UniformSampleCone, x

Percentage Accurate: 57.3% → 99.0%

Time: 18.2s

Alternatives: 18

Speedup: 2.2×

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: 57.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - ux\right) + ux \cdot maxCos\\ \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - t\_0 \cdot t\_0} \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.1%

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

   1. associate-*l* 57.1%

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

   2. sub-neg 57.1%

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

   3. +-commutative 57.1%

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

   4. distribute-rgt-neg-in 57.1%

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

   5. fma-def 57.2%

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

3. Simplified 57.3%

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

5. Taylor expanded in ux around 0 99.1%

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

   1. fma-def 99.1%

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

   2. +-commutative 99.1%

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

   3. sub-neg 99.1%

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

   4. metadata-eval 99.1%

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

   5. +-commutative 99.1%

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

   6. distribute-lft-in 99.1%

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

   7. metadata-eval 99.1%

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

   8. associate--l+ 99.1%

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

   9. mul-1-neg 99.1%

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

   10. sub-neg 99.1%

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

   11. associate-*r* 99.2%

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

   12. sub-neg 99.2%

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

   13. metadata-eval 99.2%

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

   14. +-commutative 99.2%

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

7. Simplified 99.2%

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

8. Final simplification 99.2%

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

Alternative 2: 99.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.1%

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

   1. associate-*l* 57.1%

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

   2. sub-neg 57.1%

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

   3. +-commutative 57.1%

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

   4. distribute-rgt-neg-in 57.1%

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

   5. fma-def 57.2%

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

3. Simplified 57.3%

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

5. Taylor expanded in ux around 0 99.1%

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

   1. fma-def 99.1%

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

   2. +-commutative 99.1%

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

   3. sub-neg 99.1%

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

   4. metadata-eval 99.1%

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

   5. +-commutative 99.1%

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

   6. distribute-lft-in 99.1%

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

   7. metadata-eval 99.1%

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

   8. associate--l+ 99.1%

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

   9. mul-1-neg 99.1%

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

   10. sub-neg 99.1%

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

   11. associate-*r* 99.2%

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

   12. sub-neg 99.2%

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

   13. metadata-eval 99.2%

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

   14. +-commutative 99.2%

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

7. Simplified 99.2%

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

8. Taylor expanded in uy around inf 99.1%

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

10. Simplified 99.1%

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

11. Final simplification 99.1%

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

Alternative 3: 99.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.1%

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

   1. associate-*l* 57.1%

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

   2. sub-neg 57.1%

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

   3. +-commutative 57.1%

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

   4. distribute-rgt-neg-in 57.1%

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

   5. fma-def 57.2%

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

3. Simplified 57.3%

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

5. Taylor expanded in ux around 0 99.1%

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

   1. fma-def 99.1%

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

   2. +-commutative 99.1%

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

   3. sub-neg 99.1%

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

   4. metadata-eval 99.1%

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

   5. +-commutative 99.1%

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

   6. distribute-lft-in 99.1%

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

   7. metadata-eval 99.1%

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

   8. associate--l+ 99.1%

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

   9. mul-1-neg 99.1%

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

   10. sub-neg 99.1%

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

   11. associate-*r* 99.2%

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

   12. sub-neg 99.2%

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

   13. metadata-eval 99.2%

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

   14. +-commutative 99.2%

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

7. Simplified 99.2%

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

8. Taylor expanded in uy around inf 99.1%

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

9. Final simplification 99.1%

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

Alternative 4: 96.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;maxCos \leq 3.999999989900971 \cdot 10^{-6}:\\ \;\;\;\;\cos \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux - {ux}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right) + {ux}^{2} \cdot \left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= maxCos 3.999999989900971e-6)
   (* (cos (* 2.0 (* uy PI))) (sqrt (- (* 2.0 ux) (pow ux 2.0))))
   (sqrt
    (+
     (* ux (- 2.0 (* 2.0 maxCos)))
     (* (pow ux 2.0) (* (+ maxCos -1.0) (- 1.0 maxCos)))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (maxCos <= 3.999999989900971e-6f) {
		tmp = cosf((2.0f * (uy * ((float) M_PI)))) * sqrtf(((2.0f * ux) - powf(ux, 2.0f)));
	} else {
		tmp = sqrtf(((ux * (2.0f - (2.0f * maxCos))) + (powf(ux, 2.0f) * ((maxCos + -1.0f) * (1.0f - maxCos)))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (maxCos <= Float32(3.999999989900971e-6))
		tmp = Float32(cos(Float32(Float32(2.0) * Float32(uy * Float32(pi)))) * sqrt(Float32(Float32(Float32(2.0) * ux) - (ux ^ Float32(2.0)))));
	else
		tmp = sqrt(Float32(Float32(ux * Float32(Float32(2.0) - Float32(Float32(2.0) * maxCos))) + Float32((ux ^ Float32(2.0)) * Float32(Float32(maxCos + Float32(-1.0)) * Float32(Float32(1.0) - maxCos)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (maxCos <= single(3.999999989900971e-6))
		tmp = cos((single(2.0) * (uy * single(pi)))) * sqrt(((single(2.0) * ux) - (ux ^ single(2.0))));
	else
		tmp = sqrt(((ux * (single(2.0) - (single(2.0) * maxCos))) + ((ux ^ single(2.0)) * ((maxCos + single(-1.0)) * (single(1.0) - maxCos)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if maxCos < 3.99999999e-6

