UniformSampleCone, x

Percentage Accurate: 57.4% → 99.0%

Time: 16.3s

Alternatives: 14

Speedup: 3.1×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 57.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \cos \left(\sqrt[3]{{\left(uy \cdot 2\right)}^{3} \cdot {\pi}^{3}}\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(maxCos, -2, 2\right), ux, \left(ux \cdot ux\right) \cdot \left(-{\left(1 - maxCos\right)}^{2}\right)\right)} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (cos (cbrt (* (pow (* uy 2.0) 3.0) (pow PI 3.0))))
  (sqrt
   (fma (fma maxCos -2.0 2.0) ux (* (* ux ux) (- (pow (- 1.0 maxCos) 2.0)))))))

C

float code(float ux, float uy, float maxCos) {
	return cosf(cbrtf((powf((uy * 2.0f), 3.0f) * powf(((float) M_PI), 3.0f)))) * sqrtf(fmaf(fmaf(maxCos, -2.0f, 2.0f), ux, ((ux * ux) * -powf((1.0f - maxCos), 2.0f))));
}

Julia

function code(ux, uy, maxCos)
	return Float32(cos(cbrt(Float32((Float32(uy * Float32(2.0)) ^ Float32(3.0)) * (Float32(pi) ^ Float32(3.0))))) * sqrt(fma(fma(maxCos, Float32(-2.0), Float32(2.0)), ux, Float32(Float32(ux * ux) * Float32(-(Float32(Float32(1.0) - maxCos) ^ Float32(2.0)))))))
end

Derivation

1. Initial program 55.7%

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

   1. associate-*l* 55.7%

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

   2. +-commutative 55.7%

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

   3. associate-+r- 55.8%

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

   4. fma-def 55.8%

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

   5. +-commutative 55.8%

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

   6. associate-+r- 55.7%

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

   7. fma-def 55.7%

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

3. Simplified 55.7%

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

4. Taylor expanded in ux around -inf 99.0%

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

   1. metadata-eval 99.0%

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

   2. cancel-sign-sub-inv 99.0%

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

   3. *-commutative 99.0%

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

   4. fma-def 99.0%

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

   5. cancel-sign-sub-inv 99.0%

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

   6. metadata-eval 99.0%

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

   7. +-commutative 99.0%

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

   8. *-commutative 99.0%

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

   9. fma-def 99.0%

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

   10. mul-1-neg 99.0%

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

   11. distribute-rgt-neg-in 99.0%

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

   12. unpow2 99.0%

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

   13. mul-1-neg 99.0%

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

   14. sub-neg 99.0%

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

6. Simplified 99.0%

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

   1. associate-*r* 99.0%

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

   2. add-cbrt-cube 99.0%

      \[\leadsto \cos \left(\color{blue}{\sqrt[3]{\left(\left(uy \cdot 2\right) \cdot \left(uy \cdot 2\right)\right) \cdot \left(uy \cdot 2\right)}} \cdot \pi\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(maxCos, -2, 2\right), ux, \left(ux \cdot ux\right) \cdot \left(-{\left(1 - maxCos\right)}^{2}\right)\right)} \]

   3. add-cbrt-cube 99.0%

      \[\leadsto \cos \left(\sqrt[3]{\left(\left(uy \cdot 2\right) \cdot \left(uy \cdot 2\right)\right) \cdot \left(uy \cdot 2\right)} \cdot \color{blue}{\sqrt[3]{\left(\pi \cdot \pi\right) \cdot \pi}}\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(maxCos, -2, 2\right), ux, \left(ux \cdot ux\right) \cdot \left(-{\left(1 - maxCos\right)}^{2}\right)\right)} \]

   4. cbrt-unprod 99.0%

      \[\leadsto \cos \color{blue}{\left(\sqrt[3]{\left(\left(\left(uy \cdot 2\right) \cdot \left(uy \cdot 2\right)\right) \cdot \left(uy \cdot 2\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}\right)} \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(maxCos, -2, 2\right), ux, \left(ux \cdot ux\right) \cdot \left(-{\left(1 - maxCos\right)}^{2}\right)\right)} \]

   5. pow3 99.1%

      \[\leadsto \cos \left(\sqrt[3]{\color{blue}{{\left(uy \cdot 2\right)}^{3}} \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)}\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(maxCos, -2, 2\right), ux, \left(ux \cdot ux\right) \cdot \left(-{\left(1 - maxCos\right)}^{2}\right)\right)} \]

   6. pow3 99.1%

      \[\leadsto \cos \left(\sqrt[3]{{\left(uy \cdot 2\right)}^{3} \cdot \color{blue}{{\pi}^{3}}}\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(maxCos, -2, 2\right), ux, \left(ux \cdot ux\right) \cdot \left(-{\left(1 - maxCos\right)}^{2}\right)\right)} \]

8. Applied egg-rr 99.1%

   \[\leadsto \cos \color{blue}{\left(\sqrt[3]{{\left(uy \cdot 2\right)}^{3} \cdot {\pi}^{3}}\right)} \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(maxCos, -2, 2\right), ux, \left(ux \cdot ux\right) \cdot \left(-{\left(1 - maxCos\right)}^{2}\right)\right)} \]

