UniformSampleCone, x

Percentage Accurate: 57.0% → 99.0%

Time: 12.4s

Alternatives: 10

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: 57.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

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. sub-neg 57.0%

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

   3. +-commutative 57.0%

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

   4. distribute-rgt-neg-in 57.0%

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

   5. fma-def 56.8%

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

   6. +-commutative 56.8%

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

   7. associate-+r- 56.9%

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

   8. fma-def 56.9%

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

   9. neg-sub0 56.9%

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

   10. +-commutative 56.9%

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

   11. associate-+r- 56.8%

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

   12. associate--r- 56.8%

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

   13. neg-sub0 56.8%

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

   14. +-commutative 56.8%

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

   15. sub-neg 56.8%

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

   16. fma-def 56.8%

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

3. Simplified 56.8%

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

4. Taylor expanded in ux around -inf 99.0%

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

   1. +-commutative 99.0%

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

   2. mul-1-neg 99.0%

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

   3. unsub-neg 99.0%

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

   4. associate-*r* 99.0%

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

   5. mul-1-neg 99.0%

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

   6. sub-neg 99.0%

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

   7. *-commutative 99.0%

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

   8. unpow2 99.0%

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

   9. associate-*l* 99.0%

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

   10. mul-1-neg 99.0%

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

   11. sub-neg 99.0%

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

6. Simplified 99.0%

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

7. Taylor expanded in uy around inf 99.0%

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

   1. associate-*r* 99.0%

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

   2. unpow2 99.0%

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

   3. associate-*r* 99.0%

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

   4. *-commutative 99.0%

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

   5. distribute-rgt-out-- 99.0%

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

9. Simplified 99.0%

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

10. Final simplification 99.0%

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

Alternative 2: 89.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.001500000013038516:\\ \;\;\;\;\sqrt{ux \cdot \left(2 \cdot \left(1 - maxCos\right) - ux \cdot {\left(1 - maxCos\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= uy 0.001500000013038516)
   (sqrt (* ux (- (* 2.0 (- 1.0 maxCos)) (* ux (pow (- 1.0 maxCos) 2.0)))))
   (* (cos (* uy (* 2.0 PI))) (sqrt (* 2.0 ux)))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.001500000013038516f) {
		tmp = sqrtf((ux * ((2.0f * (1.0f - maxCos)) - (ux * powf((1.0f - maxCos), 2.0f)))));
	} else {
		tmp = cosf((uy * (2.0f * ((float) M_PI)))) * sqrtf((2.0f * ux));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.001500000013038516))
		tmp = sqrt(Float32(ux * Float32(Float32(Float32(2.0) * Float32(Float32(1.0) - maxCos)) - Float32(ux * (Float32(Float32(1.0) - maxCos) ^ Float32(2.0))))));
	else
		tmp = Float32(cos(Float32(uy * Float32(Float32(2.0) * Float32(pi)))) * sqrt(Float32(Float32(2.0) * ux)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (uy <= single(0.001500000013038516))
		tmp = sqrt((ux * ((single(2.0) * (single(1.0) - maxCos)) - (ux * ((single(1.0) - maxCos) ^ single(2.0))))));
	else
		tmp = cos((uy * (single(2.0) * single(pi)))) * sqrt((single(2.0) * ux));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.00150000001