   1. Initial program 59.1%

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

      1. associate-*l* 59.1%

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

      2. sub-neg 59.1%

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

      3. +-commutative 59.1%

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

      4. distribute-rgt-neg-in 59.1%

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

      5. fma-def 59.2%

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

   3. Simplified 59.2%

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

   5. Taylor expanded in ux around 0 99.1%

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

      1. fma-def 99.1%

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

      2. +-commutative 99.1%

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

      3. sub-neg 99.1%

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

      4. metadata-eval 99.1%

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

      5. +-commutative 99.1%

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

      6. distribute-lft-in 99.1%

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

      7. metadata-eval 99.1%

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

      8. associate--l+ 99.1%

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

      9. mul-1-neg 99.1%

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

      10. sub-neg 99.1%

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

      11. associate-*r* 99.1%

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

      12. sub-neg 99.1%

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

      13. metadata-eval 99.1%

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

      14. +-commutative 99.1%

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

   7. Simplified 99.1%

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

      1. add-exp-log 99.1%

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

   9. Applied egg-rr 99.1%

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

   10. Taylor expanded in maxCos around 0 98.8%

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

       1. +-commutative 98.8%

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

       2. neg-mul-1 98.8%

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

       3. unsub-neg 98.8%

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

   12. Simplified 98.8%

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

   if 3.99999999e-6 < maxCos

   1. Initial program 46.7%

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

      1. associate-*l* 46.7%

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

      2. sub-neg 46.7%

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

      3. +-commutative 46.7%

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

      4. distribute-rgt-neg-in 46.7%

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

      5. fma-def 46.7%

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

   3. Simplified 47.2%

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

   5. Taylor expanded in ux around 0 99.1%

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

      1. fma-def 99.1%

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

      2. +-commutative 99.1%

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

      3. sub-neg 99.1%

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

      4. metadata-eval 99.1%

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

      5. +-commutative 99.1%

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

      6. distribute-lft-in 99.1%

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

      7. metadata-eval 99.1%

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

      8. associate--l+ 99.2%

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

      9. mul-1-neg 99.2%

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

      10. sub-neg 99.2%

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

      11. associate-*r* 99.3%

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

      12. sub-neg 99.3%

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

      13. metadata-eval 99.3%

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

      14. +-commutative 99.3%

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

   7. Simplified 99.3%

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

   8. Taylor expanded in uy around 0 86.9%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;maxCos \leq 3.999999989900971 \cdot 10^{-6}:\\ \;\;\;\;\cos \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux - {ux}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right) + {ux}^{2} \cdot \left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.1%

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

   1. associate-*l* 57.1%

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

   2. sub-neg 57.1%

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

   3. +-commutative 57.1%

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

   4. distribute-rgt-neg-in 57.1%

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

   5. fma-def 57.2%

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

3. Simplified 57.3%

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

5. Taylor expanded in ux around 0 99.1%

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

   1. fma-def 99.1%

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

   2. +-commutative 99.1%

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

   3. sub-neg 99.1%

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

   4. metadata-eval 99.1%

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

   5. +-commutative 99.1%

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

   6. distribute-lft-in 99.1%

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

   7. metadata-eval 99.1%

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

   8. associate--l+ 99.1%

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

   9. mul-1-neg 99.1%

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

   10. sub-neg 99.1%

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

   11. associate-*r* 99.2%

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

   12. sub-neg 99.2%

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

   13. metadata-eval 99.2%

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

   14. +-commutative 99.2%

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

7. Simplified 99.2%

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

8. Taylor expanded in maxCos around 0 97.6%

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

10. Simplified 97.6%

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

11. Taylor expanded in uy around inf 97.5%

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

    1. unpow2 97.5%

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

    2. distribute-lft-out-- 97.4%

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

    3. cancel-sign-sub-inv 97.4%

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

    4. metadata-eval 97.4%

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

    5. +-commutative 97.4%

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

    6. *-commutative 97.4%

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

    7. fma-def 97.4%

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

13. Simplified 97.4%

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

14. Final simplification 97.4%

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

Alternative 6: 96.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;maxCos \leq 3.999999989900971 \cdot 10^{-6}:\\ \;\;\;\;\cos \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right) + {ux}^{2} \cdot \left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= maxCos 3.999999989900971e-6)
   (* (cos (* 2.0 (* uy PI))) (sqrt (* ux (- 2.0 ux))))
   (sqrt
    (+
     (* ux (- 2.0 (* 2.0 maxCos)))
     (* (pow ux 2.0) (* (+ maxCos -1.0) (- 1.0 maxCos)))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (maxCos <= 3.999999989900971e-6f) {
		tmp = cosf((2.0f * (uy * ((float) M_PI)))) * sqrtf((ux * (2.0f - ux)));
	} else {
		tmp = sqrtf(((ux * (2.0f - (2.0f * maxCos))) + (powf(ux, 2.0f) * ((maxCos + -1.0f) * (1.0f - maxCos)))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (maxCos <= Float32(3.999999989900971e-6))
		tmp = Float32(cos(Float32(Float32(2.0) * Float32(uy * Float32(pi)))) * sqrt(Float32(ux * Float32(Float32(2.0) - ux))));
	else
		tmp = sqrt(Float32(Float32(ux * Float32(Float32(2.0) - Float32(Float32(2.0) * maxCos))) + Float32((ux ^ Float32(2.0)) * Float32(Float32(maxCos + Float32(-1.0)) * Float32(Float32(1.0) - maxCos)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (maxCos <= single(3.999999989900971e-6))
		tmp = cos((single(2.0) * (uy * single(pi)))) * sqrt((ux * (single(2.0) - ux)));
	else
		tmp = sqrt(((ux * (single(2.0) - (single(2.0) * maxCos))) + ((ux ^ single(2.0)) * ((maxCos + single(-1.0)) * (single(1.0) - maxCos)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if maxCos < 3.99999999e-6