9. Final simplification 99.1%

   \[\leadsto \cos \left(\sqrt[3]{{\left(uy \cdot 2\right)}^{3} \cdot {\pi}^{3}}\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(maxCos, -2, 2\right), ux, \left(ux \cdot ux\right) \cdot \left(-{\left(1 - maxCos\right)}^{2}\right)\right)} \]

Alternative 2: 99.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 55.7%

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

   1. associate-*l* 55.7%

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

   2. +-commutative 55.7%

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

   3. associate-+r- 55.8%

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

   4. fma-def 55.8%

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

   5. +-commutative 55.8%

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

   6. associate-+r- 55.7%

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

   7. fma-def 55.7%

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

3. Simplified 55.7%

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

4. Taylor expanded in ux around -inf 99.0%

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

   1. metadata-eval 99.0%

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

   2. cancel-sign-sub-inv 99.0%

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

   3. *-commutative 99.0%

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

   4. fma-def 99.0%

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

   5. cancel-sign-sub-inv 99.0%

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

   6. metadata-eval 99.0%

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

   7. +-commutative 99.0%

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

   8. *-commutative 99.0%

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

   9. fma-def 99.0%

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

   10. mul-1-neg 99.0%

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

   11. distribute-rgt-neg-in 99.0%

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

   12. unpow2 99.0%

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

   13. mul-1-neg 99.0%

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

   14. sub-neg 99.0%

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

6. Simplified 99.0%

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

7. Final simplification 99.0%

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

Alternative 3: 99.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 55.7%

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

   1. associate-*l* 55.7%

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

   2. sub-neg 55.7%

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

   3. +-commutative 55.7%

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

   4. distribute-rgt-neg-in 55.7%

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

   5. fma-def 55.7%

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

   6. +-commutative 55.7%

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

   7. associate-+r- 55.7%

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

   8. fma-def 55.7%

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

   9. neg-sub0 55.7%

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

   10. +-commutative 55.7%

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

   11. associate-+r- 55.6%

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

   12. associate--r- 55.6%

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

   13. neg-sub0 55.6%

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

   14. +-commutative 55.6%

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

   15. sub-neg 55.6%

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

   16. fma-def 55.6%

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

3. Simplified 55.6%

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

4. Taylor expanded in ux around 0 98.9%

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

   1. fma-def 98.9%

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

   2. sub-neg 98.9%

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

   3. metadata-eval 98.9%

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

   4. *-commutative 98.9%

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

   5. unpow2 98.9%

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

   6. +-commutative 98.9%

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

   7. mul-1-neg 98.9%

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

   8. sub-neg 98.9%

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

   9. metadata-eval 98.9%

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

   10. distribute-neg-in 98.9%

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

   11. metadata-eval 98.9%

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

   12. +-commutative 98.9%

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

   13. mul-1-neg 98.9%

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

   14. associate--l+ 99.0%

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

   15. mul-1-neg 99.0%

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

   16. sub-neg 99.0%

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

6. Simplified 99.0%

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

   1. pow1 99.0%

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

   2. count-2 99.0%

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

8. Applied egg-rr 99.0%

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

10. Simplified 99.0%

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

11. Final simplification 99.0%

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

Alternative 4: 90.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy 2) < 0.00159999996

   1. Initial program 55.9%

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

      1. associate-*l* 55.9%

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

      2. sub-neg 55.9%

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

      3. +-commutative 55.9%

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

      4. distribute-rgt-neg-in 55.9%

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

      5. fma-def 55.9%

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

      6. +-commutative 55.9%

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

      7. associate-+r- 55.9%

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

      8. fma-def 55.9%

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

      9. neg-sub0 55.9%

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

      10. +-commutative 55.9%

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

      11. associate-+r- 55.8%

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

      12. associate--r- 55.8%

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

      13. neg-sub0 55.8%

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

      14. +-commutative 55.8%

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

      15. sub-neg 55.8%

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

      16. fma-def 55.8%

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

   3. Simplified 55.8%

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

   4. Taylor expanded in uy around 0 55.3%

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

   5. Taylor expanded in ux around 0 58.3%

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

      1. associate--l+ 58.0%

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

      2. associate-*r* 58.0%

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

      3. *-commutative 58.0%

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

      4. sub-neg 58.0%

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

      5. metadata-eval 58.0%

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

      6. unpow2 58.0%

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

      7. fma-neg 58.0%

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

      8. associate--l+ 58.1%

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

      9. mul-1-neg 58.1%

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

      10. sub-neg 58.1%

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

      11. metadata-eval 58.1%

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

      12. distribute-neg-in 58.1%

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

      13. metadata-eval 58.1%

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

      14. +-commutative 58.1%

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

      15. sub-neg 58.1%

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

      16. metadata-eval 58.1%

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

   7. Simplified 58.1%

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

   8. Taylor expanded in ux around 0 97.8%

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

   if 0.00159999996 < (*.f32 uy 2)

