%

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

   7. fma-def 57.2%

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

3. Simplified 57.2%

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

4. Taylor expanded in ux around -inf 98.9%

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

   1. +-commutative 98.9%

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

   2. metadata-eval 98.9%

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

   3. cancel-sign-sub-inv 98.9%

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

   4. fma-def 99.0%

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

   5. cancel-sign-sub-inv 99.0%

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

   6. metadata-eval 99.0%

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

   7. +-commutative 99.0%

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

   8. *-commutative 99.0%

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

   9. fma-def 99.0%

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

   10. mul-1-neg 99.0%

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

   11. *-commutative 99.0%

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

   12. distribute-rgt-neg-in 99.0%

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

   13. mul-1-neg 99.0%

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

   14. unsub-neg 99.0%

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

   15. unpow2 99.0%

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

   16. distribute-rgt-neg-in 99.0%

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

6. Simplified 99.0%

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

7. Taylor expanded in uy around 0 78.5%

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

   1. +-commutative 78.5%

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

   2. +-commutative 78.5%

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

   3. *-commutative 78.5%

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

   4. fma-udef 78.5%

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

   5. mul-1-neg 78.5%

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

   6. *-commutative 78.5%

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

   7. distribute-rgt-neg-in 78.5%

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

   8. unpow2 78.5%

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

   9. distribute-rgt-neg-out 78.5%

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

   10. fma-udef 78.6%

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

9. Simplified 78.6%

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

10. Taylor expanded in maxCos around 0 74.8%

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

    1. +-commutative 74.8%

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

    2. *-commutative 74.8%

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

    3. mul-1-neg 74.8%

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

    4. unpow2 74.8%

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

    5. distribute-rgt-neg-out 74.8%

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

    6. distribute-lft-in 74.9%

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

    7. sub-neg 74.9%

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

12. Simplified 74.9%

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

13. Final simplification 74.9%

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

Alternative 11: 62.1% 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 57.4%

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

   1. associate-*l* 57.4%

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

   2. +-commutative 57.4%

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

   3. associate-+r- 57.4%

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

   4. fma-def 57.4%

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

   5. +-commutative 57.4%

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

   6. associate-+r- 57.2%

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

   7. fma-def 57.2%

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

3. Simplified 57.2%

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

4. Taylor expanded in uy around 0 48.2%

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

5. Taylor expanded in ux around 0 64.5%

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

6. Taylor expanded in maxCos around 0 62.3%

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

7. Final simplification 62.3%

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

Reproduce

herbie shell --seed 2023275 
(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)))))))