   1. Initial program 59.1%

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

      1. associate-*l* 59.1%

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

      2. sub-neg 59.1%

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

      3. +-commutative 59.1%

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

      4. distribute-rgt-neg-in 59.1%

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

      5. fma-def 59.2%

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

   3. Simplified 59.2%

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

   5. Taylor expanded in ux around 0 99.1%

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

      1. fma-def 99.1%

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

      2. +-commutative 99.1%

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

      3. sub-neg 99.1%

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

      4. metadata-eval 99.1%

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

      5. +-commutative 99.1%

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

      6. distribute-lft-in 99.1%

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

      7. metadata-eval 99.1%

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

      8. associate--l+ 99.1%

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

      9. mul-1-neg 99.1%

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

      10. sub-neg 99.1%

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

      11. associate-*r* 99.1%

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

      12. sub-neg 99.1%

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

      13. metadata-eval 99.1%

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

      14. +-commutative 99.1%

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

   7. Simplified 99.1%

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

   8. Taylor expanded in uy around inf 99.1%

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

   10. Simplified 99.1%

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

   11. Taylor expanded in maxCos around 0 98.8%

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

       1. mul-1-neg 98.8%

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

       2. +-commutative 98.8%

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

       3. sub-neg 98.8%

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

       4. unpow2 98.8%

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

       5. distribute-rgt-out-- 98.6%

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

   13. Simplified 98.6%

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

   if 3.99999999e-6 < maxCos

   1. Initial program 46.7%

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

      1. associate-*l* 46.7%

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

      2. sub-neg 46.7%

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

      3. +-commutative 46.7%

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

      4. distribute-rgt-neg-in 46.7%

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

      5. fma-def 46.7%

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

   3. Simplified 47.2%

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

   5. Taylor expanded in ux around 0 99.1%

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

      1. fma-def 99.1%

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

      2. +-commutative 99.1%

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

      3. sub-neg 99.1%

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

      4. metadata-eval 99.1%

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

      5. +-commutative 99.1%

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

      6. distribute-lft-in 99.1%

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

      7. metadata-eval 99.1%

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

      8. associate--l+ 99.2%

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

      9. mul-1-neg 99.2%

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

      10. sub-neg 99.2%

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

      11. associate-*r* 99.3%

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

      12. sub-neg 99.3%

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

      13. metadata-eval 99.3%

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

      14. +-commutative 99.3%

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

   7. Simplified 99.3%

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

   8. Taylor expanded in uy around 0 86.9%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;maxCos \leq 3.999999989900971 \cdot 10^{-6}:\\ \;\;\;\;\cos \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right) + {ux}^{2} \cdot \left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 94.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ux \leq 2.499999993688107 \cdot 10^{-7}:\\ \;\;\;\;\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(\left(1 + \left(1 - maxCos\right)\right) - maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= ux 2.499999993688107e-7)
   (* (cos (* uy (* 2.0 PI))) (sqrt (* ux (- (+ 1.0 (- 1.0 maxCos)) maxCos))))
   (* (cos (* 2.0 (* uy PI))) (sqrt (* ux (- 2.0 ux))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (ux <= 2.499999993688107e-7f) {
		tmp = cosf((uy * (2.0f * ((float) M_PI)))) * sqrtf((ux * ((1.0f + (1.0f - maxCos)) - maxCos)));
	} else {
		tmp = cosf((2.0f * (uy * ((float) M_PI)))) * sqrtf((ux * (2.0f - ux)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (ux <= Float32(2.499999993688107e-7))
		tmp = Float32(cos(Float32(uy * Float32(Float32(2.0) * Float32(pi)))) * sqrt(Float32(ux * Float32(Float32(Float32(1.0) + Float32(Float32(1.0) - maxCos)) - maxCos))));
	else
		tmp = Float32(cos(Float32(Float32(2.0) * Float32(uy * Float32(pi)))) * sqrt(Float32(ux * Float32(Float32(2.0) - ux))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (ux <= single(2.499999993688107e-7))
		tmp = cos((uy * (single(2.0) * single(pi)))) * sqrt((ux * ((single(1.0) + (single(1.0) - maxCos)) - maxCos)));
	else
		tmp = cos((single(2.0) * (uy * single(pi)))) * sqrt((ux * (single(2.0) - ux)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if ux < 2.49999999e-7