   1. Initial program 55.3%

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

      1. associate-*l* 55.3%

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

      2. +-commutative 55.3%

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

      3. associate-+r- 55.4%

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

      4. fma-def 55.4%

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

      5. +-commutative 55.4%

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

      6. associate-+r- 55.3%

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

      7. fma-def 55.3%

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

   3. Simplified 55.3%

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

   4. Taylor expanded in ux around 0 78.7%

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

4. Final simplification 91.3%

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

Alternative 5: 89.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.0016499999910593033:\\ \;\;\;\;\sqrt{\left(1 - maxCos\right) \cdot \left(\left(maxCos + -1\right) \cdot {ux}^{2}\right) - ux \cdot \left(2 \cdot maxCos - 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* uy 2.0) 0.0016499999910593033)
   (sqrt
    (-
     (* (- 1.0 maxCos) (* (+ maxCos -1.0) (pow ux 2.0)))
     (* ux (- (* 2.0 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 * 2.0f) <= 0.0016499999910593033f) {
		tmp = sqrtf((((1.0f - maxCos) * ((maxCos + -1.0f) * powf(ux, 2.0f))) - (ux * ((2.0f * 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 (Float32(uy * Float32(2.0)) <= Float32(0.0016499999910593033))
		tmp = sqrt(Float32(Float32(Float32(Float32(1.0) - maxCos) * Float32(Float32(maxCos + Float32(-1.0)) * (ux ^ Float32(2.0)))) - Float32(ux * Float32(Float32(Float32(2.0) * 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(2.0)) <= single(0.0016499999910593033))
		tmp = sqrt((((single(1.0) - maxCos) * ((maxCos + single(-1.0)) * (ux ^ single(2.0)))) - (ux * ((single(2.0) * 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 (*.f32 uy 2) < 0.00165

   1. Initial program 55.6%

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

      1. associate-*l* 55.6%

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

      2. sub-neg 55.6%

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

      3. +-commutative 55.6%

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

      4. distribute-rgt-neg-in 55.6%

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

      5. fma-def 55.6%

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

      6. +-commutative 55.6%

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

      7. associate-+r- 55.6%

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

      8. fma-def 55.6%

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

      9. neg-sub0 55.6%

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

      10. +-commutative 55.6%

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

      11. associate-+r- 55.5%

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

      12. associate--r- 55.5%

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

      13. neg-sub0 55.5%

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

      14. +-commutative 55.5%

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

      15. sub-neg 55.5%

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

      16. fma-def 55.5%

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

   3. Simplified 55.5%

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

   4. Taylor expanded in uy around 0 55.0%

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

   5. Taylor expanded in ux around 0 58.0%

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

      1. associate--l+ 57.7%

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

      2. associate-*r* 57.7%

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

      3. *-commutative 57.7%

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

      4. sub-neg 57.7%

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

      5. metadata-eval 57.7%

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

      6. unpow2 57.7%

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

      7. fma-neg 57.7%

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

      8. associate--l+ 57.8%

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

      9. mul-1-neg 57.8%

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

      10. sub-neg 57.8%

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

      11. metadata-eval 57.8%

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

      12. distribute-neg-in 57.8%

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

      13. metadata-eval 57.8%

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

      14. +-commutative 57.8%

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

      15. sub-neg 57.8%

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

      16. metadata-eval 57.8%

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

   7. Simplified 57.8%

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

   8. Taylor expanded in ux around 0 97.7%

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

   if 0.00165 < (*.f32 uy 2)

   1. Initial program 55.8%

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

      1. associate-*l* 55.8%

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

      2. +-commutative 55.8%

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

      3. associate-+r- 56.0%

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

      4. fma-def 56.0%

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

      5. +-commutative 56.0%

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

      6. associate-+r- 55.9%

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

      7. fma-def 55.9%

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

   3. Simplified 55.9%

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

   4. Taylor expanded in ux around 0 44.3%

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

   5. Taylor expanded in maxCos around 0 76.0%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.0016499999910593033:\\ \;\;\;\;\sqrt{\left(1 - maxCos\right) \cdot \left(\left(maxCos + -1\right) \cdot {ux}^{2}\right) - ux \cdot \left(2 \cdot maxCos - 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux}\\ \end{array} \]