   1. Initial program 58.1%

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

      1. associate-*l* 58.1%

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

      2. sub-neg 58.1%

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

      3. +-commutative 58.1%

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

      4. distribute-rgt-neg-in 58.1%

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

      5. fma-def 58.0%

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

      6. +-commutative 58.0%

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

      7. associate-+r- 58.1%

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

      8. fma-def 58.1%

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

      9. neg-sub0 58.1%

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

      10. +-commutative 58.1%

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

      11. associate-+r- 58.0%

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

      12. associate--r- 58.0%

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

      13. neg-sub0 58.0%

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

      14. +-commutative 58.0%

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

      15. sub-neg 58.0%

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

      16. fma-def 58.0%

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

   3. Simplified 58.0%

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

   4. Taylor expanded in ux around -inf 99.4%

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

      1. +-commutative 99.4%

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

      2. mul-1-neg 99.4%

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

      3. unsub-neg 99.4%

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

      4. associate-*r* 99.4%

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

      5. mul-1-neg 99.4%

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

      6. sub-neg 99.4%

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

      7. *-commutative 99.4%

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

      8. unpow2 99.4%

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

      9. associate-*l* 99.4%

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

      10. mul-1-neg 99.4%

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

      11. sub-neg 99.4%

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

   6. Simplified 99.4%

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

   7. Taylor expanded in uy around 0 96.8%

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

      1. associate-*r* 96.8%

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

      2. unpow2 96.8%

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

      3. associate-*r* 96.8%

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

      4. *-commutative 96.8%

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

      5. distribute-rgt-out-- 96.8%

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

   9. Simplified 96.8%

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

   if 0.00150000001 < uy

   1. Initial program 54.5%

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

      1. associate-*l* 54.5%

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

      2. sub-neg 54.5%

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

      3. +-commutative 54.5%

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

      4. distribute-rgt-neg-in 54.5%

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

      5. fma-def 54.2%

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

      6. +-commutative 54.2%

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

      7. associate-+r- 54.2%

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

      8. fma-def 54.2%

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

      9. neg-sub0 54.2%

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

      10. +-commutative 54.2%

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

      11. associate-+r- 54.1%

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

      12. associate--r- 54.1%

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

      13. neg-sub0 54.1%

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

      14. +-commutative 54.1%

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

      15. sub-neg 54.1%

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

      16. fma-def 54.1%

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

   3. Simplified 54.1%

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

   4. Taylor expanded in ux around 0 77.1%

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

   5. Taylor expanded in maxCos around 0 71.6%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \leq 0.001500000013038516:\\ \;\;\;\;\sqrt{ux \cdot \left(2 \cdot \left(1 - maxCos\right) - ux \cdot {\left(1 - maxCos\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux}\\ \end{array} \]

Alternative 3: 92.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

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. sub-neg 57.0%

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

   3. +-commutative 57.0%

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

   4. distribute-rgt-neg-in 57.0%

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

   5. fma-def 56.8%

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

   6. +-commutative 56.8%

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

   7. associate-+r- 56.9%

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

   8. fma-def 56.9%

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

   9. neg-sub0 56.9%

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

   10. +-commutative 56.9%

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

   11. associate-+r- 56.8%

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

   12. associate--r- 56.8%

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

   13. neg-sub0 56.8%

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

   14. +-commutative 56.8%

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

   15. sub-neg 56.8%

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

   16. fma-def 56.8%

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

3. Simplified 56.8%

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

4. Taylor expanded in ux around 0 99.0%

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

   1. +-commutative 99.0%

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

   2. fma-def 99.0%

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

   3. associate--l+ 99.0%

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

   4. mul-1-neg 99.0%

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

   5. sub-neg 99.0%

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

   6. metadata-eval 99.0%

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

   7. distribute-neg-in 99.0%

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

   8. metadata-eval 99.0%

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

   9. +-commutative 99.0%

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

   10. sub-neg 99.0%

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

   11. associate-*r* 99.0%

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

   12. *-commutative 99.0%

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

6. Simplified 99.0%

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

7. Taylor expanded in maxCos around 0 93.6%

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

   1. *-commutative 93.6%

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

   2. +-commutative 93.6%

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

   3. mul-1-neg 93.6%

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

   4. sub-neg 93.6%

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

   5. unpow2 93.6%

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

   6. distribute-rgt-out-- 93.6%

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

   7. *-commutative 93.6%

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

   8. *-commutative 93.6%

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

   9. associate-*l* 93.6%

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

9. Simplified 93.6%

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

10. Final simplification 93.6%

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

Alternative 4: 80.4% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = sqrt((ux * ((2.0e0 * (1.0e0 - maxcos)) - (ux * ((1.0e0 - maxcos) ** 2.0e0)))))
end function