   1. Initial program 14.7%

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

      1. associate-*l* 14.7%

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

      2. sub-neg 14.7%

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

      3. +-commutative 14.7%

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

      4. distribute-rgt-neg-in 14.7%

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

      5. fma-def 14.7%

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

   3. Simplified 14.7%

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

   5. Taylor expanded in ux around 0 99.2%

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

      1. mul-1-neg 99.2%

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

      2. sub-neg 99.2%

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

      3. metadata-eval 99.2%

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

      4. +-commutative 99.2%

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

   7. Simplified 99.2%

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

   if 2.49999999e-7 < ux

   1. Initial program 76.7%

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

      1. associate-*l* 76.7%

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

      2. sub-neg 76.7%

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

      3. +-commutative 76.7%

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

      4. distribute-rgt-neg-in 76.7%

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

      5. fma-def 76.9%

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

   3. Simplified 77.0%

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

   5. Taylor expanded in ux around 0 99.1%

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

      1. fma-def 99.1%

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

      2. +-commutative 99.1%

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

      3. sub-neg 99.1%

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

      4. metadata-eval 99.1%

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

      5. +-commutative 99.1%

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

      6. distribute-lft-in 99.1%

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

      7. metadata-eval 99.1%

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

      8. associate--l+ 99.1%

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

      9. mul-1-neg 99.1%

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

      10. sub-neg 99.1%

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

      11. associate-*r* 99.1%

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

      12. sub-neg 99.1%

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

      13. metadata-eval 99.1%

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

      14. +-commutative 99.1%

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

   7. Simplified 99.1%

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

   8. Taylor expanded in uy around inf 99.1%

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

   10. Simplified 99.1%

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

   11. Taylor expanded in maxCos around 0 94.6%

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

       1. mul-1-neg 94.6%

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

       2. +-commutative 94.6%

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

       3. sub-neg 94.6%

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

       4. unpow2 94.6%

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

       5. distribute-rgt-out-- 94.4%

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

   13. Simplified 94.4%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \leq 2.499999993688107 \cdot 10^{-7}:\\ \;\;\;\;\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(\left(1 + \left(1 - maxCos\right)\right) - maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 94.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ux \leq 2.499999993688107 \cdot 10^{-7}:\\ \;\;\;\;\cos \left(\pi \cdot \left(uy \cdot 2\right)\right) \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= ux 2.499999993688107e-7)
   (* (cos (* PI (* uy 2.0))) (sqrt (* ux (- 2.0 (* 2.0 maxCos)))))
   (* (cos (* 2.0 (* uy PI))) (sqrt (* ux (- 2.0 ux))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (ux <= 2.499999993688107e-7f) {
		tmp = cosf((((float) M_PI) * (uy * 2.0f))) * sqrtf((ux * (2.0f - (2.0f * maxCos))));
	} else {
		tmp = cosf((2.0f * (uy * ((float) M_PI)))) * sqrtf((ux * (2.0f - ux)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (ux <= Float32(2.499999993688107e-7))
		tmp = Float32(cos(Float32(Float32(pi) * Float32(uy * Float32(2.0)))) * sqrt(Float32(ux * Float32(Float32(2.0) - Float32(Float32(2.0) * maxCos)))));
	else
		tmp = Float32(cos(Float32(Float32(2.0) * Float32(uy * Float32(pi)))) * sqrt(Float32(ux * Float32(Float32(2.0) - ux))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (ux <= single(2.499999993688107e-7))
		tmp = cos((single(pi) * (uy * single(2.0)))) * sqrt((ux * (single(2.0) - (single(2.0) * maxCos))));
	else
		tmp = cos((single(2.0) * (uy * single(pi)))) * sqrt((ux * (single(2.0) - ux)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if ux < 2.49999999e-7

   1. Initial program 14.7%

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

   3. Taylor expanded in ux around 0 99.1%

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

   if 2.49999999e-7 < ux

   1. Initial program 76.7%

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

      1. associate-*l* 76.7%

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

      2. sub-neg 76.7%

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

      3. +-commutative 76.7%

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

      4. distribute-rgt-neg-in 76.7%

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

      5. fma-def 76.9%

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

   3. Simplified 77.0%

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

   5. Taylor expanded in ux around 0 99.1%

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

      1. fma-def 99.1%

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

      2. +-commutative 99.1%

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

      3. sub-neg 99.1%

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

      4. metadata-eval 99.1%

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

      5. +-commutative 99.1%

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

      6. distribute-lft-in 99.1%

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

      7. metadata-eval 99.1%

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

      8. associate--l+ 99.1%

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

      9. mul-1-neg 99.1%

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

      10. sub-neg 99.1%

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

      11. associate-*r* 99.1%

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

      12. sub-neg 99.1%

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

      13. metadata-eval 99.1%

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

      14. +-commutative 99.1%

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

   7. Simplified 99.1%

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

   8. Taylor expanded in uy around inf 99.1%

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

   10. Simplified 99.1%

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

   11. Taylor expanded in maxCos around 0 94.6%

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

       1. mul-1-neg 94.6%

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

       2. +-commutative 94.6%

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

       3. sub-neg 94.6%

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

       4. unpow2 94.6%

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

       5. distribute-rgt-out-- 94.4%

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

   13. Simplified 94.4%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \leq 2.499999993688107 \cdot 10^{-7}:\\ \;\;\;\;\cos \left(\pi \cdot \left(uy \cdot 2\right)\right) \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 88.7% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* uy 2.0) 0.003000000026077032)
   (sqrt (- (* ux (- 2.0 (* 2.0 maxCos))) (pow ux 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) <= 0.003000000026077032f) {
		tmp = sqrtf(((ux * (2.0f - (2.0f * maxCos))) - powf(ux, 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(0.003000000026077032))
		tmp = sqrt(Float32(Float32(ux * Float32(Float32(2.0) - Float32(Float32(2.0) * maxCos))) - (ux ^ 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(0.003000000026077032))
		tmp = sqrt(((ux * (single(2.0) - (single(2.0) * maxCos))) - (ux ^ 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) < 0.00300000003