Alternative 6: 92.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 55.7%

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

   1. associate-*l* 55.7%

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

   2. +-commutative 55.7%

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

   3. associate-+r- 55.8%

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

   4. fma-def 55.8%

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

   5. +-commutative 55.8%

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

   6. associate-+r- 55.7%

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

   7. fma-def 55.7%

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

3. Simplified 55.7%

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

4. Taylor expanded in ux around -inf 99.0%

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

   1. metadata-eval 99.0%

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

   2. cancel-sign-sub-inv 99.0%

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

   3. *-commutative 99.0%

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

   4. fma-def 99.0%

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

   5. cancel-sign-sub-inv 99.0%

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

   6. metadata-eval 99.0%

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

   7. +-commutative 99.0%

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

   8. *-commutative 99.0%

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

   9. fma-def 99.0%

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

   10. mul-1-neg 99.0%

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

   11. distribute-rgt-neg-in 99.0%

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

   12. unpow2 99.0%

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

   13. mul-1-neg 99.0%

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

   14. sub-neg 99.0%

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

6. Simplified 99.0%

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

7. Taylor expanded in maxCos around 0 94.3%

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

   1. +-commutative 94.3%

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

   2. mul-1-neg 94.3%

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

   3. unsub-neg 94.3%

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

   4. unpow2 94.3%

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

9. Simplified 94.3%

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

10. Final simplification 94.3%

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

Alternative 7: 79.7% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 55.7%

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

   1. associate-*l* 55.7%

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

   2. sub-neg 55.7%

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

   3. +-commutative 55.7%

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

   4. distribute-rgt-neg-in 55.7%

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

   5. fma-def 55.7%

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

   6. +-commutative 55.7%

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

   7. associate-+r- 55.7%

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

   8. fma-def 55.7%

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

   9. neg-sub0 55.7%

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

   10. +-commutative 55.7%

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

   11. associate-+r- 55.6%

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

   12. associate--r- 55.6%

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

   13. neg-sub0 55.6%

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

   14. +-commutative 55.6%

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

   15. sub-neg 55.6%

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

   16. fma-def 55.6%

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

3. Simplified 55.6%

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

4. Taylor expanded in uy around 0 46.9%

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

5. Taylor expanded in ux around 0 49.2%

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

   1. associate--l+ 49.0%

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

   2. associate-*r* 49.0%

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

   3. *-commutative 49.0%

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

   4. sub-neg 49.0%

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

   5. metadata-eval 49.0%

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

   6. unpow2 49.0%

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

   7. fma-neg 49.0%

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

   8. associate--l+ 49.0%

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

   9. mul-1-neg 49.0%

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

   10. sub-neg 49.0%

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

   11. metadata-eval 49.0%

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

   12. distribute-neg-in 49.0%

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

   13. metadata-eval 49.0%

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

   14. +-commutative 49.0%

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

   15. sub-neg 49.0%

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

   16. metadata-eval 49.0%

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

7. Simplified 49.0%

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

8. Taylor expanded in ux around 0 79.2%

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

9. Final simplification 79.2%

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

Alternative 8: 75.5% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 55.7%

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

   1. associate-*l* 55.7%

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

   2. sub-neg 55.7%

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

   3. +-commutative 55.7%

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

   4. distribute-rgt-neg-in 55.7%

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

   5. fma-def 55.7%

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

   6. +-commutative 55.7%

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

   7. associate-+r- 55.7%

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

   8. fma-def 55.7%

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

   9. neg-sub0 55.7%

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

   10. +-commutative 55.7%

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

   11. associate-+r- 55.6%

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

   12. associate--r- 55.6%

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

   13. neg-sub0 55.6%

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

   14. +-commutative 55.6%

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

   15. sub-neg 55.6%

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

   16. fma-def 55.6%

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

3. Simplified 55.6%

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

4. Taylor expanded in uy around 0 46.9%

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

5. Taylor expanded in ux around 0 49.2%

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

   1. associate--l+ 49.0%

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

   2. associate-*r* 49.0%

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

   3. *-commutative 49.0%

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

   4. sub-neg 49.0%

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

   5. metadata-eval 49.0%

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

   6. unpow2 49.0%

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

   7. fma-neg 49.0%

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

   8. associate--l+ 49.0%

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

   9. mul-1-neg 49.0%

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

   10. sub-neg 49.0%

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

   11. metadata-eval 49.0%

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

   12. distribute-neg-in 49.0%

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

   13. metadata-eval 49.0%

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

   14. +-commutative 49.0%

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

   15. sub-neg 49.0%

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

   16. metadata-eval 49.0%

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

7. Simplified 49.0%

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

8. Taylor expanded in maxCos around 0 75.8%

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

   1. +-commutative 75.8%

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

   2. fma-def 75.8%

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

   3. mul-1-neg 75.8%

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

   4. unpow2 75.8%

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

   5. distribute-rgt-neg-in 75.8%

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

10. Simplified 75.8%

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

11. Final simplification 75.8%

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

Alternative 9: 74.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1 + \left(ux - maxCos \cdot ux\right)\\ t_1 := \left(ux + \left(-1 - maxCos \cdot ux\right)\right) \cdot t_0\\ \mathbf{if}\;ux \leq 7.999999797903001 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1 + \left(\left(\left(1 + maxCos \cdot ux\right) - ux\right) \cdot t_0\right) \cdot t_1}{1 + t_1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ -1.0 (- ux (* maxCos ux))))
        (t_1 (* (+ ux (- -1.0 (* maxCos ux))) t_0)))
   (if (<= ux 7.999999797903001e-5)
     (sqrt (* ux (- 2.0 (* 2.0 maxCos))))
     (sqrt
      (/ (+ 1.0 (* (* (- (+ 1.0 (* maxCos ux)) ux) t_0) t_1)) (+ 1.0 t_1))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = -1.0f + (ux - (maxCos * ux));
	float t_1 = (ux + (-1.0f - (maxCos * ux))) * t_0;
	float tmp;
	if (ux <= 7.999999797903001e-5f) {
		tmp = sqrtf((ux * (2.0f - (2.0f * maxCos))));
	} else {
		tmp = sqrtf(((1.0f + ((((1.0f + (maxCos * ux)) - ux) * t_0) * t_1)) / (1.0f + t_1)));
	}
	return tmp;
}

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    real(4) :: t_0
    real(4) :: t_1
    real(4) :: tmp
    t_0 = (-1.0e0) + (ux - (maxcos * ux))
    t_1 = (ux + ((-1.0e0) - (maxcos * ux))) * t_0
    if (ux <= 7.999999797903001e-5) then
        tmp = sqrt((ux * (2.0e0 - (2.0e0 * maxcos))))
    else
        tmp = sqrt(((1.0e0 + ((((1.0e0 + (maxcos * ux)) - ux) * t_0) * t_1)) / (1.0e0 + t_1)))
    end if
    code = tmp
end function