Julia

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

MATLAB

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

Derivation

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. sub-neg 57.0%

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

   3. +-commutative 57.0%

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

   4. distribute-rgt-neg-in 57.0%

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

   5. fma-def 56.8%

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

   6. +-commutative 56.8%

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

   7. associate-+r- 56.9%

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

   8. fma-def 56.9%

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

   9. neg-sub0 56.9%

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

   10. +-commutative 56.9%

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

   11. associate-+r- 56.8%

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

   12. associate--r- 56.8%

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

   13. neg-sub0 56.8%

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

   14. +-commutative 56.8%

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

   15. sub-neg 56.8%

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

   16. fma-def 56.8%

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

3. Simplified 56.8%

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

4. Taylor expanded in ux around -inf 99.0%

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

   1. +-commutative 99.0%

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

   2. mul-1-neg 99.0%

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

   3. unsub-neg 99.0%

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

   4. associate-*r* 99.0%

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

   5. mul-1-neg 99.0%

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

   6. sub-neg 99.0%

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

   7. *-commutative 99.0%

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

   8. unpow2 99.0%

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

   9. associate-*l* 99.0%

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

   10. mul-1-neg 99.0%

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

   11. sub-neg 99.0%

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

6. Simplified 99.0%

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

7. Taylor expanded in uy around 0 78.9%

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

   1. associate-*r* 78.9%

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

   2. unpow2 78.9%

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

   3. associate-*r* 78.9%

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

   4. *-commutative 78.9%

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

   5. distribute-rgt-out-- 78.9%

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

9. Simplified 78.9%

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

10. Final simplification 78.9%

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

Alternative 5: 80.0% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

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. sub-neg 57.0%

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

   3. +-commutative 57.0%

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

   4. distribute-rgt-neg-in 57.0%

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

   5. fma-def 56.8%

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

   6. +-commutative 56.8%

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

   7. associate-+r- 56.9%

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

   8. fma-def 56.9%

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

   9. neg-sub0 56.9%

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

   10. +-commutative 56.9%

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

   11. associate-+r- 56.8%

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

   12. associate--r- 56.8%

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

   13. neg-sub0 56.8%

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

   14. +-commutative 56.8%

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

   15. sub-neg 56.8%

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

   16. fma-def 56.8%

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

3. Simplified 56.8%

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

4. Taylor expanded in ux around 0 99.0%

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

   1. +-commutative 99.0%

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

   2. fma-def 99.0%

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

   3. associate--l+ 99.0%

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

   4. mul-1-neg 99.0%

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

   5. sub-neg 99.0%

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

   6. metadata-eval 99.0%

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

   7. distribute-neg-in 99.0%

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

   8. metadata-eval 99.0%

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

   9. +-commutative 99.0%

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

   10. sub-neg 99.0%

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

   11. associate-*r* 99.0%

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

   12. *-commutative 99.0%

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

6. Simplified 99.0%

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

7. Taylor expanded in uy around 0 78.9%

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

8. Taylor expanded in maxCos around 0 78.4%

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

   1. unpow2 78.4%

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

   2. unpow2 78.4%

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

10. Simplified 78.4%

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

11. Final simplification 78.4%

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

Alternative 6: 79.5% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

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. sub-neg 57.0%

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

   3. +-commutative 57.0%

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

   4. distribute-rgt-neg-in 57.0%

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

   5. fma-def 56.8%

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

   6. +-commutative 56.8%

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

   7. associate-+r- 56.9%

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

   8. fma-def 56.9%

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

   9. neg-sub0 56.9%

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

   10. +-commutative 56.9%

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

   11. associate-+r- 56.8%

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

   12. associate--r- 56.8%

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

   13. neg-sub0 56.8%

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

   14. +-commutative 56.8%

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

   15. sub-neg 56.8%

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

   16. fma-def 56.8%

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

3. Simplified 56.8%

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

4. Taylor expanded in ux around 0 99.0%

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

   1. +-commutative 99.0%

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

   2. fma-def 99.0%

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

   3. associate--l+ 99.0%

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

   4. mul-1-neg 99.0%

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

   5. sub-neg 99.0%

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

   6. metadata-eval 99.0%

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

   7. distribute-neg-in 99.0%

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

   8. metadata-eval 99.0%

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

   9. +-commutative 99.0%

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

   10. sub-neg 99.0%

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

   11. associate-*r* 99.0%

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

   12. *-commutative 99.0%

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

6. Simplified 99.0%

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

7. Taylor expanded in uy around 0 78.9%

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

8. Taylor expanded in maxCos around 0 78.0%

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

   1. unpow2 78.0%

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

10. Simplified 78.0%

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

11. Final simplification 78.0%

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

Alternative 7: 76.1% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

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. sub-neg 57.0%

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

   3. +-commutative 57.0%

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

   4. distribute-rgt-neg-in 57.0%

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

   5. fma-def 56.8%

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

   6. +-commutative 56.8%

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

   7. associate-+r- 56.9%