   1. Initial program 57.9%

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

      1. associate-*l* 57.9%

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

      2. sub-neg 57.9%

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

      3. +-commutative 57.9%

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

      4. distribute-rgt-neg-in 57.9%

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

      5. fma-def 57.9%

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

   3. Simplified 58.0%

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

   5. Taylor expanded in ux around 0 99.4%

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

      1. fma-def 99.5%

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

      2. +-commutative 99.5%

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

      3. sub-neg 99.5%

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

      4. metadata-eval 99.5%

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

      5. +-commutative 99.5%

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

      6. distribute-lft-in 99.5%

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

      7. metadata-eval 99.5%

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

      8. associate--l+ 99.5%

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

      9. mul-1-neg 99.5%

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

      10. sub-neg 99.5%

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

      11. associate-*r* 99.5%

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

      12. sub-neg 99.5%

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

      13. metadata-eval 99.5%

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

      14. +-commutative 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in maxCos around 0 97.9%

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

   10. Simplified 97.9%

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

   11. Taylor expanded in uy around 0 94.6%

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

   if 0.00300000003 < (*.f32 uy 2)

   1. Initial program 55.1%

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

      1. associate-*l* 55.1%

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

      2. sub-neg 55.1%

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

      3. +-commutative 55.1%

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

      4. distribute-rgt-neg-in 55.1%

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

      5. fma-def 55.4%

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

   3. Simplified 55.5%

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

   5. Taylor expanded in ux around 0 76.7%

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

      1. mul-1-neg 76.7%

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

      2. sub-neg 76.7%

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

      3. metadata-eval 76.7%

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

      4. +-commutative 76.7%

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

   7. Simplified 76.7%

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

   8. Taylor expanded in maxCos around 0 73.5%

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

4. Final simplification 88.4%

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

Alternative 10: 88.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if ((uy * 2.0f) <= 0.003000000026077032f) {
		tmp = sqrtf((ux * (fmaf(maxCos, -2.0f, 2.0f) - ux)));
	} else {
		tmp = cosf((uy * (2.0f * ((float) M_PI)))) * sqrtf((2.0f * ux));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (Float32(uy * Float32(2.0)) <= Float32(0.003000000026077032))
		tmp = sqrt(Float32(ux * Float32(fma(maxCos, Float32(-2.0), Float32(2.0)) - ux)));
	else
		tmp = Float32(cos(Float32(uy * Float32(Float32(2.0) * Float32(pi)))) * sqrt(Float32(Float32(2.0) * ux)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy 2) < 0.00300000003

   1. Initial program 57.9%

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

      1. associate-*l* 57.9%

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

      2. sub-neg 57.9%

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

      3. +-commutative 57.9%

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

      4. distribute-rgt-neg-in 57.9%

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

      5. fma-def 57.9%

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

   3. Simplified 58.0%

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

   5. Taylor expanded in ux around 0 99.4%

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

      1. fma-def 99.5%

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

      2. +-commutative 99.5%

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

      3. sub-neg 99.5%

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

      4. metadata-eval 99.5%

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

      5. +-commutative 99.5%

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

      6. distribute-lft-in 99.5%

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

      7. metadata-eval 99.5%

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

      8. associate--l+ 99.5%

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

      9. mul-1-neg 99.5%

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

      10. sub-neg 99.5%

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

      11. associate-*r* 99.5%

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

      12. sub-neg 99.5%

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

      13. metadata-eval 99.5%

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

      14. +-commutative 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in maxCos around 0 97.9%

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

   10. Simplified 97.9%

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

   11. Taylor expanded in uy around 0 94.6%

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

       1. unpow2 94.6%

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

       2. distribute-lft-out-- 94.5%

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

       3. cancel-sign-sub-inv 94.5%

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

       4. metadata-eval 94.5%

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

       5. +-commutative 94.5%

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

       6. *-commutative 94.5%

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

       7. fma-def 94.5%

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

   13. Simplified 94.5%

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

   if 0.00300000003 < (*.f32 uy 2)

   1. Initial program 55.1%

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

      1. associate-*l* 55.1%

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

      2. sub-neg 55.1%

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

      3. +-commutative 55.1%

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

      4. distribute-rgt-neg-in 55.1%

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

      5. fma-def 55.4%

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

   3. Simplified 55.5%

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

   5. Taylor expanded in ux around 0 76.7%

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

      1. mul-1-neg 76.7%

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

      2. sub-neg 76.7%

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

      3. metadata-eval 76.7%

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

      4. +-commutative 76.7%

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

   7. Simplified 76.7%

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

   8. Taylor expanded in maxCos around 0 73.5%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.003000000026077032:\\ \;\;\;\;\sqrt{ux \cdot \left(\mathsf{fma}\left(maxCos, -2, 2\right) - ux\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 95.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;maxCos \leq 3.999999989900971 \cdot 10^{-6}:\\ \;\;\;\;\cos \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{ux \cdot \left(\mathsf{fma}\left(maxCos, -2, 2\right) - ux\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= maxCos 3.999999989900971e-6)
   (* (cos (* 2.0 (* uy PI))) (sqrt (* ux (- 2.0 ux))))
   (sqrt (* ux (- (fma maxCos -2.0 2.0) ux)))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (maxCos <= 3.999999989900971e-6f) {
		tmp = cosf((2.0f * (uy * ((float) M_PI)))) * sqrtf((ux * (2.0f - ux)));
	} else {
		tmp = sqrtf((ux * (fmaf(maxCos, -2.0f, 2.0f) - ux)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (maxCos <= Float32(3.999999989900971e-6))
		tmp = Float32(cos(Float32(Float32(2.0) * Float32(uy * Float32(pi)))) * sqrt(Float32(ux * Float32(Float32(2.0) - ux))));
	else
		tmp = sqrt(Float32(ux * Float32(fma(maxCos, Float32(-2.0), Float32(2.0)) - ux)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if maxCos < 3.99999999e-6