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(Float32(-1.0) + Float32(ux - Float32(maxCos * ux)))
	t_1 = Float32(Float32(ux + Float32(Float32(-1.0) - Float32(maxCos * ux))) * t_0)
	tmp = Float32(0.0)
	if (ux <= Float32(7.999999797903001e-5))
		tmp = sqrt(Float32(ux * Float32(Float32(2.0) - Float32(Float32(2.0) * maxCos))));
	else
		tmp = sqrt(Float32(Float32(Float32(1.0) + Float32(Float32(Float32(Float32(Float32(1.0) + Float32(maxCos * ux)) - ux) * t_0) * t_1)) / Float32(Float32(1.0) + t_1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	t_0 = single(-1.0) + (ux - (maxCos * ux));
	t_1 = (ux + (single(-1.0) - (maxCos * ux))) * t_0;
	tmp = single(0.0);
	if (ux <= single(7.999999797903001e-5))
		tmp = sqrt((ux * (single(2.0) - (single(2.0) * maxCos))));
	else
		tmp = sqrt(((single(1.0) + ((((single(1.0) + (maxCos * ux)) - ux) * t_0) * t_1)) / (single(1.0) + t_1)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if ux < 7.9999998e-5

   1. Initial program 32.8%

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

      1. associate-*l* 32.8%

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

      2. sub-neg 32.8%

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

      3. +-commutative 32.8%

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

      4. distribute-rgt-neg-in 32.8%

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

      5. fma-def 32.8%

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

      6. +-commutative 32.8%

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

      7. associate-+r- 32.9%

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

      8. fma-def 32.9%

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

      9. neg-sub0 32.9%

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

      10. +-commutative 32.9%

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

      11. associate-+r- 32.8%

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

      12. associate--r- 32.8%

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

      13. neg-sub0 32.8%

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

      14. +-commutative 32.8%

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

      15. sub-neg 32.8%

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

      16. fma-def 32.8%

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

   3. Simplified 32.8%

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

   4. Taylor expanded in uy around 0 29.6%

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

   5. Taylor expanded in ux around 0 32.6%

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

      1. associate--l+ 32.6%

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

      2. associate-*r* 32.6%

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

      3. *-commutative 32.6%

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

      4. sub-neg 32.6%

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

      5. metadata-eval 32.6%

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

      6. unpow2 32.6%

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

      7. fma-neg 32.6%

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

      8. associate--l+ 32.6%

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

      9. mul-1-neg 32.6%

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

      10. sub-neg 32.6%

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

      11. metadata-eval 32.6%

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

      12. distribute-neg-in 32.6%

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

      13. metadata-eval 32.6%

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

      14. +-commutative 32.6%

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

      15. sub-neg 32.6%

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

      16. metadata-eval 32.6%

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

   7. Simplified 32.6%

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

   8. Taylor expanded in ux around 0 76.6%

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

   if 7.9999998e-5 < ux

   1. Initial program 89.6%

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

      1. associate-*l* 89.6%

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

      2. sub-neg 89.6%

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

      3. +-commutative 89.6%

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

      4. distribute-rgt-neg-in 89.6%

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

      5. fma-def 89.6%

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

      6. +-commutative 89.6%

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

      7. associate-+r- 89.7%

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

      8. fma-def 89.7%

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

      9. neg-sub0 89.7%

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

      10. +-commutative 89.7%

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

      11. associate-+r- 89.6%

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

      12. associate--r- 89.6%

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

      13. neg-sub0 89.6%

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

      14. +-commutative 89.6%

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

      15. sub-neg 89.6%

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

      16. fma-def 89.6%

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

   3. Simplified 89.6%

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

   4. Taylor expanded in uy around 0 72.6%

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

      1. flip-+ 72.6%

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

      2. metadata-eval 72.6%

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

      3. associate--l+ 72.6%

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

      4. associate--l+ 72.8%

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

      5. associate--l+ 72.8%

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

   6. Applied egg-rr 72.8%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \leq 7.999999797903001 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1 + \left(\left(\left(1 + maxCos \cdot ux\right) - ux\right) \cdot \left(-1 + \left(ux - maxCos \cdot ux\right)\right)\right) \cdot \left(\left(ux + \left(-1 - maxCos \cdot ux\right)\right) \cdot \left(-1 + \left(ux - maxCos \cdot ux\right)\right)\right)}{1 + \left(ux + \left(-1 - maxCos \cdot ux\right)\right) \cdot \left(-1 + \left(ux - maxCos \cdot ux\right)\right)}}\\ \end{array} \]

Alternative 10: 74.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ux \leq 9.999999747378752 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= ux 9.999999747378752e-5)
   (sqrt (* ux (- 2.0 (* 2.0 maxCos))))
   (sqrt
    (- 1.0 (* (+ 1.0 (* ux (+ maxCos -1.0))) (- (+ 1.0 (* maxCos ux)) ux))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (ux <= 9.999999747378752e-5f) {
		tmp = sqrtf((ux * (2.0f - (2.0f * maxCos))));
	} else {
		tmp = sqrtf((1.0f - ((1.0f + (ux * (maxCos + -1.0f))) * ((1.0f + (maxCos * ux)) - ux))));
	}
	return tmp;
}