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

   8. fma-def 56.9%

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

   9. neg-sub0 56.9%

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

   10. +-commutative 56.9%

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

   11. associate-+r- 56.8%

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

   12. associate--r- 56.8%

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

   13. neg-sub0 56.8%

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

   14. +-commutative 56.8%

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

   15. sub-neg 56.8%

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

   16. fma-def 56.8%

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

3. Simplified 56.8%

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

4. Taylor expanded in ux around 0 99.0%

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

   1. +-commutative 99.0%

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

   2. fma-def 99.0%

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

   3. associate--l+ 99.0%

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

   4. mul-1-neg 99.0%

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

   5. sub-neg 99.0%

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

   6. metadata-eval 99.0%

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

   7. distribute-neg-in 99.0%

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

   8. metadata-eval 99.0%

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

   9. +-commutative 99.0%

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

   10. sub-neg 99.0%

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

   11. associate-*r* 99.0%

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

   12. *-commutative 99.0%

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

6. Simplified 99.0%

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

7. Taylor expanded in uy around 0 78.9%

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

8. Taylor expanded in maxCos around 0 76.2%

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

   1. +-commutative 76.2%

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

   2. mul-1-neg 76.2%

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

   3. sub-neg 76.2%

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

   4. unpow2 76.2%

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

10. Simplified 76.2%

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

11. Final simplification 76.2%

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

Alternative 8: 62.7% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

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. sub-neg 57.0%

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

   3. +-commutative 57.0%

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

   4. distribute-rgt-neg-in 57.0%

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

   5. fma-def 56.8%

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

   6. +-commutative 56.8%

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

   7. associate-+r- 56.9%

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

   8. fma-def 56.9%

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

   9. neg-sub0 56.9%

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

   10. +-commutative 56.9%

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

   11. associate-+r- 56.8%

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

   12. associate--r- 56.8%

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

   13. neg-sub0 56.8%

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

   14. +-commutative 56.8%

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

   15. sub-neg 56.8%

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

   16. fma-def 56.8%

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

3. Simplified 56.8%

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

4. Taylor expanded in ux around 0 76.7%

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

5. Taylor expanded in uy around 0 63.8%

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

6. Taylor expanded in maxCos around 0 62.6%

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

7. Final simplification 62.6%

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

Alternative 9: 76.1% 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 57.0%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
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. sub-neg 57.0%

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

   3. +-commutative 57.0%

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

   4. distribute-rgt-neg-in 57.0%

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

   5. fma-def 56.8%

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

   6. +-commutative 56.8%

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

   7. associate-+r- 56.9%

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

   8. fma-def 56.9%

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

   9. neg-sub0 56.9%

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

   10. +-commutative 56.9%

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

   11. associate-+r- 56.8%

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

   12. associate--r- 56.8%

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

   13. neg-sub0 56.8%

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

   14. +-commutative 56.8%

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

   15. sub-neg 56.8%

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

   16. fma-def 56.8%

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

3. Simplified 56.8%

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

4. Taylor expanded in ux around 0 99.0%

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

   1. +-commutative 99.0%

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

   2. fma-def 99.0%

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

   3. associate--l+ 99.0%

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

   4. mul-1-neg 99.0%

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

   5. sub-neg 99.0%

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

   6. metadata-eval 99.0%

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

   7. distribute-neg-in 99.0%

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

   8. metadata-eval 99.0%

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

   9. +-commutative 99.0%

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

   10. sub-neg 99.0%

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

   11. associate-*r* 99.0%

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

   12. *-commutative 99.0%

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

6. Simplified 99.0%

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

7. Taylor expanded in uy around 0 78.9%

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

8. Taylor expanded in maxCos around 0 76.2%

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

   1. +-commutative 76.2%

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

   2. mul-1-neg 76.2%

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

   3. sub-neg 76.2%

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

   4. unpow2 76.2%

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

   5. distribute-rgt-out-- 76.2%

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

10. Simplified 76.2%

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

11. Final simplification 76.2%

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

Alternative 10: 62.5% 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 57.0%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
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. sub-neg 57.0%

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

   3. +-commutative 57.0%

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

   4. distribute-rgt-neg-in 57.0%

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

   5. fma-def 56.8%

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

   6. +-commutative 56.8%

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

   7. associate-+r- 56.9%

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

   8. fma-def 56.9%

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

   9. neg-sub0 56.9%

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

   10. +-commutative 56.9%

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

   11. associate-+r- 56.8%

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

   12. associate--r- 56.8%

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

   13. neg-sub0 56.8%

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

   14. +-commutative 56.8%

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

   15. sub-neg 56.8%

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

   16. fma-def 56.8%

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

3. Simplified 56.8%

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

4. Taylor expanded in ux around 0 76.7%

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

5. Taylor expanded in uy around 0 63.8%

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

6. Taylor expanded in maxCos around 0 62.4%

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

7. Final simplification 62.4%

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

Reproduce

herbie shell --seed 2023182 
(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)))))))