   1. Initial program 59.1%

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

      1. associate-*l* 59.1%

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

      2. sub-neg 59.1%

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

      3. +-commutative 59.1%

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

      4. distribute-rgt-neg-in 59.1%

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

      5. fma-def 59.2%

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

   3. Simplified 59.2%

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

   5. Taylor expanded in ux around 0 99.1%

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

      1. fma-def 99.1%

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

      2. +-commutative 99.1%

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

      3. sub-neg 99.1%

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

      4. metadata-eval 99.1%

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

      5. +-commutative 99.1%

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

      6. distribute-lft-in 99.1%

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

      7. metadata-eval 99.1%

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

      8. associate--l+ 99.1%

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

      9. mul-1-neg 99.1%

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

      10. sub-neg 99.1%

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

      11. associate-*r* 99.1%

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

      12. sub-neg 99.1%

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

      13. metadata-eval 99.1%

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

      14. +-commutative 99.1%

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

   7. Simplified 99.1%

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

   8. Taylor expanded in uy around inf 99.1%

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

   10. Simplified 99.1%

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

   11. Taylor expanded in maxCos around 0 98.8%

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

       1. mul-1-neg 98.8%

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

       2. +-commutative 98.8%

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

       3. sub-neg 98.8%

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

       4. unpow2 98.8%

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

       5. distribute-rgt-out-- 98.6%

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

   13. Simplified 98.6%

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

   if 3.99999999e-6 < maxCos

   1. Initial program 46.7%

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

      1. associate-*l* 46.7%

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

      2. sub-neg 46.7%

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

      3. +-commutative 46.7%

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

      4. distribute-rgt-neg-in 46.7%

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

      5. fma-def 46.7%

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

   3. Simplified 47.2%

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

   5. Taylor expanded in ux around 0 99.1%

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

      1. fma-def 99.1%

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

      2. +-commutative 99.1%

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

      3. sub-neg 99.1%

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

      4. metadata-eval 99.1%

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

      5. +-commutative 99.1%

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

      6. distribute-lft-in 99.1%

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

      7. metadata-eval 99.1%

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

      8. associate--l+ 99.2%

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

      9. mul-1-neg 99.2%

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

      10. sub-neg 99.2%

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

      11. associate-*r* 99.3%

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

      12. sub-neg 99.3%

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

      13. metadata-eval 99.3%

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

      14. +-commutative 99.3%

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

   7. Simplified 99.3%

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

   8. Taylor expanded in maxCos around 0 89.8%

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

   10. Simplified 89.8%

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

   11. Taylor expanded in uy around 0 80.7%

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

       1. unpow2 80.7%

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

       2. distribute-lft-out-- 80.9%

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

       3. cancel-sign-sub-inv 80.9%

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

       4. metadata-eval 80.9%

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

       5. +-commutative 80.9%

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

       6. *-commutative 80.9%

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

       7. fma-def 80.9%

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

   13. Simplified 80.9%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;maxCos \leq 3.999999989900971 \cdot 10^{-6}:\\ \;\;\;\;\cos \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{ux \cdot \left(\mathsf{fma}\left(maxCos, -2, 2\right) - ux\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 79.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.1%

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

   1. associate-*l* 57.1%

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

   2. sub-neg 57.1%

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

   3. +-commutative 57.1%

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

   4. distribute-rgt-neg-in 57.1%

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

   5. fma-def 57.2%

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

3. Simplified 57.3%

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

5. Taylor expanded in ux around 0 99.1%

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

   1. fma-def 99.1%

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

   2. +-commutative 99.1%

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

   3. sub-neg 99.1%

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

   4. metadata-eval 99.1%

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

   5. +-commutative 99.1%

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

   6. distribute-lft-in 99.1%

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

   7. metadata-eval 99.1%

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

   8. associate--l+ 99.1%

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

   9. mul-1-neg 99.1%

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

   10. sub-neg 99.1%

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

   11. associate-*r* 99.2%

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

   12. sub-neg 99.2%

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

   13. metadata-eval 99.2%

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

   14. +-commutative 99.2%

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

7. Simplified 99.2%

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

8. Taylor expanded in maxCos around 0 97.6%

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

10. Simplified 97.6%

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

11. Taylor expanded in uy around 0 79.2%

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

    1. unpow2 79.2%

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

    2. distribute-lft-out-- 79.1%

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

    3. cancel-sign-sub-inv 79.1%

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

    4. metadata-eval 79.1%

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

    5. +-commutative 79.1%

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

    6. *-commutative 79.1%

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

    7. fma-def 79.1%

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

13. Simplified 79.1%

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

14. Final simplification 79.1%

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

Alternative 13: 75.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ux \leq 0.00019999999494757503:\\ \;\;\;\;\sqrt{2 \cdot ux + -2 \cdot \left(ux \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + \left(1 + \left(ux \cdot maxCos - ux\right)\right) \cdot \left(-1 + ux \cdot \left(1 - maxCos\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= ux 0.00019999999494757503)
   (sqrt (+ (* 2.0 ux) (* -2.0 (* ux maxCos))))
   (sqrt
    (+ 1.0 (* (+ 1.0 (- (* ux maxCos) ux)) (+ -1.0 (* ux (- 1.0 maxCos))))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (ux <= 0.00019999999494757503f) {
		tmp = sqrtf(((2.0f * ux) + (-2.0f * (ux * maxCos))));
	} else {
		tmp = sqrtf((1.0f + ((1.0f + ((ux * maxCos) - ux)) * (-1.0f + (ux * (1.0f - maxCos))))));
	}
	return tmp;
}