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    real(4) :: tmp
    if (ux <= 9.999999747378752e-5) then
        tmp = sqrt((ux * (2.0e0 - (2.0e0 * maxcos))))
    else
        tmp = sqrt((1.0e0 - ((1.0e0 + (ux * (maxcos + (-1.0e0)))) * ((1.0e0 + (maxcos * ux)) - ux))))
    end if
    code = tmp
end function

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (ux <= Float32(9.999999747378752e-5))
		tmp = sqrt(Float32(ux * Float32(Float32(2.0) - Float32(Float32(2.0) * maxCos))));
	else
		tmp = sqrt(Float32(Float32(1.0) - Float32(Float32(Float32(1.0) + Float32(ux * Float32(maxCos + Float32(-1.0)))) * Float32(Float32(Float32(1.0) + Float32(maxCos * ux)) - ux))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (ux <= single(9.999999747378752e-5))
		tmp = sqrt((ux * (single(2.0) - (single(2.0) * maxCos))));
	else
		tmp = sqrt((single(1.0) - ((single(1.0) + (ux * (maxCos + single(-1.0)))) * ((single(1.0) + (maxCos * ux)) - ux))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if ux < 9.99999975e-5

   1. Initial program 33.1%

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

      1. associate-*l* 33.1%

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

      2. sub-neg 33.1%

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

      3. +-commutative 33.1%

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

      4. distribute-rgt-neg-in 33.1%

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

      5. fma-def 33.1%

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

      6. +-commutative 33.1%

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

      7. associate-+r- 33.1%

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

      8. fma-def 33.1%

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

      9. neg-sub0 33.1%

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

      10. +-commutative 33.1%

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

      11. associate-+r- 33.1%

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

      12. associate--r- 33.1%

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

      13. neg-sub0 33.1%

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

      14. +-commutative 33.1%

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

      15. sub-neg 33.1%

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

      16. fma-def 33.1%

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

   3. Simplified 33.1%

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

   4. Taylor expanded in uy around 0 29.8%

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

   5. Taylor expanded in ux around 0 32.8%

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

      1. associate--l+ 32.8%

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

      2. associate-*r* 32.8%

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

      3. *-commutative 32.8%

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

      4. sub-neg 32.8%

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

      5. metadata-eval 32.8%

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

      6. unpow2 32.8%

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

      7. fma-neg 32.8%

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

      8. associate--l+ 32.8%

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

      9. mul-1-neg 32.8%

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

      10. sub-neg 32.8%

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

      11. metadata-eval 32.8%

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

      12. distribute-neg-in 32.8%

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

      13. metadata-eval 32.8%

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

      14. +-commutative 32.8%

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

      15. sub-neg 32.8%

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

      16. metadata-eval 32.8%

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

   7. Simplified 32.8%

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

   8. Taylor expanded in ux around 0 76.6%

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

   if 9.99999975e-5 < ux

   1. Initial program 89.8%

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

      1. associate-*l* 89.8%

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

      2. sub-neg 89.8%

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

      3. +-commutative 89.8%

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

      4. distribute-rgt-neg-in 89.8%

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

      5. fma-def 89.8%

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

      6. +-commutative 89.8%

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

      7. associate-+r- 89.8%

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

      8. fma-def 89.8%

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

      9. neg-sub0 89.8%

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

      10. +-commutative 89.8%

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

      11. associate-+r- 89.7%

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

      12. associate--r- 89.7%

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

      13. neg-sub0 89.7%

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

      14. +-commutative 89.7%

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

      15. sub-neg 89.7%

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

      16. fma-def 89.7%

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

   3. Simplified 89.7%

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

   4. Taylor expanded in uy around 0 72.6%

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

   5. Taylor expanded in ux around -inf 72.8%

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

      1. mul-1-neg 72.8%

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

      2. unsub-neg 72.8%

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

      3. mul-1-neg 72.8%

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

      4. sub-neg 72.8%

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

   7. Simplified 72.8%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \leq 9.999999747378752 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)}\\ \end{array} \]

Alternative 11: 73.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ux \leq 9.999999747378752 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \left(1 - ux\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= ux 9.999999747378752e-5)
   (sqrt (* ux (- 2.0 (* 2.0 maxCos))))
   (sqrt (- 1.0 (* (- 1.0 ux) (- (+ 1.0 (* maxCos ux)) ux))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (ux <= 9.999999747378752e-5f) {
		tmp = sqrtf((ux * (2.0f - (2.0f * maxCos))));
	} else {
		tmp = sqrtf((1.0f - ((1.0f - ux) * ((1.0f + (maxCos * ux)) - ux))));
	}
	return tmp;
}

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    real(4) :: tmp
    if (ux <= 9.999999747378752e-5) then
        tmp = sqrt((ux * (2.0e0 - (2.0e0 * maxcos))))
    else
        tmp = sqrt((1.0e0 - ((1.0e0 - ux) * ((1.0e0 + (maxcos * ux)) - ux))))
    end if
    code = tmp
end function

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (ux <= Float32(9.999999747378752e-5))
		tmp = sqrt(Float32(ux * Float32(Float32(2.0) - Float32(Float32(2.0) * maxCos))));
	else
		tmp = sqrt(Float32(Float32(1.0) - Float32(Float32(Float32(1.0) - ux) * Float32(Float32(Float32(1.0) + Float32(maxCos * ux)) - ux))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (ux <= single(9.999999747378752e-5))
		tmp = sqrt((ux * (single(2.0) - (single(2.0) * maxCos))));
	else
		tmp = sqrt((single(1.0) - ((single(1.0) - ux) * ((single(1.0) + (maxCos * ux)) - ux))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if ux < 9.99999975e-5