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    real(4) :: tmp
    if (ux <= 0.00019999999494757503e0) then
        tmp = sqrt(((2.0e0 * ux) + ((-2.0e0) * (ux * maxcos))))
    else
        tmp = sqrt((1.0e0 + ((1.0e0 + ((ux * maxcos) - ux)) * ((-1.0e0) + (ux * (1.0e0 - maxcos))))))
    end if
    code = tmp
end function

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (ux <= Float32(0.00019999999494757503))
		tmp = sqrt(Float32(Float32(Float32(2.0) * ux) + Float32(Float32(-2.0) * Float32(ux * maxCos))));
	else
		tmp = sqrt(Float32(Float32(1.0) + Float32(Float32(Float32(1.0) + Float32(Float32(ux * maxCos) - ux)) * Float32(Float32(-1.0) + Float32(ux * Float32(Float32(1.0) - maxCos))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (ux <= single(0.00019999999494757503))
		tmp = sqrt(((single(2.0) * ux) + (single(-2.0) * (ux * maxCos))));
	else
		tmp = sqrt((single(1.0) + ((single(1.0) + ((ux * maxCos) - ux)) * (single(-1.0) + (ux * (single(1.0) - maxCos))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if ux < 1.99999995e-4

   1. Initial program 34.0%

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

      1. associate-*l* 34.0%

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

      2. sub-neg 34.0%

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

      3. +-commutative 34.0%

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

      4. distribute-rgt-neg-in 34.0%

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

      5. fma-def 34.0%

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

   3. Simplified 34.0%

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

   5. Taylor expanded in uy around 0 29.9%

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

   7. Simplified 29.9%

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

   8. Taylor expanded in ux around 0 76.2%

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

   9. Taylor expanded in maxCos around 0 76.2%

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

   if 1.99999995e-4 < ux

   1. Initial program 89.3%

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

      1. associate-*l* 89.3%

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

      2. sub-neg 89.3%

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

      3. +-commutative 89.3%

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

      4. distribute-rgt-neg-in 89.3%

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

      5. fma-def 89.5%

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

   3. Simplified 89.7%

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

   5. Taylor expanded in uy around 0 75.0%

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

   7. Simplified 75.0%

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

      1. associate--l+ 75.1%

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

   9. Applied egg-rr 75.1%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \leq 0.00019999999494757503:\\ \;\;\;\;\sqrt{2 \cdot ux + -2 \cdot \left(ux \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + \left(1 + \left(ux \cdot maxCos - ux\right)\right) \cdot \left(-1 + ux \cdot \left(1 - maxCos\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 74.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ux \leq 0.00019999999494757503:\\ \;\;\;\;\sqrt{2 \cdot ux + -2 \cdot \left(ux \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + \left(1 + ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux + -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= ux 0.00019999999494757503)
   (sqrt (+ (* 2.0 ux) (* -2.0 (* ux maxCos))))
   (sqrt (+ 1.0 (* (+ 1.0 (* ux (+ maxCos -1.0))) (+ ux -1.0))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (ux <= 0.00019999999494757503f) {
		tmp = sqrtf(((2.0f * ux) + (-2.0f * (ux * maxCos))));
	} else {
		tmp = sqrtf((1.0f + ((1.0f + (ux * (maxCos + -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 <= 0.00019999999494757503e0) then
        tmp = sqrt(((2.0e0 * ux) + ((-2.0e0) * (ux * maxcos))))
    else
        tmp = sqrt((1.0e0 + ((1.0e0 + (ux * (maxcos + (-1.0e0)))) * (ux + (-1.0e0)))))
    end if
    code = tmp
end function

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (ux <= Float32(0.00019999999494757503))
		tmp = sqrt(Float32(Float32(Float32(2.0) * ux) + Float32(Float32(-2.0) * Float32(ux * maxCos))));
	else
		tmp = sqrt(Float32(Float32(1.0) + Float32(Float32(Float32(1.0) + Float32(ux * Float32(maxCos + 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(0.00019999999494757503))
		tmp = sqrt(((single(2.0) * ux) + (single(-2.0) * (ux * maxCos))));
	else
		tmp = sqrt((single(1.0) + ((single(1.0) + (ux * (maxCos + single(-1.0)))) * (ux + single(-1.0)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if ux < 1.99999995e-4