   1. Initial program 33.1%

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

      1. associate-*l* 33.1%

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

      2. sub-neg 33.1%

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

      3. +-commutative 33.1%

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

      4. distribute-rgt-neg-in 33.1%

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

      5. fma-def 33.1%

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

      6. +-commutative 33.1%

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

      7. associate-+r- 33.1%

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

      8. fma-def 33.1%

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

      9. neg-sub0 33.1%

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

      10. +-commutative 33.1%

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

      11. associate-+r- 33.1%

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

      12. associate--r- 33.1%

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

      13. neg-sub0 33.1%

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

      14. +-commutative 33.1%

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

      15. sub-neg 33.1%

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

      16. fma-def 33.1%

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

   3. Simplified 33.1%

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

   4. Taylor expanded in uy around 0 29.8%

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

   5. Taylor expanded in ux around 0 32.8%

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

      1. associate--l+ 32.8%

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

      2. associate-*r* 32.8%

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

      3. *-commutative 32.8%

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

      4. sub-neg 32.8%

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

      5. metadata-eval 32.8%

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

      6. unpow2 32.8%

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

      7. fma-neg 32.8%

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

      8. associate--l+ 32.8%

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

      9. mul-1-neg 32.8%

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

      10. sub-neg 32.8%

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

      11. metadata-eval 32.8%

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

      12. distribute-neg-in 32.8%

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

      13. metadata-eval 32.8%

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

      14. +-commutative 32.8%

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

      15. sub-neg 32.8%

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

      16. metadata-eval 32.8%

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

   7. Simplified 32.8%

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

   8. Taylor expanded in ux around 0 76.6%

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

   if 9.99999975e-5 < ux

   1. Initial program 89.8%

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

      1. associate-*l* 89.8%

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

      2. sub-neg 89.8%

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

      3. +-commutative 89.8%

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

      4. distribute-rgt-neg-in 89.8%

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

      5. fma-def 89.8%

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

      6. +-commutative 89.8%

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

      7. associate-+r- 89.8%

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

      8. fma-def 89.8%

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

      9. neg-sub0 89.8%

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

      10. +-commutative 89.8%

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

      11. associate-+r- 89.7%

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

      12. associate--r- 89.7%

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

      13. neg-sub0 89.7%

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

      14. +-commutative 89.7%

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

      15. sub-neg 89.7%

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

      16. fma-def 89.7%

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

   3. Simplified 89.7%

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

   4. Taylor expanded in uy around 0 72.6%

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

   5. Taylor expanded in maxCos around 0 69.8%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \leq 9.999999747378752 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \left(1 - ux\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)}\\ \end{array} \]

Alternative 12: 73.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ux \leq 9.999999747378752 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \left(ux + -1\right) \cdot \left(ux + -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= ux 9.999999747378752e-5)
   (sqrt (* ux (- 2.0 (* 2.0 maxCos))))
   (sqrt (- 1.0 (* (+ ux -1.0) (+ ux -1.0))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (ux <= 9.999999747378752e-5f) {
		tmp = sqrtf((ux * (2.0f - (2.0f * maxCos))));
	} else {
		tmp = sqrtf((1.0f - ((ux + -1.0f) * (ux + -1.0f))));
	}
	return tmp;
}

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    real(4) :: tmp
    if (ux <= 9.999999747378752e-5) then
        tmp = sqrt((ux * (2.0e0 - (2.0e0 * maxcos))))
    else
        tmp = sqrt((1.0e0 - ((ux + (-1.0e0)) * (ux + (-1.0e0)))))
    end if
    code = tmp
end function

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (ux <= Float32(9.999999747378752e-5))
		tmp = sqrt(Float32(ux * Float32(Float32(2.0) - Float32(Float32(2.0) * maxCos))));
	else
		tmp = sqrt(Float32(Float32(1.0) - Float32(Float32(ux + Float32(-1.0)) * Float32(ux + Float32(-1.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (ux <= single(9.999999747378752e-5))
		tmp = sqrt((ux * (single(2.0) - (single(2.0) * maxCos))));
	else
		tmp = sqrt((single(1.0) - ((ux + single(-1.0)) * (ux + single(-1.0)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if ux < 9.99999975e-5