   1. Initial program 34.0%

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

      1. associate-*l* 34.0%

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

      2. sub-neg 34.0%

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

      3. +-commutative 34.0%

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

      4. distribute-rgt-neg-in 34.0%

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

      5. fma-def 34.0%

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

   3. Simplified 34.0%

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

   5. Taylor expanded in uy around 0 29.9%

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

   7. Simplified 29.9%

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

   8. Taylor expanded in ux around 0 76.2%

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

   9. Taylor expanded in maxCos around 0 76.2%

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

   if 1.99999995e-4 < ux

   1. Initial program 89.3%

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

      1. associate-*l* 89.3%

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

      2. sub-neg 89.3%

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

      3. +-commutative 89.3%

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

      4. distribute-rgt-neg-in 89.3%

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

      5. fma-def 89.5%

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

   3. Simplified 89.7%

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

   5. Taylor expanded in uy around 0 75.0%

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

   7. Simplified 75.0%

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

   8. Taylor expanded in maxCos around 0 72.0%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \leq 0.00019999999494757503:\\ \;\;\;\;\sqrt{2 \cdot ux + -2 \cdot \left(ux \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + \left(1 + ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux + -1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 74.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ux \leq 0.00019999999494757503:\\ \;\;\;\;\sqrt{2 \cdot ux + -2 \cdot \left(ux \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + \left(1 - ux\right) \cdot \left(ux + -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= ux 0.00019999999494757503)
   (sqrt (+ (* 2.0 ux) (* -2.0 (* ux maxCos))))
   (sqrt (+ 1.0 (* (- 1.0 ux) (+ ux -1.0))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (ux <= 0.00019999999494757503f) {
		tmp = sqrtf(((2.0f * ux) + (-2.0f * (ux * maxCos))));
	} else {
		tmp = sqrtf((1.0f + ((1.0f - ux) * (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 <= 0.00019999999494757503e0) then
        tmp = sqrt(((2.0e0 * ux) + ((-2.0e0) * (ux * maxcos))))
    else
        tmp = sqrt((1.0e0 + ((1.0e0 - ux) * (ux + (-1.0e0)))))
    end if
    code = tmp
end function

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (ux <= Float32(0.00019999999494757503))
		tmp = sqrt(Float32(Float32(Float32(2.0) * ux) + Float32(Float32(-2.0) * Float32(ux * maxCos))));
	else
		tmp = sqrt(Float32(Float32(1.0) + Float32(Float32(Float32(1.0) - ux) * Float32(ux + Float32(-1.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (ux <= single(0.00019999999494757503))
		tmp = sqrt(((single(2.0) * ux) + (single(-2.0) * (ux * maxCos))));
	else
		tmp = sqrt((single(1.0) + ((single(1.0) - ux) * (ux + single(-1.0)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if ux < 1.99999995e-4

   1. Initial program 34.0%

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

      1. associate-*l* 34.0%

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

      2. sub-neg 34.0%

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

      3. +-commutative 34.0%

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

      4. distribute-rgt-neg-in 34.0%

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

      5. fma-def 34.0%

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

   3. Simplified 34.0%

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

   5. Taylor expanded in uy around 0 29.9%

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

   7. Simplified 29.9%

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

   8. Taylor expanded in ux around 0 76.2%

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

   9. Taylor expanded in maxCos around 0 76.2%

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

   if 1.99999995e-4 < ux

   1. Initial program 89.3%

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

      1. associate-*l* 89.3%

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

      2. sub-neg 89.3%

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

      3. +-commutative 89.3%

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

      4. distribute-rgt-neg-in 89.3%

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

      5. fma-def 89.5%

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

   3. Simplified 89.7%

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

   5. Taylor expanded in uy around 0 75.0%

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

   7. Simplified 75.0%

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

   8. Taylor expanded in maxCos around 0 71.7%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \leq 0.00019999999494757503:\\ \;\;\;\;\sqrt{2 \cdot ux + -2 \cdot \left(ux \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + \left(1 - ux\right) \cdot \left(ux + -1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 64.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.1%

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

   1. associate-*l* 57.1%

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

   2. sub-neg 57.1%

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

   3. +-commutative 57.1%

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

   4. distribute-rgt-neg-in 57.1%

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

   5. fma-def 57.2%

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

3. Simplified 57.3%

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

5. Taylor expanded in uy around 0 48.8%

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

7. Simplified 48.8%

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

8. Taylor expanded in ux around 0 64.3%

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

9. Taylor expanded in maxCos around 0 64.3%

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

10. Final simplification 64.3%

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

Alternative 17: 64.9% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.1%

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

   1. associate-*l* 57.1%

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

   2. sub-neg 57.1%

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

   3. +-commutative 57.1%

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

   4. distribute-rgt-neg-in 57.1%

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

   5. fma-def 57.2%

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

3. Simplified 57.3%

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

5. Taylor expanded in uy around 0 48.8%

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

7. Simplified 48.8%

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

8. Taylor expanded in ux around 0 64.3%

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

9. Final simplification 64.3%

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

Alternative 18: 62.3% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.1%

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

   1. associate-*l* 57.1%

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

   2. sub-neg 57.1%

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

   3. +-commutative 57.1%

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

   4. distribute-rgt-neg-in 57.1%

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

   5. fma-def 57.2%

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

3. Simplified 57.3%

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

5. Taylor expanded in uy around 0 48.8%

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

7. Simplified 48.8%

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

8. Taylor expanded in ux around 0 64.3%

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

9. Taylor expanded in maxCos around 0 61.0%

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

    1. *-commutative 61.0%

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

11. Simplified 61.0%

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

12. Final simplification 61.0%

    \[\leadsto \sqrt{2 \cdot ux} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024036 
(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)))))))