   1. Initial program 33.1%

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

      1. associate-*l* 33.1%

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

      2. sub-neg 33.1%

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

      3. +-commutative 33.1%

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

      4. distribute-rgt-neg-in 33.1%

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

      5. fma-def 33.1%

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

      6. +-commutative 33.1%

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

      7. associate-+r- 33.1%

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

      8. fma-def 33.1%

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

      9. neg-sub0 33.1%

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

      10. +-commutative 33.1%

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

      11. associate-+r- 33.1%

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

      12. associate--r- 33.1%

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

      13. neg-sub0 33.1%

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

      14. +-commutative 33.1%

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

      15. sub-neg 33.1%

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

      16. fma-def 33.1%

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

   3. Simplified 33.1%

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

   4. Taylor expanded in uy around 0 29.8%

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

   5. Taylor expanded in ux around 0 32.8%

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

      1. associate--l+ 32.8%

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

      2. associate-*r* 32.8%

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

      3. *-commutative 32.8%

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

      4. sub-neg 32.8%

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

      5. metadata-eval 32.8%

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

      6. unpow2 32.8%

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

      7. fma-neg 32.8%

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

      8. associate--l+ 32.8%

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

      9. mul-1-neg 32.8%

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

      10. sub-neg 32.8%

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

      11. metadata-eval 32.8%

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

      12. distribute-neg-in 32.8%

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

      13. metadata-eval 32.8%

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

      14. +-commutative 32.8%

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

      15. sub-neg 32.8%

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

      16. metadata-eval 32.8%

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

   7. Simplified 32.8%

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

   8. Taylor expanded in ux around 0 76.6%

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

   if 9.99999975e-5 < ux

   1. Initial program 89.8%

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

      1. associate-*l* 89.8%

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

      2. sub-neg 89.8%

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

      3. +-commutative 89.8%

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

      4. distribute-rgt-neg-in 89.8%

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

      5. fma-def 89.8%

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

      6. +-commutative 89.8%

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

      7. associate-+r- 89.8%

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

      8. fma-def 89.8%

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

      9. neg-sub0 89.8%

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

      10. +-commutative 89.8%

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

      11. associate-+r- 89.7%

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

      12. associate--r- 89.7%

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

      13. neg-sub0 89.7%

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

      14. +-commutative 89.7%

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

      15. sub-neg 89.7%

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

      16. fma-def 89.7%

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

   3. Simplified 89.7%

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

   4. Taylor expanded in uy around 0 72.6%

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

   5. Taylor expanded in maxCos around 0 69.5%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \leq 9.999999747378752 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \left(ux + -1\right) \cdot \left(ux + -1\right)}\\ \end{array} \]

Alternative 13: 64.7% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 55.7%

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

   1. associate-*l* 55.7%

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

   2. sub-neg 55.7%

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

   3. +-commutative 55.7%

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

   4. distribute-rgt-neg-in 55.7%

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

   5. fma-def 55.7%

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

   6. +-commutative 55.7%

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

   7. associate-+r- 55.7%

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

   8. fma-def 55.7%

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

   9. neg-sub0 55.7%

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

   10. +-commutative 55.7%

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

   11. associate-+r- 55.6%

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

   12. associate--r- 55.6%

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

   13. neg-sub0 55.6%

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

   14. +-commutative 55.6%

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

   15. sub-neg 55.6%

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

   16. fma-def 55.6%

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

3. Simplified 55.6%

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

4. Taylor expanded in uy around 0 46.9%

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

5. Taylor expanded in ux around 0 49.2%

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

   1. associate--l+ 49.0%

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

   2. associate-*r* 49.0%

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

   3. *-commutative 49.0%

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

   4. sub-neg 49.0%

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

   5. metadata-eval 49.0%

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

   6. unpow2 49.0%

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

   7. fma-neg 49.0%

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

   8. associate--l+ 49.0%

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

   9. mul-1-neg 49.0%

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

   10. sub-neg 49.0%

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

   11. metadata-eval 49.0%

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

   12. distribute-neg-in 49.0%

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

   13. metadata-eval 49.0%

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

   14. +-commutative 49.0%

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

   15. sub-neg 49.0%

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

   16. metadata-eval 49.0%

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

7. Simplified 49.0%

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

8. Taylor expanded in ux around 0 64.2%

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

9. Final simplification 64.2%

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

Alternative 14: 62.1% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 55.7%

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

   1. associate-*l* 55.7%

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

   2. sub-neg 55.7%

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

   3. +-commutative 55.7%

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

   4. distribute-rgt-neg-in 55.7%

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

   5. fma-def 55.7%

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

   6. +-commutative 55.7%

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

   7. associate-+r- 55.7%

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

   8. fma-def 55.7%

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

   9. neg-sub0 55.7%

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

   10. +-commutative 55.7%

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

   11. associate-+r- 55.6%

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

   12. associate--r- 55.6%

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

   13. neg-sub0 55.6%

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

   14. +-commutative 55.6%

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

   15. sub-neg 55.6%

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

   16. fma-def 55.6%

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

3. Simplified 55.6%

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

4. Taylor expanded in uy around 0 46.9%

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

5. Taylor expanded in ux around 0 49.2%

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

   1. associate--l+ 49.0%

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

   2. associate-*r* 49.0%

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

   3. *-commutative 49.0%

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

   4. sub-neg 49.0%

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

   5. metadata-eval 49.0%

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

   6. unpow2 49.0%

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

   7. fma-neg 49.0%

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

   8. associate--l+ 49.0%

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

   9. mul-1-neg 49.0%

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

   10. sub-neg 49.0%

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

   11. metadata-eval 49.0%

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

   12. distribute-neg-in 49.0%

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

   13. metadata-eval 49.0%

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

   14. +-commutative 49.0%

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

   15. sub-neg 49.0%

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

   16. metadata-eval 49.0%

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

7. Simplified 49.0%

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

8. Taylor expanded in ux around 0 64.2%

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

9. Taylor expanded in maxCos around 0 62.3%

   \[\leadsto \sqrt{\color{blue}{2 \cdot ux}} \]
10. Step-by-step derivation

    1. *-commutative 62.3%

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

11. Simplified 62.3%

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

12. Final simplification 62.3%

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

Reproduce

herbie shell --seed 2023255 
(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)))))))